computational model of tissue …ut27/computational modelling of...school of biomedical engineering,...

S c h o o l o f B i o m e d i c a l E n g i n e e r i n g , S c i e n c e a n d H e a l t h S y s t e m s ,

D r e x e l U n i v e r s i t y

COMPUTATIONAL MODEL OF

TISSUE INTERACTION WITH

T-RAYS

BY,

ANUP UMRANIKAR (11502070), AND

UDAYKIRAN THUMMALAPALLI (11486932)

SUBMITTED TO

DR. BAHRAD SOKHANSANJ

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1. PROBLEM STATEME�T

Electromagnetic waves having sub-millimeter wavelength are called Terahertz (THz) waves or

T-rays. They usually lie in the electromagnetic spectrum between 100 GHz and 100 THz.

Interaction of electromagnetic waves with biological matter at terahertz frequencies causes

change in the behavior of the waves. This frequency region interests because the wavelength of

the waves is comparable to the size of biological molecules and because they obey the laws of

optics. Due to the difference in the conductivity and permittivity of the tissues, the system

behaves in a nonlinear and has complex behavior. One of the most impressive features of THz

imaging is that different molecules have specific interactions with a THz beam. Thus, THz

beams can be used to molecularly distinguish the material they traverse. This is the basis of

application in the field of imaging.

Refractive index, being the measure of change in speed of the wave based on phase velocity, is a

factor which influences T-rays. The existing imaging technologies, such as X-rays and MRI, are

formulated based on the tissue density and proton density respectively. T-rays also being

electromagnetic in nature can be predicted to show the same phenomenon. This paper introduces

modeling of the interaction of T-rays with characterized tissues based on density.

2. BACKGROU�D

The THz range of frequencies is a new frontier in the field of imaging. This frequency range is

the borderline between microwave electronics and photonics and exists in the frequency bands of

molecular and lattice vibrations in gases, fluids and solids.

Since THz radiation easily passes through thin layers, it can detect and image the internal

structures. Using computer graphics, it is possible to generate 3-dimensional map of the

biological object.

2.1 Generation of T-rays

With the advent of femto-laser, the generation of electromagnetic waves with higher frequency

(100 GHz to 100 THz) has been possible.

THz pulses can be generated by means of irradiation of photoconductive antennas,

semiconductor surfaces, and quantum structures with femtosecond optical structures.

Using a coherent detection system, THz-pulsed imaging is implemented via the pump and the

probe technique used in optical spectroscopy. An ultrafast pulsed laser such as Titanium sapphire

laser is split into two beams, the pump and the probe. The pump beam is used to generate THz

pulses, whereas the probe beam is used as a reference [1].

A voltage biased photoconductor antenna or a crystal having high nonlinear susceptibility are

eliminated with femtosecond infrared lasers to produce THz rays. The THz beam is collimated

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and focused to a spot (0.5mm in diameter) using parabolic and hyperbolic mirrors. The subject

placed at the focal point casts a portrait which is dependent on the measured amplitude of the

THz electric field after interaction with the subject. The probe beam is delayed using an optical

Figure 1: Transmission Mode of THz Imaging (Courtesy: Michael Herrman et al, Terahertz

Optoelectronics)

Figure 2: Reflection mode of THz Imaging (Courtesy: Michael Herrman et al, Terahertz Optoelectronics) delay stage, before the coherence detection. Faster image acquisition can be achieved with multi-

element array detectors such as charge coupled devices and frame grabbers. The change to the

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acquired pulse is dependent on the materials through which the pulse has been propagated and

reflected.

2.2 Transmission and Reflection Imaging

The setup shown in Figure 1 is a transmission mode of operation. T-rays pass through the object

and modulated waves generate an absorption spectrum. This is the traditional way used by major

imaging modalities. This is termed as transmission imaging.

Water has high T-ray absorbance. Because of this characteristic, the penetration depth into living

tissue is just a few hundred microns, far too small for imaging of internal systems. Reflection

imaging, which uses a different geometry, can give detailed study of surface or near-surface

properties of a wide range of materials, including biological tissue [2]. Figure 2 is a

representation of reflection imaging. Because of the short coherence length of the THz radiation,

the reflection geometry presents a number of new imaging possibilities, including that of THz

tomography. The time of flight measurements in reflection imaging work in a principle similar to

ultrasonic imaging. A common problem with acoustic imaging is impedance matching between

air and solid objects. The dielectric properties of many materials are not much different from air,

so an index matching scheme is not usually required.

