Determination of the dynamical properties in turbid media using diffuse correlation spectroscopy : applications to biological tissue = Determinação das propriedades dinâmicas em meios turvos usando espectroscopia de correlação de difusão: aplicações ao te

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Rodrigo Menezes Forti

Determination of the dynamical properties in

turbid media using diffuse correlation

spectroscopy: applications to biological tissues

Determina¸c˜

ao das propriedades dinˆ

amicas de meios turvos

usando espectroscopia de correla¸c˜

ao de difus˜

ao: aplica¸c˜

oes ao

tecido biol´

ogico

Campinas

June 2015

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Universidade Estadual de Campinas

Instituto de F´ısica “Gleb Wataghin”

Rodrigo Menezes Forti

Determination of the dynamical properties in

turbid media using diffuse correlation

spectroscopy: applications to biological tissues

Determina¸c˜

ao das propriedades dinˆ

amicas de meios turvos usando

espectroscopia de correla¸c˜

ao de difus˜

ao: aplica¸c˜

oes ao tecido biol´

ogico

Disserta¸c˜ao apresentada ao Instituo de F´ısica “Gleb Wataghin” da Universidade Estadual de Campinas, para a obten¸c˜ao do T´ıtulo de Mestre em F´ısica.

Orientador: Prof. Dr. Rickson Coelho Mesquita

Este exemplar corresponde à versão final da disserta¸cão defendida pelo aluno, e orientada pelo Prof. Dr. Rickson Coelho Mesquita

Assinatura do Orientador

Campinas June 2015

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Biblioteca do Instituto de Física Gleb Wataghin Valkíria Succi Vicente - CRB 8/5398

Forti, Rodrigo Menezes,

F776d ForDetermination of the dynamical properties in turbid media using diffuse

correlation spectroscopy : applications to biological tissue / Rodrigo Menezes Forti. – Campinas, SP : [s.n.], 2015.

ForOrientador: Rickson Coelho Mesquita.

ForDissertação (mestrado) – Universidade Estadual de Campinas, Instituto de

Física Gleb Wataghin.

For1. Neuroimagem. 2. Espectroscopia de correlação de difusão. 3.

Neurociências. 4. Espectroscopia ótica de difusão. 5. Fluxo sanguíneo. I.

Mesquita, Rickson Coelho,1982-. II. Universidade Estadual de Campinas. Instituto de Física Gleb Wataghin. III. Título.

Informações para Biblioteca Digital

Título em outro idioma: Determinação das propriedades dinâmicas em meios turvos usando espectroscopia de correlação de difusão : aplicações ao tecido biológico

Palavras-chave em inglês: Neuroimaging

Diffuse correlation spectroscopy Neurosciences

Diffuse optical spectroscopy Blood flow

Área de concentração: Física Titulação: Mestre em Física Banca examinadora:

Rickson Coelho Mesquita [Orientador] José Antonio Brum

Fernando Fernandes Paiva Data de defesa: 06-04-2015

Programa de Pós-Graduação: Física

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Abstract

Determination of the dynamical properties in turbid media using diffuse correlation spectroscopy: applications to biological tissues

Rodrigo Menezes Forti Rickson Coelho Mesquita

Spectroscopic techniques based on diffuse optics are essential for determination of the optical and dynamical properties of turbid media, in which scattering predominates over absorption. Under these conditions, light propagates spherically in the medium, in an approximate diffusive regimen. Scattered light can thus be detected on the same plane of incidence, and its detection can provide information both on the optical and dynamical properties of the medium. Diffuse optical techniques are particularly useful to study the properties of biological tissue, since it behaves like a turbid medium in the near infrared region. Because diffuse optics is a relatively novel experimental technique, not much is known regarding the propagation of light in media with different geometries, particularly with relation to the dynamical properties of the medium. This project proposes a combined theoretical and experimental study of light propagation in semi-infinite and two-layered turbid media, focusing on the dynamical properties of the medium with a diffuse optical technique called diffuse correlation spectroscopy (DCS). More specifically, this project employed the semi-infinite and the two-layer geometries, testing them using Monte Carlo simulations and controlled enviroments. It was shown that by using a two-layer geometry, instead of the semi-infinite geometry, as routinely done in the literature, it is possible to significantly improve the accuracy of the recovered dynamical properties. The geometries tested in this work represent more accurate approximations for muscle and brain structures, for example, and therefore could depict different situations encountered in problems in the fields of Biology and Medicine. Last, the system was also tested in healthy subjects. The results obtained in this project have direct application in the above-cited fields, and may significantly contribute to the development of experimental techniques for diagnosis and/or monitoring of the brain and muscle in the clinic.

Keywords: Neuroimage, DCS, Neuroscience, Diffuse Optics, Blood Flow

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Resumo

Determina¸cão das propriedades dinâmicas de meios turvos usando espectroscopia de correla¸cão de difusão: aplica¸cões ao tecido biológico

Rodrigo Menezes Forti Rickson Coelho Mesquita

Técnicas de espectroscopia baseadas em óptica de difusão são essenciais para a obten¸cão das propriedades ópticas e dinâmicas em meios turvos, caracterizados pela predominância dos efeitos de espalhamento sobre a absor¸cão. Nestas condi¸cões, a luz se propaga esferi-camente no meio, num regime aproximadamente difusivo. A luz espa- lhada pode então ser detectada no mesmo plano de incidência, e sua deteçcão fornece informa¸cão das pro-priedades ópticas e dinâmicas das moléculas que compõem o meio. Em particular, a técnica encontra uma vasta aplica¸cão no estudo das propriedades do tecido biológico, uma vez que este se comporta como um meio turvo na região do infravermelho próximo. Por se tratar de uma técnica experimental relativamente recente, pouco é conhecido em rela¸cão `

a propaga¸cão da luz em meios com diferentes geometrias, principalmente em rela¸cão às propriedades dinâmicas do meio. Este projeto propôs um estudo teórico-experimental de-talhado da propaga¸cão da luz em meios turvos semi-infinitos e de duas camadas, com foco na obten¸cão das propriedades dinâmicas do meio, através de uma técnica óptica de difusão conhecida como espectroscopia de correla¸cão de difusão (DCS). Mais especificamente, esse projeto testou as geometrias de um meio semi-infinito e de duas camadas, com o uso de simula¸cões de Monte Carlo e experimentos em ambientes controlados. Foi mostrado que o uso da geometria de duas camadas, ao invés da de um meio semi-infinito, como é usualmente feito na literatura, traz melhoras significativas para a recupera¸cão das propriedades de fluxo do meio. As geometrias usadas neste trabalho representam aproxima¸cões mais precisas das estruturas muscular e cerebral, por exemplo, e retratam diferentes situa¸cões encontradas em Biologia e Medicina. Por fim, o sistema também foi testado em voluntários sadios. Os resultados obtidos neste projeto tem aplica¸cão direta nas áreas citadas, e podem contribuir significativamente para o desenvolvimento de técnicas f´ısicas para o monitoramento cerebral e muscular na cl´ınica médica.

Palavras-Chave: Neuroimagem, DCS, Neurociência, Óptica de Difusão, Fluxo San-gu´ıneo

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Posso dizer que os últimos dois anos foram os anos mais produtivos da minha vida, tanto no aspecto pessoal como no aspecto profissional. Durante meu mestrado, através de diversas discussões passando desde teoria até discussões sobre pol´ıtica, pude melhorar sig-nificativamente minha capacidade de argumenta¸cão e pensamento cr´ıtico. Diversas pessoas participaram desse processo de aprendizado e amadurecimento (pessoal e profissional), é dif´ıcil agradecer a todos de uma forma justa.

Primeiramente, quero agradecer ao meu orientador, o Prof. Dr. Rickson Coelho Mesquita, que teve papel essencial nesse processo de amadurecimento, durante os últimos quatros anos de trabalho (não somente durante o mestrado). Sem a sua paciência em apon-tar e sugerir solu¸cões para meus erros, eu não seria metade do pesquisador que sou hoje. Com a sua ajuda pude amadurecer minha percep¸cão sobre processos f´ısicos (indo além das técnicas de óptica de difusão) e, pude melhorar significativamente minha capacidade de reda¸cão de textos cient´ıficos, o que permitiu a elabora¸cão desta disserta¸cão. Por tudo, meus mais sinceros agradecimentos.

