Monte Carlo Simulations for the Interaction of Multiple Scattered Light and Ultrasound. A Thesis Presented. Luis A. Nieva

Transcription

1 Monte Carlo Simulations for the Interaction of Multiple Scattered Light and Ultrasound A Thesis Presented by Luis A. Nieva to The Department of Electrical and Computer Engineering in partial fulfillment of the requirements for the degree of Master of Science in Electrical and Computer Engineering in the field of Electromagnetics, Plasma and Optics Northeastern University Boston, Massachusetts January 22, 2003

2 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 phantoms made of titanium dioxide (TiO 2 ) suspended in polyacrylamide gel that is acoustically impedance matched with water.

3 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 Laboratory 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.

4 Contents 1 Introduction 9 2 Background: Light, Sound and Their Interaction Frequency Domain Biomedical Optics Diffuse Optical Tomography Biomedical Applications of Diffuse Optical Tomography Ultrasound as a Biomedical Tool Biomedical Imaging Using Ultrasound Acousto-Optic Effect Acousto-Photonic Effect Approaches to Explain the Interaction of Multiple Scattered Light and Ultrasound Mathematical Models for Acousto-Photonic Imaging Acoustic Modulation of the Diffuse Photon Density Waves

5 CONTENTS Temporal Light Correlation of Multiple Scattered Light and its Interaction with ultrasound Numerical Simulations: Monte Carlo Approach Monte Carlo Methods for Multiple Scattered Light Simulation Frequency Domain Monte Carlo Approach for Diffuse Optical Tomography Monte Carlo Simulations for Acousto-Photonic Imaging First Order Approximation of the Light-Ultrasound Weight Acoustic-Simulation Monte Carlo-Acoustic Simulation Ensemble Discussion and Results Experimental Methods Laser Speckle Measurements Setup Experiments and Results Acoustic Modulated Diffuse Photon Density Waves Setup Experiments and Results Conclusions and Future Work 70 A Matlab Monte Carlo-Acoustic Simulation code 72

6 CONTENTS 5 B Transport Theory 81 C Raman-Nath/Bragg Effect 84 References 89

7 List of Figures 2.1 Hemoglobin Absorption Acoustic Bragg diffraction Raman-Nath regime for the Acoustic diffraction of a light beam (Z) traveling through an acoustic beam (X) Representation of the interaction of light and sound in scattering media Frequency domain representation of the constitutive sidebands in the interaction between multiple scattered light and ultrasound Basic interaction of light, ultrasound and the particles in the medium Ultrasound simulation shows the displacement of the particles in the beam and phase variations in the focus Geometry for the simulation ensemble of multiple scattered light and ultrasound Flow diagram of Monte Carlo-Acoustic Simulation

8 LIST OF FIGURES 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 Amplitude and phase modulation of diffuse light interacting with a focussed ultrasound wave in scattering media Simulation with 1 million photons Simulation with 5 million photons Simulation with 10 million photons Simulation with 20 million photons Signal levels of the DPDW signal with respect to the CW signal Signal levels of the API signal with respect to the CW signal Setup for Speckle Contrast measurements Pressure at the focus of the ultrasound vs voltage supply 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) Speckle contrast for various sets of data Setup for DPDW experiments

9 List of Tables 2.1 Ultrasound intensities Tabulated data for speckle contrast

10 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 window 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

11 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 interaction 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 nonscattering 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 timedomain (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.

