# Investigation of Electromagnetic Properties of Multiparticle Systems in the Optical and Microwave Regions

код для вставкиСкачатьNORTHWESTERN UNIVERSITY Investigation of Electromagnetic Properties of Multiparticle Systems in the Optical and Microwave Regions A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY Field of Electrical Engineering By Wendy Yip EVANSTON, ILLINOIS August 2012 UMI Number: 3527721 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent on the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 3527721 Copyright 2012 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 UMI Number: 3527721 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent on the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 3527721 Copyright 2012 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 2 © Copyright by Wendy Yip 2012 All Rights Reserved 3 ABSTRACT Examination of Electromagnetic Properties of Multiparticle Systems in the Optical and Microwave Regions Wendy W. Yip The goal of this work is to examine the electromagnetic properties of multiple particles ensembles in optical and microwave regions. Electromagnetic scattering problems of multi-particles systems appear in many research areas, including biomedical research problems. When a particle system becomes dense, multiple scattering between the particles need to be included in order to fully describe the response of the system to an EM wave. The generalized multiparticle Mie (GMM) solution is used to rigorously solve the Maxwell’s equations for multi-particles systems. The algorithm accounts for multiple scattering effects by transforming the waves scattered by an individual particle to the incident waves of other spheres in the ensemble. In the optical region, light scattering from biological tissues can reveal structural changes in the tissues which can be a mean for disease diagnosis. A new Monte Carlo simulation method is introduced to study the effect of tissue structure on signals from two diagnostic probes, the polarization gating probe and low coherence enhanced back scattering probe (LEBS). In the microwave region, the study of electromagnetic properties with metallic nanoparticles can determine their potential as effective heating agents in microwave hyperthermia therapy. The investigation aims to study the dielectric properties of metallic nanoparticles and quantify the relationship between the characteristics of metallic nanoparticles and the heating effect. The finding should help optimize the 4 design and use of metallic nanoparticles in hyperthermia treatment. In addition, the metallic nanoparticles are studied for their potential to be contrast agents for biological tissue in the microwave region. 5 ACKNOWLEDGEMENTS I would like to thank my academic advisors Dr. Xu Li and Dr. Alan Sahakian for their guidance and support during my graduate career. I would like to thank my committee members Dr. Allen Taflove and Dr. Vadim Backman for their advice and help and for including me in their research group. I would also like to thank my fellow lab mates, in particular Ji Yi, Andrew Gomes and Ilya Mikhelson for all their help and meaningful discussions. I feel very fortunate to have worked with such amazing researchers, and they have made me a better researcher. I would also like to thank my parents Andrew Yip and Hou Hing Chow Yip, my brother William Yip, my friends including Pierre, Nikki and friends from ERC and EBC for their emotional support. 6 Contents ABSTRACT .................................................................................................................................................... 3 ACKNOWLEDGEMENTS........................................................................................................................... 5 LIST OF FIGURES ....................................................................................................................................... 7 LIST OF TABLES ......................................................................................................................................... 9 CHAPTER 1: INTRODUCTION ........................................................................................................... 10 CHAPTER 2: BACKGROUND.............................................................................................................. 12 2.1. 2.2. 2.3. INTRODUCTION ............................................................................................................................ 12 GENERALIZED MULTIPARTICLE MIE SOLUTION ........................................................................... 13 MULTIPLE VS. INDEPENDENT SCATTERING .................................................................................. 20 CHAPTER 3: MULTIPLE SCATTERING EFFECTS ON SPECTROSCOPIC SCATTERING PARAMETERS……………………………………………………………………………………………21 3.1. MULTIPLE SCATTERING EFFECTS ON SPECTROSCOPIC SCATTERING PARAMETERS ....................... 21 3.1.1. Introduction ............................................................................................................................ 21 3.1.2. Forward Problem ................................................................................................................... 22 3.1.3. Inverse problem ...................................................................................................................... 28 CHAPTER 4: INVESTIGATION OF METALLIC NANOPARTICLES IN MICROWAVE REGION……………………………………………………………………………………………………31 4.1. INTRODUCTION ............................................................................................................................ 31 4.2. MICROWAVE HEATING EXPERIMENT OF GOLD NANORODS .......................................................... 32 4.2.1 Description of test system ...................................................................................................... 32 4.2.2 Description of experiment ...................................................................................................... 33 4.2.3 Determination of heating effect .............................................................................................. 34 4.2.4 System Stability Verification and results ............................................................................... 36 4.2.5 Experiment with gold nanoparticles and p-NIPAM ............................................................... 45 4.3. NUMERICAL SIMULATION ............................................................................................................ 49 4.3.1. Numerical simulation of absorption and scattering cross section of a single gold nanoparticle .......................................................................................................................................... 49 4.3.2. Numerical simulation of an ensemble of metallic particles ................................................... 50 4.3.3. Gold nanosatellites experiment .............................................................................................. 54 4.3.4. Summary................................................................................................................................. 56 CHAPTER 5: HETEROGENEOUS MONTE CARLO SIMULATION ........................................... 57 5.1. INTRODUCTION ............................................................................................................................ 57 5.2. DESCRIPTION OF THE MONTE CARLO SIMULATION ..................................................................... 57 5.3. MONTE CARLO SIMULATION OF POLARIZATION GATING PROBE .................................................. 62 5.3.1. A study of packaging coefficient ............................................................................................ 63 Single layer simulation ........................................................................... Error! Bookmark not defined. 5.3.2. Three layer simulation ........................................................................................................... 72 5.4. MONTE CARLO SIMULATION OF LOW-COHERENCE ENHANCED BACKSCATTERING PROBE ......... 77 5.4.1. Single layer simulation ........................................................................................................... 78 5.5. SUMMARY.................................................................................................................................... 82 CHAPTER 6: CONCLUSION ................................................................................................................ 84 7 List of figures Figure 1 Optical parameters calculated by the independent scattering model and the GMM algorithm for a 1% volume fraction particle ensemble .......................................... 26 Figure 2 Optical parameters calculated by the independent scattering model and the GMM algorithm for a 20% volume fraction particle ensemble ........................................ 27 Figure 3 Comparison of optical parameters calculated by the independent scattering model and the GMM algorithm for a 20% volume fraction particle ensemble. ............... 28 Figure 4 Particle size and density reconstructions using the independent scattering model for a range of volume fractions ......................................................................................... 30 Figure 5 Experimental system for microwave heating test ............................................... 32 Figure 6 Temperature trace curves for DI water, 841 nanorods and 762 nanorods.......... 39 Figure 7 Average normalized conductivity vs. particle concentration with the 95% confidence interval ............................................................................................................ 43 Figure 8 Temperature trace curves comparing DI water, CTAB (50 mM), supernatant, gold nanoparticles (plasmon peak ~870 nm) .................................................................... 44 Figure 9 Change of absorbance as temperature increases for Gold-NIPAM................... 47 .Figure 10 The absorbance of p-NIPAM plotted for a range of temperatures .................. 47 Figure 11 The absorbance of p-NIPAM attached gold nanoparticles.............................. 48 Figure 12 Absorption cross section per unit volume for different metallic particle sizes 51 Figure 13 Scattering cross section per unit volume for different metallic particle sizes .. 51 Figure 14 Absorption cross section of an ensemble of metallic nanoparticles of 1nm radius at volume fraction from 1-20% .............................................................................. 53 Figure 15 Absorption cross section of an ensemble of metallic nanoparticles of 1micron radius at volume fraction from 1-20% .............................................................................. 53 Figure 16 Illustration of a microsphere surrounded by gold nanoparticles. .................... 55 Figure 17 Comparison of forward scattering intensity of a microsphere with comparable refractive index of biological cell in an aqueous background and an ensemble of gold nanoparticles surrounding the microsphere ...................................................................... 55 8 Figure 18 Illustration of a pair of blood vessel located at 1 mm below surface ............... 60 Figure 19 Light penetration around a pair of blood vessels.............................................. 60 Figure 20 Light penetration in a homogeneous medium with optical properties equivalent to a constant 5% blood volume medium ........................................................................... 61 Figure 21 Total reflectance measurement from a simulation where the medium has the following optical properties: μa=0 cm-1, μs=125 cm-1, g=0.9 validating the polarized heterogeneous Monte Carlo code...................................................................................... 62 Figure 22 Schematic of the polarization gating probe ..................................................... 63 Figure 23 A cross section of the blood vessel structure................................................... 66 Figure 24 A cross section of the blood vessel structure illustrating the pixelated blood vessel structure .................................................................................................................. 