Gostaria também de agradecer a todos os amigos com quem convivi nestes dois anos, desde os que conheci neste per´ıodo e os que já conhe¸co de longa data. Em espec´ıfico gostaria de agradecer a todos os que convivi dentro do espa¸co do Centro Acadêmico da F´ısica e da Rep T51. Gostaria principalmente de agradecer pelas infinitas discussões sobre religião, ciência e pol´ıtica, sem as quais não teria sido poss´ıvel evoluir meu poder de argumenta¸cão. ´

E imposs´ıvel não lembrar de todos os amigos do Laor e do Habonim Dror, que embora tenha convivido menos durante os últimos dois anos, tiveram uma influência direta na forma¸cão da pessoa que sou hoje.

Não posso deixar de agradecer às diversas pessoas que ajudaram mais ativamente no desenvolvimento deste projeto, ajuda sem a qual nada disso seria poss´ıvel. Devo agradecer a todos os membros do Laboratório de Óptica Biomédica (LOB), todos vocês ajudaram de alguma forma no projeto. Mais em espec´ıfico, gostaria de agradecer ao Renato, por toda a ajuda com o desenvolvimento e caracteriza¸cão do sistema de DCS; ao Edwin, pela ajuda com as simula¸cões de Monte Carlo e; ao Sérgio, pela ajuda na reda¸cão da disserta¸cão. Gostaria também de agradecer dois colegas de trabalho que não fazem parte do LOB: o Luis Fernando, pela a ajuda com a parte eletrônica do sistema; e o Julien, por todos os conselhos e dicas no processo de constru¸cão do sistema e no processo de aprendizado da técnica de DCS.

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Devo também agradecer à UNICAMP, mais especificamente ao Departamento de Raios Cósmicos e Cosmologia do Instituo de F´ısica “Gleb Wataghin”, por fornecer toda a infra-estrutura necessária para a realiza¸cão do projeto. Por fim, gostaria de agradecer à FAPESP pelo apoio financeiro, sem o qual este projeto não poderia ter sido realizado.

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List of Figures

2.1 Representation of the absorption of a photon by a molecule. The incident photon has an energy matching the energy of one the excited levels of the molecule. . . 6 2.2 Representation of the diffusive light propagation inside a turbid medium. . 6 2.3 Absorption coefficients of the main components of human blood. Data

avail-able from [50, 51]. Here H2O, HbO and Hb are represented in black, red and

blue, respectively. . . 7 2.4 Representation of the relevant directions and coordinates of the RTE. . . . 8 2.5 Representative siganls from the three different DOS techniques. (A)

Time-Domain (B) Frequency-Time-Domain (C) Continuous wave. . . 10 2.6 Representation of a semi-infinite medium. Here we used the method of images

to solve the diffusion equation, as described in Appendix A. . . 11 2.7 Pictorial representation of an experiment in the single scattering limit. The

particles movement will induce a change in the measured intensity. . . 15 2.8 Pictorial representation of an experiment in the multiple scattering regimen.

The movement of the scatterers will induce a change in the measured intensity. 16 2.9 Schematics of a common DCS experiment. . . 23 2.10 Comparison of the fitting of human data using both the Brownian motion

and random flow from (2.40). . . 24 3.1 Simplified schematics of a DCS experimental setup. We shine coherent light

into a medium, count the number of scattered photons some distance away from the point of illumination and read the auto-correlation function com-puted by a correlator board directly in a computer. . . 27 3.2 Setup used to measure the laser stability. . . 28 3.3 Measured output intensity of the laser over a 21-hour period. . . 29 3.4 Photon detection efficiency of the employed SPCM. This graph was taken

from the module’s data-sheet. . . 30 3.5 Picture of the output TTL of the detector. . . 32 3.6 Schematic representation of the hierarchy of the registers and levels from the

correlator board. . . 33 3.7 Pictorial example of a correlation board operation. Here we represent a

correlator board with 2 registers on the first level and one register on the subsequent levels. . . 34

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3.8 Front panel of the developed VI for data control and acquisition of the DCS system. . . 35 3.9 Picture of the built DCS system. . . 36 3.10 Schematics of the power connections of the DCS system. . . 37 3.11 Characterization results obtained for the (A) mean intensity, (B) β and (C)

BFI obtained across three different source-detector separations over a 2.5-hour period. . . 38 4.1 General diagram of the optimization routine used. . . 40 4.2 Representation of the simulated MC data geometry for the SI medium. Here,

ρ is the source-detector separation and (µa, µ0s, DB) are the optical and flow

coefficients simulated. . . 41 4.3 Sensibility of the algorithm to different initial conditions x0. Here we varied

DB0from 10−9 to 10−7cm−2, while keeping β0 = 0.5. . . 42

4.4 Error induced to the recovered rBF due to errors on either (A) µa or (B) µ0s. 42

4.5 Representation of the simulated MC data geometry for the 2L medium. ρ is the source-detector separation, (µai,µ0si, DBi) are the optical and flow

coeffi-cients of the ith layer and ` is the first layer thickness simulated. . . 43 4.6 Sensibility of the algorithm to the chosen initial conditions x0. Here we kept

β0 = 0.5 and DB10 = 1 × 10

−8 _{while varied D}0

B2 from 1 × 10

−6 _{to 3 × 10}−6_.

(A) ` = 8mm and (B) ` = 17mm. . . 44 4.7 Comparison of the recovered rBF from a two-layered media with (A) ` = 8

mm and (B) ` = 17 mm, using both the SI and the 2L solution. . . 45 4.8 Induced errors to the recovered rBF in a two-layered medium with ` = 8mm

due to errors on the optical coefficients of both layers. . . 46 4.9 Induced errors to the recovered rBF in a two-layered medium with ` = 17mm

due to errors on the optical coefficients of both layers. . . 46 5.1 Picture of the constructed liquid phantom. The phantom has dimensions of

250 × 250 × 150 mm and can mimic both SI and 2L media. . . 48 5.2 Picture of the experimental setup used in the phantom experiments. Here

we show the peristaltic pump and its pipe placement. . . 49 5.3 Comparison of the auto-correlation curve computed without the plastic film

separation (SI), and with the separation at ` = 8mm and ` = 17mm. . . 50 5.4 Recovered rBF from the SI phantom. (A) ρ = 2.0 (B) ρ = 2.5 cm (C) ρ = 3.0

cm. . . 50 5.5 Comparison of the real flow changes induced by the pump and the recovered

rBF from DCS, using the 2L and SI models. Here ` = 8 mm and (A) ρ = 2.0 (B) ρ = 2.5 cm (C) ρ = 3.0 cm. . . 52 5.6 Comparison of the real flow changes induced by the pump and the recovered

rBF from DCS, using the 2L and SI models. Here ` = 17 mm and (A) ρ = 2.0 (B) ρ = 2.5 cm (C) ρ = 3.0 cm. . . 53 5.7 Experimental setup of the arm occlusion test performed. . . 54 5.8 Results of the arm occlusion test performed. . . 54

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List of Figures xix

5.9 Results of the apnea test performed. Here we show (A) the rBF recovered from first layer using ρ = 1.0 cm, and (B) the rBF recovered from the second layer using ρ = 2.5 cm. . . 55 A.1 Representation of a semi-infinite medium, together with the representation

of the method of images to be used. . . 64 A.2 Correlation functions for a semi-infinite medium. In each graphic we varied a

different parameter, while keep the others constant at µa= 0.1 cm−1, µ0s = 11

cm−1, DB= 1 × 10−8 cm2/s and ρ = 2 cm. The different graphics represent

the changes induced by the variations of: (A) µa; (B) µ0s; (C) DB and; (D)

ρ. Here µa, µ0s and DB are the absorption, scattering and the Brownian

diffusion coefficients, and ρ is the source-detector distance. . . 66 A.3 Representation of a two-layered medium, using the extrapolated boundary

condition. Here ρ =px2_{+ y}2 _{is the planar distance. . . .} ₆₇

A.4 Correlation functions for a two-layered medium. In each graphic we varied a different parameter, while keep the others constant. The different graphics represent the changes induced by the variations of: (A) µa1; (B) µ0s1; (C)

DB1; (D) µa2; (E) µ0s2 and; (F) DB2. Here we are using ` = 8 mm and ρ = 2

cm. . . 69 A.5 Correlation function for a two-layered media with varying source-detector

separation (ρ). . . 70 A.6 Correlation functions with varying DB2/DB1, for two-layered media with (A)

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List of Tables

2.1 Comparison of some in vivo DCS validation studies. ASL-MRI: arterial spin-labelled magnetic ressonance imaging; PDT: photodynamic thepary. Table was extracted from [41]. . . 26 3.1 Power requirements of the employed SPCM, as described in the detector’s

data-sheet. . . 31 3.2 Power supplies acquired. All three supplies were custom-made. . . 31 4.1 Coefficients used in the MC simulation for the semi-infinite simulations. . . 41 4.2 Coefficients used in the MC simulation for the two-layer simulations. . . 44 5.1 Reference values for the extinction coefficients of Intralipid and India ink.