12 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

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

14 CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 13 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 statistical 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. Transport 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 problems including underwater visibility, marine biology, biomedical optics, and the propagation

15 CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 14 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, ˆΩ, t), which is defined as the number of photons per unit volume per unit solid angle traveling in direction ˆΩ at position r and time t, we can write the BTE as follows: u(r, ˆΩ, t) t = v ˆΩ u(r, ˆΩ, t) v(µ a + µ s )u(r, ˆΩ, t) + vµ s u(r, ˆΩ, t) f( ˆΩ, ˆΩ) d ˆΩ + q(r, ˆΩ, t), (2.2.1) 4π where v is the speed of light in the medium, µ a is the absorption coefficient in cm 1, µ s is the scattering coefficient in cm 1, f( ˆΩ, ˆΩ) is the phase function or the probability density of scattering a photon that travels along direction ˆΩ into direction ˆΩ, and q(r, ˆΩ, t) is the source term. q(r, ˆΩ, 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 ˆΩ. 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 ˆΩ due to the flow of photons. The second term is the loss of photons at position r and direction ˆΩ due to absorption and scattering. The third term is the gain of photons at r and ˆΩ due to scattering. Finally, the fourth term is the gain of photons due

16 CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 15 to the light sources. Some of the quantities used to describe photon transport are angular photon density u(r, ˆΩ, t), photon radiance L(r, ˆΩ, t) = vu(r, ˆΩ, t), photon density U(r, t) = 4π u(r, ˆΩ, t) dˆω, photon fluence rate Φ(r, t) = v U(r, t) and photon flux J(r, t) = 4π u(r, ˆΩ, t) ˆΩ dˆω. 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 µ a ), 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) e ikr 4πD r, (2.2.2) where D is the diffusion coefficient given by D = v/(3µ s + µ a ). The expressions for the average photon density (U DC ), and for the amplitude (U AC ) and phase (φ) of the diffuse photon density wave, derived from Eq. (2.2.2), are [5, 19]

17 CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 16 U DC (r) = P DC e r(vµ a/d) 1/2 4πD r U AC (r, ω) = P (ω) 4πD, (2.2.3) e r(vµa/d)1/2 [(1+ ω 2 v 2 µ 2 ) 1/2 +1] 1/2 a φ(r, ω) = r(vµ a /2D) 1/2 [ ( 1 + ω2 v 2 µ 2 a r ) 1/2 1, (2.2.4) ] 1/2 + φ s, (2.2.5) where φ s 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 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 relative 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

18 CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION Hemoglobin absorption spectra Hb HbO 2 Absorption coeffcient (µ a ) Wavelegth (nm) 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 properties are described by two parameters: the scattering coeffcient µ s 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 = µ s (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 µ a, therefore, we can assume that NIR light propagation is mainly due to scattering. This is one of the conditions 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

19 CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 18 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 inhomogeneities 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 depends 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 scattering 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

20 CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 19 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) + vµ a (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 µ a. 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. Alternatively, a perturbation expansion in µ a 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 theory, 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.

21 CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 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. Generally, 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.

22 CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 21 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

23 CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 22 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 2 ξ t 2 = v2 a 2 ξ x 2 (2.3.1) where ξ is the displacement or the particles and v a 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 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 Langévin 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

24 CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 23 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

25 CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 24 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 microbubbles. 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/cm 2. 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

26 CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 25 Instrument Type I SP T A Range (mw/cm 2 ) Static B-scan M-mode Dynamic B-scan sector 6-30 linear Doppler CW pulsed 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 ω a modulated by a frequency ω p which we would call the pump frequency. The typical equation of motion would for this type of oscillator is d 2 y dt 2 + γ dy dt + ( ω 2 a + α sin ω p t ) 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 ω p + ω a, where ω a 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

27 CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 26 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 sin φ in = p λ a 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 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

28 CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 27 Figure 2.2: Acoustic Bragg diffraction. corrugated as it leaves the sound field creating a set of upshifted and downshifted waves with frequency ω ±nω a and propagation in the direction k n such that k nx = nk a with angle of propagation φ n given by sin φ n = nk a k = nλ λ a (2.4.2) where φ n is the angle of diffraction of the light beam at the output of the acoustic column, k a is the acoustic wavenumber, k is the light wavenumber, n the order of the diffracted light beam, λ the optical wavelength and λ a the acoustic wavelength. This procces is shown in Fig Later on, a generalization of the work by Raman and Nath done by Van Cittert

29 CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 28 Figure 2.3: Raman-Nath regime for the Acoustic diffraction of a light beam (Z) traveling through 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.

