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Biologicals 37 (2009) 379e386www.elsevier.com/locate/biologicals

Basic principles of stability

William Egan a,*, Timothy Schofield b,1

a Pharmanet Consulting, 504 Carnegie Center, Princeton, NJ, USAb Merck Research Laboratories, 770 Sumneytown Pike, West Point, PA, USA

Received 14 July 2009; accepted 7 August 2009

Abstract

An understanding of the principles of degradation, as well as the statistical tools for measuring product stability, is essential to managementof product quality. Key to this is management of vaccine potency. Vaccine shelf life is best managed through determination of a minimumpotency release requirement, which helps assure adequate potency throughout expiry. Use of statistical tools such a least squares regressionanalysis should be employed to model potency decay. The use of such tools provides incentive to properly design vaccine stability studies, whileholding stability measurements to specification presents a disincentive for collecting valuable data. The laws of kinetics such as Arrheniusbehavior help practitioners design effective accelerated stability programs, which can be utilized to manage stability after a process change.Design of stability studies should be carefully considered, with an eye to minimizing the variability of the stability parameter. In the case ofmeasuring the degradation rate, testing at the beginning and the end of the study improves the precision of this estimate. Additional designconsiderations such as bracketing and matrixing improve the efficiency of stability evaluation of vaccines.� 2009 The International Association for Biologicals. Published by Elsevier Ltd. All rights reserved.

Keywords: Vaccine potency; Degradation; Least squares analysis; Minimum release specification; Arrhenius equation; Annual stability

1. Introduction

All molecular entities, including all vaccines, will degradeover time, resulting in the case of vaccines in a loss of potency.Under particular conditions of storage, this loss in potency overtime must be considered in setting the vaccine’s shelf life and itsassociated specifications at the time of lot release. The questionthat arises is: how should release specifications and shelf life bebest established? In this regard, it is necessary to consider thecharacterization and consequences of uncertainty and todevelop stability study designs and statistical models that bestdeal with the uncertainties; in particular, it is important to assessand appreciate the variability that is often inherent in thepotency assays that are used to characterize stability. Although

* Corresponding author. Tel.: þ1 609 951 6635.

E-mail addresses: wegan@pharmanet.com (W. Egan), timothy.schofield@

gsk.com (T. Schofield).1 Present address: GlaxoSmithKline, 2301 Renaissance Boulevard, King of

Prussia, PA 19406, USA. Tel.: þ1 610 787 3194.

1045-1056/09/$36.00 � 2009 The International Association for Biologicals. Publi

doi:10.1016/j.biologicals.2009.08.012

the storage temperature is a variable in setting vaccine shelf lifeand release specifications, for practical purposes, currently,most vaccines are stored at approximately 5 �C.

2. The boundaries for lot release and shelf life

Vaccines are not administered with arbitrary potency, butrather within a defined range that is established through clin-ical studies. The lower limit for potency is generally deter-mined as the minimum dose that is acceptable for vaccineefficacy; the upper limit is generally determined as themaximum dose that is acceptable for vaccine safety. Duringthe vaccine’s shelf life, from the time of release to its end-expiry date, it must remain within this clinically establishedpotency range.

2.1. Setting the vaccine’s shelf life

Various paradigms have been used for setting a vaccine’sshelf life. The simplest has been to monitor the vaccine’s

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380 W. Egan, T. Schofield / Biologicals 37 (2009) 379e386

potency as a function of time, with the shelf life beingestablished as the time of the last within specification potencyvalue, as illustrated in Fig. 1. Such a paradigm may be termeda ‘‘compliance model’’ due to the use of compliance to spec-ification as the basis for shelf life determination.

