Monte Carlo Simulations for the Interaction of Multiple ...4.6 Amplitude and phase modulation of...

Monte Carlo Simulations for the Interaction ofMultiple Scattered Light and Ultrasound

A Thesis Presented

by

Luis A. Nieva

to

The Department of Electrical and Computer Engineering

in partial fulfillment of the requirementsfor the degree of

Master of Science

in

Electrical and Computer Engineering

in the field of

Electromagnetics, Plasma and Optics

Northeastern UniversityBoston, Massachusetts

January 22, 2003

1

Abstract

Monte Carlo Simulations for the Interaction of

Multiple Scattered Light and Ultrasound

Acousto-Photonic Imaging is a new frequency domain optical technique for non-invasive

medical imaging. It is based on the combination of Diffuse Optical Tomography (DOT)

and focused ultrasound. Diffuse Optical Tomography, due to its diffuse nature, can not

provide good spatial resolution by itself. Therefore, the objective is to use the ultrasound

to acoustically generate optical diffuse sources at different modulation frequencies, spaced

approximately one wavelength apart in the focus of the ultrasound beam. This will improve

the spatial resolution as well as acquire the optical properties of human tissue. In addition,

the study of the physics behind this interaction is of particular interest and still is not

completely understood.

We present Monte Carlo simulations for the interaction of Near-Infrared light (NIR) and

ultrasound in dense turbid media with high albedo. The strength of the optical signals for

the continuous wave, diffuse wave, and acousto-photonic wave is computed and compared

in order to have a quantitative idea of the signals generated. Experiments based on the

speckle pattern modulation and the diffuse photon density waves modulation are described.

The experimental techniques were performed with the goal of imaging in tissue-like phan-

toms made of titanium dioxide (TiO2) suspended in polyacrylamide gel that is acoustically

impedance matched with water.

2

Acknowledgements

The completion of this thesis could not have been possible without the help, advice, and

friendship of Prof. Charles DiMarzio. I would like to thank Prof. DiMarzio for being my

friend, advisor and the motivator in my research. Through invaluable conversations not

only about optics and engineering, but also about any topic that the daily interaction had

brought up, I have learned how to be a better researcher and to be a better person. I am

deeply in debt to Chuck for his guidance.

I would also like to thank Prof. Ronald Roy. With his expertise in the field of acoustics,

he helped me have a better understanding of ultrasound wave propagation through many

meetings in which I listened to and discussed his comments with particular interest. I thank

Prof. Dana Brooks for his time, for the useful corrections in my thesis work, and for being a

member of my thesis committee. I am grateful to Dr. Gerhard Sauermann for his valuable

conversations about physics and for sharing with me his multiple experiences.

I am very thankful to the people that work and have worked at the Optical Science Labora-

tory during the last two years while completing this work. I have learned a little bit of each

of them and I hope to maintain, throughout the years, the friendship that we have built.

Finally, I wish to thank my family for all the support and the love they provide me.

Contents

1 Introduction 9

2 Background: Light, Sound and Their Interaction 11

2.1 Frequency Domain Biomedical Optics . . . . . . . . . . . . . . . . . . . . . 11

2.2 Diffuse Optical Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Biomedical Applications of Diffuse Optical Tomography . . . . . . . 16

2.3 Ultrasound as a Biomedical Tool . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Biomedical Imaging Using Ultrasound . . . . . . . . . . . . . . . . . 22

2.4 Acousto-Optic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Acousto-Photonic Effect 29

3.1 Approaches to Explain the Interaction of Multiple Scattered Light and Ul-

trasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Mathematical Models for Acousto-Photonic Imaging . . . . . . . . . . . . . 32

3.2.1 Acoustic Modulation of the Diffuse Photon Density Waves . . . . . . 32

3

CONTENTS 4

3.3 Temporal Light Correlation of Multiple Scattered Light and its Interaction

with ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Numerical Simulations: Monte Carlo Approach 40

4.1 Monte Carlo Methods for Multiple Scattered Light Simulation . . . . . . . . 41

4.2 Frequency Domain Monte Carlo Approach for Diffuse Optical Tomography 42

4.3 Monte Carlo Simulations for Acousto-Photonic Imaging . . . . . . . . . . . 45

4.3.1 First Order Approximation of the Light-Ultrasound Weight . . . . . 47

4.3.2 Acoustic-Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.3 Monte Carlo-Acoustic Simulation Ensemble . . . . . . . . . . . . . . 51

4.3.4 Discussion and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Experimental Methods 61

5.1 Laser Speckle Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.1.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.1.2 Experiments and Results . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 Acoustic Modulated Diffuse Photon Density Waves . . . . . . . . . . . . . . 66

5.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2.2 Experiments and Results . . . . . . . . . . . . . . . . . . . . . . . . 68

6 Conclusions and Future Work 70

A Matlab Monte Carlo-Acoustic Simulation code 72

CONTENTS 5

B Transport Theory 81

C Raman-Nath/Bragg Effect 84

References 89

List of Figures

2.1 Hemoglobin Absorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Acoustic Bragg diffraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Raman-Nath regime for the Acoustic diffraction of a light beam (Z) traveling

through an acoustic beam (X). . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1 Representation of the interaction of light and sound in scattering media. . . 32

4.1 Frequency domain representation of the constitutive sidebands in the inter-

action between multiple scattered light and ultrasound. . . . . . . . . . . . 46

4.2 Basic interaction of light, ultrasound and the particles in the medium. . . . 47

4.3 Ultrasound simulation shows the displacement of the particles in the beam

and phase variations in the focus. . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4 Geometry for the simulation ensemble of multiple scattered light and ultra-

sound. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.5 Flow diagram of Monte Carlo-Acoustic Simulation. . . . . . . . . . . . . . . 54

6

LIST OF FIGURES 7

4.6 Amplitude and phase modulation of diffuse light interacting with a plane

ultrasound wave in scattering media. The ultrasonic wavelength (≈ 640µm),

is well defined and modulates the optical path lengths. . . . . . . . . . . . . 55

4.7 Amplitude and phase modulation of diffuse light interacting with a focussed

ultrasound wave in scattering media. . . . . . . . . . . . . . . . . . . . . . . 56

4.8 Simulation with 1 million photons. . . . . . . . . . . . . . . . . . . . . . . . 57

4.9 Simulation with 5 million photons. . . . . . . . . . . . . . . . . . . . . . . . 57

4.10 Simulation with 10 million photons. . . . . . . . . . . . . . . . . . . . . . . 58

4.11 Simulation with 20 million photons. . . . . . . . . . . . . . . . . . . . . . . 58

4.12 Signal levels of the DPDW signal with respect to the CW signal. . . . . . . 59

4.13 Signal levels of the API signal with respect to the CW signal. . . . . . . . . 59

5.1 Setup for Speckle Contrast measurements. . . . . . . . . . . . . . . . . . . . 62

5.2 Pressure at the focus of the ultrasound vs voltage supply. . . . . . . . . . . 63

5.3 Speckle pattern with and without the prescence of the ultrasound. Notice

the bluriness of the image on the right (ultrasound on) with respect to the

one on the left (ultrasound off). . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.4 Speckle contrast for various sets of data. . . . . . . . . . . . . . . . . . . . . 66

5.5 Setup for DPDW experiments. . . . . . . . . . . . . . . . . . . . . . . . . . 67

List of Tables

2.1 Ultrasound intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.1 Tabulated data for speckle contrast . . . . . . . . . . . . . . . . . . . . . . . 65

8

Chapter 1

Introduction

Extensive research is being conducted in the field of Medical Imaging. Qualitative and

quantitative information as well as spatial resolution are the requirements to be fulfilled to

provide the medical practitioner with useful information to help diagnose the illness.

Diffuse Optical Tomography (DOT), among the various optical imaging techniques, has

been shown to be a good way to acquire information about tissue optical properties which

in turn are related to metabolic processes through the absorption of light by hemoglobin

(Hb) [1]. The non-invasive nature of this technique as well as the quantitative information

that it can provide, makes DOT an interesting field of study and a promising tool that can

work in parallel with current medical imaging methods such us MRI, X-rays, etc. Photon

migration can be explained using radiative transport theory and has been the subject of

recent and extensive research [2, 3, 4, 5]. In particular, the use of modulated near-infrared

(NIR) light in medical imaging and dignostic applications provides us with a spectral win-

dow through which is possible to get quantitative information about the absorption and

scattering properties of human tissue. The applications range from oximetry and tissue

spectroscopy to image of brain and breast tumors and functional imaging of the brain.

On the other hand, ultrasound provides a very well stablished imaging technique with good

9

CHAPTER 1. INTRODUCTION 10

spatial resolution compared with the resolution that DOT can provide. The sound wave

does not scatter as much as the light wave inside human tissue.

With the goal of obtaining the best of both imaging modalities, we have studied the inter-

action of multiple scattered light and ultrasound, which is know with the name of Acousto-

Photonic Effect, and its primarily application, the Acousto-Photonic Imaging (API). This

study is expected to lead us to imaging the optical properties of tissue with the spatial

resolution of ultrasound.

Various approaches have attempted to explain the physics behind this process. The density

changes due to the acoustic wave and therefore, the change on the density of particles,

produce index of refraction modulation and small particle displacements at the ultrasound

frequency which in turn change the photons wave vectors producing changes in the speckle

pattern, and modulation of diffuse photon density waves (DPDW).

The focus of this thesis is to investigate the feasibility of the API method through numerical

simulations and experimental techniques that try to explain the behavior and interaction

between multiple scattered light and ultrasound inside turbid media. Chapter 2 gives a

general introduction to the medical imaging applications of the two fields of study and

also describe the theories and approaches that try to explain this combination in non-

scattering media. Chapter 3 briefly reviews the mathematical theory behind the Acousto-

Photonic Effect as a matter of understanding the work done by this author. Chapter 4

presents frequency domain Monte Carlo simulations coupled with finite-difference time-

domain (FDTD) acoustic simulations that shows the expected signal level strenghts of the

interaction between diffuse waves and ultrasound. Chapter 5 presents the experimental

work done in order to study the two most important optical phenomena, which are the

laser speckle modulation and the diffuse photon density waves modulation. Finally, chapter

6 discusses the conclusions and the future work proposed for this project.

Chapter 2

Background: Light, Sound and

Their Interaction

2.1 Frequency Domain Biomedical Optics

Biomedical optics has been topic of intensive research during the last years with the goal of

developing another tool, based on the study of the optical properties of human tissue, that

can help the medical community to diagnose disease and choose the right treatment for the

patient.

Quantitative light absorption at specific wavelengths has been used since the 1930’s for

determining the oxygen content of blood, and now the scientific community is making efforts

to use this technique in imaging. In the late 1980’s the research was directed towards imaging

the transmission of light through tissue. Light in the near infrared range (wavelengths

from 700 to 1200nm) penetrates tissue and interacts with it in a random process, with

the predominant effects being absorption and scattering. Laser optical tomograpy involves

reconstruction of the amount of transmitted laser light through an object along multiple

paths. Moreover, the modulation of the light source at radio frequencies, which is the basis

11

CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 12

of diffuse optical tomography, has showed to provide information about the amplitude and

the phase of the diffuse waves, which can be used for better reconstruction of images or

more accurate measurements of optical properties of tissue. The modulation of the optical

source and its applications are usually known as frequency domain techniques.

Different applications have been developed with the use of frequency domain optical tech-

niques but we attempt to divide them in two groups. The first group would be tissue

spectroscopy and oximetry in which we treat tissue as macroscopically homogeneous and

we can use the approximations for the theory behind diffuse optical tomography that we will

review in the next subsections. Applications range from measurement of optical absorption

of tissue and Near Infrared (NIR) tissue oximetry to measurements of optical scattering in

tissues.

The second group would be related to the optical imaging of tissues. Since the objective

is to map the spatial distribution of the tissue optical properties we can not treat it as

macroscopically homogeneous anymore and then the use of the mathematical theory must

account for the spatial dependence of its constitutive parameters.

2.2 Diffuse Optical Tomography

Among the variety of optical techniques that exist to monitor and to image inside human

tissue, Diffuse Optical Tomography (DOT) has emerged as one of the most promising and

important, leading to research for different applications in the biomedical community [1, 4].

DOT has a spatial resolution of about 10mm; thus it can not provide images with the

resolution quality of X-rays, Magnetic Resonance Imaging (MRI) scans, Positron Emission

Tomography (PET) scans or ultrasound. However, this method does have a number of

practical applications even at low resolution. These include the measurement of tissue oxy-

genation for the study of muscular dystrophy [6] (which is any of a group of diseases chara-


terized by progressive wasting of muscles), tissue perfusion in the extremities for diabetic

disease [7], the detection of brain hemorrhaging [8], monitoring stroke patients [9], the study

of brain activity during specific tasks [5, 10], breast tumor detection and characterization

[5] and possibly the study of glucose concentration changes [11]. The clinical potentials of

determining the oxygenation level and functional imaging in the brain of young children has

been demonstrated [10, 12]. Beyond DOT, similar principles can be applied for fluorescense

imaging [13]: substances which play a crucial role in the body’s metabolic (energy making)

processes, such as NAD/NADH (nicotinamide adenosine diphosphate), exhibit flourescent

properties which allow detection after being excited by light. Their assessment by indirect

measurements has important potential for medical applications.