30 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

31 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 Approaches to Explain the Interaction of Multiple Scattered 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 ω c with wavelength in the medical spectral range. The ultrasound wave has the general form of S(r)cos(k a r ω a t) where S(r) is the amplitude of the acoustic wave and k a 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

32 CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 31 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 + ω a and ω ω a 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 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

33 CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 32 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 µ a 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 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

34 CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 33 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 + µ a ). Considering that the size of the particles does not change considerably in the presence of the ultrasound it was assumed that N µ s = σ s V N µ a = σ a V, (3.2.1a) (3.2.1b) where σ s and σ a are the scattering and the absorption cross sections respectively, N is the number of particles, and V is the volume. Since µ s and µ a are assumed only to be a function of the volume, the change in these parameters can be expressed as follows dµ s µ s = dv V = C (3.2.2a) dµ a µ a = 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 (1 C), (3.2.3) s + µ a )

35 CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 34 assuming that C 2 1. Similarly, the decay rate is affected by this change in volume and would take the form α = (µ a + µ a C)v = α 0 + Cα 0. (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 ultrasound travels through a change in the diffusion coefficient. Assuming a small perturbation for this model, this can be described with the following parameters: Φ = Φ 0 + Φ Φ 1 D = D 0 + D 1 α = α 0 + α 1 (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+ω a and ω ω a 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 + ω a, and neglecting higher order terms we are left only with an equation in terms of the perturbation Φ + 1 and Φ 0

36 CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 35 D 0 2 Φ + 1 Φ+ 1 t α 0 Φ D 1 2 Φ 0 + D 1 Φ 0 α 1 Φ 0 = 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 D 0 2 Φ + 1 Φ+ 1 t α 0 Φ + 1 = v Q, (3.2.7) n which is the basic equation of what we call the virtual acoustic DPWD source. Since the term Q depends on D 1 and α 1, which depend on C, Q depends on the Gaussian profile of the ultrasound beam given in the factor of compression C = Ae iωat e (x 2 +y 2 ) w 0 2(1+z2 /b 2 ) e ika(x 2 +y 2 ) 2(z+b 2 /z) e (i tan 1 ( z b )) e ikaz (3.2.8) with A in Eq. (3.2.8) given by A = i 2P πw0 2(1 + z2 /b 2 )ρ 0 va 3. (3.2.9) where P is the acoustic power, ρ 0 the density of the medium, v a the velocity of sound in the medium, w 0 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].

37 CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 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 G 1 (τ) = E(t)E (t + τ) = t P (s) E s (t)e s (t + τ) ds. (3.3.1) where E s 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 ( E s (t)es (t + τ) B = exp 2τs ) (3.3.2) τ 0 l s/l E s (t)es (t + τ) U = exp i φ j (t, τ). (3.3.3) j=1 Eq. (3.3.3) is the autocorrelation function due to the ultrasonic field along a path with s/l scatterers (s l). l is the scattering mean free path of light along succesive scatterers and s is the length of the scattering path. φ j (t) is the phase variation due to the particle displacements generated by the ultrasound. The expression for this phase variation is

38 CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 37 s/l φ j (t, τ) = j=1 s/l k j ( r j+1,j (t, τ) r j+1,j (t)), (3.3.4) j=1 where r j+1,j (t, τ) is the distance between two succesive scatterers r j+1,j (t, τ) = (r j + Asin[k a r j ω a t]) (r j+1 + Asin[k a r j+1 ω a t]), (3.3.5) and k j 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 L l ( ( sinh 6 L l ) ( ) 2 t τ 0 + (k 0 A) 2 (1 cos[ω a t])α 2 ) ( t τ 0 + (k 0 A) 2 (1 cos[ω a t])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