There are certain problems attending the use ofa ‘‘compliance model.’’ The first is that this model does notprovide an optimal estimate of the vaccine’s true shelf life. Inessence, with a ‘‘compliance model’’ the shelf life is set bya single time point, whose value is a composite of the actualloss in potency together with the variability of the assay. Inmany instances, the variability in the assay exceeds the trueloss in potency, thus uncoupling shelf life determination withdegradation. In a ‘‘compliance model’’, the other time pointmeasurements, which also contain information about thevaccine’s potency as a function of time and the variance of theassay, are disregarded. The second, and related problem, is thatthis method, being based on a single data point, does notprovide a measure of confidence in the established shelf life. Itdoes not allow an answer to the questions: how reliable is thisestimated shelf life? How certain are we of the potency of thevaccine at any point in time? A final, and probably moresignificant problem, is that a ‘‘compliance model’’ discouragesdata collection. If additional data are collected, the probabilityincreases that one or more of these additional data points, dueto assay variability, will lie outside the specification range.This increased probability of failure of additional measure-ments to comply with specification is called statistical multi-plicity. The impact of statistical multiplicity can be illustratedas follows. Suppose the true potency of a vaccine is at a levelsuch that the probability that any single measurement will fallbelow the lower limit due to assay variability is 5%. Supposefurther that the vaccine is perfectly stable, and thus thisprobability remains constant with each measurement. If twomeasurements are made on the vaccine the probability thatboth measurements will remain within specification is equal to(1 � 0.05) � (1 � 0.05) ¼ 0.95 � 0.95 ¼ 0.90. Thus theprobability that one or both of the measurements will fallbelow specification is 1 � 0.90 ¼ 0.10 (10%). This can begeneralized to the case of n-measurements:

Fig. 1. An illustrative ‘‘compliance model’’ for setting a vaccine shelf life. The

vaccine potency is monitored over time until the potency measurement falls

below the pre-established allowed lower limit (LL), the point illustrated in the

figure by the symbol, . The time for which the last potency determination

was within specifications (between the allowed upper limit (UL) and lower

limit) is considered to be the product’s shelf life.

Probðone or more failuresÞ ¼ 1� ð1� 0:05Þn ð1Þ

Table 1 shows the probabilities of failure for differing numbers ofmeasurements. Seven (7) measurements are typically made ina long-term stability study with a lot of vaccine (0, 3, 6, 9, 12, 18,and 24 months). The table shows that, in the case of 7 measure-ments, there is a 30% chance that one or more measurements willfall below specification throughout the study, even though therewas no loss of potency. This is further compounded when thereare multiple lots in the study. The failed measurements couldoccur any time throughout the study. Thus a ‘‘compliance model’’has no relationship to shelf life determination, while one ispenalized for collecting more than a minimal data set.

An alternative paradigm for the establishment of a shelf lifederives from the use of regression analysis (least squaresanalysis). A least squares analysis provides a measure of thequality of the fit of the data to the model, as well as one’sconfidence in the answer, expressed as a standard deviationand confidence interval. Additionally, with regression analysis,more data provides a better (more precise) estimate of theshelf life of the vaccine; there is a reward, not a penalty, forcollecting an increased amount of dataTable 2.

A least squares fitting of the data requires, however,a mathematical model to which the data can be fit. Thismathematical model should derive from a plausible kineticmodel. The simplest plausible kinetic model for vaccinedegradation would be one based on first order kinetics. Fora first order process, the potency of the vaccine follows a ratelaw of the form,

PðtÞ ¼ P0 eð�ktÞ ð2Þ

where,

P(t) is the potency of the vaccine at time, t.P(0) is the initial potency of the vaccine, i.e., the potency attime ¼ 0.k is the rate constant for the loss of potency.e is the base of the natural logarithm.

Eq. (2) may be linearized by taking the logarithm of both sidesof the equation. Thus,

ln PðtÞ ¼ ln Pð0Þ � kt ð3Þ

Table 1

Probability of failing one or more tests when multiple measurements are made

of a lot of vaccine.

No. measurements Probability %

1 0.05 5

2 0.10 10

3 0.14 14

4 0.19 19

5 0.23 23

6 0.26 26

7 0.30 30

Table 2

ln TCID50 values for a hypothetical live-attenuated viral vaccine.

