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
Page | 1
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
Page | 2
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
Page | 3
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
Page | 4
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
Page | 5
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.
Page | 6
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].
Page | 7
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
Page | 8
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)
Page | 9
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)
Page | 10
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)
Page | 11
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).
Page | 12
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:
Page | 13
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)
Page | 14
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.
Page | 15
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'
Page | 16
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.
Page | 17
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;
% 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];
[fit_type] = fittype('a+b*x^(1/7)+c*sin(4.65*x)','coefficients',{'a', 'b', 'c'});
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)
Page | 23
coeffvalues(cf)
goodness
output
Output:
For figure, refer Figure 5.
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
residuals: [9x1 double]
Jacobian: [9x3 double]
exitflag: 1
iterations: 2
funcCount: 9
firstorderopt: 2.3539e-007
algorithm: 'Trust-Region Reflective Newton'
A.3 MATLAB code for Model III
model_3.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]';
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]);
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 1];
[fit_type] = fittype('a+(b*x)+c/x+d*(x^2)+e/(x^2)','coefficients',{'a', 'b', 'c','d','e'});
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 6.
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
residuals: [9x1 double]
Jacobian: [9x5 double]
exitflag: 1
iterations: 4
funcCount: 25
firstorderopt: 8.2718e-008
algorithm: 'Trust-Region Reflective Newton'
A.4 MATLAB code for Refractive index vs. Wavenumber model
model_4.m
clear all;clc;
hold on;
% Input n and d and plot them
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);
%Decide the fittype. This is going to be our equation/model
start_pt=[1 1 ];
[fit_type] = fittype('a+b/(x^(1/3))','coefficients',{'a', 'b'});
fit_type %displays the equation (model)
% Fit the data to our model
[cf,goodness,output] = fit(wave,n,fit_type,'Startpoint',start_pt);
Page | 26
% 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('Wavenumber');
ylabel('Refractive Index');
coeffs = coeffnames(cf)
coeffvalues(cf)
goodness
output
Output:
For figure, refer Figure 7.
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
residuals: [13x1 double]
Jacobian: [13x2 double]
exitflag: 1
iterations: 2
funcCount: 7
firstorderopt: 6.5368e-009
Page | 27
algorithm: 'Trust-Region Reflective Newton'
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
N=a*log(b*rho)+c*sin(d*rho);
error(i)=sqrt((N-n)*(N-n)');
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
N=a*log(b*rho)+c*sin(d*rho);
error(i)=sqrt((N-n)*(N-n)');
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
N=a*log(b*rho)+c*sin(d*rho);
error(i)=sqrt((N-n)*(N-n)');
i=i+1;
d=d+0.001;
end
figure;
plot(error)
Top Related