The scattering of light can be explained using standard mathematical models. In analytical

theory we start with Maxwell’s equations and introduce the scattering and absorption of

particles, which lead us to obtain the appropiate differential or integral equations for statis-

tical quantities such as variances and correlation functions [14, 15]. This is mathematically

rigorous but in practice is impossible to obtain a closed form solution that includes all the

scattering, diffraction and interference effects. Therefore, the research community has used

transport theory (radiative transfer theory) [16] in order to explain the behavior of light

in turbid media. This mathematical approach does not start with the wave equation. It

deals directly with the transport of energy through a medium containing particles. Trans-

port theory is not mathematically rigourous and does not include diffraction or polarization

effects. It is assumed in transport theory that there is no correlation between fields, and

therefore, it uses the addition of powers rather than the addition of fields [4, 14].

Transport Theory was initialy treated by Schuster in 1903 [17] and the basic differential

equation is called the equation of transfer and is equivalent to the Boltzmann’s equation

used in kinetic theory of gases and in neutron transport theory [16, 18]. This formulation is

capable of treating many physical phenomena and has been succesfully employed for prob-

lems including underwater visibility, marine biology, biomedical optics, and the propagation


of radiant energy in the atmospheres of planets, stars and galaxies.

The Boltzmann Transport Equation (BTE) is a balance relationship that describes the flow

of particles in scattering and absorbing media. This mathematical approach can be used

to model the propagation of light in optically turbid media, where the photons are treated

as the transported particles. This theory has been investigated extensively in the literature

and a brief summary can be found in Appendix B of this thesis for the convinience of the

reader. We are going to base this formulation in the work that can be found in Ref. [5, 19].

If we denote the angular photon density with u(r, Ω, t), which is defined as the number of

photons per unit volume per unit solid angle traveling in direction Ω at position r and time

t, we can write the BTE as follows:

∂u(r, Ω, t)∂t

= −v Ω · ∇u(r, Ω, t)− v(µa + µs)u(r, Ω, t)

+ vµs

∫

4πu(r, Ω′, t) f(Ω′, Ω) dΩ′ + q(r, Ω, t), (2.2.1)

where v is the speed of light in the medium, µa is the absorption coefficient in cm−1, µs is

the scattering coefficient in cm−1, f(Ω′, Ω) is the phase function or the probability density

of scattering a photon that travels along direction Ω′ into direction Ω, and q(r, Ω, t) is the

source term. q(r, Ω, t) has units of s−1 m−3 sr−1 and represents the number of photons

injected by the light source per unit volume, per unit time, per unit solid angle at position r,

time t, and direction Ω. The left hand side of Eq. (2.2.1) represents the temporal variation

of the angular photon density. Each one of the terms on the right hand side represents a

specific contribution to this variation. The first term is the net gain of photons at position

r and direction Ω due to the flow of photons. The second term is the loss of photons at

position r and direction Ω due to absorption and scattering. The third term is the gain of

photons at r and Ω due to scattering. Finally, the fourth term is the gain of photons due


to the light sources.

Some of the quantities used to describe photon transport are angular photon density

u(r, Ω, t), photon radiance L(r, Ω, t) = vu(r, Ω, t), photon density U(r, t) =∫4π u(r, Ω, t) dΩ,

photon fluence rate Φ(r, t) = v U(r, t) and photon flux J(r, t) =∫4π u(r, Ω, t) Ω dΩ.

The BTE is a difficult differential equation to solve. Therefore, a first order approximation

is commonly used to obtain closed form solutions. The validity of this approximation relies

in the assumption that scattering is much stronger than absorption (µ′s À µa), which is

true for biological tissue. The radiance L can be expressed as an isotropic photon fluence

rate Φ plus a small directional photon flux J. This approximation takes the name of the

diffusion approximation or the diffusion equation [16, 18]. See also Appendix B.

Since DOT is based on the modulation of the light source at radio frequencies we seek

for solutions of the diffusion equation in terms of the frequency ω. The frequency domain

expression for the solution of the diffusion equation for a homogeneous, infinite medium

containing a harmonically modulated point source of power P (ω) at r = 0 is

U(r, ω) =P (ω)4πD

eikr

r, (2.2.2)

where D is the diffusion coefficient given by D = v/(3µ′s + µa). The expressions for the

average photon density (UDC), and for the amplitude (UAC) and phase (φ) of the diffuse

photon density wave, derived from Eq. (2.2.2), are [5, 19]


UDC(r) =PDC

4πD

e−r(vµa/D)1/2

r, (2.2.3)

UAC(r, ω) =P (ω)4πD

e−r(vµa/D)1/2[(1+ ω2

v2µ2a

)1/2+1]1/2

r, (2.2.4)

φ(r, ω) = r(vµa/2D)1/2

[(1 +

ω2

v2µ2a

)1/2

− 1

]1/2

+ φs , (2.2.5)

where φs is the phase of the source in radians. As we can see from Eq. (2.2.4), the oscillating

photon density is proportional to the power of the point source P (ω) and will oscillate at

the same frequency. These are scalar, damped, traveling waves. The imaginary part of the

wavenumber must be greater than zero in order to satisfy the physical condition that the

amplitude is exponentially attenuated while the wave travels through the medium.

2.2.1 Biomedical Applications of Diffuse Optical Tomography

As discussed in the introduction of this chapter we can divide the applications that use

photon migration in human tissue in two groups. The first group is tissue spectroscopy and

oximetry where we deal basically with absorption and scattering in a macroscopic view.

The absorption is mainly because of oxy-hemoglobin, deoxy-hemoglobin, and water. The

absorption spectra ranging from 300 to 1100 nm is shown in Fig. 2.1 with data compiled by

Prahl [20]. We observe that the absorption around 700 to 900 nm is low compared to other

wavelengths. This is the so-called “medical spectral” window. As a result of this, light in

this spectral range penetrates deeply into tissues, thus allowing us to perform noninvasive

investigations. This is the reason why our work is based in the use of NIR light.

The scattering properties are determined mainly by the size of the scattering particles rela-

tive to the wavelength of light, and by the refractive index mismatch between the scattering

particles and the surrounding medium. In biological tissues, the scattering is mainly forward


200 300 400 500 600 700 800 900 100010

−2

10−1

100

101

102

Wavelegth (nm)

Abs

orpt

ion

coef

fcie

nt (

µ a)

Hemoglobin absorption spectra

HbHbO

2

Figure 2.1: Hemoglobin Absorption.

directed for wavelengths in the medical spectral window because the cellular organelles and

cells have dimensions comparable to the optical wavelength. Therefore, the scattering prop-

erties are described by two parameters: the scattering coeffcient µs and the average cosine

of the scattering angle g (anisotropy). Even though each scattering event is mainly forward

directed, after a number of scattering events a photon loses memory of its original direction

of propagation. It is customary to use the reduced scattering coefficient µ′s = µs(1 − g)

which represents the reciprocal average distance over which the direction of propagation of

a photon is randomized.

When we work with human tissue µ′s is typically much larger than µa, therefore, we can

assume that NIR light propagation is mainly due to scattering. This is one of the condi-

tions for the derivation of the diffusion equation (See Appendix B). The frequency-domain

solution given by Eq. (2.2.2) provides a good quantitative description of photon migration


in an infinite medium with uniform optical properties. However, for real investigation and

experimentation we have to consider real boundary conditions. In a reflectance geometry

one typically applies the semi-infinite boundary condition. This is a reasonable assumption

if the tissue depth is greater that the optical penetration (2 - 3 cm or less) and inhomo-

geneities are small [21, 22]. This assumption is not valid in transmission geometry since the

source and the detector are located in opposite sides of the tissue. In this cases it is better

to use more appropiate boundary conditions such as a slab, cylinder or sphere geometry

[23].

The objective of tissue spectroscopy is determining certain properties of the investigated

tissue volume like the oxygenation or the hemoglobin concentration of a muscle based on

the measurement of the optical properties of the tissue. Particularly, the absorption de-

pends on the presence of different chromophores like oxy-hemoglobin, deoxy-hemoglobin,

water, cytochrome oxidase, melanin, bilirubin, and lipids. Therefore, measurements at

different wavelengths have been employed to determine the relative contributions of each

chromophore according to their concentration in the tissue of study. In many cases only

three of them are sufficient to give a good description of the absorption properties of tissues.

These three chromophores are oxy-hemoglobin, deoxy-hemoglobin, and water [24].

We can also measure the scattering properties of tissue. In the past, studies were focused in

the light absorption of tissue [24], but recent research has suggested that the reduced scat-

tering coefficient itself may provide information about physiologically relevant parameters.

For instance it has been shown that mitochondria are the main source of light scatttering in

the liver, and possibly in other tissues as well [25]. Since a number of metabolic processes

related to cellular respiration occur in the mitochondria, the reduced scattering coefficient

may be related to the cellular activity and viability.

The second group of applications as we defined them is the optical imaging of tissues. These

applications are based on the sensitivity to optical properties of tissues. The contrast in

NIR imaging originates from spatial variations in the optical absorption and scattering


properties of the tissue. These spatial variations can be due to a local change in hemoglobin

concentration or oxygen saturation, a localized change in the tissue architecture, or the

concentration of cellular organelles. It is safe to point out that the promise of diffuse optical

tomography is not in achieving a high spatial resolution, but in achieving high contrast and

specificity.

The goal of the imaging technique is to generate spatial maps that display either structural

or functional properties of tissues. Therefore, we are dealing now with inhomogeneous

media, and now we have to use the diffusion equation for inhomogeneous media

−∇ ·D(r)∇U(r, t) + vµa(r)U(r, t) +∂U(r, t)

∂t= q(r, t). (2.2.6)

Notice the dependence on r of the diffusion coefficent D and the absorption coefficient µa.

Analytical solutions for this equation are available only for a few simple geometries like

spherical or cylindrical. For arbitrary inhomogeneous cases Eq. (2.2.6) can be solved using

numerical methods such as the finite difference method or finite element method. Alter-

natively, a perturbation expansion in µa and D leads to a solution of Eq. (2.2.6) in terms

of a volume integral involving the appropiate Green’s function. A similar procedure using

perturbation techniques has been developed for the modulation of DPDW by an ultrasonic

beam [26] which will be briefly reviewed in Chapter 3 of this thesis. Besides diffusion the-

ory, the case of inhomogeneous media can also be treated with stochastic methods such as

Monte Carlo simulations [27, 28] or lattice random walk models [29].

Among the most important applications of diffuse optical imaging we have noninvasive

optical mammography and optical imaging of the human brain, specifically for the detection

of intracranial hematomas and functional imaging of the brain.


2.3 Ultrasound as a Biomedical Tool

Ultrasound is a real-time tomographic imaging modality. Ultrasound is able to produce

images of the position of reflecting surfaces like internal organs and structures, but it also

can be used to produce real-time images of tissue and blood motion.

Ultrasound denotes the use of acoustical waves at frequencies greater than 20 KHz. Gener-

ally, medical ultrasound is performed at frequencies in the range of 1 MHz. The technique

is used to determine the location of surfaces within tissues by measuring the time interval

between the production of an ultrasonic pulse and the detection of its echo resulting from

the pulse being reflected from those surfaces. By measuring the time interval between the

transmitted and detected pulse, we can calculate the distance between the transmitter and

the object.

The ultrasound pulses are both produced and detected by a piezoelectric crystal transducer.

The crystal has the property of changing its physical dimensions in response to an electric

field, and can produce an electric field if its physical shape is changed mechanically. Thus,

ultrasonic compression waves (vibrations) are produced by applying an oscillating potential

across the crystal. The reflected ultrasound imposes a distortion on the crystal, which

in turn produces an oscillating voltage in the crystal. The same crystal is used for both

transmission and reception. There are a wide variety of transducers commercially available

which can produce an acoustic wave by mechanical or electronic means. The latter is used

with arrays of piezo-electric crystals, each one producing a small acoustic wave in phase

with the other crystals in the array to produce together the complete ultrasound beam.

If a structure is stationary, the frequency of the reflected wave will be identical to that of

the impinging wave. A moving structure will cause a back-scattered signal frequency shifted

higher or lower depending on the structure’s velocity toward or away from the transducer.

Imaging based on this principle is known as Doppler ultrasound.


For example, when an impinging sound pulse passes through a blood vessel, scattering and

reflection occur from the moving red cells. In this process, small amounts of sound energy

are absorbed by each red cell, and then re-radiated in all directions. If the cell is moving

with respect to the source, the back scattered energy returning to the source will be shifted

in frequency, with the magnitude and direction proportional to the velocity of the respective

blood cell. Thus, if we use ultrasound to image the cross-sectional area of the blood vessel,

the volume of blood flow can be calculated from the area of the vessel and the average

velocities of the blood cells.

The major use of Doppler ultrasound is the study of the heart and human carotid artery

disease where imaging and frequency shift are combined to produce images of artery and

ventricle lumens. The frequency shift data is used to color the image, showing direction of

flow (e.g. carotid arteries in red and veins in blue). Obstructions to blood flow are readily

evaluated by this method using hand held scanning devices. In addition to imaging heart

valves and blood vessels, ultrasound is the most convenient and inexpensive method for

medical evaluations such as fetal monitoring and gallbladder stones. Ultrasound imaging

is also being used for monitoring therapy methods such as hyperthermia, cryosurgery, drug

injections, and as a guide during biopsies and catheter placements.

The propagation of an acoustic wave through human tissue can be fully predicted and

described if we take into account the mass and stiffness of the media, and its conformance

with basic physical laws. In the very basic process a sinusoidal wave will accelerate adjacent

tissue particles and compress that part of the medium nearest to it as it moves forward from

rest. This also is going to impart a forward momentum to the particles which is going to

be transmitted to their neighbors which were at rest. These particles in turn move closer

to their neighbors, with which they collide, and so on.