Time (months) ln TCID50

0 8.143

1 8.246

3 8.196

6 8.162

9 8.145

12 8.222

15 8.115

18 8.093

21 7.900

24 8.142

27 7.771

30 7.781

36 7.885

381W. Egan, T. Schofield / Biologicals 37 (2009) 379e386

Regression analysis (linear least squares fit) of a [time,potency] data set will provide the least squares estimates forthe slope (rate of degradation, k) and the y-intercept, (the valueof ln[potency] at time zero) of the line. A regression analysisexample, using synthetic data (see Table 2), is provided inFig. 2. In this example, the natural log of TCID50 values fora hypothetical live-attenuated viral vaccine is measured asa function of time. The equation of a straight line is fit to thedata to yield estimates for the rate of loss in potency (the slopeof the regression line) and estimated ln[potency] at the time ofrelease (the y-intercept). As with all statistical estimates eand, indeed, all measurements e the least squares estimateshave an uncertainty associated with them. This uncertaintymay be expressed, inter alia, as a confidence interval on theregression line (see Fig. 2); one may also calculate a confi-dence limit for the slope (the degradation rate). The confidenceinterval represents the range in which the true line or the trueslope is likely (i.e., with a specified degree of confidence) tofall. As discussed previously, this range is impacted by theamount (and the spread) of the data. The confidence intervalbecomes narrower, and thus the estimate becomes moreprecise, with more data. Note that the 95% confidence interval(Eq. (4)) of the regression line is bowed; this does not mean

Vaccine Stability

0 12 24 36

7.6

7.8

8.0

8.2

8.4

Time (months)

ln

T

CID

50

ln (Potency) = 8.24 – 0.0116 × time

Fig. 2. Linear least squares fit of data from Table 2. The 95% confidence limit

(dashed line) about the regression line is shown.

that the regression line may be non-linear, but rather that theprecision along the line is greatest (least variable) in the centerof the line.

CIðtÞ ¼by� t0:025;df,s,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

nþ�t� tavg

�2

SXX

sð4Þ

where

by¼baþ bb,t;ba¼estimated intercept,bb¼estimated slope,t0.025,df¼critical value from at t-distribution n,df¼degrees of freedoms¼stdev about regression,tavg¼average of time intervals,SXX¼sum of squared deviation from tavg.

Having calculated the regression equation, and assuming thatthe loss in potency remains linear, it is a simple matter toextrapolate to later times. However, there is a penalty forextrapolation, as the confidence limits on the extrapolatedvalues become wider, the further one predicts from the centerof the data (Fig. 3). In addition, there is an assumption that thedegradation process remains linear; that is, that there is nochange in mechanism. Such an assumption is not alwaysjustified. For these reasons, extrapolation may not be war-ranted, and can lead to inaccurate conclusions about thepotency at extended time intervals.

Measures of ‘‘goodness of fit’’ such as r-square are some-times used to verify the linearity of the data. R-square is notsufficient, however, to detect subtle departures from linearitythat are commonly encountered in vaccine stability. Thepotential for curvilinear kinetics, such as may occur when thedegradation rate changes throughout the course of shelf life, isbetter assessed using residual plots of the least squares fit(http://www.itl.nist.gov/div898/handbook/pmd/section4/pmd44.htm). Residual plots associated with linear kinetics aswell as curvilinear kinetics (exponential decay) are presentedin Fig. 4.

In a similar fashion, a confidence interval can be associatedwith the slope; the expression for the CI of the slope is givenin Eq. (5).

bb� ta;n�1,sffiffiffiffiffiffiffi

1SXX

qð5Þ

A confidence interval on the slope that excludes zero indicatesa statistically significant rate of degradation (P � 0.05).

With this background in hand, we can now consider anapproach to setting a vaccine shelf life and associated potencyrelease specification. As mentioned previously, the two valuese shelf life and release specifications e are intertwined.Release specifications are set in order to assure product quality(e.g., adequate potency) throughout the vaccine’s shelf life.These values, namely, the lower limit of potency at the time oflot release and the shelf life over which the vaccine is expected

Least-squares Fit with Extrapolation

0 12 24 36 48 606.5

7.0

7.5

8.0

8.5

9.0

Time (months)

ln

T

CID

50

Fig. 3. Least squares fit of the stability data provided in Table 2 along with the

associated two-sided 95% confidence limits. The regression line (and the

confidence limits) have been extended out to 60 months.