The data on the first three lines were extracted from [82], while data on the fourth line was found using a linear regression of the other lines. . . 48 5.2 Comparison of the results from the two-layered phantom (` = 8mm) using

both the SI and the 2L solutions to fit the data. . . 52 5.3 Comparison of the results from the two-layered phantom (` = 17mm) using

both the SI and the 2L solutions to fit the data. . . 53 5.4 Mean value of the averaged rBF (Figure 5.9) measured during the apnea task,

for each block, using both the SI and 2L models. The values in parenthesis are the maximum change for each trial, respectively. . . 55

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Abbreviations

2L Two - Layer

ASL Arterial Spin Labelled

CTE Correlation Transfer Equation

CW-DOS Continuous Wave - Diffuse Optical Spectroscopy DCS Diffuse Correlation Spectroscopy

DOS Diffuse Optical Spectroscopy DWS Diffuse Wave Spectroscopy

FD-DOS Frequency Domain - Diffuse Optical Spectroscopy fMRI f unctional Magnetic Ressonance Imaging

HbO Oxy Hemoglobin

Hb Deoxy Hemoglobin

LVTTL Low Voltage Transistor-Transistor Logic

MC Monte Carlo

NIR Near Infrared

NIRS Near Infrared Spectroscopy

PDT Photo Dynamic Therapy

PET Positron Emission Tomography RTE Radiative Transfer Equation

SI Semi Infinite

SPCM Single Photon Counting Module TCD Transcranial Doppler Ultrasound

TD-DOS Time-Domain - Diffuse Optical Spectroscopy TEC Thermoelectric Cooling

TTL Transistor Transistor Logic

VI Virtual Instrument

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Chapter 1

Introduction

In the clinic, there is a growing need for noninvasive measurements of oxygenation and blood flow in human tissue, as they are useful parameters to characterize tissue health. To date, there are few clinical techniques that can quantify tissue hemodynamics: 1) transcranial Doppler ultrasound (TCD) [1–3], which provides measurement of flow within large vessels; 2) functional magnetic resonance imaging (fMRI) [4, 5], which provides a signal proportional to the blood oxygenation (BOLD) with good spatial resolution (on the order of millimeters). The caveat of fMRI is due to the fact that the origin of the BOLD signal is not completely clear [6–8]; 3) Arterial spin labeling (ASL) perfusion MRI [9–14] measures blood flow with a good spatial resolution, but achieving a good signal-to-noise ratio is challenging, depending on the acquisition time [15, 16]; 4) positron emission tomography (PET) [17–19] measures tissue oxygenation and blood flow with moderate spatial resolution, but it requires the injection of a radioactive contrast. All the aforementioned techniques are too expensive, have low temporal resolution, require the relocation of the patient and/or are somewhat challenging to physically interpret their signal.

Another promising possibility to measure both blood oxygenation and blood flow changes is diffuse optical techniques: Diffuse Optical Spectroscopy (DOS) and Diffuse Corre-lation Spectroscopy (DCS). Briefly, by shining low-power near infrared light (NIR, 650−900 nm), it is possible to measure absorption and scattering coefficients of tissue (µa and µ0s,

respectively), recovering information about tissue oxygenation. It is also possible to mea-sure the mean square displacement of the particles in the medium, bringing information related to blood flow. The main advantages of diffuse optical techniques are: noninvasive-ness; moderate temporal resolution (ranging from several milliseconds to a few seconds); portability, i.e., they do not require the relocation of patients; non-ionizing radiation and;

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assessment of deeper regions of tissue (∼ 2 − 3 cm from the surface). In this dissertation, we will describe the physical and mathematical processes behind DOS and DCS, focusing on the DCS experimental methods.

When working with DOS, one measures the slow variations of light intensity emerging from tissue. As will be seen in Chapter 2, in human tissue light is propagated spherically outward from the point of illumination, carrying information about the absorption and scattering properties of the tissue. From the absorption and scattering properties, we will see that it is possible to recover the oxy- and deoxy-hemoglobin concentrations, obtaining information about tissue oxygenation (Section 2.2). DOS has been widely used to probe different types of human tissue [20–34]. However, one drawback of DOS is the fact that oxygenation changes are relatively small (on the order of 5 − 10%), making the technique not very sensitive if used as an indicator of tissue health.

On the other hand, larger changes of blood flow are expected on human tissue (∼ 30 − 40%), making it a better parameter to characterize biological tissue. DCS measures the mean square displacement of the scattering particles by assessing scattered light fluc-tuations. More specifically, DCS works with the auto-correlation function of the scattered light, from where it recovers information about blood flow changes in tissue (Section 2.3). Although DCS is relatively new, few works have been reported about DCS [22, 23, 33–44], suggesting that it is a promising blood flow measurement tool.

However, most of the published work on DCS focus on the limitations underlying the technique. Therefore, we feel that there is need of more careful validation studies that aim to quantify the actual advantages and disadvantages of DCS as a blood flow monitor in the clinic. For example, in most of the current literature, it is supposed that the human tissue is semi-infinite, which is not true, specifically for the brain. A more realistic way to model human tissue is to use a layered geometries [45, 46]. In the two-layer, for example, when working with human brain, the first layer could represent everything between the scalp and the cortex, and the second layer could represent the cortex.

Given all the above, the main goals of this dissertation are to:

• Bring a better understanding of the transport of light (and correlation) inside turbid media (i.e., media that predominantly scatters light);

• Construct and characterize a DCS system, since DCS is still a non-commercial tech-nique. In fact, we have built the first Brazilian DCS system;

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Chapter 1. Introduction 3

• Validate DCS on controlled enviroments, using a liquid phantom and;

• Propose new analysis methods to treat DCS data. For example, we propose that modeling human tissue as a two-layered medium, instead of a semi-infinite, brings significant improvements.

The most relevant contribution of this work to the literature is the use of more detailed and careful procedures to validate the semi-infinite and two-layered solutions of DCS on phantoms.

On Chapter 2, we will describe the mathematical and physical processes behind diffuse optical techniques. We will start by defining the main light-matter interactions considered on Section 2.1. On Sections 2.2 and 2.3 we describe the theory behind DOS and DCS, respectively. The operation of each component of a DCS system and its construction and characterization are described on Chapter 3. The validation work conducted, using Monte-Carlo simulations, controlled enviroments and a human subject experiment is explained in Chapters 4 and 5. Finally, on Chapter 6, we provied a brief summary of what was done and suggest possible future directions to continue this work.

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Chapter 2

Theoretical Aspects of Diffuse

Optics

In this chapter we provide some insight into the theoretical background of diffuse optical techniques, more specifically, we cover the approaches to recover the optical properties from static (Diffuse Optical Spectroscopy, DOS) and dynamic media (Diffuse Correlation Spectroscopy, DCS). On Section 2.1, we briefly describe the light-matter interactions of interest and on Sections 2.2 and 2.3 we describe the physical and mathematical formalism behind DOS and DCS, respectively.

2.1 Light-Matter Interaction

In order to appropriately employ optical techniques we must first understand the processes behind light propagation inside a medium. We will focus our discussion mainly on two kind of light-matter interactions: absorption and scattering. Other kinds of interactions are unnecessary, since we will work only with low power near infrared (NIR) light sources.

Absorption occurs when an incident photon energy matches the energy of the rotational or vibrational modes of one of the particles present in the medium (Figure 2.1), being absorbed in the process. The capacity of a medium to absorb photons can be characterized by the absorption coefficient, µa, which is a measure of the mean number of absorption

events per unit length. This coefficient depends on the concentration and the absorption cross-section of the absorber, which depends on the incident wavelength, λ.

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Rotational and Vibrational States Incident Photon Ground State Electronic States

Figure 2.1: Representation of the absorption of a photon by a molecule. The incident photon has an energy matching the energy of one the excited levels of the molecule.