39 CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 38 the following E s (t)e s (t + τ) U = s/l s/l+1 exp i φ j (t, τ) + φ nj (t, τ) j=1 j=1 (3.3.7) where φ nj (t, τ) = φ nj (t + τ) φ nj (t) and φ nj is the phase variation induced by the modulated index of refraction along the jth free path. This phase variation is φ nj (t) = lj 0 k 0 n(r j 1, s j, θ j, t)ds j, (3.3.8) where l j is the length of the jth free path, k 0 is the optical wavevector, n is the modulated index of refraction, r j is the location of the jth scatterer, s j is the distance along the jth free path, and θ j is the angle between the optical wave vector in the scattering event j and the acoustic wavevector k a. 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 v a by η = ( n/ p)ρva. 2 After expanding the variance of the phase variation into quadratic and cross terms he presents the following approximation i s/l+1 j=1 φ nj (t, τ) 2 (s/l + 1) (2n 0 k 0 A) 2 δ n [1 cos(ω a τ)], (3.3.9)

40 CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 39 where δ n = (α n1 +α n2 )η 2, in which α n1 and α n2 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 2 s/l i φ j (t, τ) j=1 s l (2n 0K 0 A) 2 δ d [1 cos(ω a τ)], (3.3.10) where δ d = 1/6. Adding these two contributions to the autocorrelation function G 1 (τ) Wang obtains the following expression for the autocorrelation function using the procedure as in Ref. [54]. G 1 (τ) = (L/l)sinh[{ε[1 cos(ω aτ)]} 1/2 ] sinh[(l/l){ε[1 cos(ω a τ)]} 1/2 ], (3.3.11) where ε = 6(δ n + δ d )(n 0 k 0 A) 2. Based on these results, Wang infers that an important difference between δ n and δ d is that the average sum of the cross terms of the variations due to the particle displacements φ j vanishes in the diffusion limit (s/l 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.

41 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 application 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. Finally, 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

42 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH Monte Carlo Methods for Multiple Scattered Light Simulation 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 sophistication 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 timeresolved 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

43 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 42 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 estimates the quantities relevant to frequency-domain measurements. The propagation of the photons having a complex weight is simulated in CW-regime and the resulting modulation 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

44 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 43 1 L(r, ˆΩ, t) v t = L(r, ˆΩ, t) µ t L(r, ˆΩ, t) +µ s L(r, ˆΩ, t) f( ˆΩ, ˆΩ) d ˆΩ + q(r, ˆΩ, t) (4.2.1) 4π where the harmonically modulated source q(r, ˆΩ, t) is given by the expression q(r, ˆΩ, t) = q DC (r, ˆΩ) [1 + R(m 0 exp (iω m t))], (4.2.2) in which m 0 is the incident modulation depth, and ω m is the light source modulation angular frequency (ω m = 2πf m where f m is the modulation frequency). Due to the linearity of Eq. (4.2.1) with respect to the radiance L(r, ˆΩ, 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, ˆΩ, t) = L DC (r, ˆΩ) + R(L AC (r, ˆΩ, ω) exp (iω m t)). (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 L AC over a certain detector area D. This functional will have the form J AC (ω) = D L AC (r, ˆΩ, ω) dr dω, (4.2.4)

45 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 44 Also we can define a functional for L DC 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 J DC = 1 N J AC = 1 N N WLg h h, (4.2.5) h=1 N ZLg h h, (4.2.6) h=1 where h represents the h-th photon in the system. W h L is the weight for the DC migration of photons and Z h L is the complex weight for the AC migration of photons which has the form Z h L = mh L exp(iϕh L ), where mh L is the intensity modulation factor of the DPDW and ϕ h L is the initial phase of the DPDW. g h 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, W0 h = mh 0 = 1 and ϕh 0 = 0. During the simulation the values of the weights are updated at each scattering event by the following relations: W h l = c W h l 1 (4.2.7) m h l = c m h l 1 (4.2.8) where c = µ s /(µ s + µ a ) is the albedo which is approximately constant for the medium. The phase ϕ h l depends on the travel direction of the photon at each scattering event and