66 Figure 25 Reflectance vs blood vessel diameter are plotted for the total reflectance, copol and cross pol signal ..................................................................................................... 68 Figure 26 Reflectance spectra comparing a homogeneous medium at 7.7% blood volume fraction, a heterogeneous blood vessel medium at 7.7% blood volume fraction with 10 micron diameter vessels, a heterogeneous blood vessel medium at 17.3% blood volume fraction with 10 micron diameter vessels and a heterogeneous blood vessel medium at 7.7% with 16 micron diameter vessels ............................................................................. 71 Figure 27 Reflectance spectra for three layer tissue structure and homogeneous tissue structure............................................................................................................................. 75 Figure 28 Comparison of reflectance from a three layer tissue structure with an increased blood density and the homogeneous structure. ................................................. 76 Figure 29 Validation of the blood vessel recovery algorithm........................................... 81 Figure 30 Illustration of the two layer model for LEBS ................................................... 83 Figure 31 Initial result of pathlength dependence on absorption cross section for twolayer control and cancerous lung tissue ............................................................................ 83 9 List of Tables Table 1 The normalized conductivity calculated at three trials to determine the stability of the system. Three samples are tested: DI water, 762 nanorods at 5 mM concentration and 841 nanorods at 5 mM concentration. ........................................................................ 38 Table 2 Summary of data comparing the normalized conductivity of DI water, 762 nanorods and 841 nanorods at 5mM, 2.5 mM and 1.25 mM ............................................ 42 Table 3 Two sample t-test comparing the normalized conductivity obtained from DI water to that obtained from 5 mM concentration of 762 nanorods................................... 42 Table 4 Two sample t-test comparing the normalized conductivity obtained from DI water to that obtained from 5 mM concentration of 841 nanorods................................... 43 Table 5 Comparison of the normalized conductivity of several background liquid with gold nanoparticles ............................................................................................................. 44 Table 6 Optical properties used in the two-vessels simulation ......................................... 61 Table 7 Packaging coefficient for different signals comparing the packaging coefficient for diffuse reflectance ....................................................................................................... 68 Table 8 Simulation parameters for the one layer model ................................................... 70 Table 9 Difference in reflectance between the different simulations at 574 nm .............. 71 Table 10 Difference in reflectance measurement at 574 nm between a homogeneous and three layer blood vessel structure ...................................................................................... 75 Table 11 Effect of increasing blood volume density at different depth in the tissue on the reflectance measurement ................................................................................................... 76 Table 12 Optical properties of the one layer LEBS simulation ........................................ 80 Table 13 Recovered blood vessel radius compared to the simulated blood vessel radius 81 Table 14 Optical properties of the two layered model at 650 nm ..................................... 83 10 Chapter 1: Introduction The study of the electromagnetic properties in multi-particle systems has many applications ranging from astrophysics, atmospheric science to biomedical engineering. In particular, with biomedical applications, multi-particle systems can be used to represent problems in imaging and therapeutic techniques. In the area of imaging, multiple scattering can be used to study and develop contrast agents to better enhance detection of structural changes in the background. At the same time, multiple scattering can be a source of noise and distortion in the imaging signal when recovering information with regards to the sample of interest. In therapeutic techniques, an ensemble of specialized scatterers could absorb and scatter electromagnetic energy more efficiently compared to the background, making them efficient heating and contrast agents. The purpose of this thesis is to examine the electromagnetic properties of multi-particle systems in both the optical and microwave region. Both numerical and experimental techniques are utilized to achieve this objective. In chapter 2, descriptions of various numerical techniques used to study electromagnetic properties of multi-particle systems are presented. After reviewing and evaluating various techniques, the generalized multi-particle Mie (GMM) solution is chosen as the method to investigate the problem outlined in chapter 3, and the method is described in further details. In chapter 3, the electromagnetic properties of multi-particle systems in the optical region are examined. The multiple scattering effects on spectroscopic parameters are discussed. In chapter 4, the dielectric properties of metallic nanoparticles in the microwave region are examined in order to evaluate their potential to 11 improve microwave hyperthermia of cancerous tissues. This is accomplished by conducting microwave heating experiments of metallic nanoparticles and performing numerical simulations of the electromagnetic wave interaction with the particles. In chapter 5, Monte Carlo simulations is used to study the impact of tissue structure on signals from two diagnostic probes, the polarization gating probe and the Low Coherence Enhanced Backscattering probe. 12 Chapter 2: Background In this chapter, a review of available light scattering methods is provided. In particular, the Generalized Multiparticle Mie theory is examined in greater detail as it is one of the methods used to investigate the impact of multiple scattering on spectroscopic parameters. Independent and multiple scattering are also described, and the need to utilize techniques to account for interaction between the particles in a dense ensemble is discussed. 2.1. Introduction Scattering methods can be classified as analytical methods and surface-based/volumebased methods. [1] The analytical method is generally based on the separation of variable approach such as the Mie theory. Mie theory is an analytical solution of Maxwell’s equations for electromagnetic scattering from a sphere. The Maxwell’s equations are used to derive a wave equation for the electromagnetic radiation in spherical coordinates with the object surface as the boundary condition. The solution becomes a separable differential equation, and the solution to the equation is an infinite series of products of spherical vector functions. The expansion coefficients to the functions are found by enforcing the boundary condition on the surface of the sphere. Mie theory is restricted to plane wave scattering by a homogeneous sphere in a lossless background. [2] Variation of the Mie theory has been developed to overcome the limitation, such as for a non-spherical particle [3] and for an anisotropic spherical shell [4]. The Generalized Multiparticle Mie solution is an extension of the Mie theory to solve for scattering properties in an ensemble of particles. [5] 13 Surface-based method enforces boundary condition on the surface of the particle, and the surface of the particle is discretized. With volume-based method, the volume of the particle and the volume the particles are placed in are discretized. Some of the methods that belong to these two categories are the T-Matrix method [6,7] , generalized multipole technique [8] and method of moments [9], finite difference time domain method [10], finite element method [11] and discrete dipole approximation [12]. In addition to scattering methods, photon transport in biological tissues can be modeled using the radiative transfer equation. [13] The numerical Monte Carlo method of photon transport belongs to this class, and it has been applied to solve biomedical imaging problems. The path a photon travels is modeled as a persistent random walk, with the direction of each step dependent on the previous step. By tracking a sufficient number of photons, physical quantities of a particle ensemble can be estimated. Monte Carlo simulation will be discussed in more detail in the next chapter. An advantage of the Monte Carlo technique is that it can solve problems involving a large geometry without a significant strain on computation resources, which is the case for full wave solution such as FDTD. However, the Monte Carlo technique requires the knowledge of statistically averaged optical parameters in order to determine photon transport, and it estimates ensemble-averaged quantities rather than providing an exact solution of EM interaction with the ensemble. 2.2. Generalized Multiparticle Mie solution Generalized Multiparticle Mie solution is used in the subsequent chapter to solve for electromagnetic interaction in multiparticle problem. The numerical solution is developed by Y.L. Xu, and the theory is given in more details in [5,14,15,16,17, 18,19]. 14 Consider non-intersecting homogeneous spheres with known radius and refractive index in a finite volume, if an incident wave which illuminates the spheres ensemble can be represented as elementary spherical waves, the scattering properties of the ensemble can be determined. To begin, consider in a source-free, isotropic and homogeneous medium, the time harmonic electric and magnetic fields are divergence free, and they satisfy the vector wave equations ∇ × ∇ × E − k 2E = 0 Equations 2-1 2 ∇×∇×H − k H = 0 where k is the wave number, and k 2 = ω 2εμ , ε is the dielectric constant and μ is the permeability of the medium. A set of linearly independent solutions to the vector wave equations in spherical coordinates is given by the vector spherical functions M mn and N mn . The vector spherical functions are given by M (mnJ ) ( ρ , θ , φ ) = [eˆ θ iπ mn (cos θ ) − eˆ φτ mn (cos θ )]zn( J ) ( ρ ) exp(imφ ) m ˆ N (J) mn ( ρ , θ , φ ) = {e r n( n + 1) Pn (cos θ ) + [eˆ θτ mn (cos θ ) + eˆ φ iπ mn (cos θ )] z n( J ) ( ρ ) Equations 2-2 ρ ( ρz n( J ) ( ρ ))' } exp(imφ ) ρ ê r , êθ , êφ are the unit vectors in spherical coordinates. ρ = kr. ( ρz n( J ) ( ρ ))' denotes the derivative with respect to ρ . z n( J ) ( ρ ) is one of four appropriately selected spherical Bessel functions. z n(1) = jn Equations 2-3 15 z n( 2 ) = y n z n(3) = hn(1) z n( 4 ) = hn( 2 ) jn is the spherical Bessel function of the first kind. y n is the spherical Bessel function of the second kind. hn(1) , hn( 2 ) are the Hankel functions of the first and second kind, respectively. Pnm (cos θ ) is the associated Legendre function of the first kind and of degree n and order m, and n and m are integers that go from 1 ≤ n < ∞ and − n ≤ m ≤ n . π mn and τ mn are angular functions defined as π mn (cos θ ) = τ mn (cos θ ) = m Pnm (cos θ ) and sin θ d m Pn (cos θ ) . dθ The internal field and the scattered field for the jth sphere in the reference frame centered at the center of the sphere can be expressed in terms of the spherical functions as ∞ n j E sca =∑ ∑ iE mn j j [amn N (mn3) ( ρ , θ , φ ) + bmn M (mn3) ( ρ , θ , φ )] n =1 m = − n j H sca = ∞ k n ∑ ∑E ωμ mn j j [bmn N (mn3) ( ρ , θ , φ ) + amn M (mn3) ( ρ , θ , φ )] n =1 m = − n ∞ n j Eint = −∑ ∑ iE j 1) j 1) [d mn N (mn ( ρ , θ , φ ) + cmn M (mn ( ρ , θ , φ )] j n mn n =1 m = − n Equations 2-4 k j H int =− ∞ ωμ j ∞ ∑ ∑E mn j 1) j 1) [cmn N (mn ( ρ , θ , φ ) + d mn M (mn ( ρ , θ , φ )] n=1 m=− n n j Einc = −∑ ∑ iE mn j 1) j 1) [ pmn N (mn ( ρ , θ , φ ) + qmn M (mn ( ρ , θ , φ )] n =1 m = − n j H inc =− k ωμ ∞ n ∑ ∑E n=1 m=− n mn j 1) j 1) [qmn N (mn ( ρ , θ , φ ) + pmn M (mn ( ρ , θ , φ )] 16 j j j j and H sca are the scattered fields of the jth sphere. E int and H int are the E sca j j and H inc are the incident field on the jth sphere internal fields of the jth sphere. E