School of Electrical and Electronic Engineering

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1 chool of Electrical and Electronic Engineering Uncertainty of Frequency Response of Human Tissues Third Year Project Report 29 th of April 2016 Name: Mohammed Hussain M. Abdulwahab University ID: upervisor: Dr Fumie Costen

2 Abstract In the Finite Difference Time Domain (FDTD) method, the Debye model of a biological tissue is used to simulate the relative permittivity and conductivity of biological tissues at a particular frequency using pre-determined Debye parameters of that tissue. According to the research group, measuring the Debye parameters of more than 50 tissues for a patient is expensive. As a result, a program is written to generate a wide range of Debye parameters of different tissues by using random number generator. However, the random number generator needs to produce infinite samples to produce ideal Gaussian and Uniform distributions. Hence, the limitation of the random number generators as well as the heterogeneous nature of biological tissues places some uncertainty in the generated Debye parameters. Therefore, the relationship between the standard deviation of the Debye parameters and the standard deviation of the output of the Debye model is investigated using Monte Carlo method and an analytical method that is based on the law of propagation of uncertainty. The relationship obtained from both methods showed a strong match with each other as long as the standard deviation of the Debye parameters is 50% of the mean value of the Debye parameters. While conducting the analysis, an algorithm has been devised taking advantage of the relationship deduced in this work in order to calculate the standard deviation in the output of Debye model for different values of standard deviation of Debye parameters. The algorithm is faster than Monte Calro method, and simpler than the law of propagation of uncertainty method. The drawback of this method is that it requires a pre-determined relationship between the standard deviation of the Debye parameters and the standard deviation of the output of the Debye model. This means that either Monte Carlo or the analytical method needs to run at least 5-6 times in order to obtain the required relationship. II

3 Table of Contents Nomenclature... VII Main Equation... XI List of Figures... XII List of Tables... XIII 1 Motivation Aims Objectives Literature Review Dielectric Properties of Biological Tissues Modelling the Complex Permittivity Factors Affecting the Dielectric Properties (εr and σ) of Biological Tissues Methods of Measuring the Dielectric Properties of Biological Tissues tatistical Concepts The Basics of tatistics Expressing Uncertainty Mathematically in Measurements Monte Carlo Method in Estimating the Propagated Relative Uncertainty in Model Equations Basics of Estimating the Propagated Uncertainty Mathematically (GUM Method) (i) Limitations of the GUM Method Determining the Debye Parameters of Different Tissues by Fitting Their Dielectric Properties to ingle Pole Debye Model Extracting the Frequency Response of Tissues Performing Data Fitting for ingle Pole Debye Model Equation Derivation of the Relative Permittivity and Conductivity Expressions Analysis of the Curve Fitting Tool Results (i) Relative Permittivity Curves (ii) Conductivity Curves III

4 7 Investigating the Relative Uncertainty in Relative Permittivity and Conductivity of Debye Model Using Monte-Carlo Method oftware (i)ource Files (ii) An Example Relating All the Terms Mentioned in 7.1(i) The Overall Flow of Program Results Analysis (i) Investigating the Effect of Frequency on the Relative Uncertainty in εr and σ (ii) The Relative Uncertainty in εr and σ for All Tissues (iii) Mean Curves (iv) Variation of Ω of ome Tissues with Frequency (v) The Relationship between x/k and Ω (vi) Using the Mean Curves to Estimate the Relative Uncertainty of Any Tissue (vii) Effect of on The Results of Relative Uncertainty (viii) Time Analysis General Discussions for Chapter Estimating the Relative Uncertainty Using GUM Method, and Comparing It with Monte Carlo Method Comparing GUM and Monte Carlo Methods relative uncertainties in εr and σ for different tissues (i) Relative Permittivity Case (ii) Conductivity case Comparing GUM and Monte Carlo Methods Relative Uncertainty in εr and σ Over Frequency (i) Relative Permittivity Case (ii) Conductivity case General Discussion of Chapter Conclusions Achievements IV

5 Further Work References Appendix A: Curve Fitted Debye Parameters Values A.1 Results of Activity in Chapter A.2 Research Group Fitted Tissues Appendix B: Analysing the Accuracy of the Fitted Debye Model Investigating The Accuracy of The Fitting Curve in The γ-dispersion Region Only Appendix C: Code for the Files in the Monte Carlo Program C.1 Main.sh C.2 Random.f C.3 Er.f C.3 ED.f C.4 Final_statistics.f C.5 tats.sh Appendix D: Investigating the Effect of Accuracy of the Random Number Generator on the Gaussian Values Obtained Using Box Muller Transform D.1 The Results of The Test D.2 Code of the Program Appendix E: Results of Monte Carlo Program for =3000 for All Tissues E.1 Conductivity E.2 Relative Permittivity Appendix F: Plots of Relative Uncertainty of Conductivity and Relative Permittivity for ome Frequencies as Input Relative Uncertainty Varies F.1 Conductivity F.2 Relative Permittivity Appendix G: Plots of Relative Uncertainty of Conductivity and Relative Permittivity for ome Tissues as Input Relative Uncertainty Varies G.1 Conductivity G.2 Relative Permittivity V

6 Appendix H: Plots of Relative Uncertainty of Conductivity and Relative Permittivity for Different as Input Relative Uncertainty Varies H.1 Conductivity H.2 Relative Permittivity Appendix I: Partial Derivatives I.1 Relative Permittivity I.2 Conductivity Appendix J: GUM Program Code J.1 GUM.sh J.2 Final.f Appendix K: The Progress Report VI

7 Nomenclature Name Description Location of Definition π The mathematical constant equal to approximately j 1. - ε o Permittivity of Free pace, and it is equal to Fm 1. - i An arbitral integer mainly used as sum operator in expressions. i Z. - ω Angular Frequency. - x An arbitrarily value that can be associated with anything. x R. - k An arbitrarily value that can be associated with anything. k R. - m An arbitral integer mainly used as sum operator in expressions. i Z. - ε Complex Permittivity. ection 4.2 ε Loss factor. ection 4.2 ε r Relative Permittivity. ection 4.2 σ Total Conductivity or simply referred as conductivity. ection 4.1 σ d Conductivity due to displacement Current. ection 4.2 σ i Conductivity due to ionic current. ection 4.2 n Order of the Debye model. ection 4.2 ε ε s The relative permittivity of a tissue at frequency when ωτ 1 1. ection 4.2 The relative permittivity of a tissue at frequency when ωτ 1 1. ection 4.2 ε 1 Magnitude of Dispersion calculated as ε s ε. ection 4.2 τ 1 Relaxation time constant of single pole Debye ection 4.2 model. α A Type of frequency dispersion. ection 4.1 VII

8 β A type of frequency dispersion. ection 4.1 γ A type of frequency dispersion. ection 4.1 ξ tandard Deviation of population. ection 5.1 Y A probability/statistical population named Y ection 5.1 y A random sample from population Y. ection 5.1 z A measurand value. ection 5.2 Z A population of measurand values. ection 5.2 ξ Y tandard Deviation of population Y. ection 5.1 y or E[Y] Mean of the population Y. ection 5.1 N Number of samples in a population. ection 5.1 Probability density function. ection 5.1 Φ Cumulative distribution function. ection 5.1 Y (y) Probability density function of population Y. ection 5.1 Φ Y (y) Cumulative distribution function of population Y. ection 5.1 u Relative uncertainty. ection 5.2 GUM Guide to the expression of Uncertainty in Measurement method. ection 5.4 u ε Relative uncertainty in ε. ection 7.1 u ε1 Relative uncertainty in ε 1. ection 7.1 u τ1 Relative uncertainty in τ 1. ection 7.1 u σi Relative uncertainty in σ i. ection 7.1 u input Relative uncertainty in Debye parameters (u ε, u ε1, u τ1 and u σi ), and an input to a software ection 7.1 program. T A number given to an arbitral tissue. ection 7.1 Number of Gaussian random samples for each T. ection 7.1 F Frequency sample point. There are 60 F points in total over the range of interest (100MHz to 6GHz). ection 7.1 R An intermediate constant used in Box Muller Algorithm. ection 7.1(i)(C) Θ An intermediate constant used in Box Muller Algorithm. ection 7.1(i)(C) U 1 and U 2 Uniformly distributed random samples. ection 7.1(i)(C) VIII

9 G 1 and G 2 Gaussian distributed random samples. ection 7.1(i)(C) ε r,f,t σ,f,t The value of ε r of a random sample of tissue T at F. The random sample is one of the samples. The value of σ of a random sample of tissue T at F. The random sample is one of the samples. ε F,T r 1 ε r s,f,t s=1 σ F,T 1 σs,f,t ξ εr F,T ξ σ F,T u F,T εr u F,T σ s=1 tandard deviation of the population ε r,f,t. The number of this population is. tandard deviation of the population σ,f,t. The number of this population is. F,T ξ εr ε F,T r ξ σ F,T σ F,T u T 1 εr F F u ε r F f=1 f,t u T 1 σ F F u σ T u εr F T u σ F ξ u T F εr ξ T u σ F F f=1 T f,t 1 T u t ε F r t=1 T 1 T u σ F t t=1 The standard deviation of population u T. The ε F r number of this population is all tissues T. The standard deviation of population u T. F The σ number of this population is all tissues T. ection 7.1(i)(B) ection 7.1(i)(B) ection 7.1(i)(D) ection 7.1(i)(D) ection 7.1(i)(D) ection 7.1(i)(D) ection 7.1(i)(D) ection 7.1(i)(D) ection 7.1(i)(E) ection 7.1(i)(E) ection 7.3(ii)(a) ection 7.3(ii)(a) ection 7.3(ii)(a) ection 7.3(ii)(a) I Input to a Fortran source file. ection 7.2 O Output from a Fortran source file. ection 7.2 IX

10 Ω x k cale factor used to scale T u or u εr F T from F σ u input = x to u input = k. ection 7.3(ii)(a) Ω T x k Ω F x k cale factor used to scale Tor u ε F r u T σ of T from F u input = x to u input = k. cale factor used to scale u ε r from u input = x to u input = k. F,Tor u σ F,T of T at F Ω A general notation for the scale factors Ω F x k, Ω T x k and Ω x k. ection 7.3(ii)(a) ection 7.3(ii)(a) ection 7.3(iv) ε r F,T σ F,T ξ εr F,T ξ σ F,T u εr F,T u σ F,T u T εr F u T σ F u T εr F u T σ F The GUM parameter that is equivalent to ε F,T r in Monte Carlo. The GUM parameter that is equivalent to σ F,T in Monte Carlo. The GUM parameter that is equivalent to ξ εr F,T in Monte Carlo. The GUM parameter that is equivalent to ξ σ F,T in Monte Carlo. The GUM parameter that is equivalent to u F,Tin εr Monte Carlo. The GUM parameter that is equivalent to u in F,T σ Monte Carlo. The GUM parameter that is equivalent to u Tin εr F Monte Carlo. The GUM parameter that is equivalent to u Tin F σ Monte Carlo. The GUM parameter that is equivalent to u T in εr F Monte Carlo. The GUM parameter that is equivalent to u T in F σ Monte Carlo. ection 8.1 ection 8.1 ection 8.1 ection 8.1 ection 8.1 ection 8.1 ection 8.1 ection 8.1 ection 8.1 ection 8.1 X

11 Main Equation u y = ξ Y y Equation (5.5) in section 5.2 XI

12 List of Figures Figure 5.1. The standard normal (Gaussian) distribution probability density function and cumulative distribution function Figure 5.2. The probability density function and its respective cumulative probability function for the uniform distribution Figure 6.1. The variations of ε r and σ over the frequency range of interest Figure 7.1. An example relating all the terms together Figure 7.2. The runtime environment of the Monte-Carlo program Figure 7.3. The flowchart of the Monte-Carlo program s Main.sh Figure 7.4. The variation of relative uncertainty in ε r over frequency for different tissues and for different relative input uncertainties Figure 7.5. The variation of relative uncertainty in σ over frequency for different tissues and for relative input uncertainties Figure 7.6. The relative uncertainty in σ of all 94 tissues for relative input uncertainies Figure 7.7. The variation in the ratio of standard deviation to the mean in ε r case Figure 7.8. The scale factor Ω for each of the 94 tissues between different relative input uncertainties Figure 7.9. The value relative uncertainty in ε r of all 94 tissues for relative input uncertainties Figure The variation in the ratio of standard deviation to the mean in σ case Figure The scale factor Ω for each of the 94 tissues between different relative input uncertainties Figure The mean fitted fourth order model curves Figure The variation of scale factor Ω over frequency for some tissues Figure The relationship between Ω k x and x/k obtained using the mean curves Figure The flowchart of the devised algorithm Figure The effect of varying on relative uncertainty in ε r and σ Figure The effect of varying on the time taken to process one tissue T Figure 8.1. Flowchart showing the steps of the GUM program Figure 8.2. Compares the relative uncertainty in ε r for some tissues using GUM and Monte Carlo methods Figure 8.3. Compares the relative uncertainty in σ for some tissues using GUM and Monte Carlo methods Figure 8.4. The variation of GUM relative uncertainty in ε r for some tissues over frequency Figure 8.5. The variation of GUM relative uncertainty in σ for some tissues over frequency XII

13 Figure D.1. The results of analysing the accuracy in Fortran random number generator used in Box Muller Transform...65 Figure F.1. The effect of varying the relative input uncertainty on relative uncertainty in σ for some tissues for some frequencies..71 Figure F.2. The effect of varying the relative input uncertainty on relative uncertainty in ε r for some tissues for some frequencies Figure G.1. The effect of varying the relative input uncertainty on the relative uncertainty in σ for some tissues...73 Figure G.2. The effect of varying the relative input uncertainty on the relative uncertainty in ε r for some tissues 73 Figure H.1. The effect of varying on the relative uncertainty in σ for all 94 tissues.74 Figure H.2. The effect of varying on the relative uncertainty in ε r for all 94 tissues 76 List of Tables Table 7.1. The trend between τ 1 and the relative uncertainty in ε r Table 7.2. The trend between σ i and the relative uncertainty in σ Table 7.3. An exmaple showing how to use the mean curves to estimate the relative uncertainty in ε r at any u input and for any tissue Table 7.4. An example showing how to use the mean curves to estimate the relative uncertainty in σ of a tissue at any frequecny point F Table A.1. The Debye parameters obtained from the curve fitting activity in chapter 6.48 Table A.2. ome of the Debye parameters that were determined by Dr F Costen research group...51 Table B.1. The values of the RM difference ratios for some tissues...52 Table B.2. The effect of eliminating the β-dispersion region from the calculations of difference ratios for some tissues 53 Table E.1. The values of relative uncertainties in σ of all 94 tissues at 6 different values of u input...67 Table E.2. The values of relative uncertainties in ε r of all 94 tissues at 6 different values of u input..69 XIII

14 1 Motivation One of the most common ways of modelling the electromagnetic propagation in human body is the Finite Difference Time Domain method (FDTD) [1]. This method is an iterative method where space is divided into small segments of magnetic and electrical fields [1]. The current research group simulates the propagation of electromagnetic pulses in the human body using the FDTD method in order to develop health care devices. The frequency pulse contains a spectrum of frequencies. Electromagnetic pulses propagate differently for different tissues and each frequency influences the same tissue differently. In other words, the complex permittivity, conductivity and relative permittivity, of human tissues are all frequency dependent. The research group s FDTD simulation software uses Debye model to simulate different relative permittivity and conductivity of each tissue specific for the patient. The Debye model can change the complex permittivity of a tissue using a pre-determined Debye parameters [1]. Ideally, this is done by measuring the Debye parameters for each tissue of the patient and uses it to generate a patient-specific healthcare device. Unfortunately, according to the research group, measuring the Debye parameters for more than 50 tissues is too expensive at this moment. As a result, an alternative way is needed to generate a range of Debye parameters for a set of human tissues. 2 Aims This project aims to produce a wide range of Debye parameters for different human tissues taking advantage of statistics and random number generators. In addition, the project then aims to analyse the relationship between the probability distributions of the Debye parameters and the probability distributions of the complex permittivity using Monte Carlo method and an analytical method called GUM that is based on the law of propagation of uncertainty. 3 Objectives Learn the basics of statistics. Conduct literature review about the topic. Extract frequency response of 102 human tissues. Learn bash shell scripting language. Learn Fortran syntax in order to perform fast statistical calculations. Curve fit the frequency response of the 102 tissue to single pole Debye model. Create a program to generate a particular number of random Debye parameters of specific tissue. Extend the program to analyse probability distributions in relative permittivity and conductivity of tissues using Monte Carlo method. 1

15 Develop a software using Fortran and shell script to implement GUM method. Analyse probability distributions in relative permittivity and conductivity of tissues using GUM method. Compare both methods and draw conclusions. Produce a relationship between the standard deviation of Debye parameters and the standard deviation of relative permittivity and conductivity. 4 Literature Review 4.1 Dielectric Properties of Biological Tissues The dielectric property of biological tissues is property of the tissues in which it expresses their behaviour when interacting with the electromagnetic spectrum [2]. Describing the behaviour of any material under electromagnetic waves requires studying of its electrical conductivity, electrical permittivity and magnetic permeability [2]. When a material experiences an electric field, it can be generally classified as either a conductor or an insulator. A conductor described in [3] as a material that allows the passage of charge through it on the application of electrical field. Whereas in insulators, their charges are restricted and not able to flow when electric field is applied [3]. The charge in a conductor will flow until the net electrical field in the material becomes zero, whereas net electric field will remain in insulators creating dipoles [3]. Many materials including biological tissues does exhibit some sort of conductive and insulating behaviour because of their heterogeneous nature [3]. This is because charge can move in them but in limited way [3]. Conductivity is related to the property of the material allowing the flow of charge when it is under the influence of electric field, while permittivity is related to the property of material to strict the movement of charge or rotation of molecular dipoles [3]. With the exception of DC fields, any AC field for any frequency follows a sinusoidal behaviour. The sinusoidal function itself is fundamentally a complex number function. [3] have combined the conductive and insulating behaviour of the dielectric material under electromagnetic fields and derived the expression of the complex permittivity (ε * ) as ε = ε r jε (4.1) where ε r is the relative permittivity, ε is referred as the loss factor, ε o is the permittivity of free space and ω is the angular frequency of the electromagnetic field [4]. 2

