chromatographic peak shape analysis and modeling

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LSU Historical Dissertations and Theses Graduate School

1990

Chromatographic Peak Shape Analysis andModeling.Mark Stephen JeansonneLouisiana State University and Agricultural & Mechanical College

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Chrom atographic peak shape analysis and m odeling

Jeansonne, Mark Stephen, Ph.D.

The Louisiana State University and Agricultural and Mechanical Col., 1990

U M I300 N. Zeeb Rd.Ann Arbor, MI 48106

CHROM ATOGRAPHIC PEAK SHAPE ANALYSIS AND

MODELING

A Dissertation

Submitted to the Graduate Faculty of the Louisiana State University and

Agricultural and Mechanical College in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

in

The Department of Chemistry

byMark Stephen Jeansonne

B.S., Louisiana State University, 1985 December 1990

D edication

In everyone's life there is a person or persons who have played a special role,

enabling one to aspire to and attain life's highest goals. In my life, these special people

have been my family, the total, unquestioning supporters of my goal of obtaining a Ph.

D. in Chemistry. Without their support and encouragement I certainly would not have

dared dream of spending the countless hours necessary for completion of this degree.

I know that they understood when I needed to apply myself to my work, even

when it meant that I might not spend as much time with them as I or they would have

liked. It is because of their unselfish dedication to my work that this dissertation is

dedicated to them, the members of my family. In this light, I consider them to be as

much an author of this dissertation as I am.

ACKNOW LEDGEM ENT

I wish to express my gratitude to Professor Joe P. Foley. His tireless dedication

his work and students has greatly inspired my work.

CONTENTS

List of Tables, ix

List of Figures, xii

Abstract, xviii

Chapter 1.

INTRODUCTION

References, 7

Chapter 2.

REVIEW OF THE EXPONENTIALLY MODIFIED GAUSSIAN

(EMG) CHROMATOGRAPHIC PEAK MODEL SINCE 1983

Introduction, 9

Development of Equations Utilizing the EMG Model, 12

Development of Equations For Single Peak Quantitation, 13

Development of Equations For Overlapping Peak Quantitation, 15

Use of the EMG Model for Study of Expected Errors in

Chromatographic Measurement, 18

Errors in the Measurement of Single Tailed Peaks, 16

Analysis of Errors in Overlapping Peaks Via the EMG Model, 21

Direct Chromatographic Application of the EMG Model, 22

Use of the EMG Model in Flow Injection Analysis, 22

Conclusion, 24

References, 31

Chapter 3. 38

IMPROVED EQUATIONS FOR THE CALCULATION OF

CHROMATOGRAPHIC FIGURES OF MERIT FOR IDEAL AND

SKEWED CHROMATOGRAPHIC PEAKS

Introduction, 39

Experimental, 40

EMG Peak Generation, 49

Method of Derivation, 41

Derivation of Equations For a, 42

Derivation of Equations for M2 ,42

Derivation of Area Equations, 43

Results and Discussion, 43

Accuracy of Equations, 43

Precision of Equations, 44

Accuracy and Preci sion of Area Equations, 44

Conclusion, 45

Acknowledgment, 45

Literature Cited, 59

Chapter 4. 60

MEASUREMENT OF STATISTICAL MOMENTS OF

RESOLVED AND OVERLAPPING CHROMATOGRAPHIC

PEAKS

v

Introduction, 62

Experimental, 64

EMG Peak Generation, 65

Real Chromatographic Peak Generation, 66

Peak Parameter Measurement, 67


Simulated Peaks Without Noise, 68

Results for Real Chromatographic Peaks, 71

Modified Width-Asymmetry Method for True Peak Deconvolution, 73

Computational Time, 74

Acknowledgment, 75

Credits, 75

References, 93

Chaper 5. 94

THE Q TRANSFORMATION: A NOVEL METHOD OF

PEAK SHAPE ANALYSIS

Introduction, 97

Theory, 98

Definition of Q Transformation, 98

Properties of the Q Transformation, 99

Peak Shape Comparison Using the Q Transformation, 100

Experimental, 101



Measurement of Q and ZQ from real data, 105

Effect of Smoothing on Q Transformation, 105

Effect of Noise, 106

Effect of the Number of Points in a Peak, 107

Comparison of Methods Using Noiseless Data, 107

Comparison of Methods Using Noisy Data, 108

Potential Applications of the Q Transform, 110

Conclusion, 111

Credit, 112


Chapter 6.

SINGLE CHANNEL PEAK IMPURITY DETECTION

USING THE Q TRANSFORMATION

Introduction, 134

Experimental, 137



Peak Shape Comparison, 140

Simulated Impure Peaks, 145

Noisy Simulated Peaks, 141

Real Impure Peaks, 142

Experimental verification of paired-t option, 143

Potential For Qualitative Analysis, 143

Conclusion, 144


Chapter 7 157

SUMMARY

APPENDICES

A. Q Transformation Program Listing and Documentation 160

B . Tables for Interconversion Between Various Chromatographic

Separation Measures 188

Vita 243

v iii

LIST OF TABLES

Page

C hapter 2.

Table 2-1 25

Chromatographic Applications of the Exponentially Modified Gaussian (EMG)

Model Since 1983.

Chapter 3.

Table 3-1 46

Coefficients, Accuracy, and Precision of Equations for Calculating Gq, and M2

of Gaussian and Exponentially Modified Gaussian (EMG) Peaks.

Table 3-2 49

Error and Uncertainty of Chromatographic Peak Parameters calculated from Gq

and M2 using the Equations in Table I.

Table 3-3 53

Peak Area Equations for EMG and Gaussian Peaks.

Table 3-4 54

Comparison of maximum errors for the CFOMs reported here and those reported

previously.

Chapter 4.

Table 4-1 76

Maximum peak overlap (percent valleys) that can be tolerated by the width-

asymmetry and summation methods for a given accuracy (<5% error)

Table 4-2 77

Peak Parameters Measured for an Isolated, Real Chromatographic Peak.

Table 4-3 78

Comparison of Summation and Width-Asymmetry Methods for Two Sets of

Overlapping, Real Peaks.

Table 4-4 80

Peak Parameters for Deconvolved Peaks of Overlapping Peak Pairs With 40%

and 67% Valleys.

Chapter 5.

Table 5-1 113

Minimum Detectable Change in T/CT For Various Methods of Peak Shape

Analysis.

Table 5-2 114

% Relative Error in ZQ Due to Baseline Errors in EMG Peaks.

x

Chapter 6.

Table 6-1 146

Minimum Resolution Necessary for the Detection of Two Overlapping EMG

Peaks.

Table 6-2 147

Simulated Impure peaks detectable by the paired t-test and the Q profile.

Table 6-3 148

Impure experimental HPLC peaks detected via the paired t-test using the Q

profile.

Table 6-4 149

Comparison of some pure peaks via the paired t-test.

Appendix B.

Table B-l 191

Table B-2 204

Table B-3 222

LIST OF FIGURES

Page

C hapter 2.

Figure 2-1 29

Illustration of the effect of %/g q on EMG peak tailing.

Figure 2-2 30

Measurement of graphical peak parameters for use in asymmetry-based

equations.

Chapter 3.

Figure 3-1 56

Graphical parameters necessary for calculation of various CFOMs.

Figure 3-2 57

Plot construction for determination of f(b/a) for calculation of Gq at r=0.25.

Note curvature at lower asymmetries.

Figure 3-3 58

Plot construction for determination of f(b/a) for calculation of M2 at r = 0.25.

Due to curvature over entire asymmetry range, two least-squares quadratics were

fit.

Chapter 4.

Figure 4-1 81

Equations used for calculation of statistical moments and other peak parameters

by a) the summation method and by b) the width-asymmetry method.

Figure 4-2 82

Effect of the data sampling rate on the measurement of peak width, asymmetry,

and peak area.

Figure 4-3 83

Measurement of graphic parameters for an overlapping pair of chromatographic

peaks. tr and hp are the retention time and peak height of respective peaks, and

hv is the height of the valley. Peak width at the desired peak height fraction is

given by ta-tb-

Figure 4-4 84

Comparison of the errors in peak area and variance occurring in the summation

and width-asymmetry methods as a function of peak overlap (percent valley) for:

(a) x/ g q =1; (b) z / g q = 2; and (c) t /o q = 4. Labels in the plot refer to: a) the

parameter, b) the peak (first [L] or second [R]) for which a parameter was

obtained; and c) the relative peak height at which the width and asymmetry were

measured (width-asymmetry only). For example, "M2, L; 75%" refers to the

variance measured for the first peak of the overlapped pair at 75% of the peak

height, while "M2, R" refers to the variance of the second peak measured by the

summation method.

Figure 4-5 88

Comparison of the errors in peak area and variance occurring in the summation

and width-asymmetry methods as a function of peak overlap for a highly skewed

pair of peaks {x / g q = 4) with a less than ideal baseline level (0.1%). Conditions

as in Fig. 4-4.

Figure 4-6 89

Comparison of the errors in peak parameters other than peak area and variance

for overlapped peak pairs with x / g q = 4 occurring in: a) summation method; and

b) width-asymmetry method. Conditions as in Fig. 4-4.

Figure 4-7 92

Visual interpretation of the width-asymmetry/deconvolution method. The solid

black line indicates the real overlapping chromatographic peaks, while the lighter

lines show the individual peaks that are predicted by the width-

asymmetry/deconvolution method.

Chapter 5.

Figure 5-1 115

Measurement of a and b used for the calculation of Q in eq 1.

Figure 5-2 116

The Q transform can be used to visualize peak shape differences without regard

to peak height or width. Peak identity: (1) Exponentially Modified Gaussian

xiv

(EMG), x/o=2.6, 0=1.0 (2) EMG, 1/0=1, 0=0.26 (3) EMG, xl 0=2.2, o=0.26

(4) EMG, t /o =2 .8, o=0.5 (5) Log-Normal, asymmetry=1.6

Figure 5-3 117

Comparison of Q profiles for various peak shapes, (a) EMG peak, ref. 13 (b)

Gamma peak, eq 4 (c) Log-Normal peak, eq 5 (d) Overlay of Q profiles from

(a) through (c). Note that peak shapes can be differentiated on the basis of XQ or

on direct point-by-point comparison of individual Q profiles.

Figure 5-4 118

Differentiation of peak shapes using Q transform between (a) a symmetric peak

(Gaussian) and a slightly asymmetric peak (EMG, t/s = 0.3) and (b) two

symmetric peaks by a redefinition of Q as discussed in the text Note: Curves in

(a) appear noisy relative to curves in (b) because of scale difference.

Figure 5-5 119

Non-superimposability of Q profiles of fundamentally different type of peaks

with identical values of XQ. Peak identities: EMG with xto = 2.8 and log

normal with an asymmetry (eq 5) of 1.6.

Figure 5-6 120

Effect of Savitsky-Golay smoothing factor (eq 6) on (a) XQ ; and (b) Q vs r for

an EMG peak with x/a =1.

Figure 5-7 121

Effect of signal to noise (S/N) ratio on the Q transformation.

xv

Figure 5-8 122

Effect of the relative sampling rate (number of points) on the measurement of

XQ. Peak shape: EMG peak, x/o=l.

Figure 5-9 123

Effect of the relative sampling rate (number of points) on the accuracy of Q.

Number of points in the peak were measured from half-height to half-height.

Figure 9e shows true Q vs r profile (search algorithm) while Figure 9f shows the

average error associated with the measurement of Q at any r.

Figure 5-10 127

Relative abilities of statistical moments, excess, skew, second derivative, XQ and

DFM to detect changes in peak shape for noiseless peaks.

Figure 5-11 128

Automatic detection by the Q transform of instrumental problems in

chromatography, as illustrated with (a) a fronted peak (indicative of sample

overload); and (b) its Q profile.

Chapter 6.

Figure 6-1 150

Effect of co-elution on overall peak profile. Impurity elutes on the tail of the

parent peak. Conditions: EMG peaks (x/c =1); area ratio = 16:1; resolution =

xv i

0.25. Note that no shoulder or other visible evidence is apparent on the overall

profile.

Figure 6-2 151

Effect of co-elution on the Q transformation in which the impurity elutes on the

tail of the parent peak. Conditions: EMG peaks ('t/a =1); area ratio =16:1;

resolution = 0.25.

Figure 6-3 152

Effect of co-elution on (a) overall peak profile; and (b) Q transformation in which

the impurity elutes on the front of the parent peak. Other conditions as in Figure

6- 2 .

Figure 6-4 153

Impure and pure n-propyl benzene peaks. Visual inspection of original peaks

fails to indicate presence of impurity.

Figure 6-5 154

Example of clustering as a function of peak identity for a series of ketones.

Figure 6-6 155

Illustration, using simulated data, of Q transformation approach to qualitative

analysis.

x v ii

ABSTRACT

Various aspects of chromatographic peak quantitation and shape characterization

are investigated in detail for single and overlapping chromatographic peaks. From the

viewpoint of providing better quantitation of real chromatographic data while minimizing

computational complexity, the results presented should be easily incorporated into

existing routine chromatographic data analysis regimes. Three topics applicable to

modem chromatographic data analysis are considered.

First, progress in the application of the exponentially modified Gaussian (EMG)

function to chromatography is reviewed. The review covers the following areas: (1)

equations derived from the model, (2) studies of inherent errors in the quantitation of

chromatographic peaks via use of the EMG model, (3) chromatographic applications

since 1983 and (4) applications to flow injection analysis. The information discussed

and the references included in this review should provide a valuable resource for those

researchers considering or already using the EMG model in their studies.

Second, improved empirical equations based on the EMG model are presented. The

equations utilize measurements of the graphical parameters peak height (hp), width (W),

and asymmetry for the calculation of the following chromatographic figures of merit

(CFOMs): t, c , area, variance, third and fourth statistical moments, excess, and skew.

The equations are shown to be more accurate than traditional numerical methods when

applied to single and overlapped chromatographic peaks that fit the EMG model.

Application to simulated and real chromatographic peaks is discussed.

Finally, a computationally simple, normalized data transformation that can be

used for peak shape analysis and comparison is introduced. Compared to slope analysis,

moment analysis, and the distribution function, the "Q transformation" performs much

better over a wide range of experimental conditions, including signal to noise (S/N)

ratios of less than 100 in which the other methods often fail. This method is shown to be

computationally simple, easy to automate, and very intuitive in nature. Application of the

Q transformation for impure chromatographic peak detection using single-channel

channel data is discussed. Its advantages over the skew, excess, slope analysis and

distribution function methods are reported using both simulated and real data.

Chapter 1

INTRODUCTION

1

2

The field of chromatography has evolved substantially since its beginning. From

large particle silica stationary phases to today's microparticulate, tended phase packed

columms and capillary columns, chromatography has become a mainstay in modem

analytical chemistry. Our understanding of the chromatographic process has progressed

such that the prediction of retention behavior for many compounds is now possible. In

fact, expert computer programs now exist that allow a chromatographer to predict with

fair accuracy the retention behavior of the compounds in a complex mixture, with only a

minimum understanding of the actual chromatographic process. By predicting retention

behavior, one should be able to predict the best separation conditions for a sample with a

minimum of work.

However, the degree of separation (and other important chromatographic

parameters) also depends on the shape of each peak. We need to know the shape and

width of the peak in order to know the minimum difference in retention times between

adjacent peaks so that they are sufficiently separated. Even with the level of

understanding available today, accurate peak shape prediction for all compounds under

given conditions is still not possible. Some general theories do exist, though, for

explaining many aspects of chromatographic peak shape.

As a single type of solute molecule emerges from a chromatographic column, a

peak indicating the distribution of the molecules with time is detected. Because this

distribution is not infinitely narrow, or not as narrow as the band when it starts at the

beginning of the column, there must be processes occuring within the column acting to

broaden the peak. According to rate theory (1,2), the three processes that contribute to

broadening a chromatographic peak symmetrically are (1) resistance to mass transfer, (2)

eddy diffusion and (3) longitudinal diffusion.

3

Broadening due to the resistance to mass transfer occurs because an instantaneous

equilibrium is not established between solute molecules in the stationary phase and those

in the mobile phase. Therefore, the solute concentration profile in the stationary phase is

always displaced slightly behind the equilibrium position and the mobile phase profile is

displaced slightly ahead of the equilibrium position. The resulting peak profile observed

at the column outlet is broadened about its center, which is located where it would have

been for an instantaneous equilibrium. Resistance to mass transfer is the dominant cause

of symmetric band broadening in chromatography.

In packed columns there are multiple possible flow paths available for solute

molecules as they flow through the column. Because each path may be of different

length, solute molecules will not emerge from the column at the same time thereby

broadening the peak.

Longitudinal diffusion is due to simple molecular diffusion in the axial direction

along the length of a column. Its contribution to symmetric band broadening will

increase with the amount of time that a solute band is inside the column.

As stated before, the aforementioned processes tend to broaden a chromatographic

peak symmetrically. However, most real chromatographic peaks are not symmetric. The

processes that act to broaden a peak asymmetrically can be divided into intracolumn

(within column) and extracolumn (outside the column) processes. Important

intracolumn contributions to peak asymmetry include incomplete resolution, slow kinetic

processes, chemical reactions, column voids, and non-linear distribution isotherms.

Important extracolumn asymmetric band broadening processes include dead-volumes or

spaces and a slow detector time constant. Although all of the intracolumn asymmetric

broadening processes do occur, for most well behaved, linear chromatographic systems,

their effects will usually be slight or even negligible. However, the extracolumn

4

contributions to peak asymmetry will affect most chromatographic peaks and tend to add

an exponential tail to them.

Most basic chromatographic theory assumes a Gaussian peak shape. However,

most practicing chromatographers realize that this assumption is normally not valid due

to the possible asymmetric band broadening processes. Therefore, application of the

theory developed for symmetric peaks to experimental peaks can result in errors. For

example, when two chromatographic peaks overlap, integration of their separate areas is

impossible. If the assumption is made that the two peaks are symmetric, simply dividing

their areas by drawing a perpendicular at the valley between them can often give

sufficient accuracy. Quantitating the areas of overlapping asymmetric peaks in this

manner will typically cause large errors, depending on the degree of asymmetry (3).

However, this error can be reduced greatly by assuming a peak shape function that is

closer to the peak shape of the experimental peaks, even if the true experimental peak

shape is not known. In this manner, a tool or model can be developed for more accurate

quantitation of the peaks. The better model can also be used to simulate and therefore

understand the errors inherent in using the less accurate method. Because

chromatographic peaks cannot always be completely separated, the development of

techniques to better quantitate them when overlap occurs has played a very important role

in the development of chromatography as a routine analysis technique.

Better models can also be developed for the quantitation of other important

chromatographic parameters when peaks are asymmetric. Perhaps the most important

descriptor of chromatographic column performance is the efficiency (N), defined as

tR2n = — —— rVariance

5

where tR is the retention time of the peak. The variance of a peak can be evaluated

statistically using numerical method techniques. But, numerical method evaluation of

peak variance has proven to be highly error prone (4). Therefore, Equation 1 is often

simplified assuming a Gaussian peak shape to give

where W is the width of the peak at the baseline. Now the variance of the peak is

of the peak. This provides a very convenient method for measuring N. However, if the

experimental peak is not Gaussian in shape, error in the value of N will occur. Again, if

one were to assume a peak shape function closer to the true experimental peak shape, a

more accurate value of N could be obtained. Of course, to be routinely useful, in this

instance, the value of the variance needed for Equation 1 should be as easy to measure as

if one had assumed a Gaussian peak shape. That is, any new models should be

sufficiently easy to use so that those not familiar with the model or its underlying

concepts can still use it in routine analyses.

The development of easy to use models and methods for more accurate quantitation

of chromatographic peaks, overlapped or no t, is the general theme of the following

chapters. This area of research has become very important in recent years with advent of

computerized data analysis. Instrument manufacturers are always striving to find easier

and more accurate methods for describing chromatographic separations, thus allowing

greater automation and subsequently greater productivity by their customers. Because

computerized data analysis is now routinely available in most laboratories, computerized

data acquisition and its implications are an integral part of the work presented. We have

strived to keep computational complexity to a minimum, however.

[2]

assumed to be equal to the variance of a Gaussian function, which is related to the width

6

To a large extent, the following chapters utilize the exponentially modified

Gaussian (EMG) function (5,6) both as a model for real chromatographic peaks and as a

means for their simulation. This function results from the convolution of a Gaussian

function with an exponential decay function. As mentioned previously, if intracolumn

processes leading to peak asymmetry are negligible, then the chromatographic column

tends to broaden a peak symmetrically, with the resulting peak having a Gaussian shape.

However, the extracolumn effects usually tend to add some degree of exponential tailing

to a chromatographic peak. Therefore, the EMG model should give a more accurate

description of real chromatographic peaks than the Gaussian model. This peak model and

its applicability to experimental peaks are described in detail in Chapter 2.

Chapter 3 presents equations for the better quantitation of real chromatographic

peaks using the EMG. These equations can be used for both single and overlapping

chromatographic peaks and allow the user to apply the EMG model without resorting to

complicated curve fitting techniques. As mentioned before, simplicity is an important

aspect for computerized chromatographic quantitation.

Chapter 4 reports the results of applying the equations from Chapter 3 to both

overlapping and single peaks. Simulated and real chromatographic peaks were

examined. It was found that accurate statistical moment measurements may be made on

overlapped peaks that fit the EMG model. A very simple procedure for peak

deconvolution is also described.

Finally, Chapters 5 and 6 introduce a new method (the Q transformation) for peak

shape analysis and the detection of severely overlapped chromatographic peaks using

only single-channel data. This work is important in that it provides a computationally

simple method for comparison of peak shapes that is more sensitive to peak shape

differences than other simple methods of peak shape comparison. It was found that the Q

transformation is more sensitive to peak shape changes than the methods of slope

7

analysis, moment analysis, and the distribution function method. It is this sensitivity to

peak shape differences that allows for the detection of overlapping peaks, even when no

visible evidence of overlap occurs. The methods introduced in these chapters also have

important implications for the field of laboratoiy automation.

REFERENCES

1) Giddings, J.C. "Dynamics of Chromatography", Part 1; Chromatogr. Science

Series, Vol. 1; Dekker, New York, 1965

2) Poole, C.F.; Schuette, S.A. "Contemporary Practice of Chromatography";

Elsevier, New York, 1984

3) Foley, J.P. J. Chromatogr. 1987, 384, 301-303

4) Anderson, D. J.; Walters, R. R J. Chromatogr. Sci. 1984, 22, 353-359

5) Sternberg, J.C. "Advances in Chromatography"; Giddings, J.C.; Keller, R.A.,

Eds.; New York: Marcel Dekker: 1966; Vol. 2, pp 205-270

6) Foley, J. P.; Dorsey, J. G. J. Chromatogr. Sci. 1984, 22, 40-46

C h a p t e r 2

REVIEW OF THE EXPONENTIALLY MODIFIED GAUSSIAN (EMG) FUNCTION SINCE 1983

8

9

INTRO DUCTIO N

Assumption of a valid peak shape model has played an important role in the

development of chromatography to date. For example, the derivations of many

frequently used fundamental equations, such as the fundamental resolution (Rs) equation

(Equation 1), depend on the assumption of Gaussian peak shape. N is the column

efficiency, k' is the capacity factor, and a is the separation factor, equal to k'2 /k'i.

Another equation often used, the equation for calculation of efficiency (Equation 2), is

also based on this assumption. In Equation 2, tR refers to the retention time of the peak

and a refers to the standard deviation of the peak. However, as many practicing

chromatographers know, the Gaussian function rarely provides an accurate model for

real chromatographic peaks. Therefore, a search for a better function to model real

chromatographic peaks has been pursued for many years.

One model that has received much attention is the exponentially modified Gaussian

(EMG) function. This function results from the convolution of a Gaussian function with

an exponential decay function, and thus, can represent both symmetrical and tailed

peaks. Its evaluation is only somewhat more difficult than that of the simple Gaussian

function, while being a much better model for real chromatographic peaks. As with the

Gaussian function, the variables of the EMG can be related to physical parameters and

[1]

[2]

1 0

the function has been justified theoretically as a good model for real chromatographic

peaks (1, 2).

These attributes have enabled the EMG model for chromatographic peaks to receive

much attention in recent years. A 1984 review (2) of the EMG for use in

chromatography provided further impetus for its increased use. In that review, the

authors described a simpler method for evaluation of the EMG function, given in

Equation 3. The function is defined by three parameters, the retention time (to), the

standard

deviation of the Gaussian component (G q ) and the time constant of the exponential decay

component (z), where z = (t - tg) / Oq - Gq/z. A is the peak area. As the ratio z/Gq

increases, the amount of tailing in the EMG peak increases (see Figure 2-1). As shown

in Equation 3, three main terms exist in this form of the equation: (PE) the pre

exponential term, (E) the exponential term and (I) the integral term. Via a change of

variable, Equation 3 can be written as

hEMG(0 = ~ exP [ \ (^)2 - (7 - ^ ) ] Jexp (-x2) dx [3]* x x -00

z /a/2

J LPE E I

z

hEMG(l)

J L JLPE E I

The integral term (I) from Equation 4 can be approximated by the product of a

polynomial and an exponential term. That is, I (z<0) = NF(z)P(q) and I (z^O) = 1-5

l(z<0), where NF(z) = exp(-z^/2) / V27t), and P(q) = ^ bnqn, q= (1+pz)'*. Then=l

parameters p ,b j , ..., b5 are constants given in Table ID of reference 2. The error

reported for this method of EMG evaluation was 1% or less for x J g q > 0.2. A short

program in BASIC using this method was listed in that report (2).

Another method for evaluating the integral from Equation 3 utilizes the error

function (3,4). By writing the integral term as

I = erf i ( - ^ + ^ ) + e r f - ^ ( ^ - 5 a ) [5]\ 2 oq x \ 2 v ctq x

and using the evaluation algorithms for the error function available in many commercial

software packages or numerical methods textbooks (5), one can evaluate it easily.

Because the first term is nearly unity when tQ/CQ > 5, the integral can also be

approximated as

I . l + e r f - U H a . S l ) . [6]V 2 x G q X

Since the publication of these evaluation methods, many studies have indicated the

validity of the EMG model in chromatography (4 - 9). Furthermore, papers by Foley (8 )

and Wu (10) have introduced simple methods for validating the EMG model for peaks of

interest.

To summarize, the attributes of the EMG model discussed above - its ability to

model tailing peaks - its relative ease of evaluation - its demonstrated validity as a model

for real peaks - have provided further incentive for studies using the EMG function since

1 2

the review of 1984. Therefore, it is prudent now to review the progress made since

then.

Although we review the progress of the EMG in only chromatography and flow

injection analysis (FLA) in this report, the results presented are applicable to other fields

in which this model is used. Because we have already briefly discussed verification of

this model for chromatography, this aspect will not be discussed further. We leave it the

interested reader if more details are desired by using the articles referenced above. Also,

we are excluding from our discussion non-linear chromatography or other forms of

chromatography where non-EMG or non-Gaussian peaks are obtained. Our intentions

are to focus mainly on the applications of the EMG model to chromatography, with a

brief review of its use in flow injection analysis (FLA). More specifically, the

organization of this review will cover four main areas into which most studies utilizing

the EMG fall: (1) use of the EMG for development of equations that quantitate

chromatographic peaks and/or their mathematical separation, (2 ) investigations of

potential errors in chromatographic measurements by using the EMG for simulation of

real peaks, (3) direct application of the EMG model to chromatography, and (4) the

application of the EMG model to FIA. Because only three reports concerning evaluation

of the EMG function have occurred since 1983 (3,11,12), this topic is not discussed

further in this report.

DEVELOPM ENT OF EQUATIONS UTILIZING THE EMG

MODEL

Due to the attributes of the EMG model listed above, many studies have

concentrated on using this model to develop other equations that can be easily and

directly applied to real chromatographic data. These studies can be broken down into

1 3

those that apply strictly to single peaks and those that are applicable to the measurement

of resolution or deconvolution of two partially resolved peaks.

Development of equations for single peak quantitation . Within this

section we begin by presenting work specifically designed to extract the fundamental

parameters, x, Gq and t^ for an EMG fit to a real chromatographic peak. Some of these

studies also presented equations for the calculation of the first (Mi), second (M2 ) and

higher statistical moments. In 1983 Foley and Dorsey (13) reported equations for the

calculation of chromatographic figures of merit (CFOM) for symmetric and asymmetric

peaks based on the EMG model. The equations enabled one to calculate the basic

parameters (x, Oq and t^) and statistical moments of an experimental peak from the

simple graphical parameters a, b and peak height, illustrated in Figure 2-2. A particularly

important equation resulting from this study is shown in Equation 7, for the

measurement of system efficiency, where Wq.i is the width of the peak at the 10% of its

height. Following that report, this equation has been referred to as one of the best

methods for calculation of efficiency (14,15).

Although the Foley-Dorsey study was noted (although only briefly) in the previous

review (2 ), we decided that a brief re-introduction of these equations was necessary due

to their rather extensive use since then by other researchers (see Table 2-1). Also, these

equations appropriately set the tone for this section.

[7]

Soon after the publication of these equations, a report by Jung et al.(16) presented

a method for the extraction of the EMG parameters via the use of normal and derivative

peak heights measured at four or five time points. Their method gave good

measurements of the fundamental EMG variables and did not require the iterative

computer searches needed to find the graphical parameters a and b. However, their

approach was more complex intuitively than that reported by Foley and Dorsey and

required solution of cubic and quartic equations after measurement of the various normal

and derivative peaks. The Foley-Dorsey equations require only the "plugging in" of the

necessary graphical peak parameters. Later, a report by Anderson and Walters (17)

presented new equations for calculation of Oq , M j , and M2 that were valid to a higher

peak asymmetry than those of Foley and Dorsey. These equations were also based on

the parameters a, b, and peak height. Based on the measurement of the same graphical

peak parameters b and a, Foley (8 ) subsequently reported accurate equations for the

measurement of peak areas for chromatographic peaks fitting the EMG model.

f t a -0.133A = 0.586 hp W0 1 ( j?) [8]

A = 0.753 hD W0.25 [9]rf t ,\+0.235

A = 1.07 hpWo.5 ^ [1 0]/]b\+0.717

A = 1.64 hp W0_75 i ' [11]

Although these equations do serve as an alternative to the numerical integration methods

for the measurement of peak areas, their greatest utility lies in their ability to predict

whether a given peak accurately fits the EMG model and for the area deconvolution of

overlapping peaks (discussed later). As an attractive alternative to least-squares curve

fitting, one can compare the area values obtained by each equation, and based on the

suggested criteria (the spread in values from Equations 8-11 should be less than ten

1 5

percent for a peak to be considered EMG), one can decide on the applicability of the

EMG model for his data.

Development of equations for overlapping peak quantitation.

Previous equations for the calculation of resolution or deconvolution of areas for

overlapping peaks have been based mainly on the Gaussian peak model. However, due

to tailing of most real peaks, these equations have limited applicability. Because the

EMG model may better represent actual chromatographic peaks, equations using this

model should offer more accuracy in the quantitation of the overlap or provide a better

description of each single peak in the pair.

An article by Frans et al. (18) in 1985 presented a method for the reiterative least

squares resolution of overlapped chromatographic peaks using multi-wavelength

detection. For a six component mixture in which resolution ranged from 0.21 to 0.36,

the retention times of all components were accurately determined with less than 1.0 %

relative error. The average relative error in Oq for the six component mixture was less

than 15%.

Because multi-channel detection is not generally available for many analyses,

methods for quantitating overlapping peaks using single-channel detection have also been

reported. Foley (19) reported in 1987 that the area equation based on graphical peak

parameters a and b measured at 75% of the peak height (see Equation 11) could be used

to deconvolute the areas of an overlapped pair of chromatographic peaks. Because the

first peak in an overlapped pair is only slightly distorted at 75% of its height, an accurate

measurement of b and a could be obtained for that peak. The author reported an accuracy

of ± 4% in the area of of the first peak in the pair by this method when the valley height

between the peaks, relative to the first peak, was less than or equal to 45%.

Furthermore, by accurately measuring the total area of the peak pair by integration, and

1 6

then subtracting the area obtained for the first peak from the total area, an accurate value

for the area of the second peak can be determined. This method has been applied

successfully to real overlapping peaks that fit the EMG model (6 ).

Soon after this report appeared, Binsheng et al.(20) reported a table of correction

factors for quantitative area determinations of overlapping peak pairs. Their method

utilized values of the errors in area obtained when the perpendicular drop method was

used for overlapping EMG peak pairs. By calculating the area of the first peak via the

perpendicular drop method and utilizing Equation 13, a more accurate estimation of the

true area, Aj ,of the first peak can be ascertained. Av is the area of the first peak

measured by the perpendicular drop method and the value of P (correction factor) needed

for Equation 13 can be obtained from tables they provided.

Although the method of Binsheng gives greater accuracy for the quantitation of

overlapping peaks, the asymmetry of both peaks in the pair is assumed to be the same.

Because Foley's method (19) does not assume equal peak asymmetries in the pair, it may

be more practical for real overlapped peaks.

Along similar lines, Haddad and Sekulic (21) reported fourth-order polynomial

equations for the calculation of percentage area overlap of a peak pair. Their approach

was to determine the value of x/Oq for the first peak in a pair (using methods such as

those described above),to measure resolution of the peak pair, and from a fourth order

polynomial equation applicable at that t/Cq and resolution,to compute the value of

percent area overlap for the peaks. In this and related papers (7,22), the authors applied

this approach for the optimization of mobile phase conditions in high-performance liquid

chromatography (HPLC). Although developed strictly for optimization purposes, this

1+ P/1001 ) [13]

approach may prove useful in other areas where quantitation of overlapping peaks is

desired.

Schoenmakers et al.(23) have recently reported a corrected resolution function for

tailed chromatographic peaks (see Equation 14). A consequence of this function is that

two different resolution values are calculated, one for each peak.

measured at the 13.5% peak height fraction of peak i, lAgj is the asymmetry of peak i

measured at 13.5% of the peak height of j, *Nj is the efficiency of peak i as given in

the same parameters for peak i measured at 13.5% of peak height relative to peak j. As

stated by Schoenmakers, if the two peaks differ in height by no more than a factor of ten,

iRsji refers to the resolution measured for the first peak, i, tj is the retention time of

peak i, tj is the retention time of peak j (second peak), lAgj is the asymmetry of peak i

Equation 15 and xNj is the efficiency of peak i as given in Equation 16.

[15]

[16]

The parameters *ai and ^ are the same as the parameters a and b shown in Figure 2-1,

for peak i measured at 13.5% of the peak height of i, while the parameters Jaj and *bj are

if JNj = xNj = N, and if ̂ j = *AS j = As, then simplified expressions for the corrected

resolution can be given by Equations 17 and 18.

1 8

4 A stj + 4tj V l + l/21n(hj/h |)(M - tj) ( l + a s)Vn

[17]

4 A sti V I + l / 2 1n(hj/hj) + 4tj

(t, - tj) (1 + A S)VN[18]

USE O F THE EMG MODEL FOR STUDY OF EXPECTED

ERRORS IN CHROM ATOGRAPHIC MEASUREM ENT

The effective study of the measurement errors expected for tailed peaks via the

EMG function stems from its accuracy and practicality as a real peak model. Again, we

can subdivide the expected errors into those for a single tailed peak and those for

overlapping tailed peaks. We suggest reading references 24,25 and 26 for good general

reviews covering many general aspects of expected errors in chromatographic

measurements.

E rro rs in the measurement of single tailed peaks. Most of the studies

appearing since 1983 utilizing the EMG for quantitation of single peaks have dealt

directly or indirectly with statistical moment measurement. For example, two of the

more important chromatographic parameters, peak area and column efficiency, can be

defmed as statistical moments or functions of statistical moments (see Equations 19 and

20).

00 stopArea = zeroth moment = Mo = J h(t) dt = £ h(t) At [19]

-oo start

1 9

M i^ tp2Efficiency = N = - j^ - - ^ [20]

Thus those studies that specifically consider errors in either parameter are in reality

considering errors in the moments. Sources of error in the measurement of statistical

moments include inappropriate choice of model (e.g., when a Gaussian derived

efficiency equation is used for tailing peaks), insufficient sampling rate,, and most

importantly, noise and associated baseline errors.

Perhaps the greatest possibility for quantitation error occurs in the measurement

of the statistical moments for tailed peaks. As shown by Anderson et al.(17), large

errors in the first and second statistical moments (calculated via summation) can occur

when a portion of a tailed peak is truncated by an improperly constructed baseline.

These large errors result from the sensitivity of moments to the data in the extreme

regions of the peak. These errors normally occur when the integrator or data system

cannot detect the proper limits for the integration because noise is obscuring the correct

limits. If the integration bounds are improperly detected, the inaccurately constructed

baseline that results causes part of the peak to be lost after baseline subtraction. Of

course, improper setting of the slope sensitivity in the integrator can also cause

improperly detected integration limits. However, errors due to improper integrator

settings are avoidable, whereas errors resulting from noise are generally not. Anderson

et al. demonstrated errors in the first moment of greater than 30% and approximately

70% in the second moment when severe baseline construction errors occurred. They

showed that the use of asymmetry based equations for the first and second moments will

greatly reduce these errors.

As mentioned above, a significant source of error when measuring efficiency (or

other parameters) occurs when an inappropriate model is chosen. Because the efficiency

2 0

is related to the distribution of a sample peak over time, a symmetrical model, such as the

Gaussian model, cannot accurately represent the distribution of a tailed peak and thus

cannot accurately measure the efficiency. Therefore, a more representative model, such

as the EMG, should enable more accurate measurement of efficiency when the actual

moments (via summation) of a peak cannot be measured accurately. Most studies agree

(7, 14, 15,24,27,28) that in the absence of large amounts of noise, column efficiency

and variance (second moment, M2 ) are best calculated via Equations 20 and 21 using

numerical computer methods. However, the asymmetry based

stopM2 = Variance = J (t-Mj)2«h(t) dt + Area = ^ (t-Mi)2»h(t) At Area [21]

-00 start

methods for measurement of efficiency and variance will provide very good accuracy

when noise is severe or numerical integration is not feasible. Because the asymmetry

based methods seem to be simpler for routine use, they may be the best overall. For

higher moments (M3 and M4 ), integration methods are normally very inaccurate and

therefore the asymmetry based methods should be used for their measurement whenever

the peak shapes are appropriate. A dissenting view was presented by Colmsjo and

Ericsson (29) who suggested that measurement of the height equivalent to a theoretical

plate (HETP) for use in van Deemter plots can be made more reproducibly by using the

Gaussian model.

Sampling rate affects the accuracy of all possible measurements concerning

digitized chromatographic peaks. Two recent papers (25,26) and a book (24) have

thoroughly evaluated the effects of sampling rate on various chromatographic

measurements. Only one paper since 1983 has addressed sampling rate errors via the

21

EMG model. In that study, Rossi (30) presented a simple program for the calculation of

expected errors in the measurement of peak area for tailing peaks.

Analysis of errors in overlapping peaks via the EMG model. The

weak link in most chromatographic analyses continues to be the accurate area quantitation

of overlapping chromatographic peaks. The popular perpendicular drop and tangent

skim methods are still used almost exclusively due to their simplicity, although large

errors are possible with tailing peaks. Therefore, the EMG, being a good model for

tailing peaks, has been used in many studies since 1983 to demonstrate the severity of

these errors and present possible alternate methods for more realistic interpretation of the

individual areas of the peaks. However, most chromatographers agree that baseline

separation of important peaks is the best way to avoid these errors.