inc which includes both the light source and the scattered light from other spheres and Emn is the normalization coefficient Emn =| E0 | i n [ (2n + 1)(n − m)! ] (n + m)! Equation 2-5 j j j j j j ( a mn , bmn ) , (cmn , d mn ) and ( pmn , qmn ) are the expansion coefficients for the scattered field, internal field and incident field, respectively. The expansion coefficients are solved by applying the boundary conditions at the surface of the jth sphere. j j j [Einc + E sca − Eint ] × eˆ r = 0 Equations 2-6 [H j inc +H j sca j int − H ] × eˆ r = 0 At the surface of the sphere r j = a j , the boundary conditions are j j j Einc ,θ + E sca ,θ = Eint,θ j j j Einc ,φ + E sca ,φ = Eint,φ Equations 2-7 H j inc,θ +H j sca ,θ =H j int,θ j j j Hinc ,φ + H sca ,φ = H int,φ The boundary conditions give rise to four linear equations j j j jn (m j x j )cmn + hn(1) ( x j )bmn = qmn jn ( x j ) Equation 2-8 j j j j j mn j j (1) n j j mn j mn j j j μ[m x jn (m x )]' c + μ [ x h ( x )]' b = q μ [ x jn ( x )]' 17 j j μm j jn (m j x j )d mnj + μ j hn(1) ( x j )amn = pmn μ j jn ( x j ) j j j [m j x j jn (m j x j )]' d mn + m j [ x j hn(1) ( x j )]' amn = pmn m j [ x j jn ( x j )]' where x j , m j are the size parameter and relative refractive index of the jth sphere, respectively, and they are x j = ka j = 2πN 0 a j λ , mj = kj Nj = 0 where λ is the k N wavelength of the incident waves in the background medium and N 0 is the refractive index of the surrounding medium. The four linear equations can be used to solve for the j j j j expansion coefficients amn in terms of the incident field coefficients , bmn , cmn , d mn j j . pmn , qmn anj , bnj , cnj , d nj are Mie coefficients for the jth sphere as described in Bohren and Huffman [2] j j j j can be rewritten in terms of the incident field coefficients amn , bmn , cmn , d mn j j pmn , qmn j j amn = anj pmn j j bmn = bnj q mn Equation 2-9 c j mn j n =c q j mn j j d mn = d nj pmn The above equations show that the scattering coefficients are simply linear modifications of the Mie coefficients by the incident field expansion coefficients. As 18 long as the incident field expansion coefficients can be found, the scattering coefficients and in turn the scattering properties of the particle can be found. The next step in the process is to find the incident field expansion coefficients. The total incident field, including the initial incident field and the scattered field from other spheres, must be expanded about the center of the jth sphere, and they can be written as Ei ( j ) = E0 ( j ) + ∑ E s (l , j ) l≠ j where E0 ( j ) the initial incident is field, and Equation 2-10 ∑ E (l, j) is the scattered field translated s l≠ j from the lth coordinate system to the jth coordinate system. ∑ E (l, j) can be expanded s l≠ j as ∞ E s (l , j ) = −∑ n ∑ iE n=1 m=− n mn l , j (1) l, j 1) [ pmn N mn + qmn M (mn ] Equation 2-11 where ∞ ν μν μν l, j l l pmn = −∑ ∑[aμν Amn (l , j ) + bμν Bmn (l , j )], (l ≠ j ) ν =1 μ =−ν ν ∞ μν μν l, j l l qmn = −∑ ∑ [aμν Bmn (l , j ) + bμν Amn (l , j )], (l ≠ j ) Equations 2-12 ν =1 μ =−ν μν μν Amn (l , j ) and Bmn (l , j ) are vector translation coefficients, which characterize the transformation of the scattered waves from lth sphere to the incident waves of the jth sphere. [5,19,20] 19 The expansion coefficients for the total incident field for the jth sphere are given by (1, L ) L p j mn =∑p l =1 l, j mn =p j, j mn −∑ l≠ j ν ∞ [aμν Aμν (l , j ) + bμν B μν (l , j )] ∑ ∑ ν μ ν l l mn =1 mn =− Equations 2-13 (1, L ) L j l, j j, j qmn = ∑ qmn = qmn −∑ l =1 l≠ j ∞ ν [aμν B μν (l , j ) + bμν Aμν (l , j )]= ∑ ∑ ν μ ν l l mn =1 mn =− j, j j, j and q mn are the initial incident waves, and the summation refers to the fields p mn j j j j j j and qmn back into amn , bmn scattered by other spheres. Substituting p mn = anj pmn = bnj q mn and rearranging yield the linear system j (1, L ) ∞ ν amn μν μν j l l = p − [aμν Amn (l , j ) + bμν Bmn (l , j )] ∑∑ ∑ mn j an l ≠ j ν =1 μ =−ν Equations 2-14 j mn j n (1, L ) ∞ ν b μν μν j l l = qmn − ∑∑ ∑ [aμν Bmn (l , j ) + bμν Amn (l , j )] b l ≠ j ν =1 μ = −ν The highest-order N necessary for convergence is determined by the Wiscombe criterion, which is N = x + 4 x1/ 3 + 2 , where x is taken to be the largest size parameter out of all the spheres if the spheres are not identical. The above linear system has 2 LN ( N + 2) unknown scattering coefficients, and hence there are 2 LN ( N + 2) equations to be solved. While it is possible to solve for the linear system with matrix inversion, however, when L and N are large, it is necessary to find more efficient method to solve for the system. [19] In the numerical code developed by Y.L. Xu, a numerical scheme described by Mackowski [19] is used making it possible to avoid the computation of the vector translation coefficients by decomposing 20 jl jl Amn μν and Bmnμν into rotational and axial translational parts. This method simplifies the problem because only the diagonal terms Aμjlnμν and Bμjlnμν (when m = μ ) exist. The iterative technique BiCGstab, a method developed by Sleijpen and Fokkema, is used to solve the linear system. 2.3. Multiple vs. Independent Scattering Since multiple scattering is of interest in this study, it is necessary to compare and contrast independent and multiple scattering. Independent scattering occurs when the density of the particles in the volume of interest is small, or the distance between the particles is large compared to the incident wavelength. The particles are not assumed to interact together, and the particles only scatter the incident light. Light scattering parameters can be calculated as a linear summation of contributions from individual particles. Multiple scattering occurs when the density of particles is high or the distance between neighboring particles is small. In multiple scattering, the incident light is considered as the original incident light as well as the light scattered from other particles. The Generalized Multiparticle Mie solution is used to provide the analytic solution to the multiple scattering problem in ensembles that resemble biological tissues and also in ensembles of metallic nanoparticles. The two problems will be discussed in further details in the next chapter. 21 Chapter 3: Multiple scattering effects on spectroscopic scattering parameters 3.1. Multiple scattering effects on spectroscopic scattering parameters 3.1.1. Introduction In a growing number of applications in biomedical optics, the spectral properties of elastic light-scattering parameters are utilized to characterize structural features of biological tissue on the micro/nano scale. In particular, there has been considerable prior research effort in recovering the size distribution and number density of cellular and subcellular particles from the measured light-scattering spectra [21,22] while assuming a discrete particle model for biological tissue scattering [23]. Specifically, the intracellular particle size distribution is approximated by an analytical function whose parameters are determined using an inverse algorithm. Most of these calculations are based on the assumption of independent scattering, which indicates that the light scattering parameters can be calculated as a linear summation of contributions from individual particles. While this assumption provides relatively convenient calculation, it obviously ignores the physical phenomenon of multiple scattering for particle ensembles. This is significant especially since intracellular particles of biological tissue can have relatively high volume fraction (e. g. 30%) [24]. Although there has been extensive research on multiple scattering models including a variety of approximation formula [32, 33], their validity for scattering media with optical parameters within the range of biological tissue has not been established. Rigorous numerical solutions of Maxwell’s Equations are used to investigate the effects of multiple scattering on the optical parameters of particle ensembles similar to biological tissue. The impact of these effects is investigated on 22 reconstructing the size and number density of intracellular particles using spectroscopic scattering parameters measured from biological tissues. 3.1.2. Forward Problem The numerical simulations are based on the Generalized Multiparticle Miesolution (GMM) described in the previous chapter. Based on the GMM simulation, the optical parameters for a range of particle aggregations representative of biological tissues are calculated. Here, plane wave illumination is generated in the simulations at seven wavelengths ranging from 810-1620 nm. Within each simulation, far-field electric field pattern in its phasor form E s (θ, φ, λ ) is calculated for an ensemble of particles randomly distributed within a cube of fixed volume with a pre-defined volume fraction within the range of 1−25%. The incident and scattered amplitudes in the far field are related to each other by [34] ⎡ E||sca ⎤ exp[ik (r − z )] ⎡S 2 ⎢E ⎥ = ⎢S − ikr ⎣ 4 ⎣ ⊥sca ⎦ S3 ⎤ ⎡ E||inc ⎤ S1 ⎥⎦ ⎢⎣ E⊥inc ⎥⎦ Equations 3-2 E||inc and E⊥inc are the incident field and E||sca and E⊥ sca are the scattered field in the far field. The parallel and perpendicular to the scattering plane is defined by the incident direction and the scattering direction. E||inc = E0 cos(φ − β p ) exp(ikz ) and E⊥inc = E0 sin(φ − β p ) exp(ikz ) 23 ⎡S || The amplitude scattering matrix is defined as ⎢ ||2 ⎣⎢S 4 S 3⊥ ⎤ ⎥ where S 2 and S 4 are S 1⊥ ⎦⎥ associated with the linear polarization state of the incident plane wave β p = φ and S 3 and S1 are β p = φ + 90 For a normal x-linearly-polarized plane wave, E⊥inc = 0 , so only the S 2 and S 4 components are used to calculate the far field. The particles’ positions are randomly generated with the Metropolis shuffling method [35] with no overlapping allowed. For all simulations, the background medium is assigned to a refractive index of n0 = 1.35 , and the particles have a refractive index of n0 = 1.42 . These parameters are chosen to match the optical properties of biological tissue scatterers [31]. In order to account for the random nature of particle distribution and to eliminate the coherent interference components, GMM simulation is repeated for N = 500 realizations with re-randomized particle locations for an ensemble with specific particle size and volume fraction. Since the particles are confined in a fixed volume in all realizations, the boundary effect from the j th realization is removed using Esj,inf = Esj − < Es > , where < E s > is the averaged E field across all realizations [36]. Then the incoherent scattering intensity for this ensemble condition is calculated by taking the average of all realizations using I M (θ , φ , λ ) = 1 N N ∑[E j s ,inf (θ , φ , λ )E sj,inf (θ , φ , λ )* ] . Here, the subscript M signifies the fact j that there are M particles in each ensemble realization, and thus the inter-particle 24 scattering for these M particles is accounted for in the calculation of I M (θ, φ, λ) . A convergence study has been performed for each ensemble condition to ensure the number of particles M is sufficient to describe all non-negligible orders of multiple scattering. The convergence study is done by repeating the calculations with a doubled number of particles 2M while keeping the volume fraction, and making sure the calculated incoherent intensity converges with that obtained from a set of simulations with a smaller number of particles, i.e. , I 2 M (θ , φ , λ ) ≅ 2I M (θ , φ , λ ) . Next, the scattering cross section of the ensemble condition σ s (λ ) |M is calculated by integrating the incoherent scattering pattern using σ s (λ ) |M = 1 k2 ∫∫ I M (θ, φ, λ ) sin θdθdφ Equations 3-3 The scattering coefficient μ s (λ) |M then can be calculated as μ s (λ) |M = σ s (λ) |M / V , where V is the total volume of the distribution space of the particles. Similarly, the anisotropy factor g (λ) |M and the reduced scattering coefficient μ′s (λ) |M are calculated according to their definitions using g (λ ) |M = ∫∫ I M (θ , φ , λ ) cosθ sin θdθdφ / ∫∫ I M (θ , φ , λ ) sin θdθdφ Equations 3-4 and μ′s (λ) |M = μ s (λ) |M (1 − g (λ) |M ) Equations 3-5 25 In order to examine the multiple-scattering effects on optical parameters of a biological tissue, μ s (λ ) |M and μ′s (λ) |M are compared with those calculated assuming an independent scattering model. In the latter case, μ s (λ ) |independent is calculated from μ s (λ ) |independent = N 0 σ s (λ ) , where N 0 is the number density of the particle distribution, and σ s (λ ) is the scattering cross section of a single particle. Parameters g (λ ) |independent and μ′s (λ ) |independent are also calculated based on the scattering of a single constituent particle according to their definition. The results of these comparisons for several ensemble conditions are summarized in Figs. 1–3. Figures 1(a) and (b) show the comparison for μ s (λ ) and μ′s (λ ) for a volume fraction of 1 % and constituent particle size of d = 400nm . The close convergence of the data between independent and multiple scattering models for this low volume fraction adds to our confidence in this calculation methodology. This result is contrasted by the comparison for a higher volume fraction of 20% shown in Fig 2(a) and 1(b). Here, optical parameters calculated with GMM simulations are significantly lower compared with those calculated with the independent scattering model. Apparently, multiple scattering effects cannot be ignored at this high volume fraction. In addition, the difference between the independent and multiple scattering models is more significant for μ s (λ ) compared to μ′s (λ ) , where the root mean square (rms) difference is 43.37% for μ s (λ ) and 20.74% for μ′s (λ ) . Figures 2(c) and 2(d) offer some insight to explain this phenomenon. Fig. 2 (c) compares the scattering pattern I ( θ, φ) at a wavelength λ = 1620nm between the independent and multiple scattering models. It indicates that one major effect of multiple scattering is the reduction of the forward scattering intensity. 