2.3 Amplitude and Phase Imaging

There is a change in the phase and amplitude of the signal which branches this imaging modality

into amplitude and phase imaging. The display modes based on the THz electric field detected

after passage through matter are characterized into amplitude and phase imaging based on

maximum magnitude and phase change in the signal.

By translating the object and measuring the transmitted THz signal for each position of the

object, one can build an image pixel by pixel. To detect a pin in a soap cake, the heterogeneity

along the path through the pin has a different response in contrast to the adjacent pixels.

The solid structure with heterogeneous composition gives a different response based on its

structure (high contrast exhibited by heavy materials). If the object has a cumbersome shape, the

‘transit time’ for the waves to reach the detector is slightly different. This is observed by

measuring the phase difference between two adjacent rays. Phase imaging determines the

thickness of the object at different points. The change in transit time ∆t is given by ∆τ = (1/c) ∫ n (z) dz, where n (z) is the refractive index sampled by the THz beam along its optical path, and

where the integral is taken along the path. For example, embossed letters on a plain surface cast a

phase change in the signal below the embossments.

2.4 Permittivity

When a material has parameters such as permittivity, conductivity, permeability, varying as a

function of frequency, it is said to be dispersive. When the permittivity has a non-zero imaginary

part (complex), it exhibits losses and is said to be dissipative. The Kramer-Kronig equations

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relate the real and imaginary parts of the permittivity through an integral over the entire

frequency range. Water, the major constituent in most tissues, is sometimes called ‘biological

water’. It is difficult to distinguish between bulk tissue water and bulk water. The permittivity

undergoes an almost monotonous decrease over the entire frequency range. The major dispersion

regions where the value of permittivity is varying strongly with frequency are alpha (kHz), beta

(100 kHz) and gamma (GHz) [4]. The alpha dispersion also results from active membrane

conductance phenomenon, charging of intracellular membrane-bound organelles that connect

with the outer cell membrane and perhaps frequency dependence in the membrane impedance

itself. Beta dispersion occurs at radiofrequencies. It arises principally from the capacitive

charging of cellular membranes in tissues. A small contribution might also come from dipolar

orientation of tissue protein at high RF. Gamma dispersion occurs due to bipolar relaxation of

water, which accounts for 80% of most soft tissues.

2.5 Conductivity

At low frequencies, a cell can conduct poorly compared to the surrounding electrolyte. The

extracellular fluid is only available to the current flow. Conductivity of soft, high water content

tissues at low frequencies is typically 0.1 or 0.2 S/m. Conductance of a material changes with

respect to a fraction of extracellular fluid, which is dependent upon physiological changes in the

cell. At higher frequencies, the cellular membranes are largely shorted out and they conduct

current. The tissues are electrically equivalent to suspensions of non-conductive protein in

electrolyte. The conductivity reaches saturation at higher frequencies.

2.6 Refractive Index (Addendum to original Part I)

Refractive index is property of an object to reduce the speed of light (electromagnetic waves) in

the medium. It can be experimentally deduced by,

Where εr and µr are the relative permittivity and permeability repetitively. For most materials, µr

is close to 1 so,

Dispersion is a property of a material whose refractive index (permittivity indirectly) changed

with increase in frequency. There are three types of dispersions in different frequency range -

alpha (~KHz), beta (~100 KHz), and gamma (~ GHz) dispersions. Charging of intracellular

membrane-bound organelles that connect with the outer cell membrane has frequency dependent

membrane impedance which is observed as alpha dispersion at very low frequencies. At high

RFs, dipolar orientation of the tissue proteins and capacitive charging of cellular membranes

contribute to beta dispersion. Gamma dispersion occurs due to dipolar relaxation of the water,

which corresponds to 80% of the most soft tissue volumes. Terahertz frequencies show a high

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effect of the gamma dispersion and high related to the content of the water in the tissue. These

waves can contrast the tissues based on the complex permittivity (refractive index) of the water

content. The polarization changes in the material causes a complex refractive index term with

real part and complex parts as refractive index and extinction coefficient (k), absorption loss in

the material [5].