A widely cited study of the possible area quantitation errors for two overlapping

peaks was published in 1987 (19). There, the author demonstrated that errors in the area

when using the perpendicular drop method for quantitation of two severely overlapped

EMG peaks, with large differences in size, could be as high as 200%. He also reported

that errors in the measurement of heights could be as high as 80%. Papas et al. in 1987

(26) also showed the possibility of large errors in overlapping peak area measurement by

some integrator systems. Later, Jeansonne and Foley (6 ) investigated the errors

resulting from the measurement of statistical moments, excess, and skew on overlapping

peaks separated by the perpendicular drop method. They also demonstrated the

feasibility of accurate statistical moment measurement for overlapping peaks by using

asymmetry based methods.

A more recent paper by Papas et al.(31) discussed overlapping, tailed peak area

measurement errors when the perpendicular drop and tangent skim methods of area

deconvolution were used. Equations obtained via multiple linear regression were used to

determine whether the tangent skim or perpendicular drop method was better for

2 2

separating a particular set of overlapping peaks. In their opinion, these two

deconvolution methods are the most reliable and simple methods available for routine use

and therefore the decision of which method to use is important Other reports (7,20)

have demonstrated indirectly the potentially large errors resulting from the use of the

perpendicular drop method.

DIRECT CHROMATOGRAPHIC APPLICATION OF THE

EMG MODEL

No review on the EMG model would be complete without reporting the practical

applications of this model. These studies have used either the EMG model directly, as

described in the introduction, or equations derived from the EMG model, such as those

presented in the previous sections. Because the number of these reports is large, they are

not discussed here in detail. Instead, we give a chronological listing in Table 2-1 of

those studies that have appeared since 1983. It is hoped that this table will facilitate easy

reference for those who wish to consider the use of the EMG model for their own

investigations. However, a note of caution is needed here. Although almost any tailed

chromatographic peak is better modeled by the EMG than the Gaussian function, EMG

validity should be checked first by the methods discussed earlier.

USE OF THE EMG MODEL IN FLOW INJECTION

A N A L Y SIS

Although not strictly a chromatographic technique, flow injection analysis (HA)

consists of a flow system similar to that found in chromatography. In fact, the

dispersion processes in FIA can be considered analogous to the extracolumn band

2 3

broadening phenomena found in chromatography. The total band broadening in terms of

variance can be described by Equation 22 (32), and is equal to that for extracolumn band

broadening in chromatography.

°^peak “ °^injection + ^ transport+ ^detection [2 2 ]

Also, because several studies (32 - 37) have demonstrated the suitability of the EMG

model for the FIA peaks, a short discussion of some results from this field is

appropriate.

Brooks et al. (32) reported several advantages for using the variance instead of

the dispersion coefficient in an FIA system. (1) The peak width can be obtained from the

variance; (2) various FIA manifolds can be compared by using variance values; (3)

variance is a more direct measure of sample throughput; and (4) the individual

contributions to total variance (including the contribution from chemical reaction) are

additive and therefore easily obtained. The EMG character of the FIA peaks was verified

in this study by using Equations 8-11. The authors considered a peak to be EMG if the

largest outlier of the four measurements was within 2 0 % of the mean area calculated by

the equations. They found that 73% of the peaks fit the EMG model to within 15% at

flow rates up to 1 .2 ml/min for flow manifolds consisting of 1 0 0 cm of straight tubing,

approximately 14 mm in diameter. At flow rates of less than 0.6 ml/min they found that

92% of the peaks fit the EMG model to within 20%. They also reported that when coiled

tubing and a flow rate of 1.04 ml/min was used, all peaks fit the EMG to within 10%.

After verification of the EMG character of their peaks, the authors found a low (6.5%)

relative standard deviation for the measurement of the total variance, without chemical

reaction. The authors concluded that moment analysis using the EMG model was a valid

2 4

method for the examination of FIA peaks. Subsequent work (33, 34) using the EMG

model for FIA peaks has also yielded excellent results under a variety of conditions.

C O N C LU SIO N

The authors of the previous EMG review (2) hoped that their simple, clear

approach for evaluation of the EMG function would contribute to its greater use by

scientists when appropriate. Based on the large number of studies published since then

for which the EMG model has been directly applied or used for development of peak

analysis methods in chromatography, that review was a success. It is likewise hoped

that this compilation will contribute to the overall use of the EMG model by providing a

valuable reference resource for those researchers interested in applying the model to their

own studies.

2 5

Table 2-1. Chrom atographic Applications of the Exponentially

M odified Gaussian (EMG) M odel Since 1983

EMG APPLICATION METHOD CHROMATOGRAPHIC R EF.

Measurement of N a

MODE

Micellar HPLC 38

Measurement of G q c HPLC 39

Measurement of N a, c Affinity 40

Generation of test peaks N/A General 41

Measurement of x and Oq c HPLC 42

Measurement of x /G q c HPLC 43

Measurement of M2, x and G q b HPLC 44

EMG curve fit N/A HPLC 45

Measurement of N,M2 a HPLC/MS 46

Measurement of CFOM a HPLC 47


Measurement of N b, c HPLC 49

Measurement of M j ,M2 d HPLC/MS 50

Measurement of N a Micellar HPLC 51

Measurement of Mj a HPLC 52

Measurement of M2 a HPLC 53

Measurement of x and G q a,b OT HPLC/MS 54

Measurement of x and G q b OTHPLC 55

EMG curve fit N/A GC/MS 56


Measurement of N a HPLC 58

EMG curve fit NA GC 59



Measurement of N,x, 0 (3,M^,M2 c HPLC 62


Measurement of N,x, Gq a HPLC 64

Measurement of x and Gq a, d HPLC 65

Measurement of M2 a HPLC 6 6


Measurement of N a Micellar HPLC 6 8


Measurement of N,x, Cq a HPLC 70

Measurement of x/Gq b GC 71


Measurement of M2 b tubular reactors 73




Measurement of M2 a, d HPLC 77




Measurement of M2 a HPLC/MS 81




2 7


Measurement of N a Chiral HPLC 8 6




Measurement of N a RP-HPLC 90

Measurement of N a, c HPLC 91


EMG curve fit N/A General 93


Measurement of CFOM a GC 95

Measurement of N,t, Oq a HPLC 96

Measurement of N,t, Oq a HPLC 97

EMG curve fit N/A General 98

Measurement of % and Oq a GLC 99

Measurement of CFOM a Micellar HPLC 1 0 0

Measurement of N a Chiral HPLC 101

Measurement of N a GLC 102

EMG curve fit N/A GC-MS 103

EMG curve fit N/A HPLC 104

Measurement of N a CSP HPLC 105




Simplex Optimization, curve fit N/A HPLC 109


2 8


Measurement of x, Oq b GAC 112




Measurement of M2 a, b HPLC 116

Measurement of N a SFC 117


Measurement of N a GC 119

Measurement of M 1 a HPLC 1 2 0

Measurement of N a Cap. electrophoresis 121



a Used method of ref 13. b Used method of ref 124. c Used method of ref 125. d Used

method of ref 17.

2 9

t / o = 0 (Gaussian)

x/ o = 2

Figure 2-1. Illustration of the effect of x/ g q on EMG peak tailing.

3 0

Peak height = hp

Peak width = W = a + b

Peak asymmetry = (b/a)

Figure 2-2. Measurement of graphical peak parameters for use in asymmetry-based

equations.

31

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3 4

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Chapter 3

IM PROVED EQUATIONS FOR THE CALCULATION OF

CHROM ATOGRAPHIC FIGURES OF MERIT (CFOMs) FOR

IDEAL AND SKEW ED CHROMATOGRAPHIC PEAKS

3 8

INTRO DUCTIO N

3 9

In a recent publication we presented an equation-based approach for the accurate

measurement of chromatographic peak statistical moments, excess, and skew (1). These

empirical equations, based on the exponentially modified Gaussian (EMG) function,

utilize the peak width, asymmetry, and in some cases, the peak height. The EMG

function, resulting from the convolution of Gaussian and exponential decay functions,

can model any tailed peak more accurately than a Gaussian function. Due to various intra

and extracolumn band broadening processes, real chromatographic peaks are seldom

symmetric, so that use of a tailed peak model should be more accurate. Hie EMG

function has been justified both theoretically (2,3,4) and experimentally (5,6) and has

been thoroughly reviewed (7,8). Because a convenient procedure for verification of the

EMG character of chromatographic peaks exists (5), these equations can be confidently

applied once EMG peak shape has been established.

A variety of approaches has been used to make this model more practical for

routine use.(5,9-15) One particularly popular approach has been to relate the graphical

parameters a and b, (Figure 3-1), measured at a particular fraction of the peak height, to

the fundamental EMG parameters, T and Gq . Relationships between a and b and the

EMG parameters have been reported in the form of both graphical curves and empirical

equations. Both techniques circumvent tedious, computer intensive curve fitting.

Equations reported by Foley and Dorsey (13) in 1983 have perhaps shown the greatest

routine applicability, based on the number of citations to that article. In that work,

equations were derived for the calculation of many important chromatographic peak

parameters (i.e., T, Gq , statistical moments, skew, excess and efficiency) that were

collectively termed chromatographic figures of merit (CFOMs). Equations based on the

measurement of b and a at 10%, 30%, and 50% of the peak height fraction were

4 0

reported, although only the equations at 1 0% were recommended for calculation of the

CFOMs. The equations at 30% and 50% were intended solely for peak modeling.

Although the CFOM equations reported by Foley and Dorsey earlier were very

accurate and precise, they were meant to be used with manually measured b and a values

where the accuracy of these measurements was the limiting factor in the accuracy of the

equations. Due to the greater accuracy available with modem electronic integrators and

data systems, the accuracy of CFOM measurement may be unnecessarily limited by the

equations themselves. Two further limitations are: (i) the somewhat narrow asymmetry

(b/a) range of the equations; and (ii) the absence of equations based on peak

measurements higher than 50% of the peak height

With regard to (ii), the ability to use b and a from higher peak height fractions is

desirable when peaks are overlapped because distortion from the adjacent peak is less at

higher peak height fractions. We have shown that accurate measurement of CFOMs

utilizing b and a measurements at 75% of the peak height is possible for both overlapping

and resolved peaks using improved equations we have not reported before.(l) In this

paper we report those improved equations (from the standpoint of the previous

limitations) for the measurement of peak area, variance (M2 ), third (M3 ) and fourth (M4 )

statistical moments, excess, and skew.

EXPERIM ENTAL

An Apple Macintosh Plus microcomputer programmed in Microsoft Basic was

used for EMG peak generation. The EMG function and universal data were calculated as

described before (7) via a search algorithm. All polynomial curve fitting was done using

commercially available software.

41

EMG Peak Generation. Equation 1 shows the form of the EMG function

used.z

hEMG(t)-A — V2 "ex p [I (S i)2 . ( i l i f i ) ]X X X

A is the peak amplitude, X is the exponential modifier, G q is the standard deviation of the

unmodified Gaussian , tQ is the retention time of the unmodified Gaussian and z = (t -

tc ) /O G - G q / t . The ratio X/Oq is used to describe the overall shape of an EMG peak; at

X/<J<3 close to zero, the peak approaches Gaussian shape, while higher T /G q values give

greater tailing. Values of A = 1, Iq = 100 and G q - 5 were used for evaluation of the

function. The times for tR (EMG peak apex), ta, and t^ (Figure 3-1) at r (peak height

fraction) = 0.10, 0.25, 0.5, 0.71, 0.73, 0.75, 0.77, and 0.79 for V G q values from 0 to

4.5 in increments of 0.05 were obtained to within 0.001 using a search algorithm. The

peak height (hp) at each V G q was determined concurrently with tR using the same search

algorithm. From these values, a, b, W (a +b), and asymmetry (b/a) could be calculated.

The values of asymmetry (b/a), W /G q , O r - 1q ) / G q and hp, termed universal data, were

used in the following derivations.

Our approach was to derive width and asymmetry based equations for G q and

M2 , to use equation 2 to determine X, and to use the equations (13) listed in column 1 of

Table 3-2 for the computation of the remaining moments, excess, and skew.

METHOD OF DERIVATION

12]

4 2

We found this approach slightly more accurate than developing asymmetry based

equations for Gq and T/Gq, and then calculating 1 from their product Due to the

inferiority of the latter approach, it is not discussed further. Equations at several peak

heights were derived, with those for 0.71 < r < 0.79 developed for purposes of

averaging to reduce uncertainty and bias.

Derivation of Equations for G q . For each r, construction of a plot of

Wj/Gq v s asymmetry (corresponding to the T/Gq range given above) gave results similar

to Figure 3-2 for r = 0.25. As shown, the plot is curved at the lower b/a values and

linear at the higher values. Therefore, a quadratic least-squares fit was used at lower

asymmetries and a linear least-squares fit was used for higher asymmetries resulting in

two equations at each r. G q could then be related to width and asymmetry as shown in

equation 3.

f(b/a) [3]

where f(b/a) is the resulting fitted equation of the form f(b/a) = Co + Cj(b/a) + C2 (b/a)“

(The coefficient C2 is 0 for f(b/a) resulting from the linear fit). Table 3-1 lists the

coefficients obtained at each r for both linear and quadratic fits and their valid asymmetry

ranges. The cut point between curved and linear fits was determined so that the relative

error in G q over the entire b/a range was minimized.

Derivation of Equations for M2 . Analogous to the derivation of G q , plots

of M2/W r 2 v s asymmetry were constructed at each r. However, as shown in Figure 3-3

for r=0.25, curvature was evident over the entire b/a range. Because an accurate

quadratic fit over the entire b/a range was not possible, two least-squares quadratic fits

were used, again choosing the cut point to minimize relative error over the entire

4 3

asymmetry range. Equation 4 shows the resulting relationship between M2 , Wf2 and

f(b/a).

M2 = Wr2f(b/a) [4]

The values of the coefficients for calculation of f(b/a) are given in Table 3-1. At r values

of 0.71 to 0.79, the plots were similar to those obtained for the G q derivations, thereby

allowing linear and quadratic fits for the two sections of curves, as before.

Derivation of Area Equations. Area equations based on width, asymmetry,

and peak height measurements at r = 0.10,0.25, 0.50, and 0.75 have been previously

reported (5) along with the method of their derivation. Additional area equations at r =

0.71,0.73,0.77, and 0.79, were derived in this report for purposes of comparison.

The area equations for these r values are reported in Table 3-3 along with the area

equations at r = 0.10, 0.25, 0.50, and 0.75 previously reported (5).

RESULTS AND DISCUSSIO N

Accuracy of CFOM Equations. Tables 3-1 and 3-2 show the maximum

errors over the T /G q range 0.3 to 4.3 for the equations derived for the various CFOMs

considered in this study. In general, the equations for G q are accurate to within 1%

while those for M2 are accurate to within 2%. Table 3-4 compares the results of the

maximum errors reported for the Foley-Dorsey (13) equations to those reported in Tables

3-1 and 3-2. Because Anderson and Walters (12) also derived equations for M j, M2,

Gq , and T, the relative errors for their equations are reported in Table 3-4 for purposes of

comparison. Only CFOMs at r = 0.10 are compared, because the Foley-Dorsey

equations at other r values are useful only for peak modeling and should not be used for

4 4

CFOM calculation. For all CFOMs considered, the accuracy of the present set of

equations is better than those derived by the previous methods. Although the Anderson -

Walters equations provide a greater valid asymmetry range, most experimental peaks will

have asymmetries within the valid range of the present equations. Because our equations

give better accuracy than the other methods while providing for the calculation of the

higher moments (whereas the Anderson - Walters equations do not), they should be the

method of choice in most instances. Also, our equations for higher r (not available

previously) allow the calculation of CFOMs when the lower peak height fractions are not

accessible due to peak overlap.

Precision of CFOM Equations. Also reported in Tables 3-1 and 3-2 are the

relative uncertainties of the CFOMs based on error propagation. Percent relative

standard deviations (%RSDs) of 0.25% and 0.5% for W and asymmetry, respectively,

were propagated through the various equations. Note that the actual precision of the

width and asymmetry measurements may be better for some electronic integration

systems. As seen in Table 3-1, the uncertainties for Oq and M2 are in general less than

2%, and less than 1% at some r values. From Table 3-2 it is evident that the values of

the higher moments, excess, and skew are slightly more imprecise. Where a range for

the uncertainty is reported, the larger number refers to the least asymmetric peak, while

the smaller number refers to most asymmetric peak. In general, the uncertainty decreases

exponentially from the larger value to the lower value.

Accuracy and Precision of Area Equations. The maximum errors and

uncertainties for the area equations derived here, along with those reported previously,

are shown in Table 3-3. The errors at all r are well within 1.5%, and are not more than

1.0% in most cases. Assuming the same %RSD values for width and asymmetry as

before, and a %RSD in peak height (hp) of 0.25%, the resulting uncertainties for all of

the area equations are less than 0 .6 %.

4 5

C O N C LU SIO N

It is hoped that the more accurate equations presented here will facilitate greater

use of the EMG peak model within electronic integrators and data systems. Because

widths and asymmetries are now commonly measured by many chromatographic data

systems, integration of these equations into existing software should be easy, while

providing better accuracy in the desired CFOMs. Also, as shown before (1), these

equations provide for the direct CFOM measurement of the first eluted peak in an

overlapped pair, even when the valley height exceeds 50% of the first peak's height.

ACKNOW LEDGEM ENT

The authors gratefully acknowledge the partial support of this work by the Computer-

Aided Chemistry Division of the Perkin-Elmer Corporation.

Table 3-1. Coefficients, Accuracy, and Precision of Equations for Calculating Oq, and M2 of Gaussian and Exponentially Modified Gaussian (EMG) Peaks.3

peak height asymmetry asymmetry coefficients maximum relativefraction, r range,t/o range, (b/a)r c 2 Cl c 0 error b uncertainty c

0 .1 0 0.30-1.15 1.03-1.46 -1.2951 6.6162 -0.9516 -0 .2 ,+0 .6 ±0.52

1.20-4.30 1.46-3.67 0 3.3139 1.11470.135 0.30-0.65 1.02-1.14 -4.1347 12.7655 -4.5742 -0.2, +0.3 ±0.53

0.65-4.30 1.14-3.42 0 3.1665 1.00510.25 0.30-1.05 1.02-1.26 -2.4352 8.8715 -3.0485 ±0 .2 ±0.52

1.10-4.30 1.26-2.85 0 2.8504 0 .6 8 8 8

0.50 0.30-1.25 1 .0 1 - 1 .21 -4.2670 12.5178 -5.8561 -0.4, +0.8 ±0 .8

1.30-4.30 1.21-2.09 0 2.4685 0.09810.71 0.30-1.85 1.01-1.23 -2.7901 9.0992 -4.6165 -0.5, +0.2 ± 1 .0

1.85-4.30 1.23-1.64 0 2.2527 -0.39830.73 1.30-1.85 1 .0 1 - 1 .2 2 -2.9116 9.3374 -4.8028 -0.5, +0.2 ± 1 .0

1.85-4.30 1.22-1.60 0 2.2449 -0.46660.75 0.30-1.25 1 .0 1 - 1 .1 2 -7.4174 18.8306 -9.8710 ±0.7 ± 1 .2

1.30-4.30 1.12-1.56 0 2.3245 -0.6604

0.77 0.30-1.85 1.01-1.19 -3.3185 10.1672 -5.3703 -0.5, +0.2 ± 1.1

1.85-4.30 1.19-1.52 0 2.2326 -0.61110.79 0.30-1.90 1.01-1.19 -3.3724 10.2546 -5.4779 -0.5, +0.2 ± 1 .2

1.90-4.30 1.19-1.49 0 2.2268 -0.6859

Table 3-1. Cont'd

M2

peak height asymmetry asymmetryfractions range,t/c range, (b/a)r C2 x 10

0 .1 0 0.30-1.15 1.03-1.46 0.1270

1.15-4.30 1.46-3.67 -0.02990.135 0.30-2.85 1.03-2.50 -0.0378

2.85-4.30 2.50-3.42 0

0.25 0.30-0.70 1.02-1.13 2.34180.70-4.30 1.13-2.85 -0.1842

0.50 0.30-1.25 1 .0 1 -1 .21 7.91611.30-4.30 1.21-2.09 -1.3369

0.71 0.30-1.30 1.01-1.14 52.7771.30-4.30 1.14-1.64 0

0.73 0.30-1.35 1.01-1.14 63.5451.35-4.30 1.14-1.60 0

0.75 0.30-1.35 1.01-1.13 84.4481.40- 4.30 1.13-1.56 0

0.77 0.30-1.70 1.01-1.17 79.7711.70-4.30 1.17-1.52 0

0.79 0.30-1.70 1.01-1.16 103.9561.70-4.30 1.16-1.49 0

coefficients maximum relativeCj x 10 Co x 10 error b uncertainty c

-0.06458 0.4766 -0 .2 , +0 .6 ±0 .6

0.3569 0.19090.4993 0.1470 -1.4, +0.5 ±0 .6

0.2493 0.5394-4.1376 2.6930 -0.2, +0.3 ±0 .8

1.6032 -0.5715-12.8196 6.6750 -0.6, +0.4 ±1.5

9.3616 -6.6407-96.293 47.101 -0 .6 , +0 .8 ±2.425.039 -22.696

116.389 56.743 -1.0, +0.5 ±2 .6

29.935 -27.601157.205 77.0333 -1.3, +0.7 ±2 .8

35.9111 -33.5484142.903 67.779 -1.3, +0.7 ±2.744.119 -41.993

188.641 89.840 -1.3, +0.7 ±2.954.353 -52.398

Table 3-1. cont'd

a General form of the equations for each parameter: G q = Wr / f(b/a); M2 = Wr 2 x f(b/a), where Wr = width of peak at the peak

height fraction given by the subscript, b/a = an asymmetry factor measured at the same peak height fraction as the width, and f(b/a) = Co + Ci(b/a) + C2 (b/a)2.

b Maximum error of the equations over the asymmetry range.

c Percent relative standard deviation (%RSD) predicted from error propagation, assuming RSDs of 0.25% and 0.5% for Wr and

b/a. Note that the precision of W and b/a can be better than what we have assumed for many data acquisdon systems. Where a range is reported, the larger number refers to the least asymmetric peak (smallest b/a value) and the smaller number refers to the most asymmetric peak (largest b/a value). The uncertainty decreases exponentially from the larger value to the lower value as b/a increases.

00

Table 3-2. Error and Uncertainty of Chromatographic Peak Parameters calculated from Oq and M2 using the Equations in Table I.a

Parameter

1.T = (M2 -(7g2)1/2

2) l/G

Peak height asymmetry asymmetry maximum relativefraction ,r range, x/a range, (b/a)r error uncertainty

0 .1 0 0.5- 4.3 1.09-3.67 -0.4, +0.2 2.4-0.3

0.135 0.5-4.3 1.08-3.42 -0.3, +0.7 2.6-0.3

0.25 0.5-4.3 1.07-2.85 ±0.4 3.0-0.30.50 0.5-4.3 1.04-2.09 -0.4, +0.6 4.2-0.4

0.71 0.5-4.3 1.03-1.64 -0.4, +0.6 5.8-0.60.73 0.5-4.3 1.03-1.60 -0.7, +0.4 6 .1-0.70.75 0.5-4.3 1.03-1.56 ±0 .8 6.5-0.70.77 0.5-4.3 1.02-1.52 -0.5, +1.0 7.0-0 .8

0.79 0.5-4.3 1.02-1.49 -0 .8 , + 1 .0 7.3-0.8

0 .1 0 0.5-4.3 1.09-3.67 -0.8, +0.3 2.5-0.6

0.135 0.5-4.3 1.08-3.42 -0.3, +0.8 2.7-0.60.25 0.5-4.3 1.07-2.85 ±0.5 3.0-0.60.50 0.5-4.3 1.04-2.09 -1 .0 , +0 .6 4.3-0.70.71 0.5-4.3 1.03-1.64 -0.6, +0.9 5.9-0.90.73 0.5-4.3 1.03-1.60 -0.6, +0.9 6.2-0.90.75 0.5-4.3 1.03-1.56 -0.9, +1.4 6 .6 - 1 .0

0.77 0.5-4.3 1.02-1.52 -0.6, +0.9 7.1-1.00.79 0.5-4.3 1.02-1.49 ±0 .8 7.4-1.1

Table 3-2. Cont'd

Parameter Peak height asymmetry asymmetryfraction/ range, t/o range, (b/a)r

3) M3

4) M4

0 .1 0 0.5-4.3 1.09-3.67

0.135 0.5-4.3 1.08-3.420.25 0.5-4.3 1.07-2.850.50 0.5-4.3 1.04-2.090.71 0.5-4.3 1.03-1.640.73 0.5-4.3 1.03-1.600.75 0.5-4.3 1.03-1.560.77 0.5-4.3 1.02-1.520.79 0.5-4.3 1.02-1.49

0 .1 0 0.5-4.3 1.09-3.67

0.135 0.5-4.3 1.08-3.420.25 0.5-4.3 1.07-2.850.50 0.5-4.3 1.04-2.090.71 0.5-4.3 1.03-1.640.73 0.5-4.3 1.03-1.600.75 0.5-4.3 1.03-1.560.77 0.5-4.3 1.02-1.520.79 0.5-4.3 1.02-1.49

maximumerror

-1.0, +0.5

-0.9, +2.2 - 1.2, + 1.1 -1.2, +1.9 -1.1, +1.7 -2.0, +1.4 -2.4, +2.5 -1.5, +2.9 -2.5, +2.9

-0.6, +0.9

-0.5, +1.2 -0.5, +0.9 ±0.9 - 1.2, + 1.6 -2.3, +1.4 -2.8, +2.3 -1.7, +1.6 -2.9, +1.7

relativeuncertainty

7.3-0.9

7.8-0.98.8-0.9 12.6- 1.217.5-2.018.3-2.019.6-2.120.9-2.322.0-2.4

3.1-1.2

3.1-1.23.5-1.24.9-1.6 6.7-2.67.0-2.77.6-2.88.0-3.0 8.4-3.2

Table 3-2. Cont'd

Parameter

5) skew = M3 / M 2 ^

6 ) excess = M4 / W ifi - 3

Peak height asymmetry asymmetryfraction,r range, x/o range, (b/a)r

0 .1 0 0.5-4.3 1.09-3.67

0.135 0.5-4.3 1.08-3.420.25 0.5-4.3 1.07-2.850.50 0.5-4.3 1.04-2.090.71 0.5-4.3 1.03-1.640.73 0.5-4.3 1.03-1.600.75 0.5-4.3 1.03-1.560.77 0.5-4.3 1.02-1.520.79 0.5-4.3 1.02-1.49

0 .1 0 0.5-4.3 1.09-3.67

0.135 0.5-4.3 1.08-3.420.25 0.5-4.3 1.07-2.850.50 0.5-4.3 1.04-2.090.71 0.5-4.3 1.03-1.640.73 0.5-4.3 1.03-1.600.75 0.5-4.3 1.03-1.560.77 0.5-4.3 1.02-1.520.79 0.5-4.3 1.02-1.49

maximum relativeerror uncertainty

-1 .0 , +0 .6 7.3-1.2

-0.7, +1.5 7.9-1.3-1.2, +0.9 8.9-1.3-1.2, +1.3 12.7-1.6-1 .0 , +1.1 17.7-2,7-1.1, +0.7 18.5-2.8-2.3, +1.8 19.8-3.0-0.4, +2.1 21.2-3.1-0.9, +1.8 22.2-3.3

-1.3, +0.8 42.1-2.5

-0.9, +2.0 45.4-2.6-1 .6 , + 1 .2 51.1-2.7-1 .6 , + 1 .8 73.0-3.4±1.4 102-5.5-1.4, +0.9 106-5.8-3.0, +2.5 105-6.1-0 .6 , +2 .8 122-6.4-1.2, +2.5 128-6.8

Table 3-2. Cont'd

a Refer to Table 3-1 for an explanation of terms.

5 3

Table 3-3. Peak Area Equations for EMG and Gaussian Peaks a

r equation %RE b %RSDC

0 .1 A = 0.586 hp W0.i (b/a)-0 1 3 3 ±0.50 0.36

0.135 A = 0.631 hp Wo. 135 (b/a)-° -106 -0.3, +0.6 0.36

0.25 A = 0.753 hp W0.25 -1 .0 , +0 .6 0.35

0.50 A = 1.07 hp W0 .5 (b/a)* 0 -236 -1 .2 , + 1 .0 0.37

0.71 A = 1.514 hp W0 .7 i (b/a)+0-591 - 1 .1 , +0 .6 0.46

0.73 A = 1.58 hp W0.73 (b/a)* 0 -661 -0.7, +1.1 0.48

0.75 A= 1.64 hp W0.75 (b/a)*0-71? -1 .1, +0 .6 0.50

0.77 A = 1.73 hp W0.77 (b/a)* 0 -763 -0.9, +0.6 0.52

0.79 A = 1.82 hp W0.79 (b/a)* 0 -836 -1.0, +0.4 0.55

a A = Area of peak, hp = peak height, W = width of peak at designated peak height

fraction, and b/a is the asymmetry factor measured at the same peak height fraction as the width. See Figure 1.

b Largest relative error in Area over the interval 0 < x/ct < 4.3, expressed as a percentage.

c Percent relative standard deviation (%RSD) of the area measurement predicted from error propagation assuming RSDs of 0.25%, 0.25%, and 0.5% for hp, W, and b/a. Note that for many data systems the precision of hp, W, and b/a can be better than

what we have assumed.

Table 3-4. Comparison of maximum errors for the CFOMs reported here and those reported previously.

Parameter3 This Report Foley and Dorsey b Anderson and Waltersc

Valid b/a range

maximumerror

Valid b/a range

maximumerror

Valid b/a range

maximumerror

<*G 1.03-3.67 -0 .2 ,+0 .6 1.09-2.76 -1.0, +0.5 1.0-5.21 -1.3, +1.6

T 1.09-3.67 -0.4, +0.2 1.09-2.76 -1.0, +3.5 1.0-5.21 -4.0, +0.4

t /o G 1.09-3.67 -0.8, +0.3 1.09-2.76 -1.0, +4.5 d d

M! d d 1.00-2.76 ± 1 .0 1.0-5.21 -0 .2 , +0 .1

m 2 1.03-3.67 -0 .2 , +0 .6 1.00-2.76 -1.5, +0.5 1.0-5.21 -0.7, +1.0

m 3 1.09-3.67 -1.0, +0.5 1.09-2.76 -2.5, +10.5 d d

M4 1.09-3.67 -0.6, +0.9 1.09-2.76 -3.0, +1.5 d d

skew 1.09-3.67 -1.0 , +0 .6 1.09-2.76 -1.0 , + 1 0 d d

excess 1.09-3.67 -1.3, +0.8 1.09-2.76 -1.5, +14 d d

014k

Table 3-4. Cont'd

a Parameters based on r = 0.10 measurements,

b From Foley, J.P.; Dorsey, J.G. Anal. Chem. 1983, 55, 730-737

c From Anderson, D J.; Walters, R.R. / . Chromatogr. Sci. 1984,22, 353-359

d Equations not derived for this parameter.

5 6

Peak height = h_pPeak width = W = a + b

Peak asymmetry = (b/a)

Figure 3-1. Graphical parameters necessary for calculation of various CFOMs.

5 7

9.0 n

8.0

7.00.25

G

6.0

5.0

4.0-

3.01.0 1.5 2.0 3.02.5

x a '0.25

Figure 3-2. Plot construction for determination of f(b/a) for calculation of <Tg at r=0.25.

Note curvature at lower asymmetries.

5 8

0.25

'0.25

0.15

0.053.02.52.01.51.0

a 7 0.25

Figure 3-3. Plot construction for determination of f(b/a) for calculation of M2 at r =

0.25. Due to curvature over entire asymmetry range, two least-squares

quadratics were fit.

5 9

LITERATURE CITED

1) Jeansonne, M.S.; Foley, J.P. J. Chromatogr. 1989, 461, 149-163

2) Grushka, E. Anal. Chem. 1972, 44, 1733-1738

3) Giddings, J.C. "Dynamics in Chromatography", Part 1; Chromatogr. Science

Series, Vol. 1; Dekker, 1965

4) Sternberg, J.C. "Advances in Chromatography”; Giddings, J.C., Keller, R.A.,

Eds.; New York: Marcel Dekker: 1966; Vol. 2, pp. 205-270

5) Foley, J.P. Anal. Chem. 1987, 59, 1984-1987

6 ) Naish, P J; Hartwell, S. Chromatographia 1988, 26, 285-296

7) Foley, J.P.; Dorsey, J.G. J. Chromatogr. Sci. 1984, 22, 40-46

8 ) Jeansonne, M.S.; Foley, J.P. 1990, Submitted to / . Chromatogr. Sci.

9) Yau, W.W. Anal. Chem. 1977, 49, 395-398

10) Barber, W.E.; Carr, P.W. Anal. Chem.. 1981, 53, 1939-1942

11) Jung, K.H.; Yun, S.J.; Kang, S.H. Anal. Chem. 1984, 56, 457-462

12) Anderson, D.J.; Walters, R.R. J. Chromatogr. Sci. 1984, 22, 353-359

13) Foley, J.P.; Dorsey, J.G. Anal. Chem. 1983, 55, 730-737

14) Haddad, P.R.; Sekulic, S. J. Chromatogr. 1988, 459, 79-90

15) Wu, N.S.; Hu, M. Chromatographia 1989,28,415-416

Chapter 4

M EASUREM ENT OF STATISTICAL MOMENTS OF

RESOLVED AND OVERLAPPING CHROMATOGRAPHIC

PEA K S

Reprinted with permission from the Journal o f Chromatography, Vol. 461, pp. 240-253,

1989.

6 0

61

Department o f C hem is try

L o u i s i a n a S t a t e U n i v e r s i t y and agricultural and mlch apical colli-clBATON ROUGE • LOUISIANA • 70803-1804 , , , ,

Mrs. M. Verhaar Elsevier Science Publishers P.O. Box 330 1000 AH Amsterdam The Netherlands

Dear Mrs. Verhaar:

I am writing to you in reference to the article entitled," Measurement of S tatis t ical Moments of Resolved and Overlapping Chromatographic Peaks", published in the Journal o f Chromatography, Vol. 461, pp. 240-253, 1989, for which I am the f i rs t author. I would like to use the manuscript as part of my Ph.D. dissertation. The completed dissertation will be submitted to University Microfilms, Incorporated. Please forward permission for reprint of the manuscript.

I appreciate your prompt reply.

Permission granted subject to permission from the author(s) and to full acknowledgement of the source.Elsevier Scienoe Publishers Physical Sciences & Engineering Div.

504/388-3361 FAX 5041388-3458

September 13, 1990

Sincerely,

Mark S. Jeansonne

6 2

INTRO DUCTIO N

The importance of statistical moment analysis to the chromatographer cannot be

overemphasized because a large amount of information can be derived from such an

analysis. Statistical moment analysis can be used not only to measure directly parameters

such as area (zeroth moment), peak centroid (first statistical moment), and variance

(second statistical moment), but also to calculate other important parameters indirectly as

well. For example, column efficiency can be calculated from N=Mj^ / M2 , where N is

the column efficiency, M j is the first statistical moment, and M2 is the variance. Other

parameters, such as the third and fourth statistical moments give information on peak

asymmetry and peak flattening, respectively. Peak skew and excess are parameters

related to statistical moments and provide a measure of the deviation of the

chromatographic peak from a Gaussian peak profile.

Traditionally, statistical moments for digitally represented chromatographic peaks

have been approximated by the simple summation of the magnitude of the peak signal at

each data point between the peak start and stop limits, as shown in Fig. 4 - l a .

However, this approach to the calculation of statistical moments has several

shortcomings when applied to real chromatographic data.

First, it has been shown that the accuracy and precision of the calculation of

statistical moments is directly affected by the amount of noise present in the

chromatogram. The noise level has been shown to affect peak start / stop

assignments, and this affects the limits of summation and, consequently, the value of the

statistical moments calculated. 3

Secondly, the accuracy of the summation method (equations shown in Fig. 4-la)

deteriorates rapidly as the peaks begin to overlap. We have recently shown 4 that errors

6 3

in peak area can exceed 1 0 0 % when the summation approach (perpendicular drop

algorithm) is applied to overlapping peaks. As we shall show later in this report, errors

in the higher moments calculated via the summation method are usually much larger

under the same circumstances.

A final drawback of the summation method is that it is computationally intensive,

requiring numerous calculations (e.g. Fig. 4-la) for every data point in the peak of

interest This is particularly true for the higher moments and related parameters.

Although this problem has been alleviated somewhat by the advances in computer

technology (faster computations), the summation method remains noticeably time-

comsuming on many commercial chromatographs with microcomputer-based data

systems.

Most, if not all, of the problems associated with the measurement of statistical

moments can be reduced or eliminated if one has an accurate model for the

chromatographic peaks of interest. A model that has been reported to be accurate for

most chromatographic peaks 5-7 is the Exponentially Modified Gaussian (EMG)

function, which is the convolution of a Gaussian and an exponential decay function.

Recently, we introduced 8 a convenient procedure for determining whether or not the use

o f the EMG model is appropriate. This procedure utilizes empirical equations for

calculating peak area based on peak width, asymmetry and peak height measurements.

Once the validity of the EMG model for a given set of peaks has been confirmed, these

equations can also be used for the accurate measurement of peak areas of overlapping

chromatographic peaks. 4. Note that this method relies on the measurement of peak

width and asymmetry for the less distorted peak of the overlapped pair (the first peak) at

a point above the valley where distortion from the second peak is low.

6 4

Although some of the problems associated with the traditional measurement of

statistical moments can be reduced or eliminated via the use of a variety of sophisticated,

curve-fitting/deconvolution procedures, these procedures also have numerous

drawbacks. First, they are nearly always even more time-consuming than the traditional

summation approach. In many cases a final summation step is required after the

preliminary curve-fitting/deconvolution procedures. Secondly, some of the procedures

require multi-channel detection which is not always available. Thirdly and most

importantly, for a variety of reasons the curve-fitting/deconvolution approaches have not

yet proven to be sufficiently reliable. For example, with iterative procedures, lack of

convergence is frequently observed. In general, these and other disadvantages have

dissuaded most, if not all, commercial manufacturers from implementing the curve-

fitting/deconvolution approaches into their chromatographic data systems.

The purpose of this paper is to report an alternative to both the traditional and

least squares/deconvolution methods for the measurement of statistical moments. Our

present approach utilizes empirical equations (Fig. 4-lb) similar to those we already

reported for peak area 4, but also includes a veiy simple deconvolution procedure for a

pair of overlapping peaks. The derivation of these equations will not be included here,

as this topic will constitute a separate paper. 9 . For the remainder of this report, we will

refer to our method of statistical moment measurement as the width-asymmetry method.

EXPERIM ENTAL

Both an Apple Macintosh and IBM PC-AT were utilized for simulated peak

generation and other calculations. All programs were written in either Microsoft BASIC

or TRUE BASIC.

6 5

EMG Peak generation. All peaks generated were based on the EMG

function^ expressed as

hEMG(0 = A — V2- exp (2S ) 2 - < ^ ) J X X X

-y 2 exp ( ~ 2 ~ ^

V27Cdy [1]

where A is the peak area, tQ is the retention time, Gq is the standard deviation of the

Gaussian function, x is the time constant of the exponential decay function convoluted

with the Gaussian function, and Z = ( t-tQ ) / o q - g q / x . As the x / g q ratio increases,

the peak in question will become more skewed, and as it decreases, the peak approaches

a Gaussian shape.

Single chromatographic peaks at x / g q ratios of 0 ,0 .5 ,1 ,2 , 3, and 4, with g q a

constant at 0 .1 min, were generated for this study, using a sampling rate of three points

per second. As shown in Fig. 4-2 (x/ g q = 2), about 30 points per peak measured from

1 0 % peak height to 1 0 % peak height were needed for < 2 % error.