26 This effect results in a decreased value for anisotropy factor g as demonstrated in Fig. 2(d). Since μ′s is defined as μ′s = μ s (1 − g ) , the reduction of g brings μ′s closer to the values calculated with the independent model compared to μ s . Figs. 3 (a) and (b) repeat the comparison for particle with d = 1μm and a volume fraction of 20%. While most of the previously observed effects are present in this case, it is noted that both μ s (λ ) and μ′s (λ ) deviate less from the independent-model calculation for this larger particle size. It is clear from Fig. 2 and Fig 3 that multiple scattering effects can have significant contribution to the optical parameters for the upper bound volume fraction of intracellular particles. Figure 1 Optical parameters calculated by the independent scattering model and the GMM algorithm for a 1% volume fraction particle ensemble with particle diameter of d = 400nm . The number of particles in the ensemble is M = 103 . (a) μ s as a function of wavelength. The insert shows the geometry for one of the realizations. (b) μ′s as a function of wavelength. 27 Figure 2 Optical parameters calculated by the independent scattering model and the GMM algorithm for a 20% volume fraction particle ensemble with particle diameter of d = 400nm . The number of particles in the ensemble is M = 385 . (a) μ s as a function of wavelength. The insert shows the geometry for one of the realizations. (b) μ′s as a function of wavelength. (d) Normalized scattering pattern I (θ) for λ = 1620nm . (c) Anisotropy factor g as a function of wavelength. 28 Figure 3 Comparison of optical parameters calculated by the independent scattering model and the GMM algorithm for a 20% volume fraction particle ensemble. The diameter of the particles is d = 1μm . The number of particles in each realization for the GMM calculation is M = 382 . (a) Scattering coefficient μ s as a function of wavelength. The insert shows the geometry for one of the realizations. ′ (b) Reduced scattering coefficient μ s as a function of wavelength 3.1.3. Inverse problem Next, the impact of these effects on the accuracy of reconstructing particle size d and number density N 0 based on the assumed independent scattering model is investigated. Here, GMM calculated μ s (λ ) and μ′s (λ ) for a specific ensemble condition are used to emulate the measurement data. The first step in solving the inverse problem is to calculate the wavelength-dependent scattering cross section σ s (λ) of single particles with a series of sizes within a reasonable range (in our case, 0.02-2.0 μm in diameter with 1nm increment). In each trial, μ s (λ) can be easily calculated using Mie theory applied to the assumed particle size. To find the particle size d in the ensemble, the normalized spectral shape of μ s (λ ) is compared to that of σ s (λ) to find the best fit using a least-squares error strategy. Subsequently, μ s (λ) is divided by σ s (λ) with the 29 best-fitting size and the result is averaged over wavelength to recover N 0 . Similarly, the inverse calculation for d and N 0 from μ′s (λ ) is performed based on single-particle scattering cross section and anisotropy. Figures 4 (a) and 4 (b) show the inverse calculation result for ensembles with actual particle size of d = 400nm and volume fractions ranging from 1% to 25%. At low volume fraction, the inverse calculation produces accurate prediction of d and N 0 . The deviation becomes substantial at higher volume fraction with the independent model under-estimating the particle size and over-estimating the number density. It is also observed that μ′s (λ ) in general produces more accurate size and number density prediction compared to μ s (λ ) . These trends are consistent with what is observed in Figs 1−3 where μ s (λ ) |M and μ′s (λ ) |M are compared with those predicted with the independent scattering model for low and high volume fractions. Figures 4(c) and 4(d) show the inverse calculation result for particle size of d = 1μm . It is also noted that the size and number density reconstruction result matches the actual values better (in an error percentage sense) for this larger particle size compared to the d = 400nm particle ensembles with same particle volume fraction. There are two potential mechanisms responsible for this improved accuracy. First, the multiple scattering effects have less contribution to the scattering parameters due to a smaller number density for fixed volume fraction. Second, as the particle size increases to be on the same order of magnitude with wavelength, there are more spectral features, including oscillation, to help identifying the particle size. This observation implies that for a multi-disperse 30 particle ensemble, which is always the case for biological tissues, multiple scattering has different levels of impact for different particle sizes. (a) (c) (b) (d) Figure 4 Particle size and density reconstructions using the independent scattering model for a range of volume fractions. (a) and (b) Particle size d = 400nm . (c) and (d) d = 1μm 31 Chapter 4: Investigation of metallic nanoparticles in microwave region 4.1. Introduction Techniques for using microwave energy to achieve absorption and hyperthermia at breast cancer sites have been studied and developed. Studies have shown that in microwave hyperthermia, the temperature in the tumor volume can reach as high as 43 C causing cell death or making the cells more vulnerable to radiation therapy or chemotherapy. [25] Previous microwave studies have shown a single metallic nanoparticle having an increased temperature effect on molecules attached to it or to the immediate area. [26,27] Metallic nanoparticles delivered to the site of cancerous cells have the potential to enhance the heating effect. The nanoparticles could do so by increasing the effective dielectric properties of the host background and hence enhancing the microwave scattering from the site. Also, microwave electromagnetic field would inductively heat the nanoparticles. These effects could potentially lead to improvement of microwave hyperthermia therapy by introducing metallic nanoparticles to the tumor sites. Previous attempts have been made to obtain data on the effective intrinsic permittivity and conductivity of isolated metallic particles but the data were compromised by a lack of control on particle size, shape and aggregation. [28,29] There is a need to quantify the relationship between the characteristics of the nanoparticles and the heating properties in order to optimize hyperthermia therapy. Experimental and numerical studies are performed to quantify the properties of metallic nanoparticles in the microwave frequencies. Specifically, the dielectric 32 properties of metallic nanoparticles embedded in a background comparable to breast cancer tissues are investigated, and the question of whether there is an enhanced heating effect with the introduction of the nanoparticles at the tumor site is addressed. In addition, numerical simulations are used to examine the potential of using metallic nanoparticles to provide better contrast to imaging biological tissue in the microwave region. 4.2. Microwave heating experiment of gold nanorods An experimental set up has been constructed to investigate the heating effects of gold nanorods in microwave frequency. The system set-up and preliminary data are described below. 4.2.1 Description of test system Microwave generator Power amplifier SWR meter Waveguide PC Temperature sensor Antenna tuner Sample Figure 5 Experimental system for microwave heating test Fig. 8 shows the experimental system used for our microwave heating test. The microwave frequency is at 4 GHz, and the output power from the amplifier is adjusted to 7 watt. At the optimal position of the tuner, the VSWR value is around 1.2. The power 33 level is monitored using a directional coupler and power meter during the heating cycle to ensure stability. A PC is used to control the on/off of the microwave source. A National Instrument AD/DA card is connected to the external port of the microwave generator with a pre-configured cable, and a visual C++ code is used to control the data acquisition. A fiber-optic temperature probe Luxtron I652 is used to measure the sample temperature. The temperature probe has been calibrated for stability. The nanoparticles are placed in Norell NMR sample tubes cut to a smaller size in order to fit the sample holder. 4.2.2 Description of experiment Each experiment trial consists of a set of three heating and cooling cycles and three trials are performed for each sample (a total of nine heating/cooling cycles for each sample). A heating cycle lasts 3 minutes, and a cooling cycle lasts 7 minute to ensure there is sufficient time for the sample to cool down to room temperature. Each experiment trial lasts for 30 minute total (each sample is tested for 90 minutes). The temperature trace curves for the heating and cooling cycles are recorded, and Matlab is used to fit the temperature trace curves to determine the equations for the temperature variation over time and heating parameters. Further description is given in the next section. The nanoparticle samples used in the experiment are prepared by Professor Messersmith’s research group at Northwestern University. The nanoparticles are gold nanorods synthesized with an aqueous seeded method, and they differ in their aspect ratios. Two kinds of nanorods are tested in the experiment. One type of nanorod has a 34 plasmon peak at 762 nm, and the other type of nanorods has a plasmon peak at 841 nm. The plasmon peak is determined by running a UV-Vis scan to determine the center wavelength of the sample. In the rest of the chapter, the two types of nanorods will be referred to as “762 nanorods” and “841 nanorods.” 4.2.3 Determination of heating effect The change in temperature of nanoparticles in solution is described by C dT kA = P − (T − T0 ) dt x Equation 4-1 where T =temperature, t =time, T0 is the room temperature, C is the heat capacity of water at room temperature and P is the absorbed power. It is assumed that the sample has the same heat capacity as that of water because the nanoparticles are suspended in water, and they are at very small concentrations. kA is related to the property of the sample tube: the thickness of the wall of the tube, x surface area, heat transfer coefficient Power is proportional to the electric field and the conductivity: P ∝| E | 2 σ , and E =electric field, σ =conductivity, The volume of liquid used in the experiment is 0.09 mL Let B = kA x For the heating phase, the change in temperature follows 35 C dT + BT − ( P + BT0 ) = 0 dt Equation 4-2 For the cooling phase, the change in temperature follows C dT + BT − BT0 = 0 dt Equation 4-3 The solution to the temperature change takes the exponential form Heating cycle: T (t ) = a1 − b1 exp(−t / τ 1 ) Cooling cycle: T (t ) = a2 + b2 exp(−t / τ 2 ) Equations 4-4 where a1 , a2 , b1 , b2 ,τ 1 ,τ 2 are parameters fitted with a Matlab program. a2 is equivalent to the room temperature, and a1 is the temperature the samples asymptotically approach during the heating cycle By substitute the heating cycle solution to heating phase equation, the power is proportional to P ∝ (a1 − T0 ) . Recall that P ∝| E | 2 σ , and conductivity is proportional to σ ∝ (a1 − T0 ) , and (a1 − T0 ) is the normalized conductivity. The normalized conductivity is a measure of the heating effect of the sample. 