382 W. Egan, T. Schofield / Biologicals 37 (2009) 379e386

to maintain a potency above the clinically established lowerlimit, may be set in the following manner; see Fig. 5. Considerthe rate of loss of potency (the regression line, illustrated asa solid line) and the associated uncertainties in the slope of theregression line and the measured potency at the time of release(the solid upward arrow); these provide a shelf life as the pointwhere the regression line crosses the lower allowable potencyvalue, represented by the dashed line in the figure and a lowerpotency value at the time of release (represented as the dia-mond in Fig. 5). It should be pointed out that this represents

Synthetic Data + NoiseLinear Fit to Exponential

25 50 75 100-25

0

25

50

75

100

Time

Co

ncen

tratio

n

Transformed Simulated DataLinear Fit

0 25 50 75 1001

2

3

4

5

Time

Co

nc

en

tratio

n

a

c

Fig. 4. Linear fit to an exponential decay curve (a); with a residual plot showing a no

(c); and a residual plot showing a random scatter about zero (d).

only one possible [shelf life, release value] set. It is possiblethat a longer shelf life may be obtained if one releases ata higher potency value or, conversely, if the vaccine is releasedat a lower potency value, the allowable shelf life would beshorter. The upper limit on the allowable potency value atrelease (and, hence, the controlling factor for the shelf life) isset by clinical safety considerations (the allowable upper levelof potency as determined during clinical trials). The potencyvalue at lot release may also be determined by manufacturingcapability. From the lower value of manufacturing capability,the uncertainty in potency from the lot release assay andregression analysis may factor in and the allowable shelf lifethen determined.

3. The effect of temperature on the rate of loss of potency

The temperature dependence of the rate of a chemicalreaction, and hence the rate of loss in vaccine potency, isgenerally described by the Arrhenius equation. According tothis equation, the rate of a reaction at any temperature isa function of two parameters, the energy of activation,designated Eact, and a pre-exponential factor (sometimestermed the frequency factor), designated A. Values of Eact andA are characteristic and approximately constant for a particularreaction. Thus,

krate ¼ A expð�Eact=RTÞ ð6Þ

where

Residuals from Linear Fit to Exponential

0 25 50 75 100

-20

-10

0

10

20

Time

Resid

uals

Residuals from Fit toTransformed Data

0 25 50 75 100-0.4

-0.2

0.0

0.2

0.4

0.6

Time

Resid

uals

b

d

n-random (curvilinear) scatter about zero (b); ln transformed data with linear fit

0 6 12 18 24 30

Time (Months)

Po

ten

cy

Regression Loss

Release Assay + Regression Loss Uncertainty

Release

24-Month Expiry

Lower Limit

22ˆ1,ttimeb̂Spec

yVariabilitCombined

Release

Assaybn ss

LossSpec

+⋅+•+=

++=

−α

Fig. 5. Pictorial model for setting lot release and shelf life (time) specifications based on the least squares estimated potency loss (and the uncertainty in that

estimate) and lot release assay variability.

383W. Egan, T. Schofield / Biologicals 37 (2009) 379e386

A is the pre-exponential (frequency) factor.Eact is the energy of activation.R is the universal gas constant.T is temperature (degrees Kelvin).

Eq. (6) may be linearized by taking the logarithm of both sidesof the equation, providing a linear relationship between thelogarithm of the rate and (1/T ); by determining the rate ofreaction at two or more temperatures, values of Eact and A maybe calculated. Thus, a plot of the log of the reaction rate, ln(k)versus (1/T ), allows one to determine Eact and A, as illustratedin Fig. 6. The slope of the regression line is Eact/R and theintercept is ln(A). From a knowledge of Eact and A, one maythen predict the reaction rate at any temperature e forexample, at a lower temperature (e.g., at 10 degrees as inFig. 6).

As can be deduced from the Arrhenius equation, recog-nizing that energies of activation are positive, reactionsproceed more rapidly at higher temperatures. Because thetemperature is contained in the exponential portion of Eq. (6),small changes in the temperature can have dramatic effects onthe rate, as is illustrated in Table 3 for a first order reaction(note that the units of A depend on the order of the reaction,and are s�1 for a first order reaction).