46 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 45 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 Fig. 4.1 shows the distribution of optical signals from a frequency domain perspective. We find the optical carrier band ω c 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 100 MHz range we get what is known as a DPDW, with frequency ω c ± ω m, 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 ω c ± ω a and ω c ± ω m ± ω a. 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

47 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 46 Figure 4.1: Frequency domain representation of the constitutive sidebands in the interaction between multiple scattered light and ultrasound. as well. The statistical weight for the API interaction is defined as follows: W nhj = J j=1 ( [A hj + a hj ] exp ik n S hj + i s hj ) k nhj e iωat + i s hj k nhj e iω at (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. W nhj 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 k nhj is defined as the vectorial difference between the wave vector

48 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 47 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 k nhj and the acoustic wave displacement s hj 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 k in and after it scatters is going to come out at a direction k out. The vectorial difference between the wavevectors in these directions gives us k nhj, 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 First Order Approximation of the Light-Ultrasound Weight The phase change due to the ultrasound that produces the API signal is defined as φ nhj = s hj k nhj. 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:

49 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 48 J W nhj [A hj + a hj ] e ik ( ns hj 1 + ishj k nhj e iω at ) ( 1 + is hj k nhje iω at ) (4.3.2) j=1 In order to find a more useful expression for Eq we are going to make use of the following mathematical derivation which has the same structure as the expression for the statistical weights: [X j + x j ] = [X 1 + x 1 ] [X 2 + x 2 ] [X 3 + x 3 ]... j X 1 X 2 X 3 + [X 2 X 3... ] x 1 + [X 1 X 3... ] x 2... = J [X j ] 1 + x j /X j. (4.3.3) j j=1 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 contributions. Reordering the expression in Eq we can express the weight with the form of Eq. (4.3.3) as: W nhj J j=1 1 + A hj 1 + J J a hj /A hj exp (ik n S hj ) j=1 j=1 J ( ishj k nhj e iωat + is hj k nhje iωat) (4.3.4) j=1

50 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 49 W nhj J J [A hj exp (ik n S hj )] 1 + a hj /A hj j=1 1 + j=1 J ( ishj k nhj e iωat + is hj k nhje iωat) (4.3.5) j=1 Assuming that the optical amplitudes are greater that the acoustic amplitudes (A hj a hj ) [ we can drop the term 1 + ] J j=1 a hj/a hj since it is approximately 1. Then, the final expression for the complex weights that include the API interaction would be: W nhj J [A hj exp (ik n S hj )] j=1 1 + J ( ishj k nhj e iωat + is hj k nhje iω at ) (4.3.6) j=1 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 s hj = s a exp (ik a r), (4.3.7) where s a 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 f a = 2.4 MHz, and the velocity of the ultrasound in tissue was considered v a = 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 finitedifference time-domain acoustic simulation was implemented which gave us information of

51 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 50 the displacement of a focussed ultrasound wave. The details of this simulation will be reviewed in the next section Acoustic-Simulation The initial Monte Carlo simulations considered a plane acoustic wave to make the assumptions 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 2 p 1 v a 2 p t 2 + 2α v a ω 2 a 3 p t 3 + β ρv 4 a 2 p t 2 = 0 (4.3.8) where p is the acoustic pressure field, v a the acoustic velocity, α is the absorption coefficient, ω a 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 = λ a /13 and δt = T a /100 where T a is the acoustic wave period. The fluid is supposed to be a homogeneous medium which, for this particular simulation, used particles of Titanium Dioxide (T io 2 ) 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 = 100KP a. and the particle displacement is shown in Fig The characteristics of the transducer are f = 2.4 MHz, radius r 1.2 cm