Overlapped chromatographic peaks at x / g q ratios of 0 .5 ,1 ,2 , 3, and 4 were

generated at resolution values of 0.625, 0.75, 0.875, 1, 1.125, 1.25, 1.375 and 1.5 by

using the same sampling rate as the single peaks. Resolution was defined as AtQ /

4(variance)l/2, where A t<3=tQ 2 - kj,l> anc* variance was defined as g q 2 + x% for an

exponentially modified Gaussian peak. The peaks were overlapped by adding two

individual, simulated peaks of equal area and x/ g q value. However, the degree of peak

overlap will be reported here as the percent valley due to the inadequacy of the resolution

parameter for fully describing tailed overlapped peaks. 4 Percent valley was defined as

hv / hp X 100%, where hv and hp are shown in Fig. 4-3.

6 6

Real Chrom atographic Peak Generation. Real single and overlapped pairs

of peaks were generated on a Series 400 liquid chromatograph (Perkin-Elmer, Norwalk,

CT, U.S.A.), using pyrene as the analyte. The mobile phase composition was 75%

aqueous acetonitrile at a flowrate of 1.5 ml/min. The column used was a Vydac pH-

stable C-8 column. A Model variable wavelength ultraviolet absoiption detector

(Isco, Lincoln, NB), set at 330 nm was used to detect pyrene. An Omega-2 data system

(Perkin-Elmer) utilizing an IBM AT computer, was used for storage of the

chromatograms.

Overlapped peaks were obtained from precise, rapid duplicate injections of a

standard solution of pyrene. This single-standard, rapid, duplicate-injection approach has

many advantages over a two-component standard solution method which would require

changing conditions to obtain different degrees of overlap. First, it allows the degree of

peak overlap to be easily controlled by simply varying the length of time between

injections. Second, this method avoids any relative change in the molar absorptivities of

two analytes in the mixture as mobile-phase conditions are changed to obtain different

degrees of overlap. Third, it permits a single peak to be obtained under the same

conditions as the overlapped peaks, thus allowing the statistical moments measured by

the summation and width-asymmetry methods for the isolated peak to be compared with

those measured for the overlapped peak pair without any concern about changes in the

peak shape and/or concentration. Finally, one can be confident that the true area ratio of

the overlapped peaks is unity, since equal amounts of the same compound are being

injected.

Two pairs of tailed, overlapping peaks with percent valleys of 40% and 67%

were generated. A single control peak with the same amount of peak tailing as the

6 7

overlapping peaks was also generated. The amount of peak tailing was adjusted by

adding or removing dead-volume ahead of the column.

Peak Param eter Measurement The equations given in Fig. 4-la, for

determining the zeroth through fourth statistical moments by summation, were applied to

both simulated and real peaks. For the simulated peaks, start/stop assignments (limits of

integration) for isolated peaks were taken as the point where the peak was determined to

be "on baseline", which depended on the " baseline level" being used for the peak.

Baseline levels of l x l 0 ' 9 and 3x10"^ were used, which corresponded to approximately

0.00 % and 0.1% of the peak height, respectively. The lower baseline was used as an

ideal baseline in order to obtains maximum level of accuracy for purposes of

comparison. The ideal baseline was chosen to be slightly above zero, since the value of Z

in Eq. 1 necessary to give a zero baseline would result in an overflow condition. This

ideal baseline was used only for peaks in which no noise was present, since noise affects

peak/start assignments in real chromatograms.

For overlapped peak pairs, the starting point for the first peak and stopping point

for the second peak were chosen as for isolated peaks. The peak stop for the first peak

and the peak start for the second peak were taken as the intersection of the baseline being

used and a perpendicular line, drawn to the minimum of the valley between the peaks

(see Fig. 4-3 ). This method for dealing with overlapped peaks is commonly referred to

as the perpendicular drop algorithm.

For the real chromatographic peaks, the baseline level and the start/stop

assignments were determined by the data system, the perpendicular drop method being

employed for overlapped peaks. The peak detection algorithms in the data system were

optimized for the types of real peaks that were generated.

68

The widths of the peaks at 10%, 25%, 50%, and 75% relative peak height were

determined by utilizing a four point least squares fit where four points on each side of the

peak, symmetric about the particular height, were used and the difference in time

between the two points ( ta - t^w as taken as the peak width (see Fig. 4-3). Four points

were used, since the accuracy of the value obtained for peak width did not increase when

more points were fit.

Peak height was obtained by subtracting the baseline value being used from the

peak maximum obtained via a quadratic least-squares curve fit of the seven highest points

in the peak. The seven-point group was selected so that the middle point had the highest

value. The time at which the maximum was calculated from the quadratic fit was used as

the retention time of the peak. Seven points were used, since this number represented a

compromise between the optimum number of points for a peak with a t/sQ ratio of 1

(mildly skewed) and a peak with a t /o q ratio of 4 (heavily skewed) for the data sampling

rate used. This compromise was selected so that the quadratic fit could be used for peaks

for which the value of t/Oq was not known, as in real chromatograms. The asymmetry

of the peak was taken as the value of b/a, where a and b were determined as shown in

Fig. 4-3 at the appropriate peak heights.


Simulated peaks without noise. The values for the area and the variance

obtained by the summation and width-asymmetry methods for simulated peaks and an

ideal baseline are compared in Fig. 4-4 as a function of the percent valley between the

peaks. For only slightly tailed peaks (x / g q = 1, Fig. 4-4 a), it appears that the

summation method for peak area and variance is fairly accurate for the noiseless peaks

69

used in this study. However, for moderately tailed peaks (x/ g q = 2, Fig. 4-4 b), the

area of the second peak and the variance of the first peak have become much less accurate

relative to the same parameters measured by using the width-asymmetry method. This

trend continues in Fig. 4-4 c for x / g q = 4. Thus, the width-asymmetry method can be

used to measure accurately both the area and variance for the left peak of a highly skewed

and overlapped pair of peaks, while the same parameters cannot be measured as

accurately for either peak when the summation method is used.

Fig. 4-5 illustrates the same results for peak area and variance as Fig. 4-4 c, but

with a less ideal baseline. Here, the variance for the first peak, as measured by the

summation method, is very inaccurate. This is due to a significant portion of the tail of

the first peak being truncated by the higher baseline, and shows the sensitivity of the

variance to baseline eirors when measured by the summation method. However, the

variance measured by our width-asymmetry method does not show this sensitivity.

The errors in the higher moments, skew, and excess are compared in Fig. 4-6 for

an overlapped pair of highly skewed peaks ( x/ g q = 4) and an ideal baseline. Since an

ideal baseline was simulated, the moments measured for the right peak by the summation

method (see Fig. 4-6 a) show a fair accuracy up to a high percent valley for these

noiseless peaks. However, results this accurate cannot be expected for real

chromatographic peaks, due to the problems outlined in the introduction for the

summation method. As expected, the higher moments, skew, and excess, measured by

the summation method for the left peak, show greater sensitivity to the truncation of the

peak tail than do the area and variance for the left peak, measured by the summation

method.

However, the higher moments, skew, and excess measured by the width-

asymmetry method for the left peak do not show this sensitivity (see Fig. 4-6 b). In fact,

7 0

the accuracy for these parameters, measured by the width-asymmetry method, is much

better than that obtained by the summation method for the right peak. This shows that the

width-asymmetry method is tetter overall for measuring the higher statistical moments,

excess, and skew for at least one peak of an overlapped pair of peaks.

Table 4-1 shows the maximum percent valleys (maximum overlap) for which the

two methods described here are in error by less than 5%. As seen there, for most of the

moments calculated by the width-asymmetry method for the left peak, the maximum

overlap that can be tolerated is higher. However, for the right peak, some of the

moments calculated by the summation method are more accurate. These results were

expected, since the tail for the right peak is fully included in the limits of integration.

Peak overlap of the peaks prevents the tail of the left peak from being included in the

summation method. This confirms the well-known results that the tail of a skewed peak is

especially important in calculating the higher moments by the summation method. Also,

the results show that the degree of distortion in the left peak is low for two overlapped

EMG peaks. Overall, these results indicate that all the statistical moments, including

excess and skew can be measured accurately for peaks that are moderately overlapped,

the more skewed peaks giving the test results. This latter trend is due primarily to the

percent valley parameter, which tends to underestimate peak overlap for symmetrical

peaks and to overestimate peak overlap for skewed peaks. (However, this measure of

peak overlap is no worse than any other parameter, and is more practical than most other

measures of peak overlap for skewed peaks 4.) Of course, Fig. 4-6 also show that for

those overlapped peaks which are not highly skewed the summation method may

occasionally give tetter results.

Although the summation method appears to be fairly accurate for the higher

moments of the second peak of a moderately to highly skewed overlapping peak pair, it

71

will generally be very inaccurate with real chromatographic data (peaks with noise). In

modem chromatographic integrators and data systems, much of the tail of even a mildly

skewed peak is often not included in the summation, due to baseline errors occurring

when the algorithm used detects a peak stop before the actual end of the peak is reached.

Many data systems rely on the first derivative, second derivative, or similar tests to detect

peak end with a slope sensitivity setting which depends on the degree of noise in the

chromatogram. ^ The slope sensitivity is set at a level higher than what might be

expected for the baseline drift. However, this setting may frequently also be higher than

the slope on the tail of a skewed chromatographic peak. On overlapped pairs of peaks,

the premature peak end would affect the second peak almost exclusively, therefore

disallowing the use of the second peak in accurate computation by the summation method

of the higher statistical moments for that peak, and often of the area and the second

moment as well. In contrast, the width-asymmetry method is relatively unaffected by this

type of truncation error. 11 Furthermore, in this example, the width-asymmetry method,

which is applied to the first peak of an overlapped pair, would be entirely unaffected by

the premature peak stop on the trailing edge of the second peak.

Results for real chrom atographic peaks. Table 4-2 shows the results for

a real isolated chromatographic peak, obtained under ideal conditions (high signal-to-

noise (S/N) ratios, no overlap, baseline resolution, etc.). As seen in Table 4-2, the zeroth

through fourth statistical moments, along with peak excess and skew for the single peak,

were found to be similar for the width-asymmetry and summation methods. Under less

ideal conditions, with a much smaller S/N ratio, the summation method would probably

give results very different from the width-asymmetry method, due to the limitations of

the summation method mentioned in the introduction.

7 2

The appropriateness of the EMG model for this real chromatographic peak is

demonstrated by the agreement obtained for the various peak parameters at different

relative peak heights. Although the agreement is not exact (there is a slight spread), the

spread in peak parameters is small compared to the error encountered when using the

summation method on most real peaks, again due to the problems outlined in the

Introduction.

The advantage of the width-asymmetry method over the summation method for

real peaks becomes apparent when overlapping peaks are examined. Table 4-3 gives the

results for the summation and width-asymmetry methods for two pairs of peaks that

overlap by different amounts. As seen for the 40% valley case, the summation method

gave relative areas of 39.5% for the left peak and 60.5% for the right peak peak. Since

the true relative areas of the peaks are 50%, the error in peak area for each peak is 10.5%

when using the summation method and perpendicular drop algorithm. In contrast, the

width-asymmetry method gave relative areas of 50.0% for each peak, i.e., exactly the

correct result.

For the more heavily overlapped peak pair (67% valley), the errors associated

with the summation method increased, whereas the width-asymmetry method again gave

results very close to the correct result (areas measured for the single peak shown in Table

4-2). The relative areas for the left and right peaks, determined by the summation

method, were in error by 17% for each peak, whereas the relative areas measured by the

width-asymmetry method were in error by only 1%. The other statistical moments,

including skew and excess for the left peak, show a large difference between those

calculated by the width-asymmetry and summation methods. Also, a comparison of these

parameters measured for the left peak by using the width-asymmetry method to those for

7 3

the single peak (see Table 4-2) shows that the width-asymmetry method gave very good

results.

Modified width-asym m etry method for true peak deconvolution. In

order to determine relative areas, the present width-asymmetry method requires

knowledge of the total area of the two overlapping peaks, as measured by the

summation method. The total area cannot be calculated by the width-asymmetry method,

because the distortion of the right peak by the tail of the left peak causes an erroneous

contribution to the total area. Thus, the width-asymmetry method, as presently

employed, is not a true peak deconvolution method, since parameters other than the peak

area are not additive and therefore cannot be determined for the second peak by

subtraction. However, the width-asymmetry method can be used to deconvolve an

overlapping pair of peaks as follows: First, the first peak of the overlapping pair is

calculated point by point via Eqn. 1 over an appropriate time interval from values of o q ,

x, and A that are estimated from peak width, asymmetry, and peak-height (area only)

measurements by using equations described elsewhere. 5 Next, tQ is estimated from Mi

- x and then adjusted so that the maxima of the calculated peak coincides exactly with the

maxima of the first peak in the overlapping pair of real peaks. Finally, the lone second

peak of the overlapped pair is obtained by subtracting, point by point, the values of the

calculated peak from the total chromatographic signal.

The accuracy of the width-asymmetry deconvolution method is evident from

Table 4-4 for real chromatographic peaks. As Tables 4-3 and 4-4 show, both the original

width-asymmetry method and the modified width-asymmetry/deconvolution method give

accurate results for all parameters for the first peak of the overlapped pair. Note,

however, that the original width-asymmetry method uses values of peak height, width,

and asymmetry obtained directly from the actual chromatogram (at 15% relative peak

7 4

height), whereas the width-asymmetry/deconvolution method uses the same values from

an artificially constructed peak.

Fig. 4-7 illustrates the results of the width-asymmetry deconvolution method,

applied to the chromatogram of overlapped peaks with the 40% valley (relative to the left

peak); superimposed on the real chromatogram are the two simulated peaks, obtained

from the width-asymmetry deconvolution method. As seen, the first simulated peak

falls directly on the actual first peak of the overlapped pair. This was expected, since

there is little distortion of the first peak in the overlapped pair from the second peak. The

distortion of the second peak in the overlapped pair, caused by the tail of the first peak, is

readily apparent, though, as the difference in both height and area of the second peak and

its corresponding simulated peak is large. This distortion is the reason why the

petpendicular drop method underestimates the area of the first peak and overestimates the

area of the second peak in an overlapped pair of peaks. It is also why, as mentioned

above and shown here, the width-asymmetry method cannot be applied directly to the

second peak of an overlapped pair of peaks.

Computational Time. As stated in the Introduction, one problem with the

summation method is that eveiy point in chromatographic peak must be involved in

moment calculations. However, most of the points in a peak of interest do not have any

calculations performed on them when the width-asymmetry method is used. When the

two methods were timed against each other, the width-asymmetiy method was found to

be about twice as fast for single-peak chromatograms and up to ten times faster for

multiple-peak chromatograms.

7 5

ACKNOW LEDGEM ENT

We gratefully acknowledge the support of this work by the Computer-Aided

Chemistry Department of the Perkin-Elmer Corporation and the National Oceanographic

and Atmospheric Administration (NOAA). Mark S. Jeansonne also thanks Jeffrey Crow

for his helpful discussion on this topic.

CREDITS

This work was presented in part at the 1988 Pittsburgh Conference and Exposition in

New Orleans, LA , February 1988, paper # 088.

Table 4-1

Maximum peak overlap (percent valleys) that can be tolerated by the width-asymmetry and summation methods for a given accuracy (<5% error)

PEAK AREA m2 m 3 m4 SKEW EXCESS

Width-asymmetry First 47 34 16 24 17 15

Method Second 4 17 8 8 8 7

Summation method/ First 9 < 2 < 2 < 2 < 2 < 2

ideal baseline* Second 8 34 5 11 5 2

Summation method/ First 8 < 2 < 2 < 2 < 2 < 2

non-ideal baseline** Second 9 24 9 < 2 9 < 2

* Peak start/stop corresponds to points where signal is 5 x 10"8 % of maximum.

** Peak start/stop corresponds to points where signal is 0.1% of maximum.

TABLE 4-2

Peak parameters measured for an isolated, real chromatographic peak (pyrene).

Height(%)

Peakarea(mV • s)

Mj M2 m3 M4 SKEW EXCESS

Summationmethod

NotApplicable

5449.6 1.718 0.0424 0.0186 0.0216 2.146 8.974

Width- 10 5370.0 1.722 0.0410 0.0149 0.0138 1.794 5.191asymmetry 25 5530.7 0.0440 0.0167 0.0159 1.807 5.242method 50 5670.8 1.892 0.0468 0.0184 0.0181 1.816 5.278

75 5802.7 0.0500 0.0205 0.0208 1.827 5.321

TABLE 4-3

COMPARISON OF SUMMATION AND WIDTH-ASYMMETRY METHODS FOR TWO SETS OF OVERLAPPING, REAL PEAKS

Overlapping peaks generated from rapid, duplicate injections of pyrene (see Experimental section). Therefore, values for all the parameters except Mj should be the same for all peaks within experimental error.

Summation method40% Valley*

Left Peak Right Peak

67% Valley*Left peak Right peak

Width-asymmetry method 40% Valley*

Left peak

Right peak

67% Valley* Left peak

* *

Right peak* *

Height (%) Peak area (mV«s)

Mj m2 m3 m4 SKEW EXCESS

Not applicable Not applicable

4481.46 8 6 6 .8

1.63292 .1 0 1 0

0.00930.0501

3.7X10' 50.0319

1.9X10-40.0469

-0.7952.839

0.04215.669

Not applicable Not applicable

3733.67347.5

1.60901.9974

0.00610.0416

-6.13X10-50.0175

5.1X10-50.0179

-1.6262.067

-0.1307.340

50%75%50%75%

5672.75676.86731.06435.0

0.04050.04190.03710.0354

0.01400.01500.01180.0113

0.01300.01410.01050.0098

1.7201.7441.6541.690

4.9075.0004.6564.794

1.9902

50%

75%50%75%

5694.27470.2 6594.9

0.04540.02180.0258

0.01700.00240.0062

0.01660 .0 0 2 0

0.0047

1.7610.7461.483

5.0631.6124.025

1.8933

TABLE 4-3 cont'd

* Valley height relative to left (first) peak.** Areas calculated as A r = (A l + AR)summation * other parameters calculated using the width-asymmetry

equations in Fig. 4-lb. Width-asymmetry equations should normally not be employed for the more distorted peak of an overlapping pair of peaks. Results are shown here for purposes of comparison only.

TABLE 4-4. PEAK PARAMETERS FOR DECONVOLVED PEAKS OF OVERLAPPING PEAK PAIRS WITH 40% AND 67% VALLEYS a

40% VALLEY** AREA0 M i m 2 m 3 M4 SKEW EXCESS

SIMULATED LEFT PEAK

5654.4 1.798 0.0415 0.0147 0.0138 1.738 4.976

SIMULATED RIGHT PEAK

5598.1 2.197 0.0411 0.0150 0.0139 1.804 5.229

67% VALLEY**

SIMULATED LEFT PEAK

5676.4 1.827 0.0450 0.0168 0.0164 1.755 5.042


5272.2 2.090 0.0356 0 .0 1 2 0 0.0104 1.780 5.138

a Values for each parameter were calculated from width-asymmetry equations atlO, 25,50 and 75% relative peak height The values obtained were then averaged for this table. The spread of values for each parameter never exceeded 5% for the left peak and 10% for the right peak.

b Valley height relative to left peak.

c Peak area given in units of mV-sec.

ooo

81

(a)oo Stop

Mq «= AREA «= Jh(t) dt - X h ( t ) At•oo start

oo StopM i= Jt*h(t) dt / Area - X l*h(t)At + Area

•oo start

oo StopM2 = J (t-Mi )2 »h(t) dt / Area - £ (t-M-j )2*h(t) At + Area

-oo start

oo stopM3 = f (t-M-j )3 »h(t) dt / Area * X (t_M1 )3 ®h(t) At + Area

start

00 stopM4 = f (t*Mi )4 *h(t) dt / Area « X (t_M1 )4*h(l) At + Area

-ed start

YS = SKEW - M3/M23/2 YE = EXCESS * M4/M22 - 3

(b)

CG - f f lg s j M2 = W 2 x f2(b/a)

M i = tR + W x f 3 (b/a) T = ' V m 2 - 0 G 2

M3 = 2 x3 M4 = 3 GG4 + 6 OG2 x2 + 9 T4

Ys = M3/M2 3/ 2 Y e = M4 /M2 2 - 3

Figure 4-1. Equations used for calculation of statistical moments and other peak

parameters by a) the summation method and by b) the width-asymmetry

method.

OBS

/ TR

UE

1.02i

1.0 0 -WIDTH

0.98-

AREA0.96-

0.94-ASYMMETR

0.920.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4 .0

LOG( # PTS,10%)

Figure 4-2. Effect of the data sampling rate on the measurement of peak width,

asymmetry, and peak area.

8 3

1 a

P,2

time

Figure 4-3. Measurement of graphic parameters for an overlapping pair of

chromatographic peaks. tr and hp are the retention time and peak height of

respective peaks, and hv is the height of the valley. Peak width at die

desired peak height fraction is given by ta-tfc.

8 4

Figure 4-4. Comparison of the errors in peak area and variance occurring in the

summation and width-asymmetry methods as a function of peak overlap

(percent valley) for: (a) x / g q = 1; (b) x /g q = 2; and (c) x /o q = 4. Labels

in the plot refer to: a) the parameter, b) the peak (first [L] or second [R]) for

which a parameter was obtained; and c) the relative peak height at which the

width and asymmetry were measured (width-asymmetry only). For

example, "M2, L; 75%" refers to the variance measured for the first peak of

the overlapped pair at 75% of the peak height, while "M2, R" refers to the

variance of the second peak measured by the summation method.

|OB

S|/T

RU

E

85

2 . 0 1

M2, L; 75%

1.5AREA, L; 75%

AREA, R: AREA, R; 75%

M2,R1.0

AREA,L

M2, R; 75%

M2,L0.5

0.00 20 40 60 80 100

% VALLEY

|OB

S| /

TRU

E

8 6

2.01

1 .5 - (b) M2, L; 75%

AREA, R; 75%AREA, R

AREA, L; 75%

1 .0 - M2, R

AREA, L

M2, R; 75%0 .5 -

M2, L

0.010060 804020

% VALLEY

anui/lsaol

87

2.0 n

AREA, R

1.5AREA, R;75%

M2, R : M2, L;75% : AREA, L;75%1.0

AREA, L0.5 M2, R; 75%

M2, L

0.00 20 40 60 80 100

% VALLEY

|OB

S| /

TRU

E

88

2.0

1.5 AREA, RAREA, R; 75%

M2, L; 75%: AREA, L; 75%1.0

M2, R

AREA, L

M2, R; 75%0.5

M2, L

0.00 20 40 60 80 100

% VALLEY

Figure 4-5. Comparison of the errors in peak area and variance occurring in the

summation and width-asymmetry methods as a function of peak overlap for

a highly skewed pair of peaks (x/ g q = 4) with a less than ideal baseline

level (0.1%). Conditions as in Fig. 4.

89

Figure 4-6. Comparison of the errors in peak parameters other than peak area and

variance for overlapped peak pairs with x/ g q = 4 occurring in: a)

summation method; and b) width-asymmetry method. Conditions as in

Fig. 4-4.

|OB

S|/T

RU

E

9 0

2 .0 1

1.5 -

M3, R ; M4, R ; SKEW, R ; EXCESS, R

1.0 -

M3, L ; M4, L0 .5 -

SKEW , L

EXCESS, L

0.00 20 40 60 80 100

% VALLEY

|OBS

| /

TRU

E

91

2.0 1

(b)

M3, L; 75% : M4, L; 75% : SKEW, L; 75% : EXCESS, L; 75%

SKEW, R; 75%

EXCESS, R; 75%0 .5 -

M3, R; 75%

M4, R; 75%

0.00 20 40 60 80 100

% VALLEY

SIG

NA

L (m

V)

9 2

400

REAL CHROMATOGRAM

SIMULATED LEFT PEAK250


100

-50

0 1 2 3TIME (MIN)

Figure 4-7. Visual interpretation of the width-asymmetry/deconvolution method. The

solid black line indicates the real overlapping chromatographic peaks, while

the lighter lines show the individual peaks that are predicted by the width-

asymmetry/deconvolution method.

9 3

REFERENCES

1. Chesler, S. N.; Cram, S. P Anal. Chem. 1971, 43, 1922 - 1933

2. Petitclerc, T . ; Guiochon, G. J. Chromatogr. Sci. 1976, 14, 531-535

3. Grushka, E.; Myers, M.N.; Schettler, P. D. Anal. Chem. 1969, 41, 889 - 892

4. Foley, J.P. J. Chromatogr. 1987, 384, 301 - 303

5. Foley, J.P.; Dorsey, J.G. J. Chromatogr. Sci. 1984, 22, 40 - 46

6 . Pauls , R.E.; Rogers, L. B. Anal. Chem. 1977, 49, 625 - 628

7. Grushka, E. Anal. Chem. 1972, 44 , 1733 - 1738

8 . Foley, J.P. Anal. Chem. 1987, 59, 1984 - 1987

9. Jeansonne, M.S.; Foley, J.P. This Dissertation, Chapter 3.

10. Colmsjo, A. L. Chromatographia 1987,23,257 - 260

11. Anderson, D.P.; Walters, R.R. / . Chromatogr. Sci. 1984,22,353-359

Chapter 5

THE Q TRANSFORM ATION: A NOVEL METHOD OF PEAK SH APE ANALYSIS.

Reprinted with permission from Burgess, S.W., Jeansonne, M.S. and Foley, J.P.;

submitted for publication in Anal. Chem. Unpublished work Copyright 1990 American

Chemical Society.

9 4

9 5

Department o f Chemistr\/

L o u i s i a n a S t a t e u n i v e r s i t y and agricultural and mechanical colleceBATON ROUGE • LOUISIANA • 70803-1804

Barbara Polanski Publications Division American Chemical Society 1 155 16th St. Northwest Washington, DC 20036

Dear Ms. Polanski,

I am writing to you in reference to the article en t i t led ," The Q Transformation: A Novel Method of Peak Shape Analysis", AC9006886, received by A nalytical Chem istry on August 24, 1990, for which I am the f i rs t author. I would like to use the manuscript as part of my Ph. D. dissertation. The completed dissertation will be submitted to University Microfilms, Incorporated. Please forward permission to reproduce the manuscript.


5041388-3361 FAX 504/388-3458

September 13, 1990

Sincerely,

Mark S. Jeansonne

American Chemical Society 9

PUBLICATIONS DIVISION 1155 SIXTEENTH STREET, N.W. W ASHINGTON, D.C. 20036 P h o n e (202) 872-4600 Fax (202) 872-6060

Septem ber 28, 1990

Mr. M ark S. Jeansonne D epartm ent o f Chem istry Louisiana S ta te University B aton Rouge, Louisiana 70803-1804

D ear Mr. Jeansonne:

Thank you for your le tter o f Septem ber 13, in which you requested permission to include in your thesis, two papers that you subm itted to Analytical Chemistry :"SingIe C hannel Peak Im purity D etection Using the Q Transform ation," and "The Q Transform ation: A Novel M ethod o f Peak Shape Analysis."

I would be happy to grant you this permission free o f charge provided that you print the required ACS copyright credit line on the first page o f your article and that your paper is published by ACS first before you submit it to UMI. Also, please inform UM I that permission is granted subject to the terms o f the ACS/UMI Agreem ent dated O ctober 1, 1984.

T he credit line we require is, "Reprinted with permission from FU LL R E FE R E N C E CITA TIO N Copyright Y EA R American Chemical Society." Please be sure to insert the appropria te inform ation in place o f the words in capital letters.

If your paper is no t accepted before you include it in your thesis, please use the following credit line: "Reprinted with permission from Burgess, S.W., (A UTHO RS NAM ES); subm itted for publication in Anal. Chem. U npublished work Copyright 1990 Am erican Chemical Society."

Thank you for your request. Please let me know if I can be o f any further help.

Sincerely,

Barbara F. Polansky Copyright A dm inistrator

9 7

INTRO DUCTIO N

In many forms of analytical chemistry the analyst must rely on a detection system

that gives a peak shaped signal as output. Most often the peak's height or area is

measured to calculate properties of the chemical or physical system being measured.

Sometimes, however, the shape of the peak holds special importance in supplying added

clues to the behavior of a particular system. For example, in chromatography the analyst

is usually interested in height, area, retention time and resolution between peaks to

characterize a sample. While the measurement of first three parameters may depend very

little on the shape of the signal peak, the accurate calculation of resolution or overlap

between adjacent peaks certainly does. Individual peak shapes in chromatography can

also give insight into many intra-column and extra-column effects that may or may not be

desirable. That is why much research has been done on peak shapes and peak shape

modelling in chromatography. In many other areas of analytical chemistry such as

nuclear magnetic resonance spectroscopy and infrared spectroscopy, chemists may also

be interested in peak shapes (1). In Mossbauer spectroscopy, for example, peak shapes

are especially important (2 ).

In this paper we wish to introduce a new method by which individual peak

shaped signals may be analyzed and/or compared. The new approach that we have

developed has become known by us as the Q transformation. In technical terms, it can

be thought of as a normalized, horizontal differentiation of a peak, in distinct contrast to

conventional "vertical" differentiation, i.e., differentiation of the signal with respect to

the abscissa.

Other peak shape analysis methods of similar computational complexity have

been introduced previously, including slope analysis (3,4), statistical moment analysis

(5), and the distribution function method (DFM) (6 -8 ). These methods have notable

98

deficiencies, however. Both the second derivative and moment methods are extremely

sensitive to noise and also, in the case of moments, baseline errors, thereby limiting their

usefulness when applied to real data (9-13). Finally, although the DFM handles noise

well, we will later show that it is not as good as the Q transformation at detecting small

changes in peak shape.

Definition of Q transformation. The Q transformation is defined and

illustrated in eq 1 and Figure 5-1,

where r is the peak height fraction (in percent) at which the measurement of Q is initiated,

b and a are graphical parameters defined in Fig.5-la, and Ab is calculated as shown in eq

2 with Aa measured similarly. By definition, the value of Q at any value of r will be less

than, equal to, or greater than unity for a fronted, symmetric, or tailed peak, respectively.

Illustrated in Figure 5-lb is the resulting Q profile (plot of Q vs r) for the peak of

Figure 5-la. As shown, the Q profile decreases monotonically toward a value of 1 as r

increases from 2 to 98%, consistent with the fact that all continuous, asymmetric peak

profiles become more symmetric as their apex is approached. Since the Q profile can

give visual indication of peak shape as a function of r, it can be used to visually

determine relative peak shape independent of time scale or sample concentration.

THEORY

[1]

[Ab]r = bj- - bj-_i [2]

9 9

As illustrated in Figure 5-2, it is very difficult to determine visually which peaks

are the same shape and which are fundamentally different. By using the Q transform,

however, it is easy to determine that peaks 1 and 2 are the same.

An important parameter, XQ, is simply the summation of Q over all r for a given

peak where r can run from 2 to 98%. As we will show, the XQ can give valuable,

compact peak shape information by compressing the Q profile data (Q vs r) into a single

number.

Properties of the Q transformation. In general, we expected the Q

transformation to be independent of both the time and concentration scales while being

very sensitive to peak shape. The results shown in Figure 5-3 for a wide range of

fundamentally different types of asymmetric peaks confirm these expectations. Shown

in Figure 5-3a are the results of applying eq 1 to single exponentially modified Gaussian

(EMG) peaks with X/(J between 0.5 and 4. The value of Q at any fraction of the peak

height depends only on the ratio X/O and not on the individual values of these width-

related parameters. Figure 5-3b shows the results for Gamma type peaks (eq 4) with a

ranging from 4 to 10. The value of Q here does not depend on the width-related

parameter (3 — only on a. The results for log-normal type peaks (Figure 5-3c, eq 5)

show that the value of Q at any height depends only on the asymmetry and not on the

value of width used in eq 5. An overlay of Figures 5-3a, 5-3b, and 5-3c, shown in

Figure 5-3d, illustrates that the shape of the Q profile also depends on the particular peak

family or class to which the Q transform is applied. These properties indicate that the Q

transformation can be used for both inter- and intra-family peak shape comparison.

A minor limitation of the Q transformation (eq 1) is its inability to distinguish

between different types of symmetric peaks (e.g., Gaussian vs Lorentzian). By

definition, any symmetric peak shape will give a value of 1 for Q at every value of r.

Note, however, that two or more perfectly symmetric peaks are seldom, if ever,

100

observed in chromatography or spectroscopy, even though many experimental peaks

may appear to be symmetric to the naked eye. Moreover the Q transform can distinguish

between a perfectly symmetric peak and a slighdy asymmetric peak. This situation is

much more probable when comparing the shapes of two peaks that appear to be

symmetric.

If the unlikely problem of comparing two symmetric peaks ever did arise, one

could simply redefine Q as Ab/Abref, and perform the comparison as shown in Figure 5-

4, where the Gaussian peak with a = 1.2 was arbitrarily used in the numerator, and the

Gaussian peak with ct= 1.0 was arbitrarily employed in the denominator (reference

peak). In this case the shapes of the two peaks are the same when one ignores scale, but

by defining Q differently, we are able to distinguish between similar symmetric peaks of

different widths. If two peaks were symmetric but of different shape (Gaussian and

Lorentzian), they could also be distinguished using this modified definition of Q. In

general, we believe such a modification of Q is unnecessary since both types of

symmetric peak shape comparisons (different width or different shape) can be performed

by simple (visual) inspection. The modification of Q is also undesirable because of the

reduction in sensitivity to peak shape differences and the lack of invariance to scale that

results. Therefore, the remainder of this paper focuses on the definition of Q given in

Eq. 1 and the comparison of peak shapes for more realistic scenarios (slightly

asymmetric to very asymmetric peaks).

Peak shape comparison using the Q transform.When comparing the

shapes of peaks, real or simulated, there can be two approaches when using the Q

transformation. Assuming that the individual peaks being compared belong to the same

family of peak shapes, the simpler approach is to compare values for XQ, the area under

the Q profile curve (Q vs. r), for the different peaks. As illustrated in Figure 5-3a, any

101

set of peaks belonging to the same class or family will have different values for XQ if

their shapes are different

Alternatively, a more sophisticated approach is to compare the values ofQ at each

r, either visually or statistically, using tests such as the paired t-test or the Wilcoxon

signed rank test (14). Such as approach would be necessary when two peaks have the

same value of XQ (within experimental error). Figure 5-5 illustrates this point for an

EMG peak and a log-normal peak. Despite their identical values for XQ (not shown),

the non-superimposability of the two Q vs r curves clearly illustrates that these peak

shapes are fundamentally different. The statistics tests mentioned above are helpful in

detecting these differences when the O profiles are "noisy" (e.g., Figure 5-9), although

they were not necessary for our comparison of the Q approach with slope analysis,

moment analysis, and the distribution function method.

EXPERIM ENTAL

An Apple Macintosh Plus personal computer was used for simulated peak

generation and other calculations. All programs were written in either Microsoft Basic or

True Basic and were copyrighted after debugging. Single, noiseless exponentially

modified Gaussian (EMG) peaks were generated with tau/sigma (t/a) ratios between 1

and 4 as described earlier (13). As x/a increases, an EMG peak becomes more tailed; as

x/a decreases, the EMG peak approaches a Gaussian shape. Eleven single EMG peaks

with x/a = 1 were generated at each signal-to-noise (S/N) ratio of 1000, 500, and 100

where S/N was defined as the signal at peak maximum (peak height) divided by the root

mean square (RMS) of the noise. We assumed that the major noise component was white

noise, and that the distribution of the amplitudes was normal. Since the peak height was

known for noiseless EMG peaks, the required RMS noise could then be calculated.

From the RMS noise calculated, a random noise factor (random number of appropriate

102

amplitude) was calculated and added to the signal as shown in eq 3, such that the RMS

of many peak data points would be the RMS needed for the correct S/N.

EMG(t) with noise = EMG(t) ± Noise factor [3]

The S/N was then checked by calculating the RMS from a large number of peak baseline

points after the noise factor had been added.

Single noiseless Gamma type peaks were generated using the following equation,

(15)

Signal(t)= t(a_1) expf-I p a ( a - l ) !

[4]

For the present study, a was varied between 3 and 10 in increments of 2 while P was

held constant at 0.2. The actual value of p did not influence the values measured for Q.

Single log-normal (16) peaks were generated using

Signal(t) = hp exp In rln2Ac [5]

JJ

where hp is the maximum height of the peak, r is from Figure 5-1, t is time, ^ is the

retention time of the peak, and W and As are the width and asymmetry of the peak

measured at r. For this study, the value of r was 0.5; the peak height was 5, the width

was 1 minute and the asymmetry was varied between 1.05 and 1.6. The value of the

width in eq 5 has no effect on the value of Q.

1 0 3

Peak param eter measurement. Values of Q for noiseless, single peaks

were measured to 5 decimal point accuracy using a search algorithm. For simulated

peaks that contained noise (EMG only), start/stop assignments (limits of integration)

were taken as the point where the signal of the peak (before noise was added) crossed the

baseline threshold level, defined as 0 .0 0 1 % of the peak height.

Peak parameters a and b were measured for the digitized peaks for r from 2 to

98% in increments of 1% of the peak height. A 4 to 10 point least-squares fit, symmetric

about the particular height, was used on each side of the peak to find the time

corresponding to the exact peak height fraction. The parameters a and b were then

calculated as shown in Figure 5-1. The number of points used in the fit was the nearest

even integer of

Nfront / 14, where Nfront is the number of points on the front of the peak (from 1% peak

height to the peak maximum). Thus, the same range of peak height was fit for all peaks,

despite any differences in Nfront. We used 1% increments for Ar, although it could have

been larger or smaller. Larger values for Ar would affect the value obtained for XQ,

depending on Ar, since XQ is simply a summation. However, smaller values of Ar

would probably have a smaller impact on the calculated value of XQ because one would

simply obtain a larger number of a and b measurements using the same line fits as

described above. Therefore, the value of Ar should only impact the XQ value as would

the increment in any summation under a curve. Of course, a more accurate method of

measuring the time corresponding to a particular height should give an even more

accurate measurement of Q at any r, but the computational complexity may be increased

also. Peak height was measured by subtracting the baseline value being used from the

peak maximum obtained via a quadratic least-squares curve fit of the seven highest points

in the peak. For peaks with finite signal to noise ratio (noise added as in eq 3), a

Savitsky-Golay polynomial smooth (17-19) was applied to the raw data prior to the

1 0 4

measurement of a and b. A smoothing factor defined in eq 6 was used to determine the

number of points employed in the smooth.

c _ # pts in Savitsky-Golay Smooth , , ,smoothing factor = # p‘, s j„ top half of pH it ~ [6)

We found that a smoothing factor of 0.2 offered a good compromise between S/N

enhancement and the peak distortion that is caused by polynomial smoothing (2 0 ).

The statistical moments, skew, and excess of the noiseless EMG peaks were

calculated from fundamental relationships (21) while the same parameters for EMG

peaks with noise were calculated after a Savitsky-Golay smooth (smoothing factor = 0.2)

using the equations in Figure la of ref. 13, except that MJ-M4 were normalized by peak

area.

The second derivative of all EMG peaks also was calculated by using the method

of Savitsky and Golay (17-19). This was done without prior smoothing because the

Savitsky-Golay method smooths the data and calculates the second derivative in a single

operation. The same number of points was used to calculate the Savitsky-Golay second

derivative as the pure smoothing performed prior to the calculation of other parameters

(Q, etc.) so that no differences in peak shape would be introduced among the different

methods of peak shape analysis. The second derivative of most peaks has two maxima

and one minimum. Grushka et al. (4) used the ratio of each maximum to the minimum

and plotted these values versus each other for peak shape comparison. Because we

wanted a single value that was related to peak shape, we decided to use the ratio of the

two maxima, and found that this was a more sensitive function of peak shape.