When an acoustic wave is propagated in the medium, several changes occur. The particles

are accelerated and as a result are displaced from their rest positions. The particle velocity

at any point is not zero except at certain instants during a cycle. The temperature at any


point will vary above and below the ambient value. Also, the pressure at any point will

vary above and below the ambient pressure. This incremental variation of pressure is called

the acoustic pressure. A pressure variation, in turn, causes a change in the density called

the incremental density. An increase in sound pressure at a point causes an increase in the

density of the medium at that point.

Ultrasound propagation through human tissue can be explained with wave theory. The

acoustic wave equation has the form

∂2ξ

∂t2= v2

a

∂2ξ

∂x2(2.3.1)

where ξ is the displacement or the particles and va is the speed of sound in the medium.

The solution to this differential equation is well known and can be found elsewhere, but in

particular, we use the solution for a Gaussian ultrasound wave rather than a plane wave

because the transducers used in our experiments radiate a Gaussian wave. This has the

purpose of maximizing the interaction between diffuse light and sound in the focus of the

ultrasound beam. See Ref. [30] for a complete treatment of ultrasound wave propagation.

2.3.1 Biomedical Imaging Using Ultrasound

Multiple applications in the biomedical field have been developed using ultrasonic waves

[31]. The discovery of the piezo-electric effect at the end of the nineteenth century, and

the development of an ultrasonic echo-sounding device in the early 1930s by Paul Langevin

and Constantin Chilowsky, formed the basis for the development of medical pulsed-echo

SONAR.

Ultrasound can be used in therapy and as a diagnostic tool. In therapy basically the thermal

energy is used when sound is propagated through tissues. Muscle and bone have been found


to absorb more energy at interfaces with other heterogeneous tissues, because at these

surfaces, the longitudinal waves of ultrasound are reflected and transformed into transverse

waves, creating a heating effect. This happens commonly in the areas in between the muscle

and bone or between the muscle and tendon. By applying ultrasonic waves to these areas,

physical therapists can take advantage of this thermal affect to reduce inflammation and

increase mobility in the joints.

As a diagnostic tool ultrasound is used primarily in imaging. Real-time ultrasonic imaging is

possible with the use of state-of-the-art piezo-electric transducers. Since there do not appear

to be any biologically significant adverse effects of ultrasound at the levels used for diagnosis,

ultrasonic imaging has become the most frequently utilized technique in obstetrics. This

helps to diagnose multiple complications that can be present during pregnancy and also to

monitor throughout the pregnancy process. For this purpose two general types of ultrasound

scanners are available, real-time scanners for depiction of structures within the body, and

scanners that are mounted in articulated arms, which, when manually moved over the

body produce static images. Most scanning studies are performed today with real-time

scanners. Other applications are renal and urological imaging, echocardiography to examine

the structure and functioning of the heart for abnormalities and disease, and pediatric

imaging among others.

The use of ultrasound with biological tissue has to take into account potential damage and

bioeffects. There are two primary mechanisms by which ultrasound can produce biological

effects: heat and cavitation. Attenuation in tissues is, on the average, 90 to 95 percent

absorption, specifically conversion to heat. Temperature rise in a particular spot where

the acoustic wave is radiated depend upon the ultrasound intensity, frequency, specific

heat and thermal conductivity of the tissue, and vascularization. These factors combine

in complicated ways to determine the ultimate temperature rise at a given site exposed

to an ultrasound beam. A higher frequency results in a higher absorption coefficient but

greater attenuation so that a given source intensity would result in a lower intensity at a


given depth. Some concern is warranted with pulsed ultrasound Doppler and flow imaging

equipment in color where high power levels and time-averaged intensities may result in large

values in thermal index.

On the other hand, cavitation is the generation, growth, and dynamics of bubbles in a

medium. It can be generated in media that have cavitation nuclei, which are microbub-

bles. Cavitation is divided in two types: stable and transient. Stable cavitation describes

the situation in which bubbles are oscillating repeatedly in an acoustic field. Transient

cavitation refers to the situation in which cavities reach resonance size, at which violent

nonlinear dynamics occur, with the cavitiy collapsing and producing pressure shock waves

and extreme temperature gradients.

The American Institute for Ultrasound in Medicine (AIUM) has multiple publications in

which we can find information about the biological effects and safety of diagnostic ultrasound

[32]. In these references we can find that in the low megahertz frequency range there have

been no independently confirmed significant biological effects in tissue for Spatial Peak

Temporal Average (SPTA) intensites below 100mW/cm2. The above information is for

scanning imaging systems. Doppler instrument outputs can be significantly higher than

those for imaging. Typical intensities according to the National Council on Radiation

Protection and Measurements (NCRP) and the AIUM are presented in Table 2.1.

We can say that there is presently no identified risk to the use of ultrasound at intensity

levels used commonly in diagnosis. However, a prudent approach should always be employed

with the use of this technique.

2.4 Acousto-Optic Effect

The first type of interaction between light and sound was studied in non-scattering media.

Acousto-optic interaction occurs in all optical media when an acoustic wave and a ray


Instrument Type ISPTA Range (mW/cm2)Static B-scan 10 - 170

M-mode 10 - 160Dynamic B-scan

sector 6 - 30linear 0.01 - 12

DopplerCW 0.6 - 2500

pulsed 50 - 1945

Table 2.1: Ultrasound intensities

of light are present in the medium. When an acoustic wave is launched into the optical

medium, it generates a refractive-index wave that behaves like a sinusoidal grating. An

incident laser beam passing through this grating will diffract the laser beam into several

orders. Its angular position is linearly proportional to the acoustic frequency, so that the

higher the frequency, the larger the diffracted angle.

Various attempts to explain this phenomena have been developed during this century [33,

34]. We can treat this as a parametric process in which the optic and the acoustic wave

are mixed via the elasto-optic effect. This works as a oscillator system with frequency ωa

modulated by a frequency ωp which we would call the pump frequency. The typical equation

of motion would for this type of oscillator is

d2y

dt2+ γ

dy

dt+

(ω2

a + α sinωpt)y = 0.

In the case of the acousto-optic effect the pump frequency would be the light wave frequency

and the signal wave that that gets frequency shifted due the pump wave has a frequency

of ωp + ωa, where ωa is the acoustic wave frequency. This model does not provide a clear

explanation of the physical behavior of the acousto-optic effect; therefore, it has not been


further developed.

The approach that has proven to be successful in the treatment of this phenomena is the

one based on scattering theory in which the light wave is treated as photons and sound

waves as particles. This leads us to two regimes for the acousto-optic effect: The Bragg

regime [35] and the Raman-Nath regime [36, 37].

The Bragg regime was first discovered by Brillouin in the 1920’s by a set of basic experiments

using a sound column and light waves propagating with angles of incidence that were chosen

to ensure constructive interference of the light beams reflected off the crests of the sound

wave. This is also the condition for x-ray diffraction. This leads to critical angles of incidence

sinφin = pλa

2λ(2.4.1)

which are called the Bragg angles. The angle of reflection on the crest of the acoustic beam

is twice the Bragg angle. This is shown in Fig. 2.2. Brillouin makes the observation that

the sound wave is a sinusoidal grating and that therefore we should only expect two critical

angles given by the relation in Eq. (2.4.1), for p = +1 and p = −1. This is only valid for a

thick column, in contrast to the Raman-Nath regime. The validity for this assumption and

the limits will be explained in the next section.

The Raman-Nath regime owes its name to the Indian researchers C. V. Raman and N. S.

N. Nath, who based their work on the previous work done by Brillouin, Lucas and Biquard,

and Debye and Sears, which showed that the existence of numerous modes was due to the

fact that the interaction length was too small (considering a thin sound column), but in a

more direct way for the mathematical explanation of the acousto-optic interaction. This

work considers a very thin acoustic column as a phase grating which the light rays traverse

in straight lines. Because of the phase shift suffered by each ray, the total wavefront is


Figure 2.2: Acoustic Bragg diffraction.

corrugated as it leaves the sound field creating a set of upshifted and downshifted waves

with frequency ω±nωa and propagation in the direction kn such that knx = nka with angle

of propagation φn given by

sinφn =nka

k=

nλ

λa(2.4.2)

where φn is the angle of diffraction of the light beam at the output of the acoustic column,

ka is the acoustic wavenumber, k is the light wavenumber, n the order of the diffracted light

beam, λ the optical wavelength and λa the acoustic wavelength. This procces is shown in

Fig. 2.3. Later on, a generalization of the work by Raman and Nath done by Van Cittert


Figure 2.3: Raman-Nath regime for the Acoustic diffraction of a light beam (Z) travelingthrough an acoustic beam (X).

[38], which considers a thick film of sound as a collection of thin films, confirms the Bragg

regime and the expectation of only two critical angles. A more complete treatment of the

Acousto-optic effect can be found in Ref. [39] or in Appendix C.

Chapter 3

Acousto-Photonic Effect

Although DOT and photon migration based imaging modalities in general have proved to

be a feasible technique among the multiple imaging methods and are in the proccess of

becoming comercially available for the medical community, they lack the spatial resolution

for measurements made deep into biological tissue. This fact led our research group and a

few other optical research groups to suggest the combination of multiple scattered light and

ultrasound with the goal of enhancing the spatial resolution and provide a more complete

medical imaging tool. The combination between multiple scattered light and ultrasound

for imaging purposes was initially reported by Brooksby et.al. [40, 41] in 1993 where they

showed a basic interaction and postulate the tagging of light with ultrasound. Later on in

1995, Leutz et.al. [42], Wang et.al. [43, 44, 45] and Kempe et.al. [46] combined continuous

wave light with ultrasound reporting one and two dimensional images using single detectors.

Bocarra et.al. [47, 48] applied the same technique but used paralled detection to study the

modulation of the speckle pattern generated by the ultrasound. In 1996 Gaudette and

DiMarzio [49], postulated the modulation of diffuse photon density waves by a focused

ultrasound field which is the basis of the Acousto-Photonic Imaging (API) technique. The

subject of this thesis is the continuation of this challenging work of trying to explain the

foundations of API.

29

CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 30

Along with the experimental techniques, various mathematical models have been developed

to explain, and to find the best way to achieve, this interaction. There are mathematical

methods based on the temporal correlation of the electric fields [42, 53]. These methods

rely on the index of refraction modulation of the medium due to the ultrasound, which in

turn, modulates the laser speckle generated by the diffusing medium. There are also other

methods based purely on transport theory that try to explain the acoustic generated diffuse

photon density waves [26].

The author of this thesis has developed a numerical simulation for the API technique based

on a frequency domain Monte Carlo method [50] and finite difference time domain (FDTD)

simulations of a focused ultrasound wave [51], which will be extremely useful to corroborate

the experimental results as well as to establish the viability of this imaging technique. This

work will be presented in Chapter 4.

3.1 Approaches to Explain the Interaction of Multiple Scat-

tered Light and Ultrasound

Laser Speckle Modulation

The first idea we present to explain this phenomena is to study the behavior of the particles

as independent scatterers in the presence of an ultrasound beam. For this purpose it is

assumed that the acoustic wave does not scatter in the medium. The light source has a

carrier frequency ωc with wavelength in the medical spectral range. The ultrasound wave

has the general form of S(r)cos(ka · r − ωat) where S(r) is the amplitude of the acoustic

wave and ka is the acoustic wavevector.

The presence of this oscillating mechanical field will produce variations on the index of

refraction of the medium following the ultrasound amplitude and, therefore, modulate the

optical path lengths of the light interacting with the acoustic wave. This will cause a change


in the speckle pattern formed by the multiple scattered light leaving the tubid media. Also

due to this mechanical wave the particles in the medium will oscillate at the frequency of

the acoustic wave. This will produce variations in the optical phase which will vary the

speckle pattern produced by the light exiting the medium. Leutz and Maret [42] as well as

Mahan [52] studied these particle displacements and developed mathematical models trying

to provide an explanation of the phenomena. Wang [53] developed a mathematical model

in which he took into account this variation of the index of refraction together with the

contributions to the modulated signal by the particle displacements. Wang’s model is based

on the temporal correlation of the multiple scattered light. This mathematical models will

be reviewed in the following sections.

Diffuse Photon Density Waves Modulation

Another approach to the combination of light and sound is the one developed by Gaudette

[26] in which he proposed that the combination of DPDW and ultrasound will provide a new

way of obtaining information using this technique. This approach is based on the creation

of virtual acoustic generated diffuse photon density waves due to a change in the density

of the medium, at the combination frequencies ω + ωa and ω − ωa where ω is the intensity

modulation frequency of the monochromatic light. This technique considers a DPDW from

source to receiver near the surface as show in Fig. 3.1. The received signal will contain

information about the optical properties of the media along a path parallel to the surface

at small depths. With the use of the ultrasound we expect to receive this virtual diffuse

sources, which due to the quantum noise and DC optical levels, will be significantly small

compared with the pure diffuse optical signal, but will provide information about the paths

that the diffuse waves have traveled, since we know the position of the ultrasound wave, we

can map these paths and provide enhanced resolution of the tissue in study.

This density modulation model is based on the diffusion equation which was stated at the

begining of the chapter, and not in Maxwell’s equations. Therefore, this does not take


Figure 3.1: Representation of the interaction of light and sound in scattering media.

into account the coherent characteristic of light which is reasonable since human tissue

is a strongly scattering media with µ′s ≈ 10cm−1 and µa ≈ 0.1cm−1. This mathematical

approach, which is the basis of what we call Acousto-Photonic Imaging, and its implications

will be revised in the next section which is dedicated to the mathematical models dealing

with this phenomena.