16 The loss factor ε is related to the (total) conductivity σ, and the expression is given in [4] as follows σ = ε o ωε (4.2) Furthermore, the (total) conductivity is related to the electrical energy dissipated [4]. The total conductivity in biological tissues is due to the displacement current (σ d ) and ionic (σ i ) current [4]. Therefore, the total conductivity (σ) of a biological tissue is expressed in [4] as σ = σ d + σ i (4.3) Biological tissues are assumed materials having permeability the same as the permeability of free space, therefore magnetic permeability is not considered when analysing the behaviour of tissues in electromagnetic fields [2]. Consequently, the relative permittivity ε r and conductivity σ are called the dielectric properties of the biological tissues [2]. When analysing the dielectric properties of tissues, the complex permittivity is investigated [4]. The existence of complex numbers in equation (4.1) indicates that the dielectric properties of tissues are frequency dependents. This means that it varies as the frequency varies. The variation of relative permittivity due to frequency changes is referred to as dispersion [3]. The dielectric properties also depend on the physiological state of the tissue [5, 6]. Different dispersions result from different polarisation mechanisms [7, 8]. When an electric field is applied, it will take some amount time to polarise [8]. This amount of time is referred to as relaxation time τ [8]. Different polarization mechanisms have different relaxation time [3]. The frequency dispersion broadly falls into three main regions depending on the magnitude of frequency. The first region is α-dispersion region which occurs at the site of cellular membrane because of the ionic diffusion process [4, 5]. This region corresponds to the largest values of permittivity, and it occurs at the low end of the frequency spectrum (typically below a few kilohertz) [3, 4, 5, 9]. The β-dispersion region occurs for 3-4 frequency decades (300kHz-300MHz) after the α-dispersion [9]. The β-dispersion is resulted from the polarisation of the cellular membrane and other organic polymers [4, 5]. The γ-dispersion region is the last region which is dominant at frequencies exceeding approximately 100MHz [9]. The γ-dispersion is related to the polarisation of water molecules [4, 5]. 4.2 Modelling the Complex Permittivity The different kinds of dispersions/relaxations requires the complex permittivity to be expressed in model equations of different parameters. One of the most important models in expressing the complex permittivity in terms of relaxation is the Debye model. A multipole Debye model is expressed in [7] and [8] as 3

17 n m=1 ] + 1+jωτ m ε (ω) = ε + [ Δε m σ i jωε o (4.4) where n represents the number of dispersion regions, i.e, the order of the Debye model [8], Δε n is the magnitude of dispersion [7], ε is the relative permittivity at frequency when ωτ m 1 [4], τ m is the relaxation time constant for m-dispersion region [4], σ i is the conductivity due to ionic current [7, 8], and Δε n is expressed in [7] and [8] as Δε m = ε s,m ε (4.5) where ε s,m is the relative permittivity at frequency when ωτ m 1 [4, 7, 8]. The frequency range of interest in this work is between 100MHz-6GHz. At this frequency range, γ- dispersion is dominant so one relaxation time constant is needed, and therefore only a single pole Debye model is considered. Also, it has been shown by [10] that a single pole model is sufficient for frequency range between 0.5 GHz-20 GHz. As a result, a single Debye model that is used throughout this work is ε (ω) = ε + Δε 1 1+jωτ 1 + σ i jωε o (4.6) Finally, it worth emphasising that the real part of (4.6) is the relative permittivity ε r, and the imaginary part is the loss factor in which (4.2) can be used to calculate the conductivity σ. 4.3 Factors Affecting the Dielectric Properties (ε r and σ) of Biological Tissues As it has been described earlier, the dominant dispersion in the frequency range of interest (100MHz-6GHz) is the γ-dispersion. This means that water content of the tissue is the main factor in determining the dielectric properties of the tissue in this frequency range of interest. Not to mention, water is one of the main compounds present in biological tissues [6]. The water content of a tissue is affected by age, gender, tissue type and physiological state [6]. There is no doubt also that the complex permittivity varies with temperature [2, 8]. Using fundamental physics knowledge, kinetic energy of the atoms and the ions is affected by temperature, and therefore, relaxation time should be affected. The key factor that plays a big role in causing the variations in the complex permittivity is the water content of a tissue [1, 5]. The total water content of a tissue is affected by the following factors: a) Tissue Type A study has shown that different types of tissues have different water content [11]. [7] have carried out measurements on many tissue types including : Grey Matter, Cornea, Dura, Retina, Bone and many others. The measurements are made in the range of 10Hz 20 GHz. From [7] results, it can be seen that hard tissues, such as Bone, that have low water content in their compositions tend to have much lower Cole-Cole ε s parameter. 4

18 The Cole-Cole model is a small modification of Debye model [4, 7], and hence, it is also expected that the Debye ε s parameter of hard or low water content tissues to have smaller values of ε s. This is because ε s parameter and water content of a tissue are related to each other [7]. b) Age According to [2], it is found out that the total body water content decreases with age. The study that was carried out in [2] showed a general decrease in the dielectric properties with age. The decrease in water content because of increasing age was more noticeable in various head tissues and skin [2]. c) Gender The total water body (TBW) content is different between males and females of the same age [8]. [8] has provided a simplified equation to calculate the total body water content of males and females as a function of age A, mass M and height h. The equations are as follows Females TBW = h m (4.7) Males TBW = A h m (4.8) where TBW is measured in litres. Notice that (4.7) is not a function of age. This is most likely because the changes in body water content in females due to age is not significant compared to height and mass. Furthermore, human body develop with age differently for each gender [8]. Also, note that equations (4.7) and (4.8) are simplified expressions used to estimate TBW [8]. d) Body hapes Different bodies have different water contents. Bodies with more fats tend to have lower water contents [4]. According to equations (4.7) and (4.8), the body water content depends on height and mass which are factors affecting body shape. As a result, body shape contributes to the variations in the dielectric properties of the human tissues. 4.4 Methods of Measuring the Dielectric Properties of Biological Tissues An open ended coaxial cable transmission line technique is one of the main methods used when dielectric properties of tissues are to be measured [4]. Many studies use this technique when measuring the dielectric properties of tissues including [4, 9, 12, 13]. In order to measure the dielectric properties of a tissue, two more additional pieces of hardware are needed. The first hardware is a probe in which it will be in direct contact with the tissue sample [4, 13]. A Vector Network Analyser (VNA) is used to sweep different electromagnetic signals, and measures the reflection from the tissue sample [4, 13]. Modern VNAs have software embedded in them that can be used for dielectric measurements by calculating admittance or reflection coefficient [4]. The reflection coefficient of open-ended coaxial cable is related to the dielectric 5

19 Y (y) Φ Y (y) properties of the tissue that is in contact with it, and it is used by the software in VNA to calculate the dielectric properties of the tissue being measured [13]. This technique is suitable for frequency range from 10Hz to 20GHz [12]. 5 tatistical Concepts 5.1 The Basics of tatistics In order to carry out the objectives in this project, some statistical concepts need to be highlighted. These concepts include probability distributions and standard deviation. The standard deviation ξ is a measure of the spread of a distribution/population. Variance ξ 2 is the square of the standard deviation. For a random variable y, the variance is calculated in [14] as ξ 2 Y = 1 N (y N i=1 i y ) 2 (5.1) where y is the mean of the population Y, N is the number of random samples present in the population Y in which for ideal distributions N = and y i is the value of one random sample from the population Y. In statistics, the term population is a group of random samples of a quantity [14, 15]. For example, population Y is a group of random samples y of a quantity referred to as Y. It is worth noting that y and E[Y] (Expected value of population Y) represent the same quantity, i.e., the mean [14, 15]. This is because the mean is the most expected value of the population Y, and statistically, the expected value is expressed as E[Y]. y % a) probability density function b) cumulative distribution function Figure 5.1. The standard normal (Gaussian) distribution probability density function and cumulative distribution function. The y-axis for (a) has been normalised. The plots were generated using spreadsheet software. y 34% The standard normal (Gaussian) distribution has a mean of zero and a standard deviation of 1 unit [14]. The standard normal distribution is presented in figure 5.1. The probability density function y

20 (y) of population Y, Y (y), represents how likely a random variable y to take a specific value compared to the other values in the distribution [14]. For example, from figure 5.1(a), the value of 0 is the most likely value to occur for y. This is because at y = 0, Y has the highest magnitude compared to other values in the population Y. for standard normal distribution with zero mean and standard deviation of 1 unit (figure 5.1(a)) is expressed in [14] as Y (y) = 1 2π e 0.5y2 (5.2) The cumulative distribution function Φ(y) of Y, Φ Y (y), represents the probability that the population Y produces a random value which lies within the range of to y [14]. For example, Φ Y (0) is equal 50% because from figure 5.1(a) half of the population Y lies between and 0. The same information can be extracted from figure 5.1(b) in which the y-axis value is read at y=0. It is found out that for symmetrical distributions such as Gaussian in figure 5.1(a), 68% of the population lies within 1 unit of standard deviation from the mean [14]. A symmetrical distribution is a type of probability density function such as Gaussian in figure 5.1(a) in which a population has a mirror reflection around the mean value. It is also worth mentioning that 95% of the population lies within 2 units of standard deviation from the mean [14]. The general relationship between (y) and Φ(y) is expressed in [14] as follows Φ(y) = (y)dy The integral form of (y) for the normal distribution does not exist mathematically, and it has to be looked up in statistical tables [14, 15]. (5.3) Another important distribution is the uniform distribution. The probability density function shown in figure 5.2(a) for the uniform distribution is constant in the population range and zero out of it, meaning that each point in the population range has the exact same probability of being selected compared to other values in the population [14]. The cumulative density function of uniform distribution is linear in the population range and flat outside the range as shown in figure 5.2(b) [14]. Note that the uniform distribution (figure 5.2) is different from normal (Gaussian) distribution (figure 5.1). 7

21 (y) Φ (y) y y a) probability density function b) cumulative probability function Figure 5.2. The probability density function and its respective cumulative probability function for the uniform distribution. The plots were generated using spreadsheet software. 5.2 Expressing Uncertainty Mathematically in Measurements The term uncertainty means the doubts that exist in measured parameters [16, 17]. An error on the other hand is the amount in which a measurand (measured quantity) deviates from its true value [16, 17]. The term uncertainty is broad and it will be explained more specifically in this section. When the measured system is free from systematic error (offset/bias), the main contributor of the uncertainty of the dielectric properties of the human tissues are the random variations that arises from measurements and the inhomogeneous nature of the biological tissues [4, 18]. The variations in the probability distribution of ε r and σ of biological tissues due to the heterogeneous nature of biological tissues follow the normal (Gaussian) distribution [4]. When expressing measured data, it is often desired to state the standard deviation of the parameter based on the population of the measured samples. In the context of measurement, the standard deviation is called the standard uncertainty in the measurand. However, in order to avoid confusion, the standard deviation term will be used rather than standard uncertainty throughout this work. As outlined earlier in this section, ε r and σ take the form of Gaussian distribution, and the Debye model parameters are most likely to have the same distribution. Hence, a population of measured (random) parameter Z is expressed in [17] as Z = z ± ξ z (5.4) When comparing different types and magnitudes of measured parameters, a more meaningful way of showing the uncertainty in the measured data is expressing it into relative uncertainty u. The relative uncertainty u is expressed as u z = ξ z z (5.5) 8

22 where z is the mean of the population Z [17]. z in equation (5.5) can be any of the Debye parameters (ε, ε 1, τ 1 and σ i ) which will be obtained from chapter 6 curve fitting activity. The values of mean and standard deviation of Gaussian distribution are the only information needed in order to draw the probability distribution of a quantity and conduct statistical analysis on the quantity. The uncertainty concept is usually associated with measurements. In literature, the calculation of standard deviation in the outputs of model equations like ε r and σ in Debye model is usually described in the context of relative uncertainty propagation, and therefore the term will be used so that this work can be compared with other work in literature. Recall that one of the aims of this work is investigate the relationship between the distribution of the random samples of Debye parameters and the distribution of ε r and σ of the Debye mode. Furthermore, the term propagation represents how the relative uncertainty in the input of model equations like the Debye parameters in Debye model propagates to (resulted in) the outputs of the model equations like ε r and σ in Debye model. Finally, note that equation (5.5) is very important equation, and will be used extensively throughout the rest of the work. 5.3 Monte Carlo Method in Estimating the Propagated Relative Uncertainty in Model Equations The Monte Carlo method is easy to implement and widely used [19]. The Monte Carlo method considers all the standard deviations of the input parameters regardless the amount in which each parameter affects the output of a model equation [20, 21]. In case of single pole Debye model, the input parameters are the Debye model parameters (ε, ε 1, τ 1 and σ), and the output parameters are the relative permittivity ε r and conductivity σ. In other words, the relative uncertainty of each Debye parameter is propagated to (resulted in) ε r and σ. As outlined earlier, the randomness of the dielectric properties of biological tissues is Gaussian [4]. When Monte Carlo trials are carried out, values from the probability distribution of ε, ε 1, τ 1 and σ are randomly chosen, and ε r and σ are calculated [20, 21]. This procedure is carried out for many times [19]. Generally, the more trials are performed, the greater the confidence in the results. [21] states that in order to have a certain convergence probability p, the number of trials M should satisfy the following M > p Hence, when 90% convergence is needed, at least 100,000 calculation trials are required. (5.6) When all the calculation trials are performed, an estimate of the output quantity (ε r and σ) is calculated by finding the mean of all the trials [21]. The relative uncertainty can be calculated by 9

23 working out the standard deviation of the output quantity for all trials then equation (5.5) is applied. Finally, when the number of Monte Carlo calculations becomes very large, the distribution of the output of the model equation will have a Gaussian distribution regardless the probabilistic distributions of the input parameters [19]. 5.4 Basics of Estimating the Propagated Uncertainty Mathematically (GUM Method) Another way of estimating u is using the GUM method (Guide to the Uncertainty estimation in Measurements method) [16]. The GUM method is derived from the law of propagation of uncertainty [16, 21]. The GUM method is preferred over Monte Carlo as it provides an exact mathematical expression for the standard deviation of the output of model equations without the need of conducting very large numbers of trials as it is the case of Monte Carlo [16, 21]. The exact standard deviation in ε r and σ is calculated as follows ξ Y = L ( Y ) 2 2 m=1 ξ z zm + 2 L 1 L ( Y ) ( Y i=1 m=i+1 ) ξ m z m z zi ξ zm ρ zi z m (5.7) i where i and m are sum operators, Y can be either ε r or σ, L is the number of variables in the model equation in which for a single pole Debye model L = 4, z 1, z 2, z 3 and z 4 are ε, ε 1, τ 1 and σ respectively and ρ zi z m represents the correlation coefficient between two Debye parameters (ε, ε 1, τ 1 and σ) [6, 16, 21, 22]. The correlation (and covariance) between two variables is a measure of the dependence between these two variables [14]. Mathematically, the correlation coefficient ρ is expressed in [14] as follows ρ zi z m = ξ z i zm ξ zi ξ z m (5.8) where ξ zi z m is the covariance between two parameters. The correlation coefficient can take any value between -1 and 1 [14]. If z i and z m are statistically independent to each other, then ρ zi z m = 0 [14]. tatistical independence means that changes in z i cannot be used to predict changes in z m [14]. For many applications, the correlation term is very small and can be practically approximated to zero [6]. As a result, equation (5.7) has been simplified by [6, 16, 21, 22] to give the following expression ξ Y = L ) 2 2 m=1 ξ z zi (5.9) m ( Y Equation (5.9) is valid for many applications. Once the standard deviation can be calculated using (5.9), the relative uncertainty u in ε r and σ can easily be determined using (5.5). 10

24 As it can be seen, it is complicated to work out the exact relative uncertainty propagated in a multi parameter model such as the Debye model. It is even more complicated if the variables involved are not independent, i.e., correlation terms exist. 5.4(i) Limitations of the GUM Method The GUM method (5.7 and 5.9) is based on first order Taylor series approximations, and hence it assumes linear expressions [16, 21, 22, 6]. Unfortunately, this is not true for Debye model (4.6). Furthermore, the Debye parameters are correlated to each other [7], and this will be shown in chapter 8. As a consequence, some doubts will exist around the accuracy in using (5.9), and this will be investigated in chapter 8. 6 Determining the Debye Parameters of Different Tissues by Fitting Their Dielectric Properties to ingle Pole Debye Model 6.1 Extracting the Frequency Response of Tissues In order carry out the aim of this work by generating different random values of Debye parameters of different tissues, the single pole Debye prameters of each tissue T needs to be calculated. The first step is to extract the frequency responses of ε r and σ for 102 tissues from [23]. The frequency range used is between 100MHz and 6GHz. In this frequency range, 60 frequency samples F of ε r and σ were taken for each tissue at 100 MHz interval. This means F is related to ω (rads -1 ) as ω = 2πF 10 8 (6.1) The data extracted from this source will be referred as IT I data for the rest of the report. 6.2 Performing Data Fitting for ingle Pole Debye Model Equation A program provided by Dr F Costen research group was used to fit the frequency response of ε r and σ for each tissue to single pole Debye model (4.6). Unfortunately, the program can only accept one tissue at a time, and since the activity of curve fitting is not trivial compared to the activity in section 6.1, a shell script program has been developed to automate the task. The shell script code can be view in the appendix of the progress report. The progress report is attached as Appendix K. For the curve fitting to be performed as desired, two parameters needed to be adjusted. The first parameter is the value of tolerance. The tolerance is the difference between the value of the previous iteration and the current iteration. The second parameter that needs to be adjusted is the initial values of the Debye parameters. This also has a great deal of importance since large deviations of the initial values of the Debye parameters from the true values cause the program to behave in uncontrolled manner, and the results may not converge. Even if it converges, it takes high number of iterations for the convergence to complete which is still not desired. 11

25 In the progress report, different combinations of Debye parameters (ε, ε 1, τ 1 and σ) initial values were tested. The program managed to calculate the Debye parameters of 92 tissues for just one combination of initial values. The other 10% required two further runs with two other different combinations of initial values. Only two tissues were fitted from the two additional runs, Thymus and Invertible Disc. The remaining eight tissues could not be fitted by the software because their ε r or σ values were zero for the whole frequency range of interest (100MHz-6GHz). These eight tissues are Air, Blood Plasma, Bronchi Lumen, Esophagus Lumen, Lymph, Pharynx, Trachea Lumen and Water. These eight tissues were not fitted to the Debye Model, and they will not be used in the activities in chapters 7 and 8. As a result, the relative uncertainty analysis will be carried out for only 94 tissues. It is important to verify the accuracy of the fitted results of the 94 tissues. A comparison was made with 30 tissues fitted by Dr F Costen s research group. Most tissues had matching values for ε, ε 1 and σ i with the ones fitted by the research group. Nonetheless, the relaxation constant for the 30 tissues did not match with the research group results. This is likely due to the fact that the frequency response of ε r and σ was obtained from different source, the fitting software algorithm has been modified or that [23] database has been updated by the time this work has been conducted. The matching values of ε, ε 1 and σ i with the research group results increases the confidence in the fitted Debye parameters of the 94 tissues. ee Appendix A for complete comparison and Debye parameters of all fitted tissues. Moreover, because the Debye parameters are correlated to each other, there is no correct combination for the Debye parameters of a specific tissue [7]. As long as the Debye model behaviour matches the true behaviour of the frequency response of ε r and σ for a tissue, the fitted Debye parameters for that tissue are acceptable. An analysis will be shown in section Derivation of the Relative Permittivity and Conductivity Expressions The Debye Model in (4.6) can be separated into real part and imaginary parts: ε = ε + ε 1 1+(ωτ 1 ) 2 j [ ωτ 1 ε 1 1+(ωτ 1 ) 2 + σ i ωε o ] (6.3) where j = 1 and ε, ε 1, τ 1 and σ i are the input parameters of the model. The relative permittivity ε r is the real part of the complex permittivity ε, and it is expressed as ε r = ε + ε 1 1+(ωτ 1 ) 2 The conductivity σ is a bit more complicated than just extracting the imaginary part. As mentioned in section 4.2, σ is expressed as (6.3) σ = ωε o ε = jωε o j [ ωτ 1 ε 1 1+(ωτ 1 ) 2 + σ i ωε o ] = ε oω 2 τ 1 ε 1 1+(ωτ 1 ) 2 + σ i (6.4) 12