52 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 51 Radial direction Displacement in the direction of propagation X direction x Radial direction Phase of the acoustic pressure in the focus X direction Figure 4.3: Ultrasound simulation shows the displacement of the particles in the beam and phase variations in the focus. and focal length fl 3.5 cm. The transducer surface is S = 5.25x10 4 m 2. Using the formula P ower = (Sp 2 )/(ρv a ) with rho = 998kg/m 3 and v a = 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 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

53 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 52 ( s hj ) 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 TiO 2 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

54 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 53 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-

55 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 54 sidered for this grid were: reduced scattering coeffcient µ s = 10cm 1, absorption coefficient µ a = 1cm 1 and anisotropic coefficient g = 0.9. The wavelength λ of the light source is 690 nm with the power output modulated at 72.4 MHz. 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 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

56 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 55 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 10 6 photons were used. The optical properties of the tissue were µ s = 10cm 1 and µ a = 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 Y direction (cm) Y direction (cm) X direction (cm) X direction (cm) 3 Figure 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. The next step was to include the information about the particle displacement provided by the acoustic simulation which is shown in Fig 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. 4x10 7 photons were used. The last version of the code has a computational time of approximatelly 30 minutes for 10 6 photons in a Pentium IV 1.9 GHz.

57 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 56 Y axis (cm) Magnitude and Phase of Modulated Diffuse Wave 40 million photons X axis (cm) Y axis (cm) Phase X axis (cm) 2 3 Figure 4.7: Amplitude and phase modulation of diffuse light interacting with a focussed ultrasound wave in scattering media. Figure 4.7 presents the results for the simulation using 4x10 7 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

58 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 57 that the figures are showing the interaction inside the ultrasound beam. Y axis (cm) Magnitude db X axis (cm) Y axis (cm) Phase X axis (cm) 2 3 Figure 4.8: Simulation with 1 million photons. Y axis (cm) Magnitude db X axis (cm) Y axis (cm) Phase X axis (cm) 2 3 Figure 4.9: Simulation with 5 million photons. In order to get more realistic results, instead of computing and averaging the 3D computational 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 1mm 2. 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 and For this simulations 2x10 7 photons were used and the results are shown in Fig and Fig

59 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 58 Y axis (cm) Magnitude db X axis (cm) Y axis (cm) Phase X axis (cm) 2 3 Figure 4.10: Simulation with 10 million photons. Y axis (cm) Magnitude db X axis (cm) Y axis (cm) Phase X axis (cm) 2 3 Figure 4.11: Simulation with 20 million photons. Fig 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 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

60 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 59 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 Modulation depth Distance between source and detector (cm) Figure 4.12: Signal levels of the DPDW signal with respect to the CW signal. 1 x Modulation depth Distance between source and detector (cm) 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

61 CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 60 in the field of Computer Engineering at Northeastern University, achieving promissing results 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 optimization 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].

62 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 = σ I I, (5.1.1) 61

63 CHAPTER 5. EXPERIMENTAL METHODS 62 where σ I 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 Setup The setup for these measurements is as shown in Fig 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 manufactured by NTD Systems and driven at a frequency of 2.3 MHz. 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 The usual pressure used in these experi-

64 CHAPTER 5. EXPERIMENTAL METHODS 63 ments is 1 MPa, and the diameter of the beam focus is approximately 1.4λ a. Therefore the power per square centimeter in the focus is 67.7W/cm 2, which is higher than the maximum safety value given in Table 2.1. Once a complete understanding of this phenomena is obtained, the interaction should be optimized in order to reduce these values to the diagnostic levels. 4.5 Pressure at the transducer focus Pressure (MPa) Voltage ENI (Vpp) 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 focussed 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.

65 CHAPTER 5. EXPERIMENTAL METHODS 64 Ultrasound OFF Ultrasound ON pixels pixels pixels pixels Figure 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) 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 SC on = and SC off = where SC on stands for speckle contrast with the ultrasound on and SC off 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