Scattering is the process in which light changes direction upon interaction with a par-ticle. Similar to the absorption case, scattering can be quantified by a scattering coefficient, µs, which is again a measure of the mean number of scattering events per unit length. Its

value depends on: the relative size of the particles; the incident wavelength; the medium’s refractive index and; in general, on the scattering cross-section. The photon’s angular dis-tribution carries useful information about, for example, the mean scattering angle, which is directly related to the anisotropy factor (g = hcos θi), a parameter that describes the anisotropy of a medium. Another useful parameter for the scattering process is the reduced scattering coefficient, defined as µ0_s= µs(1 − g), which is mainly used for describing

scatter-ing processes in the multiple scatterscatter-ing limit. In this dissertation, I will neglect non-elastic scattering events, since we are only working with low intensity NIR light.

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Chapter 2. Theoretical Aspects of Diffuse Optics 7

In the formalism presented here, the scattering and absorption coefficients are sufficient to characterize the light propagation inside a medium. More specifically, we will focus on turbid media, i.e., highly scattering media where µ0_s µa. In these media, as will be

shown later, electromagnetic radiation spreads across the medium in an approximately diffusive regimen. This diffusive nature of light propagation inside the medium allows us to eventually detect the photons at some distance away from the light source (Figure 2.2), obtaining information about the optical coefficients (µa and µs) of the medium, including

deep regions.

Biological tissue can be considered as turbid medium in the NIR spectra (∼ 700 − 900 nm). In this range, tissue has a small absorption coefficient with a considerable level of scattering (µ0_s ≈ 100µ_a) [47, 48]. Furthermore, for human tissue, NIR light is mainly absorbed by oxy- and deoxy-hemoglobin, respectively HbO and Hb, (Figure 2.3) and is mainly scattered by red blood cells, if we ignore the static scatterers. The low absorption of NIR light allows it to penetrate deeply inside the medium, carrying information about deep tissue, eventually allowing us to non-invasively recover information about oxygenation and blood flow changes in deep regions. The technique employed to characterize the static contributions (mainly the chromophores concentrations) is known as Diffuse Optical Spec-troscopy (DOS), whereas the one employed to infer about the dynamical properties of the medium (perfusion) is known as Diffuse Correlation Spectroscopy (DCS) [44, 49].

500 600 700 800 900 1000 10−2 10−1 100 101 102 λ(nm) µa (c m − 1 ) H₂O H bO H b

Figure 2.3: Absorption coefficients of the main components of human blood. Data avail-able from [50, 51]. Here H2O, HbO and Hb are represented in black, red and blue,

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2.2 Diffuse Optical Spectroscopy

2.2.1 The Radiative Transport Equation and its Diffusive Approximation

There are many ways to model light propagation inside turbid media. In the classical approach one should directly use Maxwell’s equations and the wave properties of light. However, this method would require to account for every single interaction of light with the medium’s particles, resulting in a huge computational problem when working in the multiple scattering limit, which is the main interest of this dissertation.

A more practical approach is to model light propagation using the concept of quanta of light and the scalar theory of linear transport. This model only accounts for the interactions between the “particles” of light (photons) and the medium’s particles, ignoring polarization effects. The problem of light transport inside a turbid medium is then described by the radiative transport equation (RTE), or Boltzmann transport, which is an approximation to Maxwell’s equations [52–56]: 1 v ∂L(~r, bΩ, t) ∂t + bΩ.∇L(~r, bΩ, t) = Q(~r, bΩ, t) + µs I L(~r, bΩ, t)f (bΩ, bΩ0)dbΩ0− µ_tL(~r, bΩ, t), (2.1) where L(~r, bΩ, t) is the radiance, defined as the average power at ~r and instant t across the area oriented in the direction of the unit vector bΩ (Figure 2.4); v is the speed of light inside the medium; µt= µa+ µsis the transport coefficient, whereas µaand µsare the absorption

and scattering coefficients, respectively; f (bΩ, bΩ0) represents the scattering phase function

𝜃

Ω

′

Ω

Incident Beam x y z Scattered Beam Scattering Medium r

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(i.e., the probability of a photon coming from the direction bΩ to be scattered into bΩ0) and; Q(~r, bΩ, t) is a source term.

Note that (2.1) is basically an energy conservation equation, i.e., it says that the number of photons leaving a specific region and direction in time t inside the medium (left side of the equation) is equal to what is lost via scattering to other directions and/or absorption (represented by the third term in the right side of (2.1)) plus what comes into it from any sources and/or scattering from other directions (first and second terms of the right side of (2.1), respectively).

It is important to mention the assumptions behind the RTE: 1) intensity is transferred element-wise (based on the principle of energy conservation); 2)the backscattering is in-significant when compared to the scattering to other directions; 3) the scattering particles are non-interacting; 4) the wave spectrum of the incident source has a Dirac δ distribution function and; 5) the scattering particles are distant from each other, so that we can use the far-field approximation [57].

Analytical solutions to the RTE are available only for very simple and highly symmetric media, which is not the case of biological tissue (such as the human brain, for example). We must then work with approximations of the RTE. The PN approximation is the most

commonly used. In the PN approximation, the radiance is expanded in terms of the spherical

harmonics, Ynm, up to n = N .

If we keep our attention to the P1 approximation, we arrive at a diffusion equation for

the photon propagation. In this approximation, we expand the radiance in terms of the spherical harmonics up to the Y1m term, obtaining [52]:

L(~r, bΩ, t) = 1 4πφ(~r, t) + 3 4π ~ J (~r, t) · bΩ, (2.2)

where φ(~r, t) is called the fluence and is defined as the integral of the radiance over all angles. The fluence represents the number of photons per unit area per unit time. ~J (~r, t) · bΩ is the photon flux leaving ~r in the direction bΩ at instant t.

By substituting (2.2) into (2.1) and further assuming that: 1) the scattering function is independent of the angle between the incident and scattered directions, i.e., f (bΩ, bΩ0) = f (bΩ · bΩ0); 2) the photon flux varies slowly and; 3) all the sources term are isotropic, we obtain the diffusion equation for the photon fluence [43, 54–56, 58]:

∂φ(~r, t)

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where S(~r, t) represents an isotropic source term and D(~r) is the diffusion coefficient, defined as D(~r) = v/(3µ0_t(~r)) = v/(3(µa(~r)+µ0s(~r))), where µ0sis the reduced scattering, or

random-walk, coefficient. Under appropriate boundary conditions it is possible to analytically solve (2.3), making it straightforward to recover the optical coefficients. On Appendix A we show the solutions of (2.3) for some selected geometries.

The diffusive approximation has proven to be accurate to describe light propagation inside biological tissues, considering macroscopic distances greater than 5mm [59–61]. In fact, with this approach it is possible to measure the optical coefficients of the tissue, which carries information about tissue oxygenation, for example. On the next section we will briefly describe some experimental methods to obtain the optical coefficients, and eventually tissue properties, from the diffusion equation (2.3).

2.2.2 Experimental Methods of Diffuse Optical Spectroscopy (DOS)

In DOS we work with the Diffusion Equation (2.3) to recover the “static” optical properties of a given medium. We measure slow variations (of the order of milliseconds) of the ab-sorption and scattering coefficients by shining a low power NIR light source in the medium and collecting the scattered intensity some distance away from the point of incidence. To obtain the desired optical properties, we must first specify the boundary conditions of the medium, i.e., its geometry, and then solve (2.3) for further comparisons with the measured values. The solution for some selected geometries are shown in Appendix A.

DOS can be separated into three different experimental techniques: 1) time domain (DOS); 2) frequency domain (FD-DOS) and; 3) continuous wave (CW-DOS). The TD-DOS formalism collects information from the medium by shining a pulsated laser, on the

Input Output Time (ns) 𝜃 _Input Output Time (ns) 𝐼0 Input Output Time 𝐼(𝜌)

(A)

(B)

(C)

In tensity In ten sity Log (In tensity )

Figure 2.5: Representative siganls from the three different DOS techniques. (A) Time-Domain (B) Frequency-Time-Domain (C) Continuous wave.

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order of nanoseconds, and by measuring the broadened pulse (Figure 2.5-A), which carries information about the absorption and scattering coefficients. This technique is not part of the goals of this dissertation, so its more delicate experimental and analytical procedures are omitted.