The distribution function was calculated for all peaks using equations published

elsewhere (6 -8 ). For the peaks containing noise, a Savitsky-Golay smooth was applied

using the same smoothing factor as above before the DFM calculation.

1 0 5


M easurement of Q and XQ from real data. Application of the Q

transformation on real (noisy, digitized) data requires the consideration of several data-

handling factors, such as (i) noise, (ii) smoothing, and (iii) the number of points in a

peak. Since each of these factors may potentially affect the measurement of Q and XQ, it

is appropriate to discuss them here.

Effect of smoothing on Q transformation. Depending on the degree of

noise, one may need to smooth the data prior to applying eq 1. However, smoothing of

peak data usually results in distortion of the peak shapes being measured. Therefore,

one should smooth the data in a manner that maximizes the accuracy of the measurements

being made while minimizing any peak shape distortion due to smoothing (2 0 ).

Fortunately, this is fairly easy to accomplish via judicious selection of the smoothing

parameters. Furthermore, when the objective is peak shape comparison of similar peaks,

any slight errors caused by smoothing process itself will be virtually eliminated

(cancelled) so long as the smoothing is performed consistently for all peaks.

Based on the above considerations and our prior experience with

chromatographic peak characterization (13,21,22), we selected the efficient smoothing

algorithm of Savitsky and Golay (17-19). As with all smoothing methods, some

distortion of the peak shape, however slight, is inevitable. Figure 5-6 shows the

resulting errors in XQ introduced into the data for noiseless EMG peaks with T/(7=l.

Since the peaks contain no noise, the errors are assumed to be due solely to the distortion

of the peak's shape. Because the errors are so small (< 0.25%), we believe that if one

uses the same smoothing factor (eq 6) for aR the peaks being compared (unknowns and

standards), no error between peaks due to distortion caused by smoothing should be

1 0 6

observed. In addition to errors in XQ, one can expect some error in the individual

points within the Q profile. In general, however, the errors in Q at any value of r are so

slight that Q profiles of the same peak calculated with different smoothing factors are

virtually superimposable. Therefore to illustrate just how small this effect of smoothing98

is, we calculated the normalized residual sum, ^

r= 2

the smoothing factor. The results, shown in Figure 5-6b, demonstrate that similarly to

the effect of smoothing on XQ, the effect of smoothing on Q itself is entirely negligible if

an appropriate degree of smoothing is employed.

Effect of noise. Whether or not the real (noisy) data are smoothed, they will

still contain some noise which may-have an effect on the measurement of Q and XQ.

Note that since Q represents a data transformation in which both the signal and the noise

are transformed, the effect of noise on Q cannot be predicted intuitively based on

conventional S/N wisdom. Figure 5-7 shows the relative error of Q (averaged over r = 2

to 98%) for various signal to noise (S/N) ratios. A smoothing factor of 0.2 was applied

to the data before measuring Q and calculating the errors. This smoothing factor was

chosen because it offers a good compromise between peak distortion (see Figure 5-6)

and S/N enhancement (17), while also requiring only a small number of convolute points

for the smooth, thereby reducing computation time. Although the relative error for each

point occasionally exceeds 6 % as shown in Figure 5-7, the errors appear to be randomly

distributed. Therefore the value of XQ will be only slightly affected by the magnitude of

the noise. For a S/N of 100 (same data as before), the relative error in XQ is only 0.4%;

larger S/N ratios resulted in even smaller relative errors.

Because a least squares line fit is used to determine a and b accurately at the

desired height, some additional smoothing may be introduced by this operation.

However, the effect will be slight, as the peak has already been smoothed via the

* as a function of

1 0 7

Savitsky-Golay method. As shown before (20), the fraction of noise remaining after

multiple smooths varies approximately with the inverse eighth root of the number of

smooths.

Effect of the num ber of points in a peak. The final consideration when

measuring Q is the number of points within the peak(s) being investigated. Because we

are dealing with digitized data, we can expect increased accuracy in the measurement of

the parameters a and b, and subsequently greater accuracy in the values of Q, as the

number of points in a peak increases. Figure 5-8 shows the accuracy of XQ as a

function of the number of points in the top half of a peak. Instead of a monotonically

changing curve as one would expect, the data seems to be discontinuous, but with all

values of relative error less than 0.2%. We believe this scatter is due to the complex

relationship between the number of points in the peak, the shape of the peak, and the

changing number of points used in the line fit for measurement of the parameters b and a

(see Experimental Section). As seen in Figure 5-8, very few points are needed for

accurate measurement of XQ, in sharp contrast to the moment method which needs at

least 4 to 5 times as many points for accurate quantitation (12).

Although the error in XQ remains small as the number of points in a peak

decreases from 250 to 20 in Figure 5-8, the error in Q at all relative peak heights (r)

increases (Figure 5-9). As illustrated in Figure 5-9f, the error in Q (averaged over all r

values) increases almost exponentially from 0.1 to 3.5% as the number of points in a

peak decreases. This error is due to the errors in the measurement of the parameters a

and b as discussed in ref. 13. Again, as shown in Figure 5-8, these errors tend to cancel

when XQ is evaluated.

Comparison of methods using noiseless data. To compare the abilities

of the four methods (XQ, DFM, moments and second derivative) for detecting peak

shape changes or differences, we first calculated the sensitivity of each method for

1 0 8

noiseless EMG peaks. We defined this noiseless sensitivity as the percent change in

each peak shape parameter (ZQ, etc.) relative to its value at %Jo=\ as the asymmetry of

the peak is increased from at x/a=l. The basis of peak shape measurement for the DFM

was simply the area under the curve while the ratio of the peak height maxima was used

for the second derivative. As mentioned in the Experimental Section, we found that the

percent change in the second derivative maxima ratio was greater than either the first

maximum to minimum or second maximum to minimum ratio.

Shown in Figure 5-10 are the sensitivities of the methods for detecting peak

shape changes in noiseless data. Apparently, the third and fourth statistical moments are

the most sensitive in terms of percent change per change in peak shape. The sensitivities

of the other methods (in descending order) are excess, skew, second derivative, ZQ and

DFM for noiseless EMG peaks.

Comparison of methods using noisy data. When noise is introduced

into the EMG peaks, the relative abilities of the methods to detect differences in peak

shape change dramatically. The relative ability of each method, defined in terms of the

minimum detectable change in peak shape, relative to the shape at x/a= 1, was calculated

using a t-test (14). Rearranging equation 7, for calculation of t for the comparison of

two means

we obtain equation 8 . The parameter S is the pooled standard deviation for the two sets

of data, n corresponds to the number of points in each data set, and x is equal to mean of

parameter being considered (i.e. ZQ).

[7]

1 0 9

x j - x2 = tSni + n2

n l n 2[8]

Equation 8 allows one to calculate the minimum difference in means between two data

sets needed for the means to be considered significantly different at the desired

confidence interval. For this work we chose the 99% confidence interval, nj and n2

were equal to eleven, as that was the number of peaks simulated for each S/N (see

Experimental section). Equation 8 can then be reduced to

parameter of interest (ZQ, DFM, etc.), we can determine the corresponding peak shape

change ('t/a increased) from a reference value (t/a=l) that would give this difference

between means. For this work we assumed that the necessary significant difference in

means was equal to the difference in the true values, thereby significantly reducing the

number of simulated peaks needed. S was calculated from the eleven peaks at the desired

S/N, assuming that the standard deviations of the reference value and the difference value

were equal. Shown in Table 5-1 are the results for ZQ, DFM, second derivative, and

all the moment-based parameters at three different S/N ratios. Obviously, the Q

transform is superior to the other methods when noise is included in the data, especially

at the lower S/N ratios, while the second moment and DFM methods compare favorably

to the ZQ method at higher S/N ratios. We arbitrarily chose 't/a =1 as the reference peak

shape. However, the relative abilities of the various methods to detect peak shape

differences should not change with a different reference peak shape. Note that because

we used ideal baseline levels in the calculations for all the methods, the adverse effects of

low S/N ratios are not completely reflected in the results for the statistical moments in

Ax = 1.2131 S [9]

Therefore, by knowing the necessary significant difference between means for the

110

Table 5-1, i.e.,the performance o f all the moments, including the excess and skew,

would have been much poorer had more realistic baseline levels been used. This is due

to the extreme sensitivity of the statistical moments to baseline errors (9-11,23). Shown

in Table 5-2 is the performance of the Q method when baseline errors are introduced;

these results are significantly better than those obtained for the moments of EMG peaks

described elsewhere (23).

Although the performance of the DFM and XQ methods are comparable at

moderate to high S/N ratios, the XQ method is significantly better for low S/N ratios. In

addition, the Q transform approach inherently provides well established graphical

parameters (a, b) that can also be used to calculate other important chromatographic

figures of merit such as plate count (2 1 ), peak area (2 2 ), and statistical moments (13,

14). Given the similar moderate computational complexity of the XQ and DFM

methods, the Q data transformation approach to peak shape analysis and comparison

arguably represents a significant improvement over the DFM method, and certainly over

the slope and moment-based single-channel methods.

Potential applications of Q the transform . The Q transformation may be

employed in other ways in chromatography and analytical chemistry. For example,

based on preliminaty unreported gas chromatographic data, we have found that the XQ

can be very useful for solute identification when retention times alone are insufficient.

Because each compound interacts with the stationary phase differently, their peak shapes

may be different enough to identify them based on their XQ value even when their

retention times are similar. We have been able to distinguish between solutes with

identical retention times whose XQ values are different. In the paper immediately

following this issue, we examine the ability of the Q transform to detect the presence of

impure chromatographic peaks when the degree of overlap between two overlapping

111

peaks is so severe that no shoulder or other visible evidence exists to indicate this

coelution.

Because many times peak shapes can give indications of instrumental problems,

the Q transformation may also be useful for automatic instrument troubleshooting. For

example, consistently negatives values of Q vs r would signal the presence of a fronted

peak in chromatography, which in turn is often indicative of sample overload. As

illustrated in Figure 5-11, a fronted chromatographic peak will give a Q profile {small

positive values of Q, Q < 1) and/or EQ value that is different from normal peaks. Also,

peaks with excessive tailing {large positive values of Q) may indicate that (i) the detector

time constant is excessive for the present conditions, (ii) retention is due to two or more

mechanisms, or (iii) active sites in the chromatographic column are present.

C O N C LU SIO N

For asymmetric peaks, such as the exponentially modified Gaussian peaks

employed primarily in this study, the Q tranformation approach introduced here appears

to be superior to slope analysis, moments, and the distribution function method for the

measurement and characterization of peak shape, including the detection of peak shape

changes, especially at low S/N ratios. At higher S/N ratios the differences are somewhat

less significant, particularly between EQ and DFM. The relative insensitivity of the latter

methods to baseline errors and their computational simplicity would appear to make both

suitable for peak shape difference measurement As we will show in the article

immediately following this one, however, the Q transform is more sensitive than the

DFM to peak shape changes caused by the overlap of two peaks and is therefore better

for the single-channel detection of impure chromatographic peaks. Given this advantage,

the lower sensitivity of the EQ method to low S/N ratios, and the fact that it inherently

112

provides values for graphical parameters (a and b) that can be used to calculate other

important chromatographic figures of merit, however, we believe that the XQ approach

is moderately superior to the DFM. The Q transform may also provide additional

advantages for other applications that we have not yet explored.

CREDIT

This work was supported in part by the National Oceanographic and Atmospheric

Administration, NOAA-50-ABNC-7-00100.

1 1 3

TABLE 5-1. MINIMUM DETECTABLE CHANGE IN (x/c) FOR VARIOUS

METHODS OF PEAK SHAPE ANALYSIS a

S/N

1 0 0 500 1 0 0 0

XQ 0.047 0.018 0.016

Slope Analysis 0 .2 0 0.16 0.067

m 2 0.074 0.026 0.018

m 3 0.36 0 .1 0 0.091

M4 0.56 0.17 0.096

EXCESS b 1 .0 2 0.48

SKEW 2.32 0.23 0 .1 2

DFM 0 .2 2 0 .0 2 0 0.018

a t/ct for the reference peak is 1 .0 .

b No change in peak shape detectable in the range 1.0 < 't/a < 4.0 at this signal-to-noise

by the specified method.

1 1 4

TABLE 5-2. % RELATIVE ERROR IN I Q DUE TO BASELINE ERRORS IN EMG

PEAKS.

HORIZONTAL BASELINE a

BASELINE HEIGHT x/a=l i /o=3

2% 1.74 2.86

4% 3.16 5.30

6 % 4.39 7.52

8 % 5.50 9.57

10% 6.51 11.51

SLOPING BASELINE b

BASELINE HEIGHT x/o=l x/a=3

2% 6.79 8.80

4% 11.59 15.20

6 % 15.40 20.34

8 % 18.60 24.69

10% 21.38 28.48

a Baseline was a line drawn horizontally at indicated fraction of peak height. It was

then subtracted from the peak.

b Baseline was drawn from beginning of peak(approximately 0.001 % of peak height)

to a point on the tail of the peak nearest indicated baseline height (fraction of original

peak height) and then subtracted from peak.

1 1 5

Q TRANSFORMATION

r-1r-1

imi i i i i | i i i i im i | in i i i i i | i i i i i i i i i |80 100

Figure 5-1. Measurement of a and b used for the calculation of Q in eq 1.

1 1 6

WHICH TWO PEAKS HAVE THE SAME SHAPE?

ANSWER : PEAKS 1 AND 2

1 EMG ( T / a =2.6)2 EMG ( T / 0 = 2.6)3 EMG ( X / a = 2.2)4 EMG ( T / a = 2.8)5 LOG-NORMAL ( Asymmetry = 1.6)

Figure 5-2. The Q transform can be used to visualize peak shape differences without

regard to peak height or width. Peak identity: (1) Exponentially Modified

Gaussian (EMG), T/0 =2.6, 0=1.0 (2) EMG, T/0=1, 0=0.26 (3) EMG,

T/0=2.2, 0=0.26 (4) EMG, T/0=2.8, 0=0.5 (5) Log-Normal,

asymmetry=1 .6

1 1 7

1311

9

7

Q

5

3

0.51

1006040206040200 '0,0/

EMG

GAMMAASYMMETRYLOG-NORMAL

1.4

10040

Figure 5-3. Comparison of Q profiles for various peak shapes, (a) EMG peak, ref. 13

(b) Gamma peak, eq 4 (c) Log-Normal peak, eq 5 (d) Overlay of Q

profiles from (a) through (c). Note that peak shapes can be differentiated

on the basis of LQ or on direct point-by-point comparison of individual Q

profiles.

A b

/ (A

b,a=

1)

1 1 8

1 .5 -

GAUSSIAN, <7 = 1.2

1.0 .

GAUSSIAN, 0 = 1.0

0 - 5 ■ ' • ■'' ‘ r — ■ ■ ■ ■ I • ■ ■ ""> I ■' » " "» ............ ........ ' I- - » """l

20 40 60 80 100f fOA

Figure 5-4. Differentiation of peak shapes using Q transform between two symmetric

peaks by a redefinition of Q as discussed in the text.

1 1 9

1 1 1LOG-NORMAL PEAK SHAPE ’Asym = 1.6

Q

EMG (tau/sigma = 2.8)

80 1000 20 40 60

Figure 5-5. Non-superimposability of Q profiles of fundamentally different type of

peaks with identical values of XQ. Peak identities: EMG with x/o = 2.8

and log-normal with an asymmetry (eq 5) of 1.6.

120

0 .22-1

0 .2 0 -

owulCC 0 .1 8 -

0 .16

0.14*1 "i—r -T —r "'"| - t i i i | i i i i " » i i | i i i i | i i i i |

0. 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6

SMOOTHING FACTOR

O

<=>q(/)h iCCoUJNZj<sCCoz

0 .04

0 .0 3

0.02

0.01

0.000.2 0 .3 0 .5 0.60.0 0.1 0 .4

SMOOTHING FACTOR

Figure 5-6. Effect of Savitsky-Golay smoothing factor (eq 6 ) on (a) Z Q ; and (b) Q vs

r for an EMG peak with T/G =1.

AVER

AGE

% RE

, Q

121

8

7

6

5

4

3

2

1

0200015001000500

S/N

Figure 5-7. Effect of signal to noise (S/N) ratio on the Q transformation.

122

0 .20 -l

0.15-

0.05-

0.002.51.5 2.01.0

LOG (#PTS, 50%)

Figure 5-8. Effect of the relative sampling rate (number of points) on the measurement

of IQ . Peak shape: EMG peak, T/G=l.

1 2 3

Figure 5-9. Effect of the relative sampling rate (number of points) on the accuracy of

Q. Number of points in the peak measured from half-height to half-height.

Figure 5-9e shows true Q vs r profile (obtained via search algorithm) while

Figure 5-9f shows the average error associated with the measurement of Q

at any r.

1 2 4

3.01

2.5-

2.0

1008040 600 20

3.0 ~|

2.5

2.0

1.5

60 10020 40 600

1 2 5

Q

3.0-I

2.5

2.0

80 10040 60f fo,

3.0 n

2.5

2 .0 -

80 10040 60I* / o>

AVERAGE % RE, Q

Us- 0H0 )

01o

rooo

cn

roo

roUl

o

o cno cn oo

AO

o

00o

126

LOG

(% C

HA

NG

E)

1 2 7

SLOPE ANALYSIS

EXCESS

SKEW

IQ

DFM

3

t/ ct

Figure 5-10. Relative abilities of statistical moments, excess, skew, second derivative,

I Q and DFM to detect changes in peak shape for noiseless peaks.

SIG

NA

L

128

0.5-1

0.4

0.3

0.2

0.1

0.066420

TIME (MIN)

1.0i

0.8

0.6

Q0.4

0.2 :

0.0100806020 400

r (%)

Figure 5-11. Automatic detection by the Q transform of instrumental problems in

chromatography, as illustrated with (a) a fronted peak (indicative of sample

overload); and (b) its Q profile.

1 2 9

LITERATURE CITED

1) Brown, Steven D.; Barker, Todd Q.; Larivee, Robert J.; Monfre, Stephen L.;

Wilk, Harlan R. Anal. Chem, 1988, 60, 252R-273R.

2) Stevens, J. G.; Bowen, L. H.; Whatley, K. M. Anal. Chem, 1988, 60, 90R-

106R.

3) Grushka, Eli. Anal. Chem., 1972, 44, 1733-1738.

4) Grushka, Eli.; Monacelli, G. C. Anal. Chem., 1972,44,484-489.

5) Grushka, Eli.; Myers, M. N.; Giddings, J. C. Anal. Chem, 1970, 42, 21-26.

6 ) Rix, H. Anal. Chim. Acta, 1986, 191, 467-472.

7) Rix, H. J. Chromatogr., 1981, 204, 163-165.

8 ) Excoffler, J.-L.; Jaulmes, A.; Vidal-Madjar, C.; Guiochon, G. Anal. Chem.,

1982, 54, 1941-1947.

9) Chesler, S. N.; Cram, S. P. Anal. Chem, 1971, 43, 1922-1933.

10) Petitclerc, T.; Guiochon, G. J. Chromatogr. Sci., 1976, 14, 531-535.

11) Grushka, Eli.; Myers, M. N.; Schettler, P. D. Anal. Chem., 1969, 41, 889-892.

12) Eikens, David I.; Carr, Peter W. Anal Chem., 1989, 61, 1058-1062.

13) Jeansonne, Mark S.; Foley, Joe P. J. Chromatogr., 1989, 461, 149-163.

14) Miller, J.C.; Miller, J.N., Statistics For Analytical Chemistry, 2nd ed.; Horwood:

1988; Chapters 3 and 6 .

15) Mendenhall, William; Scheaffer, Richard L ., Mathematical Statistics with

Applications, 1st ed.; Duxbury Press: California, 1973; Chapter 4.

16) Grimalt, J.; Iturriaga, H.; Thomas, X. Anal. Chim. Acta, 1982, 139, 155-166.

17) Savitsky, A.; Golay, M. J. E. Anal. Chem., 1964, 36, 1627-1639.

18) Steinier, J.; Termonia, Y.; Deltour, J. Anal. Chem., 1972, 44, 1906-1909.

19) Madden, H. H. Anal. Chem., 1978, 50, 1383-1386.

1 3 0

20) Enke, C. G.; Nieman, T. A. Anal. Chem., 1976, 48, 705A-709A.

21) Foley, Joe P.; Dorsey, John G. Anal. Chem., 1983,55,730-737.

23) Anderson, David.J.; Walters, Rodney.R.; J. Chromatogr. Sci., 1984, 22, 353-359

22) Foley, Joe P. Anal. Chem., 1987, 59, 1984-1987.

Chapter 6

SINGLE CHANNEL PEAK IM PURITY DETECTION USING

THE Q TRANSFORMATION

Reprinted with permission from Burgess, S.W., Jeansonne, M.S. and Foley, J.P.;

submitted for publication in Anal. Chem. Unpublished work Copyright 1990 American

Chemical Society.

13 1

1 3 2

Department o f Chem is try

L o u i s i a n a S t a t e U n i v e r s i t y and agricultural and mechanical collegeBATON ROUGE • LOUISIANA ■ 70803-1804 , « 7

Barbara Polanski Publications Division American Chemical Society 1 155 16th St. Northwest Washington, DC 20036

Dear Ms. Polanski,

1 am writing to you in reference to the article en t i t led ," Single Channel Peak Impurity Detection Using the Q Transformation ", AC900689Y, received by A nalytical Chem istry on August 24, 1990, for which J am the f irs t author. I would like to use the manuscript as part of my Ph. D. dissertation. The completed d isserta t ion will be submitted to University Microfilms, Incorporated. Please forward permission to reproduce the manuscript.


S04/388-3361 FAX 5041388-3458

September 13, 1990

Sincerely,

'Prof. Joe P. FMeyT toA jtw d i

Mark S. Jeansonne

American Chemical Society 133

PUBLICATIONS DIVISION 1155 SIXTEENTH STREET, N.W. WASHINGTON, D.C. 20036 P h o n e (202) 872-4600 Fax (202) 872-6060

Septem ber 28, 1990

Mr. M ark S. Jeansonne D epartm ent o f Chemistry Louisiana S ta te University B aton Rouge, Louisiana 70803-1804

D ear Mr. Jeansonne:

Thank you for your le tter o f September 13, in which you requested permission to include in your thesis, two papers that you subm itted to Analytical Chemistry :"Single Channel Peak Im purity D etection Using the Q Transform ation," and "The Q Transform ation: A Novel M ethod o f Peak Shape Analysis."

I would be happy to grant you this permission free o f charge provided that you print th e required ACS copyright credit line on the first page o f your article and that your paper is published by ACS first before you submit it to UMI. Also, please inform UM I that perm ission is granted subject to the terms o f the ACS/UM I Agreem ent dated O ctober 1, 1984.

The credit line we require is, "Reprinted with permission from FU LL R E FE R E N C E CITA TIO N Copyright Y EA R American Chemical Society." Please be sure to insert the appropria te inform ation in place o f the words in capital letters.

If your paper is not accepted before you include it in your thesis, please use the following credit line: "Reprinted with permission from Burgess, S.W., (A U TH O RS NAM ES); subm itted for publication in Anal Chem. U npublished work Copyright 1990 Am erican Chem ical Society."

Thank you for your request. Please let me know if I can be o f any further help.

Sincerely,

Barbara F. Polansky Copyright A dm inistrator

1 3 4

INTRO DUCTIO N

The problem of overlapping peaks is .frequently encountered in chromatographic

analysis. In many instances, the peaks are sufficiently resolved that the overlap is

obvious from visual inspection and detection of the overlapping peaks can be easily

accomplished visually and/or automatically. In other situations, however, the resolution

of peaks may be low enough that no shoulder or other easily detectable distinguishing

characteristics appear, i.e., the composite peak is indistinguishable from a single peak as

illustrated in Figure 6-1.

Modem multichannel, hyphenated techniques such as GC/MS, GC/FTIR,

HPLC/photodiode array and HPLC/ICP can be used to assess peak purity, but are not

always available or directly amenable for use with present general-puipose laboratory

single-channel data systems (1). Also, these multichannel techniques usually assume

that the characteristics of the compounds forming the composite peak (mass spectrum,

UV spectrum, e tc .,) are different. As pointed out before, the co-elution of two

compounds is frequently due to their very similar structures, and therefore their spectra

could also be too similar for the multichannel techniques to be successful (2). Hence,

various single-channel methods for peak purity determination have been pursued,

including slope analysis (3,4), moment analysis (5) and the distribution function method

(DFM) (6-8).

Slope analysis utilizes the second derivative of a chromatographic peak to

determine peak purity by comparing properties of the second derivative for pure and

impure peaks. By comparing the ratios of the peak's second derivative maxima and

minimum to those for a known pure peak, a determination of purity can be made.

However, there are two major drawbacks to this approach. First, in the method

described by Grushka and co-workers (3,4), a very accurate peak model must be used

1 3 5

or assumed. Unfortunately, real chromatographic peaks are not always described with

sufficient accuracy by any given model. Secondly, the second derivative is very

sensitive to noise. Because accurate values of the peak's second derivative minimum and

maxima must be measured, noise can cause a large error.

Moment analysis is a mathematical peak purity determination method in which the

skew and excess of pure peaks are compared to the skew and excess of impure peaks.

Here, the excess is plotted versus the skew for pure modeled or reference peaks in order

to obtain a pure reference curve. When the plotted excess versus skew point for other

peaks do not fall on the curve, the peaks can be suspected of being impure. As with

slope analysis, this method must assume a peak shape model or use internal standards

with exactly the same peak shape as those in the chromatogram to be examined.

Assuming that an accurate model or reference peak(s) is/are available for comparison

with the real chromatographic peaks, yet another challenge awaits. The skew and excess

for the real peaks must now be measured very accurately. As seen in eqs 1 and 2, the

skew and excess depend on the second, third, and/or fourth central moments.

Skew = Ys = j^S/2 I‘]

Excess = 7e - M ^2 *_ j PJ

The inability to measure accurately the third and fourth statistical moments for a real

chromatographic peaks due to noise (9-11) and baseline errors (12,13) results in a

significant propagation of errors to the skew and excess. Thus, the comparison of the

skew and excess of real peaks to reference values obtained either via mathematical

models or internal standards can be quite error prone, and represents a major limitation of

moment analysis for peak purity determinations.

1 3 6

In the distribution function method (DFM), the distribution of area under a peak is

plotted as a function of time and the resulting distribution is compared to a standard or a

reference. Although this method is sensitive to peak shape changes (6-8,14), we will

show later in this paper that it is not as sensitive to peak impurities as the method we are

introducing.

Due to the shortcomings of the above methods, we believe that another method for

the single-channel detection of co-eluting peaks is needed. In this current project, our

goal was to develop such an alternative that circumvents the problems associated with the

second derivative, moment analysis, and DFM while retaining their computational

simplicity.

In the preceding chapter, we reported a mathematical transformation for data

typically associated with digitized chromatograms or spectra. The signal vs. time data

are transformed into a shape factor measured from 2 to 98% of the peak height in

increments of 1%. We term this the "Q transformation", defined in eq 1 of the preceding

chapter (14). Also introduced were the "Q vs r" profile and XQ, the latter simply the

summation of Q over all r for a given peak.

Whereas the previous chapter focused on the benefits of this new data

transformation for the analysis and comparison of pure peaks, the purpose of the present

article is to demonstrate that this same approach can be used for the single-channel

detection of severely overlapped peaks (no valley, shoulder, or other visible artifact).

The advantages of this approach over multi-channel approaches are: 1) the universal

availability of single-channel chromatographic detection in any laboratory; and 2) the

ability to work even when the impurity and major peak have very similar or even

identical absorption or fluorescence spectra. Potential advantages of this approach over

the second derivative, moment analysis, and DFM methods are as follows: 1) no prior

assumption of a particular peak shape model; 2) less vulnerability to noise than the

1 3 7

second derivative; 3) and more sensitivity than the DFM to peak shape changes caused

by co-eluting peak impurities.

EXPERIM ENTAL

An Apple Macintosh Plus personal computer was used for simulated peak

generation and other calculations. All programs were written in either Microsoft Basic or

True Basic. An Omega-4 data system (Perkin-Elmer, Norwalk, CT, USA) utilizing an

IBM PC-AT computer was used for chromatogram collection and storage. Liquid

chromatographic (LC) data was collected from a Series 4 liquid chromatograph (Perkin

Elmer, Norwalk, CT, USA) using a reversed phase C8 column. Detection was achieved

using a Model variable wavelength ultraviolet absorption detector (Isco, Lincoln,

NB, U.S.A.). Injections were made manually using a Model 7125 standard six-port

injection valve (Rheodyne, Cotati, CA) with 20 |iL loop.

Individual liquid chromatographic peaks of toluene, ethylbenzene, isopropyl

benzene, n-propyl benzene, n-butyl benzene, 4-methyl cyclohexanone, 3-heptanone, 4-

heptanone and 5-nonanone were generated using a mobile phase consisting of 60%

HPLC grade acetonitrile (ACN) and 40% water. Individual peaks of toluene,

ethylbenzene, n-propyl benzene, and n-butylbenze were generated using a mobile phase

of 70% HPLC grade methanol (MeOH) and 30% water. Individual liquid

chromatographic peaks of valerophenone and acetophenone were generated using a

mobile phase of 50% HPLC grade ACN and 50% water. Individual peaks of n-propyl

benzene and toluene were obtained with a gradient of 5% ACN/min while individual n-

butyl benzene and ethyl benzene peaks were generated using a gradient of 5%

MeOH/min. All peaks were generated with a flow rate of 2.0 ml/min with detection of

the alkyl benzenes at 254 nm and the ketones at 272 nm. The water used was double

1 3 8

distilled and deionized. All of the compounds used were at least 99% pure and the

sample solvent was identical to the mobile phase in all cases. This was done to minimize

any effects on peak shape caused by the delayed-injection method of generating co

eluting peaks (vide infra), whereby the solvent or void peak passes through a previously

injected analyte peak(s). Except when using the delayed-injection method, an approach

not applicable to real samples, our experience and the results of others have shown that

composition of sample solvent has little effect on peak shape provided that its solvent

strength is less than or equal to that of the mobile phase (15).

Overlapping real chromatographic peak pairs consisting of n-propyl benzene :

toluene, isopropyl benzene : ethylbenzene, and valerophenone : acetophenone were

generated using a delayed injection technique. That is, in each case the compound with

the greater retention time was injected, a time approximately equal to the difference in

retention time between the two was allowed to pass, and the second compound injected.

In this way, overlapping peaks could be generated with various resolutions, depending

on the delay time used. However, the actual difference in retention times could only be

estimated to within 0.5 second due to the inherent imprecision of manual injections.

Due to experimental difficulties, we were not able to generate any impure peaks by

the double-injection method when employing a gradient. Therefore, we artificially

contaminated a real peak by mathematically adding to it another peak obtained under the

same conditions, after the size of the impurity peak had been mathematically reduced.

Single, noiseless exponentially modified Gaussian (EMG) peaks were generated

with tau/sigma (x /a) ratios between 1 and 4 as described previously (12). The EMG

function results from the convolution of a Gaussian (standard deviation of O) and an

exponential decay function (standard deviation of X). As the T/O ratio increases, the peak

in question will become more skewed, and as it decreases, the peak approaches a

1 3 9

Gaussian shape. Eleven single EMG peaks of x /a =1 were generated at Signal-to-Noise

(S/N) ratios of 100, 500, and 1000 (14).

Simulated overlapping EMG peaks were generated by simple addition of the major

and impurity peaks; both elution orders were considered, i.e., the impurity could elute on

either the front or tail of the major peak. The area ratio of major peak to impurity was

16/1, with both peaks having a T/O ratio of unity. The resolution was varied from

0.03125 to 0.5 in increments of 0.0625. Resolution was defined as AtR /4 VM2 , where

M2 , the variance, is equal to Oq2 + x2 for an EMG peak.

Peak param eter measurement.For simulated peaks, start/stop assignments

(limits of integration) were taken as the point where the signal height of the peak (before

noise was added) crossed the baseline threshold level, arbitrarily defined as r = 0.001%

(see Figure 5-la of preceding article for definition of r). The parameters a and b for

calculation of Q, skew, excess, and second derivative were measured by methods

explained before (14). The distribution function was calculated using equations

published elsewhere (6-8).

The second derivative of most peaks has two maxima and one minimum. Grushka

et al. (4) used the ratio of each maxima to the minimum and plotted these values versus

each other for simulated peaks using a model they thought appropriate. They then

compared this plot to those points determined for sample peaks in order to detect peak

impurities. We used either the ratio of maxi or max2 to the minimum, whichever was

more sensitive. In general, the maxl/min ratio will more sensitive to impurities eluting in

front of a peak while the max2/min ratio will be more sensitive to impurities eluting in the

tail of a peak.

1 4 0


Peak shape comparison. When comparing the shapes of peaks, real or

simulated, there can be two approaches when using the Q transformation. First, one can

compare values for XQ, which corresponds to the area under the Q profile (plot of Q vs

r). If the assumption is made that the individual peaks being compared belong to the

same family or class of peak shapes (for example, EMG peak shapes), then any

difference in peak shape will translate into a difference in area under the Q profile or XQ

(14). One can make this assumption for either of the following scenarios: (1) a sample

peak is compared to a reference peak; or (2 ) the compounds are known to be pure and

elute with the same retention mechanism, such as compounds of a homologous series,

where any differences in peak shapes may be attributed only to differences in

intracolumn band broadening.

The second approach is to compare point by point the shapes of the individual Q

profiles either visually or statistically using a test such as the paired t-test (16,17). This

approach is very useful, as shown later, for the detection of impure peaks with only one

reference peak and one sample peak.

Simulated im pure peaks. In general, the deviation of the Q profile measured

for an impure peak will be influenced by location of the impurity within the peak as

shown in Figures 6-2 and 6-3. In these two examples, the areas under the Q profiles for

simulated, noiseless standard and sample peaks were compared for peak purity

determination. Figure 6-1 shows a simulated impure peak composed of two overlapping

EMG peaks, each with X/G = 1 at a resolution of 0.25 while Figure 6-2 shows the

corresponding Q profiles for the pure and impure peaks from Figure 6-1. The important

feature to note in Figure 6-2 is the positive deviation of the Q profile for the impure peak

from that of the pure peak for r = 15 to 98% without any compensating negative

141

deviation from r = 2 to 14%. This difference enables an impure peak to be detected by

comparing XQ for both peaks.

Figure 6-3a again illustrates a simulated impure peak composed of two overlapping

EMG peaks, each with T/G = 1 at a resolution of 0.25, except that the impurity elutes on

the front of the major peak. In this case the resulting Q profile for the impure peak

deviates negatively from that for the pure peak, again allowing one to use XQ for

impurity detection while also allowing one to judge whether the impurity is eluting on the

front or the back of the major peak.

Noisy simulated peaks. When random noise is present in the peaks (as in real

data), one must treat the data statistically in a way similar to that reported in the preceding

paper, except that the resolution between co-eluting peaks is changed instead of the

shape of the single pure peak. The ability of each method to detect co-eluting peaks was

evaluated as follows: For a given signal to noise (S/N) ratio, the necessary change in the

parameter of interest (XQ, XDFM, etc.) was calculated depending on the standard

deviation of eleven single noisy peaks, as described in the previous chapter. Another

smaller peak was then superimposed on the original, and the resolution was then

increased until the value of the parameter of interest (XQ, DFM, etc.) exceeded the

necessary change for the 99% confidence level. The resolution at which this occur ed is

reported in Table 6-1 for the various methods as a function of S/N and elution order

(major and minor (impurity) peak).

Importantly, Table 6-1 shows that the ZQ method is better at detecting peak

impurities than any o f the other methods. Note also that the skew and excess methods

failed to detect any of the impurities co-eluting with the major peaks over the resolution

range shown here, even for large S/N values. Finally, although in the preceding paper

(14) we found the DFM to be almost as good as the Q transform approach for the

142

detection of overall changes in peak shape, as Table 6-1 shows the DFM is not nearly as

sensitive as the XQ approach to peak shape differences caused by impurities.

Real impure peaks. In order to determine peak purity in a noisy,

multicomponent chromatogram, one can use a procedure similar to that described above,

provided that the identity of the analytes in question are known or can be assumed. Once

their identity has been determined, replicate injections of both the sample and standard

peaks will provide two sets of XQ points. A t-test can then be used to determine whether

the two sets of XQ points are significantly different at the 99% confidence interval.

Because the t-test for comparison of means performs better as the number of injections is

increased, one may need many XQ points for both the standard and sample. For routine

use this multi-injection approach is somewhat laborious. Another possible statistical test

exists that requires only one injection of the standard and one injection of the sample. By

comparing the individual Q profiles via a paired t-test (16,17) one can determine whether

the two profiles are significantly different. The results of applying this procedure to

simulated peaks are shown in Table 6-2. As seen there, this test is not quite as sensitive

as the multiple injection method, although the results are comparable. Because only

discreet resolutions were used, the results may be better than those presented had smaller

resolution increments been considered. Nevertheless, the paired t-test using Q profiles

allows better detection of impure peaks than the multiple injection method used in

conjunction with the DFM,slope analysis, excess, and skew methods . When the paired

t-test was used with DFM and slope analysis, no impurites could be detected over the

resolution range used. In fact, the DFM always found that two simulated pure peaks

with the same x/a and same S/N were significantly different! The paired t-test cannot be

used with the excess and skew methods because those methods only give a single value

pertaining to peak shape.

1 4 3

To summarize, the analyst has two options for comparing standard peaks and

sample peaks. First, by injecting the the standard many times and the sample several

times and comparing the resulting two sets of XQ points via a t-test. If time and labor

are not large concerns, this option will give superior results. In fact, by using a larger

number of peaks than were used for determination of the values in Table 6-1, one may

achieve even better results than those presented there. Second, one can inject the

standard and sample peaks each once and utilize a paired t-test to compare the individual

Q profiles. Although this option is slightly less sensitive to impurities than the first

option, it may be better in terms of time and labor.

Experim ental verification of paired-t option. Because the t-test for

comparison of means discussed above is not amenable for most routine analyses, we

present experimental verification for only the paired t-test in this report Shown in Table

6-3 are the results of applying the paired t-test to detect impure peaks under various

mobile phase conditions in high performance liquid chromatography (HPLC). Note that

the paired t-test worked over a range of mobile phase conditions and S/N ratios. Both

the second derivative and DFM methods were unable to detect the impurities. Figure 6-4

illustrates the appearance of the pure and impure n-propyl benzene peaks that were used

for the data in row four of Table 6-3. How does the paired t-test treat identical pure

peaks obtained under the same chromatographic conditions? Table 6-4 shows that groups

of pure peaks, all obtained under the same experimental conditions, are not detected as

impure by the paired t-test.

Potential for qualitative analysis. Another possible application of the Q transform

is qualitative analysis. Presently, most qualitative analysis in chromatography is

performed using retention time windows. As illustrated in Figure 6-5, XQ data obtained

for a given compound during different runs (but otherwise identical chromatographic

conditions) will cluster together when its XQ values are plotted vs log t^ (Figure 6-6).

1 4 4

From preliminary data not presented in this article, we have observed that ZQ data for

pure compounds with similar or identical retention times (same set of chromatographic

conditions) will often cluster in different regions because their peak shapes are different.

As seen in Figure 6-5, peaks of 3-heptanone and 4-heptanone cluster slightly apart from

each other even though their retention times are almost identical. These peak shape

differences could be the result of differences in the sites at which retention occurs

(differences in solute-stationary phase interactions) or some other phenomenon.