26 Conductivity/(/m) Relative Permittivity conductivity/(/m) Relative Permittivity Conductivity/(/m) Relative Permittivity Conductivity/(/m) Relative Permittivity Conductivity/ (/m) Relative Permittivity Conductivity/(/m) Relative Permittivity 6.4 Analysis of the Curve Fitting Tool Results Measured Conductivity Debye Model Conductivity Measured Permittivity Debye Model Permittivity Measured Conductivity Debye Model Conductivity Measured Permittivity Debye Model Permittivity Frequency/GHz Frequency/GHz a) Bone Marrow (Yellow) b) Eye (Cornea) 40 1 Measured Conductivity Debye Model Conductivity Measured Permittivity Debye Model Permittivity Measured Conductivity Debye Model Conductivity Measured Permittivity Debye Model Permittivity Frequency/GHz c) Fats d) Pons Measured Conductivity Debye Model Conductivity Measured Permittivity Debye Model Permittivity Frequency/GHz Measured Conductivity Debye Model Conductivity Measured Permittivity Debye Model Permittivity Frequency/GHz Frequency/GHz e) Tooth f) White Matter Figure 6.1. The variations of ε r and σ over the frequency range of interest (100MHz-6GHz). The fitted Debye model is compared with the IT I measured data. ix tissues are used to analyse the accuracy of fitting IT I data to single pole Debye model (4.6). Pons tissue is chosen because it is one of the tissues that have the highest Debye parameters, whereas Bone Marrow is chosen because it is one of the tissues that have the lowest values of 13

27 Debye parameters. The other four tissues were chosen at random. 100MHz equally spaced intervals over the frequency range of interest (100MHz-6GHz) were used to plot the curves in figure 6.1. As a result, there are 60 frequency points F used to plot each curve for the tissues in figure (i) Relative Permittivity Curves Figure 6.1 shows six fitted tissues to a single pole Debye model over the frequency range 100MHz- 6GHz. As it can be seen, there are two main dispersions in the frequency range of interest. These dispersions are β-dispersion and γ-dispersion. Inspecting the relative permittivity curves of IT I (the measured curves) shows the β-dispersion ends approximately at 300 MHz which confirms the information mentioned in literature review. The choice of fitting the IT I data to a single Debye model is appropriate since the range is mostly dominated by one relaxation process that is the γ-dispersion. This can be easily observed from the IT I measured relative permittivity curves. From the relative permittivity results in figure 6.1, the relaxation constant τ 1 obtained from the curve fitting tool is the average for the β-dispersion and the γ-dispersion, and this is might be the reason why τ 1 obtained from this activity does not match the value of τ 1 determined by Dr F Costen research group. The higher the discrepancy between the two dispersions the less accurate the relative permittivity and conductivity curves of the Debye model becomes. This can be observed by comparing figure 6.1(a) and figure 6.1(d) together. 6.4(ii) Conductivity Curves The parameter that affects the accuracy of the Debye model fitting the most is τ 1. τ 1 appears in both relative permittivity and conductivity Debye model expressions (equations (6.3) and (6.4)). As expected, the conductivity plot of the Debye model for all tissues is most accurate at DC (0Hz) frequency. The τ 1 parameter has no effect on the value of conductivity in DC because it is a coefficient of ω in conductivity expression. Therefore, the term ω2 τ 1 ε 1 in (6.4) will be zero. If the 1+(ωτ 1 ) 2 tissue s dielectric properties (relative permittivity ε r and conductivity σ) were not fitted accurately to Debye model then the conductivity curve will start to deviate from IT I data as frequency increases. The ε r Debye and IT I curves for Pons and Bone Marrow tissues appear to be similar, so it might be thought that the two tissues were fitted in the same accuracy. However, once the conductivity curves are inspected for both tissues in figure 6.1, it can be seen that Bone Marrow figure 6.1(a) had been fitted to the Debye model much more accurately, whereas the pons tissue Debye σ curve in figure 6.1(d) deviates from IT I conductivity curve. This deviation is most likely due to the rapid decrease in IT I ε r curve of pons tissue in the β-dispersion region (100MHz-300MHz). The relaxation time τ 1 determined is just a mean of the two dispersion regions (β and γ). As a result, 14

28 the inaccuracy was reflected in the Debye σ and ε r curves for Pons tissue. For Pons tissue, a second order Debye model is recommended because of the large discrepancy between the magnitudes of β-dispersion and γ-dispersion as it can be seen in figure 6.1. Finally, further analysis on the accuracy of curve fitting can be found in Appendix B. 7 Investigating the Relative Uncertainty in Relative Permittivity and Conductivity of Debye Model Using Monte-Carlo Method 7.1 oftware The aim of this work is to produce a wide range of Debye parameters for different tissues in order to be used in FDTD simulations, and investigate the relationship between the relative uncertainty of Debye parameters (u ε, u ε1, u τ1 and u σi ) and the relative uncertainty of relative permittivity ε r and conductivity σ. The definition of relative uncertainty u is discussed in section 5.2. The Monte Carlo program generates number of random Debye parameters of each tissue T of the 94 tissues. This means there will be random tissues of tissue T. The program is written in shell script, and it calls five different source files (small programs) written in Fortran 95. The five source files will be described in section 7.1(i). The algorithm and the runtime environment of the Monte Carlo software will be explained in section 7.2. There are two parameters that need to be specified by the user as an input. The first parameter is the relative uncertainty in the Debye parameters (u ε, u ε1, u τ1 and u σi ), and since these values are an input to the program, they will be called the relative input uncertainty u input of the Monte Carlo program. The other input parameter is the number of random samples of each Debye parameter required to be generated for each tissue T. In order to calculate u for the Debye parameters and use equation (5.5), the mean of each Debye parameter of each tissue T needs to be identified. The mean of tissue T s Debye parameters are the results obtained from the activity in chapter 6 of curve fitting IT I data to Debye model. Also note that u for all Debye parameters (u ε, u ε1, u τ1 and u σi ) will have the same value, and each cannot be assigned a specific value, since there is only one input for u in the Monte Carlo program. For example, if u input = 0.1 (10%) then u ε = 0.1, u ε1 = 0.1, u τ1 = 0.1 and u σi = 0.1. The reason of doing this is to simplify the program, and because the fact that the Debye parameters of the same tissue are most likely to have the same u. 15

29 7.1(i)ource Files This section will describe each source file used in the Monte Carlo program. ee Appendix C for the complete code of each program. A) Main.sh The main shell script manages and runs each small program. Full details of main are presented as a flowchart in figure 7.3 in section 7.2. B) Values: This text file contains all Debye parameters of the 94 tissues obtained by fitting IT I data to a single pole Debye model. The Values file has a similar layout to table A.1 in Appendix A.1, but without the headers and the tissue number column. The Debye parameters obtained from chapter 6 activity will be used as the mean value when calculating the standard deviation for each Debye parameter using equation (5.5). C) Random.f95: This source file accepts the four fitted Debye parameters of a tissue T as inputs, and generates number of Gaussian distributed random samples for each of the four fitted Debye model parameters as specified by the user in main.sh. Random.f95 program produces the Gaussian distributed samples of each fitted Debye parameter for tissue T using the Box Muller algorithm. For each of the four fitted Debye parameter, values are printed out in which the main.sh directs the output to the tissue T directory. More details about the runtime environment of the Monte Carlo program will be discussed in section 7.2. Converting a Uniformly Distributed Value to Gaussian Distributed Value Using Box Muller Transform In Fortran, there are two intrinsic (built-in) floating point pseudo random generators routines (methods) [24]. Both methods generate uniformly distributed values between 0 and 1 [24]. The first method is RAND() [24]. The second method is RANDOM_NUMBER(), and it uses a better algorithm [24]. As a result, the latter method will provide more realistic representation of uniform distribution. The intrinsic methods in Fortran produces only uniformly distributed values between 0 and 1. Unfortunately, the required distribution is Gaussian distribution with a specified mean and standard deviation. Consequently, Box-Muller transform will be used to convert a uniform distribution to the required Gaussian distribution with the desired mean (expected value) and standard deviation [25, 26]. The proof of the algorithm can be found in [25]. The Box-Muller transform is one of the easily implemented algorithms to produce a set of Gaussian distributed values from uniformly distributed values [26]. Moreover, the Box Muller transform is an 16

30 exact algorithm [26]. In other words, if the uniform distribution used in the conversion is ideal (flat horizontal line without fluctuations), then the standard Gaussian distribution produced by Box Muller will also be ideal. Even if the uniform distribution is not ideal, the algorithm is quite robust. An analysis on the accuracy of the Gaussian values obtained using Fortran s non-ideal random number generator can be found in Appendix D. In the first step of the algorithm, Random.f95 source file produces number of random values using RANDOM_NUMBER() function, and stores the samples in an array ψ [26]. ubsequently, the algorithm accepts two uniformly distributed random values U 1 and U 2 from ψ, and generates two normally (Gaussian) distributed values G 1 and G 2. The next step of the algorithm is to convert U 1 and U 2 into two intermediate constants. The first constant is calculated as follows R = 2ln (U 1 ) (7.1) where R is an intermediate constant [25, 26]. The other constant is calculated as Θ = 2πU 2 (7.2) where Θ is another intermediate constant [25, 26], and π is the mathematical constant with a value of approximately The values of R and Θ are then used to produce two standard Gaussian distributed values G 1 and G 2 [25, 26]. As outlined earlier, the Box Muller algorithm transforms uniformly distributed values to standard Gaussian distributed values (mean is zero and standard deviation is 1 unit). The two produced Gaussian values are calculated in [25] and [26] as follows G 1 = Rcos(Θ) (7.3) G 2 = Rsin(Θ) (7.4) The conversion of standard Gaussian distributed values G 1 and G 2 to non-standard Gaussian distributed values such as the Debye parameters is trivial, and it is explained in [14] as follows z = G ξ z + z (7.5) where G is either G1 or G2, z is any of the Debye parameters, z is the respective fitted Debye parameter obtained from the curve fitting activity in chapter 6, and ξ z is the standard deviation of the respective Debye parameter calculated using (5.5). The standard deviation of each Debye parameter (ξ ε, ξ ε1, ξ τ1 and ξ σi ) is calculated from (5.5) using the value of u input (u ε, u ε1, u τ1 and u σi ) specified by the user. For example, if the user specifies u input = 0.1 and ε of the fitted tissue T is 50, then ξ ε is 5. ubsequently, ε and ξ ε is substituted with z and ξ z in (7.5) respectively to obtained a Gaussian distributed sample of ε. 17

31 C) Er.f95: This small program calculates the values of ε r and σ over the frequency range of interest (100MHz- 6GHz) using 60 frequency samples F for one set of Debye parameters of tissue T. The values of ε r,f,t and σ for one sample of the samples of tissue T at F will be denoted as ε r and σ,f,t,f,t respectively. ε r and σ,f,t values are obtained by substituting the Debye parameters of the random tissue of tissue T in equations (6.3) and (6.4) respectively. The random Debye parameters of tissue T are passed to this source file from main.sh. The results of ε r,f,t and σ,f,t are printed out in which the main.sh will store them as appropriate. ee figure 7.2 and figure 7.3 for more details on how main.sh interacts with other source files. D) ED.f95: The purpose of this program is to calculate the mean and the standard deviation of ε r,f,t and σ,f,t. The mean of all the values of ε r,f,t for tissue T at frequency point F is calculated as follows ε F,T r = 1 ε s=1 r s,f,t (7.6) In (7.6), σ,f,t,f,t can be substituted with ε r to find σ F,T. Additionally, ED.f95 calculates the standard deviation of σ,f,t and ε r,f,t, and they will be detonated as ξ εr F,T and ξ σ F,T respectively. ED.f95 then prints out the results (ε F,T r, σ F,T, ξ F,T εr and ξ F,T σ ) in which the main.sh will place it in the statistics folder of the tissue T (see figure 7.2). Furthermore, the program uses (5.5) to calculate u F,T and u εr F,T. ED.f95 will run 60 times for σ the frequency range of interest producing 60 results of ε F,T r, σ F,T, ξ F,T εr and ξ F,T σ that can be used to calculate u F,T and u εr F,T at each F for tissue T. These 60 results of ε F,T σ r, σ F,T can also be used to plot the mean frequency response of tissue T over the frequency region of interest (see figure 7.1(b)). E) final_statistics.f95: This source files calculates the average of the 60 values of u F,T and u over the 60 frequency εr F,T σ points F for tissue T producing u T and u Trespectively. Mathematically, Tis u calculated εr F σ F εr F as follows u εr F F u ε f,t r Also, u F,T can be substituted with u σ ε F,T to calculate u T. r σ F T = 1 F f=1 (7.7) 18

32 F) tats.sh This program is an extension to the main.sh program that can be commented out. The main purpose of this extension is to group the results of final_statistics.f95 for the 94 tissues in one file in order to conduct the analysis of this work easily. 7.1(ii) An Example Relating All the Terms Mentioned in 7.1(i) uppose the Monte Carlo program was run for = 3. Then for tissue T, the Random.f95 file will use the fitted Debye parameters that were calculated in chapter 6 as the mean in equation (5.5) to calculate the standard deviation for each Debye parameter of T. After that, Random.f95 applies Box Muller algorithm to generate (3 in this case) Gaussian random samples of Debye parameters of tissue T. Er.f95 file is then used to calculate the values of σ,f,t,f,t and ε r in the frequency range of interest (100MHz-6GHz) at 100 MHz interval (60 F points) for the (3 in this case) random tissues of T.,F,T The result of Er.f95 calculations for ε r are shown in figure 7.1(a). Notice that the mean value of Debye parameters is regarded as one of the three samples, and hence, it is plotted in figure 7.1(a).,F,T Figure 7.1. An example relating ε r and ε F,T r together. ED.f95 needs to run 60 times to generate 60 points similar to point P in figure 7.1(b). The value of P is ε F,T r of tissue T at 2 GHz (F=20). The standard deviation of the 3 samples of tissue T at point F (for example, 2GHz) is denoted as ξ εr F,T. Finally, ED.f95 uses equation (5.5) to calculate u F,T of tissue T at each F (u εr ε r F,T = ξ εr F,T ). There will be 60 points of u over the frequency F,T εr F,T ε r range of interest (100MHz-6GHz). final_statistics.f95 will calculate the average of these 60 points using (7.7) producing T. u The terms σ F,T εr F, ξ F,T σ, u F,T and u T σ F are calculated the same σ way as ε F,T r, ξ F,T εr, u F,T and u T respectively. εr ε F r 19

33 7.2 The Overall Flow of Program 𝜎 𝑆,𝐹,𝑇 and 𝜀𝑟 𝑆,𝐹,𝑇 for a random sample of 𝑆 at all 60 𝐹. (𝑆 files) 𝜎 𝑆,𝐹,𝑇 and 𝜀𝑟 𝑆,𝐹,𝑇 for all 𝑆 at 𝐹. (60 files) Figure 7.2. The runtime environment of the Monte-Carlo program. The figure needs to be read in conjunction with figure 7.3. Figure 7.3. The flowchart of the Monte-Carlo program s Main.sh. 20

34 To help visualise the actual run time environment of main.sh, figure 7.2 is presented. In figure 7.2, the numbers above some of the arrows represents the step number, I represents input to a file, and O represents an output from a file. This is needed in order to simplify the description in figure 7.3. Figure 7.3 is a flowchart representing the flow of main.sh as it interacts with the different Fortran programs. The main.sh creates a directory to organise the results of the calculations. The directory s name is Tissues as shown in figure 7.2. The tissue specification folder is created by joining randomly generated Debye parameters of all the 94 tissues together creating tissue specification files. Each tissue specification file will contain a random set of Debye parameters for each of the 94 tissues. An example of tissue specification file is Appendix A.1. By creating tissue specification files one of the main aims of this report has been accomplished. 7.3 Results Analysis The main purpose of the Monte-Carlo program is to investigate the effect of varying u ε, u ε1, u τ1 and u σi (u input ) on the resulted relative uncertainty in ε r and σ (u F,T, u F,T, εr σ u T εr F and u T) for all 94 tissues. The analysis is broken down into different sections to σ F investigate the effect of varying different parameters as well as observing similar patterns, and trying to explain any differences. As mentioned earlier, two input parameters can be changed by the user in the Monte Carlo program described in section 7.1. The program was run for u input = 0.1, 0.3, 0.5, 0.7, 0.9 and 1.2. The program can accept one value of u input at a time. Therefore, the Monte Carlo program was run 6 times for the same number of. This is in order to investigate the effect of varying u input on u F,T, u εr σ T F,T, u ε F r and u T. Note that u F F,T, u σ εr σ T F,T, u ε F r and u T σ F terms are very important and will be used extensively in the rest of the report. To recall their meaning refer to section 7.1(i) and the example in 7.1(ii). Theoretically, the higher the value of, the more accurate the results of the Monte Carlo calculations are. The program was run for = 200,1000,2000 and For each value, the Monte Carlo method was run for u input = 0.1, 0.3, 0.5, 0.7, 0.9 and 1.2. The time taken to run the program increases with the increase of. Therefore, it is also desirable to investigate the effect of increasing on the results of the Monte Carlo program. This will be addressed in sections 7.3(vii) and 7.3(viii). Finally, the plots that will be shown in sections 7.3(i) to 7.3(vi) are obtained by running the Monte Carlo program for = 3000 and for u input = 0.1, 0.3, 0.5, 0.7, 0.9 and

35 7.3.(i) Investigating the Effect of Frequency on the Relative Uncertainty in ε r and σ a) The Relative Uncertainty in ε r as Frequency Varies u F,T ε r Frequency/GHz a) Fat b) Dura u F,T ε r Frequency/GHz u F,T ε r Frequency/GHz c) White Matter Figure 7.4. The variation of u over frequency for different tissues and for different values of εr F,T u input (u ε, u ε1, u τ1 and u σi ). The legends in all plots represent the values of u input. From figure 7.4, there is a general increase in u of the tissues as the frequency increases. The εr F,T pattern seems to be more apparent as the u input (u ε, u ε1, u τ1 and u σi ) gets larger. For example, when u input = 0.1, the increase in u can be hardly observed. On the other hand, when εr F,T u input = 0.7, the increase in u as frequency increases is more apparent. εr F,T The white matter tissue showed the greatest change in u εr F as frequency increases. Inspecting the Debye parameters (Appendix A.1) of the three tissues in figure 7.4 suggests that τ 1 is the main reason why white matter showed the greatest increase in value of u F,T. εr 22