FD-DOS employs modulated lasers (typically in the 70-140 MHz region), and mea-sures the scattered amplitude and phase (Figure 2.5-B). Mathematically, it uses the Fourier transform of (2.3), which is basically a wave equation:

∇2φ(~r, ω) − k2φ(~r, ω) = S0(~r, ω), (2.4) where k2 = vµa+iω

D is the wave vector of the diffusive propagation and S

0_(~_{r, ω) = −}v

DS(~r, ω)

is the isotropic source term in the frequency domain. Note that in (2.4) we consider the optical coefficients to be homogenous, i.e., µ0_t(~r, t) = µ0_t(t). From Appendix A.3 we know that the solution to (2.4) for a semi-infinite medium (Figure 2.6) is:

φG(~r, ~r0, ω) = v 4πD e−kr1 r1 − e −kr2 r2 , (2.5)

where r1 = p(z − l0t)2+ ρ2; r2 = p(z + 2zb+ l0t)2+ ρ2, and; zb and lt are defined in

Figure 2.6. If we consider that we are measuring far away from the source, i.e., ρ >> lt+2zb,

we obtain: φG(~r, ~r0, ω) = v 4πD e−kReρ ρ2 2k(zblt+ z 2 b | {z } A(ρ) e θ(ρ) z }| { −ikImρ, _(2.6) Turbid Medium 𝜙 = 0 𝑧 = −(𝑧₀+ 2𝑧_𝑏) 𝑧 = −𝑧_𝑏 𝑧 = 𝑧₀ Image Source Real Source 𝑧 = 0 Detector Fiber Source Fiber 𝜌 𝑟₁ 𝑟₂ 𝑧 𝜌 𝑛𝑖𝑛 𝑛𝑜𝑢𝑡

Figure 2.6: Representation of a semi-infinite medium. Here we used the method of images to solve the diffusion equation, as described in Appendix A.

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where kReand kImare respectively the real and imaginary parts of the wave vector k. From

the phase, θ(ρ), and the logarithm of the amplitude times ρ2, ln A(ρ)ρ2, we obtain:

ln ρ2A(ρ) = −kReρ + ln A0 (2.7a)

θ(ρ) = −ikImρ + θ0. (2.7b)

Note that both (2.7a) and (2.7b) follow a linear relationship with the source-detector dis-tance (ρ), allowing us to recover the real and imaginary parts of the wave vector k by measuring the fluence at different distances and simply using a linear fitting procedure. Moreover, after some manipulations we can write the absorption and scattering coefficients in terms of the real and imaginary parts of k as:

k2 = vµa+ iω D ⇒    µa= ω(−k2 Re+k 2 Im) 2vkRekIm µ0_s= −v2kRekIm 3ω . (2.8)

Equations (2.7) and (2.8) show how to compute the absolute optical properties of a turbid, semi-infinite medium by employing the FD-DOS.

CW-DOS is the limit of ω → 0 in FD-DOS. The main difference is the fact that the wave vector k is real, i.e., k2 = vµa/D. In practice, this means that we have access

only to an effective transport coefficient, defined as µef f =p3µa(µa+ µ0s) when working

with continuous wave sources. Thus, CW-DOS methods are unable to separate the effects of absorption and scattering without any extra information. What is usually done to overcome this problem is to work with absorption changes (∆µa), while considering that µ0s remains

constant, as will be seen in the next subsection [62, 63].

2.2.3 DOS applied to Human Tissue

On the previous section we demonstrated how to compute the theoretical optical coefficients using the diffusion equation formalism. In this subsection we will describe how to use the optical coefficients to recover information about human tissue properties. More specifically, we will demonstrate how to recover the oxy- and deoxy-hemoglobin concentrations, cHbO2

and cHb, respectively, from optical measurements.

In the FD-DOS case, it is straightforward to recover the absolute concentrations of the absorbers inside the medium, since we can write the absorption coefficient of the medium

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as a linear superposition of the absorption coefficients of each separate chromophore:

µa(λ) =

X

εi(λ) ci = εHbO2(λ) cHbO2 + εHb(λ) cHb+ εH2O(λ) cH2O, (2.9)

where εi and ci are the extinction coefficient and the concentration of the ith chromophore,

respectively. The second equality is valid for human tissue, and is based in the human tissue absorption spectra, shown in Figure 2.3. Thus, from (2.9) we have a method for obtain-ing the oxy- and deoxy-hemoglobin concentrations from measurements of the absorption coefficient.

However, since we are not able to directly measure µa with CW-DOS, we need to

develop a new formalism. In this case we must work with the optical density (OD), which is the negative of the logarithm of the change in the incident intensity caused by absorption. Moreover, if we use the Beer-Lambert Law, that states that the optical density in a detector is proportional to the absorption coefficient of the medium, we can write:

OD (t, λ, µa) ≡ − ln

I (t, λ) I0(t, λ)

= µa(t, λ) L, (2.10)

where I (t, λ) and I0(t, λ) are the measured and incident intensities, respectively, and L is

the linear photon pathlength inside the medium.

We need a further modification on (2.10) since we are not yet accounting for scattering. We must consider that the OD also depends on the scattering coefficient, i.e., OD(t, µa) ⇒

OD(t, µa, µ0s), where we are omitting the λ dependence for simplicity. Assuming that after

some time the absorption coefficient changes (µa⇒ µa+∆µa) while the scattering coefficient

remains constant, we can write OD(t0, µa, µ0s) and OD(t, µa+ ∆µa, µ0s) as the initial and

measured ODs, respectively:

∆OD = OD(t, µa+ ∆µa, µ0s) − OD(t0, µa, µ0s) =

=hhh hhhh OD(t, µa, µ0s) + ∂OD(t, µa, µ0s) ∂µa ∆µa+ *0 O(2) −hhh_hhh h OD(t0, µa, µ0s), (2.11)

where we have expanded the measured OD in terms of the variation ∆µa. Neglecting the

higher order terms and using (2.10) we arrive at the Modified Beer-Lambert Law (MBLL) [64]: ∆OD(t) = − ln I (t) I (t0) = ∂OD(t, µa, µ 0 s) ∂µa | {z } lDP F ∆µa, (2.12)

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where lDP F is the differential pathlength of the photons, which is basically a correction to

the non-linearity of the photon path due to scattering. With (2.12) we show that, although we can’t directly measure µaand µ0swith CW-DOS, we are still able to measure absorption

changes ∆µa, and consequently, changes in chromophore concentrations can be obtained

as: − 1 lDP F ln φ (t, λ) φ (t0, λ) = ∆µa(λ) = X εi(λ) ∆ci= εHbO2(λ) ∆cHbO2 + εHb(λ) ∆cHb, (2.13) where we considered that the water concentration remains constant and made explicit the λ dependence again. Note that we used the proportionality of the fluence and the inten-sity, considering that the coupling constant is time invariant, therefore being canceled out. Although CW-DOS can only measure concentrations changes, it is still a useful technique when working with functional changes in tissue [21, 24, 30, 31, 65, 66]. This methodology is known in the literature as Near-Infrared Spectroscopy (NIRS).

2.3 Diffuse Correlation Spectroscopy

In this chapter we will describe Diffuse Correlation Spectroscopy, a technique capable of measuring the dynamical properties of a turbid medium. By accessing light intensity fluc-tuations, due to particle motion inside the medium, it is possible to measure the mean square displacement of the particles, which can be related to perfusion and flow properties of the medium. We will show the formalism behind DCS using two different methods: 1) the Diffusing Wave Spectroscopy (DWS) theory and some concepts from (2.3), as done by Zhou et al. [34] and; 2) the correlation transfer formalism, as done by Ackerson et al., Boas et al. and Dougherty et al. [43, 57, 67]. On Sections 2.3.1 and 2.3.2 we will summarize the theory behind DCS, for both aforementioned methods, respectively, and on Section 2.3.3 we will describe some of the experimental procedures for DCS.