36 4.2.4 System Stability Verification and results System stability is verified by repeating the heating and cooling trials with DI water using the same volume. Three trials were tested for DI water to ensure there is repeatability. Figure 9a provides the illustration for three trials of the heating and cooling cycle curves for DI water. The heating and cooling curves do not overlap exactly between different trials. That is due to a slight fluctuation in the room temperature, as can be seen in the starting point of the heating cycle. The room temperature can fluctuate by one degree depending on the time of the day. However, conductivity is related to the temperature difference between the room temperature and the temperature the samples asymptotically approach to during the heating cycle. As long as the temperature difference can be shown to be consistent, the trials are shown to be repeatable. Table 1 shows the normalized conductivity calculated for three trials, the average value obtained over the three trials. The standard deviation over the average value is less than 2% of the average value, which shows that the trials are repeatable, and that the system is sufficiently stable. Subsequently, three trials were performed for both the 5mM concentration of the 841 nanorods and 5 mM concentration of the 762 nanorods for repeatability. The temperature trace curves are shown in Figure 9b and 9c. Again, table 1 shows the normalized conductivity calculated for the three different trials for both samples. For both samples, the standard deviation of the normalized conductivity is less than 2% of the average value, similar to that of DI water. This reassures the system stability. In addition, the average values for the normalized conductivity are both higher compared to 37 that of DI water, which perhaps indicates an enhanced heating effect for the samples over water. 38 Samples Trial 1 Trial 2 Trial 3 Standard deviation/Average 8.832 0.0185 Average DI Water 8.665 8.840 8.991 762 nanorods 11.195 11.105 11.127 11.142 0.0042 5mM 841 nanorods 10.068 10.108 9.772 9.983 0.0184 5mM Table 1 The normalized conductivity calculated at three trials to determine the stability of the system. Three samples are tested: DI water, 762 nanorods at 5 mM concentration and 841 nanorods at 5 mM concentration. Table 1 shows the temperature difference is fairly consistent for all samples during three separate experiment trials. The data fitting used to retrieve the temperature information is described in the next part. 39 Figure 6 Temperature trace curves for DI water, 841 nanorods and 762 nanorods. Three trials of heating and cooling cycles are performed to ensure repeatability. Once the stability of the system is verified, different concentrations of the nanorods are tested to see whether there is a correlation between concentrations of the nanorods and change in normalized conductivity. 5mM concentrations of the gold nanorods are diluted with DI water to obtain lower concentrations of the nanorods. The heating and cooling trials are repeated for the different dilutions of nanorods. Again, to ensure repeatability, more than one trial (3 heating/cooling cycles) is performed for each dilution. Table 2 shows the average normalized conductivity of all the trials taken for each kind of samples, standard deviation between the trials and the 95% confidence interval. Figure 10 shows a plot of the average normalized conductivity for all the trials vs. 40 concentration for each sample, and the 95 % confidence interval is also included. The general trend is that both samples of nanorods have a higher normalized conductivity than DI water and hence have a more efficient heating effect than DI water. A two sample t-test has been performed to compare the mean of the normalized conductivity from water and from the 5mM concentration of each kind of nanorods. Three trials were performed for each sample. The mean and standard deviation of the trials were determined for them, and the two-sample t-test is used to determine whether the trial means indeed differ. Table 3 shows the two samples t-test performed for the normalized conductivity comparing DI water with 5 mM concentration of 762 nanorods, and Table 4 shows the ttest comparing DI water with 5 mM concentration of 841 nanorods. For the t-test comparing DI water with the 762 nanorods, the two-tailed p-value is less than 0.0001. Generally, if the p-value is less than 0.05, the two sample means difference can be concluded to be statistically significant. It can be concluded that the normalized conductivity mean is different between DI water and 762 nanorods, and that the 762 nanorod sample has a higher normalized conductivity and hence heats more efficiently. For the t-test comparing DI water with the 841 nanorods, the two-tailed p-value is 0.0013, which is also lower than the p-value threshold for statistical significance. It can also be concluded that the normalized conductivity mean is different between DI water and 841 nanorods. 41 The gold nanoparticles used in the experiments were surrounded by a surfactant called CTAB and gold salts after the synthesis. In order to isolate the heating effect of the nanoparticles from the other constituents in the sample solution, the experiments were repeated to compare the heating effects of DI water, a solution of CTAB and the supernatant the particles are suspended in and a 5 mM concentration of nanorods. Table 5 shows the normalized conductivity of the different background liquids in different trials. The normalized conductivity shows that the background constituents liquid heat up almost as efficiently as the 5mM nanoparticles. It cannot be concluded that the heating effect observed in Figure 6 and Figure 7 is entirely due to the gold nanoparticles. 42 Number of trials Samples DI Water 762 nanorods 5mM 762 nanorods 2.5mM 762 nanorods 1.25mM 841 nanorods 5mM 841 nanorods 2.5mM 841 nanorods 1.25mM 3 3 3 2 3 3 3 Average Normalized conductivity (degrees) 8.832 11.142 Standard deviation (degrees) 95% confidence interval 0.163 0.047 10.630 0.107 0.184 0.054 0.122 10.162 0.040 9.983 0.184 9.535 0.071 9.514 0.106 0.055 0.208 0.081 0.121 Table 2 Summary of data comparing the normalized conductivity of DI water, 762 nanorods and 841 nanorods at 5mM, 2.5 mM and 1.25 mM Normalized conductivity (degrees) Samples Trial 1 Trial 2 Trial 3 DI Water 762 nanorods 5mM 8.665 8.840 8.991 Mean of the trials 8.832 11.195 11.105 11.127 11.142 T p-value degrees of freedom Standard deviation 0.163 0.047 23.6 <0.0001 4 Table 3 Two sample t-test is performed comparing the normalized conductivity obtained from DI water to that obtained from 5 mM concentration of 762 nanorods. The p-value shows that the two means indeed differ. 43 Normalized conductivity (degrees) 8.665 Trial 2 8.840 Trial Mean of the 3 trials 8.9908 8.832 10.068 10.108 9.772 Sample Trial 1 DI Water 841 nanorods 5mM Standard deviation 0.163 0.184 9.983 T 8.1130 p-value 0.0013 degrees of 4 freedom Table 4 Two sample t-test is performed comparing the normalized conductivity obtained from DI water to that obtained from 5 mM concentration of 841 nanorods. The p-value shows that the two means indeed differ. Comparison of Normalized Conductivity 11.5 DI water 841 nanorods 762 nanorods Normalized conductivity (degrees) 11 10.5 10 9.5 9 8.5 0 1 2 3 4 5 6 Particle Concentration (mM) Figure 7 Average normalized conductivity vs. particle concentration with the 95% confidence interval 44 Figure 8 Temperature trace curves comparing DI water, CTAB (50 mM), supernatant, gold nanoparticles (plasmon peak ~870 nm) Sample DI Water CTAB Supernatant Gold nanoparticles Trial 1 11.055 12.637 13.467 Trial 2 13.34 12.379 16.862 12.912 14.179 Trial 3 Average conductivity 12.198 12.508 15.164 13.919 13.67 Std/average conductivity 0.132 0.0146 0.158 0.0489 Table 5 Comparison of the normalized conductivity of several background liquid with gold nanoparticles 45 4.2.5 Experiment with gold nanoparticles and p-NIPAM It is hypothesized that since the volume fraction of the gold nanoparticles in the solution is very small, the heating effect of the gold nanoparticles can be masked by the heating effect of the background constituent. An experiment to test for localized heating of the golden nanoparticles is conducted by attaching a polymer called Nisopropylacrylamide or p-NIPAM to the gold nanoparticle. p-NIPAM has a phase transition temperature of around 30 degrees Celsius. When p-NIPAM undergoes a phase transition, the polymers aggregate together which can be detected by a change in the absorption spectrum. The absorption spectrum would broaden and its peak would shift. [30] The p-NIPAM is bonded to the gold nanorods by a researcher in Professor Phil Messersmith’s group. If the gold nanoparticles have a localized heating effect, the pNIPAM bonded to the gold nanorods would perhaps undergo the phase transition at a lower temperature than p-NIPAM alone. First, a solution of the gold nanoparticles attached with p-NIPAM is tested to see if the phase transition can be observed. The solution is heated in a waterbath with a spectrometer attached, and the absorbance is recorded as the temperature of the water bath increases incrementally. In Figure 12, the absorbance of the solution of the gold nanoparticles attached pNIPAM is plotted for different temperatures ranging from 28.8 degrees to 33.7 degrees Celsius. The range of temperature is chosen so that it is slightly below and above the phase transition temperature of p-NIPAM. The shift of the absorbance peak of the goldNIPAM solution indicates the phase transition has occurred, and it shows that the phase 46 transition of the gold nanoparticles attached p-NIPAM can be observed. In the next step, the phase transition temperatures of the p-NIPAM solution and that of the gold-NIPAM are compared. Both of the solutions are individually heated in the waterbath with a spectrometer attached in order to record the change in absorbance as the solutions are heated. .Figure 10 and Figure 11 compare the difference between the absorbance of pNIPAM and gold nanoparticles attached p-NIPAM. It shows that the absorbance peak shifts and broadens for the p-NIPAM at a lower temperature than the p-NIPAM attached gold nanoparticles. From this experiment, it does not seem that the gold nanoparticles produce a localized heating effect as hypothesized because when it is attached to the pNIPAM, a lower phase transition temperature is not observed. It is possible that by attaching the gold nanoparticles to the p-NIPAM, the property of the new molecule is different from that of p-NIPAM and that the phase transition temperature would be different for the new molecule, but the experiment does not prove that the gold nanoparticles produce a localized heating effect. 47 Gold-NIPAM 3/25/10 1.8 28.8 29.8 30.7 31.8 32.6 33.2 33.7 1.6 1.4 1.2 abs 1 0.8 0.6 0.4 0.2 0 200 300 400 500 600 700 wavelength (nm) 800 900 1000 1100 Figure 9 Change of absorbance as temperature increases for Gold-NIPAM measured in spectrometer of Messersmith lab, waterbath is used to change the temperature of the sample .Figure 10 The absorbance of p-NIPAM is plotted for a range of temperatures showing the shift of the absorbance peak as the phase transition occurs in the solution 48 Figure 11 The absorbance of p-NIPAM attached gold nanoparticles is plotted and compared to the absorbance of p-NIPAM. The solution of p-NIPAM shows a shift in the absorbance at a lower temperature than the solution of gold nanoparticles attached p-NIPAM. 