Provided that differences in peak shape among compounds exist, it should be possible,

in instances where two or more compounds of interest have the same retention time but it

is known that only one of them is present, to identify the unknown analyte on the basis

of its peak shape.

C O N C LU SIO N

Our goal in this project was to develop a method for the single-channel detection of

peak impurities. We also wanted an approach that was computationally simple, but

could detect impurities at a lower S/N ratios, lower resolution, and lower contamination

levels than other methods of similar complexity, i.e., slope analysis, moments, and the

distribution function. There are other methods for the detection of impure

chromatographic peaks using single-channel data (18,19), but these methods appear to

be much too complex for general use.

The Q transformation approach that we reported here appears to fulfill all of these

objectives, although a minor drawback of all of these methods including the Q approach

is the need for some type of peak shape reference. Note, however, that our approach to

the detection of co-eluting peaks requires only one injection of a reference

multicomponent solution. In most routine labs where quantitative analysis is done, a

1 4 5

standard or mixture of standards is injected at least once per day, and in many instances,

before every sample. This standardization could easily include a peak shape calibration

as well.

Although the Q transformation approach to peak shape analysis is still in its infancy

at this time, we feel that further investigation into its properties and applications is

warranted. In fact, improvements in the measurement of the graphical parameters of a

and b or in the statistical analyses of Q transform data may result in substantial

improvements in the ability to detect peak impurities. As of this writing we know of no

other peak shape analysis techniques that can characterize peaks in the way that the Q

transform does. This coupled with its computational simplicity, and its easy automation

should provide ample incentive for further investigation.

1 4 6

TABLE 6-1. Minimum Resolution Necessary for the Detection of Two Overlapping

EMG Peaks. a

First Peak 16 Times Larger Than Second Peak.

S/N

1 0 0 500 1 0 0 0

IQ .23 .13 .1 1

Slope Analysis b .36 .24

XDFM b .42 .375

Skew b b b

Excess b b b

Second Peak 16 Times Larger than First Peak.

S/N

1 0 0 500 1 0 0 0

IQ 0.14 0.07 0.04

Slope Analysis b 0.24 0.175

XDFM 0.36 0 .2 0 0.18

Skew b b b

Excess b b b

a t / a of EMG peaks was 1.0.

b Impurity not detectable by this method at specified Signal-to-Noise before a shoulder

appears on the peak.

1 4 7

Table 6-2. Simulated Impure peaks detectable by the paired t-test and the Q profile.

Rs Ara 10 0

S/N

500 1 0 0 0

0.125 1/16 no no yes

0.1875 16/1 no no yes

0.1875 1/16 no yes yes

0.25 16/1 no yes yes

0.25 1/16 yes yes yes

0.3125 16/1 yes yes yes

a Area ratio of first peak to the second peak

1 4 8

Table 6-3. Impure experimental HPLC peaks detected via the paired t-test using the Q

profile.

Peak combination (Pure peak/impurity)

Rs Ar S/N Mobile phase

Valerophenone/Acetophenone 0 .1 16/1 2 0 0 0 Isocratic, 50% ACN

Valerophenone/Acetophenone 0 .2 16/1 2 0 0 0 Isocratic, 50% ACN

n-propyl benzene/ toluene 0 .1 2 0 /1 2 0 0 0 Isocratic, 70% MeOH

n-propyl benzene/ toluene 0 .1 2 0 /1 1 0 0 Isocratic, 60% ACN

isopropyl benzene/ethylbenzene 0.15 14/1 1 0 0 Isocratic, 60% MeOH

n-propyl benzene/n-propyl benzene a 0 .2 2 0 /1 1 0 0 Gradient, 5% ACN/min

n-butyl benzene/ethylbenzene a 0 .1 16/1 2 0 0 0 Gradient, 5%

MeOH/min

a Peaks obtained via mathematical addition of single peaks as described in the

Experimental Section.

1 4 9

Table 6-4. Comparison of some pure peaks via the paired t-test.

peaks compared mobile phase t value calculated S/N

1 & 2 Isocratic, 60% ACN 0.981 1 0 0

1&3 Isocratic, 60% ACN 1.488 1 0 0

2&3 Isocratic, 60% ACN 0.923 1 0 0

1 & 2 Isocratic, 70% MeOH 0.082 2 0 0 0

1&3 Isocratic, 70% MeOH 0 .1 0 2 0 0 0

2&3 Isocratic, 70% MeOH 0 .2 2 0 0 0

a Ethyl benzene peaks injected sequentially under given mobile phase conditions,

b Tabulated t-value for 99% confidence was 2.576. Curves are not significantly

different if calculated t is less than the tabulated value.

1 5 0

0.4-I

0.3

0.2

0.0

PURE PEAK-

IMPURITY

IMPURE PEAK (PURE PEAK + IMPURITY)

7.0 9.0 11.0 13.0 15.0 17.0

TIME (MINUTES)

Figure 6-1. Effect of co-elution on overall peak profile. Impurity elutes on the tail of the

parent peak. Conditions: EMG peaks (x/cr = 1); area ratio = 16:1; resolution

= 025. Note that no shoulder or other visible evidence is apparent on the

overall profile.

151

2.5

Q 2.0

IMPURE PEAK

1.5

PURE PEAK

1.080 10020 40 600

r (%)

Figure 6-2. Effect of co-elution on the Q transformation in which the impurity elutes on

the tail of the parent peak. Conditions: EMG peaks (x/o =1); area ratio =

16:1; resolution = 0.25.

SIG

NA

L

1 5 2

0 .41

0.3

IMPURE PEAK PURE PEAK + IMPURITY0.2

IMPURITYPURE PEAK

0.07.0 9.0 11.0 13.0 15.0 17.0

TIME (MINUTES)

2.5

2.0

PURE PEAK

IMPURE PEAK

1.00 20 40 60 100

r (%)

Figure 6-3. Effect of co-elution on (a) overall peak profile; and (b) Q transformation in

which the impurity elutes on the front of the parent peak. Other conditions as

in Figure 6-2.

1 5 3

IMPURE n-PROPYL BENZENE

PURE n-PROPYL BENZENE

Figure 6-4. Impure and pure n-propyl benzene peaks. Visual inspection of original

peaks fails to indicate presence of impurity.

1 5 4

170 i

160A

1

150

IQ140

130

2 .

8

(1) 4-Methyl Cyclohexanone(2) 3-Heptanone(3) 4-Heptanone(4) 5-Nonanone

120

110

1000.0 0.1

> I " " '

0.2' i '0.3

"r ~ T ~ r

0.4> i ■0.5 0.6

LOG t:R

Figure 6-5. Example of clustering as a function of peak identity for a series of ketones.

1 5 5

HOMOLOGOUS SERIES

DIFFERENT COMPOUNDS, SAME t'R

LOG t'R

Figure 6 -6 . Illustration, using simulated data, of Q transformation approach to

qualitative analysis.

LITERATURE CITED

1 5 6

1) Papas, A. N. CRC Crit. Rev. Anal. Chem., 1989, 20, 359-404.

2) Schieffer, G.W. J. Chromatogr., 1985, 319, 387-391.

3) Grushka, Eli; Monacelli, G. C. Anal. Chem., 1972, 44, 484-489.

4) Grushka, Eli. Anal. Chem., 1972, 44, 1733-1738.

5) Grushka, Eli.; Myers, M. N.; Giddings, J. C. Anal. Chem., 1970, 42, 21-26.

6) Rix, H. Anal. Chim. Acta, 1986, 191, 467-472

7) Rix, H. J. Chrom., 1981, 204, 163-165

8 ) Excoffler, J.-L.; Jaulmes, A.; Vidal-Madjar, C.; Guiochon, G. Anal. Chem.,

1982, 54, 1941-1947

9) Chesler, S. N.; Cram, S. P. Anal. Chem., 1971, 43, 1922-1933.

10) Petitclerc, T.; Guiochon, G. J. Chromatogr. Sci., 1976, 14, 531-535.

11) Eikens, David I.; Carr, Peter W. Anal Chem., 1989, 61, 1058-1062.

12) Jeansonne, Mark S.; Foley, Joe P. J. Chromatogr., 1989, 461, 149-163.

13) E. Grushka, M.N. Myers, and P.D. Schettler, Anal. Chem., 1969, 41, 889-892.

14) Jeansonne, Mark S.; Foley, Joe P. Manuscript submitted to Anal. Chem.

15) Snyder, L.R.; Kirkland, J.J., Introduction to Modern Liquid Chromatography,

2nd ed.; Wiley: 1979; Chapter 7

16) Miller, J.C.; Miller, J.N., Statistics For Analytical Chemistry, 2nd ed.; Horwood:

1988; Chapters 3 and 6 .

17) Harris, D.C., Quantitative Chemical Analysis; W.H. Freeman: 1982; Chapter 4

18) Sybrandt, L.B. ; Perone, S.P., Anal Chem, 1972, 44, 2331-2339

19) Pichler, M.A.; Perone, S.P., Anal. Chem., 1974, 46, 1790-1798

Chapter 7

SUM M ARY

1 5 7

1 5 8

As mentioned in the Introduction, the field of chromatography has advanced greatly

since its inception. Of particularly fast growth has been the use of computerized data

analysis. In fact, computer data analysis has now become a routine part of

chromatography, due, in part, to the computer’s ability to process large amounts of data

quickly and efficiently. Quicker and more efficient processing of data thus allows more

information to be obtained than would have been feasible without the computer. The

availabilty of more information should result in more productivity, because trial and error

techniques become less necessary. Enhancing productivity is a major goal of computer

data analysis.

Recently, the trend in chromatographic automation has teen the development of

expert systems. It is believed that these systems will allow those with only limited

chromatographic background to do more difficult and complex types of analyses.

However, the expert system needs large amounts of information in order to make its

decisions. In the opinion of this author, chromatographic data analysis techniques like

those presented in the preceding chapters are necessary to supply the needed information.

Of particular importance will be peak shape information. Because chromatographic

peak shapes in chromatography are somewhat unpredictable, yet reveal much about the

processes occurring within the column, efficient methods of peak shape measurement

and comparison are necessary. Therefore, the information presented in Chapters 2 - 4

should enable more accurate measurement of important chromatographic parameters via

use of the EMG model because it describes real chromatographic peaks tetter than the

frequently used Gaussian model. Although methods more mathematically rigorous than

those presented in chapters 2-4 may be available, their use for routine data analysis is

often limited by their complexity and time consuming calculations. The easier and more

1 5 9

intuitive approaches presented in those chapters will encourage greater use of the more

accurate EMG model for routine chromatographic analyses.

Similarly, the Q transformation, discussed in Chapters 5 and 6 should encourage

more routine use of peak shape comparison with single-channel data in order to extract

more information during routine analyses. Again, more sophisticated techniques are

available, but their complexity often discourages their routine use. Also, because the Q

transformation is better than other methods of similar complexity for comparing peak

shapes, it should be the technique of choice when peak shape comparison is necessary.

Increasing the information available and shortening analysis times through the use

of the techniques presented here will play a major role in the future development of

chromatography. The importance of these techniques cannot be overemphasized.

Appendix A

Q TRANSFORM ATION PROGRAM DOCUM ENTATION AND

LISTIN G

1 6 0

161

Q T r a n s f o r m — O V E R V I E W

Q TRANSFORM is a program that can be used for peak shape analysis. Q is measured at fractions of the peak height (r), between 2 and 98% in increments of 1%. The total or £Q over an entire peak gives a single number that can indicate peak shape. This program will take a user specified data file and topographical file and output a Q file in ASCII format. The Q file output contains (i) the value of Q measured at peak height fractions between 2 and 98%; and (ii) XQ, the sum of these Q values that indicates peak shape in a single number. The interpretation of the Q output data which permits the detection of co-eluting peaks is described in the accompanying manuscript preprint "Single Channel Peak Impurity Detection...".

When Q TRANSFORM is started, the user will be presented with a choice of whether to "CONFIGURE", "RUN Q" or "STOP". Choosing "CONFIGURE" enables the user to set up the RUNDATA.TXT file (described below), while choosing "RUN Q" will start the calculations for the chromatogram files that have been listed in RUNFILES.TXT (see INPUT section below). To stop the program after choosing "RUN Q", simply press uppercase Q on the keyboard. It may take a few seconds, but the program will eventually quit

The file SAMPLEDATA. 1 included on your disk is a simulated chromatogram that is used for the sample chromatogram and topographical files discussed below. You can run Q TRANSFORM on this data in order to verify that the program is working. The resulting data will be printed at the end of the Q output file already present on the disk. Any time Q TRANSFORM is run on a data file for which a Q output file already exists, the ouput data will simply be added to the end of the existing Q ouput file.

I N P U T

The Q TRANSFORM program is designed to allow users to input their own chromatograms or other peak shaped data. Two ASCII (text) files are required for each chromatogram:1. User-named raw data file, with the extension ".txt". Example: "Sampledata.l.txt"2. User-named topographical file, with extension ".top". Example:

"Sampledata. l.top"

1 6 2

Two other ASCII files are required each time Q Transform is run.3. "Runfiles.txt"4. "Rundata.txt."Note that for a given run of Q Transform, all files must have the same filepath (be in the same folder).

The first three files should be set up by the user before running Q Transform. The last file is set up automatically while running Q Transform by using the CONFIGURE routine. The purpose and format of each of these files is given below. Two sets of raw data and topographical files are included on the demonstration disk: Sampledata.l.txt, Sampledata. 1.top, Sampledata.2.txt, and Sampledata.2.top

"Runfiles.txt" is an ASCII (text) file that contains the name(s) of the data file(s) on which the Q tranform will be run. Open this file using any word processing program (Word, MacWrite, etc.) to verify the file name(s) BEFORE running Q Transform.

"Rundata.txt" is an ASCII (text) file that contains user-specified parameters for the subsequent running of Q Transform. This file is generated from within Q Transform by running the CONFIGURE routine. The next section describes this in detail.

R U N D A TA .TX T FILE

This file is read by the Q TRANSFORM program when it is run. When run, a menu is presented that allows the user the option of running Q TRANSFORM or configuring the Rundata.txt file. Rundata.txt is in ASCII format so that the user can modify the contents using his own programs instead of configuring it from the CONFIGURE option in Q TRANSFORM. Each line in the file gives a parameter that Q TRANSFORM uses. There can be no extra lines or blank lines in the file.

Line 1 in this file consists of the path to the data files to be used. For example, to examine a file on the Q TRANSFORM disk not in any subdirectories or folders the path would simply be Q TRANSFORM: for Macintosh computers or Q TRANSFORM\ for IBM personal computers.

Line 2 gives the smoothing factor used by the program. There must be a number here. There cannot be a blank line or non-numerical character. A smoothing factor of 0 will tell Q TRANSFORM not to smooth the data.

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Unless a smoothing factor of 0 is input by the used, the Q TRANSFORM will perform a weighted moving average quadratic smooth on the data as described by Savitsky and Golay L The smoothing factor is defined as the number of points in the moving window divided by the number of points in the peak from half-height on the front of the peak to half-height on the back of the peak. For example, a peak that has 25 points from half-height to half-height would have a 25 point moving average window applied to it if the smoothing factor was 1 , while a peak with 11 points from half-height to half-height would have an 11 point moving average window applied to it with a smoothing factor of 1. Therefore, peaks that do not have the same number of points can still be smoothed equally. Moving windows of between 5 and 25 points are available with the program and if a smoothing factor is used that indicates more than 25 points should be used then 25 points will be used. Likewise, if the smoothing factor indicates that less than 5 points should be used in the moving window, then 5 will be used, unless the smoothing factor is 0 .

Line 3 is a "Y" or "N" depending on whether the user would like each individual peak to be plotted on the screen.

Line 4 is again a "Y" or "N" depending on whether the user would like a bell to ring after each chromatogram is finshed.

Note: When using the CONFIGURE option from Q TRANSFORM, the answer in square brackets given after each question is the default answer. Simply pressing the return key will enter this answer.

DATA FILE FORMAT

The name of a chromatogram data file to be used with this program is always followed by the ".TXT" extension. Case does not matter.

The file must be in ASCII format. These files consist of one number per line with no leading or trailing spaces on each line. The numbers are the signal heights for the chromatographic data. The sampling interval or times at which each point occurs is included within the topographical file, described below. Due to the type of smoothing

1 A. Savitsky and M. J. E. Golay, "Smoothing and Differentiation of Data by Simplified Least Squares Procedures", Analytical Chemistry 36,1627 (1964)

1 6 4

employed by this program, a minimum of 16 data points must be present in this file before the first peak's data actually starts, while 16 points must also be included after the data for the last peak in the chromatogram ends. These "extra" points may be added as dummy points, set to 0 , if needed.

TOPOGRAPHICAL FILE FORMAT

The name for the topographical file corresponding to a data file is the same as the root name for the data file with the extension ".TOP" instead of ".TXT". Again, case does not matter.

The topographical files to be used are in ASCII format Each line contains 3 comma delimited numbers. The first number in each line is a code number that tells Q TRANSFORM what the second and third numbers refer to.

Sam ple Topographical File

o, 0 .0000, 3.000006 , 2 .0 000 , 1.000001, 2.0050, 3.000002 , 2.7310, 3.076595, 3.1150, 14.83301, 5.0350, 3.001087, 54.000, 7.000002 , 5.6320, 3.085055, 6.1440, 8.640751, 8 .0210 , 3.000827, 56.000, 4.000002 , 8.4480, 3.066515, 9.1310, 12.37851, 11.264, 3.000607, 6 6 .0 00 , 6.00000

CODE 0

All topographical files begin this way. The second number in the line refers to the chromatogram start time in minutes, e.g., the time at which the first point in the chromatogram occurs. Most of the time this number is 0. The third number in this line is

1 6 5

a code referring to the sample interval for the chromatogram. Note: the sample interval must be constant for the entire chromatogram in order for this version of Q TRANSFORM to run correctly. The following chart explains the sample interval codes.

Sam pling Code Sampling interval(S econ d s)

3 2.564 1.285 0.646 0.327 0.168 0.089 0.0410 0 .0211 0.0112 0.313 0.214 0.115 0.05

CODE 6

This line should always be the same as shown in the sample topographical file.

CODE 1

This line tells Q TRANSFORM where a basepoint occurs. The program draws straight lines between basepoints, so that the baseline level can be subtracted from each peak before Q is calculated at each value of r. The second number is the time (in minutes) at which the signal level is "on baseline" and the third number is the signal height at that time. The sample chromatogram in Fig. 1 below shows the 4 basepoints that are given in the sample topographical file above.

1 6 6

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8.0 10.0 12.00*0 4.0 6.02*0

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CODE 2

Indicates a peak start. The second number in the line indicates time of peak start while the third number indicates the height of the signal at the peak start Can also indicate a peak end(peak start) if a valley occurrs between two peaks and the signal level is not yet at the baseline level. Normally, a code of 1 will indicate a peak end if two peaks are well separated.

CODE 5

Indicates peak apex. The second number is the retention time of the peak while the third number is the signal height at that retention time.

CODE 7

1 6 7

Second number indicates the number of points in the peak and the third number indicates the area of the peak. This code always comes after the peak end that this line applies to whether the peak end code is indicated by a 1 or 2 .

OTHER CONSIDERATIONS

1) There must not be any extra blank lines in the topographical file, either before the first line or between lines.

2) There must not be a line feed and/or carriage return after the last number in the last line.

O U T P U T

Q O UTPUT FILE FORM AT

The name for the Q file that is output is derived from the name of the data file used by the program. For example, the file Sampledata.l.txt results in an output file named Sampledata.l.Q.

A sample Q output file is included on the Q TRANSFORM disk. You can open this file with any text editing program. The first line gives information about whether the data was smoothed prior to calculating Q .what the smoothing factor was and the date and time the Q analysis was done. The time given is in ANS standard format "hh:mm:ss", where hours are measured by a 24 hour clock (midnight is 00:00:00). This format allows one to do multiple Q TRANSFORM runs on the same data file and store all runs in the same .Q file with each subsequent run stored after prior runs. More about the smoothing factor is given later. The second line in the file consists of the column names. In the sample file (Sampledata. l.Q), you will see a column for r, Q l, Q2, and Q3. These columns correspond to columns for the r values used, and the Q value at each r for each peak in the chromatogram. In this case there are 3 peaks, so 3 Q columns are seen, one for each peak. The last line in the file shows the XQ value for each peak. Columns are tab delimited for this file making it easy for the user to open the file directly into most popular spreadsheet and graphing programs.

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Note: If a valley between two peaks is high enough then Q TRANSFORM will not be able to calculate Q for every r. If there are zero's for Q for a peak at the lower r values this indicates that two peaks were not resolved enough to calculate Q at these lower r values. The file Sampledata.2.Q illustrates this for two partially overlapping peaks.

USE OF Q OUTPUT

The preprint of an article describing the use of the Q TRANSFORM data for the detection of impure peaks in chromatography is included with the demo disk. The data from the ”.Q” file and the ".TOP" file are needed for the methods given in the article.

TIPS ON USING Q TRANSFORM

1) The same smoothing factor should be used for both the reference peak(s) and sample peak(s).

2) For best results, there should be at least 30 points from half-height to half-height in the peaks used. The signal-to-noise (defined as the height of peak divided by the root-mean-square noise in the baseline) should be greater than 2 0 0 for best results. We have used this program down to signal-to-noise values of less than 1 0 0 although the ability to detect impurities decreases as the signal-to-noise decreases.

3) When running Q TRANSFORM for the first time, set the "plot peak on screen" option in RUNDATA.TXT to "Y" in order to make sure that the program is reading in peaks correcdy. Be aware that 16 extra points will be plotted before and after each peak. These extra points are used in the smoothing routine but not in the actual calculations of Q.

4) Sample peak(s) and reference peak(s) must be obtained under the same chromatographic conditions because different conditions can introduce differences in peak shape.

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RUNNING Q TRANSFORM

Two versions of Q TRANSFORM have been written—one for the Macintosh Personal Computer and one for the IBM Personal Computer. Please refer to individual owner guides for information on how to start an application program.

Once Q TRANSFORM has started, the user will be presented with a choice of whether to "CONFIGURE", "RUN Q" or "STOP". Choosing "CONFIGURE" enables the user to set up the RUNDATA.TXT file, while choosing "RUN Q" will start the calculations for the chromatogram files that have been listed in RUNFILES.TXT. If the user wishes to stop the program after choosing "RUN Q" simply press uppercase Q on the keyboard. It may take a few seconds, but the program will eventually quit

Q TRANSFORM ATION PROGRAM4c 4c * * 4c * * * * * * * 4c 4c >|c * * * * 4c * * * * * * * 4c 4c * # 4c * * 4c * * * * * * * * 4> * * * * * * * * * * * * * * * *

! Copyright 1990! by! Mark S. Jeansonne and Joe P. Foley! Department of Chemistry, 232 Choppin Hall! Louisiana State UniversityI Baton Rouge, LA 70803i

LIBRARY "Pictlib*"DIM M$(3)LET COL$='"'OPEN #2:SCREEN 0,.01,0,.01 DOCALL STARTUP CLOSE #9 RESTORE LOOPSUB STARTUP

CLEARCALL set_Frame(.l,.8,.3,.8)

CALL draw_pictfile("Copyright notice",0)CALL MENU_SET("TOP",COL$,3,10,MENU$,#9)MAT READ M$DATA CONFIGURE,RUN Q.STOPCALL MENU_ALL(M$,3,""ANS,MENU$,#2,#9)

LETT1=TIME SELECT CASE ANS CASE 1

CLOSE #2OPEN #2:SCREEN 0,1,0,1 CALL CONFIG CLOSE #9 RESTORE CALL STARTUP

CASE 2OPEN #l:Name "runfiles.txt",access input,organization text DO while more #1

INPUT #l:file$LET a$=E$&file$CALL Qtransform3Mult(a$)

LOOP CASE 3

STOP CASE ELSE

STOP END SELECT

END SUB LETT2=TIME

171

END

EXTERNALSUB Qtransform3Mult(FILE$)

LIBRARY "sgsmooth*"CLEAR

DIM TIM(3000), TEMPTIM(3000), RT(50), RTHEIGHT(50), PSTART(50), PEND(50),TPEND(50)

DIM A(50,100), B 1(50,100), PEAK(3000), TEMPPEAK(3000), temp(50)DIM NUMSAMP(3000), TEMP_HALF_HEIGHT(50), START(50),PTEND(50),

point(50)DIM X(50,2), Y(50,l), Xt(2,50), XtX(2,2),XtY(2,l), XtXi(3,3), B(2,l)DIM Xraw(4,l), Yraw(4,l),TA(100),TB(100)DIM AREA 1(1),AREA2( 1 )rAREA(8 )JHEIGHT( 100)DIM PERCENTHT(8 )DIM TPSTART(50),TBSPOINT(50),BSPOINT(50),BSLNSLP(50)DIM XC(60,3), YC(60,1), XTC(3,60), XTXC(3,3), XTYC(3,1), XTXIC(3,3),

BC(3,1)DIM XRAWC(60,1 ),YRAWC(60,1 ),H 1 $(20),H2(50),H3(50),H4(50)DIM H5(50),H6(50),H7(50),H8(50)DIM changeina(100),changeinb(100)DIM Q(50,100),SUMQ(50)OPEN #4: Name "rundata.txt".access input,organization text LINE INPUT #4 :5 LET f$=trim$(f$)LET INFO$=HLE$OPEN #1 : screen 0, 1, .8 , 1IINPUT prompt "Name of data file ? ":file$LET file$=f$&file$! INPUT #4:answer$INPUT #4:smoothfact IINPUT #4:Threshold 1INPUT #4:Start_Q IINPUT #4 :End_Q IINPUT #4:a6$INPUT #4:a7$INPUT #4:a8$INPUT #4:all$CALL CONTINU SUB CONTINU

IF KEY INPUT THEN GET KEY KEY IF KEY=81 THEN STOP

END IF END SUB

OPEN #7:NAME FILE$&".Q",CREATE NEWOLD, ACCESS OUTPUT,ORGANIZATION TEXT !******

SET #7:POINTER END SET #7:MARGIN 2000j****************************************************************

CLEAROPEN #2:name file$&".txt",access INput,ORGANIZATION textOPEN #3 : screen 0, 1,0, .8WINDOW #3OPEN #5:PRINTERSET MARGIN 70PRINT "NOW READING IN TOPOGRAPHICAL DATA FROM ";INFO$ PRINTOPEN #6 :NAME file$&".TOP",ORGANIZATION TEXT ACCESS INPUT LET NUMPEAK=0 LET K=1DO WHILE MORE # 6

WHEN error inINPUT #6:H4(k),H6(k),H8(K)

USELET h4(K)=20

END WHEN LETK=K+1

LOOPFORL=l TO K-l

IF H4(L)=6 THEN LET BSCODE=H6 (L) -

END IF

IF H4(L)=0 THEN LET RUNTIME=H6 (L)SELECT CASE H8 (L)CASE 3

LET DT=2.56/60 CASE 4


LET DT=.64/60 CASE 6


LET DT=.l 6/60 CASE 8








1 7 3


LET DT=0.001/60 CASE 17

LET DT= 15/60 CASE 18

LET DT=4/60 CASE 19

LET DT=8/60 CASE 20


LET DT=6/60 CASE 22


LET DT=18/60 CASE 24

LET DT=20/60 CASE 25

LET DT=9/60 CASE 26

LET DT=11/60 END SELECT

END IFIF H4(L)=1 THEN

LET NUMBSPT=NUMBSPT +1 LET TB SPOINT(NUMB SPT)=H6 (L)LET BSPOINT(NUMBSPT)=H8 (L)

END IF NEXT LFOR L=1 TO K-l

IF H4(L)=5 THEN IF h8 (L)>=threshold then

LET NUMPE AK=NUMPE AK+1 LET PEAKNUM=NUMPEAK LET RTHEIGHT(PEAKNUM)=H8 (L) LET RT(PEAKNUM)=H6 (L)IF H4(L-1)=2 OR H4(L-1)=1 THEN

LET TPSTART(PEAKNUM)=H6 (L-1) LET PS TART (PE AKNU M )=H 8 (L-1)

ELSEIF H4(L-2)=2 OR H4(L-2)=1 THEN LET TPSTART(PEAKNUM)=H6(L-2) LET PSTART(PEAKNUM)=H8(L-2)

END IFIF H4(L+1)=7 THEN

LET NUM S AMP(PE AKNUM)=H6 (L+1) ELSEIF H4(L+2)=7 THEN

LET NUMSAMP(PEAKNUM)=H6(L+2) END IF

END IF END IF

174

NEXT LCALL CONTINU LET J=1LET T=RUNTIME LET K=1 LET FLAG=0

LETT1=TIME FOR PEAKNUM=1 TO NUMPEAK

LET START(PEAKNUM)=round((TPSTART(PEAKNUM)-T-16*DT)/DT) LET

PTEND(PEAKNUM)=round(START(PEAKNUM)+NUMSAMP(PEAKNUM)+32) NEXT PEAKNUMPRINT "NOW READING IN PEAK DATA FROM ";INFO$PRINTDO WHILE MORE #2

LINE INPUT #2: RAW$FOR kk=l to numbspt

IF J>round((tbspoint(kk)-T)/dt)-3 and J<round((tbspoint(kk)-T)/dt)+3 then LET temp(kk)=temp(kk)+val(raw$)! PRINT val(raw$)

END IF NEXT kk LET J=J+1

LOOP

FOR kk=l to numbspt LET bspoint(kk)=temp(kk)/5

NEXT kk

LET HOLDJ=l RESET #2:BEGIN

FOR PEAKNUM=1 TO NUMPEAKMATTA=0MATTB=0

MAT PEAK=TEMPPEAK MAT TIM=TEMPTIM LET J=HOLDJ DOLINE INPUT #2: RAW$

IF J > START(PEAKNUM) then IF J < PTEND(PEAKNUM) THEN

LET JJ=J-START(PEAKNUM)+ADDON LET PEAK( J J)=V AL(RA W$)LET tim(jj)=(jj-l)*dt+tpstart(peaknum)-16*dt

END IF END IFIF J>START(PEAKNUM+1) AND PEAKNUMoNUMPEAK THEN

LET II=J-START(PEAKNUM+1)LET TEMPPEAK(II)=V AL(RA W $)

1 7 5

LET TEMPTTM(II)=(II-1 )*DT+TPSTART(PEAKNUM+1)-16*DT END IFCALL CONTINU LET J=J+1 LET HOLDJ=J

LOOP until J> PTEND(PEAKNUM)LET T2=TTME

CALL CONTINU CALL initialize CLEARCALL CONTINU CALL smooth CALL CONTINU CALL BSLN.SUBTRACT CALL CONTINU CALLMAXFIT CALL CONTINU CALL QCALC CALL CONTINU CALL CONTINU

NEXT PEAKNUM CALL PRINTQ

IF al 1$="Y" then CALL ring_bell

SUBMAXFIT LET XMEAN=0 LET YMEAN=0 LET XSUM=0 LET YSUM=0 MAT XRAWC=0 MAT YRAWC=0 MAT XC=0 MAT YC=0 MAT XTC=0 MAT XTXC=0 MATXTYC=0 MAT XTXIC=0 MATBC=0 LET N=7LET JJ=(RT(PEAKNUM)-tpstart(peaknum))/DT+1+16 FOR 1=1 TO N

LET XRAWC(l,l)=tim((JJ-ROUND(N/2)+I))LET YRAWC(1,1 )=PEAK((J J-ROUND(N/2)+I))1PRINT USING .#########": XRAWC(I,1),YRAWC(I,1)

NEXT I FOR 1=1 TO N

LET XSUM=XSUM+XRAWC(I,1)LET YSUM=YSUM+YRAWC(I,1)

NEXT ILET XMEAN=XSUM/N LET YMEAN=Y SUM/N

1 7 6

FOR 1=1 TO N LET XC(I,1)=1LET XC(I,2)=XRA W C(1,1 )-XMEAN LET XC(I,3)=XC(I»2)A2

LET Y C(1,1 )=YRA WC(1,1)-YMEAN NEXT IMAT XTC=TRN(XC)! MAT PRINT XTC;MAT XTXC=XTC*XC MAT XTXIC=INV(XTXC)1PRINT "DET(XTXC)=";DETMAT XTYC=XTC*YC!MAT PRINT XTYCMAT BC=XTXIC*XTYCLET TEMPXMAX=-BC(2,1 )/(2*B C(3,1))LET TEMPYMAX=BC( 1,1 )+BC(2,1 )*TEMPXMAX+BC(3,1 )*TEMPXMAXA2 LET XMAX=TEMPXMAX+XMEAN LET YMAX=TEMPYMAX+YMEAN LET RTHEIGHT(PEAKNUM)=YMAX LET RT(PEAKNUM)=XMAX

END SUB

SUB initialize FOR k=l to (numbspt-1)

IF rt(peaknum)>tbspoint(k) and rt(peaknum)<tbspoint(k+l) then LET point(peaknum)=kLET bslnslp(peaknum)=(bspoint(k+1 )-bspoint(k))/(tbspoint(k+1 )-tbspoint(k))

END IF NEXT kLETTPEND(PEAKNUM)=TPSTART(PEAKNUM)+(NUMSAMP(PEAKNUM)-

1)*DTLETBSLN=BSLNSLP(peaknum)*(RT(PEAKNUM)-

tbspoint(point(peaknum)))+bspoint(point(peaknum))LETTEMP_HALF_HEIGHT(PEAKNUM)=.5*(RTHEIGHT(PEAKNUM)-

BSLN)END SUB

SUB smooth LET N=17FOR I=N to (N+Numsamp(peaknum))

LET bsln=bslnslp(peaknum)*(tim(i)- tbspoint(point(peaknum)))+bspoint(point(peaknum))

LET temp_pointl=peak(I)-bsln LET bsln=bslnslp(peaknum)*(tim(i-1 )-

tbspoint(point(peaknum)))+bspoint(point(peaknum))LET temp_point2=peak(I-1 )-bslnIF tim(I)<Rt(peaknum) and temp_pointl >TEMP_HALF_HEIGHT(PEAKNUM)

and temp_point2<TEMP_HALF_HEIGHT(PEAKNUM) then LET tl=tim(i)

END IFIF tim(I)>rt(peaknum) and temp_point 1 <TEMP_HALF_HEIGHT(PEAKNUM)

and temp_point2>TEMP_HALF_HEIGHT (PEAKNUM) then

177

LET t2=tim(i)END IF

NEXT iLET ptshalfHt=round((t2-tl)/dt)LET hold=int(SMOOTHFACT*ptshalfht)IF mod(hold,2)oO then

LET smoothwidth=int(hold)ELSE

LET smoothwidth=int(hold-l)END IFIF SMOOTHWIDTH<=0 THEN

LET SMOOTHWIDTH=0 ELSEIF S MOOTHWIDTH>0 AND SMOOTHWIDTH<5 THEN

LET SMOOTHWIDTH=5 ELSEIF SMOOTHWIDTH>25 THEN

LET SMOOTHWIDTH=25 END IF PRINTIF SMOOTHWIDTHoO THENPRINT "SMOOTHING PEAK # ";STR$(PEAKNUM)PRINTPRINT "SMOOTHING FACTOR = ";SMOOTHFACT PRINTPRINT "NUMBER OF POINTS IN SMOOTH = ";SMOOTHWIDTH

SELECT CASE smooth width CASE 5

CALL sgsmooth5(PEAK,(NUMSAMP(PEAKNUM)+32)) CASE 7



CALL sgsmooth 11 (PEAK,(NUMSAMP(PEAKNUM)+32)) CASH 13

CALL sgsmooth 13(PEAK,(NUMS AMP(PEAKNUM)+32)) CASE 15

CALL sgsmooth 15 (PEAK,(NUMS AMP(PEAKNUM)+32)) CASE 17

CALL sgsmooth 17(PEAK,(NUMSAMP(PEAKNUM)+32)) CASE 19

CALL sgsmooth 19(PEAK,(NUMSAMP(PEAKNUM)+32)) CASE 21



CALL sgsmooth25(PEAK,(NUMSAMP(PEAKNUM)+32)) END SELECT ELSEPRINT "NO SMOOTH APPLIED TO THIS DATA”

END IFFOR 1=1 TO NUMSAMP(PEAKNUM)+32

1 7 8

IF TIM(I)=TBSPOINT(POINT(PEAKNUM)) THEN LET BSPOINT(POINT(PEAKNUM))=PEAK(I)

END IFIF TIM(I)=TBSP0INT(P0INT(PEAKNUM)+1) THEN

LET B SPOINT (POINT (PEAKNUM)+1 )=PE AK(I)END IF

NEXT I END SUB

SUB BSLN_SUBTRACT LET MAX=0LET BSLNSLP(PEAKNUM)=(BSPOINT(POINT(PEAKNUM)+l)-

B S POINT (POINT (PE AKNUM)) )/(TB S POINT (POINT (PEAKNUM)+1 )- TBSPOINT(POINT(PEAKNUM)))

LET N=17FOR I=N TO (NUMSAMP(PEAKNUM)+N-1)

LET BSLN=(BSLNSLP(peaknum)*((tim(i))- tbspoint(point(peaknum)))+bspoint(point(peaknum)))

LET PEAK(I)=PEAK(I)-BSLN !PRINT #5:I,PEAK(I,PEAKNUM)IF PEAK(I)>MAX THEN

LET MAX=PEAK(I)- LET RT(PEAKNUM)=tim(i)

END IF NEXT ILET PEND(PEAKNUM)=peak((16+numsamp(peaknum)))LET PSTART(PEAKNUM)=peak(17)IF a7$="Y" then

CLEAR SET WINDOW

TB SPOINT (POINT(PEAKNUM)),TB SPOINT (POINT (peaknum)+1 1 *max,MAX FOR 1=17 TO (NUMS AMP(PEAKNUM)+16)

PLOT LINES: tim(i),PEAK(I);tim(i-l), Peak(I-l)NEXT I PLOT

LINES:TBSPOINT(POINT(PEAKNUM)),0;TBSPOINT(POINT(PEAKNUM)+1),0 PAUSE 5

END IF END SUB

SUB QCALC CLEARSET WINDOW 0,1,0,1PRINT "CALCULATING Q FOR PEAK # ”;STR$(peaknum)PRINTFOR KK=1 TO 99

!***DETERMINE SIGNAL HEIGHT AT WHICH PEAK FRACTIONS OCCUR***

LET HEIGHT(KK)=(KK/100)*(RTHEIGHT(PEAKNUM))NEXT KKLET complete=0LET increment=.5102041

1 7 9

PLOT TEXT, AT .27,.8:"% complete"LET place=98 LET COUNT2=98LET HOLD=RTHEIGHT(PEAKNUM)LET N=17 LET tempO l=le200 LET temp02=le200

LET PLACE=1 LET PLACE2=98

FOR I=N TO (N+NUMSAMP(PEAKNUM))IF RTHEIGHT(PEAKNUM)>THRESHOLD THEN

IF TIM(I)<RT(PEAKNUM) THEN FOR COUNT=PLACE TO 98

IF PEAK(I)>HEIGHT (COUNT) THEN IF PEAK((I-l))<HEIGHT(COUNT) THEN caU CONTINU

LET complete=complete+increment BOX CLEAR .2,.25,.7,.87

PLOT TEXT, AT .2,.8:str$(int(complete))IF COUNT=l THEN

CALL FITFACTCALC CALL Linefit

END IFIF COUNT<91 THEN

IF PEAK(I)oTEMP01 AND PEAK((I-l))oTEM P02 THEN CALL LINEFIT

END IFLETTA(COUNT)=(HEIGHT(COUNT)-B(l,l))/B(2,l)