2.3.1 The Diffusing Wave Spectroscopy

2.3.1.1 Dynamic Light Scattering

Let us begin with a simpler case: suppose we shine a light beam into an optically dilute system, where light is scattered no more than once before leaving (Figure 2.7). If we collect

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Figure 2.7: Pictorial representation of an experiment in the single scattering limit. The particles movement will induce a change in the measured intensity.

the scattered light in a point detector making an angle θ to the line of incidence, we can write the scattered electric field from a single particle as [34]:

~ Ei(t) =beE0F (θ)e −iωt ei~kin·[~ri(t)− ~Rs]ei~kout·~ri(t) ₌ =_beE0F (θ)e−iωtei( ~_k_out_{· ~}_R d−~kin· ~Rs)e−~q·~ri(t)_, (2.14)

wheree is a unitary vector representing the polarization of the scattered wave, which is in_b general different from the incident beam polarization; F (θ) is the form factor of the particles and it depends on their shape, size and refractive index and also on the scattering angle θ; E0, ω and λ are the beam light amplitude, frequency and wavelength, respectively, which is

assumed to be monochromatic (i.e., single wavelength and infinite coherence) and stationary (E0 is independent of time); ~kinand ~koutare the incident and scattered wave vectors, where

kin = kout = k0 = 2πn/λ; n is the refractive index of the medium; ~q = ~kout− ~kin is the

momentum transfer, with amplitude q = 2k0sin(θ/2); ~Rs, ~Rd and ~ri(t) are the position of

the source, detector and the ith scattering particle, respectively.

The total electrical field measured at the detector will just be a superposition of all the scattered fields from the N individual particles. For an ensemble of identical particles:

~ ET(t) = N X i=1 ~ Ei(t) =beE0F (θ)e −iωt ei(~kout· ~Rd−~kin· ~Rs) N X i=1 e−~q·~ri(t) ! . (2.15)

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phase of the total scattered electrical field depends on the momentum transfer and on the position of each individual particle, which is considered to be independent. Therefore, by measuring the phase changes of the total electric field we can access information about the dynamics of the scattering particles in the medium. To extract information out of (2.15), we generally use the temporal auto-correlation function of the scattered field:

g1(τ ) =

h ~ET(t) · ~ET∗(t + τ )i

h ~ET(t) · ~E_T∗(t)i

, (2.16)

where τ is the time delay and hi represents an ensemble average. We can analytically compute (2.16) from (2.15) if we consider that the particles are randomly positioned and that their movements are random and non-correlated. In this case [34, 43]:

g1(τ ) = eiωte−

1 6q

2_h∆r2_{(τ )i}

, (2.17)

where h∆r2(τ )i is the mean square displacement of the scattering particles. 2.3.1.2 The Multiple Scattering Limit

If we now increase the density of scattering particles in the medium, light will be scattered multiple times before leaving the medium. In this case, there are several possible photon paths and each path will contribute to the total electric field. Moreover, if the particles in the medium are moving, some of those paths will have their lengths changed, leading to a variation in the measured intensity (Figure 2.8). Mathematically, we can write the

Particle at time 𝑡 Particle at time 𝑡 + 𝜏 Detector Laser 𝑡0𝑡 + 𝜏 𝐼0 𝐼(𝑡) 𝑡

Figure 2.8: Pictorial representation of an experiment in the multiple scattering regimen. The movement of the scatterers will induce a change in the measured intensity.

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measured electric field due to one of the paths after N scattering events as [34]: ~ Eonepath(t) =eEb 0e −iωt_ei~k1·[~r1(t)− ~Rs]F (θ 1)ei~k2·[~r2(t)−~r1(t)]F (θ2)ei~k3·[~r3(t)−~r2(t)]· . . . ·F (θ_{N −1})ei~kN·[~rN(t)−~rN −1(t)]_{F (θ} N)ei~kN +1·[ ~ Rd(t)−~rN(t)] = =_beE0e−iωtei( ~_k N +1· ~Rd−~k1· ~Rs) ·   N Y j=1 F (θj)     N Y j=1 e−i~qj·~rj  , (2.18)

where F (θj), θj, ~qj = ~kj+1− ~kj and ~rj(t) are the form factor, scattering angle, momentum

transfer vector and the position of the jth scattering event, respectively. Both products are over the N scattering events along the photon path. The total electric field is just the sum over all the possible paths, i.e., ~ET(t) =

PAllpaths _~

Eonepath(t).

Assuming that the electric fields from each path are non-correlated, we can write the total auto-correlation function as a sum over the correlation function from each individual path, i.e., g1(τ ) = All paths X k=1 Pk· g1(τ )kth path, (2.19)

where Pk is the probability of the kth photon path. Now supposing that: i) the medium is

homogenous; ii) that all the scattering events are independent, and; iii) that the particles movement are non-correlated (which is valid for media with no particle-particle interac-tions), we can show that [34]:

g1(τ ) = eiωτe−

1

6h∆r

2_{(τ )i}PN

j=1q2j _{= e}iωτ_e−61h∆r2(τ )i2k20Y_, _(2.20)

where we used that q2_j = 2k2₀(1 − cos θj) and defined Y = PNj=1(1 − cos θj). For highly

scattering media we can write Y as:

Y =

N

X

j=1

(1 − cos θj) = N (1 − hcos θiN) , (2.21)

where hcos θiN is a mean over all N scattering events along the path. For large N this is

just the anisotropy factor, i.e.:

lim

N →∞hcos θiN = hcos θi = g. (2.22)

The scattering and random-walk lengths can be written as the inverse of the respective coefficients, i.e., ls = 1/µs and l0s = 1/µ0s, which combined leads to l0s = ls/(1 − g). For

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large N we can write the total length of a photon path as s = N ls, since lsis the mean free

path of a photon before it gets scattered and N is the total number of scattering events. From the relation between the scattering and random-walk lengths and the definition of s, we can write N = s/ls = s/ [ls0(1 − g)]. With this relation, along with the limit of large N

in (2.21), we can show that Y = s/l0_s. Substituting this into (2.20) and summing over all possible infinitesimal paths, we arrive at:

g1(τ ) = eiωτ Z ∞ 0 P (s)e− s 3l0sk 2 0h∆r2(τ )i ds, (2.23)

where P (s) is the probability of a photon to travel a path of length s. We substituted the summation to an integral in (2.23) since we are dealing with infinitesimal path lengths on the limit N → ∞. This equation is valid for turbid homogenous medium, composed of uniformly distributed non-interacting particles, that is illuminated by an infinitely coherent source (or at least with a correlation length much longer than the mean photon path length). Note that (2.23) is essentially a Laplace transformation of P (s), where P (s) can be computed from the theory of Section 2.2, more specifically from Equation (2.3), if the scattering length is shorter than the dimensions of the sample and the absorption length. If we consider we have a point source S(~r, t) = S0δ(~r − ~Rs, t) and remember that after t = s/v

the photons travel a distance s, we can write:

P (s) = φ ~r, t = s v R∞ 0 φ(~r, t)dt = φ ~r, t = s v I(~r) , (2.24)

where the denominator is the light intensity measured at ~r. If we now define p = _3l10 sk 2 0h∆r2(τ )i, then: g1(τ ) = eiωτ Z ∞ 0 P (s)e−spds = e iωτ I(~r) Z ∞ 0 φ~r, t = s v e−spds. (2.25) Multiplying both sides of the DOS equation (2.3) by eiωte−sp, integrating from 0 to ∞ and changing t → s_v: eiωτ Z ∞ 0 v∂φ(~r, s v) ∂s e −sp ds + eiωτ Z ∞ 0 vµaφ(~r, s v)e −sp ds −eiωτ Z ∞ 0 ∇ ·hD(~r)∇φ(~r, s v) i e−spds = eiωτ Z ∞ 0 vS0δ(~r − ~Rs, s v)e −sp_ds. (2.26)

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Since the integrations are in respect to s = vt, the spatial derivatives ∇ can go outside the integrals: veiωτ Z ∞ 0 ∂φ(~r,s_v) ∂s e −sp ds + vµaeiωτ Z ∞ 0 φ(~r,s v)e −sp ds −∇ · D(~r)∇ eiωτ Z ∞ 0 φ(~r, s v)e −sp_ds = veiωτ Z ∞ 0 S0δ(~r − ~Rs, s v)e −sp_ds (2.27)

The term on the right side of (2.27) can be calculated as:

eiωτ Z ∞ 0 δ(~r − ~Rs, s v)e −sp_{ds = S} 0eiωτδ(~r − ~Rs). (2.28)

And the 1st term on the left side yields (integrating by parts):

eiωτ Z ∞ 0 ∂φ(~r,s_v) ∂s e −sp_{ds = e}iωτ " :0 e−spφ(~r,s v) ∞ 0 + p Z ∞ 0 φ(~r,s v)e −sp_ds # . (2.29)

If we recall that the normalized auto-correlation function is defined as g1(~r, τ ) = G1_I(~(~r,τ )_r) , we

can use (2.25) to show that G1(~r, τ ) = eiωτ

R∞

0 φ(~r, s v)e

−sp_ds.