49 4.3. Numerical simulation In addition to the microwave heating and p-NIPAM experiments, numerical simulations are used to study the dielectric properties of an ensemble of gold nanoparticles embedded in a biological background in the microwave region. Numerical simulation can help further the understanding of the heating properties of the nanoparticles with respect to their size range and volume fraction. 4.3.1. Numerical simulation of absorption and scattering cross section of a single gold nanoparticle The absorption and scattering cross sections per unit volume of gold nanoparticles in the microwave region are calculated using Mie theory. The particles simulated are spherical. The absorption cross section is calculated to determine what particle size is the most efficient heating agent, and the scattering cross section is used to determine the most efficient contrast agent. The illuminating source is a x polarized, z-propagating wave at 2GHz frequency. The dielectric property of the particle is the bulk metal property with σ= 106. The study of absorption cross section would help us understand the potential of the particles as heating agents, and the study of scattering cross section would help us understand the potential of the particles as contrast agents. The skin depth of the metallic particle = 1 πfμ 0 μ Rσ , where f =frequency, μ r =relative permeability, μ 0 =permeability, σ =conductivity, and it is calculated to be around 11 micron. Figure 12 andFigure 13 show the absorption cross section and scattering cross section of a single metallic particle. Particle sizes ranging from 1nm to 10 mm in radius are simulated. 50 Figure 12 shows that the absorption cross section per unit volume is highest when the particle is 1 mm, an order of magnitude larger than the skin depth at 1 mm. Figure 13 shows the scattering cross section per unit volume is at 10 mm. This study shows that the gold nanoparticles are not efficient scattering and contrast agents. 4.3.2. Numerical simulation of an ensemble of metallic particles In the simulation in the previous section, the absorption cross section of a single gold nanoparticle is examined. In this section, the absorption cross section of an ensemble of spherical gold nanoparticles is examined. It is discovered in the previous section that the absorption cross section per unit volume of a nanoparticle is small, perhaps an ensemble of gold nanoparticles would interact together and the multiple scattering effect would produce an enhanced heating effect. In order to test this, ensembles of gold nanoparticles are simulated with volume fraction ranging from 1%-20%. The simulation is performed with the Generalized Multiparticle Mie solution as described in previous chapter, and the ensemble is illuminated by a 2GHz source. Particles with radius= 1nm and radius =1 micron are both tested in the simulations. Figure 14 is the result of the absorption cross section of an ensemble of nanoparticles of 1nm radius at volume fraction from 1-20%. Figure 15 is the result of an ensemble of nanoparticles of 1micron radius at volume fraction from 1-20%. For both types of particles, the absorption cross section of the ensemble has a linear relationship with volume fraction. The linear relationship shows that the multiple scattering effect is insignificant in the ensemble of nanoparticles and microspheres because the absorption cross section of the ensemble increases proportionally to the number of particles in the ensemble. The single sphere simulation 51 described in the previous section and numerical experiment of the multiple particle shows that the metallic particles are not efficient heating agents. Absorption Cross section/Volume, 2GHz Absorption cross section/Volume (m -1) 10 9 8 7 6 5 4 3 2 1 0 -9 10 10 -8 10 -7 -6 10 -5 10 -4 10 10 -3 radius (m) Figure 12 Absorption cross section per unit volume for different metallic particle sizes -3 8 Scattering c ross sec tion/Volume x 10 - Scattering cross section/volume (m1) 7 6 5 4 3 2 1 0 -9 10 -8 10 10 -7 -6 10 radius (m) 10 -5 -4 10 -3 10 Figure 13 Scattering cross section per unit volume for different metallic particle sizes 52 53 1 x 10 -28 absorption cross section, radius=1 nm 0.8 m-2 0.6 0.4 0.2 0 0 5 10 volume fraction (%) 15 20 Figure 14 Absorption cross section of an ensemble of metallic nanoparticles of 1nm radius at volume fraction from 1-20% 1.2 x 10 -16 absorption cross section radius=1000 nm 1 m-2 0.8 0.6 0.4 0.2 0 0 5 10 volume fraction (%) 15 20 Figure 15 Absorption cross section of an ensemble of metallic nanoparticles of 1micron radius at volume fraction from 1-20% 54 4.3.3. Gold nanosatellites experiment In this experiment, metallic nanoparticles used as contrast agents for biological tissue in the microwave region are studied. Gold nanoparticles have been shown to be effective contrast agents for optical imaging of cells and tissue phantoms. [31, 32, 33] The simulation is performed with the Generalized Multiparticle Mie Solution. The background is simulated to have a refractive index matched to water n=1.33. A 5 micron diameter sphere with refractive index n=1.05 relative to the background is simulated, which is a realistic representation for biological tissue. 1000 gold nanoparticles with 50 nm diameter surround the microsphere at a distance 5 nm from the surface of the core sphere. The positions of the nanoparticles are randomly generated and uniformly distributed. The forward scattering intensity of the microsphere and nanoparticles is calculated for several frequency from 500 MHz to 5 GHz, and it is compared to the forward scattering intensity of the microsphere alone. Figure 16 is an illustration of the microsphere and nanoparticles ensemble. It shows the nanoparticles randomly distributed over the surface of the core sphere. Figure 17 shows a comparison of the forward scattering intensity of the microsphere vs the microsphere surrounded by the gold nanoparticles. The forward scattering intensity is compared to see if the addition of the gold nanoparticles would change the scattering intensity and have the potential to be a contrast agent. The simulation shows that the forward scattering intensity is very similar with and without the gold nanoparticles. The simulation suggests that the gold nanoparticles would not be good contrast agents. 55 Figure 16 Illustration of a microsphere surrounded by gold nanoparticles. The nanoparticles are randomly and uniformly distributed over the surface of the microsphere. Figure 17 Comparison of forward scattering intensity of a microsphere with comparable refractive index of biological cell in an aqueous background and an ensemble of gold nanoparticles surrounding the microsphere 56 4.3.4. Summary Microwave heating experiments and numerical experiments have been performed to evaluate the potential of metallic nanoparticles as heating agents and contrast agents. Microwave heating experiments show that gold nanoparticles suspended in solutions do not heat up more efficiently than its background constituents. Numerical simulation of single metallic nanoparticle and an ensemble of metallic nanoparticles further demonstrate that metallic nanoparticles are not efficient heating agents. Numerical simulation is used to determine whether gold nanoparticles can provide contrast to biological tissue. A microsphere with comparable refractive index to a biological cell is surrounded by gold nanoparticles, and the forward scattering is compared to the forward scattering of the microsphere alone. The result shows that the nanoparticles do not provide very much contrast to the microsphere. 57 Chapter 5: Heterogeneous Monte Carlo Simulation 5.1. Introduction Monte Carlo simulations have been used to simulate optical reflectance measurement from diagnostic tools including the LEBS (Low-coherence enhanced backscattering) probe and polarization gating probe. [37, 38] Reflectance measurements are used to detect changes in physical parameters that could help differentiate healthy and cancerous tissues. Monte Carlo simulations have been used to understand how different physical parameters can impact the reflectance measurements, but previously only homogeneous mediums with the equivalent optical properties have been simulated. Tissues have complex structure and are composed of different layers. For example, colon tissue is made up of different layers (epithelial, mucosa, submucosa and muscle) and a study of the impact of tissue structure on the diagnostic probes signals would help understand whether the recovered physical parameters from probe measurement is accurate. A new Monte Carlo code is developed that allows the light propagation in heterogeneous structure to be modeled, and the technique allows a more realistic simulation of the diagnostic probe measurements to be studied. 5.2. Description of the Monte Carlo Simulation The Monte Carlo method used in this work is adapted from combining the previous works of Ramella-Roman [34] et al. and Chen et al [35]. Ramella-Roman developed a Monte Carlo simulation code that tracks the polarization of each photon as it travels through a medium. The Mie theory is used to calculate the optical properties of 58 the medium and the phase function of the medium. The photon is initially launched randomly within a circular area centered at the middle of the surface of the medium to simulate an illuminating beam. The Stokes vector of each photon is tracked during the simulation, and the code is modified to record the location where the photon exits from the medium and the exit angle. Mueller Matrix multiplication is used to obtain the output co-polarized and cross-polarized intensity, the intensity if a polarizer is oriented 0 degrees and 90 degrees to the incident polarization respectively. The feature in RamellaRoman’s program that allows polarization to be tracked is integrated to another Monte Carlo program written by Chen that allows each voxel in the simulation to have individual optical properties, the absorption coefficient(μa), scattering coefficient(μs) and anisotropy factor (g). In Chen’s code, the 3D simulation space is broken up into individual voxels. This method allows heterogeneous structure to be modeled using a Monte Carlo method. The Chen code was used by Jacques [36] to simulate blood vessels in dermis. To illustrate the difference in light penetration between homogeneous and heterogeneous tissue medium, Monte Carlo simulation is performed simulating a pair of blood vessels in a dermis background located 1 mm below the surface illuminated by a 8mm collimated beam. The optical properties of the blood vessel and the background medium are listed in Table 6 , and they are from Jacques et al. [36] Figure 19 shows the light penetration around the blood vessels and this is compared to the light penetration in an equivalent homogeneous medium in Figure 20. In the blood vessel simulation, the light penetration decreases further within the blood vessel center, and this is different from the case in the homogeneous medium where there is a gradual decrease of light penetration depth increases within the medium. 