3.2 Mathematical Models for Acousto-Photonic Imaging

3.2.1 Acoustic Modulation of the Diffuse Photon Density Waves

Physical considerations

This approach was developed by Gaudette [26] where he postulated that the primary source

of interaction was the modulation of DPDW due to the density modulation of the medium

by the ultrasound. This model is based on the diffusion equation and not in any type of

correlation of the electric fields. It only takes into account the fluence rate. This model

postulates that the interaction of the ultrasound beam with the DPDW is caused by the


time-harmonic change of the number of scatterers and absorbers per unit volume. Therefore,

this change in the number of particles produces a change in density, and this implies a change

in volume, where it is assumed that the change in scattering and absorption of the media

is only due to this fact. In consequence, this will produce a change in the photon diffusion

coefficient D = v/(3µ′s + µa).

Considering that the size of the particles does not change considerably in the presence of

the ultrasound it was assumed that

µs = σsN

V(3.2.1a)

µa = σaN

V, (3.2.1b)

where σs and σa are the scattering and the absorption cross sections respectively, N is the

number of particles, and V is the volume.

Since µs and µa are assumed only to be a function of the volume, the change in these

parameters can be expressed as follows

dµs

µs= −dV

V= C (3.2.2a)

dµa

µa= −dV

V= C, (3.2.2b)

where C is the compression factor due to the ultrasound wave. This will produce a change

on the diffusion coefficient expressed by

D =v

3(µ′s + µa)(1− C), (3.2.3)


assuming that C2 ¿ 1.

Similarly, the decay rate is affected by this change in volume and would take the form

α = (µa + µaC)v = α0 + Cα0. (3.2.4)

The compression factor C can be modeled as a Gaussian wave since the ultrasound beam

used experimentally is a Gaussian beam.

Development of the Acoustic Generated Diffusion Equation

Based on the previous analysis it is assumed that the DPDW in the presence of the ultra-

sound travels through a change in the diffusion coefficient. Assuming a small perturbation

for this model, this can be described with the following parameters:

Φ = Φ0 + Φ+1 + Φ−1

D = D0 + D1

α = α0 + α1 (3.2.5)

where the subscript 0 corresponds to the original DPDW and the subscript 1 is due to

the particle modulation. The + and − signs refer to the ultrasound modulated DPDW at

frequencies ω+ωa and ω−ωa for Φ+1 and Φ−1 respectively. If we expand the diffusion equation

using Eq. (3.2.5), considering only one sideband at frequency ω +ωa, and neglecting higher

order terms we are left only with an equation in terms of the perturbation Φ+1 and Φ0


D0∇2Φ+1 −

∂Φ+1

∂t− α0Φ+

1 + D1∇2Φ0 +∇D1 · ∇Φ0 − α1Φ0 = 0, (3.2.6)

The first three terms of Eq. (3.2.6) look like a normal diffusion equation, therefore, the

second part is treated as a source term. Thus, the diffusion equation for the perturbed

wave is

D0∇2Φ+1 −

∂Φ+1

∂t− α0Φ+

1 =v

nQ, (3.2.7)

which is the basic equation of what we call the virtual acoustic DPWD source. Since the

term Q depends on D1 and α1, which depend on C, Q depends on the Gaussian profile of

the ultrasound beam given in the factor of compression

C = Aeiωate

− (x2+y2)

w20(1+z2/b2)

e

ika(x2+y2)

2(z+b2/z)

e(i tan−1( zb))eikaz (3.2.8)

with A in Eq. (3.2.8) given by

A = −i

√2P

πw20(1 + z2/b2)ρ0v3

a

. (3.2.9)

where P is the acoustic power, ρ0 the density of the medium, va the velocity of sound in

the medium, w0 is the waist radius of the beam and b is the Rayleigh range. The solution

for the diffusion equation for the virtual DPDW source and a complete treatment of this

mathematical model can be found in Ref. [26].


3.3 Temporal Light Correlation of Multiple Scattered Light

and its Interaction with ultrasound

Another approach to explain this interesting physical phenomena is the use of the temporal

autocorrelation functions of the scattered light, based on the use of Maxwell’s equations

and probability distribution functions of the scatterers. A basic theoretical model was first

developed by Leutz and Maret [42], in which they took into account the autocorrelation

function of the electric fields defined by

G1(τ) = 〈E(t)E∗(t + τ)〉 =∫ ∞

tP (s)〈Es(t)E∗

s (t + τ)〉ds. (3.3.1)

where Es is the scattered electric field along a path s and P (s) is the distribution function

of the fraction of the incident intensity scattered into paths of length S. For this analysis

they consider Brownian and ultrasonic motion as the principal sources of modulation of

the laser speckle produced by the scattered light. The average field correlation for this two

cases are

〈Es(t)E∗s (t + τ)〉B = exp

(−2τs

τ0l

)(3.3.2)

〈Es(t)E∗s (t + τ)〉U =

⟨exp

−i

s/l∑

j=1

∆φj(t, τ)

⟩. (3.3.3)

Eq. (3.3.3) is the autocorrelation function due to the ultrasonic field along a path with

s/l scatterers (s À l). l is the scattering mean free path of light along succesive scatterers

and s is the length of the scattering path. ∆φj(t) is the phase variation due to the particle

displacements generated by the ultrasound. The expression for this phase variation is


s/l∑

j=1

∆φj(t, τ) =s/l∑

j=1

kj(∆rj+1,j(t, τ)−∆rj+1,j(t)), (3.3.4)

where ∆rj+1,j(t, τ) is the distance between two succesive scatterers

∆rj+1,j(t, τ) = (rj + Asin[ka · rj − ωat])

−(rj+1 + Asin[ka · rj+1 − ωat]), (3.3.5)

and kj is the light wavevector after the jth scattering event. Assuming that the phase

change at each scattering event are independent variables with a Gaussian distribution and

transmission of light through a slab of thickness L, Leutz obtained the following expression

for the field correlation function using the procedure as in Ref. [54].

G(t) =

√6

(Ll

2) (

tτ0

+ (k0A)2(1− cos[ωat])α)

sinh

(√6

(Ll

2)(

tτ0

+ (k0A)2(1− cos[ωat])α)) . (3.3.6)

A similar approach, considering only the displacement of the particles due to the ultrasound

is the one developed by Kempe [46].

Leutz does not consider the modulation of the index of refraction, which is considered in

the mathematical model developed by Wang [53], in which he postulates that there is also

a phase change produced by the change of the index of refraction due to the acoustic wave.

This will add a second term in Eq. (3.3.3) and the expression that Wang presents as the

principal one responsible for the interaction between multiple scattered light and sound is


the following

〈Es(t)E∗s (t + τ)〉U =

⟨exp

−i

s/l∑

j=1

∆φj(t, τ) +s/l+1∑

j=1

∆φnj(t, τ)

⟩(3.3.7)

where ∆φnj(t, τ) = φnj(t + τ) − φnj(t) and φnj is the phase variation induced by the

modulated index of refraction along the jth free path. This phase variation is

φnj(t) =∫ lj

0k0∆n(rj−1, sj , θj , t)dsj , (3.3.8)

where lj is the length of the jth free path, k0 is the optical wavevector, ∆n is the modulated

index of refraction, rj is the location of the jth scatterer, sj is the distance along the jth

free path, and θj is the angle between the optical wave vector in the scattering event j and

the acoustic wavevector ka.

Wang assumes that the modulated index of refraction is due to an acoustic plane wave

and the piezooptical properties of the medium expressed by η which is related to the

piezooptical coefficient of the material ∂n/∂p, the density ρ and the acoustic velocity va by

η = (∂n/∂p)ρv2a.

After expanding the variance of the phase variation into quadratic and cross terms he

presents the following approximation

⟨−i

s/l+1∑

j=1

∆φnj(t, τ)

2⟩≈ (s/l + 1) (2n0k0A)2δn × [1− cos(ωaτ)], (3.3.9)


where δn = (αn1 +αn2)η2, in which αn1 and αn2 are related to the averages done to simplify

the expansions into the quadratic and cross terms. A similar procedure is followed to derive

an approximate expression for the first term in Eq. (3.3.7) which is

⟨−i

s/l∑

j=1

∆φj(t, τ)

2⟩≈ s

l(2n0K0A)2δd[1− cos(ωaτ)], (3.3.10)

where δd = 1/6. Adding these two contributions to the autocorrelation function G1(τ)

Wang obtains the following expression for the autocorrelation function using the procedure

as in Ref. [54].

G1(τ) =(L/l)sinh[ε[1− cos(ωaτ)]1/2]sinh[(L/l)ε[1− cos(ωaτ)]1/2]

, (3.3.11)

where ε = 6(δn + δd)(n0k0A)2. Based on these results, Wang infers that an important

difference between δn and δd is that the average sum of the cross terms of the variations

due to the particle displacements ∆φj vanishes in the diffusion limit (s/l À 1) which means

that the contributions from the displacement by different scattering events are independent

but the contributions from the index of refraction by different free paths are correlated.

The complete derivation of this mathematical model can be found in Ref. [53].

It is important to remark that this mathematical approach assumed the use of continuous

wave light; consequently, there is no presence of a DPDW.

Chapter 4

Numerical Simulations: Monte

Carlo Approach

This chapter covers the numerical simulations performed in order to study the Acousto-

Photonic effect. The first part gives an overview of the Monte Carlo method and its appli-

cation to the simulation of scattered light. The second part studies the specific application

of the Monte Carlo algorithm in the simulation of frequency domain optical techniques. Fi-

nally, the remaining sections of this chapter are devoted to explaining the main goal of this

simulation, which is to combine this frequency domain Monte Carlo with a finite difference

time domain simulation of a focussed ultrasound beam. This will lead to an important tool

that will help us determine the feasibility of this interaction for its future use in medical

imaging.

40

CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 41

4.1 Monte Carlo Methods for Multiple Scattered Light Sim-

ulation

The Monte Carlo algorithm is a very useful method for simulating random processes and

in particular, light propagation in tissue. Monte Carlo refers to a technique first proposed

by Metropolis and Ulam to simulate physical processes using a stochastic model [55]. In a

radiative transport problem, the Monte Carlo method consists of recording photon’s travel

histories as they are scattered and absorbed. Monte Carlo programs with great sophis-

tication have been developed for different applications that deal with multiple scattered

light and that take into account the different geometries and boundary conditions that the

specific problem has. The Monte Carlo method is being used by the biomedical research

community to model laser tissue interactions, evaluate scattering and absorption properties,

and in general as a tool to solve inverse problems.

The simulation is based on the random walks that photons make as they travel through

tissue, which are chosen by statistically sampling the probability distributions for step size

and angular deflection per scattering event. After propagating many photons, the net

distribution of all the photon paths yields an accurate approximation to reality.

There are a variety of ways to implement Monte Carlo simulations of light transport. One

approach is to predict steady-state light distributions. Another approach is to predict time-

resolved light distributions. A third approach is to implement Monte Carlo in the frequency

domain to predict amplitude and phase information, which is specially useful when working

with DPDW.

The basic producedure is as follows: after setting up the intial conditions and launching the

photons in the medium, a photon is moved a distance s where it may be scattered, absorbed,

propagated ballistically, internally reflected, or transmitted out of the tissue. The photon

is repeatedly moved until it either escapes from or is absorbed by the tissue. If the photon


escapes from the tissue, the reflection or transmission of the photon is recorded. If the

photon is absorbed, the position of the absorption is recorded. This process is repeated until

the desired number of photons have been propagated. The recorded reflection, transmission,

and absorption profiles will approach true values (according to the optical parameters of

the tissue being studied) as the number of photons propagated approaches infinity. The

exact formulas and procedures to implement this method are well explained in the literature

[27, 56].

4.2 Frequency Domain Monte Carlo Approach for Diffuse

Optical Tomography

The use of frequency domain techniques for near-infrared spectroscopy has led to new ways

to do numerical computations. Therefore, one of the approaches is to use the Monte Carlo

algorithm to directly predict the modulation and phase for a frequency domain measurement

without storing data related to temporal events. For this purpose, Yaroslavsky [50] reduced

the time-dependent radiative transport equations to a stationary one for the case of a

harmonically modulated radiation source.

This technique avoids the tracking of the time-histories of each individual photon and esti-

mates the quantities relevant to frequency-domain measurements. The propagation of the

photons having a complex weight is simulated in CW-regime and the resulting modula-

tion and phase, respectively, are computed directly. This method reduces the amount of

computational time and decreases the amount of information that needs to be stored.

Basically, if we work with the transport equation as in Eq. (2.2.1), rewritten here in terms

of radiance for convenience


1v

∂L(r, Ω, t)∂t

= −∇ · L(r, Ω, t)− µtL(r, Ω, t)

+µs

∫

4πL(r, Ω′, t) f(Ω′, Ω) dΩ′ + q(r, Ω, t) (4.2.1)

where the harmonically modulated source q(r, Ω, t) is given by the expression

q(r, Ω, t) = qDC(r, Ω) [1 + <(m0 exp (iωmt))] , (4.2.2)

in which m0 is the incident modulation depth, and ωm is the light source modulation angular

frequency (ωm = 2πfm where fm is the modulation frequency).