36 u F,T εr Tissue Name τ 1 /ps White Matter Dura Fat Table 7.1. The values of u F,T taken from figure 7.4 at F=60 (6GHz) for u εr input =0.5. Table 7.1 shows an example that when τ 1 for a tissue is small, u will be smaller compared to εr F,T the other tissues. The other Debye parameters (ε, ε 1 and σ i ) did not show a visible trend with the decreasing value of u F,T. The pattern in table 7.1 suggests that τ εr 1 affect u of a tissue the εr F,T most. Recall that τ 1 is the main parameter that is responsible for ε r behaviour of a tissue over frequency. Higher values in τ 1 of tissue T compared to other tissues results in larger changes in ε r of tissue T as frequency increases. Larger changes in ε r should increase ξ F,T εr at each frequency point F of tissue T. Refer to chapter 6 for more details about the effect of varying τ 1 on the value of ε r. Recall that relative uncertainty u is related to the ratio of a standard deviation of a population to the population mean value (equation (5.5)). Hence, u F,T at each F is proportional to ξ F,T εr ε r. Therefore, the rise of ξ F,T εr because of the rise of τ 1 results u to have larger magnitude at εr F,T each F as shown in table 7.1. Finally, it is worth noting that figure 7.4 can be plotted with u input as x-axis and F points as legend resulting into 60 available curves. This is impractical if all the 60 points needs to be considered at once. ome of these 60 plots can be found in Appendix F.2. b) The Relative Uncertainty in σ as Frequency Varies u F,T σ Frequency/GHz a) Fat b) Dura u F,T σ Frequency/GHz

37 u F,T σ Frequency/GHz c) White Matter Figure 7.5. The variation of u over frequency for different tissues and for different values F,T σ of u input. The legends in all plots represent the values of u input From figure 7.5, the pattern in which the u varies with frequency is rather more complex than F,T σ that of u F,T presented in figure 7.4. Nonetheless, u εr σ F,T also increases with increasing u input. For all the tissues in figure 7.5, there is a notch between 1GHz and 2GHz. The behaviour of u beyond the notch (when frequency is larger than 2GHz) is different for each F,T σ tissue. This difference can be explained using the conductivity σ expression (6.4). Inspecting the Debye parameters of the three tissues in figure 7.5 shows that σ i is the main parameters responsible for the differences. Tissue Name σ i /m -1 u σ F,T Dura White Matter Fat Table 7.2. The values of u F,T taken from figure 7.5 at F=60 (at 6GHz) for u σ input =0.5. From table 7.2, σ i seems to be the main parameter responsible for the magnitude difference of u F,T between the three tissues. The larger σ σ i is, the shallower the notch is. This can be confirmed by the fact that Fat tissue in figure 7.5 has the sharpest notch of all three tissues because it has the smallest value of σ i. On the contrary, Dura and White Matter tissues have shallower notches compared to Fat as it can be seen in figure 7.5. This can be explained by the fact that Dura and White Matter tissues have higher value of σ i compared to Fat tissue. 24

38 It is worth mentioning at this point that u T σ F calculated using equation (7.7). The terms T u εr F sections. and u Tfor a certain value of u εr F input can be and u T F σ 7.3.(ii) The Relative Uncertainty in ε r and σ for All Tissues will be needed in the upcoming In the previous section, the variation of u F,T and u with frequency was analysed and εr σ F,T observed for three different tissues. For the same tissue, the variation of u with frequency εr F,T (figure 7.4) is different from the variation of u with frequency (figure 7.5). The results of F,T σ u T and u εr F T were collected for the 94 tissues. Each tissue was assigned a number based on its F σ alphabetical order. The numbering of each tissue can be found in Appendix A.1. a) The Relative Uncertainty in σ Case for All Tissues The value of u T for each tissue is plotted in figure 7.6 for various input relative uncertainties F σ u input (u ε, u ε1, u τ1 and u σi ). As mentioned earlier, the results shown in the main report are when = Plots for = 200,1000 and 2000 cases can be found in Appendix H u T σ F Ω T Ω T Ω Ω T Ω T (0.5) u σ F T (0.3) u σ F T (0.1) u σ F Tissue Number T Figure 7.6. u Tof all 94 tissues when u F σ input =0.1, 0.3, 0.5, 0.7, 0.9 and 1.2. In figure 7.6, as u input increases, u T of all tissues seems to increase by the same factor. For σ F example, when u input =0.1, u T σ F increased three times to become 0.3. The value of u T σ F of most tissues is approximately Now when u input is for each tissue T is increased approximately by a factor of 3 so that u T value for many tissues is approximately In F σ another perspective, the curves for different u input values in figure 7.6 are to be thought as the same 25

39 curve but scaled by different amounts. For example, the curve for u input = 0.3 in figure 7.6 is a scale of the curve for u input = 0.1. In figure 7.6, the curves for u input = 0.7, 0.9 and 1.2 appear to be shifted by an offset rather than scaled. Nonetheless, the curves are still a scale to each other, but the magnitude of the scale is close to one. For example, the scale between 0.7 curve and 0.9 curve is 1.29 which is close to 1, and hence, 0.9 appear to be shifted when compared with 0.7 curve rather than scaled. The same argument can be applied between curves 0.9 and 1.2 ( 1.2 = ). 0.9 The mean of u T for all the 94 tissues in figure 7.6 at a specific value of u F σ input is calculated as follows T = 1 T t T t=1 (7.8) u σ F u ε F r The standard deviation of 94 values of u T for a specific u F σ input is calculated as )/% F T 100 (ξ T/u uσ F σ T = 1 T (u T ε F t u T ) t=1 F 2 (7.9) r σ ξ u F σ u input Figure 7.7. The variation in the ratio of ξ T to u T u σ F F as u σ input changes. The maximum difference between any two points in figure 7.7 is 1.7%. This very small change of percentage in figure 7.7 proves that u T values of all the tissues with increasing u σ F input retain their relative spread to one another. In other words, when the mean value u T at a specific u F σ input increases by a factor Ω, then the standard deviation ξ u (the spread) also increases by a factor of T F σ Ω. This can be seen in figure 7.6, and deduced from figure 7.7. Mathematically, it can be proved with the following argument ξ u T(0.3) Ω ξ T(0.1) u F σ T u (0.3) = F σ Ω F T (0.1) u σ F σ = ξ u T(0.1) F σ T u (0.1) F σ (7.10) 26

40 Ω where 0.1 and 0.3 in ξ T and u T u σ F σ F represent their values at u input = 0.1 and 0.3, and Ω is calculated as Ω = T u (0.3) σ F T u (0.1) σ F (7.11) The scale factor used to scale u T of tissue T from u F σ input = k curve to u input = x curve will be denoted as Ω T, and it is calculated as follows k x Ω T k x = u T(x) σ F T(k) u σ F (7.12) The terms Ω k x and Ω T have been marked in figure 7.6 in order to draw a clear distinction k x between the two terms L L 2 Point Ω Tissue Number T Ω T Ω T Ω T Ω T Ω T Figure 7.8. The scale factor Ω (Ω T k x ) for each of the 94 tissues between different u input. Point 95 of each curve represents Ω k x. An example is shown in the figure. Note that there is no tissue number 95. Point 95 is added for comparison of Ω k x with Ω T k x. Figure 7.8 shows that Ω T k x that is required to scale u T σ F of tissue T from u input = k curve to u input = x curve is roughly the same for all tissues. For example, in figure 7.8, Ω T of tissue number 20 (L 1 ) is approximately the same as Ω T of tissue number 50 (L 2). Nonetheless, the concept of scale is just an approximation, and this approximation will not have good accuracy if Ω T k x > 3 as it can be seen in figure 7.8. This is because Ω Tk x when Ω T k x > 3, and the approximation of Ω Tk x longer valid. starts to fluctuate significantly being the same for all tissues becomes no 27

41 Inspecting figure 7.8 further shows that Ω T k x of most tissues is approximately the same as Ω k x (point 95) provided that Ω 3. As a result, equation (7.12) can be modified to Ω T k x = u T(x) T u (x) σ F T(k) σ F u σ F u T σ F (k) = Ω k x (7.13) This concept of scale factor Ω (equation (7.13)) is powerful, and it will be used to calculate u T F σ for any tissue at any value of u input instead of running the Monte Carlo calculations, since Ω T k x is the same for all tissues. Additionally, Ω T k x is approximately the same as Ω k x provided that Ω 3. b) The relative uncertainty in ε r case for all tissues The value of u T for each tissue is plotted in figure 7.9 for various values of u εr F input. As metioned ealier, the results shown in the main report are obtained by running the Monte Carlo program for = Plots for = 200,1000 and 2000 can be found in Appendix H u T F ε r Tissue Number T Figure 7.9. The value of T u of all 94 tissues when u εr F input = 0.1, 0.3, 0.5, 0.7, 0.9 and 1.2. It is clear that figure 7.9 is different from figure 7.6. This is expected because of the fact that the expression of conductivity σ (6.4) is different from that of relative permittivity ε r (6.3). Inspecting figure 7.6 and figure 7.9 shows that u T εr F usually have a lower value than u T σ F for the same u input. The reason is that the variation of u over frequency is much more significant than the σ F,T variation of u over frequency, and this can be deduced by comparing figures 7.4 and 7.5 εr F,T together. Although some tissues have higher u T than u T, u εr F σ F T is less than u εr F T for any F σ u input. u T is also calculated the same way as u εr F T was calculated using equation (7.8). σ F 28

42 100 (ξ T/u T )/% uε F F r εr Ω u input Figure The variation in the ratio of ξ T to T u u as u F εr ε F input changes. r The six curves in figure 7.9 appear to be the same but scaled by some factor of Ω. The scale factor Ω for ε r is defined the same way as in the equation (7.11) and (7.12). Figure 7.10 also proves that all tissues are scaled by the same amount from the fact that the ratio of ξ u T to u T hardly F F εr ε r changes as u input increases Ω Ω T Ω T Ω T Ω T Ω T Tissue Number T Figure The scale factor Ω (Ω T k x ) for each of the 94 tissues between different u input. Point 95 of each curve represents Ω k x. An example is shown in the figure. Note that there is no tissue number 95. This point is added for comparison of Ω k x with Ω T k x. From figure 7.11, the value of Ω T k x also remains the same for all tissues in the case of ε r. It is actually more stable even when Ω > 3. Additionally, there is hardly any difference between Ω T k x and Ω k x. Furthermore, the Ω k x for T u and u εr F T are almost the same when Ω 3. This can F σ be confirmed by comparing figures 7.11 and 7.8 together. As a result, there is really no need to distinguish between Ω for ε r cases and Ω for σ cases when Ω is discussed. 29

43 u 7.3.(iii) Mean Curves T T u = x x x x σ F u εr F = x x x x u T σ F T u F εr Curve Fitted T u (4 σ F th order) Curve Fitted T u (4 F th order) εr u input Figure The variation of u T and u εr F T as u σ F input varies. The variable x in the curve fitted equations represents u input. It is desirable to investigate the variations in u T and u εr F T for all tissues as u σ F input of the Monte Carlo program varies. The variation of u T εr F and u Tas u σ F input varies for each tissue follows a similar pattern to figure These plots can be found in Appendix G. Hence, it is most sensible to plot T u and u εr F T as u F σ input varies for the purpose of calculating Ω k x, and hence Ω T k x. The value of Ω k x is sufficient because Ω k x is approximately the same as Ω T. It is also worth k x noting that Ω k x in σ cases is approximately the same as Ω k x in ε r cases. Nevertheless, Ω k x in σ cases will be calculated from only u T values, and the same for Ω F σ k x in ε r cases. This is in order to have high number of accurate decimal places. However, if one or two decimal places are needed then Ω k x obtained from one curve is sufficient to calculate either u T or u εr F T. F σ In figure 7.12, u T and u εr F T curves are fitted to model equations using a curve fitting tool. A σ F fourth order model is quite accurate for both u T and u εr F T, and higher orders are not really F σ needed. The purpose of finding a model equation for T u and u εr F T so that the scale factor F σ Ω k x can be calculated using software programs. Recall that Ω T k x required to move u T or ε F r u T of tissue T from one u σ F input = k to u input = x is approximately the same for all tissues, and therefore, Ω k x is approximately equal to Ω T (equation (7.13)). As a result, the curves in figure k x 30

44 Ω Ω Ω Ω 7.12 will be used to calculate Ω k x and hence, Ω T. Refer to section 7.3(ii)(a) for the definitions k x of Ω k x and Ω T k x. 7.3(iv) Variation of Ω of ome Tissues with Frequency Ω F Ω F Ω F Ω F Ω F Ω F Frequency/GHz Frequency/GHz a) Dura u F,T εr b) Dura u F,T σ Ω F Ω F Ω F Ω F Ω F Ω F Frequency/GHz Frequency/GHz c) Fat u F,T εr d) Fat u σ F,T Figure The variation of scale factor Ω of u F,T and u over frequency for some tissues. εr σ F,T From figure 7.13, the scale factor Ω is hardly affected by the values of u F,T and u as the εr F,T σ frequency varies. Ω F k x at a frequency point F for u is calculated as follows ε F,T r u Ω F k x = εr F,T (x) u F,T (k) εr (7.14) Equation (7.14) also applies to u σ F,T. Comparing figures 7.13(a) and 7.13(c) with figure 7.11, and figures 7.13(b) and 7.13(d) with figure 7.8 leads to the following approximation in case of σ and ε r Ω k x and Ω T k x Ω F k x Ω Tk x Ω k x (7.15) are defined in equations (7.11) and (7.12) respectively. Also, the plots in figure 7.13 shows that Ω F k x for σ and ε r cases are approximately the same. For example, Ω F in figure 7.13(a) is approximately the same as Ω F in figure 7.13(b) at any frequency

45 It has been discussed in the previous sections that when the value of Ω > 3, Ω fluctuates significantly (see figures 7.8 and 7.11). Hence, the approximation in (7.14) will also not be valid when Ω > 3. As a consequence, any value of Ω F >3 has not been considered in figure 7.13 plots. k x The approximation in (7.14) is quite powerful and convenient when dealing with large numbers of tissues. As a consequence, a simple and much faster software than the Monte Carlo software described in section 7.1 can be constructed to estimate u F,T and u for any tissue and at any εr σ F,T frequency point F. The next sections will describe the algorithm of such software. 7.3(v) The Relationship between x/k and Ω The purpose of figure 7.14 plot is to show that the value of x/k is not the same as Ω k x, in particular at values far greater than 1. x and k represents the values of u input at the desired and current u input respectively. Refer to section 7.3.(ii)(a) for more explaination. The plot in figure 7.14 has been generated using the mean curves plotted in figure It can be seen that the gradient of the fitted linear lines is not 1. However, the fact that a linear relationship exist between x/k and Ω k x provides a quick verification tool to verify the values of k and x in Ω k x. 3.5 Ω 3 k x = x k Ω k x = x k Ω k x ε r case σ case Linear fit for ε r case Linear fit for σ case x/k Figure The relationship between Ω k x and x obtained using the mean curves in figure k Additionally, Ω k x should not be approximated using x/k linear relationship shown in figure This is because the approximation will introduce doubts in Ω k x accuracy by the fact the linear fitted curves shown in figure 7.14 are not fitted as accurate as the fitted mean curves shown in figure Furthermore, it can be deduced from this plot that Ω k x for ε r is approximately the same for σ, and that is why there is no distinction made when Ω k x is discussed. However, in order to have high number of accurate decimal places, Ω k x for each case will be calculated from its respective fitted curve in figure

46 7.3(vi) Using the Mean Curves to Estimate the Relative Uncertainty of Any Tissue This section is one of the most important sections of this work, and it will provide a method of estimating the value of T u εr F and u T for tissue T, the value of F T u σ εr F and u T of any tissue at any u F σ input. In order to estimate u T εr F and u T for one u F σ input needs to be known. The other information needed is the scale factor Ω k x that is required to scale T u εr F and u T from the F σ known value of u input = k to the desired value of u input = x. The scale factor Ω k x will be calculated using the fitted model equations in figure An example will be shown shortly. The values of u T and u εr F T for u σ F input = 0.5 is a good point to be chosen and stored. T u εr F and u T for u F σ input = 0.5 can be determined by Monte Carlo method using the program described in section 7.1, or the GUM method that will discussed in chapter 8. The range of u input used to conduct the analysis in this section (section 7.3) is from 0.1 to 1.2. Recall that the Monte Carlo program in section 7.1 was run for u input = 0.1, 0.3, 0.5, 0.7, 0.9 and 1.2. u input = 0.5 is chosen because it is one of the closest points to the midpoint in the range of u input used. This is to reduce the value of Ω as much as possible. If figures 7.8 and 7.11 were to be inspected, the value of Ω fluctuates more significantly when it is larger than 3 compared to the values when Ω is less than 3. A B C D E=D/B F G=E A u u εr F εr F u at at εr F u input = u input = u input = x from 0.5 x x Monte-Carlo u T εr F at u input = 0. 5 u Tat u εr F input = x from estimation Tissue Name Ω 0.5 x Bladder Pons Fat Dura Table 7.3. hows how to use the mean curves (figure 7.12) to estimate T u for the desired εr F u input = x of tissue T. In table 7.3, column A represents the values of u T of tissue T at u εr F input = 0.5. As mentioned earlier, u T of all tissues for u εr F input = 0.5 needs to be obtained and stored using Monte Carlo Method or the GUM method. 33

47 The next step is to specify the desired relative input uncertainty x in which Tneeds u to be εr F estimated at. The value of T u for u εr F input = 0.5 and x needs to be calculated using the fitted model equation in figure The two values of u T at u εr F input = 0.5 and x are used in equation (7.11) to calculate Ω 0.5 x. Using the assumption in (7.13), Ω 0.5 x Ω T. As a result, column E in table 7.3 is multiplied 0.5 x by column A to obtain the estimation in column G. As it can be observed in table 7.3, column F and G are approximately equal to each other. This process can be repeated for u T cases using the σ F mean curve model of u T in figure σ F The algorithm can be extended to find u F,T or u εr σ F,T at any F and for any u input. The value of u F,T and u εr σ F,T at u input = k for each F also needs to be known and stored so that the algorithm can be applied. For the range of u input used, k (the stored u input value) is preferred to be 0.5. A B C D E F=E/C G H=B F T T Value of F/GHz u at σ F,T u σ F at u input = 0.5 x u σ F at u input = x u at σ F,T u input = x from Monte- Carlo u at σ F,T u input = 0. 5 Ω 0.5 x u input = x from estimation Table 7.4. hows some examples of how to use the algorithm to estimate u of Dura tissue for F,T σ different u input values. The algorithm can sometimes become inaccurate at estimation of u F,T or u in particular for εr F,T σ u F,T case. Nonetheless, it still provides a good estimation of u σ ε F,T or u over frequency. Note r σ F,T that the assumption of Ω 0.5 x Ω F 0.5 x is not as accurate as the assumption Ω 0.5 x Ω T 0.5 x. This can be confirmed from the slightly inaccurate value shaded in table 7.4, or from the fact that Ω F k x Ω T k x changes more significantly over frequency (see figure 7.13(b)) compared the change of as T number changes (see figures 7.8 and 7.11). In order to increase the accuracy of the estimation, Ω k x needs to be as small as possible. Consequently, it might be a good idea to store u Tand u Tat multiple values of u F σ εr F input, and not just at 0.5. The linear relationship between x k and Ω k x discussed in section 7.3(v) can easily help in choosing the right value of u input depending on the application this algorithm is used for. 34