1000/T

50 oC

30 oC

10 oCln(k

)

( ) ⎟⎠⎞⎜

⎝⎛⋅+=⎟

⎠⎞⎜

⎝⎛⋅−=

T

1ba

1)ln(ln

TR

EAk act

T

Fig. 6. Hypothetical Arrhenius plot (with rates determined at 55, 45, 35, and

25 �C) showing extrapolation to 5 �C (red circle).

The principle utility of studying reactions at highertemperatures (and evaluating the Arrhenius parameters, Eact

and A) than those intended for storage is that the reactionshappen more quickly; thus, there is a greater observablechange per unit of time that will be observed at highertemperatures than at a low temperature. This is illustrated bythe example calculations that are presented in Table 3, where itcan be seen that the rates of the reaction at 55 �C (328 �K) areapproximately 2000e3000 times those at 5 �C (278 �K). Thisbehavior is useful in developing vaccine formulations or indemonstrating product comparability after a manufacturingchange. Vaccine formulations are developed to maximizevaccine stability; one can study various formulations at hightemperature and select those that are predicted to offer the beststability at the intended storage condition. Followinga manufacturing change, a demonstration of comparableArrhenius parameters (or, simply, reaction rate at a highertemperature) is indicative of comparable products and may bedetermined more rapidly than, for example, carrying out realtime stability studies at the intended storage temperature.Additionally, as noted previously, knowledge of the Arrheniusparameters permits one to estimate the rate and extent ofchange at any given temperature. Thus, one can determine theeffect of a particular temperature excursion, as might occurduring shipping or storage, on vaccine stability.

In principle, the high temperature data may be extrapolatedto the intended storage temperature to calculate a shelf life. Asmentioned previously, there are two major reasons why, inpractice, this is not generally done. The first is that formolecular systems Arrhenius behavior is only an

Table 3

Reaction rate constants as a function of temperature for two values of the

activation energy; the pre-exponential factor was set at 1 � 1012 s�1.

Temperature (�K) Eact ¼ 120 kJoules (day�1) Eact ¼ 110 kJoules (day�1)

278 2.4 � 10�6 1.8 � 10�4

288 14.7 � 10�6 9.5 � 10�4

298 79.5 � 10�6 45 � 10�4

308 380 � 10�6 190 � 10�4

318 1640 � 10�6 734 � 10�4

328 6650 � 10�6 2680 � 10�4

384 W. Egan, T. Schofield / Biologicals 37 (2009) 379e386

approximation, albeit a good approximation. There are devi-ations from true linearity. The second, and perhaps morecritical, is that the determined Arrhenius parameters containerror e they are estimates e and the confidence limits aboutthe regression line in an Arrhenius plot become increasinglywider the further the plot is extrapolated. This is illustrated inFig. 7. At 5 �C, the 95% confidence interval spans anapproximately 30-fold difference in rate constants; such anuncertainty would be unacceptable in setting a shelf life.

4. Annual stability studies

Regulatory agencies generally require e or stronglyencourage e manufacturers to have an annual stabilityprogram in place. In an annual stability program, at least onelot of each vaccine that is manufactured, as well as eachpresentation of that vaccine, is randomly chosen to be placedon stability. The goal of such a stability program is, presum-ably, to assess product and manufacturing process consistency.Given such a goal, two questions immediately arise: the first is‘‘what is the appropriate stability metric for monitoringconsistency?’’ and the second is ‘‘what is the best design forattaining that goal?’’. A ‘‘compliance model’’ approach iscommonly taken to annual stability studies; the vaccine ismonitored over the shelf life to determine whether the vaccineremains within specification at each selected time point.

However, an alternate suggestion for the best metric for anannual stability study is the slope of the stability regressionline, i.e., the rate of degradation of the vaccine (loss ofpotency). One may assess the equivalence of the slopes (withinsuitable boundaries). Assuming that it is the slope of theregression line that is desired and that the degradation processis linear, it can be shown that for a fixed number of data points,the error in the slope is minimized by taking half the datapoints at the beginning and half the data points at the end.