Due to the linearity of Eq. (4.2.1) with respect to the radiance L(r, Ω, t), the solution will

have the general form of a wave composed of a DC and an AC component, as was shown

in Eq. (2.2.3) and Eq. (2.2.4), given by

L(r, Ω, t) = LDC(r, Ω) + <(LAC(r, Ω, ω) exp (iωmt)). (4.2.3)

which are written in terms of radiance for the convenience of our formulation.

Once the frequency domain Monte Carlo is implemented the objective is to estimate a

functional for the oscillating radiance LAC over a certain detector area D. This functional

will have the form

JAC(ω) =∫ ∫

DLAC(r, Ω, ω) dr dΩ, (4.2.4)


Also we can define a functional for LDC of the same form as Eq. (4.2.4). Based on these

expressions we can define in our algorithm complex weights for the photons, leading us to

find directly the information for the amplitude and the phase of the DPDW. The functionals

needed are function of the weight and the number of photons launched into the system and

will have the form

JDC =1N

N∑

h=1

W hLgh, (4.2.5)

JAC =1N

N∑

h=1

ZhLgh, (4.2.6)

where h represents the h-th photon in the system. W hL is the weight for the DC migration

of photons and ZhL is the complex weight for the AC migration of photons which has the

form ZhL = mh

L exp(iϕhL), where mh

L is the intensity modulation factor of the DPDW and

ϕhL is the initial phase of the DPDW. gh is 1 if the h-th photon has reached the detector

or 0 otherwise. L is the number of scattering events that the photons had gone through

before reaching the detector. The values for the initilization of these parameters are L = 0,

W h0 = mh

0 = 1 and ϕh0 = 0.

During the simulation the values of the weights are updated at each scattering event by the

following relations:

W hl = c W h

l−1 (4.2.7)

mhl = c mh

l−1 (4.2.8)

where c = µs/(µs + µa) is the albedo which is approximately constant for the medium.

The phase ϕhl depends on the travel direction of the photon at each scattering event and


is randomly updated according to the step size and direction of the new scattering event.

In particular, for the Acousto-Photonic effect the harmonic displacements of the particles

due to the ultrasound play an important role in this change of phase. This idea and its

implementation in the Monte Carlo simulation will be addressed in detail in the next section.

4.3 Monte Carlo Simulations for Acousto-Photonic Imaging

In order to simulate the interaction between the sound and the optical fields, we have

implemented this frequency domain approach to the Monte Carlo method, taking into

account the presence of the acoustic field inside the medium. We define our statistical

weights including the effect of the ultrasound as small harmonic displacements about the

rest position of the particles in the medium. These displacements are going to produce

a small change on the optical path lengths of the optical carrier and its sidebands. The

latter are due to the diffuse waves generated at 72.4 MHz. This frequency was chosen by

convenience since it is the same frequency used in the experiments, but we could have used

a different frequency in the MHz range. See Fig. 4.1.

Fig. 4.1 shows the distribution of optical signals from a frequency domain perspective. We

find the optical carrier band ωc in the THz range which is going to produce the DC photon

migration. If we now modulate the power of the optical source with a radio-frequency signal

around the 100MHz range we get what is known as a DPDW, with frequency ωc ± ωm,

where the subscript m stands for modulated. After this, if the ultrasound is present in the

medium at the same time as the light, it is going to modulate the optical signals creating

acoustic generated sidebands at frequencies ωc ± ωa and ωc ± ωm ± ωa. The first set are

the frequencies that the acoustic modulated carrier is going to have and the latter, are the

frequencies that the acoustic modulated DPDW will have. Thus, in the frequency domain

Monte Carlo-Acoustic simulation our goal is to find the amplitude and phase of acoustic

modulated DPDW signals, and if necessary, information about the other optical sidebands


Figure 4.1: Frequency domain representation of the constitutive sidebands in the interactionbetween multiple scattered light and ultrasound.

as well.

The statistical weight for the API interaction is defined as follows:

WnhJ =J∏

j=1

[Ahj + ahj ] exp(iknShj + i~shj ·∆~knhje

iωat + i~s∗hj ·∆~knhje−iωat

)(4.3.1)

where the capital letters represent static quantities due to the optical field, and the lower

case letters represent variations at the acoustic frequency which in fact, are smaller than

the pure optical quantities. WnhJ is the statistical weight for one of the n = 9 frequency

domain sidebands that are generated in the process necessary to get the API signal. This

weight represents the value of the weight for the photon h at frequency n and scattering

event j. The value of ∆knhj is defined as the vectorial difference between the wave vector


Figure 4.2: Basic interaction of light, ultrasound and the particles in the medium.

of the incoming photon and its wave vector after the scattering event, which is going to be

random due to the diffuse nature of the process. The dot product between ∆knhj and the

acoustic wave displacement ~shj is responsible for the interaction between the ultrasound

and the scattered light. We can see a graphical representation of this process in Fig. 4.2,

where we show a photon h at frequency n and at scattering event j. The incoming photon

has certain direction kin and after it scatters is going to come out at a direction kout. The

vectorial difference between the wavevectors in these directions gives us ∆knhj , that along

with the ultrasound induced displacement of the particles, is going to give us the change in

the optical path length for the photons in the diffusive medium.

4.3.1 First Order Approximation of the Light-Ultrasound Weight

The phase change due to the ultrasound that produces the API signal is defined as ∆φnhj =

~shj ·∆~knhj . Now, if we expand in Taylor series the exponentials containing the information

about the ultrasound in Eq. (4.3.1), considering only the first order term we have:


WnhJ ≈J∏

j=1

[Ahj + ahj ] eiknShj(1 + ishj ·∆knhje

iωat) (

1 + is∗hj ·∆knhje−iωat

)(4.3.2)

In order to find a more useful expression for Eq. 4.3.2 we are going to make use of the

following mathematical derivation which has the same structure as the expression for the

statistical weights:

∏

j

[Xj + xj ] = [X1 + x1] [X2 + x2] [X3 + x3] . . .

≈ X1X2X3 · · ·+ [X2X3 . . . ]x1 + [X1X3 . . . ] x2 . . .

=∏

j

[Xj ]

1 +

J∑

j=1

xj/Xj

. (4.3.3)

The terms involving the lower case letters have been neglected since they are going to be

small compared to the mixing between optical (upper case letters) and acoustical contribu-

tions. Reordering the expression in Eq. 4.3.2 we can express the weight with the form of

Eq. (4.3.3) as:

WnhJ ≈

J∏

j=1

Ahj

1 +

J∑

j=1

ahj/Ahj

×

J∏

j=1

exp (iknShj)

1 +

J∑

j=1

(ishj ·∆knhje

iωat + is∗hj ·∆knhje−iωat

) (4.3.4)


WnhJ ≈J∏

j=1

[Ahj exp (iknShj)]×1 +

J∑

j=1

ahj/Ahj

1 +

J∑

j=1

(ishj ·∆knhje


) (4.3.5)

Assuming that the optical amplitudes are greater that the acoustic amplitudes (Ahj À ahj)

we can drop the term[1 +

∑Jj=1 ahj/Ahj

]since it is approximately 1. Then, the final

expression for the complex weights that include the API interaction would be:

WnhJ ≈J∏

j=1

[Ahj exp (iknShj)]×1 +

J∑

j=1

(ishj ·∆knhje


) (4.3.6)

Eq. (4.3.6) is implemented in the Matlab code listed in Appendix A as the weights defining

the expression for the functional for the DPDW.

For a preliminary model we used a plane ultrasonic wave defined as

~shj = ~sa exp (i ~ka · ~r), (4.3.7)

where ~sa is amplitude of the displacement of the particles due to the acoustic wave. The

amplitude of this displacement was considered 1µm, the ultrasound frequency was fa =

2.4MHz, and the velocity of the ultrasound in tissue was considered va = 1480m/s. The

results of this simulation showed that this basic interaction was possible and are going to

be presented in the next sections. However, in order to get more realistic results a finite-

difference time-domain acoustic simulation was implemented which gave us information of


the displacement of a focussed ultrasound wave. The details of this simulation will be

reviewed in the next section.

4.3.2 Acoustic-Simulation

The initial Monte Carlo simulations considered a plane acoustic wave to make the assump-

tions simpler. In order to get a more realistic and accurate numerical simulation of the

interaction between multiple scattered light and ultrasound, the pressure field of a focussed

circular transducer was simulated using a two-dimensional axisymmetric finite-difference

time-domain (FDTD) code developed by Manneville (see Ref. [51]). This simulation uses

as a basis the nonlinear propagation equation for an acoustic field inside an absorbing

medium, which is

∇2p− 1va

∂2p

∂t2+

2α

vaω2a

∂3p

∂t3+

β

ρv4a

∂2p

∂t2= 0 (4.3.8)

where p is the acoustic pressure field, va the acoustic velocity, α is the absorption coefficient,

ωa is the acoustic angular frequency, β the nonlinearity coefficient and ρ is the density of

the medium.

Eq. (4.3.8) was solved in cylindrical coordinates (r, z) on a grid with δr = δz = λa/13 and

δt = Ta/100 where Ta is the acoustic wave period. The fluid is supposed to be a homogeneous

medium which, for this particular simulation, used particles of Titanium Dioxide (TiO2)

suspended in water. The fact that the medium is considered homogeneous neglects the

influence of the suspension on the pressure field. The simulation is run for a time long

enough to reach steady state over the whole computational domain.

The pressure on the transducer surface was p = 100KPa. and the particle displacement is

shown in Fig. 4.3. The characteristics of the transducer are f = 2.4MHz, radius r ≈ 1.2 cm


0

1

2

3

4

5

x 10−8

X direction

Rad

ial d

irect

ion

Displacement in the direction of propagation

200 400 600 800 1000 1200 1400

100

200

300

400

500

600

−3

−2

−1

0

1

2

3

X direction

Rad

ial d

irect

ion

Phase of the acoustic pressure in the focus

50 100 150 200 250 300 350 400

20

40

60

80

100

Figure 4.3: Ultrasound simulation shows the displacement of the particles in the beam andphase variations in the focus.

and focal length fl ≈ 3.5 cm. The transducer surface is S = 5.25x10−4m2. Using the

formula Power = (Sp2)/(ρva) with rho = 998kg/m3 and va = 1480m/s, the acoustic

power is 3.55 watts.

The results of this acoustic simulation provide us with information about the axial and

radial displacement, along with the corresponding phase of the particles. This simulation

also provides information about the streaming of the particles due to the pressure level used

in the simulation.

4.3.3 Monte Carlo-Acoustic Simulation Ensemble

One of the objectives of this work is to combine these two numerical simulations into a single

one. For this purpose the information regarding the axial and radial particle displacement


(~shj) due to the ultrasound wave was taken into account in the weights defined for the optical

field in Eq. (4.3.6). These displacements are complex numbers with information about the

amplitude and phase of the displacement of the particles due to the acoustic field and were

implemented in the Matlab code together with the Monte Carlo simulation. Assuming that

the ultrasound beam is symmetric with respect to its propagation axis, we can consider the

radial information the same for the azimuthal angle which gives us the possibility to make

a three dimensional simulation. The displacements due to the streaming provided by the

acoustic simulation were not used considering that in our experimental setup we are using

Acrylamide gel phantoms to simulate human tissue. In these gels the particles have fixed

positions. Therefore, the particles of TiO2 are not subject to streaming. This fact will be

explained in Chapter 5.

Figure 4.4: Geometry for the simulation ensemble of multiple scattered light and ultrasound.

In order to explain in detail how the ensemble simulation takes into account the acoustic


information and the propagation of the light it is important to understand the geometry

of the problem. Fig. 4.4 shows the ultrasound beam in an arbitrary postion in the Monte

Carlo computational grid. The light source launches the photons in the direction specified

by the initial conditions stated at the end of section 4.2. The light source is positioned at

the origin of the coordinate system launching the photons in the +Z direction. In order to

simplify the computations, this simulation uses cartesian and cylindrical coordinates. This is

useful because the random steps of the photons are easily computed in cartesian coordinates

but the calculation of the acoustic interaction is less difficult if cylindrical coordinates are

used. Moreover, the computational time is reduced by combining both coordinates systems.

This simulation gives the possibility to explore the API signal for different positions of the

ultrasound beam with respect to the light source. For these simulations the light source was

positioned off-axis with respect to the ultrasound beam radiating in the positive Z direction.

The simulation considers each photon as an individual element in a row vector and the

information about its position and weight is computed for each scattering event. A three

dimensional array is defined with size equal to the size of the discretized volume that the

ultrasound simulation occupies. This array is initialized with weight 0, since there is no

API signal at the begining. Basically, each time that a photon randomly arrives to one

voxel of the computational grid, it contributes to the complex weight of that particular

voxel, giving information about the amplitude and the phase change which is stored in the

corresponding position of the three dimensional array. This volume array is shown in Fig.

4.4 where the red lines illustrate the trayectories that the photons will follow during the

simulation and the box in the middle illustrates the position of the ultrasound focus with

respect to the light source. It is important to mention that the photons were assumed to

travel in an infinite turbid media with defined optical characteristics, but only the weight

histories of the photons that interacted with the ultrasound were stored. This reduces the

amount of data and the computational time that we have to deal with.

Fig. 4.5 shows the flow diagram for the ensemble simulation. The optical parameters con-


sidered for this grid were: reduced scattering coeffcient µ′s = 10cm−1, absorption coefficient

µa = 1cm−1 and anisotropic coefficient g = 0.9. The wavelength λ of the light source is 690

nm with the power output modulated at 72.4MHz. N is the number of scattering events

and is increased until we get stable results as we update the statistical weights for the DC,

DPDW and API signals.