ELSELET M=(PEAK(I)-PEAK((I- 1)))/(TIM(I)-TIM((I-1)))LET TA(COUNT)=(HEIGHT(COUNT)-PEAK((I-l))+M*TIM((I-

1)))/MEND IF

LET TEMPO 1=PEAK(I)LET TEMP02=PEAK((I-1))LET PLACE=PLACE+1•PRINT COUNT;" ";FITFACT;" ";HEIGHT(COUNT);" ”;TIM(I);"

”;TIM((I-1));" ’VTA^'iTACCOUNT)END IF

END IF NEXT COUNT

END IFIF TIM(I)>RT(PEAKNUM) THEN

FOR COUNT2=PLACE2 TO 1 STEP -1 IF PEAK(I)<HEIGHT (COUNT2) THEN

IF PEAK((I-1 ))>HEIGHT(COUNT2) THEN CALL CONTINU

LET complete=complete+increment BOX CLEAR .2,.25,.7,.87

PLOT TEXT, AT .2,.8:str$(int(complete))IF COUNT2<91 THEN

1 8 0

IF count2=90 then CALL linefitIF TEM P03oPEAK(I) AND TEM P04oPEAK((I-1)) THEN

CALL LINEFIT END IFLET TB (COUNT2)=(HEIGHT (COUNT2)-B (1,1 ))/B(2,1)

ELSELET M=(PEAK(I)-PEAK((I-1 )))/(TIM(I)-TIM(a-1)))LET TB (COUNT2)=(HEIGHT(COUNT2)-PE AK((I-

1 ))+M*TIM((I-1 )))/MEND IF

LET TEMP03=PEAK(I)LET TEMP04=PEAK((I-1))!PRINT COUNT2;" ";FITFACT;" ”;HEIGHT(COUNT2);" ,,;TIM(I);,,

”;TIM((I-1));" ";"TB=";TB(COUNT2)LET place2=place2-l

END IF END IF

NEXT COUNT2 END IF

END IF NEXT I

!DETERMINE PEAK WIDTH AND ASYMMETRY AT DESIRED PEAK HEIGHT FRACTIONS

FOR KK =1 TO 98 IF TA(KK)>0 AND TB(KK)>0 THEN

LET A(PEAKNUM,KK)=RT(PEAKNUM)-TA(KK)LET B1 (PEAKNUM,KK)=TB(KK)-RT(PEAKNUM)

ELSELET A(PEAKNUM,KK)=0 LET B 1 (PEAKNUM,KK)=0 END IF NEXT KK FOR JJ=2 TO 98IF A(PE AKNUM, J J)>0 AND A(PEAKNUM,JJ-1)>0 AND

B 1 (PEAKNUM,JJ)>0 AND B1(PEAKNUM,JJ-1)>0 THENLET changeina(ij)=a(PEAKNUM jj- l)-a(PEAKNUM jj)LET changeinb(jj)=bl (PEAKNUM ,jj-l )-bl (PEAKNUM jj)IF CHANGEDMA(J3)>0 AND CHANGEINB(JJ)>0 THEN

LET Q(PEAKNUM,JJ)=changeinB(JJ)/changeinA(jj)ELSE

LET Q(PE AKNUM, J J)=0 END IF

END IFLET sumQ(peaknum)=SumQ(peaknum)+Q(peaknumjj)

NEXT JJ

END SUB

SUB FITFACTCALC

! ** NMPTSHALFPEAK=# PTS FROM PEAK START TO PEAK APEX

LET NMPTSHALFPE AK=(RT(PEAKNUM)-TIM(I))/DT +1 LET FITFACT=ROUND (NMPTS HALFPEAK/14)IF MOD(FITFACT,2)>0 THEN LET FITFACT=FITFACT-1 IF FITFACT>10 THEN LET FITFACT-10 IF FTTFACT<2 THEN LET FITFACT=2 LET FITFACTHOLD=FITFACT

END SUB

SUB LINEFIT MAT x=0 MAT y=0 MAT xt=0 MAT xtx=0 MAT xtxi=0 MAT b=0 LET COUNTER=0FOR L=I-(FITFACT/2) TO I+(FITFACT/2-l)

LET COUNTER=COUNTER+1 LET X(COUNTER, 1 )=1 LET X(COUNTER,2)=TIM(L)LET Y (COUNTER, 1 )=PEAK(L)

NEXT LMAT XT=TRN(X)MAT XTX=XT*X MAT XTY=XT*Y MAT XTXi=inv(xtx)MAT B = XtXi*XtY

END SUB

SUB PRINTQ CLEARPRINT "PRINTING Q TO FILE ";INFO$&".Q"PRINTLET MONTH$=DATE$[5:6]LET YEAR$=DATE$[ 1:4]LET DAY$=DATE$[7:8]

PRINT #7:"SMOOTHFACT=";SMOOTHFACT;" DATE ";MONTH$;"/”;DAY$;'‘/u;YEAR$;1’ TIME: ";TIME$

PRINT #7:"r";Chr$(9);If a8 $="y" or a8 $="Y" then FOR Peaknum=l to Numpeak

PRINT #7:"Q”; STR$(peaknum); chr$(9);"b";str$(peaknum); chi$(9);"a";str$(peaknum); chr$(9);

NEXT peaknumelse

FOR Peaknum=l to Numpeak PRINT #7:"Q";STR$(peaknum);chr$(9);

NEXT peaknumend if

PRINT #7

1 8 2

If a8$="y" or a8 $="Y" then FOR jj=l to 98

PRINT #7:str$(jj);FOR peaknum=l to numpeak

PRINT #7:CHR$(9); Str$(Q(peaknumjj)); chr$(9); str$(bl(peaknumjj)); chr$(9);str$(a(peaknumjj));

NEXT peaknum PRINT #7

NEXTjj

FOR jj=2 to 98 PRINT #7:str$(jj);FOR peaknum=l to numpeak

PRINT #7:CHR$(9);Str$(Q(peaknumjj));NEXT peaknum PRINT #7

NEXTjjend if

PRINT #7 PRINT #7:"XQ";FOR peaknum=l to numpeak

PRINT #7:chr$(9);str$(sumQ(peaknum));NEXT peaknum PRINT #7

END SUB

SUB ring_bell FOR RING=1 TO 2

FOR 1=1 TO 15 SOUND 600, .03 SOUND 2000, .02

NEXT I PAUSE 1

NEXT RING END SUB

END SUB

SUB CONFIGOPEN #l:Name "Rundata.txt",create newold,access outin,organization text WHEN error in

INPUT #l:a$USEEND WHEN PRINT A$LINE INPUT prompt "Change path to data?[N]":a0$LET a0$=trim$(a0$)IF a0$="y" or a0$="Y” then

INPUT prompt "New Path:":al2$IF a 12$="" then LET al2$=a$

ELSE LET a!2$=a$

END IFLINE INPUT prompt "Input smoothing factor:[0.3] ":a2$LET a2$=trim$(a2$)IF a2$="" then LET a2$="0.3"LINE INPUT prompt "Plot individual peaks on the screen?[N] ":a7$

LET a7$=trim$(a7$)IF a7$="" then LET a7$="N"Line Input prompt "Print a and b to data file?[N] ":a8 $ let a8 $=trim$(a8$)If a8 $="" then let a8 $="N"LINE INPUT prompt "Ring bell when program is finished?[Y] ":al 1$ LET all$=trim $(all$)IF al 1$="" then LET al 1$="Y" f r a s f #1PRINT #l:Ucase$(al2$)PRINT #l:Ucase$(a2$)PRINT #l:ucase$(a7$)Print #1 :ucase$(a8 $)PRINT #l:ucase$(all$)

END SUB

Menu routines

a True BASIC(tm), Inc. product

ABSTRACT Library of portable menu routines for use in user programs.

Copyright (c) 1985 by True BASIC, Inc.

SUB menu_set(where$,c$,maxent,maxlen,menu$,#9)

! Where$: top, bottom, left, or right ! C$ : color choice, ignored in MAC version ! maxent: maximum items in menu ! Maxlen: maximum length of menu item ! Menu$: for internal use ! #9 window

! Compute parameters

LET w$ = lcase$(where$) ! Screen positionIF w$="left" or w$="right" then LET vert = 1 ! Vertical windowLET m = min(maxent,10) ! Max entriesLET ml = min(maxlen,10) ! Max lengthLET fill = 3 ! Button, spacesLET zone = ml + fill ! ZonewidthLET chars = 70 ! Chars per line

1 8 4

IF vert = 0 then ! Horizontal windowIF m>5 then LET lines=3 else LET lines=l

END IF

! Pack them

LET menu$ = ""CALL packb(menu$, 1,1, large)CALL packb(menu$,2,1,vert)CALL packb(menu$,3,1,extra)CALL packb(menu$,4,1,inverse)CALL packb(menu$,5,4,m)CALL packb(menu$,9,4,ml)CALL packb(menu$,13,4,perline)

! Open window

IF vert = 1 then ! Vertical windowLET lines = 2*m LET yl = (20-lines)/20 LET y2 = 1LET width = max(zone,12)/chars IF where$ = "left" then

LETxl = 0 LET x2 = width

FT SFLET xl = 1 - width LET x2 = 1

END IFOPEN #9: screen xl,x2,yl,y2 SET WINDOW 0,5,20*yl,20

ELSE ! Horizontal windowLETxl = 0 LET x2 = 1IF m<=5 then LET width = .1 else LET width = .2 IF where$ = "top" then

LET yl = .95 - width LET y2 = .95

ELSE LET yl - 0 LET y2 = width

END IFOPEN #9: screen xl,x2,yl,y2 IF m>5 then LET ym=0 else LET ym=1.75 SET WINDOW 0,25,ym,3

END IF

SET COLOR "black/white"CLEAR

END SUB

SUB menu_show(M$(),ml,menu$,#9)

WINDOW #9LET large = unpackb(menu$, 1,1)LET vert = unpackb(menu$,2,l)LET extra = unpackb(menu$,3,l)LET inverse = unpackb(menu$,4,l)LET m = unpackb(menu$,5,4)LET m = min(ml,m)LET ml = unpackb(menu$,9,4)LET perline = unpackb(menu$,13,4)

IF vert=0 then FOR i = 1 to m

LET x$ = using$("############",m$(i))IF i<=5 then

DRAW button(x$) with shift(5*i-4.75,2) ELSE

DRAW button(x$) with shift(5*i-29.75,.25) END IF

NEXT i ELSE

FOR i = 1 te mLET x$ = using$("############",m$(i)) DRAW button(x$) with shift(.25,20.5-2*i)

NEXT i END IF

END SUB

PICTURE button(s$)PLOT .25,1;4.25,1;4.5,.75;4.5,.25;PLOT 4.25,0;.25,0;0,.25;0,.75;.25,1

PLOT TEXT, AT .25,.25: s$END PICTURE

SUB menu_ask(ml,z,menu$,#9)

LET ml = unpackb(menu$,5,4)LET m = min(ml,ml)LET vert = unpackb(menu$,2,l)WINDOW #9

DO ! Force right answerDO

GET MOUSE x,y,s LOOP until s=2 IF vert=0 then

LET v = int(x/5)+l IF y>=1.75 and y<=3 then

LET z = v ELSEEF y<1.75 then

LET z = v+5 ELSE

LET z = 0 END IF

ELSE LET v = int(19.5 - y)IF mod(v,2)=l then

LET z=0 ELSE

LET z = v/2 + 1 END IF

END IFIF z<l or z>m then LET z = 0

LOOP until z>0 CALL blink(z,vert)

SUB blink(i,v)IF v=0 then

IF i<=5 then L E T xl =5*i-4.5 LET x2 = 5*i-.5 LET yl =2.1 LET y2 = 2.9

ELSE L E T xl =5*i-29.5 LET x2 = 5*i-25.5 L E T yl = .35 LET y2= 1.15

END IF ELSE

LE T xl = .5 LET x2 = 4.5 LET yl = 20.5 - 2*i + .1 LET y2 = yl + .8

END IFBOX KEEP xl,x2,yl,y2 in k$BOX CLEAR xl,x2,yl,y2P A T T C F y

BOX SHOW k$ at x l,y l END SUB

END SUB

SUB menu(M$(),m,a,menu$,#9) ! Show menu, get answer

CALL menu_show(M$,m,menu$,#9)CALL menu_ask(m,a,menu$,#9)CLEAR

END SUB

SUB menu_all(M$(),m,prompt$,ans,menu$,#l ,#9)

1 8 7

WINDOW #1 PRINT prompt$;M?CALL menu(M$,m,ans,menu$,#9) WINDOW #1 PRINT MS(ans)

END SUB

Appendix B

TABLES FOR INTERCONVERSION BETW EEN VARIOUS

CHROM ATOGRAPHIC SEPARATION MEASURES

1 8 8

1 8 9

The following tables are a compilation of various separation measures employed in

chromatography. They were developed by systematic variation of resolution between

two overlapped EMG peaks. As the true resolution (RS)True> Equation 1), was varied

from 0.5 to 1.5 in increments of 0.1, the height ratio between the two peaks was varied

from 1 to 8 in increments of 1. Also, x/a for the two peaks was varied from 0 to 3 in

increments of 1, both peaks having the same x/a. Thus, many possible combinations of

overlapping peaks were considered.

R s= AtG [1 ]4 V variance

The discrimination factor for each peak was defined as

h n j ■ h y j

[ 2 ]

where hp is the height of the peak, hv is the valley height, and the subscript i refers to

peak in question.

The measured resolution, Rs m, based on the Gaussian peak model, was calculated

via Equations 3 and 4 for width (W) measured at 50% or 75% of the peak height,

respectively, where the subscripts 1 and 2 refer to the first or second peak in the pair.

( lR .2 - tR ,l)Rs,m,50% - 1-16 (W lj50% + W2>50%) 13 3

(*R,2 ~ tR .l) RS,m,7 5 % = 0.759 (W l j 0 7 5 + W2 0 .75) [ 4 ]

1 9 0

The percent relative area overlap (%RO) is defined as

%ROj = a [ X 100% [ 5 ]

where i refers to the peak in question, Ajj refers to the common area of the two

overlapped peaks, and Aj refers to the total area of the ith peak, before overlap.

The overlap integral (£2, ref. 1) was defined as

O O

( JcAcB dt)2W — ^ -------= -------- [6]

|C A2 d t JC B 2 dt

where Qa and Cb are their signal heights at time L

The parameters b, a, and width (W), d0 >j, do 2 , and tR were measured using a

search algorithm (See Figure 2-1), while %RQ and £2 were measured from digitized

overlapping peaks with a sampling rate of 50 points/GQ.

To use the tables one must first measure dG j , dQ 2 and the parameters a and b

(for both peaks), at 25% (Table B-l), 50% (Table B-2), or 75% (Table B-3) of the peak

height The values of a and b are measured with respect to each peak individually and

therefore, in some situations one may be able to obtain b and a for one peak at the desired

peak height fraction, but not for the other peak. Width (W) for each peak is is the sum of

a and b for the respective peak. Secondly, by matching as close as possible the measured values of ^ ( ^ ) 2 , do l and do 2 , one can arrive at an

approximate value for %RO or Rs,true- Alternatively, one can arrive at Rs,true by using

Rs,m . b, and a from both peaks. The tables are arranged in acending order of the values

of dQ j. Therefore, one should match the values of dGj and do2 first

Table B -l. W, a and b measured at the 25% peak height fraction.