So, the complex permittivity can be deduced as

(1)

(2)

εr’ and εr’’ can be calculated from the Kramer Kronig’s relation, while integrating over the entire

frequency range as

where is the permittivity at infinity,

is the permittivity at zero frequency and

is the angular frequency.

And hence, n can be rewritten as a function of angular frequency ω. This paper discusses the dependency of the refractive index on density of the material exciting different materials at same

frequency (THz) [8,9].

2.7 Polarization (Addendum to original Part I)

The relative permittivity of biological tissues decreases as frequency increases. Three basic

phenomenons exist in the dielectric characterization of a tissue – Dipolar orientation, Interfacial

relaxation, Ionic diffusion. These relaxation processes are responsible for the dielectric

properties of the tissues.

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An individual atom or molecule has negative and positive charges irrespective of its dipole

moment. This can be modeled as a harmonic oscillator, taking energy from the EM field, at the

resonant frequency if the field perturbs. This is also called as optical or electronic polarization.

The surrounding dipoles creates a field adds up a dipole density to the equation.

Dipolar orientation: The alignment of the molecule dipolar moment due to applied field is called

dipolar orientation and it is a slow phenomenon. It is well described by the first-order equation of

the Debye model.

Where ε0 vacuum dielectric constant, εα high frequency dielectric constant, N is dipole density, µ dipole moment, k Boltzman’s constant, T is temperature. The time taken for the polarization to

get saturated is called relaxation time τ.

Interfacial relaxation: Charges appear on the interfaces (boundary) within the material which

dominate the dielectric properties of the colloids and emulsions. The bulk permittivity and

conductivity of the composite material at the contact surface can be calculated by Maxwell-

Wagner model. Dispersion succeeds after this effect and has a contributing value at higher

frequencies (100-1000MHz) [10].

2.8 Biomedical Applications

T-ray imaging has the following potential medical applications. It could be used for transmission

imaging of the skin, preferably through the web of skin in the space between thumb and first

finger or a pinch on the back of the hand. This would be useful in the assessment of skin diseases

and their treatment, and for measuring the effect of products such as moisturizers.

Reflection imaging of skin would allow assessment of skin on any part of the body. The ability

to visualize this boundary gives an indication of the depth to which features such as tumors may

be seen, which is important for treatment and prognosis.

Terahertz imaging has a potential application in the detection of early dental caries, by helping

visualization of demineralized areas. The microscopic porosities caused by demineralization can

be detected with THz imaging because they fill with saliva which is more absorbing than enamel.

Also, to detect dental caries, reflection imaging of the enamel dentine boundary in the tooth

depends on the ability that demineralized regions could be correctly assigned to lying within

enamel or dentine.

3. MODEL DEVELOPME�T

The refractive index (η) of the tissue is dependent on the frequency of the T-ray beam and the

density (ρ) of the material. Table 1 and 2 show the results obtained by R. E. Miles et al and M. E.

Thomas et al and using standard values established by the National Institute of Standards and

Technology [5].

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3.1 Relationship between Refractive Index and Density

The relationship between refractive index and density is obtained by curve fitting using the

‘Trust-Region Reflective Newton’ algorithm.

Observing the successive data points in Figure 3, it can be noted that, as the density increases,

the refraction phenomenon also increases. So the basis function is a growth term like a

logarithmic or polynomial or inverse sine hyperbolic function.

Table 1: Values for broadband refractive index (η) and broadband linear attenuation coefficient (µ). Mean values are shown ± one standard deviation. N

η and N

µ were the number of

measurements used to calculate the mean values for refractive index and linear attenuation

coefficient respectively

3.1.1 Model I

Consider the basis function in logarithmic form. Let

η = a.log (b.ρ) (3)

This generates the plot shown in Figure 3.

We observe that the residuals like on either sides of the zero line. This suggests that an

oscillatory term must be introduced in the model. Hence, we come up with the following

equation

η = a.log (b.ρ) + c.sin (d.ρ) (4)

The plot of this model is shown in Figure 4.