The error in the slope of a regression line, sb, is given byEq. (7):

Fig. 7. Illustrative Arrhenius plot: ln(k) versus 1/T. A least squares fitting of the

data allow estimates of Eact and A, from which reaction rates may be predicted

at differing temperatures.

sb ¼ s

ffiffiffiffiffiffiffiffiffiffi1

SXX

r; ð7Þ

where s¼standard error of the regression, and

SXX¼ S x2i � ðS xiÞ2=n

We may use Eq. (7) to calculate the error in the slope fordifferent selections of time points for a vaccine with a 36-month shelf life. Let’s consider the following four cases, eachhaving 8 data points:

Case 1. ICH: (0, 3, 6, 9, 12, 18, 24, and 36 months).Case 2. Symmetric: (0, 6, 12, 18, 21, 24, 30, and 36 months).Case 3. The ends: (0, 0, 0, 0, 36, 36, 36, and 36 months).Case 4. Extended: (0, 0, 0, 0, 48, 48, 48, and 48 months).

Assuming that the standard error of the regression is equalto 1, then the errors in the slopes may be calculated and are:

Case 1. 0.0315.Case 2. 0.0314.Case 3. 0.0196.Case 4. 0.0147.

As can be seen, the error in the slope is minimized by takingthe measurement points at the ends and, in this particular case,comparing Case 1 and Case 3, can be seen to be 38% less thanwhen the time points are selected as suggested by ICH. In fact,further precision can be achieved by extending the duration ofthe study. The reduction in variability is 53% when measure-ments have been made in an extended range (48 months) fromshelf life. Our main point here is that due consideration shouldbe given to the placement of the time-points.

A further suggestion can be made to better evaluate productand manufacturing consistency and that is to measure stabilityat an elevated temperature where the extent of change per unittime is greater. When considering errors in potency measure-ments, evaluating consistency at an elevated temperatureallows for a better assessment of whether there is a change inslope, or not. Allowable differences (e.g., the allowablemargins in equivalence testing) in slopes would need to beestablished on a case-by-case basis.

Other design factors could be incorporated into the stabilityprotocol. For example, if the inter-session precision of theassay is significantly more than that of any single session, thenall time point samples could be stored frozen and analyzed atone time (providing, of course, that there is no loss of potencyon freezing and subsequent thawing). However, as mentionedbefore, the determination of the optimal experimental designdepends on the goals and purposes of the experiment. In thisregard, Guidelines from the Regulatory Agencies would bewelcome.

USLLSL

Fig. 8. Areas under normal curves representing risk of falling outside the

specification range (LSL, USL) for a more variable distribution (blue) and less

variable distribution (red).

385W. Egan, T. Schofield / Biologicals 37 (2009) 379e386

5. Additional statistical considerations

Statistical considerations in evaluation of vaccine stabilityrelate to the goals of each study, which in turn determine theoptimal design and analysis of the study. Following are someprinciples that should be adhered to in order to obtain reliableinformation regarding vaccine stability.

1. The goals of stability evaluation of vaccines change overthe life cycle of the product. The goal during earlydevelopment is to estimate the potency of materials usedduring the course of a clinical study, whereas in latedevelopment it is shelf life and release potency determi-nation. The goal after license approval is maintenance ofproduct stability. The design and analysis of vaccinestability studies should strategically address the goal of thestudy at that phase of development or commercialization.Generally the study should be designed to achieve the best(most precise) estimate of the appropriate stabilitycharacteristic.For example, studies supporting clinical programs shouldbe designed to obtain precise estimates of vaccine potencythroughout the course of the clinical study. Stabilitymeasurements should be concentrated over the period ofuse of the clinical program, and regression should beutilized to obtain precise estimates of the clinical material.Studies supporting shelf life release potency determinationshould be designed to obtain precise estimates of thedegradation rates at key temperatures. As discussedbefore, this is ideally accomplished by making measure-ments at the beginning and the end of the storage period,and beyond. Likewise studies supporting product

Time 0 1 3 6 9 12 18 24

Lot 1 X X X X X

Lot 2 X X X X

Lot 3 X X X X X

Lot 4 X X X X

Fig. 9. Matrix design with 4 lots of vaccine tested in fewer numbers of assays

(75%) than the full design using 3 lots.

maintenance post licensure should be designed to obtainthe best estimate of the stability characteristic, such as theslope, and thereby provide a measure of manufacturingand product consistency. Accelerated stability should beutilized whenever possible, to obtain timely and preciseinformation regarding vaccine product stability.

