Broadband time-domain diffuse optics for clinical diagnostics, and diffuse Raman spectroscopy

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Politecnico di Milano

Physics Department

Doctoral Programme In Physics

Broadband Time-Domain Diffuse Optics for

Clinical Diagnostics, and Diffuse Raman

Spectroscopy

Doctoral Dissertation of:

Sanathana Konugolu Venkata Sekar

Supervisor:

Prof. Antonio Pifferi

Assistant Supervisor:

Prof. san

Tutor:

Prof. Paola Taroni

The Chair of the Doctoral Program:

Prof. Paola Taroni

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to Amma (Sai Radha K V), Nanna (Venkata Sekar K),

Akka (Sai Deevana K V),

Ardhangi.

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List of Abbreviations

AOTF Acousto-Optical Tunable Filter APD Avalanche Photo Diode

BFI Blood Flow Index BMD Bone Mineral Density BMI Body Mass Index CCD Charge Coupled Device C Calcaneus

CW Continuos Wavelength DA Diffusion Approximation

DEXA Dual-Energy X-ray Absorptiometry DCR Dark Count Rate

DCS Diffuse Correlation Specroscopy DGC Double Gate Configuration DOT Diffuse Optical Tomography DXA Dual X-ray Absorptiometry EBC Extrapolated Boundary Condition FD Frequency Domain

fNIRS functional Near Infra Red Spectroscopy FNA Fine Needle Aspiration

FORS Frequency Offset Raman Spectroscopy FWHM Full Width Half Maximum

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Hb Deoxygenated Hemoglobin HbO₂ Oxygenated (O₂) Hemoglobin ICCD Intensified Charge Coupled Device IRF Instrument Response Function MCP Micro Channel Plate

ME Memory Effect

MLT Mellin - Laplace Transform NA Numerical Aperture

NIR Near-InfraRed OD Optical Density

PDMS Polydimethylsiloxaneone PMT Photo Multiplier Tube PSC Polarization Splitting Cube QE Quantum Efficiency

RTE Radiative Transfer Equation RD Radius Distal

RP Radius Proximal

SDD Source- Detector Distance SiPM Silicone PhotoMultiplier

SORS Spatial Offset Raman Spectroscopy SNR Signal-to- Noise Ratio

SPAD Single Photon Avalanche Diode SWIR Short Wave nearInfra Red T Trochanter

TD Time Domain

TD-DOS Time Domain Diffuse Optical Spectroscopy TD-DRS Time Domain Diffuse Raman Spectroscopy TOF Time Of Flight

TCSPC Time - Correlated Single- Photon Counting TR Time- Resolved

UD Ulna Distal UP Ulna Proximal

VOA Variable Optical Attenuator ZBC Zero Boundary Condition

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Abstract

The mystery of light-tissue interaction explored by the scientific evolution can be classified into few set of basic phenomena like absorption of photons, Raman effects, fluorescence effect, stimulated emission, two photon absorption, etc. These inventions played significant role in enhancing the specificity of molecular detection, thereby revolutionizing the field of in vitro tissue diagnostics and microscopy. However, the understanding of scattering in tissues is still at its evolutionary stage where steadfast attempts have been made by the scientific community to demystify the secrets of light propagation in diffusive media like human tissues. In the recent decade, diffuse optics has been successful in advancing the studies on light-tissue interaction in diffusive media.

This PhD thesis stands at this juncture of scientific revolution, contributing to the in-vention and revelation of advance techniques to understand the interaction and propagation of photons in diffusive media and there by propelling the next-gen non-invasive diagnosis of biological tissues. The research leading to this PhD dissertation has been carried out in the physics department of Politecnico di Milano (Milan, Italy), and two secondments at The Institute of Photonic Sciences (ICFO - Barcelona, Spain), in collaboration with other organizations (PicoQuant - Berlin,Germany) and hospitals(Hospital Universitari Germans Trias i Pujol - Badalona, and HospitalClinic Barcelona - Barcelona,Spain)

The essence of the thesis can be broken into two parts:

1. Time-domain Diffuse Optical Spectroscopy (TD-DOS) for Clinical Diagnostics. 2. Diffuse Raman Spectroscopy (DRS).

The key achievements of thesis are summarized below: Though diffuse optical studies are possible by continuous wave (CW) or modulated light sources. Time-domain diffuse optics is an advanced tool among them, as it can naturally disentangle absorption from scattering, providing high spatial and depth resolution. The entire DOS part of the thesis

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was carried out with time-domain techniques. Chapter 1 serves as an introduction and provides preliminary basis for the rest of the thesis. Various theoretical models of diffusion exploited by this thesis for data analysis are elaborated on chapter 2.

Chapter:3 Development of broadband (500-1350 nm) TD-DOS spectrome-ter Typically, broadband TD-DOS on human tissues are carried out over 600-1000 nm. Though the tissue spectrum has interesting features beyond 1100 nm, the absence of broad-band detectors coupled with stringent requirements of time-domain optics and theoretical modelling demand a generous spectral extension. During this thesis, a successful exten-sion was achieved by adapting key strategies on source and detection chain optics, fibers, responsivity extension with two detectors and drift and distortion compensation by IRF acquisition. Additionally, this system was designed and validated in the view of clinical application, thus making it a unique masterpiece and an invaluable instrument in the field of TD-DOS.