Figure 4.5: Flow diagram of Monte Carlo-Acoustic Simulation.

4.3.4 Discussion and Results

The results of this numerical simulations are presented in this section. Once the information

of the weights for the different sidebands is computed, the data is presented as a two

dimensional figure or as the amount of generated signal with respect to the continuous

wave value, captured by an optical detector.

For the case of the 2-D figure, depending on the position of the ultrasound, we averaged


all the non zero voxels of the three dimensional weight array in a direction perpendicular

to the ultrasound propagation. We can consider each voxel of the computational grid as a

single detector so that the 2-D figure will represent an snapshot of an array of detectors.

This shows how the diffuse wave gets modulated by the ultrasound. The amplitude of the

wave is presented decibels.

Fig. 4.6 presents the results for the very basic simulation of a plane ultrasound wave

and light in a 2x2x2 cm computational grid with voxel size of 100x100x100 µm. For this

simulation 106 photons were used. The optical properties of the tissue were µ′s = 10cm−1

and µa = 0.1cm−1. The number of photons in the Monte Carlo simulation is very small

for the quantity of photons needed to have results close to the real physical interaction,

however, this gave an insight that we were in the right track.

−20

−10

0

10

20

30

40

50

60

70

80

X direction (cm)

Y d

irect

ion

(cm

)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1−3

−2

−1

0

1

2

3

X direction (cm)

Y d

irect

ion

(cm

)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 4.6: Amplitude and phase modulation of diffuse light interacting with a plane ultra-sound wave in scattering media. The ultrasonic wavelength (≈ 640µm), is well defined andmodulates the optical path lengths.

The next step was to include the information about the particle displacement provided by

the acoustic simulation which is shown in Fig. 4.3. In order to use all the information we

included the axial as well as the radial displacement of the particles with the same optical

parameters in the medium. 4x107 photons were used. The last version of the code has a

computational time of approximatelly 30 minutes for 106 photons in a Pentium IV 1.9 GHz.


−100

−90

−80

−70

−60

−50

−40

X axis (cm)

Y a

xis

(cm

)

Magnitude and Phase of Modulated Diffuse Wave − 40 million photons

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.2

0.4

0.6

−3

−2

−1

0

1

2

3

X axis (cm)

Y a

xis

(cm

)

Phase

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.2

0.4

0.6

Figure 4.7: Amplitude and phase modulation of diffuse light interacting with a focussedultrasound wave in scattering media.

Figure 4.7 presents the results for the simulation using 4x107 photons. We have found that

as we increase the number of photons in the media the generated API signal gets weaker.

One possible explanation of this fact is that the increase in the number of photons increase

the probabilty of photons being scattered in random ways, which in terms of the radiance,

increases the amount of DC light in the system. Another possibility is that the displacement

of the particles due to the ultrasound is not big enough to mantain the signal characteristics.

The process of generating the signal by increasing the number of photons in the simulation

is shown in Figs. 4.8, 4.9, 4.10 and 4.11 where 1, 5, 10 and 20 million photons were

used respectively. A decrease in specificity of the pattern is observed but the amplitude

information shows that the optical signal is increasing in that part of the medium where the

ultrasound is located. The levels of the signal are 60 dB and below this value with reference

to the DC signal which is assumed to be 0 dB. Another characteristic of this simulation is


that the figures are showing the interaction inside the ultrasound beam.

−130

−120

−110

−100

−90

−80

−70

−60

X axis (cm)Y

axi

s (c

m)

Magnitude dB

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.2

0.4

0.6

−3

−2

−1

0

1

2

3

X axis (cm)

Y a

xis

(cm

)

Phase

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.2

0.4

0.6

Figure 4.8: Simulation with 1 million photons.

−120

−110

−100

−90

−80

−70

−60

X axis (cm)

Y a

xis

(cm

)

Magnitude dB

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.2

0.4

0.6

−3

−2

−1

0

1

2

3

X axis (cm)

Y a

xis

(cm

)

Phase

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.2

0.4

0.6


In order to get more realistic results, instead of computing and averaging the 3D com-

putational grid, we considered a semi-infinite media in which the photons traveled in the

presence of the ultrasound. We assumed a source at position 0 and a detector fiber at the

boundary with collection area 1mm2. The signal strengths were computed for the CW,

DPDW, and API signals for different distances between the source and detectors providing

similar results to those previously shown in Figs. 4.8 4.9 4.10 and 4.11. For this simulations

2x107 photons were used and the results are shown in Fig. 4.12 and Fig. 4.13.


−120

−110

−100

−90

−80

−70

−60

−50

X axis (cm)

Y a

xis

(cm

)

Magnitude dB

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.2

0.4

0.6

−3

−2

−1

0

1

2

3

X axis (cm)

Y a

xis

(cm

)

Phase

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.2

0.4

0.6


−120

−110

−100

−90

−80

−70

−60

−50

X axis (cm)

Y a

xis

(cm

)

Magnitude dB

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.2

0.4

0.6

−3

−2

−1

0

1

2

3

X axis (cm)

Y a

xis

(cm

)

Phase

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.2

0.4

0.6


Fig. 4.12 shows the signal strength of the pure optical diffuse wave with respect to the

original modulated signal. The strength of the modulated optical source was normalized

to 1 in order to show the decrease in the modulation depth on the DPDW signal as we

increase the separation the source and detector optical fibers. These results were fitted

to the analytical solution of the diffusion equation considering the appropiate boundary

conditions for a semi-infinite medium [22]. Fig. 4.13 shows the signal strength of the

acoustic generated diffuse wave (API signal) with respect to the modulated optical source.

We can see that the levels of the API signals are at least 60 dB lower than the pure optical


DPDW signals which are set to 0 dB. The values of these signals were computed using

Eq. (4.2.5). Since the model developed by Gaudette [26] does not consider a semi-infinite

medium for the solution of the diffusion equation, a decaying exponential function was fitted

to the values computed by the numerical simulation in order to show its decaying behaviour.

This exponential function has the same form as the analytical solution used for the results

on Fig. 4.12.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance between source and detector (cm)

Mod

ulat

ion

dept

h

Figure 4.12: Signal levels of the DPDW signal with respect to the CW signal.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−6

Distance between source and detector (cm)

Mod

ulat

ion

dept

h

Figure 4.13: Signal levels of the API signal with respect to the CW signal.

One of the disadvantages of the Monte Carlo method is the lack of efficiency in terms of

computational time. In order to improve this efficiency we worked in parallel with experts


in the field of Computer Engineering at Northeastern University, achieving promissing re-

sults in reducing the computational times for this simulation. The simulation times were

reduced by approximately 90 %. See the work done by Ashouei in Ref. [57]. This opti-

mization was based on the parallelization of the code originally written in Matlab. This

code was converted to C and the time-consuming functions of Matlab were re-written and

optimized for the calculations that the Monte Carlo Simulation needed. The fact that the

computational optimization was done for general purpose functions within matlab and for

algorithms commonly used in computer science makes possible to use this computational

techniques in the simulation of different physical processes. A detailed explanation can be

found in Ref. [57].

Chapter 5

Experimental Methods

This chapter presents the two types of experiments performed in this project to study

the interaction between multiple scattered light and ultrasound. In the first setup the

modulation of the speckle pattern due to the ultrasound is investigated using a continuous

wave light source. Measurements of the change of the speckle contrast with and without

the ultrasound were done. In the second set of experiments the modulation of the DPDW

was studied using a power modulated laser source and lock-in measurement techniques.

5.1 Laser Speckle Measurements

One of the most important effects when diffuse light is in the presence of an ultrasound

beam is the change of the speckle pattern [47]. One parameter that can be measured is the

speckle contrast [58] which is defined as

SC =σI

〈I〉 , (5.1.1)

61

CHAPTER 5. EXPERIMENTAL METHODS 62

where σI is the standard deviation of the speckle pattern and I is the intensity. Even though

the detection of the modulation of a single speckle is extremely difficult, the calculation of

the speckle contrast, and the decrease in its value when the ultrasound is present, shows

that this speckle pattern is being changed by acoustic field.

5.1.1 Setup

The setup for these measurements is as shown in Fig. 5.1. We are using a 633 nm Helium-

Neon Melles-Griot laser with 15 mW of optical power. The detector is an 8-bit NEC

TI-23EX CCD camera, 484x512 pixels of resolution. The ultrasound transducer is manu-

factured by NTD Systems and driven at a frequency of 2.3MHz. The diameter is 1 inch

and has a spherical focus of 1.5 inches.

Figure 5.1: Setup for Speckle Contrast measurements.

The modulation for the transducer is provided by the ENI power amplifier model 325LA.

The measured pressure response vs. the input voltage in the amplifier for the transducer

used in our experiments is presented in Fig. 5.2. The usual pressure used in these experi-


ments is 1 MPa, and the diameter of the beam focus is approximately 1.4λa. Therefore the

power per square centimeter in the focus is 67.7W/cm2, which is higher than the maximum

safety value given in Table 2.1. Once a complete understanding of this phenomena is ob-

tained, the interaction should be optimized in order to reduce these values to the diagnostic

levels.

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Pre

ssur

e (M

Pa)

Voltage ENI (Vpp)

Pressure at the transducer focus

Figure 5.2: Pressure at the focus of the ultrasound vs voltage supply.

The output of the CCD camera is connected to a Cortex-1 frame grabber manufactured

by Imagenation Inc. and redirected to a TV screen. The combination between the fo-

cussed ultrasound beam and the light takes place in an acrylamide gel phantom which has

titanium dioxide particles as scatterers. The use of these phantoms is useful because its

acoustic impedance is matched with that of the water and therefore, close to human tissue

characteristics. The ultrasound transducer and the gel phantom are immersed in water for

better impedance coupling.


pixels

pixe

ls

Ultrasound OFF

50 100 150 200 250 300 350 400 450

50

100

150

200

250

300

350

400

pixels

pixe

ls

Ultrasound ON

50 100 150 200 250 300 350 400 450

50

100

150

200

250

300

350

400

Figure 5.3: Speckle pattern with and without the prescence of the ultrasound. Notice thebluriness of the image on the right (ultrasound on) with respect to the one on the left(ultrasound off).

5.1.2 Experiments and Results

The measurements of the speckle contrast were done taking sets of 10 images with the

frame grabber and evaluating Eq. (5.1.1). We observed a decrease on the speckle contrast

when the ultrasound was present and was somewhat noticiable in the TV screen for forward

scattering and the CCD detector camera off-axis of the laser beam.

Fig. 5.3 shows the speckle pattern as was seen on the TV screen. The picture on the

left was obtained without the presence of the ultrasound while the one on the right was

obtained with the presence of the ultrasound. This data was obtained by capturing sets of

5 images for each case with the CCD camera and the frame grabber. The change on speckle

contrast (SC) calculated for this set, averaging the corresponding images, was SCon =

0.2010 and SCoff = 0.2130 where SCon stands for speckle contrast with the ultrasound on

and SCoff stands for speckle contrast with the ultrasound off. This represents a change

of approximatelly 2%, which agrees with the percentage change observed by Li et.al. [59].

The picture on the right shows that when the ultrasound is present the speckle pattern gets

blurred. This is mainly due to the change of the optical path lengths that the photons have


to travel to exit the scattering media while the ultrasound is harmonically modulating the

particle positions.

The values for similar experiments are tabulated in Table 5.1 and shown in Fig. 5.4. We

can notice from Fig. 5.4 that the amount in the change of the value of the speckle contrast

is approximately constant for all the measurements. In Fig. 5.4 the vertical axis shows the

value of the speckle contrast calculated for pairs of images with and without the presence

of the ultrasound. The experiments consisted in obtaining sets of 50 pictures. An element

of each set is a pair of images of the speckle pattern with and without the ultrasound. Fig.

5.4 show 8 sets obtained in the experiments.

In Fig. 5.4 the surface on top represents the speckle contrast measurements without the

presence of the ultrasound and the surface in the bottom with the presence of the ultrasound

for 8 sets of pictures. The values obtained in Table 5.1 were obtained averaging the 50

images obtained from the frame grabber. Also for these experiments the position of the

ultrasound was optimized to get the greater change in speckle contrast obtaining a change

up to approximately 12% in some of the cases. This shows us that there is an effective

change in the optical signal due to the ultrasound.

SC UltrasoundOFF

SC UltrasoundON

Percentage ofchange %

0.4372 0.3821 12.600.3164 0.2819 10.880.5046 0.4469 11.430.4835 0.4283 11.410.4561 0.4178 08.400.4340 0.3885 10.480.3584 0.3212 10.400.3112 0.2644 15.05

Table 5.1: Tabulated data for speckle contrast


010

2030

4050

12

34

56

78

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Number of pictures per setNumber of sets

spec

kle

cont

rast

Figure 5.4: Speckle contrast for various sets of data.

5.2 Acoustic Modulated Diffuse Photon Density Waves

On the other hand, based on the idea developed by Gaudette [26] we conducted experiments

using a modulated diode laser to create a DPDW in the acrylamide phantoms and combine

them with the ultrasound beam. This process is complicated due to the high scattering in

the medium which attenuates the virtual diffuse waves that are created from this combina-

tion. In the next sections we present a detailed description of the experimental setup and

the experiments conducted to pursue this optical phenomena.

5.2.1 Setup

The setup is shown as a block diagram in Fig. 5.5 and uses a 690nm diode laser manufac-

tured by Melles Griot with approximately 25 mW of output power. The electronic setup is

more elaborate than the one to measure the speckle contrast.