59 The new Monte Carlo code combining the features of Chen’s and RamellaRoman’s Monte Carlo codes is validated by simulating a homogeneous medium with the following optical properties: μa=0 cm-1, μs=125 cm-1, g=0.9. The result of the new heterogeneous polarized Monte Carlo code is compared with that of Ramella-Roman’s polarized Monte Carlo code. The total intensity collected from photons exiting a collection area of 800 microns centered at the medium’s center is compared between the two codes. Figure 21 shows the total intensity from a simulation using the new heterogeneous polarized Monte Carlo code and a simulation using Ramella-Roman’s polarized Monte Carlo code. The intensity is plotted as a function of radial distance from the center of the collection area. One million photons are used in both of the simulations. It validates the polarized heterogeneous Monte Carlo code can generate results that match the established polarized Monte Carlo code. The cross and co-polarized measurement from both Monte Carlo codes are also compared and they converge in the homogeneous case. 60 Figure 18 Illustration of a pair of blood vessel located at 1 mm below surface Figure 19 Light penetration around a pair of blood vessels 61 Figure 20 Light penetration in a homogeneous medium with optical properties equivalent to a constant 5% blood volume medium Tissue Dermis with 0.2 % blood Whole blood μa (cm-1) 0.34 μs (cm-1) 315 G 0.9 168 631 0.95 Table 6 Optical properties used in the two-vessels simulation 62 Figure 21 Total reflectance measurement from a simulation where the medium has the following optical properties: μ a=0 cm-1, μ s=125 cm-1, g=0.9 validating the polarized heterogeneous Monte Carlo code (red) converge to the established polarized Monte Carlo (blue) in a homogeneous medium In order to more realistically simulate the scattering in biological tissue, the Whittle-Mattern phase function is implemented in the heterogeneous polarized Monte Carlo code. The two parameters needed to calculate the phase function is the anisotropy factor g and m which dictates the shape of the function that relates the refractive index fluctuation in the medium with the Whittle-Mattern correlation function. [37] 5.3. Monte Carlo Simulation of polarization gating probe The polarization gating probe allows for depth selective measurement of biological tissue. [38]. A schematic of the polarization gating probe developed by the Backman lab is shown in Figure 22. The idea of polarization gating is that light undergoes depolarization as it propagates in scattering medium. Photons that experience many 63 scattering events undergo a randomization of the polarization state. Three additional reflectance signals can be obtained from the polarization gating probe, the co-polarized signal, cross-polarized signal and differential polarized signal. The co-polarized signal (I|| )has a polarization axis of scattered light that is parallel to that of the incident light. The cross polarized signal (I) has a polarization axis perpendicular to that of the incident light, and it is dominated by photons that have undergone multiple scattering effects and have traveled further within the tissue. The co-polarized signal comes from the photons that have undergone few scattering events and also photons that have undergone multiple scattering. The difference between the two or the differential polarized signal (ΔI= I||- I) comes from the photons that have retained their polarization or the photons that have the shallowest penetration depth. Figure 22 Schematic of the polarization gating probe 5.3.1. A study of packaging coefficient Heterogeneous Monte Carlo simulation is performed to analyze the effect of discrete absorbers inside blood vessels. Other groups have developed theoretical models to analyze the packaging effect of the discrete absorbers [39, 40, 41, 42], and Rajaram et 64 al. utilized microfluidic channels to study the validity of the theoretical model. [43] However, these models are based on the assumption that equal contribution from vessels at different depths within the tissue. In this study, the polarized Monte Carlo simulation is used to study the packaging effect of vessels at different depth. Vessel structure is modeled by simulating cylinders running perpendicular to the incident direction. The vessel location is randomly distributed throughout the medium. illustrates a cross section of a vessel structure. In the simulation, a layer of blood vessels with 10 micron diameter represented by pixels of 4x4 microns is simulated. Figure 23 illustrates a cross section of the blood vessels randomly distributed throughout the sample space. Figure 24 is an enlarged image of the pixelated vessel. Photon is launched at z=0 and is scattered in the medium according to the optical property of the material. The absorption coefficient of the blood vessel is of whole blood with 75% oxygen saturation. The absorption coefficient for blood at a particular wavelength is calculated as described on the Oregon Medical Laser Center’s website. [44] The absorption coefficient for the background where the vessels are situated is assumed to be negligible. The scattering coefficient and anisotropy for the background medium are calculated from Mie theory by assuming a scatterer to be collagen. The scattering coefficient for the blood vessel in different wavelength is calculated from Mie theory using scatterer size and volume fraction found in literature. [45] For the simulations, the scattering coefficient and anisotropy of the vessel is the same as that of the background in order to understand the effect of discrete areas of absorption, and the parameter m is set to be 1.5. The specification of the polarization gating probe is implemented in the simulation. The illumination area of the probe has a radius of 4 micron, and it has a collection radius of 4 65 micron and a collection angle of 18 degrees. The illumination area is implemented by allowing photons to enter the medium at z=0 and the x and y position are randomized but such that the radial distance from the surface center is less than or equal to 4 micron. The collection radius and angle are implemented such that only photons that are scattered out of the medium at collection radius less than 4 micron and at an angle less than 18 degrees are recorded. 66 absorption coefficient (cm -1) 0.02 0.04 y (cm) 0.06 0.08 0.1 0.12 0.14 0.16 0.02 0.04 0.06 z (cm) 0.08 0.1 Figure 23 A cross section of the blood vessel structure Figure 24 A cross section of the blood vessel structure illustrating the pixelated blood vessel structure 67 Simulations are performed for a constant 5% blood volume with blood vessels randomly distributed in a single layer. Blood vessels with diameter ranging from 561064 micron are simulated. Each simulation contains only one blood vessel size, and the reflectance from each blood vessel size simulation is recorded. The absorption coefficient of the blood vessel is taken to be that of blood with 75% oxygen saturation at 574 nm at μa= 272.21 cm -1 . The other optical properties are g=0.79, μs= 195.06 cm -1 In addition, a 5% blood volume homogeneous medium is simulated. The co-pol and cross-pol and differential signals are also recorded. The reflectance vs diameter of blood vessel for the co-pol, cross pol, differential and total reflectance signal, and an exponential curve is fitted for each signal to the form R = Rbaseline + ΔR(1 − exp(− packagingcoefficient * μ a * diameter)) . Figure 25 shows the reflectance vs the diameter data for the co-pol, cross-pol and total reflectance signal and the exponential curve fitted to the data. Table 7 shows the packaging coefficient obtained from the total reflectance, co-pol, cross pol and differential signals. The packaging coefficient for diffuse reflectance from Jacques et al., [36] is also listed and compared to the coefficient obtained from the different signals. The study shows that the packing effect is different at different penetration depth, with the effect being the largest at the shallowest penetration depth. This is significant for the inverse algorithm which relies on using a correction factor to account for discrete absorbers used to recover blood vessel diameter to recover blood vessel size from reflectance measurement. 68 Figure 25 Reflectance vs blood vessel diameter are plotted for the total reflectance, co-pol and cross pol signal Diffuse Reflectance (Jacques et al.) Co-pol Cross-pol Differential Total reflectance Packaging coefficient 0.232 0.329 0.262 0.364 0.299 Table 7 Packaging coefficient for different signals comparing the packaging coefficient for diffuse reflectance 69 A single layer simulation is performed in order to analyze the difference in reflectance measurement due to a change in blood vessel size and blood vessel density. Four cases are examined. Table 8 shows the simulation parameters for the four cases. The blood vessel size and density are taken from literature [45]. In the control case, a tissue medium with blood vessel 10 micron in diameter at 7.7% blood volume fraction is simulated. It is compared to a homogeneous medium with the equivalent blood volume fraction. Next, a medium with an increased blood vessel diameter 16 micron is simulated keeping the blood volume fraction at 7.7%. Finally, a medium with a 17.33 % blood volume fraction with 10 micron diameter blood vessel is simulated. The blood vessels are randomly placed in the medium. Figure 26 compares the reflectance spectra of the four simulated medium, and Table 9 summarizes the difference in the reflectance spectra at 574 nm. The wavelength is chosen because there is a strong absorption from blood at that wavelength which would create a larger difference between the different scenarios compared to at a wavelength when there is low absorption. The result shows that between homogeneous and blood vessel structure, there is a 11 % difference in the reflectance value. For constant blood volume fraction of 7.7%, an increase of blood vessel diameter from 10 micron to 16 micron yields an increase of almost 7% in the reflectance value. 70 Vessel size-Control Vessel densityControl 10 microns diameter 7.7% blood volume fraction Vessel sizeCancerous 16 microns diameter Vessel densityCancerous 17.33% blood volume fraction Blood vessel density (control) 7.7% blood volume fraction Blood vessel density (cancerous) 17.33% blood volume fraction Table 8 Simulation parameters for the one layer model 71 5 x 10 4 4.5 4 homogeneous normal increase density increase size reflectance 3.5 3 2.5 2 1.5 1 0.5 500 550 600 650 wavelength (nm) 700 750 Figure 26 Reflectance spectra comparing a homogeneous medium at 7.7% blood volume fraction, a heterogeneous blood vessel medium at 7.7% blood volume fraction with 10 micron diameter vessels, a heterogeneous blood vessel medium at 17.3% blood volume fraction with 10 micron diameter vessels and a heterogeneous blood vessel medium at 7.7% with 16 micron diameter vessels At 574 nm % diff between homogeneous medium and control blood vessel structure % diff between control blood vessel medium and increase blood vessel size structure 11.6% 6.58% % diff between control blood vessel structure and increase blood vessel density structure 45.3% Table 9 Difference in reflectance between the different simulations at 574 nm which has a high blood absorption coefficient 72 5.3.2. Three layer simulation Colon tissue is made up of different tissue layers, the epithelium, mucosa, submucosa and muscle layer. In this study, the effect of the different vessel sizes at different depth on the reflectance measurement is studied. In all simulations, 10,000,000 photons are launched. The μs and g are the same whether in the vessel or the background. In all simulations , the dimensions of the simulated tissue is 4000 microns (width) x 4000 microns (length) x 2050 microns (depth) Three layers of tissue mucosa, submucosa and muscle layers are simulated with the first layer=450 micron in depth, second layer=700 micron and third layer =900 micron. The table below shows the descriptions of the six simulations performed. Simulation Healthy Homogeneous Increase mucosa density Descriptions of blood vessel size and density -3 layers all with vessel structures -Layer 1 7.7% blood volume, vessel diameter=11 micron -Layer 2 7.7% blood volume, vessel diameter=25 micron -Layer 3 9.5% blood volume, vessel diameter=50 micron -3 homogeneous layers, no vessel structure -Layer 1 7.7% blood volume -Layer 2 7.7% blood volume -Layer 3 9.5% blood volume -3 layers all with vessel structures -Layer 1 17.3% blood volume, vessel diameter=11 micron -Layer 2 7.7% blood volume, vessel diameter=25 micron -Layer 3 9.5% blood volume, vessel diameter=50 micron 73 Increased mucosa density homogeneous -3 homogeneous layers, no vessel structure -Layer 1 17.3% blood volume -Layer 2 7.7% blood volume -Layer 3 9.5% blood volume Increased sub mucosa density -3 layers all with vessel structures -Layer 1 7.7% blood volume, vessel diameter=11 micron -Layer 2 17.3% blood volume, vessel diameter=25 micron -Layer 3 9.5% blood volume, vessel diameter=50 micron Increased sub mucosa density homogeneous -3 homogeneous layers, no vessel structure -Layer 1 7.7% blood volume -Layer 2 17.3% blood volume -Layer 3 9.5% blood volume Figure 27 and Figure 28 illustrate the difference between homogeneous tissue medium and their respective counterpart using blood vessel structure. Similar to the one layer simulation, the reflectance measurement at 574 nm is compared between the homogeneous and layer simulation. Table 10 shows the percentage difference between the three layer homogeneous medium and the three layer medium with blood vessel density corresponding to the blood volume density at the different layers. It shows that the difference in reflectance can be significant, up to 30% between the total intensity data. Table 11 summarizes the result of increasing the submucosa and the mucosa layer. For the comparisons of increasing the mucosa layer, the cross pol signal yields the most difference at-58.8%. Since the differential signal is composed of photons from the shallowest depth, it was expected the differential signal would yield the biggest difference between two signals. With an increase in blood volume 74 fraction, there should be a more significant difference between the signal from a homogeneous medium vs a heterogeneous medium, however the differential signal yielded the lowest error. For the comparison of increasing the deeper layer, the submucosa layer, the cross pol signal comes from photons with the deeper penetration into the medium, and the cross-pol signal yields a 45% difference between the signal which is expected. Three simulations are run to obtain the average reflectance for each cases. More simulations are needed to draw a more precise conclusion on the trend of the data because the data is noisy from looking at the standard deviation of the average reflectance. 