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

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NORTHWESTERN 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
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