As a detector we use the H6573 modulated photomultiplier tube module manufactured


Figure 5.5: Setup for DPDW experiments.

by Hamamatsu. For this experiment we need to use three frequency generators. The first

generator at a fixed frequency of 72.3 MHz drives the modulation in the diode laser to create

the DPDW and also gets combined with a fixed frequency generator at 70 MHz, from which,

after passing through a low pass filter stage, we get the 2.3 MHz frequency needed to feed

the power amplifier for the ultrasound transducer. These two fixed frequency generators

are manufactured by Wilmanco and have a power output of +17 dBm. The third generator

is a Hewlett Packard 8647A variable signal generator which is operated at 70.025 MHz and

feeds the Minicircuits power amplifier with a gain of +33 dBm which is connected to the

photomultiplier, modulating the second dynode voltage supply. This modulation of the

photomultiplier allows us to down-convert the API signal coming out from the interaction

between the scattered light and the ultrasound which is 70 MHz. Thus, this 70 MHz is

mixed with the 70.025 MHz giving out a 25KHz signal which can be handled by the final

stages of the photomultiplier since these do not have a good frequency response. Again the


ultrasound and the light get combined in an acrylamide gel phantom.

The output of the photomultiplier is conected to a transimpedance amplifier which is

conected to a Lock-in amplifier. The reference signal for the Lock-in is obtained from

the mixing of the 70 MHz and the 70.025MHz signals.

5.2.2 Experiments and Results

Different experiments were conducted in order to obtain the desired effect of combining

DPDW and ultrasound in highly scattering media. At this stage of the research we have

not been able to obtain the desired interaction between the DPDW and the ultrasound

but we are still making efforts to optimize the setup. We have considered a number of

improvements that may have to be done.

We have to perform changes in the position of the ultrasound beam with respect to the light

source in order to optimize the position in which we would have the maxium interaction

between the two fields. Since the expected API signals are weak compared to the original

DPDW this is a disadvantage for our purposes. The Monte Carlo simulation will help us to

solve this problem since we have the flexibility to position the light source in any position

near the ultrasound beam.

The detection system, although we are using a photomultiplier tube which gives us more

gain, also gives us more noise in the low frequencies, and since we are down-converting the

signals to the KHz range, this is a disadvantage for the detection process since the API signal

that we are expecting is very small. A solution has been suggested by our collaborators

at Boston University which involves the use of a Photorefractive Crystal (PRC) which will

work as a phase conjugator of the the signal buried into the speckle pattern. We will be

able to combine the diffuse radiance coming of from the turbid media with a reference beam

inside the PRC in order to generate phase conjugate waves that will enhance the levels of

the expected signal.


On the other hand, we do not have a complete characterization of the optical properties

of the gel phantom. Although it gives us good acoustical properties that resemble those of

human tissue, we can not infere that the optical properties are optimized to match those of

biological tissue.

Chapter 6

Conclusions and Future Work

In this thesis we have presented a numerical simulation based on the Monte Carlo method

with the addition of an acoustic simulation that will help us determine the feasibility of the

interaction between diffuse photon density waves and ultrasound.

This numerical simulation is very flexible and provides us with different ways to examine

the relative positions between the ultrasound and the light source. We can also use it

considering a single detector in order to find information about the amplitude and phase

of the diffuse wave in fixed points within the medium. This simulation considers a semi-

infinite medium and the reflection of the photons in tissue air interface. However, it still has

parameters that have not been considered. We have to take into account the elimination

of photons when the statistical weights are small. This is a secondary problem since the

value of these weights approach to zero as the number of scattering events increase, but it

will help to reduce the computational time of the simulation. The results show that the

interaction between ultrasound and diffuse photon density waves, although is very weak

and around 70 dB less than the pure optical signals, is possible.

Another situation that has to be studied is what happens when the acoustic modulated

diffuse wave leaves the ultrasound beam. As we can see the interaction is small and we can

70

CHAPTER 6. CONCLUSIONS AND FUTURE WORK 71

not assure that when the diffuse wave leaves the region of interaction it is going to remain

strong enough to come out from the medium since these diffuse waves are highly damped.

Also, with the use of these simulations we need to obtain signal levels for all the optical

sidebands in order to compare the strength of the API signal with the common DPDW

and DC photon migration. We expect the API signal to be small compared to the other

two, but the numerical simulations will help us to determine if the API signal is within the

range in which we would be able to detect it or it will be buried in the optical noise and

the remaining stronger signals. Also, we can specify different modulation frequencies either

for the light source and the ultrasound in order to optimize the devices that we should use

with this technique.

Current research shows that the speckle modulation phenomena is one of the most impor-

tant characteristics of the ultrasound modulated optical tomography. The development of

mathematical models with more insight in the acoustic part has to be considered in order

to have full understanding of the medium in which these diffuse waves are going to travel

and how the medium is going to affect their behavior.

A variety of experiments have been performed with the objective of modulating the DPDW,

and even though we have not yet been able to obtain optimum results there is still work

to be done. We need to optimize the optical properties of the medium in order get our

phantoms to resemble human tissue. In parallel, the numerical simulations as well as the

mathematical models will help us to determine which way to follow in order to obtain the

maximum benefits from this imaging technique.

We can implement coherent detection systems using a chopper and averaging the signals

that contain the information of the interaction between DPDW and ultrasound. This will

help us to increase the SNR. On the other hand, our collaborators at Boston University are

working using a Photorefractive Crystal to collect the coherent information of the diffuse

wave modulated at 2.3 MHz by a pulsed focused ultrasound beam.

Appendix A

Matlab Monte Carlo-AcousticSimulation code

% Monte-Carlo program for API%% by Alex Nieva - Chuck DiMarzio,% Optical Science Laboratory% Northeastern University, August 2001%%%% Coordinate axis%% ------------------> X% |% |% |% |% |% |% Z%%%% NP is the number of photons launched in the system%%

ticglobal CHANCE CRIT_ANG

72

APPENDIX A. MATLAB MONTE CARLO-ACOUSTIC SIMULATION CODE 73

CHANCE = 0.1;CRIT_ANG = 0.8509;

rand(’state’,sum(100*clock));s=load(’disp5’);acouwave = s.dis_z; acourad = s.dis_r;clear s

% setup parameterslambda = 0.69e-4; % optical wavelength (cm)fm = 72.4e6; % modulation frequencyc = 2.99e10; % cm/sg = 0.9; % Anisotropyus = 100; % Scattering coefficient cm^-1ua = 0.1; % Absorption coefficient cm^-1ut = ua + us; % Total interaction coefficientalbedo = us/ut;alph = 2.5e-3;

% Calculated parametersf0=c/lambda; % optical carrierk0=2*pi*f0/c;fm1=f0-fm; % lower sidebandkm1=2*pi*fm1/c;fp1=f0+fm; % upper sidebandkp1=2*pi*fp1/c;

% Grid sizearraystep=5e-3; % cm (50 microns)

% Definition of volume of interaction with ultrasoundminx = 0; miny = -1.5; minz = 0;maxx = 3; maxy = 1.5; maxz = 4;

% Initial position and direction - x,y,z is position,% u,v,w direction% x(i) is x coordinate of photon number i, etc.% NP is number of photons for the simulation.NP = 100000; N = NP;x = zeros(NP,1);y = x;z = x;u = zeros(size(x));


v = zeros(size(x));w = ones(size(x));

% Initial photon weight for 9 frequency bandsthe = zeros(size(x));dpp10 = exp(i.*the);

%weightm1m1 = zeros(size(x));%weightm10 = 0.5*ones(size(x));%weightm1p1 = zeros(size(x));

%weight0m1 = zeros(size(x));weight00 = ones(size(x));%weight0p1 = zeros(size(x));

weightp1m1 = zeros(size(x));weightp10 = ones(size(x));%weightp1p1 = zeros(size(x));

weightamp = albedo; %sqrt(albedo);%factor to reduce weight as field interactionweightampdpdw = albedo/sqrt(1+alph^2);

% weightamp = albedo; factor to reduce weight as power interaction% Power --> Wnew = Wold*albedo;% Field --> Wnew = Wold*sqrt(albedo);

% Photon step size using Prahl et. al.

s = photonstep(x,ut);

% move (first step)xold = x; yold = y; zold = z;x=x+s.*u; y=y+s.*v; z=z+s.*w;

offsetx = (maxx + minx)/2;offsety = (maxy + miny)/2;offsetz = minz;

% Loop on scattering events. Scattering events last until% all the photons had gone outside of the volume of interacion or% had died due to ruse roulette.

n = 0; %counter for # of iterations


modDC = 0; modAC = 0; modAPI = 0;

for n=1:800%while N >= 1,

number_of_photons = Nn = n + 1

% New directions[unew,vnew,wnew] = deflect(g,x,u,v,w);deltak = (2*pi/lambda)*[sqrt((unew-u).^2+(vnew-v).^2),(wnew-w)];u = unew; v = vnew; w = wnew;clear unew vnew wnew

% Here we are going to find the box in which we have to% put the weights% We also find out which of the photons are inside the% area of interaction% for the pure optical signalsinx = find(x<maxx & x>minx);iny = find(y<maxy & y>miny);inz = find(z<maxz & z>minz);photonin = intersect(intersect(inx,inz),iny);

clear inx iny inz

% Here we find the photons that have reached the detector.% match is the variable that has the position in the vector of photons% of the photons that have crossed the boundary.match = find(z<0);if length(match)~=0

[a,b] = check_detector(match,s(match),u(match),v(match),...z(match),xold(match),zold(match),yold(match));

modDC = modDC + sum(weight00(a));modAC = modAC + sum(weightp10(a));

modAPI = modAPI + sum(weightp1m1(a));

weight00(a) = 0;weightp10(a) = 0;

weightp1m1(a) = 0;

[w_new,ref,trans] = fresnel(b,w(b));weight00(trans) = 0;

weightp10(trans) = 0;weightp1m1(trans) = 0;w(b) = w_new;


new_s = (s(ref).*zold(ref))./(zold(ref) - z(ref));z(ref) = w(ref).*(s(ref)-new_s);

end

clear match new_s w_new ref trans a b

posr = ceil(sqrt((x-offsetx).^2 + (y-offsety).^2)./arraystep);posz = ceil(z./arraystep);

auxr = find(posr==0); posr(auxr) = 1; clear auxrauxr = find(posr>300); posr(auxr) = 300; clear auxrauxz = find(posz==0); posz(auxz) = 1; clear auxzauxz = find(posz>300); posz(auxz) = 300; clear auxz

% acoustic interactiondeltasa = zeros(NP,2);for h = 1:length(photonin)

deltasa(photonin(h),1) = acourad(posz(photonin(h)),...posr(photonin(h)));

deltasa(photonin(h),2) = acouwave(posz(photonin(h)),...posr(photonin(h)));

endclear posr posz

% recalculate weight%dpm1m1 = exp(i*km1*s).*(-i*dot(deltak,deltasa,2));%dpm10 = exp(i*km1*s);%dpm1p1 = exp(i*km1*s).*(i*dot(deltak,deltasa,2));

%dp0m1 = exp(i*k0*s).*(-i*dot(deltak,deltasa,2));dp00 = exp(i*k0*s);%dp0p1 = exp(i*k0*s).*(i*dot(deltak,deltasa,2));

%dpp10 = exp(i*kp1*s);the = atan((sin(the) - alph.*cos(the))./...

(cos(the) + alph.*sin(the)));dpp10 = exp(i.*the);

dpp1m1 = dpp10.*(-i*dot(deltak,deltasa,2));%dpp1p1 = exp(i*kp1*s).*(i*dot(deltak,deltasa,2));

%weightm1m1 = weightm1m1 + weightm10*weightamp.*dpm1m1;%weightm1p1 = weightm1p1 + weightm10*weightamp.*dpm1p1;%weightm10 = weightm10*weightamp.*dpm10;

%weight0m1 = weight0m1 + weight00*weightamp.*dp0m1;


%weight0p1 = weight0p1 + weight00*weightamp.*dp0p1;weight00 = weight00.*weightamp;%.*dp00;

weightp1m1 = weightp1m1 + weightp10.*weightamp.*dpp1m1;%weightp1p1 = weightp1p1 + weightp10*weightamp.*dpp1p1;weightp10 = weightp10.*weightampdpdw.*dpp10;

% ruse rouletteweight00 = roulette(abs(weight00));

N = length(find(weight00));

% Calculate next step size of photonss = photonstep(x,ut);

% New position of the photonsxold = x; yold = y; zold = z;

x=x+s.*u; y=y+s.*v; z=z+s.*w;

clear r h photonin aux deltasaend;

time = tocsave(’name’,’modDC’,’modAC’,’modAPI’,’time’)clear

% Calculation of the photon step at each scattering% event%%% Usage:% s = photonstep(x,ut)% where:% x is used to let the function know the number of photons,% and ut is the total interaction coefficient ut = us + ua.%% Original code in standard C by Wang-Jacques.% "Monte Carlo Modeling of Light Transport in% Multi-layered Tissues in Standard C" 1995

function s = photonstep(x,ut)

s = -log(rand(size(x)))/ut;


return

% Calculation of the deflection angle THETA and% PSI for photon scattering, as well as the new% direction cosines.% When g = 0, we aproximate cos(theta)=2*rand -1% When g > 0, we use the Henyey-Greenstein function.%% Usage:% [unew,vnew,wnew] = deflect(g,x,u,v,w)% where:%% unew,vnew,wnew are the new direction cosines% and u,v,w are the old ones.% g is the anisotropy and we use x just to let the% function know the size of the matrix that has the% photons.%% Original code in standard C by Wang-Jacques.% "Monte Carlo Modeling of Light Transport in% Multi-layered Tissues in Standard C" 1995

function [unew,vnew,wnew] = deflect(g,x,u,v,w)

if g==0costheta = 2*rand(size(x)) - 1;

elseaux = ((1-g^2)./(1-g+2*g*rand(size(x)))).^2;costheta = (1+g^2-aux)/(2*g);

end

psi = 2*pi*rand(size(x));

sintheta = sin(acos(costheta));sinpsi = sin(psi);cospsi = cos(psi);

if min(abs(w))>0.9999 % In the future we have to consider% each photon at a time.