75 Figure 27 Reflectance spectra for three layer tissue structure and homogeneous tissue structure. Healthy denotes three layer tissue structure At 574 nm Total intensity Co-pol signal Cross-pol signal Differential signal % difference between three layer structure and homogeneous =( three layer structurehomogeneous)/three layer structure -30% 26.9% 38% 20.3% Table 10 Difference in reflectance measurement at 574 nm between a homogeneous and three layer blood vessel structure 76 Figure 28 The figure on the left illustrates the difference between the reflectance from a three layer tissue structure where the top layer has an increased in blood density and the homogeneous structure. The figure on the right illustrates the difference in the case where the middle layer has an increase in blood volume density at 574 nm Total intensity Cross-pol signal Co-pol signal Differential signal Increase mucosa density-% difference between vessel and homogeneous structure =( heterogeneous structurehomogeneous)/heterogeneous structure -36.9% -58.8% -31.5% -24% Increase sub mucosa density% difference between vessel and homogeneous structure =( heterogeneous structurehomogeneous)/heterogeneous structure 45.7% 45.3% 41.1% 31.4% Table 11 Effect of increasing blood volume density at different depth in the tissue on the reflectance measurement 77 5.4. Monte Carlo Simulation of Low-Coherence enhanced backscattering probe Heterogeneous Monte Carlo Simulation is used to study the path length dependence on the Low-Coherence enhanced backscattering probe. The Low-Coherence enhanced backscattering (LEBS) [46] utilizes the enhancement of intensity in the backward direction of the incident light in the partial spatial coherence regime. LEBS has shown potential in the early detection of colon and pancreatic cancer [47,48,49]. The heterogeneous Monte Carlo simulation is used to validate blood vessel diameter recovery algorithm in single layer model. 78 5.4.1. Single layer simulation LEBS probe specifications, which include the collection and illumination areas, are implemented into the heterogeneous Monte Carlo code described in the previous section. A single layer of constant blood vessels is simulated. Blood vessels sizes of radius=12.6, 27 and 53.6 micron are simulated. The absorption coefficient of the blood vessel is calculated at 75 % oxygen saturation for wavelength 506 nm-700 nm. The scattering coefficient and anisotropy of the background medium are g=0.9 and μs=950 cm-1 at 650 nm, and m=2. The optical properties at the other wavelength are obtained by using the following scaling μ x' ∝ λ2 m−4 and μ s ∝ λ−2 . The absorption coefficient of the background medium is set to 0, and the scattering coefficient and anisotropy factor of the blood vessels are set to the values of the background medium. The simulation for each blood vessel size is repeated up to 30 times in order to increase the signal to noise ratio. The blood vessel size recovery algorithm is used to recover information from the reflectance measurement from the tissue sample from LEBS. The algorithm is constructed by deriving the modified Beer-Lambert law for the medium by constructing a relationship between the path length that photons travel in the tissue medium and the scattering properties of the medium. The path length relationship is derived from Monte Carlo simulation of tissue medium and tracking the path length of the photons propagated in the tissue. [50] From curve fitting, the absorbance of the tissue medium follows the form absorbance = ((2 * (0.13 * μ s' ) −0.228 (−1.48 0.13μ a ,effective ) − 1) * exp(1.61 − 1.48 0.13μ a ,effective ) + 4.5679(0.13μ s' ) −0.228 79 where μa,effective= μa*packaging factor Packaging factor is described in van Veen et al. [42] which is C packing = [1 − exp(−2 μ a rvessel ) / 2μ a rvessel ] , μa is the wavelength dependent absorption of blood, and rvessel is the radius of the blood vessel that is a parameter to be fitted for in the algorithm given a reflectance spectrum Table 12 shows the optical properties used in the simulation. Table 13 and Figure 29 show the result of the simulated blood vessel radius and the recovered blood vessel radius. The result shows that the recovered blood vessel is within 15% of the actual blood vessel size. The result provides the first validation of the blood vessel recovery algorithm for LEBS probe measurement. 80 Wavelength (nm) 506 540 564 568 574 586 610 630 700 μs (cm-1) 1567.65 1376.46 1261.80 1244.09 1218.22 1168.84 1078.68 1011.27 819.13 g 0.939 0.930 0.925 0.924 0.922 0.919 0.912 0.906 0.884 μa of blood vessel (cm-1) 112.88 278.51 205.63 226.08 272.21 152.08 18.85 9.42 3.60 Table 12 Optical properties of the one layer LEBS simulation 81 Figure 29 Validation of the blood vessel recovery algorithm. Blood vessels of three different sizes are simulated with the heterogeneous Monte Carlo code, and the blood vessel size algorithm is used to recover the blood vessel size from the reflectance results to see if there is a match Simulated blood vessel radius (micron) 12.6 27 53.8 Recovered blood vessel radius (micron) 14 31 54 % diff 11% 14.8% 0.37% Table 13 Recovered blood vessel radius compared to the simulated blood vessel radius 82 5.5. Summary A polarized heterogeneous Monte Carlo code was developed to model reflectance measurement from Polarized gating and LEBS probes. Simulations of depth selective signals with polarization gating show that the packaging coefficient is different for different depth penetration signal with a greater packaging coefficient for shallower penetration. Single layer simulation shows that the difference in reflectance measurement from homogeneous and heterogeneous structure. In addition, the heterogeneous Monte Carlo code is used to validate the algorithm for recovering the blood vessel diameter from LEBS reflectance measurement. The next step in the study would be to develop a two layered model for LEBS. A two-layered model could help to study and explain the phenomenon of early decrease in blood supply observed in lung cancer tissue sample for LEBS. Preliminary work has been done to construct a relationship between the average pathlength of the medium and absorption coefficient for a control medium and a cancerous medium. Figure 30 illustrates the two-layer medium with the optical properties of each layer listed in Table 14. It is hypothesized that the observation in the early decrease in blood supply is due to the increase in μs’ in the epithelium layer, and Figure 31 shows the initial work in deriving a relationship between the pathlength to the absorption coefficient of the medium for the control and cancer model. 83 Figure 30 Illustration of the two layer model, the epithelium layer has a thickness of 270 micron and the stroma layer has a 20000 micron. The epithelium layer has smaller blood vessels with diameter at 10 micron and the stroma has blood vessels with diameter at 25 micron Epithelium Stroma Thickness (micron) g 270 20000 0.9 0.9 μs’ control (cm -1 ) at 650 nm 31.25 95 μs’ cancer (cm-1) at 650 nm 80 95 m (control) m (cancer) 2 2 2 2 Table 14 Optical properties of the two layered model at 650 nm Figure 31 Initial result of pathlength dependence on absorption cross section for two-layer control and cancerous lung tissue 84 Chapter 6: Conclusion The purpose of this thesis is to examine the electromagnetic properties with multiparticles systems in the optical and microwave regions. Chapter 2 provides an overview of available numerical electromagnetic scattering techniques. Chapter 3 focuses on how multiple scattering can impact the recovery of information about tissue structure in optical imaging. Results show that the multiple scattering effects can significantly alter spectroscopic scattering parameters calculations and hence affect the optical characterization of biological tissues. Chapter 4 provides a rigorous study of the potential for gold nanoparticles to be used as heating agents and contrast agents in the microwave region. The study shows that gold nanoparticles are not efficient heating agents and that they do not provide a significant contrast to biological material at microwave frequency. Chapter 5 describes the development of a polarized heterogeneous Monte Carlo simulation used to study the effect of tissue structure on signals from two different diagnostic probes, the polarization gating probe and low coherence enhanced backscattering (LEBS) probe. The study of the signal from polarization gating probe illustrates that the packaging effect from blood vessels is different for different light penetration depth, which can impact the accuracy of the inverse algorithm used to recover blood vessel diameter from probe measurement. The polarization gating study also illustrates the difference in reflectance measurement between a homogeneous and heterogeneous tissue sample. The heterogeneous Monte Carlo simulation is used to validate the inverse algorithm used to recover blood vessel diameter for LEBS probe 85 The studies of the above problems provide improvements that can lead to better and more accurate interpretation of results from disease diagnostic tools for differentiation of healthy and diseased tissue. 86 List of publications Peer-reviewed publications W. Yip, X. Li, “Multiple Scattering Effects on Optical Characterization of Biological Tissue Using Spectroscopic Scattering Parameters,” Optics Letters, Vol. 33, Issue 23, December 2008, pp. 2877-2879 (Selected for publication in the Virtual Journal of Biomedical Optics) Conference Presentations W.Yip, A. Gomes, A. Sahakian, V. Backman, “Polarized Monte Carlo Simulation of Blood Vessel Structure in Colon Tissues,” SPIE Photonics West, Jan 2012 W.Yip, A. Gomes, A. Sahakian, “Polarization Gating in Resolving Blood Vascular Structure,” United States National Committee for the International Union of Radio Science Meeting, January 2011 W.Yip, X. Li, “Numerical investigation of dielectric properties of metallic microspheres in the microwave frequency based on a volume integral approach” Progress in Electromagnetics Research Symposium, July 2010 J. Yi, W. Yip, X. Li, Numerical Investigation of Spectral Optical Coherence Tomography Based on Full-Wave Solution of Maxwell’s Equations, Progress in Electromagnetics Research Symposium, March 2009 W. Yip, X. Li, “Numerical Investigation of Multiple Scattering Effects of Biological Tissues”, IEEE International Symposium on Antennas and Propagation, July 2008 (Invited) W. Yip, X. Li, “Multiple Scattering Effects on Particle Sizing in Optical Characterization of Biological Tissues,” Optical Society of America - Frontiers in Optics Conference , September 2007 87 References [1] T. Wriedt, "A Review of Elastic Light Scattering Theories," Particle and Particle Systems Characterization, vol. 15, pp. 67-74, 1997. [2] C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, 1983. [3] R. J. Martin, "Mie scattering formulae for non-spherical particles," Journal of Modern Optics, vol. 40, pp. 2467-2494, 1993. [4] D. K. Hahn and S. R. 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