3.1.2 Model II

The model y = b. ρ 1/n has a rising form. Eventually, an oscillatory term, sin (d.ρ) creates a much

better fit. Hence, the model is derived as

Material �ηηηη <ηηηη> �

µ <µ>/cm

-1 <ρρρρ>Density

(g/cm3)

Deionized water 16 2.04 ± 0.07 13 225 ± 21 0.998

Tooth enamel 44 3.06 ± 0.09 44 62 ± 7 2.9

Tooth dentine 72 2.57 ± 0.05 72 70 ± 7 2.5

Skin 36 1.73 ± 0.29 36 121 ± 18 1.1

Adipose tissue 37 1.50 ± 0.47 37 89 ± 23 0.92

Striated muscle 37 2.00 ± 0.35 37 164 ± 17 1.04

Cortical bone 59 2.49 ± 0.07 59 61 ± 3 1.85

Vein 33 1.58 ± 0.49 33 110 ± 43 1.04

Nerve 12 1.95 ± 0.46 12 246 ± 27 1.02

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η = a + b. ρ 1/n + c.sin (d.ρ) (5)

The curve obtained from Model II is depicted in Figure 5.

0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

Density

Refractive Index

Figure 3: Plot of the refractive index vs. density for the model in equation (3)

0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

Density

Refractive Index

Figure 4: Plot of the refractive index vs. density for the final Model I in equation (4)

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0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

Density

Refractive Index

Figure 5: Plot of the refractive index vs. density for the Model II in equation (5)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.80

0.5

1

1.5

2

2.5

3

3.5

Density

Refractive Index

Figure 6: Plot of the refractive index vs. density for the Model III in equation (6)

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3.1.3 Model III

We know the general curve for a first-order linear equation. To employ ‘Least Square

Polynomial Curve Fitting’, using a polynomial equation of second-order, we get the following

model which represents our data as shown in Figure 6.

η = a + b. ρ + c/ρ + d.(ρ2) + e/(ρ2) (6)

3.2 Relationship between Refractive Index and Wavenumber

To establish the relationship between refractive index of the tissue and the wavelength of the

THz radiation, we employ the following data set obtained by M. E. Thomas et al. This data set

considers wavenumber (ν’) instead of frequency (ω). However, we use the relation ω = 2π.ν’ to aid us [7].

Table 2: Comparison of the calculated values of the real index of refraction for sapphire at 295K

Wave

Number Refractive

Index

SD for refractive

index

1800 1.5921 0.0041

1900 1.6112 0.0034

2000 1.6271 0.0029

2100 1.6403 0.0026

2200 1.6516 0.0022

2300 1.6612 0.002

2400 1.6696 0.0018

2500 1.6768 0.0016

2600 1.6832 0.0014

2700 1.6888 0.0013

2800 1.6938 0.0012

2900 1.6983 0.0011

3000 1.7023 0.001

3.2.1 Model

Clearly, the refractive index vs. wavenumber (ν’) data corresponds to a cubic relationship, which can be interpreted in the form of the equation below

η = a + b/(ν’1/3) (7)

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1600 1800 2000 2200 2400 2600 2800 3000 32001.56

1.58

1.6

1.62

1.64

1.66

1.68

1.7

1.72

Wavenumber

Refractive Index

Figure 7: Plot of the refractive index vs. wavenumber for the data in table 2 and for the model in

equation (7)

Assuming that the frequency of excitation is independent of density and the refractive index

η(ω) does not contain a mixed term F (ρ,ω).

η(ω) = a.log (b. ρ) + c. sin (d. ρ) + e + f. (ω)1/3 (8) Squaring and Subtracting Eq (2) from (1), we have,

(9)

Substituting the n (w,ρ) and µ (w, ρ) in the above equation, permittivity can be transformed into

terms of density and angular frequency (at terahertz frequency).

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PART II BEGI�S FROM HERE (Sections 2.6, 2.7 and References are added to Part I)

4. PARAMETER ESTIMATIO� A�D COEFFICIE�T OF

DETERMI�ATIO�

Optimum parameters have to be chosen for a model to make it more realistic. This approach is

referred to as ‘Parameter Estimation’.

The approach to modeling that we adopted was to plot the given experimental data set and to fit a

good curve to it using regression methods.