unew = sintheta.*cospsi;vnew = sintheta.*sinpsi;


wnew = sign(w).*costheta;else

den = sqrt(1 - w.^2);coef = ((sintheta)./den);unew = coef.*(u.*w.*cospsi-v.*sinpsi) + u.*costheta;vnew = coef.*(v.*w.*cospsi+u.*sinpsi) + v.*costheta;wnew = -sintheta.*cospsi.*den + w.*costheta;

end

return;

% Function to check if the photon has reached the% detector and in what iteration.% The output is the position in the vector of photons,% of the photons that have reached the detector.

function [a,b] = check_detector(match,s,u,v,z,xold,zold,yold)

detxmin = 1.3; detxmax = 1.4;detymin = -0.05; detymax = 0.05;radius = (detxmax - detxmin)/2;

new_s = (s.*zold)./(zold - z);xzplane = xold + u.*new_s;yzplane = yold + v.*new_s;

rzplane = sqrt((xzplane - (detxmax - detxmin)/2).^2 ...+ (yzplane - (detymax - detymin)/2).^2);

aa = find(rzplane<radius);bb = find(rzplane>=radius);a = match(aa); b=match(bb);

return

% Internal reflection of photons% using Fresnel reflection coefficient

function [w_new,ref,trans] = fresnel(b,w_new)

global CRIT_ANG


% This part finds which photons have been internally reflected% because of the critical angle.R = zeros(size(b));phi_i = acos(abs(w_new));aux_i = find(phi_i >= CRIT_ANG);R(aux_i) = 1;

aux_t = find(phi_i < CRIT_ANG);aux_phi_i = phi_i(aux_t);phi_t = asin(1.33.*sin(aux_phi_i));

aux_R = 0.5*((sin(aux_phi_i - phi_t).^2)./(sin(aux_phi_i + phi_t).^2) + ...(tan(aux_phi_i - phi_t).^2)./(tan(aux_phi_i + phi_t).^2));

R(aux_t) = aux_R;

test = rand(size(R));aux_test = find(test>R);R(aux_test) = 0; % This means that the photon escaped the tissue.R = ceil(R); % R = 1 when the photon is internally reflected.

% 0 otherwise.reflected = find(R); transmitted = find(R==0);

w_new(reflected) = -w_new(reflected);ref = b(reflected); trans = b(transmitted);return

% This is the russian roulette algorithm to kill% the photons that have a weight lower that the% pre-defined CHANCEfunction new_weight=roulette(weight)

global CHANCE

a = find(weight<0.001 & weight~=0);rand_num = rand(size(a));b = find(rand_num>CHANCE);rand_num(b) = 0;weight(a) = weight(a).*rand_num;new_weight = weight;

return

Appendix B

Transport Theory

The angular photon density was denoted with u(r, Ω, t), and the BTE written as follows:

∂u(r, Ω, t)∂t

= −v Ω · ∇u(r, Ω, t)− v(µa + µs)u(r, Ω, t)

+ vµs

∫

4πu(r, Ω′, t) f(Ω′, Ω) dΩ′ + q(r, Ω, t), (B.0.1)

The BTE can also be written in terms of the radiance L(r, Ω, t) = vu(r, Ω, t)

1v

∂L(r, Ω, t)∂t

= −∇ · L(r, Ω, t)− µtL(r, Ω, t)

+µs

∫

4πL(r, Ω′, t) f(Ω′, Ω) dΩ′ + q(r, Ω, t), (B.0.2)

where µt = µa +µs, is the total interaction coefficient or transport coefficient, and the otherquantites remain as defined before. The photon fluence rate is given by:

Φ(r, t) =∫

L(r, Ω, t) dΩ. (B.0.3)

The photon flux, or current density, is given by

J(r, t) =∫

L(r, Ω, t) Ω dΩ. (B.0.4)

81

APPENDIX B. TRANSPORT THEORY 82

The derivation of the diffusion approximation is well documented in many other works(see Ref. [5, 19]). Basically, the method used for this approximation is know as the PN

approximation [18]. This method is based in the expansion of the radiance, phase function,and source term in spherical harmonics Yl,m, which is a mathematical tool that is based onthe solutions of the general Helmholtz’s equation. For the case of human tissue, the albedo,which is defined as W = µs/(µs + µa), is close to unity, therefore the P1 approximation isquite good and this lead us to the standard diffusion equation given by

−D∇2Φ(r, t) + vµaΦ(r, t) +∂Φ(r, t)

∂t= vq0(r, t), (B.0.5)

or in terms of the photon density

−D∇2U(r, t) + vµaU(r, t) +∂U(r, t)

∂t= q0(r, t), (B.0.6)

where D = v/(3µ′s + µa) is the photon diffusion coefficient and µ′s is the reduced scatteringcoefficient defined as µ′s = µs(1 − g), which is defined because of the usefulness of thisquantity in the mathematical derivations for the diffusion approximation. The scatteringcoefficient and the scattering anisotropy do not explicitly appear in the P1 equation northe diffusion equation but instead appear together as the reduced scattering coefficient.q0(r, t) is the monopole (isotropic) moment of the source. We can also express the diffusioncoefficient as D = v/(3µ′s) considering that µ′s À µa.

The standard photon diffusion equation is obtained assuming an isotropic source which istrue also for collimated sources displaced one mean free path into the scattering media fromthe collimated source.

We also can write the photon diffusion equation (see Eq. (B.0.6)) in the frequency domainwhich takes the form of the Helmholtz equation

(∇2 + k2)U(r) =

−q0(r)D

, (B.0.7)

where the wavenumber is complex and takes the form

k2 =−vµa + iω

D. (B.0.8)

The solution for Eq. (B.0.7) in the frequency domain for a homogeneous, infinite mediumcontaining a harmonically modulated point source of power P (ω) at r = 0 is

APPENDIX B. TRANSPORT THEORY 83

U(r, ω) =P (ω)4πD

eikr

r, (B.0.9)

with the following expressions for the average photon density (UDC), and for the amplitude(UAC) and phase (φ) [5, 19]

UDC(r) =PDC

4πD

e−r(vµa/D)1/2

r, (B.0.10)

UAC(r, ω) =P (ω)4πD

e−r(vµa/D)1/2[(1+ ω2

v2µ2a

)1/2+1]1/2

r, (B.0.11)

φ(r, ω) = r(vµa/2D)1/2

[(1 +

ω2

v2µ2a

)1/2

− 1

]1/2

+ φs , (B.0.12)

Appendix C

Raman-Nath/Bragg Effect

In this section we are going to give an overview of the Raman-Nath regime and the derivationof the Raman-Nath equations, followed by the considerations that make this effect to beconsidered in the Bragg regime.

Let us represent the pressure of the sound field in two dimensions by the expression

s(x, z, t) = <[S(x, z) exp jωat] (C.0.1)

where S(x, z) is a phasor with associated frequency ωa; and the electric field of light wavewith the expression

e(x, z, t) = <[E(x, z) exp jωt] (C.0.2)

where E(x, z) is a phasor with associated frequency ω. If we consider the acoustic wavetraveling in the postive +X direction and the light wave traveling in the positive +Zdirection the following relations hold as shown in Fig. 2.3.

ka =ωa

va(C.0.3)

k =ω

c=

n0ω

cν= n0kν , (C.0.4)

where the subscript ν represent vacuum values and we assume the the medium is nonmag-netic (µν = µ0).

84

APPENDIX C. RAMAN-NATH/BRAGG EFFECT 85

Next we assume that the polarization in the medium is related to the electric field by asound-induced time-varying electric susceptibility

χ(t) = χ0 + χ1 cos (ωat− kax + φa). (C.0.5)

where χ1 is real and may be negative. The relative dielectric constant is given by

εr0 = (1 + χ0) = n20 (C.0.6)

so that

n(t)2 = [1 + χ(t)] = [1 + χ0 + χ1 cos (ωat− kax + φa)]. (C.0.7)

Assuming that |χ1| ¿ 1, we may write n(t) expanding Eq. (C.0.7) in Taylor series

n(t) ≈ n0 ± |∆n| cos (ωat− kax + φa) (C.0.8)

|∆n| =χ1

2n0. (C.0.9)

Therefore, we can define

δn(x, z, t) = |∆n| cos (ωat− kax + φa) = b(t) cos (β(t)− kax). (C.0.10)

It is important to write δn in terms of b(t) and β(t) as in Eq. (C.0.10) because the temporalvariation of b and β is assumed to be extremely slow compared to the transit time of thelight through the sound field. Therefore, the analysis is going to consider the sound fieldstationary during which b and β are constant. Also, as for the frequencies of sound andlight, it will be assumed throughout this thesis that ωa/ω ¿ 1, and that ka/k ¿ 1.

Let us assume normal incidence of the light wave on the acoustic field (z = 0). Since we areassuming that we can treat the optical field as in geometrical optics, the total phase shiftof the light ray will be given by

θ(x, L, t) = −kν

∫ L

0δn(x, z, t)dz − kL. (C.0.11)


Using Eq. (C.0.10) we find that

θ(x, L, t) = −kνLb(t) cos (β(t)− kax)− kL. (C.0.12)

Thus, the electric field at z = L is given by

E(x, L, t) = Eie−jkL e−jkνLb(t) cos (β(t)−kax). (C.0.13)

Eq. (C.0.13) represents a spatially corrugated wavefrom coming out the thin acousticfilm. Also this equation results in many sidebands due to the phase modulation, and theamplitudes are given by Bessel functions. Therefore, it is shown that

E(x, L, t) = e−jkLn=+∞∑n=−∞

(−j)nEiJn(kνLb(t)) e(jnβ(t)−jnkax) (C.0.14)

where Jn denotes the nth order Bessel function.

The physical interpretation is that the nth term causes a plane wave to propagate in thedirection kn such that knx = nka as said in the introduction to the acousto-optic effect andthe angle of propagation is that of Eq. (2.4.2). If we consider paraxial propagation thisangle may be written as

φn ≈ nka

k=

nλ

λa. (C.0.15)

Now if we use Eq. (C.0.14) and Eq. (C.0.10) we find that the nth order contribution to thetotal exit field is

En(x, L, t) = Ene−jknx−jknz+jnωat, (C.0.16)

where

knx = k sin (φn) (C.0.17a)knz = k cos (φn) (C.0.17b)En = (−j)nejnφs)Jn(ν)Ei , (C.0.17c)


where ν is called the Raman-Nath parameter and is defined as

ν = kνL|∆n|. (C.0.18)

Then we can write the complete expression for the electric field plane wave in the regionz ≥ l

en(x, z, t) = <Enej(ω+nωa)t−jkx sin (φn)−jkz cos (φn). (C.0.19)

It is seen from Eq. (C.0.19) that the nth order is shifted in frequency by nωa. This is aDoppler shift. If an observer sees the sound-induced radiating dipoles moving upward withsound velocity V , the velocity component in his direction is given by V sin (φn), hence usingEq. (2.4.2) we find for the Doppler shift

∆ω =(ω

c

)V sin (φn) = k

(ωa

ka

)sin (φn) = nωa. (C.0.20)

This mathematical approach describes Raman-Nath diffraction and a similar procedure canbe done for oblique incidence of the light wave (see Ref. [39]). For oblique incidence theangle of diffraction of the optical wave is given by

φn = φ0 +nλ

λa= φ0 +

nka

k. (C.0.21)

The considerations for the Raman-Nath regime were that the sound field is thin enoughto ignore optical diffraction effects and that the sound is weak enough to ignore opticalray-bending effects. The condition for this to occur is that

L ¿ λ2a

2πλ(C.0.22)

where L is the thickness of the acoustic beam; or using the Klein-Cook parameter Q = Lk2a/k

[39].

Q ¿ 1 (C.0.23a)Qν ¿ 1 (C.0.23b)


where ν is defined in Eq. (C.0.18).

If the interaction is long enough we can consider the sound column as an ensemble ofthin gratings. This will create a phase mismatch in the exiting optical field, preventingany cumulative contributions from neighboring orders and the net effect would be that nodiffraction occurs at all. However, there exist two conditions where there is phase matchingbetween light impinging at an angle φ0 and one neighboring order. This is when φ0 = φ1, inwhich the +1 order interacts sinchronously with the incident light and or when φ0 = φ−1,in which case, the interaction takes place with the −1 order. This is know as the Braggregime.

Using Eq. (C.0.21) we see that the first case for the order +1 is satisfied when

φ0 =−ka

2k= −φin (C.0.24)

and for the case of order −1 we have

φ0 =+ka

2k= +φin. (C.0.25)

This is shown in Fig. 2.2. For this regime the conditions that must hold are

Q À 1 (C.0.26a)Q/ν À 1. (C.0.26b)

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