48 Figure The flowchart of the algorithm. Finally, the flow chart in figure 7.15 summaries the algorithm mentioned in this section. The values of u T and u εr F T for all tissues at u F σ input = 0.1,0.3,0.5,0.7,0.9 and 1.2 is provided in Appendix H if the algorithm needs to be tested with other combinations not discussed in table 7.3. Not to mention, Ω 0.5 x in ε r case and σ case are close to each other as it can be seen in figure 7.14, so one value of Ω 0.5 x can also be used if less number of decimal places needs to be matched with the Monte Carlo results. 7.3(vii) Effect of on The Results of Relative Uncertainty u T F ε r u T σ F Number of amples Number of amples a) ε r case b) σ case Figure The effect of varying on u T and u εr F T for u σ F input = 0.1,0.3,0.5,0.7,0.9 and 1.2. The results displayed in sections 7.3(i) to 7.3(vi) are obtained by running the Monte Carlo program described in section 7.1 for = Refer to section 7.1 for the definition of. The program in 35

49 Time taken to process one tissue T/seconds section 7.1 was run further for = 200,1000 and 2000 samples with the same combinations of u input = 0.1,0.3,0.5,0.7,0.9 and 1.2. Ideally and according to (5.6), needs to be infinite in order to have 100% convergence probability for T u and u εr F T at any u F σ input. This is because the larger becomes, the more accurate the Gaussian distribution and any other probability distribution can be modelled. From figure 7.16, u T and u εr F T values remain roughly constant when > 1000 for any u F σ input values. This means that computation time can be significantly reduced since there will be no need for large values of. Time analysis will be discussed in more details in the next section. 7.3(viii) Time Analysis It is useful to know how the Monte Carlo software described in section 7.1 behaves as the number of samples varies. The results in this section can be used to compromise between accuracy and performance, and predict the time in which the calculations of the Monte Carlo program finishes. It is worth mentioning that Linux Mint is the system that is used to run the Monte Carlo software described in section 7.1, and obtain the results shown in section 7.3. Linux Mint was run on a virtual machine. The machine hardware had 16 GB of DDR3 RAM clocked at 1600MHz, and a quad core processor clocked at 2.7 GHz. The virtual machine was given access to two cores and 4GB of RAM Number of amples Figure The effect of varying on the time taken to process one tissue T, i.e., the time taken for main.sh to run from step1 to step11 as shown in figure 7.3. The time taken to process one tissue represents the time taken for main.sh to run from step 1 to step 11 once in figure 7.3. The main.sh will run step 1 to step times, processing all the 94 tissues. The average of the 94 runs for steps1-11 is the y-axis in figure

50 From figure 7.17, the time taken to process one tissue is approximately directly proportional to. The more samples are there, the more time is needed to process one tissue. The time taken does vary from system to system and from hardware to hardware. However, the relationship between and processing time of one tissue should remain the same provided all the other conditions are kept constant. 7.4 General Discussions for Chapter 7 The Monte-Carlo analysis for the relative uncertainty u in ε r and σ is one of the most reliable methods in engineering and scientific applications to estimate u and ξ in model equations such as Debye model. A program has been written to apply Monte Carlo technique in estimating u in ε r and σ at different levels (u F,T, u εr σ T, u T, F T u and u σ εr F T ). The program generates number of F σ F,T, u ε F r normally (Gaussian) distributed random samples of Debye parameters for each of 94 tissues, and hence, fulfilling one of the aims of this work. is one of the inputs specified by the user. The other input is the relative uncertainty in the Debye parameters (u ε, u ε1, u τ1 and u σi ) which was simply referred to in section 7.1 as u input. Furthermore, it was shown that frequency has very little effect on u for most tissues. On the εr F,T other hand, the variation of u with frequency is more rigorous over the frequency range σ F,T 100MHZ to 6GHz. In addition, the relationship between the relative uncertainty in the Debye parameters u ε, u ε1, u τ1 and u σi (u input ), and T u and u εr F T has been established by fitting the relationship σ F to 4 th order model equations shown in figure Moreover, the scale factor Ω T k x for u T or u ε F T is the same for all tissues. As a result, an F r σ algorithm has been provided to take an advantage of this deduction. The algorithm provides accurate estimation of u T or u εr F T of any tissue for any value of u σ F input. The algorithm can be extended to calculate u F,T and u εr σ F,T of any tissue at any F and for any u input. Nevertheless, it is not very accurate because of the significant behaviour of Ω F over frequency, but the algorithm k x still provide an acceptable estimation for u F,T and u εr σ F,T. The drawback of this algorithm is that an accurate fitted model equation of the relationship between u input, and u T and u εr F T is σ F 37

51 needed. This means that the Monte Carlo method needs to run at least 5-6 times to obtained a relationship similar to the 4 th order model equations in figure The value of used in the Monte Carlo computations is clearly important to determine the confidence in the results. This was also mentioned in section 5.3. When > 1000, T u and εr F T remains approximately unchanged. This is important because the time taken to process one u σ F tissue is directly proportional to the number of samples, and hence a compromise can be made between accuracy and performance. 8 Estimating the Relative Uncertainty Using GUM Method, and Comparing It with Monte Carlo Method A simple program has been written in order to use the GUM method to estimate the relative uncertainties in ε r and σ for different tissues and different values of u input. The GUM program has two inputs. The first input is the fitted Debye parameters (ε, ε 1, τ 1 and σ i ) of a tissue T. The second input is the relative uncertainty in Debye parameters (u ε, u ε1, u τ1 and u σi ) which will also F,T be referrred to as u input. Note that the GUM method provides an analyitical expression for ξ σ and ξ εr F,T without the need of performing number of trials at each F like the Monte Carlo program in chapter 7. o when expressing the GUM results, the superscript will drop from all the parameters. The first step in the GUM method is to find the paratial derivatives of ε r and σ with respect to each Debye parameter. The mathematical expressions of found in Appendix I. σ, σ, σ, σ, ε r, ε r, ε r and ε r ε ε 1 τ 1 σ i ε ε 1 τ 1 σ i can be ubsequently, the values of ξ σ F,T and ξ εr F,T are calculated using equation (5.9). The values of ε r F,T and σ F,T of T at each F can be calculated by substituting the fitted Debye parameters of T obtained from chapter 6 in expressions (6.3) and (6.4) respectively. Upon obtaining ξ σ F,T, ξ εr F,T, ε r F,T and σ F,T, equation (5.5) can be applied to calculate u εr F,T and u σ F,T at F. This process is repeated for the other 60 values of F. The 60 values of u εr F,T and u σ F,T can then be used to find u T εr F and u T σ F of tissue T. A flowchart of the GUM program is displayed in figure 8.1. The code of the GUM program can be found in Appendix J. 38

52 Figure 8.1. Flowchart showing the steps of the GUM program. 8.1 Comparing GUM and Monte Carlo Methods relative uncertainties in ε r and σ for different tissues. 8.1(i) Relative Permittivity Case In figure 8.2, the gradient of the GUM curve is almost the same as the linear region of the Monte Carlo curves for the three tissues. The linear region of Monte Carlo curves is when 0.1 u input 0.5. The GUM curve of Dura and Fat tissues in figure 8.2 almost coincide with the linear region of their respective Monte Carlo curve. This is a very strong indication that the Monte Carlo program in section 7.1 has been written correctly. The GUM curve of Pons tissue in figure 8.2 is slightly shifted downward. In chapter 6, it was mentioned that the Pons tissue is one of the least accurately fitted tissue to single pole Debye model (4.6). Nonetheless, the fact that even the gradient of the GUM curve matches the gradient of the linear region of the Monte Carlo curve for Pons tissue gives a strong confidence in the Monte Carlo results. 39

53 Relative Uncertainty Relative Uncertainty Relative Uncertainty u input a) Dura b) Fat c) Pons Figure 8.2. Compares Tin u Monte Carlo with u Tin εr F εr F GUM for some tissues at different values of u input. Monte Carlo GUM Monte Carlo GUM u input As mentioned in section 5.4, assuming independence between the Debye parameters of the Debye model forces the covariance between them to be zero. As a result, equation (5.7) was simplified to (5.9), and that is why the GUM method curves in figure 8.2 has linear responses. Recall that u input is proportional to standard deviation of the Debye parameters (equation (5.5)). Generally, the higher the standard deviation the larger the covariance [14]. Therefore, the value of correlation coefficient ρ (equation (5.8)) between any two Debye parameter increases when u input increases u input Monte Carlo GUM The fact that Monte Carlo curves in figure 8.2 are non-linear suggests that the Debye parameters are correlated to each other, and significant values of negative correlation coefficients exist between some if not all Debye parameters. [7] has also stated that the Debye parameters are correlated to each other. In addition, the Monte Carlo curves in figure 8.2 become non-linear when u input 0.5 because the correlation coefficient values become significantly negative when u input 0.5. Recall that the correlation coefficient can take a value between -1 and 1. 40

54 Relative Uncertainty Relative Uncertainty Relative Uncertainty In addition, it was mentioned in section 5.5 that the non-linearity in ε r expression limits the accuracy of GUM results. As a result, it is difficult to decide which method is more accurate in the linear region (0.1 u input 0.5). Luckily, both methods strongly agree with each other when u input is between 0.1 and (ii) Conductivity case From figure 8.3, there is a strong match between the GUM and Monte Carlo curves in the linear region of Monte Carlo curve (0.1 u input 0.5). The gradient of the GUM curve is generally the same as the linear region gradient of the Monte Carlo curves in figure 8.3. Additionally, significant correlation exist between the Debye parameters. Otherwise, the Monte Carlo curve would also be linear. The correlation coefficients between the Debye parameters become more negative as u input increases. As a result, the Monte Carlo curve becomes nonlinear when u input high enough (>0.5) as shown in figure Monte Carlo 1.2 Monte Carlo 1 GUM 1 GUM u input a) Dura b) Fat u input c) Pons Conductivity Figure 8.3. Compares u Tin Monte Carlo with u Tin σ F σ F GUM for some tissues at different values of u input Monte Carlo GUM u input Additionally, it is worth noting that the non-linearity in σ expression makes it difficult to decide which method is more accurate. GUM provides an exact term of the standard deviation, but lack of 41

55 information about the correlation between the Debye parameters as well as the non-linearity in σ expression places some doubts in the degree of accuracy in u T. σ F Luckily, both methods (GUM and Monte Carlo) generally match each other when 0.1 u input 0.5, where the value of correlation coefficients between Debye parameters are not significantly negative. Even when both methods do not match each other in the linear region, the difference between both methods is usually small. From figures 8.2 and 8.3 this difference does not exceed approximately The 0.06 difference happens at the boundary between the linear and non-linear region, i.e., when u input = 0.5, and it decreases as u input decreases. 8.2 Comparing GUM and Monte Carlo Methods Relative Uncertainty in ε r and σ Over Frequency 8.2(i) Relative Permittivity Case u ε r F,T u ε r F,T Frequency/GHz Frequency/GHz a) Dura b) Fat Figure 8.4. The variation in GUM u εr F,T over frequency for some tissues at different values of u input. From figure 8.4, the variation in GUM u εr F,T as frequency varies is very close to the behaviour resulted from the Monte Carlo trials shown in figure 7.4. If u input 0.5 then GUM u εr F,T will approximately have the same value as Monte Carlo u F,T. On the contrary, GUM u εr ε F,T r for u input > 0.5 curves is larger than Monte Carlo u εr F,T. The reason of this difference is mainly because of the correlation coefficient between the Debye parameters becomes significantly negative when u input > 0.5. Consequently, the Monte Carlo response deviates from the GUM response as outlined in section

56 u σ F,T 8.2(ii) Conductivity case a) Fat b) Dura Figure 8.5. The variation in GUM u σ F,T over frequency for some tissues at different values of u input Frequency/GHz The effect of frequency variation on u σ F,T in figure 8.5 is also very similar to the Monte Carlo results in figure 7.5. In particular, the notch that is between 1GHz and 2GHz. This notch exists in both figure 8.5 and figure 7.5. Furthermore, when the frequency is larger than 2GHz, the shape of the plots in figure 8.5 and figure 7.5 strongly agree with each other. Obtaining similar plots from entirely two different methods (Monte Carlo and GUM) is a strong indication that both methods has been implemented correctly u σ F,T Frequency/GHz Nonetheless, the values of the u σ F,T in figure 8.5 have higher values than their respective Monte- Carlo results shown in figure 7.5 when u input > 0.5. This also because of the first order approximations of the GUM expression that assumes no significant correlation between Debye parameters. 43

57 8.3 General Discussion of Chapter 8 To sum up, the GUM method is another way of calculating the relative uncertainty u in ε r and σ. The GUM method is faster and relatively involves much fewer calculations when it is compared to Monte Carlo. The GUM method takes around 2 seconds to complete all the steps in figure 8.1. By completing all the steps in figure 8.1, one tissue has been processed. The timing code for GUM program can be found in GUM.sh shell script file in Appendix J.1. On the contrary, the Monte Carlo program required around 210 seconds to process one tissue for = 3000 by completing steps 1 to 11 shown in figure 7.3. The timing code for Monte Carlo program can also be found in main.sh shell script file in Appendix C.1. Clearly, the GUM method is much faster than the Monte Carlo method. This conclusion is expected because GUM method does not repeat the calculations times. Additionally, the GUM method is quite accurate for regions where u input is low (typically below 0.5) by the fact it agrees with the Monte Carlo results in this range. This is because the correlation terms can be ignored when calculating the standard deviation using equation (5.9). However, care should be taken when using the GUM method by ensuring that the correlation coefficient between any two Debye parameters is not significantly negative. Otherwise, the GUM linear estimation (equation (5.9)) does not become valid. The large confirmation between the results obtained from Monte Carlo and GUM makes it possible to use GUM to work out an accurate point of T u and u εr F T at u σ F input = 0.5 then apply the algorithm in section 7.3(vi) to work out u T εr F and u T for any tissue at any u F σ input. Consequently, the calculation intensive Monte Carlo method can be completely avoided. Nevertheless, the Monte Carlo method is a very robust and easy method, and can be used with many applications without the need of working out the correlation between the model parameters. 44

58 9 Conclusions In the FDTD method, the Debye model is used to simulate the relative permittivity and conductivity of a specific patient. The Debye parameters are patient specific due to body shape, size, age and heterogeneous nature of biological tissues. Measuring the Debye parameters is an expensive process so alternatively random number generators can be used to generate a wide range of Debye parameters for FDTD simulations. Using the random number generators places doubts (uncertainty) in the true values of the Debye parameters of the patient and the randomised Debye parameters assigned to the patient. As a result, it was desirable to investigate how the standard deviation in the Debye parameters affects the standard deviation in the relative permittivity and conductivity using the GUM and Monte Carlo methods. Achievements The first aim of this work has been met by creating a program in order to generate a certain number of randomly Gaussian distributed Debye parameters of a specific tissue. ubsequently, using Monte Carlo method, a relationship between the standard deviation (and u) of the Debye parameters and the standard deviation (and u) of ε r and σ was established. This relationship has been fit to 4 th order model equation. Moreover, the GUM method results showed strong agreement with Monte Carlo method when the relative uncertainty u in the Debye parameters is less than 50%. Therefore, the usage of the slow Monte Carlo method can be completely avoided in this range. At this stage, the other main aim of this work has been met. However, while conducting the investigation, an algorithm has been devised utilising the fitted 4 th order model in order to estimate the standard deviation in ε r and σ of any tissue. This algorithm is faster than Monte Carlo and simpler than GUM method. The drawback of this algorithm that it requires at least 5-6 runs from either Monte Carlo or GUM method to establish a relationship similar to the 4 th order fitted model equations. Further Work One of the main areas that can be improved in this work is to optimise the code used to generate the random Debye parameters as well as the code used to conduct the Monte Carlo analysis. Although the Box Muller algorithm is exact, it is not recommended when very large number of Monte Carlo trials are needed because it relies heavily on the non-linear sine and logarithmic functions. With 3000 samples, the program needed 210 seconds to process one tissue. Using the linear relationship established earlier, 7000 seconds (1.94 hours) are needed to process one tissue if 10 5 samples are used. If 0.01 seconds can be reduced from each trial, then the performance can be reduced by 1000 seconds (16 minutes). However, in this work, accuracy was more important than performance. Another area can be tackled is assigning different values of u to each Debye parameter (u ε, u ε1, u τ1 or u σi ), and investigate its effect on the relative uncertainties of ε r and σ. Finally, the steps performed in this work that lead to the design of the simple algorithm ustilising the fitted 4 th order model equations can be repeated with other models, not just the Debye model. 45

59 References [1] M. Lazebnik and J. H. Booske, Highly Accurate Debye Models for Normal and Malignant Breast Tissue Dielectric Properties at Microwave Frequencies, IEEE, [2] M. Ibrani, E. Hamiti and L. Ahma, The Age-Dependence of Microwave Dielectric Parameters of Biological Tissues, [3] D. Miklavcic and N. Pavselj, Electric Properties of Tissues, University of Ljubljana, Ljubljana, [4] A. Peyman,. Holden and C. Gabriel, Dielectric Properties of Tissues at Microwave Frequencies, [5] A. Peyman, Dielectric properties of tissues; variation with age and their relevance in exposure of children to electromagnetic fields; state of knowledge, Oxon, [6] M. J. chroeder, A. adasiva and R. M. Nelson, An Analysis on the Role of Water Content and tate on Effective Permittivity Using Mixing Formulas, Biomechanics, Biomedical and Biophysical Engineering, vol. 2, no. 1, [7]. Gabriel, R. W. Lau and C. Gabriel, The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues, King's College, London, [8] G. C. R. Melia, Electromagnetic Absorption by the Human Body from 1 to 15 GHz, The University of York, York, [9] C. Gabriel, Dielectric Properties of Biological Tissue: VariationWith Age, London, [10] M. Lazebnik, M. C. Converse, J. H. Booske and. C. Hagness, Ultrawideband temperaturedependent dielectric properties of animal liver tissue in the microwave frequency range, Institution of Physics, [11] R. F. Reinoso, B. A. Tefler and M. Rowland, Tissue Water Content in Rats Measured by Dessiccation, [12]. Gabriel, R. W. Lau and C. Gabriel, The dielectric properties of biological tissues: II.Measurements in the frequency range 10 Hz to 20 GHz, London, [13] J. Vorlicek, L. Oppl and J. Vrba, Measurement of Complex Permittivity of Biological Tissues, Czech Technical University, [14]. Ross, A FIRT COURE IN PROBABILITY, Upper addle River: Pearson Prentice Hall, [15] D. J. Rumsey, tatistics Essentials For Dummies, Hoboken: Wiley Publishing,