However, nonlinear regression produces numerous outputs that can fit the data. A key step in

modeling our system was to determine the best fit to the data – to find parameter values that

make the curve go close to the data points and also to check that the results we obtained are

scientifically plausible.

Coefficient of Determination (R2) is an important factor in nonlinear regression analysis, used

here, to fit the data. The value of R2 quantifies the goodness of fit. R

2, a unitless coefficient, is

has a value in the range 0.0 and 1.0. Higher values of R2 indicate that the curve comes close to

the data. R2 = 1.0 implies that all the points lie exactly on the curve with no scatter [6].

However, the value of R2 can often land the system into a trap, because although a higher R

2

indicates that the curve came close to the points (a highly desired feature), it tells nothing about

the physical realization of the fit. The crux of obtaining a good fit was, hence, to find the best fit

with highest value of R2, along with a physical sense interpretation of the system; rather than

obtaining R2 = 1.0.

Table 3: R2 values for the three models (ref App.A)

Model I 0.8697

Model II 0.8704

Model III 0.8696

Although these values are not close to 1.0, they indicate a high correlation between the model

and the data set. These models were selected since they made physical sense along with high R2

values.

The parameter values obtained from Parameter Estimation using codes mentioned in Appendix A

(MATLAB7.5 Curve Fitting Toolbox) are as follows

Model I:

η = a.log (b.ρ) + c.sin (d.ρ) a = 2.0538, b = 1.8567, c = 0.5174 and d = 1.8255.

Model II:

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η = a + b. ρ 1/n + c.sin (d.ρ) a = –4.0266, b = 5.9300 and c = 0.1317.

Model III:

η = a + b. ρ + c/ρ + d.(ρ2) + e/(ρ2) a = 28.9971, b = –11.5348, c = –25.6276, d = 1.8269 and e = 8.1069.

5. SE�SITIVITY A�ALYSIS

Sensitivity analysis determines the effect on the model with respect to parameter variation and to

changes (deviation) in the model equation. It aids the designer in understanding the dynamics of

the system. By demonstrating how the model behavior responds to changes in the parameter

values, sensitivity analysis can be used as a tool in model building and model evaluation.

In this paper, we focus on the parameter sensitivity of Model I. Sensitivity of the system is the

response of the output to changes in the parameters of the system. Hence, if we change the

parameters of the system and calculate the difference in the new output and the old output

(error), it will reflect the sensitivity of the system to that parameter.

Root Mean Square (RMS), a quadratic mean, is a statistical measure of the magnitude of

deviation from the natural behavior of the system or experimental quantities. It is a useful tool

for sensitivity analysis. Hence, we calculated the RMS of the error between actual refractive

index (obtained from experimental data) and the calculated refractive index (based on Model I)

Sensitivity analysis was performed using MATLAB7.5 (ref. App A).

n = Experimental refractive index values

N = Refractive index values derived from the Model

(N-n) = Difference between Experimental and model derived values (Error)

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Table 4: Uphill and Downhill slopes from Sensitivity Analysis

Parameter Uphill Slope Downhill Slope

a 0.003 -0.003

b 0.0016 -0.0085

c 0.0026 -0.0027

d 0.0014 -0.0007666

Here, ‘a’ and ‘c’ are the scaling parameters (amplification) for the model; whereas ‘b’ and‘d’

represent the parameters that describe the variation in physical behavior of the model. b,

represents the growth constant of the ‘log’ term and the minimum value of d evaluates the

resonant frequency at which the dipoles under EM field vibrate at maximum amplitude.

At terahertz frequencies, gamma dispersion is much prominent than any other and water exhibits

the maximum contribution to the tissue permittivity. The number of dipoles formed

exponentially increases as the frequency increased after 1THz; this is due to the formation of

new dipoles of the non polar molecules (by absorbing energy) and adds to the dipolar moment.

As the density of the material is increased, the molecules of the tissue exhibits an oscillatory

behavior in the measured refractive (indirectly by permittivity) which causes the ‘Sin’ term.

There is a resonant frequency at which the maximum refractive index is observed and the

parameters take the optimum value. This phenomenon may be due to sudden change in the EM

field may not allow the dipoles to completely relax and a pattern of oscillations are seen in the

observed results.