2. Lower variability is associated with lower risk. This ismost simply illustrated using release testing, as in Fig. 8.Here the areas outside the lower specification limit (LSL)and the upper specification limit (USL) represent the riskof an out-of-specification (OOS) result. The areas associ-ated with the more variable curve are greater than the lessvariable curve. Thus, lower variability of release potencymeasurements translates into lower risk of an OOS result.The variability is managed through design; number ofreplicates in a release assay, and number and duration ofstability measurements.

3. By contrast, and as discussed previously, there is a disin-centive to collecting more measurements, and thereforedesigning a stability study to reduce the variability of thestability parameter of interest, when individual measure-ments are held to specifications. The goals of stabilityevaluation of vaccines should be directed towards thestability characteristics of intermediates and final product.Those characteristics are best represented by a statisticalmodel such as a line, a non-linear model, or a parameter ofstability such as the slope. Stability should not be evalu-ated by holding stability measurements to specifications.The individual stability measurements are strictly a meansto an end, similar to individual release measurements,which are averaged to obtain a release potency.

4. Statistical conventions such as matrixing (ICH Q1D)provide an effective way to balance efficiency and repre-sentativeness. Fig. 9 illustrates a matrix design using 4rather than 3 lots of vaccine, which has approximately thesame precision as a full design in 3 lots, with fewernumbers (75%) of stability measurements.

6. Conclusions

Key to a proper assessment of vaccine stability is a back-ground in the fundamentals of degradation and clear vision ofthe goals of stability evaluation throughout the vaccine lifecycle. During vaccine development the goals of stability are tobetter understand the properties of clinical materials (i.e., toknow what the subjects received), and to obtain precise esti-mate of stability parameters in order to establish an appro-priate shelf life and release strategy for the product.

The fundamentals of degradation start with understandinghow vaccines degrade, and establishing a model, whichapproximates the degradation process. Once a model has beenestablished, statistical methods can be utilized to fit the modelto data from a suitably designed stability study. For mostvaccines, a first order kinetic model is an accurate represen-tation of the degradation process, and linear least squares isutilized to fit this model to the data. The accuracy can be

386 W. Egan, T. Schofield / Biologicals 37 (2009) 379e386

assessed using standard tools such as residual analysis. Oncesatisfied that the model is a good fit to the data, additionalstatistical calculations provide an assessment of the uncer-tainty in the estimate of the stability parameter. Confidenceintervals on the line, or on the slope of the line, reveal wherethe true parameter is likely to fall.

The effect of temperature on the rate of loss of potency ofa vaccine is generally described by the Arrhenius equation.The Arrhenius principle can be used to: (1) screen develop-ment formulations to optimize the vaccines stability; (2)compare stability of materials made by a modified process orin a new facility; and (3) monitor stability post licensure. Thisprovides a rapid means of assessing vaccine stability, whichcan be verified through follow up long-term studies. TheArrhenius equation is generally not used to forecast shelf lifeor derive release specifications due to the unquantifiable risksassociated with the assumptions of the method.

Statistical considerations in the evaluation of vaccinestability speak to variability and the associated risk of inter-preting the data. The variability associated with estimating

stability parameters, and thereby their uncertainty, is associ-ated with the risk of drawing the wrong conclusions from thestudy. A fundamental underlying principle of stability evalu-ation is that measurements are collected over time to estimateimportant stability parameters, not to assess compliance ofstability measurements to specifications. With this under-standing stability studies can be designed with a suitablenumber of replicates and time intervals to reduce the vari-ability of a stability parameter, and thereby its uncertainty. Thenet result is reduced risk to the manufacturer and customeralike. Measuring uncertainty together with a stability param-eter puts the onus upon the manufacturer to use designs, whichreduce variability and risk. This includes using efficient designsuch as matrix designs, which can be formulated to includemore information (such as more lots) at a reduced burden tothe testing laboratory.

Understanding the basic principles of vaccine stability,including the statistical properties of stability evaluation, willassist manufacturers and regulators, to bring high quality lifesaving vaccines to the worldwide community.