60 [16] JCGM, Evaluation of measurement data Guide to the expression of uncertainty in measurement, JCGM, [17]. Bell, A Beginner's Guide to Uncertainty of Measurement, National Physical Laboratory, Middlesex. [18] C. Gabriel and A. Peyman, Dielectric measurement: error analysis and assessment of uncertainty, Berkshire: Institution of Physics, [19] I. A. Macdonald, J. A. Clarke and P. A. trachan, Assessing Uncertainty in Building imulation, University of trathclyde, Glasgow. [20] JCGM, Evaluation of measurement data upplement 1 to the Guide to the expression of uncertainty in measurement Propagation of distributions using a Monte Carlo method, JCGM, [21] P. R. G. Couto, J. C. Damasceno and. P. d. Oliveira, Monte Carlo imulations Applied to Uncertainty in Measurement, INTECH, [22] M. Azpurua, C. Tremola and E. Paez, Comparison of The GUM and Monte Carlo Methods for the Uncertainty Estimation in Electromagnetic Compatibility Testing, Caracas, [23] P. Hasgall, F. Di Gennaro, C. Baumgartner, E. Neufeld, M. Gosselin, D. Payne, A. Klingenböck and N. Kuster, IT I Database for thermal and electromagnetic parameters of biological tissues, Zurich: IT'I Foundation, [24] GCC, The GNU Fortran Compiler, GCC, [Online]. Available: [Accessed 22 April 2016]. [25] J. Goodman, Lecture Notes on Monte Carlo Methods. Chapter 2: imple ampling of Gaussians, Courant Institute of Mathematical ciences, New York, [26] D. B. Thomas, W. Luk, P. H. W. Leong and J. D. Villasenor, Gaussian Random Number Generators, ACM Computing urveys,

61 Appendix A: Curve Fitted Debye Parameters Values A.1 Results of Activity in Chapter 6 Tissue Number Tissue Name ε ε 1 τ 1 /ps σ i /m -1 1 Adrenal Gland Bile Blood Blood erum Blood Vessel Wall Bone (Cancellous) Bone (Cortical) Bone Marrow (Red) Bone Marrow (Yellow) Brain Brain (Grey Matter) Brain (White Matter) Breast Fat Breast Gland Bronchi Cartilage Cerebellum Cerebrospinal Fluid Cervix Commissura Anterior Commissura Posterior Connective Tissue Diaphragm Ductus Deferens Dura Epididymis Esophagus Eye (Cornea) Eye (Lens) Eye (clera) Eye (Vitreous Humor) Eye Lens (Cortex) Eye Lens (Nucleus) Fat Fat (Average Infiltrated) Fat (Not Infiltrated) Gallbladder Heart Lumen Heart Muscle Hippocampus Hypophysis Hypothalamus Invertebral Disc

62 44 Kidney Kidney (Cortex) Kidney (Medulla) Large Intestine Large Intestine Lumen Larynx Liver Lung Lung (Deflated) Lung (Inflated) Lymph node Mandible Medulla Oblongata Meniscus Midbrain Mucous Membrane Muscle Nerve Ovary Pancreas Penis Pineal Body Placenta Pons Prostate alivary Gland AT (ubcutaneous Fat) eminal vesicle kin mall Intestine mall Intestine Lumen pinal Cord pleen tomach tomach Lumen Tendon\Ligament Testis Thalamus Thymus Thyroid Gland Tongue Tooth Tooth (Dentine) Tooth (Enamel) Trachea Ureter\Urethra Urinary Bladder Wall

63 91 Urine Uterus Vagina Vertebrae Air N/A N/A N/A N/A 96 Blood Plasma N/A N/A N/A N/A 97 Bronchi lumen N/A N/A N/A N/A 98 Esophagus Lumen N/A N/A N/A N/A 99 Lymph N/A N/A N/A N/A 100 Pharynx N/A N/A N/A N/A 101 Trachea Lumen N/A N/A N/A N/A 102 Water N/A N/A N/A N/A Table A.1. The Debye parameters obtained from curve fitting activity in chapter 6. Note that this is the Values file that is used by the Monte Carlo program described in section

64 A.2 Research Group Fitted Tissues Tissue Number Tissue Name ε ε 1 τ 1 /ps σ i /m -1 1 White Matter Cerebellum Midbrain Cornea Thalamus Hypothalamus Tongue Cerebrospinal Fluid Urinary Bladder Colon(Large Intestine) Esophagus Gallbladder Right kidney Liver Pancreas mall Intestine pleen tomach Thyroid Gland Trachea Fat (Not Infiltrated) Muscle kin Diaphragm eminal Vesicle pinal cord testicle Left kidney alivary gland Value matches with respective value in table A.1 Value does not match with respective value in table A.1 Table A.2. ome of the Debye parameters that were determined by Dr F Costen research group. 51

65 Appendix B: Analysing the Accuracy of the Fitted Debye Model It is important to know discrepancy between the IT I Foundation and Debye model σ and ε r curves at each of the 60 frequency points F in figure 6.1. The difference ratio D X at each frequency point is calculated as D X F,T = X value of IT I at F X value of Debye at F X value of IT I at F where X in this case is either relative permittivity ε r or conductivity σ. (B.1) ince D F,T εr and D F,T σ can be positive and negative, it is more meaningful to calculate the root-meansquare (RM) rather than performing a simple average calculation of D F,T εr and D F,T σ over all F points. The smaller the RM value of D F,T εr and D F,T σ for T, the more accurate it is fitted to the Debye model. The RM value of D F,T εr and D F,T σ over the frequency range is calculated as follows f,t ) 2 F,T D RM X = 1 F (D F f=1 X (B.2) where X in this case is either relative permittivity ε r or conductivity σ. Tissue Name F,T D RM εr D σ F,T RM Fat Eye (Cornea) Bone Marrow (Yellow) Tooth (Dentine) Brain (White Matter) Pons Average F,T Table B.1. The values of D rms F,T εr and D rms σ for some tissues. F,T In table B.1, the pons tissue has the highest values of D rms F,T εr and D rms σ. This can be justified from the fact that it is the tissue in which the Debye ε r and σ curves deviate from IT I ε r and σ curves the most as it was shown in figure 6.1. The most accurate data fitted tissue, fat, has the F,T smallest values of D rms F,T εr and D rms σ. Theoretically, fat tissue should show the smallest variation in the γ-dispersion because it has the least amount of water content [4]. Therefore, this results also increases the confidence in the fitted Debye parameters values. 52

66 Investigating The Accuracy of The Fitting Curve in The γ-dispersion Region Only F,T This section investigates the effect of calculating D rms F,T εr and D rms σ in the γ-dispersion range F,T (300MHz-6GHz). When calculating D rms εr was omitted. Tissue Name F,T D rms εr and D σ F,T rms, any frequency point below 300 MHz Percentage Decrease in F,T D rms εr /% D σ F,T rms Percentage Decrease in D σ F,T rms /% Fat Eye (Cornea) Bone Marrow (Yellow) Tooth (Dentine) Brain (White Matter) Pons Average F,T Table B.2. The effect of eliminating β-dispersion region on D rms F,T εr and D rms σ for some tissues. The percentage decrease was calculated as Percentage Decrease = D F,T rms F,T X with β and γ D rms X with γ only F,T D rms X with β and γ 100 (B.3) where X in this case is either relative permittivity ε r or conductivity σ. F,T Eliminating the β-dispersion region from the calculations of D rms εr for each tissue reduced their values by an average of 27.2% as shown in table B.2. However, D σ F,T rms slightly changes by a F,T maximum of ±2% which is insignificant compared to D rms F,T εr changes. D rms σ is hardly affected because σ usually have the highest accuracy in the β-region (frequency less than 300MHz) as it was shown in figure

67 Appendix C: Code for the Files in the Monte Carlo Program C.1 Main.sh #!/bin/bash clear ##########!!THI I THE ONLY TAGE THAT NEED TO BE CHANGED!!!!!!!!!!!## sample=10 #Change number of samples uncertainty=0.9 #change the percentage uncertainty in the Debye Parameters. #Please do not use % value. For example, use 0.3 and not 30%. ########################################################################### ##############################Initialisations################################## rm -r Tissues n="integer, PARAMETER :: n = $sample" #number of samples s="integer, PARAMETER :: line = $sample" #number of samples head="frequency\tpermitivity\tconductivity" head_result="frequency\tpermitivity_sd\tconductivity_sd\tuncertainty_p\tuncertainty_c" U="uncertainty=$uncertainty" #percentage needed sed -i "6i $n" Random.f95 sed -i "7d" Random.f95 #pecifying the number of samples to be generated. sed -i "11i $U" Random.f95 sed -i "12d" Random.f95 #pecifying the percentge uncertainty sed -i "6i $s" ED.f95 sed -i "7d" ED.f95 #pecifying the number of samples. mkdir./tissues mkdir./tissues/all mkdir./tissues/phantoms #Creating Directories. gfortran Er.f95 -o Er gfortran ED.f95 -o ED gfortran final_statistics.f95 -o final_statistics gfortran Random.f95 -o Random ########################################################################### #PART 1 of the program# #############################TART OF THE MAIN LOOP######################## while read l #This first loop extract the Debye parameters for each tissue 54

68 do d=$(date +"%T") ###Timing Code echo -e "tart \t$d" >> Time ###Timing Code echo $l rm parameters echo $l > parameters./random #Randomise the paramters according to the specified percentages. The output #file of this program is the tissue name. name=$(awk '{print $1}' parameters) #get tissue name from the parameters mkdir./tissues/$name #create a directy for that tissue ############################################### #INNER LOOP TART# i=0 while read line do #Calculate the frequency response for each random sample of the looped tissue. echo $line >RandomTissue #store the random tissue for working echo -e $head>fieldnames # prepares the header, can use sed as alternative way or echo #with awk./er # calculates freq response for that random tissue cat RandomTissue fieldnames>header # create the header to be placed. The header consists of two lines # 1st line: Paramaters Values with tissue names. #2nd : Header for columns cat header fort.10 > $i #stores the result with the sample number (ranges from 0 to #(sample-1)) mv $i./tissues/$name #move the file to the respective directory i=$((i+1)) #increment rm RandomTissue # remove that the temp files done < $name #End of the First INNER LOOP# ############################################## ########################################################################### #PART 2 of the program# 55

69 #The following part of the program makes a file for each frequency used, and places this #specific frequency respnose of all samples in one file. mkdir./tissues/$name/statistics rm Result ######################################## #NEW INNER LOOP# #The Following Loop (1st) separate the frequecncy response of all the samples in a way that #each resulted file holds the for((linen=3;linen<63;linen+=1)) do #collect each frequency response of every sample in one file. #linen represents 100MHz with increment of 100MHz. o when linen=0, frequency = 100MHz ######################################## #NEW INNER INNER LOOP# for((number=0;number<sample;number+=1)) do #ADD the specific frequency respnose of the current sample to the specific frequency #file. awk 'NR=='"$lineN"'{print}'./Tissues/$name/$number>>./Tissues/$name/statistics/$lineN done ######################################### #END of INNER INNER LOOP# cp./tissues/$name/statistics/$linen./extract #make a copy to the parent dir echo $linen #show which line is being processed currently./ed>>result #do stat analysis for each freq done ######################################### #END of INNER LOOP# #This section makes the final statistics of all the grouped results and for all frequencies# sed -i -e "1i $head_result" Result #add header to the result echo >>Result #new line echo "Final tatistics:">> Result echo -e "mean_of_permitivity_sd\tsd_of_permitivity_sd\tmean_of_conductivity_sd\tsd_of_conductivity_sd \tmean_uncertainty_p\tsd_uncertainty_p\tmean_uncertsinty_c\tsd_uncertainty_c">> Result./final_statistics>>Result # final stats on the whole result file mv Result./Tissues/$name/statistics 56

70 cp $name./tissues/$name mv $name./tissues/all d=$(date +"%T") ###Timing Code echo -e "tart \t$d" >> Time ###Timing Code done < Values ##############################END OF MAIN LOOP############################# ########################################################################### ##############CREATING TIUE PECIFICATION ####################### for((k=0;k<(sample+1);k+=1)) do #create human phantoms. Number of phantoms = number of samples. #Again parameters are stored in the following order : #TissueNames,ep_infty,Dlt_ep_1,tau_1_nl,sg_0 for f in $(find./tissues/all -type f) do #search for all the files in Tissues folder and open each one, one at a time. awk 'NR=='"$k"'{print}' $f #Each lines in every file represents a phantom value of that tissue. awk 'NR=='"$k"'{print}' $f> temp cat temp >> $k #phantom number done done rm 0 for((x=1;x<(sample+1);x+=1)) do #move the phantoms to the phantoms folder cp $x./tissues/phantoms/$x rm $x done ###########################################################################./tats.sh #extraxct final stats results for each tissue and place them into one file for further #analysis on changing uncertainty. #############################END OF PROGRAM############################### 57

71 C.2 Random.f95 Program Random implicit none CHARACTER (len=50) :: TissueNames, p INTEGER, PARAMETER :: n = 10 INTEGER :: i, IO REAL::ep_infty,Dlt_ep_1,tau_1_nl,sg_0, temp_ep_infty, temp_dlt, temp_tau, temp_sg, negative REAL:: random_ep_infty(n),random_tau_1_nl(n), random_dlt_ep_1(n), random_sg_0(n) REAL :: uncertainty uncertainty=0.9 OPEN(UNIT=7,FILE="parameters",TATU='OLD',ACTION='READ',IOTAT=IO) READ(7,*)TissueNames,ep_infty,Dlt_ep_1,tau_1_nl,sg_0 CLOE(7) print*, TissueNames p="ep_infty" call GenRandom(random_ep_infty,ep_infty, uncertainty, n, p) p="dlt_ep_1" call GenRandom(random_Dlt_ep_1,Dlt_ep_1, uncertainty,n,p) p="tau_1_nl" call GenRandom(random_tau_1_nl,tau_1_nl, uncertainty,n,p) p="sg_0" call GenRandom(random_sg_0, sg_0, uncertainty, n,p) OPEN(UNIT=10,FILE=TissueNames,TATU='NEW',ACTION='WRITE',IOTAT=IO) Do i=1,n if (random_ep_infty(i)<0) then negative=1 temp_ep_infty=0-random_ep_infty(i) else temp_ep_infty= random_ep_infty(i) end if if (random_dlt_ep_1(i)<0) then negative=1 temp_dlt= 0-random_Dlt_ep_1(i) else temp_dlt= random_dlt_ep_1(i) 58

72 end if if (random_tau_1_nl(i)<0) then negative=1 temp_tau= 0 - random_tau_1_nl(i) else temp_tau=random_tau_1_nl(i) end if if (random_sg_0(i)<0) then negative=1 temp_sg = 0- random_sg_0(i) else temp_sg = random_sg_0(i) end if write(10,*)tissuenames,temp_ep_infty,temp_dlt,temp_tau,temp_sg!,negative negative=0 end do Close(10) end program Random ubroutine GenRandom (uniform,average, error, k, para) Implicit none CHARACTER (len=50) :: para REAL,PARAMETER::pi= INTEGER :: j,k REAL:: R, theta REAL ::x,stdev, uniform(k), gaussian(k),average, error stdev=error*mean CALL RANDOM_NUMBER(unifrom)!Applying Box Muller Transform do j = 1, k-1,2 R= QRT(-2.0*LOG(uniform(j))) theta=2*pi*uniform(j+1) x = stdev*r * CO(theta) + average gaussian(j+1) = stdev * R * IN(theta)) + average gaussian(j) = x end do end subroutine GenRandom 59

73 C.3 Er.f95 program Er implicit none!calculate Frequency Response Program COMPLEX, external :: ep_cal_0 CHARACTER (len=50) :: TissueNames CHARACTER (len=2000) :: line REAL,PARAMETER::pi=4.0*ATAN(1.0),ep_0=8.854e-12!!physical constants DOUBLE PRECIION :: Random INTEGER :: i,j, IO INTEGER,DIMENION(60)::frequency=(/(i,i=1,60)/)!!!!when we need frecuency REAL::omg(60),omg_0,ep(60,50),sg(60,50),ep_imag(60,50) REAL::ep_infty,Dlt_ep_1,tau_1_nl,sg_0 COMPLEX::ep_cmplx(60,50),ep_cal(60,50),d_ep(60,50),g(60,4) OPEN(UNIT=7,FILE="RandomTissue",TATU='OLD',ACTION='READ',IOTAT=IO) READ(7,*)TissueNames,ep_infty,Dlt_ep_1,tau_1_nl,sg_0 CLOE(7) i=1 j=1 DO i=1,60 DO j=1,1 omg(i)=2.0*pi*frequency(i)*1e8 omg_0=2.0*pi*1e9 END DO end do Do i=1,60 ep_cal(i,j)=ep_cal_0(omg(i),omg_0,ep_0,ep_infty,dlt_ep_1,tau_1_nl,sg_0) write(10,*)frequency(i)*1e8,real(ep_cal(i,j)),-ep_0*omg(i)*aimag(ep_cal(i,j))!freq, permitvity,conductivity end do end program Er 60

74 !!This function obtained from Debye.F90 file that was provided by the research group!!! Note omg that is multiplied with tau_nl_1 is scaled down by omg_0. COMPLEX FUNCTION ep_cal_0(omg,omg_0,ep_0,ep_infty,dlt_ep_1,tau_1_nl,sg_0) REAL::ep_infty,Dlt_ep_1,tau_1_nl,sg_0 REAL::omg,omg_0,ep(60,50),sg(60,50),ep_imag(60,50) ep_cal_0=(cmplx(ep_infty,0.0)+cmplx(dlt_ep_1,0.0)/cmplx(1.0,tau_1_nl*omg/omg_0))& & +(cmplx(sg_0,0.0)/cmplx(0.0,omg*ep_0)) END 61

75 C.3 ED.f95 program ED implicit none!program to calculate the mean and standard deviation. INTEGER, PARAMETER :: line = 10 INTEGER :: i,j, IO REAL:: frequency(line), permitivity(line), conductivity(line)! The value of line is!automatically changed by the shell scripting program REAL:: meanp,meanc, sdp, sdc!mean of permitivity, mean of conductivity, standard deviation of permitivity, standard deviation of conductivity. OPEN(UNIT=7,FILE="extract",TATU='OLD',ACTION='READ',IOTAT=IO) do i=0,line-1 READ(7,*)frequency(i), permitivity(i), conductivity(i) end do CLOE(7) meanp = UM(permitivity)/line meanc = UM(conductivity)/line! sdp = QRT(UM((permitivity - meanp)**2)/line) sdc = QRT(UM((conductivity - meanc)**2)/line) write(*,*)frequency(1),sdp,sdc,sdp/meanp,sdc/meanc end program ED 62