Substituting the above results on (2.27), remembering that p = _3l10 sk

2

0h∆r2(τ )i and

rearranging some terms, we arrive at the diffusion equation for the unnormalized field auto-correlation: v µa+ 1 3l0 s k2₀h∆r2(τ )i G1(~r, τ ) − ∇ · [D(~r)∇ (G1(~r, τ ))] = vS0eiωτδ(~r − ~Rs). (2.30)

2.3.2 The Correlation Transfer Equation and its Diffusive Approximation

On the preceding discussion of DWS we implicitly assumed that different paths do not interfere with each other. This is similar to the element wise transfer of energy considered in the development of the radiative transport in Equation (2.1):

1 v ∂L(~r, bΩ, t) ∂t + bΩ.∇L(~r, bΩ, t) + µaL(~r, bΩ, t) + µsL(~r, bΩ, t) = = Q(~r, bΩ, t) + µs I L(~r, bΩ, t)f (bΩ, bΩ0)dbΩ0, (2.31)

where the radiance L(~r, bΩ, t) can be defined as:

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and hi is an ensemble average. The time-dependent correlation function (G1(~r, bΩ, t, τ )) is

defined in a similar fashion:

G1(~r, bΩ, t, τ ) = h ~E(~r, bΩ, t) ~E∗(~r, bΩ, t + τ )i. (2.33)

In both definitions, ~E(~r, bΩ, t) is the total electric field measured at the detector.

Ackerson et al. proposed a Correlation Transfer Equation (CTE) using the similarities between DWS and the RTE, together with the aforementioned definitions for the field auto-correlation and the radiance [67]. In a slightly different notation, their CTE can be written as: 1 v ∂G1(~r, bΩ, t, τ ) ∂t + bΩ.∇G1(~r, bΩ, t, τ ) + µaG1(~r, bΩ, t, τ ) + µsG1(~r, bΩ, t, τ ) = = Q(~r, bΩ, t) + µs I G1(~r, bΩ, t, τ )g1(bΩ, bΩ0, τ )f (bΩ, bΩ0)dbΩ0, (2.34)

where g1(bΩ, bΩ0, τ ) is the single-scattering auto-correlation function from Section 2.3.1.1; µa

and µsare the absorption and scattering coefficients, respectively, and; f (bΩ, bΩ0) is the same

scattering phase function from (2.1).

From (2.32) and (2.33) we see that in the limit of τ → 0 the radiance and the auto-correlation function are functionally the same. Thus, we should expect the CTE (Equa-tion (2.34)) to reduce to the RTE (Equa(Equa-tion (2.31)) in this limit, which is easily shown to be true. Here we implicitly assumed that the average field auto-correlation is performed instantaneously for each τ . For nonzero τ , different particles positions will lead to different path sizes, decreasing the correlation, just like in Section 2.3.1.

We can derive the CTE by making the appropriate changes in the RTE. First we must note that the correlation is not conserved, as opposed to the radiance. In addition, we know that correlation is only lost due to scattering, i.e., after each scattering event less and less correlation is “transferred”. Therefore the two terms we must change in (2.31) are the two terms related to scattering: the fourth term, which accounts for the loss of radiance due to scattering to other directions and; the last term, which accounts for the scattering coming from other directions.

The fourth term describes the amount of radiance or correlation scattered out of bΩ, which will give no further contributions in that direction. Thus, this term should be the same for both the RTE and the CTE, meaning that the fraction of scattered paths out of b

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The difference of the CTE and the RTE will thus arrive from the last term. In the RTE, what is scattered into bΩ comes from other directions bΩ0, such that energy is conserved. On the other hand, correlation is not conserved after a scattering event and the amount of transferred correlation depends upon the phase function, the cross section and the single particle correlation function, where the last one decreases for increasing τ . Thus, correlation may be scattered out of one direction, but is transferred less and less completely into the b

Ω direction as τ increases. The CTE then takes the form of (2.34). Here we must assume that the particles are sufficiently well separated to act as single independent scatterers.

The argument cited above is only qualitative. The formal derivation of the CTE, from a more fundamental perturbative solution to the Maxwell’s equations, was done by Dougherty et al. in 1994 and will not be a topic of this dissertation, as it is out of the scope of this work [57].

The main advantage of the CTE formalism over the DWS formalism is that the first does not make any assumptions about the medium’s properties. That is, the CTE should be valid from symmetrical to heterogeneous and from optically dilute to optically dense media, whereas DWS is only valid for symmetrical and optically dense media [34, 43, 44, 57, 67]. We must note, however, that the turbid media is the regime we typically work in biomedical applications.

Since the RTE and the CTE are formally similar, one should be able to arrive at a diffusion equation for the field auto-correlation function, just like the one we found for the fluence of photons (Equation (2.3)). Focusing on the continuous wave (CW) case, we can drop out the time dependence from (2.34) and arrive at (refer to Appendix A.2 from [43]):

v µa+ 1 3µ 0 sk20h∆r2(τ )i G1(~r, τ ) − ∇ · [D(~r)∇ (G1(~r, τ ))] = vS(~r), (2.35)

where v is the speed of light; µaand µ0sare the absorption and reduced scattering coefficients;

k0 is the incident wave vector; D(~r) = v/(3µ0t) is the photon diffusion coefficient and;

h∆r2_{(τ )i is the mean square displacement of the scattering particles. In his thesis, D. A.}

Boas derived the diffusion equation (2.35) using the P1 approximation of the CTE (refer to

Appendix A.2 of this thesis) [43]. There he had to assume that the scattering phase function (f (bΩ, bΩ0)) and the single scattering auto-correlation function (g1(bΩ, bΩ0, τ )) only depends on

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2.3.3 Experimental Methods of Diffuse Correlation Spectroscopy and its Application to Human Tissue

On the previous sections we briefly described the physical processes behind the correlation diffusion equation, using the multiple scattering limit of the Dynamic Light Scattering and also the Correlation Transfer theory described by (2.34). In both cases we arrived at a Correlation Diffusion Equation similar to the static case of Diffuse Optical Spectroscopy of Section 2.2.

We can recast (2.35) as a Helmholtz equation for the field auto-correlation function:

∇2− K2(τ ) G₁(~r, τ ) = −vS Dδ

3_(~_{r − ~}_r

s), (2.36)

where K2(τ ) = v µa+1₃µ0sk02h∆r2(τ )i /D; the light source is considered to be point

like, continuous and located at position ~rs. Note that there are two loss terms, µa and 1

3µ 0

sk20h∆r2(τ )i, where the first one is just the amount of light lost due to absorption and

the second accounts for the loss of correlation due to scattering, as briefly described in Section 2.3.2.

Before applying (2.36) to human tissues, one last modification must be made. So far, we considered that all scattering particles were moving, whereas in human tissue there are both moving and static scatterers, and the static scatterers do not contribute to the loss of correlation. To account for this problem we introduce a new parameter α such that K2(τ ) = v µa+ α1₃µ0sk02h∆r2(τ )i /D. This new parameter is just the probability that a

photon is scattered by a moving scatterer, and can be defined as:

α = µ 0 s(moving) µ0 s(moving) + µ0s(static) , (2.37)

where µ0_s(moving) and µ0_s(static) are the respectively reduced scattering coefficients of the moving and static portions of the tissue.

Comparing the optical diffusion equation (2.4) and the correlation diffusion equation (2.36), we see that both are mathematically similar. The only difference is in the definition of the wave vector. Therefore, we should be able to use the same mathematical methods shown in Appendix A to solve both equations. For example, for a semi-infinite medium we arrived at: G1(~r, τ ) = v 4πD " e−K(τ )r1 r1 −e −K(τ )rb rb # , (2.38)

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where r1 =p(z − lt)2+ ρ2, rb =p(z + 2zb+ lt)2+ ρ2, and both are defined in Figure 2.6.

Note that G1(~r, τ ) depends on µa,µ0s and αh∆r2(τ )i.