do.l do ,2 ^s.true %ROi %R0 2 n

0.751 0.878 0 .8 8.948 4.474 3.395E-03

0.752 0.876 1.1 3.895 1.947 6.252E-05

0.759 0.883 0.7 10.792 5.396 6.724E-03

0.770 0.986 1.3 3.321 0.208 1.344E-06

0.770 0.885 1 .0 5.475 2.738 4.910E-04

0.771 0.986 0.9 8.438 0.527 1.388E-03

0.778 0.791 0.7 9.853 9.853 6.724E-03

0.779 0.787 0 .8 7.898 7.898 3.395E-03

0.779 0.972 1.1 5.210 0.651 1.589E-04

0.787 0.947 1 .2 3.173 0.793 9.930E-06

0.791 0.974 0.9 7.648 0.956 1.388E-03

0.798 0.987 0 .8 9.309 0.582 2.893E-03

0.811 0.976 0 .8 8.685 1.086 2.893E-03

0.811 0.953 0.9 6.892 1.723 1.388E-03

0.812 0.814 1 .0 4.480 4.480 4.910E-04

*>1 alW i W j

0.6471 0.3529

0.5047 0.4953

0.6932 0.3068

0.5000 0.5000

0.5673 0.4327

0.6340 0.3660

0.6899 0.3101

0.6349 0.3651

0.5618 0.4382

0.5007 0.4993

0.6317 0.3683

0.6880 0.3120

0.6879 0.3121

0.6308 0.3692

0.5588 0.4412

b2 a 2w2 w2

0.6145 0.3855

0.4634 0.5366

0.6722 0.3278

0.4825 0.5175

0.5436 0.4564

0.6289 0.3711

0.6465 0.3535

0.5888 0.4112

0.5522 0.4478

0.4851 0.5149

0.6278 0.3722

0 .6 8 6 6 0.3134

0.6855 0.3145

0.6256 0.3744

0.5299 0.4701 191

Table B-l. Cont’d

d0,l do ,2 Rs,true %ROi %R0 2 Q

0.815 0.954 1 .1 4.371 1.093 1.589E-04

0.822 0.822 1 .1 2.781 2.781 6.252E-05

0.822 0.822 1 .1 2.781 2.781 6.252E-05

0.823 0.956 0 .8 8.067 2.017 2.893E-03

0.825 0.978 1.3 2.470 0.309 1.344E-06

0.829 0.916 0.9 6.171 3.085 1.388E-03

0.830 0.989 1 .2 4.005 0.250 5.132E-05

0.835 0.919 0 .8 7.452 3.726 2.893E-03

0.843 0.922 1 .2 2.299 1.150 9.930E-06

0.845 0.990 1 .0 5.734 0.358 5.675E-04

0.846 0.923 1 .1 3.639 1.819 1.589E-04

0.847 0.853 0 .8 6.838 6.838 2.893E-03

0.847 0.851 0.9 5.480 5.480 1.388E-03

0.856 0.982 1 .2 3.400 0.425 5.132E-05

0.858 0.982 1 .0 5.213 0.652 5.675E-04

b l alWi Wj

0.5570 0.4430

0.5105 0.4895

0.5105 0.4895

0.6878 0.3122

0.5001 0.4999

0.6303 0.3697

0.5551 0.4449

0.6878 0.3122

0.5014 0.4986

0.6300 0.3700

0.5551 0.4449

0.6877 0.3123

0.6301 0.3699

0.5542 0.4458

0.6300 0.3700

^ 2 a 2w2 w2

0.5509 0.4491

0.4895 0.5105

0.4895 0.5105

0.6832 0.3168

0.4934 0.5066

0.6207 0.3793

0.5531 0.4469

0.6782 0.3218

0.4939 0.5061

0.6292 0.3708

0.5482 0.4518

0.6659 0.3341

0.6090 0.3910

0.5527 0.4473

0.6286 0.3714

19

2

Table B-l. Cont'd

do,l do ,2 &s,true %ROi %R02 Q

0.862 0.991 1.4 1.845 0.115 1.550E-07

0.863 0.991 0.9 6.357 0.397 1.245E-03

0.869 0.967 1.3 1.811 0.453 1.344E-06

0.871 0.984 0.9 5.945 0.743 1.245E-03

0.871 0.968 1 .0 4.716 1.179 5.675E-04

0.873 0.874 1.1 3.001 3.001 1.589E-04

0.876 0.752 1 .1 1.947 3.895 6.252E-05

0.878 0.970 1 .2 2 .8 6 8 0.717 5.132E-05

0.879 0.970 0.9 5.538 1.384 1.245E-03

0.883 0.942 1 .0 4.241 2 .1 2 0 5.675E-04

0.887 0.944 0.9 5.133 2.567 1.245E-03

0 .8 8 8 0 .8 8 8 1 .2 1.639 1.639 9.930E-06

0 .8 8 8 0 .8 8 8 1 .2 1.639 1.639 9.930E-06

0.890 0.993 1.3 2.589 0.162 1.656E-05

0.895 0.898 0.9 4.731 4.731 1.245E-03

*>1 a lW j Wi

0.5000 0.5000

0.6877 0.3123

0.5002 0.4998

0.6877 0.3123

0.6299 0.3701

0.5542 0.4458

0.5047 0.4953

0.5538 0.4462

0.6877 0.3123

0.6299 0.3701

0.6877 0.3123

0.5028 0.4972

0.5028 0.4972

0.5536 0.4464

0.6877 0.3123

t>2 a 2w2 w2

0.4973 0.5027

0.6870 0.3130

0.4970 0.5030

0.6863 0.3137

0.6272 0.3728

0.5420 0.4580

0.4634 0.5366

0.5520 0.4480

0.6848 0.3152

0.6243 0.3757

0.6817 0.3183

0.4972 0.5028

0.4972 0.5028

0.5532 0.4468

0.6747 0.3253

19

3

Table B-l. Cont’d

do,l do,2 Rs,true %ROi %R02 Q

0.895 0.897 1.0 3.787 3.787 5.675E-04

0.895 0.993 1.1 3.889 0.243 2.320E-04

0.896 0.987 1.4 1.365 0.171 1.550E-07

0.897 0.797 1.1 2.449 4.899 1.589E-04

0.898 0.949 1.2 2.403 1.201 5.132E-05

0.903 0.816 0.9 4.328 8.655 1.245E-03

0.904 0.988 1.1 3.547 0.443 2.320E-04

0.905 0.952 1.3 1.309 0.655 1.344E-06

0.906 0.988 1.3 2.209 0.276 1.656E-05

0.906 0.819 1.0 3.352 6.704 5.675E-04

0.907 0.994 1.0 4.336 0.271 5.357E-04

0.912 0.989 1.0 4.064 0.508 5.357E-04

0.912 0.978 1.1 3.219 0.805 2.320E-04

0.915 0.916 1.2 1.996 1.996 5.132E-05

0.917 0.979 1.0 3.795 0.949 5.357E-04

*>1 a lW! W j

0.6299 0.3701

0.6299 0.3701

0.5000 0.5000

0.5538 0.4462

0.5536 0.4464

0.6877 0.3123

0.6299 0.3701

0.5004 0.4996

0.5535 0.4465

0.6299 0.3701

0.6877 0.3123

0.6877 0.3123

0.6299 0.3701

0.5535 0.4465

0.6877 0.3123

*>2 a2W2 w 2

0.6178 0.3822

0.6295 0.3705

0.4987 0.5013

0.5247 0.4753

0.5505 0.4495

0.6557 0.3443

0.6291 0.3709

0.4985 0.5015

0.5530 0.4470

0.6005 0.3995

0.6872 0.3128

0.6868 0.3132

0.6282 0.3718

0.5474 0.4526

0.6858 0.3142

19

4

Table B-l. Cont'd

d0>l do ,2 Rs,trae

0.920 0.960 1 .1

0.920 0.980 1.3

0.921 0.995 1.5

0.922 0.843 1 .2

0.923 0.962 1 .0

0.923 0.981 1.4

0.928 0.929 1 .0

0.928 0.929 1 .1

0.929 0.996 1.4

0.929 0.996 1 .2

0.931 0.862 1 .2

0.932 0.932 1.3

0.932 0.932 1.3

0.933 0.967 1.3

0.933 0.871 1 .0

%ROi %R0 2 Q

2.906 1.453 2.320E-04

1.873 0.468 1.656E-05

0.986 0.062 1.523E-08

1.150 2.299 9.930E-06

3.529 1.764 5.357E-04

0.996 0.249 1.550E-07

3.264 3.264 5.357E-04

2.607 2.607 2.320E-04

1 .6 6 8 0.104 5.342E-06

2.634 0.165 9.486E-05

1.643 3.287 5.132E-05

0.932 0.932 1.344E-06

0.932 0.932 1.344E-06

1.578 0.789 1.656E-05

3.000 6 .0 0 0 5.357E-04

*>1 alW j Wj

0.6299 0.3701

0.5534 0.4466

0.5000 0.5000

0.5014 0.4986

0.6877 0.3123

0.5000 0.5000

0.6877 0.3123

0.6299 0.3701

0.5534 0.4466

0.6299 0.3701

0.5535 0.4465

0.5007 0.4993

0.5007 0.4993

0.5534 0.4466

0.6877 0.3123

b2 a 2w2 w2

0.6264 0.3736

0.5526 0.4474

0.4995 0.5005

0.4939 0.5061

0.6838 0.3162

0.4994 0.5006

0.6796 0.3204

0.6225 0.3775

0.5533 0.4467

0.6296 0.3704

0.5402 0.4598

0.4993 0.5007

0.4993 0.5007

0.5518 0.4482

0.6697 0.3303 19

5

Table B-l. Cont'd

do.l do ,2 Rs,true %ROi %R02 n

0.935 0.992 1 .2 2.408 0.301 9.486E-05

0.935 0.874 1.1 2.321 4.642 2.320E-04

0.937 0.996 1 .1 2.954 0.185 2.305E-04

0.938 0.770 1 .0 2.735 10.938 5.357E-04

0.939 0.992 1.4 1.429 0.179 5.342E-06

0.940 0.993 1.1 2.774 0.347 2.305E-04

0.940 0.985 1 .2 2.192 0.548 9.486E-05

0.941 0.993 1.5 0.726 0.091 1.523E-08

0.943 0.780 1.1 2.047 8.189 2.320E-04

0.944 0.986 1.1 2.596 0.649 2.305E-04

0.944 0.778 1 .2 1.339 5.355 5.132E-05

0.944 0.944 1.3 1.320 1.320 1.656E-05

0.944 0.972 1.4 0.718 0.359 1.550E-07

0.946 0.973 1 .2 1.986 0.993 9.486E-05

0.947 0.787 1 .2 0.793 3.173 9.930E-06

*>1 alW i Wj

0.6299 0.3701

0.6299 0.3701

0.6877 0.3123

0.6877 0.3123

0.5534 0.4466

0.6877 0.3123

0.6299 0.3701

0.5000 0.5000

0.6299 0.3701

0.6877 0.3123

0.5534 0.4466

0.5534 0.4466

0.5001 0.4999

0.6299 0.3701

0.5007 0.4993

*>2 a 2w2 w2

0.6294 0.3706

0.6137 0.3863

0.6874 0.3126

0.6344 0.3656

0.5532 0.4468

0.6871 0.3129

0.6288 0.3712

0.4998 0.5002

0.5850 0.4150

0.6865 0.3135

0.5175 0.4825

0.5501 0.4499

0.4997 0.5003

0.6277 0.3723

0.4851 0.5149 19

6

Table B-l. Cont’d

d0,l ^o ,2 fts.true %ROi %R0 2 Q

0.947 0.974 1.1 2.421 1 .2 1 0 2.305E-04

0.948 0.987 1.4 1.218 0.304 5.342E-06

0.951 0.951 1 .1 2.247 2.247 2.305E-04

0.951 0.951 1 .2 1.789 1.789 9.486E-05

0.952 0.997 1.3 1.781 0 .1 1 1 3.878E-05

0.952 0.905 1.3 0.655 1.309 1.344E-06

0.954 0.908 1.3 1.095 2.189 1.656E-05

0.954 0.910 1 .1 2.073 4.147 2.305E-04

0.955 0.997 1.5 1.071 0.067 1.723E-06

0.956 0.913 1 .2 1.601 3.202 9.486E-05

0.956 0.995 1.3 1.632 0.204 3.878E-05

0.956 0.978 1.4 1.031 0.516 5.342E-06

0.957 0.989 1.5 0.528 0.132 1.523E-08

0.957 0.997 1 .2 2 .0 1 1 0.126 9.920E-05

0.957 0.837 1.1 1.900 7.600 2.305E-04

bl alWi W i

0.6877 0.3123

0.5534 0.4466

0.6877 0.3123

0.6299 0.3701

0.6299 0.3701

0.5004 0.4996

0.5534 0.4466

0.6877 0.3123

0.5534 0.4466

0.6299 0.3701

0.6299 0.3701

0.5534 0.4466

0.5000 0.5000

0.6877 0.3123

0.6877 0.3123

*>2 a 2w2 w2

0.6852 0.3148

0.5530 0.4470

0.6826 0.3174

0.6253 0.3747

0.6297 0.3703

0.4985 0.5015

0.5465 0.4535

0.6768 0.3232

0.5534 0.4466

0.6203 0.3797

0.6296 0.3704

0.5525 0.4475

0.4999 0.5001

0.6875 0.3125

0.6620 0.3380 19

7

Table B-l. Cont'd

do.l ^o,2 Rs.true %RO\ %R0 2 Q

0.959 0.995 1.2 1.892 0.236 9.920E-05

0.960 0.990 1.3 1.490 0.373 3.878E-05

0.960 0.960 1.4 0.511 0.511 1.550E-07

0.960 0.960 1.4 0.511 0.511 1.550E-07

0.960 0.846 1.2 1.421 5.685 9.486E-05

0.961 0.995 1.5 0.921 0.115 1.723E-06

0.962 0.990 1.2 1.774 0.444 9.920E-05

0.962 0.849 1.3 0.900 3.598 1.656E-05

0.963 0.963 1.4 0 .8 6 8 0 .8 6 8 5.342E-06

0.963 0.982 1.3 1.354 0.677 3.878E-05

0.964 0.982 1.2 1.658 0.829 9.920E-05

0.966 0.966 1.2 1.543 1.543 9.920E-05

0.966 0.992 1.5 0.788 0.197 1.723E-06

0.967 0.967 1.3 1.225 1.225 3.878E-05

0.967 0.869 1.3 0.453 1.811 1.344E-06

b l alWi Wi

0.6877 0.3123

0.6299 0.3701

0.5002 0.4998

0.5002 0.4998

0.6299 0.3701

0.5534 0.4466

0.6877 0.3123

0.5534 0.4466

0.5534 0.4466

0.6299 0.3701

0.6877 0.3123

0.6877 0.3123

0.5534 0.4466

0.6299 0.3701

0.5002 0.4998

*>2 a 2w 2 w 2

0.6873 0.3127

0.6292 0.3708

0.4998 0.5002

0.4998 0.5002

0.6077 0.3923

0.5533 0.4467

0.6869 0.3131

0.5379 0.4621

0.5516 0.4484

0.6285 0.3715

0.6861 0.3139

0.6844 0.3156

0.5532 0.4468

0.6270 0.3730

0.4970 0.5030 19

8

Table B-l. Cont'd

d0 ,l do ,2 Rs,true %ROi %R0 2 Q

0.968 0.998 1.4 1.202 0.075 1.586E-05

0.968 0.938 1.2 1.429 2.858 9.920E-05

0.969 0.984 1.5 0.380 0.190 1.523E-08

0.969 0.939 1.4 0.725 1.449 5.342E-06

0.969 0.759 1.3 0.731 5.849 1.656E-05

0.970 0.940 1.3 1.101 2.202 3.878E-05

0.970 0.996 1.4 1.105 0.138 1.586E-05

0.971 0.886 1.2 1.315 5.262 9.920E-05

0.971 0.998 1.3 1.367 0.085 4.268E-05

0.971 0.986 1.5 0.671 0.336 1.723E-06

0.972 0.944 1.4 0.359 0.718 1.550E-07

0.972 0.997 1.3 1.288 0.161 4.268E-05

0.973 0.993 1.4 1.011 0.253 1.586E-05

0.973 0.893 1.3 0.982 3.930 3.878E-05

0.973 0.796 1.2 1.202 9.612 9.920E-05

*>1 alW i Wj

0.6299 0.3701

0.6877 0.3123

0.5000 0.5000

0.5534 0.4466

0.5534 0.4466

0.6299 0.3701

0.6299 0.3701

0.6877 0.3123

0.6877 0.3123

0.5534 0.4466

0.5001 0.4999

0.6877 0.3123

0.6299 0.3701

0.6299 0.3701

0.6877 0.3123

t>2 a2w 2 w 2

0.6298 0.3702

0.6808 0.3192

0.4999 0.5001

0.5496 0.4504

0.5045 0.4955

0.6240 0.3760

0.6297 0.3703

0.6727 0.3273

0.6876 0.3124

0.5529 0.4471

0.4997 0.5003

0.6875 0.3125

0.6295 0.3705

0.6171 0.3829

0.6486 0.3514 19

9

Table B-l. Cont'd

do,l d0 ,2 Rs.true %ROi %R0 2 Q

0.974 0.994 1.3 1 .2 1 0 0.303 4.268E-05

0.975 0.899 1.4 0.600 2.401 5.342E-06

0.975 0.988 1.4 0.922 0.461 1.586E-05

0.975 0.988 1.3 1.134 0.567 4.268E-05

0.976 0.812 1.3 0.869 6.953 3.878E-05

0.976 0.976 1.5 0.568 0.568 1.723E-06

0.977 0.977 1.3 1.058 1.058 4.268E-05

0.977 0.977 1.4 0.836 0.836 1.586E-05

0.978 0.978 1.5 0.270 0.270 1.523E-08

0.978 0.978 1.5 0.270 0.270 1.523E-08

0.978 0.825 1.3 0.309 2.470 1.344E-06

0.978 0.999 1.5 0.811 0.051 6.483E-06

0.978 0.957 1.3 0.983 1.966 4.268E-05

0.979 0.836 1.4 0.492 3.938 5.342E-06

0.979 0.959 1.4 0.755 1.509 1.586E-05

*>1 alW i W j

0.6877 0.3123

0.5534 0.4466

0.6299 0.3701

0.6877 0.3123

0.6299 0.3701

0.5534 0.4466

0.6877 0.3123

0.6299 0.3701

0.5000 0.5000

0.5000 0.5000

0.5001 0.4999

0.6299 0.3701

0.6877 0.3123

0.5534 0.4466

0.6299 0.3701

^ 2 a 2w2 w2

0.6872 0.3128

0.5454 0.4546

0.6290 0.3710

0.6867 0.3133

0.5984 0.4016

0.5524 0.4476

0.6856 0.3144

0.6281 0.3719

0.5000 0.5000

0.5000 0.5000

0.4934 0.5066

0.6299 0.3701

0.6833 0.3167

0.5351 0.4649

0.6262 0.3738 200

Table B-l. Cont'd

<*0,1 do,2 Rs.true %ROj %R02 n

0.980 0.960 1.5 0.477 0.954 1.723E-06

0.980 0.921 1.3 0.908 3.633 4.268E-05

0.980 0.998 1.5 0.746 0.093 6.483E-06

0.980 0.999 1.4 0.929 0.058 1.837E-05

0.981 0.923 1.4 0.249 0.996 1.550E-07

0.981 0.998 1.4 0.876 0.110 1.837E-05

0.981 0.926 1.4 0.677 2.707 1.586E-05

0.981 0.857 1.3 0.834 6.669 4.268E-05

0.982 0.995 1.5 0.685 0.171 6.483E-06

0.982 0.996 1.4 0.825 0.206 1.837E-05

0.983 0.992 1.5 0.626 0.313 6.483E-06

0.983 0.933 1.5 0.398 1.592 1.723E-06

0.983 0.869 1.4 0.602 4.817 1.586E-05

0.983 0.992 1.4 0.774 0.387 1.837E-05

0.984 0.984 1.4 0.724 0.724 1.837E-05

*>1 alWi Wj

0.5534 0.4466

0.6877 0.3123

0.6299 0.3701

0.6877 0.3123

0.5000 0.5000

0.6877 0.3123

0.6299 0.3701

0.6877 0.3123

0.6299 0.3701

0.6877 0.3123

0.6299 0.3701

0.5534 0.4466

0.6299 0.3701

0.6877 0.3123

0.6877 0.3123

t>2 a2w2 w2

0.5513 0.4487

0.6785 0.3215

0.6298 0.3702

0.6876 0.3124

0.4994 0.5006

0.6876 0.3124

0.6222 0.3778

0.6667 0.3333

0.6296 0.3704

0.6874 0.3126

0.6293 0.3707

0.5491 0.4509

0.6127 0.3873

0.6870 0.3130

0.6863 0.3137 201

Table B-l. Cont'd

d0,l do,2 Rs,true

0.984 0.969 1.5

0.985 0.985 1.5

0.985 0.772 1.4

0.985 0.971 1.4

0.986 0.770 1.3

0.986 0.972 1.5

0.986 0.889 1.5

0.986 0.946 1.4

0.987 0.999 1.5

0.987 0.896 1.4

0.987 0.900 1.4

0.987 0.949 1.5

0.987 0.998 1.5

0.988 0.997 1.5

0.988 0.820 1.4

%ROi %R02 Q

0.190 0.380 1.523E-08

0.570 0.570 6.483E-06

0.531 8.491 1.586E-05

0.675 1.349 1.837E-05

0.208 3.321 1.344E-06

0.516 1.032 6.483E-06

0.329 2.631 1.723E-06

0.626 2.502 1.837E-05

0.630 0.039 7.903E-06

0.171 1.365 1.550E-07

0.577 4.613 1.837E-05

0.465 1.858 6.483E-06

0.596 0.075 7.903E-06

0.562 0.140 7.903E-06

0.528 8.443 1.837E-05

*>1 alW i

0.5000 0.5000

0.6299 0.3701

0.6299 0.3701

0.6877 0.3123

0.5000 0.5000

0.6299 0.3701

0.5534 0.4466

0.6877 0.3123

0.6877 0.3123

0.5000 0.5000

0.6877 0.3123

0.6299 0.3701

0.6877 0.3123

0.6877 0.3123

0.6877 0.3123

*>2 a2w2 w2

0.4999 0.5001

0.6288 0.3712

0.5804 0.4196

0.6849 0.3151

0.4825 0.5175

0.6276 0.3724

0.5442 0.4558

0.6819 0.3181

0.6877 0.3123

0.4987 0.5013

0.6751 0.3249

0.6251 0.3749

0.6876 0.3124

0.6875 0.3125

0.6571 0.3429 202

Table B-l. Cont'd

d0,l <*0 ,2 ^s,true %ROi %R0 2 Q

0.989 0.909 1.5 0.416 3.324 6.483E-06

0.989 0.994 1.5 0.528 0.264 7.903E-06

0.989 0.820 1.5 0.269 4.308 1.723E-06

0.989 0.957 1.5 0.132 0.528 1.523E-08

0.989 0.989 1.5 0.495 0.495 7.903E-06

0.990 0.840 1.5 0.369 5.897 6.483E-06

0.990 0.980 1.5 0.462 0.925 7.903E-06

0.991 0.963 1.5 0.430 1.720 7.903E-06

0.991 0.931 1.5 0.398 3.182 7.903E-06

0.991 0.862 1.4 0.115 1.845 1.550E-07

0.992 0.874 1.5 0.366 5.851 7.903E-06

0.993 0.941 1.5 0.091 0.726 1.523E-08

0.995 0.921 1.5 0.062 0.986 1.523E-08

bi alW i Wi

0.6299 0.3701

0.6877 0.3123

0.5534 0.4466

0.5000 0.5000

0.6877 0.3123

0.6299 0.3701

0.6877 0.3123

0.6877 0.3123

0.6877 0.3123

0.5000 0.5000

0.6877 0.3123

0.5000 0.5000

0.5000 0.5000

b2 a 2w2 w2

0.6198 0.3802

0.6873 0.3127

0.5317 0.4683

0.4999 0.5001

0 .6 8 6 8 0.3132

0.6063 0.3937

0.6859 0.3141

0.6840 0.3160

0.6799 0.3201

0.4973 0.5027

0.6703 0.3297

0.4998 0.5002

0.4995 0.5005

20

3

Table B-2. W, a and b measured at the 50% peak height fraction.*>1Wj

alW i

*>2w2

a 2w2do,l do,2 Rs,trae Rs.meas %ROi %R0 2 Q

0.501 0.875 1 .0 0.921 8.703 2.176 3.355E-04 0.5016 0.4984 0.4425 0.5575

0.504 0.969 0.7 0.982 18.161 1.135 8.304E-03 0.6077 0.3923 0.5835 0.4165

0.508 0.939 0.9 0.981 1 2 .0 1 2 1.502 1.507E-03 0.5672 0.4328 0.5327 0.4673

0.508 0.757 0 .8 0.849 1 2 .1 0 1 6.051 4.550E-03 0.5752 0.4248 0.5215 0.4785

0.536 0.598 0.5 0.788 20.203 20.203 3.632E-02 0.6381 0.3619 0.5806 0.4194

0.543 0.585 0 .6 0.794 16.151 16.151 2.030E-02 0.5988 0.4012 0.5365 0.4635

0.550 0.944 0.7 1 .0 0 0 16.338 2.042 8.304E-03 0.5919 0.4081 0.5822 0.4178

0.564 0.945 1 .1 1.059 7.215 0.902 6.252E-05 0.5003 0.4997 0.4788 0.5212

0.566 0.973 0 .6 1.019 19.870 1.242 1.563E-02 0.6313 0.3687 0.6281 0.3719

0.587 0.897 0.9 1.004 9.965 2.491 1.507E-03 0.5448 0.4552 0.5310 0.4690

0.594 0.950 0 .6 1.018 18.437 2.305 1.563E-02 0.6303 0.3697 0.6267 0.3733

0.595 0.900 0.7 1 .0 0 0 14.595 3.649 8.304E-03 0.5879 0.4121 0.5795 0.4205

0.604 0.615 0 .8 0.854 9.721 9.721 4.550E-03 0.5471 0.4529 0.5029 0.4971

0.605 0.605 0.9 0.841 7.186 7.186 1.534E-03 0.5227 0.4773 0.4773 0.5227

0.605 0.605 0.9 0.841 7.186 7.186 1.534E-03 0.5227 0.4773 0.4773 0.5227 20

4

Table B -2. Cont’d

d0,l do,2 Rs.true

0.607 0.976 1 .0

0.622 0.908 0 .6

0.625 0.812 1 .0

0.633 0.977 1 .2

0.639 0.825 0.7

0.650 0.833 0 .6

0.660 0.831 0.9

0.662 0.979 0 .8

0 .6 6 6 0.917 1 .1

0.667 0.958 1 .0

0.678 0.707 0 .6

0.681 0.700 0.7

0.693 0.962 0 .8

0.704 0.982 0.7

0.707 0.513 0 .6

Rs.meas %RO] %R02

1.127 9.442 0.590

1.015 17.011 4.253

0.965 6.363 3.182

1.171 5.743 0.359

0.993 12.927 6.463

1.006 15.583 7.791

1.005 8.179 4.089

1.155 12.393 0.775

1.073 5.353 1.338

1.130 7.941 0.993

0.986 14.142 14.142

0.974 11.327 11.327

1.154 11.192 1.399

1.193 13.611 0.851

0.916 12.670 25.340

n blWi

alWi

b2w2

a2w2

4.910E-04 0.5403 0.4597 0.5338 0.4662

1.563E-02 0.6299 0.3701 0.6238 0.3762

3.355E-04 0.5032 0.4968 0.4847 0.5153

9.930E-06 0.5000 0.5000 0.4908 0.5092

8.304E-03 0.5863 0.4137 0.5737 0.4263

1.563E-02 0.6297 0.3703 0.6175 0.3825

1.507E-03 0.5390 0.4610 0.5275 0.4725

3.395E-03 0.5851 0.4149 0.5839 0.4161

6.252E-05 0.5005 0.4995 0.4913 0.5087

4.910E-04 0.5371 0.4629 0.5334 0.4666

1.563E-02 0.6296 0.3704 0.6026 0.3974

8.304E-03 0.5855 0.4145 0.5596 0.4404

3.395E-03 0.5849 0.4151 0.5831 0.4169

6.724E-03 0.6295 0.3705 0.6286 0.3714

1.563E-02 0.6295 0.3705 0.5404 0.4596 20

5

Table B -2. Cont'd

<*o,l do, 2 Rs,true

0.717 0.965 1 .2

0.721 0.931 1 .0

0.722 0.508 0.7

0.722 0.966 0.7

0.723 0.931 0 .8

0.725 0.729 0.9

0.729 0.729 1 .0

0.729 0.729 1 .0

0.740 0.984 1 .1

0.741 0.936 0.7

0.751 0.878 0 .8

0.752 0.876 1 .1

0.759 0.883 0.7

0.770 0.986 1.3

0.770 0.885 1 .0

Rs.meas %ROi %R02

1.177 4.304 0.538

1.130 6.624 1.656

0.899 9.786 19.572

1.191 12.665 1.583

1.151 10.046 2.511

0.996 6.632 6.632

0.970 4.550 4.550

0.970 4.550 4.550

1.247 6.167 0.385

1.188 11.727 2.932

1.145 8.948 4.474

1.078 3.895 1.947

1.182 10.792 5.396

1.278 3.321 0.208

1.128 5.475 2.738

Q

9.930E-06

4.910E-04

8.304E-03

6.724E-03

3.395E-03

1.507E-03

3.355E-04

3.355E-04

1.589E-04

6.724E-03

3.395E-03

6.252E-05

6.724E-03

1.344E-06

4.910E-04

bl alWj W i

0.5001 0.4999

0.5356 0.4644

0.5851 0.4149

0.6295 0.3705

0.5848 0.4152

0.5365 0.4635

0.5068 0.4932

0.5068 0.4932

0.5349 0.4651

0.6295 0.3705

0.5848 0.4152

0.5010 0.4990

0.6295 0.3705

0.5000 0.5000

0.5349 0.4651

*>2 a 2w 2 w 2

0.4958 0.5042

0.5325 0.4675

0.4970 0.5030

0.6277 0.3723

0.5815 0.4185

0.5194 0.4806

0.4932 0.5068

0.4932 0.5068

0.5340 0.4660

0.6258 0.3742

0.5779 0.4221

0.4960 0.5040

0.6219 0.3781

0.4982 0.5018

0.5306 0.4694 206

Table B-2. Cont'd

do ,l do ,2 Rs.true Rs.meas %ROi %R02

0.771 0.986 0.9 1.301 8.438 0.527

0.778 0.791 0.7 1.168 9.853 9.853

0.779 0.787 0 .8 1.132 7.898 7.898

0.779 0.972 1 .1 1.247 5.210 0.651

0.782 0.578 0.9 0.965 5.303 10.606

0.787 0.947 1 .2 1.180 3.173 0.793

0.791 0.974 0.9 1.300 7.648 0.956

0.796 0.640 0.7 1.135 8.902 17.805

0.798 0.987 0 .8 1.364 9.309 0.582

0.805 0.640 0 .8 1 .1 0 0 6.889 13.778

0.811 0.976 0 .8 1.363 8.685 1.086

0.811 0.953 0.9 1.298 6.892 1.723

0.812 0.814 1 .0 1 .1 2 1 4.480 4.480

0.812 0.625 1 .0 0.965 3.182 6.363

0.815 0.954 1.1 1.246 4.371 1.093

a blWi

alWi

b2w2

a2w2

1.388E-03 0.5847 0.4153 0.5842 0.4158

6.724E-03 0.6295 0.3705 0.6133 0.3867

3.395E-03 0.5848 0.4152 0.5701 0.4299

1.589E-04 0.5346 0.4654 0.5338 0.4662

1.507E-03 0.5354 0.4646 0.4955 0.5045

9.930E-06 0.5001 0.4999 0.4980 0.5020

1.388E-03 0.5847 0.4153 0.5837 0.4163

6.724E-03 0.6295 0.3705 0.5907 0.4093

2.893E-03 0.6295 0.3705 0.6289 0.3711

3.395E-03 0.5848 0.4152 0.5496 0.4504

2.893E-03 0.6295 0.3705 0.6283 0.3717

1.388E-03 0.5847 0.4153 0.5826 0.4174

4.910E-04 0.5346 0.4654 0.5265 0.4735

3.355E-04 0.5032 0.4968 0.4847 0.5153

1.589E-04 0.5344 0.4656 0.5333 0.4667 20

7

Table B -2. Cont'd

do,l do,2 Rs,true

0.822 0.822 1 .1

0.822 0.822 1 .1

0.823 0.956 0 .8

0.825 0.978 1.3

0.829 0.916 0.9

0.830 0.989 1 .2

0.835 0.919 0 .8

0.843 0.922 1 .2

0.845 0.990 1 .0

0.846 0.923 1 .1

0.847 0.853 0 .8

0.847 0.851 0.9

0.849 0.704 1 .0

0.856 0.982 1 .2

0.858 0.982 1 .0

Rs.meas %ROi %R02

1.079 2.781 2.781

1.079 2.781 2.781

1.360 8.067 2.017

1.280 2.470 0.309

1.294 6.171 3.085

1.362 4.005 0.250

1.356 7.452 3.726

1.181 2.299 1.150

1.446 5.734 0.358

1.244 3.639 1.819

1.346 6.838 6.838

1.285 5.480 5.480

1.106 3.621 7.242

1.361 3.400 0.425

1.445 5.213 0.652

n

6.252E-05

6.252E-05

2.893E-03

1.344E-06

1.388E-03

5.132E-05

2.893E-03

9.930E-06

5.675E-04

1.589E-04

2.893E-03

1.388E-03

4.910E-04

5.132E-05

5.675E-04

bl alW! W i

0.5019 0.4981

0.5019 0.4981

0.6295 0.3705

0.5000 0.5000

0.5847 0.4153

0.5343 0.4657

0.6295 0.3705

0.5003 0.4997

0.5847 0.4153

0.5343 0.4657

0.6295 0.3705

0.5847 0.4153

0.5344 0.4656

0.5343 0.4657

0.5847 0.4153

b2 a 2w2 w2

0.4981 0.5019

0.4981 0.5019

0.6271 0.3729

0.4992 0.5008

0.5805 0.4195

0.5341 0.4659

0.6246 0.3754

0.4990 0.5010

0.5844 0.4156

0.5322 0.4678

0.6193 0.3807

0.5759 0.4241

0.5168 0.4832

0.5340 0.4660

0.5841 0.4159 208

Table B -2. Cont'd

d0,l do,2 Rs,true

0.859 0.740 0 .8

0.862 0.991 1.4

0.863 0.991 0.9

0.865 0.743 0.9

0.869 0.967 1.3

0.871 0.984 0.9

0.871 0.968 1 .0

0.871 0.561 0 .8

0.873 0.874 1.1

0.875 0.501 1 .0

0.876 0.752 1 .1

0.878 0.970 1 .2

0.879 0.970 0.9

0.882 0.571 0.9

0.882 0.544 1 .0

Rs.meas %ROi %R0 2

1.326 6 .2 2 1 12.441

1.379 1.845 0.115

1.535 6.357 0.397

1.266 4.818 9.636

1.280 1.811 0.453

1.534 5.945 0.743

1.444 4.716 1.179

1.267 5.591 22.364

1.240 3.001 3.001

0.921 2.176 8.703

1.078 1.947 3.895

1.361 2 .8 6 8 0.717

1.532 5.538 1.384

1.213 4.182 16.727

1.059 2 .8 8 6 11.545

Q

2.893E-03

1.550E-07

1.245E-03

1.388E-03

1.344E-06

1.245E-03

5.675E-04

2.893E-03

1.589E-04

3.355E-04

6.252E-05

5.132E-05

1.245E-03

1.388E-03

4.910E-04

bl alW i W i

0.6295 0.3705

0.5000 0.5000

0.6295 0.3705

0.5847 0.4153

0.5000 0.5000

0.6295 0.3705

0.5847 0.4153

0.6295 0.3705

0.5343 0.4657

0.5016 0.4984

0.5010 0.4990

0.5343 0.4657

0.6295 0.3705

0.5847 0.4153

0.5343 0.4657

t>2 a 2w2 w2

0.6071 0.3929

0.4997 0.5003

0.6291 0.3709

0.5652 0.4348

0.4996 0.5004

0.6287 0.3713

0.5834 0.4166

0.5687 0.4313

0.5300 0.4700

0.4425 0.5575

0.4960 0.5040

0.5337 0.4663

0.6279 0.3721

0.5324 0.4676

0.4846 0.5154 20

9

Table B •2. Cont'd

do,l do ,2 ^s.true

0.883 0.942 1 .0

0.887 0.944 0.9

0 .8 8 8 0 .8 8 8 1 .2

0 .8 8 8 0 .8 8 8 1 .2

0.890 0.993 1.3

0.895 0.898 0.9

0.895 0.897 1 .0

0.895 0.993 1 .1

0.896 0.987 1.4

0.897 0.797 1 .1

0.898 0.949 1 .2

0.903 0.816 0.9

0.904 0.988 1.1

0.905 0.952 1.3

0.906 0.988 1.3

^s.meas %ROj %R0 2

1.441 4.241 2 .1 2 0

1.529 5.133 2.567

1.181 1.639 1.639

1.181 1.639 1.639

1.475 2.589 0.162

1.523 4.731 4.731

1.435 3.787 3.787

1.591 3.889 0.243

1.379 1.365 0.171

1.232 2.449 4.899

1.360 2.403 1 .2 0 1

1.509 4.328 8.655

1.590 3.547 0.443

1.280 1.309 0.655

1.475 2.209 0.276

Q

5.675E-04

1.245E-03

9.930E-06

9.930E-06

1.656E-05

1.245E-03

5.675E-04

2.320E-04

1.550E-07

1.589E-04

5.132E-05

1.245E-03

2.320E-04

1.344E-06

1.656E-05

*>1 alWi W i

0.5847 0.4153

0.6295 0.3705

0.5005 0.4995

0.5005 0.4995

0.5343 0.4657

0.6295 0.3705

0.5847 0.4153

0.5847 0.4153

0.5000 0.5000

0.5343 0.4657

0.5343 0.4657

0.6295 0.3705

0.5847 0.4153

0.5001 0.4999

0.5343 0.4657

^ 2 a 2W2 w 2

0.5821 0.4179

0.6263 0.3737

0.4995 0.5005

0.4995 0.5005

0.5342 0.4658

0.6230 0.3770

0.5792 0.4208

0.5845 0.4155

0.4998 0.5002

0.5253 0.4747

0.5331 0.4669

0.6158 0.3842

0.5843 0.4157

0.4998 0.5002

0.5341 0.4659 210

Table B -2. Cont'd

^0 ,1 do ,2 Rs,true

0.906 0.819 1 .0

0.907 0.994 1 .0

0.911 0.679 0.9

0.912 0.989 1 .0

0.912 0.978 1.1

0.915 0.916 1 .2

0.917 0 .6 6 6 1.1

0.917 0.690 1 .0

0.917 0.979 1 .0

0.918 0.679 1.1

0.920 0.960 1.1

0.920 0.980 1.3

0.921 0.995 1.5

0.922 0.843 1 .2

0.923 0.962 1 .0

Rs.meas %ROi %R0 2

1.423 3.352 6.704

1.706 4.336 0.271

1.476 3.920 15.680

1.705 4.064 0.508

1.589 3.219 0.805

1.357 1.996 1.996

1.073 1.338 5.353

1.394 2.935 11.739

1.704 3.795 0.949

1 .2 1 2 1.975 7.901

1.587 2.906 1.453

1.475 1.873 0.468

1.478 0.986 0.062

1.181 1.150 2.299

1.702 3.529 1.764

n *>1Wi

alWi

*>2w2

a2w2

5.675E-04 0.5847 0.4153 0.5731 0.4269

5.357E-04 0.6295 0.3705 0.6292 0.3708

1.245E-03 0.6295 0.3705 0.5980 0.4020

5.357E-04 0.6295 0.3705 0.6290 0.3710

2.320E-04 0.5847 0.4153 0.5839 0.4161

5.132E-05 0.5343 0.4657 0.5319 0.4681

6.252E-05 0.5005 0.4995 0.4913 0.5087

5.675E-04 0.5847 0.4153 0.5581 0.4419

5.357E-04 0.6295 0.3705 0.6285 0.3715

1.589E-04 0.5343 0.4657 0.5138 0.4862

2.320E-04 0.5847 0.4153 0.5830 0.4170

1.656E-05 0.5343 0.4657 0.5339 0.4661

1.523E-08 0.5000 0.5000 0.4999 0.5001

9.930E-06 0.5003 0.4997 0.4990 0.5010

5.357E-04 0.6295 0.3705 0.6274 0.3726 Ki

Table B ■2. Cont'd

<*0 ,1 do ,2 Rs.true

0.923 0.981 1.4

0.928 0.929 1 .0

0.928 0.929 1 .1

0.929 0.996 1.4

0.929 0.996 1 .2

0.931 0.862 1 .2

0.932 0.932 1.3

0.932 0.932 1.3

0.933 0.967 1.3

0.933 0.871 1 .0

0.935 0.992 1 .2

0.935 0.874 1 .1

0.936 0.508 1 .1

0.937 0.996 1 .1

0.938 0.770 1 .0

Rs.meas %ROi %R0 2

1,379 0.996 0.249

1.697 3.264 3.264

1.583 2.607 2.607

1.589 1 .6 6 8 0.104

1.736 2.634 0.165

1.352 1.643 3.287

1.281 0.932 0.932

1.281 0.932 0.932

1.474 1.578 0.789

1.687 3.000 6 .0 0 0

1.735 2.408 0.301

1.575 2.321 4.642

1.135 1.570 12.557

1.876 2.954 0.185

1.667 2.735 10.938

a

1.550E-07

5.357E-04

2.320E-04

5.342E-06

9.486E-05

5.132E-05

1.344E-06

1.344E-06

1.656E-05

5.357E-04

9.486E-05

2.320E-04

1.589E-04

2.305E-04

5.357E-04

*>1 alWj W]

0.5000 0.5000

0.6295 0.3705

0.5847 0.4153

0.5343 0.4657

0.5847 0.4153

0.5343 0.4657

0.5001 0.4999

0.5001 0.4999

0.5343 0.4657

0.6295 0.3705

0.5847 0.4153

0.5847 0.4153

0.5343 0.4657

0.6295 0.3705

0.6295 0.3705

*>2 a2w2 w2

0.4999 0.5001

0.6253 0.3747

0.5813 0.4187

0.5342 0.4658

0.5846 0.4154

0.5294 0.4706

0.4999 0.5001

0.4999 0.5001

0.5336 0.4664

0.6209 0.3791

0.5845 0.4155

0.5776 0.4224

0.4629 0.5371

0.6293 0.3707

0.6108 0.3892 212

Table B -2. Cont'd

^0 ,1 do, 2 Rs.true

0.939 0.992 1.4

0.940 0.993 1 .1

0.940 0.985 1 .2

0.941 0.993 1.5

0.943 0.780 1 .1

0.943 0.606 1 .0

0.944 0.986 1.1

0.944 0.778 1 .2

0.944 0.944 1.3

0.944 0.972 1.4

0.945 0.564 1 .1

0.946 0.973 1 .2

0.947 0.787 1 .2

0.947 0.974 1 .1

0.948 0.987 1.4

Rs.meas %ROi %R0 2

1.589 1.429 0.179

1.876 2.774 0.347

1.734 2.192 0.548

1.478 0.726 0.091

1.557 2.047 8.189

1.613 2.465 19.721

1.875 2.596 0.649

1.341 1.339 5.355

1.473 1.320 1.320

1.379 0.718 0.359

1.059 0.902 7.215

1.733 1.986 0.993

1.180 0.793 3.173

1.874 2.421 1 .2 1 0

1.589 1.218 0.304

Q. t>lW j

alW i

^ 2W2

a 2w 2

5.342E-06 0.5343 0.4657 0.5342 0.4658

2.305E-04 0.6295 0.3705 0.6291 0.3709

9.486E-05 0.5847 0.4153 0.5842 0.4158

1.523E-08 . 0.5000 0.5000 0.5000 0.5000

2.320E-04 0.5847 0.4153 0.5693 0.4307

5.357E-04 0.6295 0.3705 0.5828 0.4172

2.305E-04 0.6295 0.3705 0.6288 0.3712

5.132E-05 0.5343 0.4657 0.5239 0.4761

1.656E-05 0.5343 0.4657 0.5329 0.4671

1.550E-07 0.5000 0.5000 0.5000 0.5000

6.252E-05 0.5003 0.4997 0.4788 0.5212

9.486E-05 0.5847 0.4153 0.5837 0.4163

9.930E-06 0.5001 0.4999 0.4980 0.5020

2.305E-04 0.6295 0.3705 0.6281 0.3719

5.342E-06 0.5343 0.4657 0.5341 0.4659 213


do,l do,2 Rs.true

0.950 0.628 1 .1

0.951 0.951 1 .1

0.951 0.951 1 .2

0.952 0.997 1.3

0.952 0.905 1.3

0.954 0.908 1.3

0.954 0.910 1.1

0.955 0.997 1.5

0.955 0.651 1 .2

0.956 0.913 1 .2

0.956 0.995 1.3

0.956 0.978 1.4

0.957 0.989 1.5

0.957 0.997 1 .2

0.957 0.837 1.1

^s.meas %ROj %R02

1.513 1.784 14.274

1.870 2.247 2.247.

1.730 1.789 1.789

1.880 1.781 0 .1 1 1

1.280 0.655 1.309

1.470 1.095 2.189

1.864 2.073 4.147

1.702 1.071 0.067

1.316 1.077 8.613

1.725 1.601 3.202

1.880 1.632 0.204

1.588 1.031 0.516

1.478 0.528 0.132

2.047 2 .0 1 1 0.126

1.850 1.900 7.600

Q

2.320E-04

2.305E-04

9.486E-05

3.878E-05

1.344E-06

1.656E-05

2.305E-04

1.723E-06

5.132E-05

9.486E-05

3.878E-05

5.342E-06

1.523E-08

9.920E-05

2.305E-04

*>1 alWi Wj

0.5847 0.4153

0.6295 0.3705

0.5847 0.4153

0.5847 0.4153

0.5001 0.4999

0.5343 0.4657

0.6295 0.3705

0.5343 0.4657

0.5343 0.4657

0.5847 0.4153

0.5847 0.4153

0.5343 0.4657

0.5000 0.5000

0.6295 0.3705

0.6295 0.3705

t>2 a 2w2 w2

0.5472 0.4528

0.6268 0.3732

0.5825 0.4175

0.5847 0.4153

0.4998 0.5002

0.5316 0.4684

0.6240 0.3760

0.5343 0.4657

0.5100 0.4900

0.5803 0.4197

0.5846 0.4154

0.5339 0.4661

0.5000 0.5000

0.6294 0.3706

0.6179 0.3821 214


do ,l do,2 Rs,true

0.959 0.995 1 .2

0.960 0.990 1.3

0.960 0.960 1.4

0.960 0.960 1.4

0.960 0.846 1 .2

0.961 0.714 1.1

0.961 0.995 1.5

0.962 0.990 1 .2

0.962 0.849 1.3

0.963 0.963 1.4

0.963 0.982 1.3

0.964 0.982 1 .2

0.964 0.522 1.1

0.965 0.717 1 .2

0.965 0.734 1 .2

Rs,meas %ROi %RC>2

2.047 1.892 0.236

1.879 1.490 0.373

1.379 0.511 0.511

1.379 0.511 0.511

1.714 1.421 5.685

1.818 1.725 13.800

1.702 0.921 0.115

2.046 1.774 0.444

1.463 0.900 3.598

1.587 0 .8 6 8 0 .8 6 8

1.879 1.354 0.677

2.045 1.658 0.829

1.711 1.546 24.743

1.177 0.538 4.304

1 .6 8 8 1.249 9.989

a

9.920E-05

3.878E-05

1.550E-07

1.550E-07

9.486E-05

2.305E-04

1.723E-06

9.920E-05

1.656E-05

5.342E-06

3.878E-05

9.920E-05

2.305E-04

9.930E-06

9.486E-05

*>1 alW i Wi

0.6295 0.3705

0.5847 0.4153

0.5000 0.5000

0.5000 0.5000

0.5847 0.4153

0.6295 0.3705

0.5343 0.4657

0.6295 0.3705

0.5343 0.4657

0.5343 0.4657

0.5847 0.4153

0.6295 0.3705

0.6295 0.3705

0.5001 0.4999

0.5847 0.4153

*>2 a 2w2 w2

0.6293 0.3707

0.5844 0.4156

0.5000 0.5000

0.5000 0.5000

0.5754 0.4246

0.6035 0.3965

0.5342 0.4658

0.6291 0.3709

0.5287 0.4713

0.5335 0.4665

0.5840 0.4160

0.6286 0.3714

0.5486 0.4514

0.4958 0.5042

0.5641 0.4359 215

Table B-2. Cont'd

do,l do,2 Rs.true

0.966 0.966 1 .2

0.966 0.992 1.5

0.967 0.967 1.3

0.967 0.869 1.3

0.968 0.998 1.4

0.968 0.938 1 .2

0.969 0.984 1.5

0.969 0.939 1.4

0.969 0.558 1 .2

0.969 0.759 1.3

0.970 0.940 1.3

0.970 0.996 1.4

0.971 0 .8 8 6 1 .2

0.971 0.998 1.3

0.971 0.986 1.5

Rs.meas %ROi %RC>2

2.043 1.543 1.543

1.702 0.788 0.197

1.877 1.225 1.225

1.280 0.453 1.811

2.025 1 .2 0 2 0.075

2.038 1.429 2.858

1.478 0.380 0.190

1.585 0.725 1.449

1.613 1.083 17.322

1.450 0.731 5.849

1.873 1 .1 0 1 2 .2 0 2

2.025 1.105 0.138

2.029 1.315 5.262

2.218 1.367 0.085

1.702 0.671 0.336

f i

9.920E-05

1.723E-06

3.878E-05

1.344E-06

1.586E-05

9.920E-05

1.523E-08

5.342E-06

9.486E-05

1.656E-05

3.878E-05

1.586E-05

9.920E-05

4.268E-05

1.723E-06

*>1 alWj Wi

0.6295 0.3705

0.5343 0.4657

0.5847 0.4153

0.5000 0.5000

0.5847 0.4153

0.6295 0.3705

0.5000 0.5000

0.5343 0.4657

0.5847 0.4153

0.5343 0.4657

0.5847 0.4153

0.5847 0.4153

0.6295 0.3705

0.6295 0.3705

0.5343 0.4657

b2 a 2w2 w2

0.6277 0.3723

0.5342 0.4658

0.5833 0.4167

0.4996 0.5004

0.5847 0.4153

0.6259 0.3741

0.5000 0.5000

0.5328 0.4672

0.5277 0.4723

0.5223 0.4777

0.5820 0.4180

0.5846 0.4154

0.6221 0.3779

0.6294 0.3706

0.5341 0.4659

Table B-2. Cont’d

do,l d0>2 Rs,true

0.972 0.944 1.4

0.972 0.997 1.3

0.973 0.993 1.4

0.973 0.893 1.3

0.973 0.796 1 .2

0.974 0.994 1.3

0.975 0.899 1.4

0.975 0.988 1.4

0.975 0.647 1 .2

0.975 0.988 1.3

0.976 0.812 1.3

0.976 0.622 1.3

0.976 0.976 1.5

0.977 0.977 1.3

0.977 0.633 1 .2

Rs.meas %ROi %R0 2

1.379 0.359 0.718

2.218 1.288 0.161

2.024 1 .0 1 1 0.253

1 .8 6 6 0.982 3.930

2.009 1 .2 0 2 9.612

2.217 1 .2 1 0 0.303

1.582 0.600 2.401

2.024 0.922 0.461

1.959 1.086 17.377

2.216 1.134 0.567

1.849 0.869 6.953

1.417 0.587 9.385

1.701 0.568 0.568

2.215 1.058 1.058

1.171 0.359 5.743

n

1.550E-07

4.268E-05

1.586E-05

3.878E-05

9.920E-05

4.268E-05

5.342E-06

1.586E-05

9.920E-05

4.268E-05

3.878E-05

1.656E-05

1.723E-06

4.268E-05

9.930E-06

1>1 alW i Wi

0.5000 0.5000

0.6295 0.3705

0.5847 0.4153

0.5847 0.4153

0.6295 0.3705

0.6295 0.3705

0.5343 0.4657

0.5847 0.4153

0.6295 0.3705

0.6295 0.3705

0.5847 0.4153

0.5343 0.4657

0.5343 0.4657

0.6295 0.3705

0.5000 0.5000

b2 a 2w 2 w 2

0.5000 0.5000

0.6293 0.3707

0.5845 0.4155

0.5790 0.4210

0.6138 0.3862

0.6292 0.3708

0.5312 0.4688

0.5843 0.4157

0.5922 0.4078

0.6289 0.3711

0.5725 0.4275

0.5052 0.4948

0.5338 0.4662

0.6283 0.3717

0.4908 0.5092


<*0,1 <*o,2 ^s.true

0.977 0.977 1.4

0.978 0.978 1.5

0.978 0.978 1.5

0.978 0.825 1.3

0.978 0.999 1.5

0.978 0.957 1.3

0.979 0.679 1.3

0.979 0.836 1.4

0.979 0.959 1.4

0.980 0.960 1.5

0.980 0.921 1.3

0.980 0.998 1.5

0.980 0.999 1.4

0.981 0.923 1.4

0.981 0.998 1.4

Rs.meas %ROj %RC>2

2.023 0.836 0.836

1.478 0.270 0.270

1.478 0.270 0.270

1.280 0.309 2.470

2.170 0.811 0.051

2.212 0.983 1.966

1.811 0.760 12.165

1.574 0.492 3.938

2.020 0.755 1.509

1.700 0.477 0.954

2.205 0.908 3.633

2.170 0.746 0.093

2.388 0.929 0.058

1.379 0.249 0.996

2.388 0.876 0.110

Q

1.586E-05

1.523E-08

1.523E-08

1.344E-06

6.483E-06

4.268E-05

3.878E-05

5.342E-06

1.586E-05

1.723E-06

4.268E-05

6.483E-06

1.837E-05

1.550E-07

1.837E-05

bl alWi Wi

0.5847 0.4153

0.5000 0.5000

0.5000 0.5000

0.5000 0.5000

0.5847 0.4153

0.6295 0.3705

0.5847 0.4153

0.5343 0.4657

0.5847 0.4153

0.5343 0.4657

0.6295 0.3705

0.5847 0.4153

0.6295 0.3705

0.5000 0.5000

0.6295 0.3705

b2 a 2w 2 w 2

0.5839 0.4161

0.5000 0.5000

0.5000 0.5000

0.4992 0.5008

0.5847 0.4153

0.6272 0.3728

0.5565 0.4435

0.5279 0.4721

0.5830 0.4170

0.5334 0.4666

0.6248 0.3752

0.5847 0.4153

0.6294 0.3706

0.4999 0.5001

0.6294 0.3706 218

Table B -2. Cont'd

do,l do,2 Rs.true

0.981 0.926 1.4

0.981 0.857 1.3

0.982 0.995 1.5

0.982 0.996 1.4

0.983 0.746 1.3

0.983 0.992 1.5

0.983 0.933 1.5

0.983 0.869 1.4

0.983 0.992 1.4

0.983 0.737 1.4

0.984 0.984 1.4

0.984 0.969 1.5

0.985 0.985 1.5

0.985 0.772 1.4

0.985 0.971 1.4

Rs.meas %ROj %R02

2.015 0.677 2.707

2.192 0.834 6.669

2.169 0.685 0.171

2.388 0.825 0.206

2.162 0.759 12.139

2.169 0.626 0.313

1.698 0.398 1.592

2.005 0.602 4.817

2.388 0.774 0.387

1.558 0.399 6.387

2.386 0.724 0.724

1.478 0.190 0.380

2.168 0.570 0.570

1.982 0.531 8.491

2.384 0.675 1.349

Q *>1 al *>2 a2W i W j w 2 w 2

1.586E-05 0.5847 0.4153 0.5811 0.4189

4.268E-05 0.6295 0.3705 0.6196 0.3804

6.483E-06 0.5847 0.4153 0.5846 0.4154

1.837E-05 0.6295 0.3705 0.6293 0.3707

4.268E-05 0.6295 0.3705 0.6079 0.3921

6.483E-06 0.5847 0.4153 0.5845 0.4155

1.723E-06 0.5343 0.4657 0.5326 0.4674

1.586E-05 0.5847 0.4153 0.5772 0.4228

1.837E-05 0.6295 0.3705 0.6291 0.3709

5.342E-06 0.5343 0.4657 0.5203 0.4797

1.837E-05 0.6295 0.3705 0.6287 0.3713

1.523E-08 0.5000 0.5000 0.5000 0.5000

6.483E-06 0.5847 0.4153 0.5842 0.4158

1.586E-05 0.5847 0.4153 0.5685 0.4315

1.837E-05 0.6295 0.3705 0.6280 0.3720

I!

Table B•2. Cont'd

do,l do,2 Rs,true

0.986 0.770 1.3

0.986 0.972 1.5

0.986 0.889 1.5

0.986 0.946 1.4

0.987 0.999 1.5

0.987 0.896 1.4

0.987 0.900 1.4

0.987 0.949 1.5

0.987 0.998 1.5

0.988 0.997 1.5

0.988 0.820 1.4

0.989 0.909 1.5

0.989 0.994 1.5

0.989 0.820 1.5

0.989 0.957 1.5

Rs,meas %ROi %r o 2

1.278 0.208 3.321

2.166 0.516 1.032

1.694 0.329 2.631

2.380 0.626 2.502

2.559 0.630 0.039

1.379 0.171 1.365

2.371 0.577 4.613

2.163 0.465 1.858

2.559 0.596 0.075

2.559 0.562 0.140

2.351 0.528 8.443

2.156 0.416 3.324

2.558 0.528 0.264

1.685 0.269 4.308

1.478 0.132 0.528

Q blW i

alW j

b2w 2

a2w 2

1.344E-06 0.5000 0.5000 0.4982 0.5018

6.483E-06 0.5847 0.4153 0.5836 0.4164

1.723E-06 0.5343 0.4657 0.5308 0.4692

1.837E-05 0.6295 0.3705 0.6264 0.3736

7.903E-06 0.6295 0.3705 0.6295 0.3705

1.550E-07 0.5000 0.5000 0.4998 0.5002

1.837E-05 0.6295 0.3705 0.6232 0.3768

6.483E-06 0.5847 0.4153 0.5825 0.4175

7.903E-06 0.6295 0.3705 0.6294 0.3706

7.903E-06 0.6295 0.3705 0.6294 0.3706

1.837E-05 0.6295 0.3705 0.6162 0.3838

6.483E-06 0.5847 0.4153 0.5801 0.4199

7.903E-06 0.6295 0.3705 0.6293 0.3707

1.723E-06 0.5343 0.4657 0.5269 0.4731

1.523E-08 0.5000 0.5000 0.5000 0.5000 220


d0,l do,2 Rs.true

0.989 0.989 1.5

0.990 0.840 1.5

0.990 0.980 1.5

0.991 0.963 1.5

0.991 0.931 1.5

0.991 0.862 1.4

0.992 0.874 1.5

0.993 0.941 1.5

0.995 0.921 1.5

^s,meas %ROi %R02

2.558 0.495 0.495

2.142 0.369 5.897

2.556 0.462 0.925

2.553 0.430 1.720

2.547 0.398 3.182

1.379 0.115 1.845

2.534 0.366 5.851

1.478 0.091 0.726

1.478 0.062 0.986

n *>1 al *>2 a2Wi w2 w2

7.903E-06 0.6295 0.3705 0.6290 0.3710

6.483E-06 0.5847 0.4153 0.5750 0.4250

7.903E-06 0.6295 0.3705 0.6285 0.3715

7.903E-06 0.6295 0.3705 0.6275 0.3725

7.903E-06 0.6295 0.3705 0.6254 0.3746

1.550E-07 0.5000 0.5000 0.4997 0.5003

7.903E-06 0.6295 0.3705 0.6211 0.3789

1.523E-08 0.5000 0.5000 0.5000 0.5000

1.523E-08 0.5000 0.5000 0.4999 0.5001

221

Table B-3. W, a and b measured at the 75% peak height fraction.