As expected, changing the value of the parameters in the model does make some difference in

the behavior of the output. Also, the results indicate that some parameter changes result in

‘greater’ or significant changes in the output.

We now interpret the results of the sensitivity tests in detail. Note that sensitivity of ‘b’ being

0.0016 indicates an error of 0.0016 in the output in the simulated model, for every unit change in

the parameter ‘b’ value. Also, the sensitivity of parameter‘d’ is 0.0014; indicating that for every

unit change in the value of‘d’, the output deviates by an error of 0.0014.

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1 2 3 4 5 6 7 8 91.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

Density

Refractive Index

(1) b = 1.8567

1 2 3 4 5 6 7 8 9

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

Density

Refractive Index

(2) b = 1.6567

Figure 8: Simulated and actual data for different values of parameter 'b'

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1 2 3 4 5 6 7 8 91.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

Density

Refractive Index

(1) d = 1.8255

1 2 3 4 5 6 7 8 91.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

Density

Refractive Index

(2) d=1.6255

Figure 9: Simulated and actual data for different values of parameter 'd'

These values indicate that the system is not too sensitive to changes in the parameter values,

which is a strong point of the model.

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

6

7

8

9

Parameter "a"

Error

(a) Parameter ‘a’ (interval [0,5])

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

25

30

35

40

45

50

Parameter "b"

Error

(b) Parameter ‘b’ (interval [0,5])

Page | 18

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

Parameter "c"

Error

(a) Parameter ‘c’ (interval [-2,3])

-1 0 1 2 3 4 50.5

1

1.5

2

2.5

3

Parameter "d"

Error

(a) Parameter ‘a’ (interval [-1,5])

Figure 10: Sensitivity Analysis for parameters 'a', 'b', 'c', 'd' of Model I

Page | 19

6. ROBUST�ESS

A common goal that designers strive to attain is a robust model. Robustness is the ability of the

system to continue to operate correctly withstanding changes in the procedure or circumstances.

A system or a model is considered to be robust if it is capable of coping well with variations in

its output with minimal alteration or loss of functionality.

Robustness is, thus, an add-on to the sensitivity analysis of the system. Changes in the values of

parameters ‘b’ and‘d’ does not result in major changes in the output of the system. This indicates

that the model developed is robust.

7. JUSTIFICATIO� OF SIMULATIO� METHOD A�D

RELEVA�CE OF SIMULATIO� RESULTS TO SYSTEM

After developing a model, it is necessary for the developer to verify that the model generated fits

the experimental data. Hence, simulation methods have to be used to find the accuracy and

coherence of the model with the experiments.

Since our modeling approach was to develop equations/models that best fit the data set we have,

we plotted the data and then tried to fit the curve using nonlinear regression analysis methods

mentioned earlier. Hence, simulation of the model is an unnecessary step in our approach as the

model has been generated from the data and the goodness of fit parameters have already been

discussed.

8. DETERMI�ATIO� OF FOLLOW-UP EXPERIME�TS TO

IMPROVE THE MODEL

The data set that we had for the values of refractive index and density for different body tissues

(presented in Table I), considers only 9 tissues. Improved modeling can be achieved if the data

set that we have has more number of points. An insufficient data set is a major hindrance to

effective modeling. Hence, we suggest that experiments for different tissues be carried out to aid

modeling of the system.

Although the data considers important tissues from the point of view of T-rays imaging, such as

skin, adipose tissue, cortical bone, striated muscle, vein and nerve, if data for tissues/materials

with higher density is obtained, more number of points can be plotted for higher densities (where

we currently faced a shortage). This will help in improving the model.

Page | 20

9. REFERE�CES (Addendum to Part I as well as Part II)

[1] Sensing with Terahertz Radiation; Daniel Mittleman; Springer 2003.

[2] Terahertz Sources and Systems; R.E.Miles, P.Harrison; Kluwer Academic Out

publishers; 2001.

[3] Terahertz Optoelectronics; Kiyomi Sakai; Springer 2005.