Experimentally, however, we only have access to the intensity and its correlation func-tion G2= hI(t)I(t + τ )i. We must then use the Siegert Relation (Appendix B):

g2(τ ) = 1 + β|g1(τ )|2, (2.39)

where g2(τ ) = G2(τ )/hI(0)i2 and g1(τ ) = G1(τ )/hI(0)i are the normalized intensity and

field auto-correlation functions, respectively, and; β is an experimental parameter that depends on the laser and on the collection optics used. For example, as will be seen later, in our experiments we use mono-mode fibers and a non-polarized long coherence laser, in which case β = 0.5, theoretically (refer to Appendix B for a more detailed discussion on β). In Figure 2.9, we show a scheme of the procedure for collecting data from DCS mea-surements. We shine a long coherent laser into a sample (a human head, for example), and collect the scattered light some distance ρ away from the illumination point, using a mono-mode fiber and a photon counter. We then feed this measured counts into a physical correlator that computes the intensity auto-correlation function, g2, reading it directly in

a computer. Using a non-linear fit procedure (refer to Section 4.1, the chosen boundary

𝛽 and Δ𝑟

2

𝜏

Measuring Site Laser Photon Counter Correlator Non-Linear Fit (𝜇𝑎e 𝜇𝑠′known a priori) Data Data Fit

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conditions (Appendix A) and the Siegert relation (2.39) from Appendix B), we can estimate the experimental values for β and αhr2(τ )i.

We must also define the type of random movement of the scatterers are undergoing inside the tissue. There are two main possibilities, respecting the underlying conditions imposed by Section 2.3.1 and 2.3.2: Brownian motion and random flow. The mean square displacement for these types of movements can be written as [34, 43, 44]:

Brownian Motion: h∆r2(τ )i = 6DBτ

Random Flow: h∆r2(τ )i = hV iτ2.

(2.40)

Experimentally, the Brownian motion was shown to provide the best fits to data from human tissue (Figure 2.10). Note that the Brownian diffusion coefficient DBin (2.40) is a few orders

of magnitude larger than the traditional Einstein’s Brownian movement coefficient [34, 44]. Since we can’t directly measure α we define a Blood Flow Index (BFI) as:

BF I = αDB (2.41)

The BFI is in fact a blood flow index since the main moving scatterers of human tissue are the red blood cells. There is still one minor problem when working with human tissues: the

10

−6

10

−4

10

−2

1

1.2

1.4

1.6 τ

( s)

g

2

(

τ

)

Human Data

Random Flow

Brownian Motion

Figure 2.10: Comparison of the fitting of human data using both the Brownian motion and random flow from (2.40).

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units of the measured BFI is cm2/s whereas the usual units for blood flow is ml/min/100g. Thus, we typically work with relative changes of blood flow (rBF), defined as:

rBF = BF I BF I0

[%] (2.42)

where BF I0 is the BFI measured during a baseline period and rBF is typically expressed

in %. Measurements of rBF from DCS have been shown to be highly correlated to other gold standard techniques, as can be seen in Table 2.1 [41], which reinforce the potential of DCS as a blood flow measurement tool in medical and biological applications.

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T able 2.1: Comparison of some in vivo DCS v alidation studies. ASL-MRI: arterial spin-lab elled magnetic ressonance imaging; PDT: photo dynamic thepary . T able w as extracted from [41 ]. Sample P erturbation M o dalit y Correlation Co efficien t Slop e DCS /Mo d Reference mouse femoral artery o cclusion laser Doppler > 0 .8 0 .96 − 1 .07 [68 ] mouse tumor an ti-v ascular therap y c on trast-enhanced ul-trasound n.a. agreemen t [69 ] mouse tumor PDT Doppler ultrasound n.a. agreemen t [39 ] mouse tumor PDT p o w er Doppler ultra-sound n.a. 0 .97 [70 ] rat h yp ercapnia ASL-MRI 0 .81 − 0 .86 0 .75 [71 ] rat h yp o c ap nia laser Doppler 0 .94 1 .3 [72 ] neonatal piglet traumatic brain injury fluorescen t micro-spheres 0 .63 0 .4 [73 ] premature neonates ab sol ute baseline transcranial Doppler 0 .53 n.a. [74 ] term neonate h yp ercapnia ASL-MRI 0 .7 0 .85 [75 ] premature infan t absolute baseline transcranial Doppler 0 .91 0 .9 [76 ] h uman m uscle cuff -inflat ion/deflation ASL-MRI > 0 .77 1 .5 − 1 .7 [77 ] adult h uman pressors and h yp erv en-tilation xenon-CT 0 .73 1 .1 [78 ] adult h uman ace taz olamid e transcranial Doppler n.a. agreemen t [79 ]

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Chapter 3

Design of a Diffuse Correlation

Spectroscopy System

Broadly speaking, a Diffuse Correlation Spectroscopy System consists of three main compo-nents: a light source, a photon counter and a correlator board. In a typical DCS experiment we shine coherent light into a medium, and we count the number of scattered photons some distance away from the point of illumination. By using a correlator board and the measured

Figure 3.1: Simplified schematics of a DCS experimental setup. We shine coherent light into a medium, count the number of scattered photons some distance away from the point of illumination and read the auto-correlation function computed by a correlator board

directly in a computer. 27

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photon counts we then compute the intensity auto-correlation function, g2(τ ), reading it

directly in a computer (Figure 3.1).

On Sections 3.1, 3.2 and 3.3 we succinctly describe the operation of each one of the sys-tem components and on Section 3.4 we show the operation of the data acquisition software. The characterizations of each component are shown on their respective sections. Finally, on Section 3.5, we describe the construction of the DCS system, along with the whole system characterization results.

3.1 Light Source

We know from Chapter 2 that the light source of a DCS experiment must be a NIR, point-like, continuous and infinitely coherent source. In practice, we must require that the coherence length of the laser must be much larger than the mean photon path length, which in biological tissue typically requires the laser to have a coherence length much larger than 1m.

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Chapter 3. DCS System 29 0 200 400 600 800 1000 1200 0.475 0.48 0.485 0.49 0.495 0.5 Time (min) In te n si ty (a .u .)

Figure 3.3: Measured output intensity of the laser over a 21-hour period.

Moreover, for practical experimental reasons, we must also require that the light source:

• Has an output power of 60 − 100mW , so that after all the experimental losses, due to light coupling and absorption of light inside the tissue, it is still possible to measure the scattered wave;

• Is highly stable over long periods of time, making it possible to continuously monitor patients over long periods. Ideally the laser must oscillate less than 1% over 24 hours; • Has an optical isolator, such that the back-reflections from the tissue and the fiber

coupling do not damage the laser;

• Must be a compact laser, allowing the system to be portable for clinical applications.

To account for the all the requirements mentioned above, we acquired a 808nm laser, with a nominal power of 100mW (IRM808SA-100FC, LaserCentury). We measured the output stability of the laser using the setup shown in Figure 3.2, obtaining a relative stan-dard deviation of 0.25% over a 21-hour period. The measured output of the laser is shown in Figure 3.3.

Determination of the dynamical properties in turbid media using diffuse correlation spectroscopy : applications to biological tissue = Determinação das propriedades dinâmicas em meios turvos usando espectroscopia de correlação de difusão: aplicações ao te

Rodrigo Menezes Forti

Determination of the dynamical properties in

turbid media using diffuse correlation

spectroscopy: applications to biological tissues

Determina¸c˜

ao das propriedades dinˆ

amicas de meios turvos

usando espectroscopia de correla¸c˜

ao de difus˜

ao: aplica¸c˜

oes ao

tecido biol´

ogico

Campinas

June 2015

Universidade Estadual de Campinas

Instituto de F´ısica “Gleb Wataghin”

Rodrigo Menezes Forti

Determination of the dynamical properties in

turbid media using diffuse correlation

spectroscopy: applications to biological tissues

Determina¸c˜

ao das propriedades dinˆ

amicas de meios turvos usando

espectroscopia de correla¸c˜

ao de difus˜

ao: aplica¸c˜

oes ao tecido biol´

ogico

Abstract

Resumo

Contents

“The only real mistake is the one from which we learn

nothing.”

Henry Ford

Agradecimentos

List of Figures

List of Tables

Abbreviations

Chapter 1

Introduction

Chapter 2

Theoretical Aspects of Diffuse

Optics

2.1

Light-Matter Interaction

2.2

Diffuse Optical Spectroscopy

𝜃

Ω

Ω

(A)

(B)

(C)

2.3

Diffuse Correlation Spectroscopy

𝛽 and Δ𝑟

𝜏

10

10

10

1

1.2

1.4

1.6

τ

( s)

g

(

τ

)

Human Data

Random Flow

Brownian Motion

Chapter 3

Design of a Diffuse Correlation

Spectroscopy System

3.1

Light Source