do.l do,2 Rs.true Rs,meas %ROi %R02 Q blW i

alWi

*>2w 2

a2w 2

0.253 0.623 0.8 0.689 15.255 7.628 5.9760E-03 0.5094 0.4906 0.4123 0.5877

0.258 0.655 0.5 0.694 26.592 13.296 4.9457E-02 0.6020 0.3980 0.5355 0.4645

0.265 0.265 0.7 0.527 16.151 16.151 1.9841E-02 0.5849 0.4151 0.4151 0.5849

0.265 0.265 0.7 0.527 16.151 16.151 1.9841E-02 0.5849 0.4151 0.4151 0.5849

0.280 0.955 0.6 0.899 26.539 1.659 2.0300E-02 0.5738 0.4262 0.5495 0.4505

0.294 0.822 0.9 0.840 13.630 3.408 1.5338E-03 0.5016 0.4984 0.4541 0.5459

0.297 0.912 0.8 0.897 17.969 2.246 4.5500E-03 0.5515 0.4485 0.5194 0.4806

0.309 0.662 0.7 0.759 17.688 8.844 1.3305E-02 0.5560 0.4440 0.5100 0.4900

0.347 0.919 0.6 0.912 23.774 2.972 2.0300E-Q2 0.5598 0.4402 0.5484 0.4516

0.352 0.441 0.5 0.689 22.864 22.864 4.9457E-02 0.5668 0.4332 0.5133 0.4867

0.363 0.920 1.0 0.969 11.639 1.455 3.3546E-04 0.5002 0.4998 0.4769 0.5231

0.365 0.961 0.5 0.943 28.960 1.810 3.6315E-02 0.5804 0.4196 0.5767 0.4233

0.404 0.852 0.8 0.916 14.844 3.711 4.5500E-03 0.5328 0.4672 0.5180 0.4820

0.407 0.927 0.5 0.941 26.782 3.348 3.6315E-02 0.5791 0.4209 0.5755 0.4245

0.413 0.857 0.6 0.912 21.129 5.282 2.0300E-02 0.5548 0.4452 0.5462 0.4538

222

Table B-3. Cont'd

do,l do,2 Rs.true Rs,meas %ROi %RC>2

0.422 0.964 0.9 1.043 14.336 0.896

0.443 0.470 0.7 0.755 14.076 14.076

0.447 0.447 0.8 0.743 10.961 10.961

0.447 0.447 0.8 0.743 10.961 10.961

0.448 0.965 1.1 1.085 9.544 0.597

0.450 0.867 0.5 0.936 24.609 6.152

0.457 0.728 0.9 0.874 10.028 5.014

0.479 0.752 0.6 0.903 18.591 9.296

0.493 0.764 0.5 0.925 22.423 11.212

0.501 0.875 1.0 0.986 8.703 2.176

0.504 0.969 0.7 1.079 18.161 1.135

0.508 0.939 0.9 1.047 12.012 1.502

0.508 0.757 0.8 0.916 12.101 6.051

0.536 0.598 0.5 0.900 20.203 20.203

0.543 0.585 0.6 0.880 16.151 16.151

o *>1 al *>2 a2W i W j w2 W2

1.5066E-03 0.5280 0.4720 0.5204 0.4796

1.3305E-02 0.5344 0.4656 0.4948 0.5052

5.9760E-03 0.5218 0.4782 0.4782 0.5218

5.9760E-03 0.5218 0.4782 0.4782 0.5218

6.2522E-05 0.5001 0.4999 0.4892 0.5108

3.6315E-02 0.5784 0.4216 0.5729 0.4271

1.5338E-03 0.5034 0.4966 0.4839 0.5161

2.0300E-02 0.5526 0.4474 0.5414 0.4586

3.6315E-02 0.5781 0.4219 0.5677 0.4323

3.3546E-04 0.5006 0.4994 0.4903 0.5097

8.3040E-03 0.5511 0.4489 0.5499 0.4501

1.5066E-03 0.5243 0.4757 0.5201 0.4799

4.5500E-03 0.5263 0.4737 0.5150 0.4850

3.6315E-02 0.5780 0.4220 0.5556 0.4444

2.0300E-02 0.5516 0.4484 0.5302 0.4698 223

Table B-3. Cont'd

<*o,l do,2 Rs.true

0.550 0.944 0.7

0.564 0.945 1.1

0.566 0.973 0.6

0.581 0.361 0.5

0.587 0.897 0.9

0.594 0.950 0.6

0.595 0.900 0.7

0.604 0.615 0.8

0.605 0.605 0.9

0.605 0.605 0.9

0.607 0.347 0.6

0.607 0.976 1.0

0.622 0.908 0.6

0.623 0.253 0.8

0.625 0.812 1.0

Rs.meas %RO\ %R02

1.078 16.338 2.042

1.093 7.215 0.902

1.136 19.870 1.242

0.839 17.904 35.807

1.047 9.965 2.491

1.134 18.437 2.305

1.074 14.595 3.649

0.905 9.721 9.721

0.881 7.186 7.186

0.881 7.186 7.186

0.818 13.790 27.580

1.169 9.442 0.590

1.130 17.011 4.253

0.689 7.628 15.255

0.993 6.363 3.182

Q 1>1W i

alWi

t>2w 2

nw 2

8.3040E-03 0.5508 0.4492 0.5492 0.4508

6.2522E-05 0.5001 0.4999 0.4950 0.5050

1.5627E-02 0.5778 0.4222 0.5770 0.4230

3.6315E-02 0.5779 0.4221 0.5193 0.4807

1.5066E-03 0.5225 0.4775 0.5193 0.4807

1.5627E-02 0.5778 0.4222 0.5763 0.4237

8.3040E-03 0.5507 0.4493 0.5478 0.4522

4.5500E-03 0.5235 0.4765 0.5081 0.4919

1.5338E-03 0.5072 0.4928 0.4928 0.5072

1.5338E-03 0.5072 0.4928 0.4928 0.5072

2.0300E-02 0.5510 0.4490 0.4941 0.5059

4.9097E-04 0.5215 0.4785 0.5206 0.4794

1.5627E-02 0.5778 0.4222 0.5747 0.4253

5.9760E-03 0.5094 0.4906 0.4123 0.5877

3.3546E-04 0.5011 0.4989 0.4954 0.5046 224

Table B -3. Cont'd

do,l do,2 Rs.true

0.633 0.977 1.2

0.639 0.825 0.7

0.650 0.833 0.6

0.660 0.831 0.9

0.662 0.979 0.8

0.666 0.917 1.1

0.667 0.958 1.0

0.678 0.707 0.6

0.681 0.700 0.7

0.691 0.416 0.8

0.693 0.962 0.8

0.704 0.982 0.7

0.707 0.513 0.6

0.717 0.965 1.2

0.721 0.931 1.0

Rs,meas %ROi %R02

1.198 5.743 0.359

1.066 12.927 6.463

1.121 15.583 7.791

1.044 8.179 4.089

1.235 12.393 0.775

1.097 5.353 1.338

1.169 7.941 0.993

1.104 14.142 14.142

1.050 11.327 11.327

0.870 7.678 15.356

1.234 11.192 1.399

1.326 13.611 0.851

1.065 12.670 25.340

1.199 4.304 0.538

1.168 6.624 1.656

o bi al t>2 a2W i W i w2 w2

9.9295E-06 0.5000 0.5000 0.4978 0.5022

8.3040E-03 0.5506 0.4494 0.5449 0.4551

1.5627E-02 0.5778 0.4222 0.5714 0.4286

1.5066E-03 0.5216 0.4784 0.5177 0.4823

3.3952E-03 0.5506 0.4494 0.5501 0.4499

6.2522E-05 0.5001 0.4999 0.4976 0.5024

4.9097E-04 0.5212 0.4788 0.5204 0.4796

1.5627E-02 0.5778 0.4222 0.5642 0.4358

8.3040E-03 0.5506 0.4494 0.5386 0.4614

4.5500E-03 0.5221 0.4779 0.4889 0.5111

3.3952E-03 0.5506 0.4494 0.5497 0.4503

6.724IE-03 0.5778 0.4222 0.5773 0.4227

1.5627E-02 0.5778 0.4222 0.5466 0.4534

9.9295E-06 0.5000 0.5000 0.4989 0.5011

4.9097E-04 0.5210 0.4790 0.5199 0.4801 22

5

Table B -3. Cont'd

<*o,l <*o,2 ^s,true

0.722 0.508 0.7

0.722 0.966 0.7

0.723 0.931 0.8

0.725 0.729 0.9

0.728 0.457 0.9

0.729 0.729 1.0

0.737 0.257 0.6

0.740 0.984 1.1

0.741 0.936 0.7

0.751 0.878 0.8

0.752 0.876 1.1

0.759 0.883 0.7

0.763 0.252 0.7

0.770 0.986 1.3

0.770 0.885 1.0

**s,meas %ROi %R02

1.014 9.786 19.572

1.325 12.665 1.583

1.231 10.046 2.511

1.036 6.632 6.632

0.874 5.014 10.028

0.994 4.550 4.550

0.928 11.128 44.511

1.287 6.167 0.385

1.322 11.727 2.932

1.226 8.948 4.474

1.099 3.895 1.947

1.316 10.792 5.396

0.870 8.291 33.162

1.300 3.321 0.208

1.165 5.475 2.738

Q b lWi

alWi

*>2w2

a2w2

8.3040E-03 0.5506 0.4494 0.5226 0.4774

6.724IE-03 0.5778 0.4222 0.5768 0.4232

3.3952E-03 0.5505 0.4495 0.5488 0.4512

1.5066E-03 0.5212 0.4788 0.5141 0.4859

1.5338E-03 0.5034 0.4966 0.4839 0.5161

3.3546E-04 0.5022 0.4978 0.4978 0.5022

1.5627E-02 0.5778 0.4222 0.4609 0.5391

1.5892E-04 0.5208 0.4792 0.5207 0.4793

6.7241E-03 0.5778 0.4222 0.5758 0.4242

3.3952E-03 0.5505 0.4495 0.5470 0.4530

6.2522E-05 0.5003 0.4997 0.4988 0.5012

6.7241E-03 0.5778 0.4222 0.5736 0.4264

8.3040E-03 0.5506 0.4494 0.4309 0.5691

1.3438E-06 0.5000 0.5000 0.4996 0.5004

4.9097E-04 0.5209 0.4791 0.5190 0.4810 226

Table B-3. Cont'd

d0,l ^o,2 ^s,tnie ^s.meas %ROi %R02 Cl b lW i

alW i

*>2w2

a2w2

0.771 0.986 0.9 1.390 8.438 0.527 1.3881E-03 0.5505 0.4495 0.5503 0.4497

0.778 0.791 0.7 1.303 9.853 9.853 6.7241E-03 0.5778 0.4222 0.5692 0.4308

0.779 0.787 0.8 1.215 7.898 7.898 3.3952E-03 0.5505 0.4495 0.5432 0.4568

0.779 0.972 1.1 1.286 5.210 0.651 1.5892E-04 0.5208 0.4792 0.5206 0.4794

0.782 0.578 0.9 1.017 5.303 10.606 1.5066E-03 0.5210 0.4790 0.5060 0.4940

0.787 0.947 1.2 1.200 3.173 0.793 9.9295E-06 0.5001 0.4999 0.4995 0.5005

0.791 0.974 0.9 1.389 7.648 0.956 1.3881E-03 0.5505 0.4495 0.5500 0.4500

0.796 0.640 0.7 1.277 8.902 17.805 6.724IE-03 0.5778 0.4222 0.5593 0.4407

0.798 0.987 0.8 1.516 9.309 0.582 2.8934E-03 0.5778 0.4222 0.5775 0.4225

0.805 0.640 0.8 1.191 6.889 13.778 3.3952E-03 0.5505 0.4495 0.5346 0.4654

0.811 0.976 0.8 1.515 8.685 1.086 2.8934E-03 0.5778 0.4222 0.5772 0.4228

0.811 0.953 0.9 1.387 6.892 1.723 1.3881E-03 0.5505 0.4495 0.5494 0.4506

0.812 0.814 1.0 1.160 4.480 4.480 4.9097E-04 0.5208 0.4792 0.5172 0.4828

0.812 0.625 1.0 0.993 3.182 6.363 3.3546E-04 0.5011 0.4989 0.4954 0.5046

0.815 0.954 1.1 1.286 4.371 1.093 1.5892E-04 0.5208 0.4792 0.5203 0.4797 22

7

Table B -3. Cont'd

do,l do,2 Rs.true

0.815 0.417 0.7

0.822 0.822 1.1

0.822 0.822 1.1

0.822 0.294 0.9

0.823 0.956 0.8

0.825 0.978 1.3

0.829 0.916 0.9

0.830 0.989 1.2

0.832 0.423 0.8

0.832 0.371 0.9

0.835 0.919 0.8

0.843 0.922 1.2

0.845 0.990 1.0

0.846 0.923 1.1

0.847 0.853 0.8

Rs.meas %ROi %R02

1.214 7.923 31.690

1.099 2.781 2.781

1.099 2.781 2.781

0.840 3.408 13.630

1.513 8.067 2.017

1.301 2.470 0.309

1.384 6.171 3.085

1.404 4.005 0.250

1.135 5.915 23.662

0.969 4.170 16.680

1.509 7.452 3.726

1.200 2.299 1.150

1.545 5.734 0.358

1.284 3.639 1.819

1.500 6.838 6.838

a t>lW i

alWi

b2w 2

a2w 2

6.7241E-03 0.5778 0.4222 0.5321 0.4679

6.2522E-05 0.5006 0.4994 0.4994 0.5006

6.2522E-05 0.5006 0.4994 0.4994 0.5006

1.5338E-03 0.5016 0.4984 0.4541 0.5459

2.8934E-03 0.5778 0.4222 0.5765 0.4235

1.3438E-06 0.5000 0.5000 0.4998 0.5002

1.388 IE-03 0.5505 0.4495 0.5484 0.4516

5.1316E-05 0.5208 0.4792 0.5208 0.4792

3.3952E-03 0.5505 0.4495 0.5106 0.4894

1.5066E-03 0.5209 0.4791 0.4813 0.5187

2.8934E-03 0.5778 0.4222 0.5751 0.4249

9.9295E-06 0.5001 0.4999 0.4997 0.5003

5.675 IE-04 0.5505 0.4495 0.5504 0.4496

1.5892E-04 0.5208 0.4792 0.5198 0.4802

2.8934E-03 0.5778 0.4222 0.5723 0.4277 228

Table B-3. Corn'd

d0,l do,2 Rs.true

0.847 0.851 0.9

0.849 0.704 1.0

0.856 0.982 1.2

0.858 0.982 1.0

0.859 0.740 0.8

0.862 0.991 1.4

0.863 0.991 0.9

0.865 0.743 0.9

0.869 0.967 1.3

0.871 0.984 0.9

0.871 0.968 1.0

0.871 0.561 0.8

0.873 0.874 1.1

0.875 0.501 1.0

0.876 0.752 1.1

^s,meas %RO\ %R02

1.376 5.480 5.480

1.149 3.621 7.242

1.404 3.400 0.425

1.544 5.213 0.652

1.481 6.221 12.441

1.401 1.845 0.115

1.706 6.357 0.397

1.361 4.818 9.636

1.301 1.811 0.453

1.706 5.945 0.743

1.543 4.716 1.179

1.441 5.591 22.364

1.281 3.001 3.001

0.986 2.176 8.703

1.099 1.947 3.895

f l *>1Wi

alWi

b2w2

a2w2

1.3881E-03 0.5505 0.4495 0.5460 0.4540

4.9097E-04 0.5208 0.4792 0.5131 0.4869

5.1316E-05 0.5208 0.4792 0.5206 0.4794

5.6751E-04 0.5505 0.4495 0.5502 0.4498

2.8934E-03 0.5778 0.4222 0.5663 0.4337

1.5498E-07 0.5000 0.5000 0.4999 0.5001

1.2450E-03 0.5778 0.4222 0.5776 0.4224

1.3881E-03 0.5505 0.4495 0.5410 0.4590

1.3438E-06 0.5000 0.5000 0.4999 0.5001

1.2450E-03 0.5778 0.4222 0.5774 0.4226

5.675 IE-04 0.5505 0.4495 0.5498 0.4502

2.8934E-03 0.5778 0.4222 0.5521 0.4479

1.5892E-04 0.5208 0.4792 0.5188 0.4812

3.3546E-04 0.5006 0.4994 0.4903 0.5097

6.2522E-05 0.5003 0.4997 0.4988 0.5012 229

Table B-3. Cont'd

<*0,1 do,2 Rs.true Rs.meas %RO\ %RC>2

0.878 0.970 1.2 1.403 2.868 0.717

0.879 0.970 0.9 1.704 5.538 1.384

0.882 0.571 0.9 1.326 4.182 16.727

0.882 0.544 1.0 1.125 2.886 11.545

0.883 0.942 1.0 1.540 4.241 2.120

0.884 0.315 0.8 1.330 4.936 39.488

0.887 0.944 0.9 1.701 5.133 2.567

0.888 0.888 1.2 1.200 1.639 1.639

0.888 0.888 1.2 1.200 1.639 1.639

0.890 0.993 1.3 1.521 2.589 0.162

0.895 0.898 0.9 1.695 4.731 4.731

0.895 0.897 1.0 1.535 3.787 3.787

0.895 0.993 1.1 1.700 3.889 0.243

0.896 0.987 1.4 1.401 1.365 0.171

0.897 0.797 1.1 1.275 2.449 4.899

QbiWi

alWj

*2w2

a2w2

5.1316E-05 0.5208 0.4792 0.5206 0.4794

1.2450E-03 0.5778 0.4222 0.5769 0.4231

1.3881E-03 0.5505 0.4495 0.5291 0.4709

4.9097E-04 0.5208 0.4792 0.5034 0.4966

5.675 IE-04 0.5505 0.4495 0.5491 0.4509

2.8934E-03 0.5778 0.4222 0.5037 0.4963

1.2450E-03 0.5778 0.4222 0.5761 0.4239

9.9295E-06 0.5002 0.4998 0.4998 0.5002

9.9295E-06 0.5002 0.4998 0.4998 0.5002

1.6558E-05 0.5208 0.4792 0.5208 0.4792

1.2450E-03 0.5778 0.4222 0.5743 0.4257

5.675 IE-04 0.5505 0.4495 0.5477 0.4523

2.3202E-04 0.5505 0.4495 0.5504 0.4496

1.5498E-07 0.5000 0.5000 0.5000 0.5000

1.5892E-04 0.5208 0.4792 0.5166 0.4834 23

0


d0,l do,2 Rs.true

0.898 0.949 1.2

0.898 0.330 0.9

0.903 0.816 0.9

0.904 0.988 1.1

0.905 0.952 1.3

0.906 0.988 1.3

0.906 0.819 1.0

0.907 0.994 1.0

0.909 0.328 1.0

0.911 0.679 0.9

0.912 0.989 1.0

0.912 0.978 1.1

0.915 0.916 1.2

0.917 0.666 1.1

0.917 0.690 1.0

Rs.meas %ROi %RC>2

1.402 2.403 1 .2 0 1

1.232 3.566 28.526

1.682 4.328 8.655

1.699 3.547 0.443

1.301 1.309 0.655

1.521 2.209 0.276

1.525 3.352 6.704

1.896 4.336 0.271

1.059 2.262 18.092

1.655 3.920 15.680

1.896 4.064 0.508

1.698 3.219 0.805

1.401 1.996 1.996

1.097 1.338 5.353

1.503 2.935 11.739

Q bl al b2 a2W i Wl w2 w2

5.1316E-05 0.5208 0.4792 0.5202 0.4798

1.3881E-03 0.5505 0.4495 0.4888 0.5112

1.2450E-03 0.5778 0.4222 0.5705 0.4295

2.3202E-04 .0.5505 0.4495 0.5503 0.4497

1.3438E-06 0.5000 0.5000 0.4999 0.5001

1.6558E-05 0.5208 0.4792 0.5207 0.4793

5.675 IE-04 0.5505 0.4495 0.5447 0.4553

5.3573E-04 0.5778 0.4222 0.5777 0.4223

4.9097E-04 0.5208 0.4792 0.4705 0.5295

1.2450E-03 0.5778 0.4222 0.5622 0.4378

5.3573E-04 0.5778 0.4222 0.5775 0.4225

2.3202E-04 0.5505 0.4495 0.5501 0.4499

5.1316E-05 0.5208 0.4792 0.5197 0.4803

6.2522E-05 0.5001 0.4999 0.4976 0.5024

5.675 IE-04 0.5505 0.4495 0.5380 0.4620 231

Table B -3. Cont'd

do,l do,2 Rs.true

0.917 0.979 1.0

0.918 0.679 1.1

0.919 0.471 0.9

0.920 0.363 1.0

0.920 0.960 1.1

0.920 0.980 1.3

0.921 0.995 1.5

0.922 0.843 1.2

0.923 0.962 1.0

0.923 0.981 1.4

0.928 0.929 1.0

0.928 0.929 1.1

0.928 0.493 1.0

0.929 0.996 1.4

0.929 0.996 1.2

Rs.meas %RO| %R02

1.895 3.795 0.949

1.262 1.975 7.901

1.593 3.502 28.016

0.969 1.455 11.639

1.696 2.906 1.453

1.521 1.873 0.468

1.501 0.986 0.062

1.200 1.150 2.299

1.893 3.529 1.764

1.401 0.996 0.249

1.888 3.264 3.264

1.693 2.607 2.607

1.452 2.533 20.262

1.638 1.668 0.104

1.854 2.634 0.165

Q blWi

alWi &2

w2a2w2

5.3573E-04 0.5778 0.4222 0.5772 0.4228

1.5892E-04 0.5208 0.4792 0.5120 0.4880

1.2450E-03 0.5778 0.4222 0.5411 0.4589

3.3546E-04 0.5002 0.4998 0.4769 0.5231

2.3202E-04 0.5505 0.4495 0.5496 0.4504

1.6558E-05 0.5208 0.4792 0.5206 0.4794

1.5230E-08 0.5000 0.5000 0.5000 0.5000

9.9295E-06 0.5001 0.4999 0.4997 0.5003

5.3573E-04 0.5778 0.4222 0.5767 0.4233

1.5498E-07 0.5000 0.5000 0.5000 0.5000

5.3573E-04 0.5778 0.4222 0.5755 0.4245

2.3202E-04 0.5505 0.4495 0.5487 0.4513

5.6751E-04 0.5505 0.4495 0.5208 0.4792

5.3418E-06 0.5208 0.4792 0.5208 0.4792

9.4861E-05 0.5505 0.4495 0.5505 0.4495 232


d0,l do,2 Rs.true

0.931 0.862 1.2

0.932 0.932 1.3

0.932 0.932 1.3

0.933 0.967 1.3

0.933 0.871 1.0

0.935 0.992 1.2

0.935 0.874 1.1

0.936 0.508 1.1

0.937 0.996 1.1

0.938 0.770 1.0

0.939 0.992 1.4

0.940 0.993 1.1

0.940 0.985 1.2

0.941 0.993 1.5

0.943 0.780 1.1

Rs.meas %ROi %RC>2

1.397 1.643 3.287

1.301 0.932 0.932

1.301 0.932 0.932

1.520 1.578 0.789

1.879 3.000 6.000

1.854 2.408 0.301

1.686 2.321 4.642

1.231 1.570 12.557

2.086 2.954 0.185

1.861 2.735 10.938

1.638 1.429 0.179

2.086 2.774 0.347

1.853 2.192 0.548

1.501 0.726 0.091

1.672 2.047 8.189

n blW i

alW!

b2w 2

a2W2

5.1316E-05 0.5208 0.4792 0.5185 0.4815

1.3438E-06 0.5000 0.5000 0.5000 0.5000

1.3438E-06 0.5000 0.5000 0.5000 0.5000

1.6558E-05 0.5208 0.4792 0.5205 0.4795

5.3573E-04 0.5778 0.4222 0.5731 0.4269

9.4861E-05 0.5505 0.4495 0.5504 0.4496

2.3202E-04 0.5505 0.4495 0.5469 0.4531

1.5892E-04 0.5208 0.4792 0.5003 0.4997

2.3053E-04 0.5778 0.4222 0.5777 0.4223

5.3573E-04 0.5778 0.4222 0.5680 0.4320

5.3418E-06 0.5208 0.4792 0.5208 0.4792

2.3053E-04 0.5778 0.4222 0.5776 0.4224

9.4861E-05 0.5505 0.4495 0.5503 0.4497

1.5230E-08 0.5000 0.5000 0.5000 0.5000

2.3202E-04 0.5505 0.4495 0.5429 0.4571 233


do.l do,2 Rs.true

0.943 0.606 1.0

0.944 0.986 1.1

0.944 0.778 1.2

0.944 0.944 1.3

0.944 0.972 1.4

0.945 0.564 1.1

0.946 0.973 1.2

0.947 0.787 1.2

0.947 0.974 1.1

0.948 0.987 1.4

0.949 0.372 1.0

0.950 0.628 1.1

0.951 0.951 1.1

0.951 0.951 1.2

0.951 0.283 1.1

Rs,meas %ROi %R02

1.821 2.465 19.721

2.085 2.596 0.649

1.389 1.339 5.355

1.519 1.320 1.320

1.401 0.718 0.359

1.093 0.902 7.215

1.852 1.986 0.993

1.200 0.793 3.173

2.083 2.421 1.210

1.638 1.218 0.304

1.719 2.186 34.983

1.640 1.784 14.274

2.080 2.247 2.247

1.850 1.789 1.789

1.133 1.226 19.609

n 1>1Wi

alW i

^2w 2

a2w 2

5.3573E-04 0.5778 0.4222 0.5564 0.4436

2.3053E-04 0.5778 0.4222 0.5774 0.4226

5.1316E-05 0.5208 0.4792 0.5160 0.4840

1.6558E-05 0.5208 0.4792 0.5202 0.4798

1.5498E-07 0.5000 0.5000 0.5000 0.5000

6.2522E-05 0.5001 0.4999 0.4950 0.5050

9.4861E-05 0.5505 0.4495 0.5499 0.4501

9.9295E-06 0.5001 0.4999 0.4995 0.5005

2.3053E-04 0.5778 0.4222 0.5771 0.4229

5.3418E-06 0.5208 0.4792 0.5207 0.4793

5.3573E-04 0.5778 0.4222 0.5222 0.4778

2.3202E-04 0.5505 0.4495 0.5338 0.4662

2.3053E-04 0.5778 0.4222 0.5763 0.4237

9.4861E-05 0.5505 0.4495 0.5494 0.4506

1.5892E-04 0.5208 0.4792 0.4524 0.5476 23

4

Table B -3. Cont'd

do.l do,2 ^s,true

0.952 0.997 1.3

0.952 0.905 1.3

0.954 0.908 1.3

0.954 0.910 1.1

0.955 0.997 1.5

0.955 0.651 1.2

0.956 0.913 1.2

0.956 0.995 1.3

0.956 0.978 1.4

0.956 0.406 1.1

0.957 0.989 1.5

0.957 0.997 1.2

0.957 0.837 1.1

0.959 0.995 1.2

0.960 0.990 1.3

Rs.meas %ROi %R02

2.009 1.781 0.111

1.301 0.655 1.309

1.517 1.095 2.189

2.074 2.073 4.147

1.755 1.071 0.067

1.373 1.077 8.613

1.845 1.601 3.202

2.009 1.632 0.204

1.638 1.031 0.516

1.562 1.530 24.484

1.501 0.528 0.132

2.276 2.011 0.126

2.061 1.900 7.600

2.276 1.892 0.236

2.008 1.490 0.373

n *>1W i

alW i

*>2w 2

nw 2

3.8783E-05 0.5505 0.4495 0.5505 0.4495

1.3438E-06 0.5000 0.5000 0.4999 0.5001

1.6558E-05 0.5208 0.4792 0.5195 0.4805

2.3053E-04 0.5778 0.4222 0.5747 0.4253

1.7232E-06 0.5208 0.4792 0.5208 0.4792

5.1316E-05 0.5208 0.4792 0.5106 0.4894

9.486IE-05 0.5505 0.4495 0.5483 0.4517

3.8783E-05 0.5505 0.4495 0.5505 0.4495

5.3418E-06 0.5208 0.4792 0.5206 0.4794

2.3202E-04 0.5505 0.4495 0.5077 0.4923

1.5230E-08 0.5000 0.5000 0.5000 0.5000

9.9195E-05 0.5778 0.4222 0.5778 0.4222

2.3053E-04 0.5778 0.4222 0.5716 0.4284

9.9195E-05 0.5778 0.4222 0.5777 0.4223

3.8783E-05 0.5505 0.4495 0.5504 0.4496 23

5

Table B -3. Cont'd

do,l do,2 Rs.true

0.960 0.960 1.4

0.960 0.960 1.4

0.960 0.846 1.2

0.961 0.714 1.1

0.961 0.995 1.5

0.962 0.990 1.2

0.962 0.849 1.3

0.963 0.963 1.4

0.963 0.982 1.3

0.964 0.982 1.2

0.964 0.522 1.1

0.965 0.717 1.2

0.965 0.734 1.2

0.965 0.470 1.2

0.965 0.448 1.1

Rs.meas %ROj %R02

1.401 0.511 0.511

1.401 0.511 0.511

1.835 1.421 5.685

2.035 1.725 13.800

1.755 0.921 0.115

2.275 1.774 0.444

1.512 0.900 3.598

1.637 0.868 0.868

2.007 1.354 0.677

2.274 1.658 0.829

1.975 1.546 24.743

1.199 0.538 4.304

1.815 1.249 9.989

1.334 0.853 13.649

1.085 0.597 9.544

n bl alWi

b2w2

a2w2

1.5498E-07 0.5000 0.5000 0.5000 0.5000

1.5498E-07 0.5000 0.5000 0.5000 0.5000

9.4861E-05 0.5505 0.4495 0.5458 0.4542

2.3053E-04 0.5778 0.4222 0.5646 0.4354

1.7232E-06 0.5208 0.4792 0.5207 0.4793

9.9195E-05 0.5778 0.4222 0.5776 0.4224

1.6558E-05 0.5208 0.4792 0.5182 0.4818

5.3418E-06 0.5208 0.4792 0.5204 0.4796

3.8783E-05 0.5505 0.4495 0.5502 0.4498

9.9195E-05 0.5778 0.4222 0.5773 0.4227

2.3053E-04 0.5778 0.4222 0.5478 0.4522

9.9295E-06 0.5000 0.5000 0.4989 0.5011

9.486IE-05 0.5505 0.4495 0.5405 0.4595

5.1316E-05 0.5208 0.4792 0.4964 0.5036

6.2522E-05 0.5001 0.4999 0.4892 0.5108 236

Table B-3. Cont'd

<*o,l do,2 Rs.true Rs.meas %ROj %R02 Q t>lW i

alW i

t>2w 2

a2W2

0.966 0.966 1 .2 2.272 1.543 1.543 9.9195E-05 0.5778 0.4222 0.5768 0.4232

0.966 0.992 1.5 1.755 0.788 0.197 1.7232E-06 0.5208 0.4792 0.5207 0.4793

0.967 0.967 1.3 2.006 1.225 1.225 3.8783E-05 0.5505 0.4495 0.5498 0.4502

0.967 0.869 1.3 1.301 0.453 1.811 1.3438E-06 0.5000 0.5000 0.4999 0.5001

0.968 0.998 1.4 2.164 1 .2 0 2 0.075 1.5856E-05 0.5505 0.4495 0.5505 0.4495

0.968 0.938 1 .2 2.267 1.429 2.858 9.9195E-05 0.5778 0.4222 0.5759 0.4241

0.969 0.984 1.5 1.501 0.380 0.190 1.5230E-08 0.5000 0.5000 0.5000 0.5000

0.969 0.939 1.4 1.636 0.725 1.449 5.3418E-06 0.5208 0.4792 0.5201 0.4799

0.969 0.558 1 .2 1.770 1.083 17.322 9.4861E-05 0.5505 0.4495 0.5278 0.4722

0.969 0.759 1.3 1.503 0.731 5.849 1.6558E-05 0.5208 0.4792 0.5153 0.4847

0.970 0.940 1.3 2.003 1 .1 0 1 2 .2 0 2 3.8783E-05 0.5505 0.4495 0.5491 0.4509

0.970 0.996 1.4 2.163 1.105 0.138 1.5856E-05 0.5505 0.4495 0.5505 0.4495

0.971 0.886 1 .2 2.259 1.315 5.262 9.9195E-05 0.5778 0.4222 0.5738 0.4262

0.971 0.998 1.3 2.466 1.367 0.085 4.2684E-05 0.5778 0.4222 0.5778 0.4222

0.971 0.986 1.5 1.755 0.671 0.336 1.7232E-06 0.5208 0.4792 0.5207 0.4793 237

Table B -3. Cont'd

do,l

c-i©•o

Rs.true

0.972 0.944 1.4

0.972 0.997 1.3

0.973 0.993 1.4

0.973 0.893 1.3

0.973 0.796 1.2

0.974 0.994 1.3

0.975 0.899 1.4

0.975 0.988 1.4

0.975 0.647 1.2

0.975 0.988 1.3

0.976 0.812 1.3

0.976 0.622 1.3

0.976 0.976 1.5

0.977 0.977 1.3

0.977 0.633 1.2

Rs,meas %ROi %RC>2

1.401 0.359 0.718

2.466 1.288 0.161

2.163 1.011 0.253

1.996 0.982 3.930

2.241 1.202 9.612

2.465 1.210 0.303

1.633 0.600 2.401

2.163 0.922 0.461

2.202 1.086 17.377

2.464 1.134 0.567

1.983 0.869 6.953

1.483 0.587 9.385

1.755 0.568 0.568

2.463 1.058 1.058

1.198 0.359 5.743

n blWi

alW i

*>2w 2

a2w 2

1.5498E-07 0.5000 0.5000 0.5000 0.5000

4.2684E-05 0.5778 0.4222 0.5777 0.4223

1.5856E-05 0.5505 0.4495 0.5504 0.4496

3.8783E-05 0.5505 0.4495 0.5475 0.4525

9.9195E-05 0.5778 0.4222 0.5695 0.4305

4.2684E-05 0.5778 0.4222 0.5776 0.4224

5.3418E-06 0.5208 0.4792 0.5193 0.4807

1.5856E-05 0.5505 0.4495 0.5503 0.4497

9.9195E-05 0.5778 0.4222 0.5599 0.4401

4.2684E-05 0.5778 0.4222 0.5775 0.4225

3.8783E-05 0.5505 0.4495 0.5444 0.4556

1.6558E-05 0.5208 0.4792 0.5089 0.4911

1.7232E-06 0.5208 0.4792 0.5206 0.4794

4.2684E-05 0.5778 0.4222 0.5772 0.4228

9.9295E-06 0.5000 0.5000 0.4978 0.5022 23

8


<*0,1 d0,2 **s,true

0.977 0.977 1.4

0.978 0.978 1.5

0.978 0.978 1.5

0.978 0.825 1.3

0.978 0.999 1.5

0.978 0.957 1.3

0.979 0.679 1.3

0.979 0.836 1.4

0.979 0.959 1.4

0.980 0.960 1.5

0.980 0.921 1.3

0.980 0.998 1.5

0.980 0.999 1.4

0.981 0.923 1.4

0.981 0.998 1.4

Rs,meas %ROi %R02

2.161 0.836 0.836

1.501 0.270 0.270

1.501 0.270 0.270

1.301 0.309 2.470

2.318 0.811 0.051

2.460 0.983 1.966

1.954 0.760 12.165

1.628 0.492 3.938

2.159 0.755 1.509

1.754 0.477 0.954

2.454 0.908 3.633

2.318 0.746 0.093

2.655 0.929 0.058

1.401 0.249 0.996

2.655 0.876 0.110

Q blWi

alWi

b2w2

a2w2

1.5856E-05 0.5505 0.4495 0.5501 0.4499

1.5230E-08 0.5000 0.5000 0.5000 0.5000

1.5230E-08 0.5000 0.5000 0.5000 0.5000

1.3438E-06 0.5000 0.5000 0.4998 0.5002

6.4826E-06 0.5505 0.4495 0.5505 0.4495

4.2684E-05 0.5778 0.4222 0.5765 0.4235

3.8783E-05 0.5505 0.4495 0.5373 0.4627

5.3418E-06 0.5208 0.4792 0.5178 0.4822

1.5856E-05 0.5505 0.4495 0.5496 0.4504

1.7232E-06 0.5208 0.4792 0.5204 0.4796

4.2684E-05 0.5778 0.4222 0.5752 0.4248

6.4826E-06 0.5505 0.4495 0.5505 0.4495

1.8367E-05 0.5778 0.4222 0.5778 0.4222

1.5498E-07 0.5000 0.5000 0.5000 0.5000

1.8367E-05 0.5778 0.4222 0.5777 0.4223 239


<*o,l <*o,2 **s,true

0.981 0.926 1.4

0.981 0.857 1.3

0.982 0.995 1.5

0.982 0.996 1.4

0.983 0.746 1.3

0.983 0.992 1.5

0.983 0.933 1.5

0.983 0.869 1.4

0.983 0.992 1.4

0.983 0.737 1.4

0.984 0.984 1.4

0.984 0.969 1.5

0.985 0.985 1.5

0.985 0.772 1.4

0.985 0.971 1.4

**s,meas %ROi %R02

2.155 0.677 2.707

2.441 0.834 6.669

2.318 0.685 0.171

2.655 0.825 0.206

2.415 0.759 12.139

2.317 0.626 0.313

1.752 0.398 1.592

2.146 0.602 4.817

2.655 0.774 0.387

1.616 0.399 6.387

2.653 0.724 0.724

1.501 0.190 0.380

2.317 0.570 0.570

2.128 0.531 8.491

2.651 0.675 1.349

Q blWi

alWi

*>2w2

a2w2

1.5856E-05 0.5505 0.4495 0.5487 0.4513

4.2684E-05 0.5778 0.4222 0.5725 0.4275

6.4826E-06 0.5505 0.4495 0.5505 0.4495

1.8367E-05 0.5778 0.4222 0.5777 0.4223

4.2684E-05 0.5778 0.4222 0.5666 0.4334

6.4826E-06 0.5505 0.4495 0.5504 0.4496

1.7232E-06 0.5208 0.4792 0.5200 0.4800

1.5856E-05 0.5505 0.4495 0.5467 0.4533

1.8367E-05 0.5778 0.4222 0.5776 0.4224

5.3418E-06 0.5208 0.4792 0.5146 0.4854

1.8367E-05 0.5778 0.4222 0.5774 0.4226

1.5230E-08 0.5000 0.5000 0.5000 0.5000

6.4826E-06 0.5505 0.4495 0.5503 0.4497

1.5856E-05 0.5505 0.4495 0.5425 0.4575

1.8367E-05 0.5778 0.4222 0.5770 0.4230 240

Table B-3. Cont’d

d0fl d0,2 Rs.true

0.986 0.770 1.3

0.986 0.972 1.5

0.986 0.889 1.5

0.986 0.946 1.4

0.987 0.999 1.5

0.987 0.896 1.4

0.987 0.900 1.4

0.987 0.949 1.5

0.987 0.998 1.5

0.988 0.997 1.5

0.988 0.820 1.4

0.989 0.909 1.5

0.989 0.994 1.5

0.989 0.820 1.5

0.989 0.957 1.5

Rs.meas %ROi %R02

1.300 0.208 3.321

2.315 0.516 1.032

1.749 0.329 2.631

2.647 0.626 2.502

2.845 0.630 0.039

1.401 0.171 1.365

2.639 0.577 4.613

2.312 0.465 1.858

2.845 0.596 0.075

2.845 0.562 0.140

2.621 0.528 8.443

2.307 0.416 3.324

2.845 0.528 0.264

1.742 0.269 4.308

1.501 0.132 0.528

n blW i

alWi

*>2W2

a2w 2

1.3438E-06 0.5000 0.5000 0.4996 0.5004

6.4826E-06 0.5505 0.4495 0.5499 0.4501

1.7232E-06 0.5208 0.4792 0.5191 0.4809

1.8367E-05 0.5778 0.4222 0.5761 0.4239

7.9032E-06 0.5778 0.4222 0.5778 0.4222

1.5498E-07 0.5000 0.5000 0.5000 0.5000

1.8367E-05 0.5778 0.4222 0.5744 0.4256

6.4826E-06 0.5505 0.4495 0.5494 0.4506

7.9032E-06 0.5778 0.4222 0.5778 0.4222

7.9032E-06 0.5778 0.4222 0.5777 0.4223

1.8367E-05 0.5778 0.4222 0.5707 0.4293

6.4826E-06 0.5505 0.4495 0.5481 0.4519

7.9032E-06 0.5778 0.4222 0.5777 0.4223

1.7232E-06 0.5208 0.4792 0.5174 0.4826

1.5230E-08 0.5000 0.5000 0.5000 0.5000

241

Table B -3. Cont’d

do,l do,2 ^s.true

0.989 0.989 1.5

0.990 0.840 1.5

0.990 0.980 1.5

0.991 0.963 1.5

0.991 0.931 1.5

0.991 0.862 1.4

0.992 0.874 1.5

0.993 0.941 1.5

0.995 0.921 1.5

Rs.meas %ROi %R02

2.844 0.495 0.495

2.294 0.369 5.897

2.842 0.462 0.925

2.839 0.430 1.720

2.834 0.398 3.182

1.401 0.115 1.845

2.822 0.366 5.851

1.501 0.091 0.726

1.501 0.062 0.986

n *>1Wi

alWi

*>2w2

a2w2

7.9032E-06 0.5778 0.4222 0.5776 0.4224

6.4826E-06 0.5505 0.4495 0.5456 0.4544

7.9032E-06 0.5778 0.4222 0.5773 0.4227

7.9032E-06 0.5778 0.4222 0.5767 0.4233

7.9032E-06 0.5778 0.4222 0.5756 0.4244

1.5498E-07 0.5000 0.5000 0.4999 0.5001

7.9032E-06 0.5778 0.4222 0.5732 0.4268

1.5230E-08 0.5000 0.5000 0.5000 0.5000

1.5230E-08 0.5000 0.5000 0.5000 0.5000

VITA

Name:

Bom:

Marital Status:

Children:

Formal Education:

Mark S. Jeansonne

May 12,1962 in Alexandria, Louisiana

Married

None

B.S., 1985, Louisiana State University

Major: Chemistry

Ph. D., 1990, Louisiana State University

Major: Chemistry (Chromatography)

Dissertation Title: " Chromatographic Peak Shape Analyis

and Modeling”, supervised by Professor Joe P. Foley.

2 4 3

DOCTORAL EXAMINATION AND DISSERTATION REPORT

Candidate: M ark S. J e a n s o n n e

Major Field: C h e m is try

Title of Dissertation: C h ro m a to g ra p h ic P eak S hape A n a ly s is and M o d e lin g

Approved:

M ajo r P ro fe sso r ai lha irm an

D ean of th e G ra d u a te School

EXAMINING COMMITTEE:

Date of Examination:

O c to b e r 15, 1990

chromatographic peak shape analysis and modeling

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