[4] RF/Microwave Interaction with biological tissues; Andre Vander Vorst, Arye Rosen,

Youji Kotsuka, Wiley-Interscience 2006. [5] Berry, E. and Fitzgerald, A.J. and Zinov'ev, N.N. and Walker, G.C. and Homer-

Vanniasinkam, S. and Sudworth, C.D. and Miles, R.E. and Chamberlain, J.M. and Smith,

M.A. (2003) Optical properties of tissue measured using terahertz pulsed imaging.

Proceedings of SPIE: Medical Imaging 2003: Physics of Medical Imaging, 5030. pp. 459-

470.

[6] Fitting Models to Biological Data Using Linear and Nonlinear Regression: A Practical

Guide to Curve Fitting. Harvey Motulsky and Arthur Christopoulos. 2004. [7] Michael E. Thomas, Stefan K. Andersson, Frequency and Temperature Dependence of

the Refractive Index of Sapphire; Infrared Physics and Technology; 39(1998) 235-49. [8] R.Pathick. Dielectric properties of body tissues. Clin. Phys. MEAS. (1987); 8A 5-12. [9] C.M.Alabaster Permittivity of human skin in millimeter waveband; Electronic Letters;

39(21) 2003.

[10] G.L.Hey – Shipton. The complex permittivity of human tissue at microwave

frequencies. Phys. Med. Biology.; 27(8) 1067-71;1982

APPE�DIX

APPE�DIX A: MATLAB CODES

A.1 MATLAB code for Model I

model_1.m

clear all;clc;

hold on;

% Input n and d and plot them

n=[1.5 2.04 1.95 2 1.58 1.73 2.49 2.57 3.06]';

d=[0.92 0.998 1.02 1.04 1.06 1.1 1.85 2.5 2.9]';

en=[0.47 0.07 0.46 0.35 0.49 0.29 0.07 0.05 0.09]';

axis([0.5 3 0 3.5]);

errorbar(d,n,en,'Linestyle','none');

h = line(d,n,'Linestyle','none', 'LineWidth',1,'Marker','.', 'MarkerSize',5);

%Decide the fittype. This is going to be our equation/model

start_pt=[1 1 1 1];

Page | 21

[fit_type] = fittype('(a*log(b*x)+c*sin(d*x))','coefficients',{'a', 'b', 'c','d'});

fit_type %displays the equation (model)

% Fit the data to our model

[cf,goodness,output] = fit(d,n,fit_type,'Startpoint',start_pt);

% Plot the fit and output all the important things

h = plot(cf,'fit',0.95);

legend off;

set(h(1),'LineStyle','-', 'LineWidth',2,'Marker','none', 'MarkerSize',6);

xlabel('Density');

ylabel('Refractive Index');

coeffs = coeffnames(cf)

coeffvalues(cf)

goodness

output

Output:

For figure, refer Figure 4.

fit_type =

General model:

fit_type(a,b,c,d,x) = (a*log(b*x)+c*sin(d*x))

coeffs =

'a'

'b'

'c'

'd'

ans =

2.0538 1.8567 0.5174 1.8255

goodness =

sse: 0.2733

rsquare: 0.8697

dfe: 5

adjrsquare: 0.7916

rmse: 0.2338

output =

Page | 22

numobs: 9

numparam: 4

residuals: [9x1 double]

Jacobian: [9x4 double]

exitflag: 1

iterations: 12

funcCount: 61

firstorderopt: 9.1414e-006

algorithm: 'Trust-Region Reflective Newton'

A.2 MATLAB code for Model II

model_2.m

clear all;clc;

hold on;


n=[1.5 2.04 1.95 2 1.58 1.73 2.49 2.57 3.06]';

d=[0.92 0.998 1.02 1.04 1.06 1.1 1.85 2.5 2.9]';

en=[0.47 0.07 0.46 0.35 0.49 0.29 0.07 0.05 0.09]';

axis([0.5 3 0 3.5]);




start_pt=[1 1 1];

[fit_type] = fittype('a+b*x^(1/7)+c*sin(4.65*x)','coefficients',{'a', 'b', 'c'});






legend off;


xlabel('Density');



Page | 23

coeffvalues(cf)

goodness

output

Output:


fit_type =

General model:

fit_type(a,b,c,x) = a+b*x^(1/7)+c*sin(4.65*x)

coeffs =

'a'

'b'