76 C.4 Final_statistics.f95 program ED implicit none INTEGER :: i,j, IO, line=60 REAL,DIMENION(60)::frequency, permitivity, conductivity, u_p, u_c REAL::meanp,meanc, sdp, sdc,meanup,meanuc,sdup,sduc!mean of permitivity,!mean of!conductivity, standard deviation of permitivity, standard deviation of conductivity,!mean of uncertainty of permitivity, mean of uncertainty of conductivity, standard deviation!of!uncertainty of permitivtiy, standard deviation of uncertainty of conductivity. OPEN(UNIT=7,FILE="Result",TATU='OLD',ACTION='READ',IOTAT=IO) READ(7,*) do i=0, line-1 READ(7,*) frequency(i), permitivity(i), conductivity(i), u_p(i), u_c(i) end do CLOE(7) meanup= UM(u_p)/line meanuc= UM(u_c)/line meanp = UM(permitivity)/line meanc = UM(conductivity)/line sdup= QRT(UM((u_p - meanup)**2)/line) sduc= QRT(UM((u_c - meanuc)**2)/line) sdp = QRT(UM((permitivity - meanp)**2)/line) sdc = QRT(UM((conductivity - meanc)**2)/line) write(*,*) meanp,sdp,meanc,sdc,meanup,sdup,meanuc,sduc end program ED 63

77 C.5 tats.sh #!/bin/bash rm./tissues/all_stats rm all_stats for direc in./tissues/* #for direc in $(find./tissues/ -type d) do #echo $direc p=$(echo "$direc" awk -F/ '{print $3}') #get the tissue name #echo $p if [! -z "$p" -a "$p"!= " " ]; then # check if p is not no null or space. Needed because the for #loop is a bit crude. awk -v var=$p 'NR==65 {print var, $0} ' OF="\t"./Tissues/$p/statistics/Result >> all_stats fi #ADD the final stat line which holds the D of all the frequency statistical analysis. done sed -i -e "1i Name\tmean_sd_P\tsd_of_P_sd\tmean_sd_C\tsd_of_C_sd\tmean_uncert_P\tsd_uncert_P\tmean_un cert_c\tsd_uncert_c" all_stats #Header to be imported to excel. mv all_stats./tissues 64

78 Number of times this sample value occurred, i.e. Number of times this sample value occurred, i.e. Appendix D: Investigating the Effect of Accuracy of the Random Number Generator on the Gaussian Values Obtained Using Box Muller Transform D.1 The Results of The Test A program has been written to investigate how the accuracy of Fortran s uniform random number generator is reflected on the results of the Box Muller algorithm. Using =3000, 3000 uniform randomly distributed values were generated using RANDOM_NUMBER() method. a) Uniform Distribution b) tandard Gaussian (Normal) Distribution Figure D.1. The results of the test. The y-axis has not been normalised because the number of samples are not infinite, and it is known, i.e., =

79 Inspecting figure D.1(a) shows that the uniform random number generator of Fortran is quite accurate. Notice that the plot in figure D.1(a) is not ideal uniform distribution. Nevertheless, the plot is an acceptable representation of the uniform distribution. As a result, the Box Muller algorithm produced an acceptable standard Gaussian distribution as shown in figure D.1(b). D.2 Code of the Program Program InvestRandom implicit none INTEGER:: k=3000 INTEGER:: i,j REAL:: array(3000), G(3000) REAL,PARAMETER::pi=4.0*ATAN(1.0) REAL:: x!!!!!!!!!!!!!!!!!get 3000 Uniform Random amples!!!!!!!!!!!!! CALL RANDOM_NUMBER(array) Do i=1,k-1 write(9,*) array(i) end do!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!box Muller Algorithm!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! DO j = 1, k-1, 2 x = QRT(-2.0*LOG(array(j))) * CO(2*pi*array(j+1)) G(j+1) = QRT(-2.0*LOG(array(j))) * IN(2*pi*array(j+1)) G(j) = x Write (8,*)G(j+1) Write (8,*)G(j) END DO!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! end Program 66

80 Appendix E: Results of Monte Carlo Program for =3000 for All Tissues E.1 Conductivity u input Name Adrenal Gland Bile Blood Blood erum Blood Vessel Wall Bone (Cancellous) Bone (Cortical) Bone Marrow (Red) Bone Marrow(Yellow) Brain Brain (Grey Matter) Brain (White Matter) Breast Fat Breast Gland Bronchi Cartilage Cerebellum Cerebrospinal Fluid Cervix Commissura Anterior Commissura Posterior Connective Tissue Diaphragm Ductus Deferens Dura Epididymis Esophagus Eye(Cornea) Eye(Lens) Eye Lens(Cortex) Eye Lens(Nucleus) Eye(clera) Eye(Vitreous Humor) Fat Fat(AverageInfiltrated) Fat(Not Infiltrated) Gallbladder Heart Lumen Heart Muscle Hippocampus Hypophysis Hypothalamus Invertebral Disc Kidney Kidney(Cortex)

81 Kidney (Medulla) Large Intestine Large Intestine Lumen Larynx Liver Lung Lung(Deflated) Lung(Inflated) Lymph node Mandible Medulla Oblongata Meniscus Midbrain Mucous Membrane Muscle Nerve Ovary Pancreas Penis Pineal Body Placenta Pons Prostate alivary Gland AT(ubcutaneousFat) eminal vesicle kin mall Intestine mall Intestine Lumen pinal Cord pleen tomach tomach Lumen Tendon Ligament Testis Thalamus Thymus Thyroid Gland Tongue Tooth Tooth(Dentine) Tooth(Enamel) Trachea Ureter Urethra Urinary Bladder Wall Urine Uterus Vagina Vertebrae Mean, u σ F Table E.1. hows the values of u for all tissues as well as u σ F different u σ F input. 68

82 E.2 Relative Permittivity u input Name Adrenal Gland Bile Blood Blood erum Blood Vessel Wall Bone (Cancellous) Bone (Cortical) Bone Marrow (Red) Bone Marrow(Yellow) Brain Brain (Grey Matter) Brain (White Matter) Breast Fat Breast Gland Bronchi Cartilage Cerebellum Cerebrospinal Fluid Cervix Commissura Anterior Commissura Posterior Connective Tissue Diaphragm Ductus Deferens Dura Epididymis Esophagus Eye(Cornea) Eye(Lens) Eye Lens(Cortex) Eye Lens(Nucleus) Eye(clera) Eye(Vitreous Humor) Fat Fat(AverageInfiltrated) Fat(Not Infiltrated) Gallbladder Heart Lumen Heart Muscle Hippocampus Hypophysis Hypothalamus Invertebral Disc Kidney Kidney(Cortex) Kidney (Medulla) Large Intestine Large Intestine Lumen

83 Larynx Liver Lung Lung(Deflated) Lung(Inflated) Lymph node Mandible Medulla Oblongata Meniscus Midbrain Mucous Membrane Muscle Nerve Ovary Pancreas Penis Pineal Body Placenta Pons Prostate alivary Gland AT(ubcutaneousFat) eminal vesicle kin mall Intestine mall Intestine Lumen pinal Cord pleen tomach tomach Lumen Tendon Ligament Testis Thalamus Thymus Thyroid Gland Tongue Tooth Tooth(Dentine) Tooth(Enamel) Trachea Ureter Urethra Urinary Bladder Wall Urine Uterus Vagina Vertebrae Mean, u T εr F Table E.2. hows the values of u T for all 94 tissues as well as u εr F T at different u εr F input. 70

84 Appendix F: Plots of Relative Uncertainty of Conductivity and Relative Permittivity for ome Frequencies as Input Relative Uncertainty Varies F.1 Conductivity F,T u F,T u σ σ MHz MHz 2.9GHz 0.2 5GHz u input a) Dura Tissue MHz MHz 2.9GHz 0.2 5GHz u input b) Fat Tissue Figure F.1. The effect of varying u input on u for some F points. F,T σ 71

85 F.2 Relative Permittivity u F,T ε r MHz 900MHz 2.9GHz 5GHz u input a) Dura Tissue u F,T ε r MHz 900MHz 2.9GHz 5GHz u input b) Fat Tissue Figure F.2. The effect of varying u input on u for some F points. εr F,T 72

86 Appendix G: Plots of Relative Uncertainty of Conductivity and Relative Permittivity for ome Tissues as Input Relative Uncertainty Varies G.1 Conductivity F T u T F u ε r σ u input Figure G.1. The effect of varying u input on u T for some tissues. This plot was generated from σ F table E.1. G.2 Relative Permittivity Dura Fat Kidney Pons 0.2 Dura Fat 0.1 Kidney Pons u input Figure G.2. The effect of varying u input on u T for some tissues. This plot was generated from εr F table E.2. 73

87 Appendix H: Plots of Relative Uncertainty of Conductivity and Relative Permittivity for Different as Input Relative Uncertainty Varies H.1 Conductivity u T σ F Tissue Number T a) =3000 samples u T σ F Tissue Number T b) =2000 samples 74

88 u T σ F Tissue Number T c) =1000 samples u T σ F Tissue Number T d) =200 samples Figure H.1. Plots of u T for all 94 tissues at u F σ input = 0.1,0.3,0.5,0.7,0.9 and 1.2. Each plot is obtained by running the Monte Carlo program with different. 75

89 H.2 Relative Permittivity u T F ε r Tissue Number T a) =3000 samples u T F ε r Tissue Number T b) =2000 samples 76

90 u T F ε r Tissue Number T c) =1000 samples u T F ε r Tissue Number T d) =200 samples Figure H.2. Plots of u T for all tissues at u εr F input = 0.1,0.3,0.5,0.7,0.9 and 1.2. Each plot is obtained by running the Monte Carlo program with different. 77

91 Appendix I: Partial Derivatives I.1 Relative Permittivity ε r ε = 1 ε r ε 1 = 1 1+(ωτ 1 ) 2 ε r τ 1 = 2 ε 1ω 2 τ 1 (1+(ωτ 1 ) 2 ) 2 ε r σ i = 0 (I.1) (I.2) (I.3) (I.4) I.2 Conductivity σ ε = 0 σ ε 1 = ε oω 2 τ 1 1+(ωτ 1 ) 2 (I.5) (I.6) σ = ε o ε 1 ω 2 (1 (ωτ 1 ) 2 ) τ 1 (1+(τ 1 ω) 2 ) 2 (I.7) σ σ i = 1 (I.8) 78

92 Appendix J: GUM Program Code The GUM program written consists of two files. The first one is a shell script file which calls the second source file Final.f95 several times. J.1 GUM.sh #!/bin/bash d=$(date +"%T") ###Timing Code echo -e "tart \t$d" >> Time ###Timing Code uncertainty=0.1 #change the percentage uncertainty in the Debye Parameters. ################################################################################ rm U2 gfortran Final.f95 -o Final for((j=0;j<60;j+=1)) do rm U echo -e "Fat\t \t \t \t E-02\t$j"> U./Final >> U2./Final done d=$(date +"%T") ###Timing Code echo -e "tart \t$d" >> Time ###Timing Code 79

93 J.2 Final.f95 Program Final implicit none CHARACTER (len=50) :: TissueNames INTEGER, PARAMETER :: n = 2000 REAL,PARAMETER::pi=4.0*ATAN(1.0),ep_0=8.854e-12, omg_0=2.0*4.0*atan(1.0)*1e9 INTEGER :: i, IO REAL::ep_infty,Dlt_ep_1,tau_1_nl,sg_0, negative, omg,f REAL:: sd_inf, p, var_p, p_2, sd_dlt, sd_tau, sd_sg, par_inf, par_dlt, par_tau COMPLEX:: ep_cal REAL:: c, c_2, c_par_sg, c_par_dlt, c_tau, var_c, inter REAL:: random_ep_infty(n),random_tau_1_nl(n), random_dlt_ep_1(n), random_sg_0(n) REAL :: uncertainty uncertainty= 1.2 OPEN(UNIT=7,FILE="U",TATU='OLD',ACTION='READ',IOTAT=IO) READ(7,*)TissueNames,ep_infty,Dlt_ep_1,tau_1_nl,sg_0, f CLOE(7) omg= 2*pi*f*1e8 ep_cal=(cmplx(ep_infty,0.0)+cmplx(dlt_ep_1,0.0)/cmplx(1.0,tau_1_nl*omg/omg_0))& & +(cmplx(sg_0,0.0)/cmplx(0.0,omg*ep_0))!!! Note omg that is multiplied with tau_nl_1 is scaled down by omg_0. p= real(ep_cal) p_2= p*p par_inf=1 par_dlt=1/(1+ ((omg*omg)*(tau_1_nl*tau_1_nl))/(omg_0*omg_0)) par_tau=(-1*dlt_ep_1)*( (1/(1+(omg*tau_1_nl/omg_0)**2) )**2 * tau_1_nl*2*(omg**2/omg_0**2) ) sd_inf=(uncertainty*ep_infty) sd_dlt= uncertainty*dlt_ep_1 sd_tau= uncertainty*tau_1_nl sd_sg= uncertainty*sg_0 80

94 var_p = ((par_inf*sd_inf)**2)+ ((par_dlt*sd_dlt)**2)+((par_tau*sd_tau)**2) c= -ep_0*omg*aimag(ep_cal) c_par_dlt=(ep_0*tau_1_nl*omg**2)/(omg_0*(1+(omg*tau_1_nl/omg_0)**2)) inter= (1+(omg*tau_1_nl/omg_0)**2)*(ep_0*Dlt_ep_1*omg**2/omg_0)- (ep_0*tau_1_nl*dlt_ep_1*omg**2/omg_0)*(tau_1_nl*2*(omg/omg_0)**2) c_tau= inter/((1+(omg*tau_1_nl/omg_0)**2)**2) c_par_sg= 1 var_c= (c_par_dlt*sd_dlt)**2 + (c_tau*sd_tau)**2 + (c_par_sg*sd_sg)**2 write(*,*) f, "U_P=",QRT(var_p)/p, p, "U_C=", QRT(var_c)/c, c, uncertainty end Program 81

95 Appendix K: The Progress Report CHOOL OF ELECTRICAL AND ELECTRONIC ENGINEERING Progress Report Uncertainty of Frequency Response of Human Tissues Name: Mohammed Hussain M Abdulwahab ID Number: upervised by: Dr. Fumie Costen 82

96 Table of Contents 1 Introduction Motivation Aims Objectives Literature Review Background Uncertainty Uncertainty in imulations Practical Progress Theory The basics of statistics Programming & Analysis Extract Data from website Data Fitting Program and hell cripting File Analysis of the results Comparison of the results with previously calculated ones Problems Encountered Future Work Conclusion References Appendix A) Table of Uncertainties for ome Tissues B) Basics of hell cripting For Beginners C) Advanced hell cripting D) Graph for other Data Fitted Tissues E) hell Program Code to Automate the Data Fitting Activity F) Project Plan G) Technical Risk Analysis H) chool of EEE Health & afety Risk Assessment

97 1 Introduction 1.1 Motivation One of the most common ways of modelling the electromagnetic propagation in human body is the Finite Difference Time Domain method (FDTD). This method is an iterative method where space is divided into small segments of magnetic and electrical fields. The current research group simulate the propagation of electromagnetic pulses in the human body using the FDTD method in order to develop health care devices. The frequency pulse contains a spectrum of frequencies. Electromagnetic pulses propagates differently for different tissues and each frequency influences the same tissue differently. In other words, the complex permittivity, conductivity and permittivity, of human tissues are all frequency dependent. The FDTD method uses a general single pole Debye model to generate four Debye parameters for each tissue specific for the patient. Ideally, this is done by measuring the complex permittivity of each tissue of the patient and uses it to generate a patientspecific healthcare device. Unfortunately, measuring the frequency response for more than 50 tissues is too expensive at this moment. As a result, an alternative way is needed to generate a range of Debye parameters for a set of human tissues. 1.2 Aims This project aims to produce a wide range of Debye parameters for different human tissues taking advantage of statistics and random number generators. 1.3 Objectives Learn how to calculate standard deviation and analyzing normal distribution curves. Investigate the uncertainty of frequency response of human tissues and the size of the human body. Extract frequency response of 102 human tissues. Learn how to use bash shell scripting in order to automate the data fitting of the 102 tissues. Explore the basics of Fortran syntax in order to perform statistical calculations. Generate random Debye parameters using the statistical results. 2 Literature Review 2.1 Background When a dielectric experiences an electric field, the dielectric will take a certain amount of time to polarise, and this time is referred to as relaxation time. The permittivity is complex since the real 84

98 part of the permittivity represents the polarisation component in the direction of the field where as the imaginary part represents the polarisation component that is normal to the applied field. [1] This nonlinear behavior can be described by the relaxation model. The model that is used by the data fitting software in this project is a single-pole (first order) Debye model. The model is represented as follows: [1] ε = ε + ε 1 1+jωτ 1 + σ i jωε o (1) ε is the complex permittivity. Δε1 = (ε - εs), where ε (infinity permittivity) is the high frequency limits of the permittivity at ωτ1>>1, εs (static permittivity) is the low frequency limit (i.e. steady state) at ωτ1<<1, τ1 is the time constant of the relaxation and σi is the static ionic conductivity of the tissue. [1][2] It is also worth mentioning that the permittivity of the tissue decreases non-linearly as the frequency increases, and the conductivity increases as the frequency increases due to polarization as it has been discussed previously. [2] 2.2 Uncertainty Where the measuring system is free from bias (systematic errors), the main component of the uncertainty of the dielectric properties of the human tissues are the random variations that arises from sampling and the inhomogeneous nature of the biological tissues. This is due to the fact that human bodies are not identical.[3] It is found out that the probability distribution of the Debye parameters variation follows the normal (Gaussian) distribution. When the measurements are repeated again with the exact conditions, the most probable result is the mean value. The mean itself tend to approach the theoretical value as the number of measurements become very large. The uncertainty in the mean value can be found by calculating the standard deviation of the mean, and hence the uncertainty will decrease if the number of samples increases. [2] The uncertainty from random errors in many tissues do not exceed 10%. This 10% tend to be more apparent at high frequencies greater than 10 GHz.[2] The uncertainty in conductivity tends to be higher for many tissues due to the fact that its magnitude is small (in the order of ). This conclusion will be useful when different sets of Debye parameters are needed to be generated. ee Appendix A for more about uncertainty of complex permittivity in some tissues. 85