'c'

ans =

-4.0266 5.9300 0.1317

goodness =

sse: 0.2718

rsquare: 0.8704

dfe: 6

adjrsquare: 0.8273

rmse: 0.2128

output =

numobs: 9

numparam: 3



exitflag: 1

iterations: 2

funcCount: 9



A.3 MATLAB code for Model III

model_3.m

clear all;clc;

hold on;


n=[1.5 2.04 1.95 2 1.58 1.73 2.49 2.57 3.06]';

Page | 24

d=[0.92 0.998 1.02 1.04 1.06 1.1 1.85 2.5 2.9]';

en=[0.47 0.07 0.46 0.35 0.49 0.29 0.07 0.05 0.09]';

axis([0.5 3 0 3.5]);




start_pt=[1 1 1 1 1];

[fit_type] = fittype('a+(b*x)+c/x+d*(x^2)+e/(x^2)','coefficients',{'a', 'b', 'c','d','e'});






legend off;


xlabel('Density');



coeffvalues(cf)

goodness

output

Output:


fit_type =

General model:

fit_type(a,b,c,d,e,x) = a+(b*x)+c/x+d*(x^2)+e/(x^2)

coeffs =

'a'

'b'

'c'

'd'

'e'

ans =

Page | 25

28.9971 -11.5348 -25.6276 1.8269 8.1069

goodness =

sse: 0.2736

rsquare: 0.8696

dfe: 4

adjrsquare: 0.7392

rmse: 0.2615

output =

numobs: 9

numparam: 5



exitflag: 1

iterations: 4

funcCount: 25



A.4 MATLAB code for Refractive index vs. Wavenumber model

model_4.m

clear all;clc;

hold on;


wave=[1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000]';

n=[1.5921 1.6112 1.6271 1.6403 1.6516 1.6612 1.6696 1.6768 1.6832 1.6888 1.6938 1.6983 1.7023]';

en=[0.0041 0.0034 0.0029 0.0026 0.0022 0.002 0.0018 0.0016 0.0014 0.0013 0.0012 0.0011 0.001]';

errorbar(wave,n,en,'Linestyle','none');

h = line(wave,n,'Linestyle','none', 'LineWidth',1,'Marker','.', 'MarkerSize',5);


start_pt=[1 1 ];

[fit_type] = fittype('a+b/(x^(1/3))','coefficients',{'a', 'b'});



[cf,goodness,output] = fit(wave,n,fit_type,'Startpoint',start_pt);

Page | 26



legend off;


xlabel('Wavenumber');



coeffvalues(cf)

goodness

output

Output:


fit_type =

General model:

fit_type(a,b,x) = a+b/(x^(1/3))

coeffs =

'a'

'b'

ans =

2.2868 -8.3292

goodness =

sse: 2.9786e-004

rsquare: 0.9797

dfe: 11

adjrsquare: 0.9778

rmse: 0.0052

output =

numobs: 13

numparam: 2



exitflag: 1

iterations: 2

funcCount: 7


Page | 27


A.5 MATLAB code for Sensitivity Analysis of Model I

sensitivity.m

clear all;

clc;

n=[1.5 2.04 1.95 2 1.58 1.73 2.49 2.57 3.06];

rho=[0.92 0.998 1.02 1.04 1.06 1.1 1.85 2.5 2.9];

%Sensitivity Analysis for parameter 'a'

b=1.8567;

c=0.5174;

d=1.8255;

a=0; i=1;

while a<5

N=a*log(b*rho)+c*sin(d*rho);

error(i)=sqrt((N-n)*(N-n)');

i=i+1;

a=a+0.001;

end

figure;

plot(error)

%Sensitivity Analysis for parameter 'b'

a=2.0538;

c=0.5174;

d=1.8255;

b=0; i=1;

while b<5



i=i+1;

b=b+0.001;

end

figure;

plot(error)

Page | 28

%Sensitivity Analysis for parameter 'c'

a=2.0538;

b=1.8567;

d=1.8255;

c=-2; i=1;

while c<3



i=i+1;

c=c+0.001;

end

figure;

plot(error)

%Sensitivity Analysis for parameter 'd'

a=2.0538;

b=1.8567;

c=0.5174;

d=-1; i=1;

while d<5



i=i+1;

d=d+0.001;

end

figure;

plot(error)

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