99 2.2.1 Uncertainty in imulations It is important to introduce uncertainty in the numerical simulations. Monte Carlo uncertainty analysis is relatively easy to apply and commonly used, and hence it will be used as an example to show how uncertainty is handled in context of simulation. [5] The Monte Carlo technique take into account all the uncertainties of the input parameters regardless how much each parameter contribute to the output. In other words, the uncertainties of each input parameter is propagated. The first step in the simulations is defining the input parameters of the system. After doing so, it is then important to define the model equation of the system, and in this case, it is the single pole Debye model. [5] As mentioned before, the randomness of the parameters follows the normal (Gaussian) probability distribution. When Monte Carlo trials are performed, values from the probability distribution are randomly selected and simulated. This process is repeated for many times.[4] In general, the more trials are performed, the greater the confidence in the results. As a general rule, [5] states that the number of trials M should satisfy the following condition: M > p Where p is the selected coverage probability. o, when about 95% convergence is needed, then the simulation must run at least 200,000 times. (2) After running the simulation trials, an estimate of the output quantity is taken by finding the mean of all the trials. The uncertainty in the output is the standard deviation of the output. When displaying the results, it is recommended to state convergence intended and the upper and the lower limits of the results. [5] It is also worth mentioning that when the number of trials is very large, the output of uncertainty in the result will have a normal distribution regardless of the input parameters probabilistic distributions.[4] For computer simulations to be as real as possible, the randomness of the inputs in programs needs to be as realistic as to its distribution. This means that a good random number generator with smart algorithms is needed to be used in the computations. [5] 86

100 3 Practical Progress 3.1 Theory The basics of statistics In order to carry out the objectives in this project, some statistical concepts are needed to be studied. These include the normal distribution, standard deviation and continuous data analysis. The first concept studied is the standard deviation (σ). The standard deviation indicate the spread of the distribution. It is found out that for symmetrical distributions, 68% of the population lie within 1 standard deviation of the mean. Moreover, 95% of the population lies within 2 standard deviation of the mean. Variance is the square of the standard deviation and it is calculated as follows: [6] σ 2 = 1 (x N i x ) 2 f i (3) Where the mean x = 1 xf, the total frequency N = f and x N i is the value of a variable.[6] ince the uncertainty in the complex permittivity follows the normal distribution, then the normal distribution needed to be studied. The standard normal distribution has been explored and it was found out that it has a mean of zero and a standard deviation of 1 unit. [6] Additionally, the use of standard normal distribution table in order to find the probability has been understood. Finally, the concept of standardising a variable from a general normal distribution in order to use the standard normal distribution table has been appreciated. The standardisation is done as follows: Z = X μ σ Where X is the variable, µ is the mean and σ is the standard deviation. [6] (4) 3.2 Programming & Analysis Extract Data from website The dielectric properties of 102 tissues have been extracted from [7]. The complex permittivity of each tissue has been saved in a separate file. Each Tissue file consists of three columns: Frequency, Permittivity and then conductivity. All the tissues have been saved in a folder called "Tissues" in the data fitting program folder Data Fitting Program and hell cripting File After extracting the data from the website, a shell script was needed to be written to pass each tissue to the data fitting program one at a time. The data fitting program is written in Fortran. A shell script has been written to automate this activity. The script loops over all the tissues in the Tissues 87

101 folder then splits the permittivity and the conductivity of each tissue into two different files. After that, it saves the data into a specific format required by the data fitting program. The program is then compiled and executed. Finally, the results of the currently processed tissue is added to a file containing the results of all the previously processed tissues. ee Appendix B & C for more on shell scripting Analysis of the results The data fitting program Debye.F90 generates the 4 parameters for one pole Debye model. The data fitting is done using the iterative approach. The data fitting depends on the initial values of the four Debye parameters. o the following line of code needed to be changed depending on the tissue: A= (/ , , , /). Where A= [ep_infty (ε ), Dlt_ep_1 ( ε 1 ), tau_1_nl (τ 1 ), sg_0 (σ i )] respectively. In addition, the convergence of the calculation depends on the following line of code in the program: Do while (error>= ). These two line of codes were needed to be changed carefully in order to have an acceptable accuracy of the data fitting. Otherwise, the calculation will not finish, and the terminal keep on printing the results for infinite time. The value of tolerance (error) deemed to be small when dealing with a large numbers of tissues. It was challenging to come up with an algorithm to change the initial values of the Debye parameters depending on the tissue. The results were reasonably accurate and, therefore, it was not needed to invest a lot of time carrying out more simulation trials and come up with the algorithm. As a consequence, the tolerance was changed to (increased by a factor of 10). Then the simulation was carried out with three sets of initial values. The first initial sets of values were A= (/ (2), (5), (0.1), (0.1)/). The simulation gave good convergence results (more on comparison in the next section). However, it failed to calculate the Debye parameters for two tissues: invertible disc and Thymus. The results for these tissues showed either infinity or NaN (Not a Number). As a consequence, the simulation was repeated twice to obtain the parameters for the remaining two tissues as well as to compare the convergence values of the other tissues. The Debye parameters for the invertible disc were found out using A= (/ (2), (7), (0.1), (0.1)/) as initial values. Then, the Debye parameters for the Thymus for were calculated using A= (/ (2), (10), (0.08), (0.1)/) as initial values Comparison of the results with previously calculated ones The 4 Debye parameters for 50 tissues has been calculated previously by someone else a while ago before the start of this project. A comparison will be done on some of the tissues between the old results and the three simulation trials. Results are shown here for Cornea, Grey Matter, White 88

102 Matter, Liver and pinal Cord. General conclusions will be easily drawn after analysing this sample of the results. Table 1: hows the results of the three simulation trials for some tissues Table 2: hows the difference between the simulation results As it can be seen from Table 1 and Table 2, the initial values of the Debye parameters at the start of the simulation did not have any effect on the final values of the Debye parameters of all the tissues. The Difference between any two sets of initial values is zero as it is shown in Table 2. The values calculated in table 2 are the difference between all simulation results. The calculation was done as follows: Difference = (simualtion1-simualtion2)-(simualtion1-simualtion3)-(simualtion2- simualtion3). It can be inferred that the initial values only determine whether the calculation would finish or no, as it was described previously by the fact that the simulation needed to be repeated another two times in order to determine the parameters for the Invertible Disc and the Thymus. Old imulation Table 3: hows the percentage difference between old and current simulation results Table 3 compare the results between the old simulation and the simulation carried out in this project. According to the calculations, the maximum percentage variations are only 0.23%. This proves that the data fitting has been done within an acceptable accuracy. The difference in the results is probably due to the fact that the dielectric properties have been obtained a long time ago, and it is most likely that [33] has updated its database. The other reason of this difference is that the old experiment was conducted with a different value of tolerance. 89

103 Value (Conductivity is in /m, Permittivity is dimensionless) In order to investigate the effect of changing the value of tolerance in the program. Three simulation trials have been conducted to investigate the effect of changing the tolerance by a factor of 20 db and 40 db. The results are displayed in Table 4. The initial values of the 4 Debye parameters have been fixed for the three trials. The values were: A= (/ (2), (5), (0.1), (0.1)/). Table 4: hows the results of the three simulation trials Table 5: hows the percentage difference between the least and highest tolerances As it can be seen from Table 5, the effect of increasing the tolerance by a factor of 40 db did only cause a maximum variation of 0.04%. For the purpose of this project, the results with tolerance are very accurate. This experiment showed that the tolerance value did not cause a significant effect on the accuracy of the results. However, varying the tolerance value did have some effect on the results, unlike varying the initial values of the 4 Debye parameters which had no effect on the results. The data fitting software also generates the frequency response of the tissues using the 4 Debye calculated parameters. A gnuplot 1 has been made for both White Matter and Cornea. Frequency/Hz Figure 1: The plot of the frequency response of Cornea tissue 1 A plotting tool in Linux that is similar to MATLAB. 90

104 Value (Conductivity is in /m, Permittivity is dimensionless) According to figure 1 and figure 2, the frequency response of both tissues follows the same shape. The deviation between the measured and the data fitted results looks the same for both cases. The accuracy of the data fitted conductivity is very high. There is only a slight variation at higher frequencies around 6GHz region. However, the variation between data fitted permittivity and measured permittivity is much higher. There is a maximum ± 5 unit change for any frequency higher than 1GHz. There is around 15 unit change at low frequencies. This is probably due to the fact that the model is very simple and higher order Debye model is needed for better accuracy. The ε n 1+jωτ n term in the Debye model (from equation (1)) represent the nth order of the model. This is analogous to Taylor series where expressing a function in higher order gives more realistic approximation of the actual function. If this term is investigated, it can be seen that it is related to permittivity only, and that is why the accuracy of conductivity is high. ee Appendix D for more plotted graphs for different tissues. Frequency/Hz Figure 2: The plot of the frequency response of White Matter tissue To conclude, the initials values of the Debye parameters does not have any effect on the final values of the calculated Debye parameters. These values determines the ability of the program on whether it will finish the calculation or run for indefinite time. The tolerance value does not only contributes in the determination of whether the calculation would finish or not, but also affect the accuracy to some extent. Higher orders of Debye model is needed for more realistic representation of the dielectric properties of the human tissues. 4 Problems Encountered A couple of problems have been encountered during the practical progress. It was the first time ever to explore a Fortran program, so it took some time to figure out how to compile and run it using the MakeFile. This was done by looking into the documentations of Fortran. 91

105 Unfortunately, it was extremely difficult to automate the extraction of the dielectric properties of tissues from [7] using a shell script, since knowledge in HTML, XML and Javacript was needed. As a result, the process had to be done manually which took around three hours to do so. Another problem was the fact that prior to the start of the project, it was the first time to experience shell scripting, so the fundamentals needed to be learned. The syntax was totally different compared to other high level languages like C or Java. After learning the basics and some advanced features of scripting languages, the use of shell scripting to solve problems have been successfully appreciated and implemented. Finally, many simulation trials have been carried out in order to understand how the data fitting software works. In many cases, the calculations of the data fitting did not converge and the program continued to run until it was manually stopped. It took some to understand the effects of the initial values and the tolerance value. Eventually, the problem has been solved. 5 Future Work After performing data fitting and finding the 4 Debye parameters for 105 tissues. tatistics will be applied to generate a set of Debye parameters for each tissue in order to achieve the project s first aim. ee Appendix G for more about the project s plan. Also, see Appendix H for Health & afety Risk Analysis, and Appendix G for Technical Risk Analysis. 6 Conclusion The FDTD method is an extremely useful way to study the dielectric properties of the human tissues. The project aims and objectives have been discussed in this report. In addition, some important concepts have been emphasised from the literature review. Furthermore, the practical progress so far have been discussed and conclusions have been drawn from the results of the data fittings. The project plan will be carried out. By the end of the plan, a final report will be written explaining and documenting everything happened during the project s life time. 92

106 7 References [1] G. C. R. Melia, "Electromagnetic Absorption by the Human Body from 1 to 15 GHz," The University of York Department of Electronics, York, [2] A. Peyman,. Holden and C. Gabriel, "Dielectric Properties of Tissues at Microwave Frequencies," MTHR cientific Co-ordination Team, Didcot, [3] C. Gabriel and A. Peyman, "Dielectric measurement: error analysis and assessment of uncertainty," INTITUTE OF PHYIC, Newbury, [4] I. A. Macdonald, J. A. Clarke and P. A. trachan, "Assessing Uncertainty In Building imulation," University of trathclyde, Glasgow. [5] P. R. G. Couto, J. C. Damasceno and. P. de Oliveira, "Monte Carlo imulations Applied to Uncertainty in Measurement," InTech, [6] "IT'I material parameter database EM tissue parameters thermal tissue parameters EM simulations material properties dielectric tissue properties tissue densities tissue heat capacities tissue thermal conductivities tissue heat transfer ra," IT'I Foundation, [Online]. Available: [Accessed 26 October 2015]. [7] D. J. Rumsey, tatistics Essentials For Dummies, Hoboken: Wiley Publishing, Inc, [8] W. E.. Jr., The Linux Command Line: A Complete Introduction, an Francisco: William Pollock, [9] B. Barnett, "Top Ten Reasons not to use the C shell," [Online]. Available: [Accessed 4 November 2015]. 93

107 8 Appendix A) Table of Uncertainties for ome Tissues Frequency range Tissues Permittivity ε' (±%) Conductivity σ (±%) MHz Grey Matter Cornea Long Bone White Matter Liver Cartilage >300 MHz 10 Grey Matter GHz Cornea Long Bone White Matter Liver Cartilage > GHz Grey Matter Cornea Long Bone White Matter Liver Cartilage Table A: shows the dielectric properties of a select of tissues over three frequency divisions. [2] The frequencies have been dived into three sections because the sensitivity of the measuring probe or the network analyser used to carry out the measurements changes for specific frequency region which results in different variations. The uncertainty here has been calculated for every frequency region. [2] 94

108 B) Basics of hell cripting For Beginners A shell script is a file that contains all the commands that the user intends to execute in command line terminal. The shell scripting language is extremely useful, since it is easier to debug the file than to check the typed history in the command line. As a result, these series of commands can be easily re-executed again in the same order. Furthermore, there are many types of shell scripting languages. The most common ones are C-shell (csh) and Bash (Bourne Again hell). In order to choose which type to learned and study, further researches were needed to be carried out. [8] discussed the top 10 disadvantages of using csh over Bourne hell, the shell in which Bash is derived from. According to the author s many years of experience in using csh, one of its main issues is that its syntax is really fussy, especially the parenthesis in which even a space would matter. As a consequence, it has been decided to learn bash. The first thing to learn is that the shell file needs to start with #!/bin/bash. This line is needed in order to specify the type of shell that is being used in the file. [9] provided an example and explained how to write it in the terminal. In order to make a shell file executable, chmod +x [FILENAME].sh has to be typed in the terminal. Writing./[FILENAME].sh will execute the script. Further tutorials and guides were read in order to become competent in bash scripting. As a result, usage of echo, cat, less, if, while and for commands has been explored. [9] has also provided an introduction to variables and parameters, and how to manipulate them. 95

109 C) Advanced hell cripting Learning basics of bash scripting was not enough to perform even the first task in this project. Hence, some advanced shell techniques and concepts had to be learned. [9] has provided a beneficial guide in order to start accomplishing the objectives. The first thing was to read in depth about the use of some special characters. One of the most frequently used characters in this project is the piping character " ". This character passes the output of the first command to the input of the second command. The second character which considered important in this project is the redirection operator ">" which allows the output of a file, command or program to be directed to another. Another important character is "#" which is used to write comments in the program. Other characters will be discussed when they appear in context. After learning special characters, some of the advanced shell commands needed to be learned. The commands which will be extensively used are awk, sed, paste and read. Awk is a text processing language which has a similar syntax to C. It is capable of filtering and editing tables in a text file. ed is defined in the bash terminal manual as a stream editor. This means that it reads one line at a time and performs a specific task. In many cases, awk and sed can be used to achieve the same task. However, awk tend to have much easier syntax when processing multiple lines. ed will be only used in this project when there is a need to delete, add or edit a specific line in a large text file. Other commands will be discussed when they appear in the code. 96

110 Value (Conductivity is in /m, Permittivity is dimensionless) Value (Conductivity is in /m, Permittivity is dimensionless) D) Graph for other Data Fitted Tissues Frequency/Hz Figure A: The plot of the frequency response of pinal Cord tissue Frequency/Hz Figure B: The plot of the frequency response of Grey Matter tissue 97

111 E) hell Program Code to Automate the Data Fitting Activity #!/bin/bash #Indicates which shell version to use. Clear #Clear the screen of the terminal. rm FinalValues rm f rm filename rm P rm C rm P2 rm C2 rm FileName rm ready rm variables rm TissueName rm current #Removes all the working files resulted from the previous tissue. for f in $(find./tissues -type f) #Loop through all the tissue files in Tissues folder and open each one, one at a time. Do #tart of the for loop equivalent to { in C. This for loop execute with same number as there are tissues in the Tissue folder. echo $f # Print out the name of the current file being processed along with its own directory echo $f >> current # tores the name of the current file directory. This line helps in debugging when data fitting does not converge for the current tissue. echo $f > f #ave the file directory in f (a file) awk -F/ '{print $3}' f > filename # Extract current tissue name and save it into another file. awk '{ printf("%.6f\t%.6f\t%.6f\t%.6f\n",(2),(10),(0.08),(0.1))}' > variables # Awk is used to store 4 doubles in 6 decimal places in a file called variables awk '{print "A=(/"$1","$2","$3","$4"/)"}' variables > ready # Make the parameters ready for substitution in the program Debye.F90 at line 314. This is where the initial parameters of the Debye model are defined. read line < ready # tore the initial values in an argument by reading the line in ready. sed -i "314i $line" Debye.F90 #Insert the line into line number 314 in the data fitting program sed -i "315d" Debye.F90 #Delete the line 315. This line had the older values of the initial parameters. 98

112 awk 'FNR>2{printf("%.6f\n", $2)}' $f > P # Extract and store the second column, Permittivity of the current tissue, readings awk 'FNR>2{printf ("%.6f\n", $3)}' $f > C #Extract and store the third column, Conductivity of the current tissue, readings paste Template P > P2 # Merge the columns of the two files together. The Template file contains the frequency column expressed in MHz. This is a requirement by the program. paste Template C > C2 # Merge the two files together cat header C2 > conductivity # Add a header and rename the file so that the program can use it. The cat commands opens the first file and place the second one beneath it. This is another requirement so that the program can process the file. cat header P2 > permitivity #Add a header an rename the file so that the program can use it. make clean make make run #compile and run the Fortran program. paste filename fort.30 > TissueName # Fort.30 is generated by the program each time a data fitting is performed. The file contains the 4 Debye parameters. Another column containing the current tissue name is created and merged with the 4 parameters, and saved in the file TissueName. awk '{print $1"\t\t\t"$3"\t"$4"\t"$5"\t"$6}' TissueName >>FinalValues #This command stores the results of the Debye parameters of all the tissues processed so far. The ">>" operator is used to add a new line to the FinalValues file without deleting its contents. If ">" is used then the previous content will be deleted (overwritten). Done # End of the for loop. Equivalent to } in C. 99

113 F) Project Plan Figure G1: hows emester 1 Project Plan A Project plan has been set up for the first semester as shown in figure G1. The Christmas break is shown as well as a float period to improve or finish up any pending activity. There will be no activity carried out in the exam period as shown in figure G1. M0: Represents the first milestone which is the academic malpractice course followed by test, declaration form and tutorial questions. M1: Represents the second milestone which is the progress report, and it is due Friday 6 th of November

114 Figure G2: hows emester 2 Project Plan Figure G2 shows the project plan for semester two including the float period of Easter Break. The first week of Easter period will be used to carry out any pending activity. The rest of the Easter break will be used for writing up at least 60% of the final report and create the project s poster. The last two weeks in the semester will be used to make final adjustments in the poster, and prepare for the viva voce. M2: Represents the third milestone which is the final report, and it is due Friday 29 th of April M3: Represents the fourth milestone which is the project s poster, and it is to be submitted at the viva voce at some point, TBA, between week 11 and week 12. M4: Represents the fifth and last milestone which is the viva voce and it will be carried out at some point between week 11 and week