Chapter:4 Discovery and characterization of new tissue constituents Four tissue constituents namely: collagen, elastin, tyrosine and thyroglobulin were characterized during this thesis period. Moreover, high scattering, fluorescence, boundary effect and bandwidth effects were some of the problems tackled in this work by means of customized probes, Monte Carlo models, bandwidth simulations and fluorescence filters, to extract absorption spectra. The characterized collagen and elastin spectra (500-1700 nm) can be key to new exploration in the beyond range (>1100 nm); breast tissue diagnosis, elasticity assessment being a few potential areas of application. Tyrosine and Thyroglobulin are tissue constituents specific to thyroid organ, and the work done on these constituents has direct impact on non-invasive diagnosis of thyroid pathologies.

Chapter:5 Broadband in vivo studies on: Human bone, Thyroid, and Ab-domen The broadband clinical system developed during this thesis was employed in mul-tiple hospitals. A well deliberated protocol was carried out to measure non-invasively the above-mentioned locations. The presence of collagen in bone tissue revealed by our studies emphasized its relevance to the global diagnosis of bone pathologies like osteoporosis. The high absorption of thyroid organ led us to the discovery of new tissue constituents (thy-roglobulin, tyrosine) that were absent in our data analysis, thus unlocked the new methods in thyroid diagnosis. The preliminary results of the abdomen studies for fat distinction, revealed its potential to distinguish multi-layered structure of abdomen tissues.

Chapter:6 Invention of Frequency Offset Raman Spectroscopy for deep tis-sue Raman Spectroscopy Raman spectroscopy of diffusive media has been explored in the recent decade to extract Raman spectrum of deep tissues. The method based on multi-distance approach, Spatial Offset Raman Spectroscopy (SORS) has revolutioned the CW diffuse Raman spectroscopy. In this dissertation, an alternative technique, Frequency Off-set Raman Spectroscopy (FORS) to probe deep into tissue was proposed and demonstrated on tissue mimicking solid phantoms. The proposed technique is performed at multiple ex-citation wavelengths and utilizes variations in optical properties (absorption, scattering) to probe deep into tissues. The figure depicts the Raman probe and the principle of FORS. FORS has been found to have superior spatial resolution and low signal to noise ratio as compared to SORS thereby making it as a valid alternative to the existing SORS.

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Contents

Chapter:7 Time-domain Diffuse Raman Spectroscopy (TD-DRS)The aim of this work was to transfer the know-how developed in the last 20 years of Time-Domain Diffuse Optics (TD-DO) to the realm of Raman spectroscopy and to lay a strong foundation by the theoretical and experimental realization of TD-DRS. A new analytical model in collaboration with the University of Florence was developed to answer the propagation of Raman photons in diffusive media. Unfortunately, the experimental realization of diffuse Raman is complex due to very low signal (factor of 10-9 as compared to Tyndall) scattering. However, a customized spectrometer and probe were built to enable the sequential detection of time resolved Raman photons thereby opening up new horizons in the branch of TD-DRS. Furthermore, a new detector technology developed by PicoQuant was exploited to make parallel acquisition of TD-Raman curves. Preliminary results revealed feasibility of the TD-DRS measurement with single-photon counting and demonstrated depth sensitivity of the approach.

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CHAPTER

1

Diffuse Optics and Biomedical In Vivo

Studies - An Introduction

In this chapter, the key properties of light-matter interaction in a diffu-sive media and its relevance to biological tissues are elucidated. Mainly three different techniques are discussed: Diffuse optical spectroscopy (DOS), Diffuse correlation spectroscopy (DCS) and Spatial Offset Ra-man Spectroscopy (SORS). A brief overview on various techniques, existing systems and methodologies are briefly introduced. This chap-ter serves as background for the rest of the thesis.

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Ch 1. Diffuse Optics and Biomedical In Vivo Studies - An Introduction

1.1 Transparent vs Diffusive Media

Figure 1.1: Light propaga-tion in a clear medium

Figure 1.2: Light propaga-tion in a diffusive medium

In a given transparent medium, due to negligible scatter-ing, light photons propagate in a straight line. As early as 1729 [1] Beer-Lambert law was proposed to explain absorp-tion in such medium. Fig.1.1 shows the phenomenon of light propagation in a transparent medium. When the number of scattering centers in a medium increases, the photons en-counter scattering along with absorption. In particular, the medium is said to be diffuse when scattering is relatively higher than absorption. Fig.1.2 illustrates an example of light propagation in a diffusive medium. The scattering cen-ters in diffuse media scramble the photon paths in random directions. Importantly, at microscopic level these scatter-ing centers are particles with size comparable to the wave-length of light, irregular shapes, different optical properties (absorption, refractive index) compared to its background. This inhomogeneity at microscopic level in a scale compa-rable to the wavelength of light gives rise to diffusive spread of light at macroscopic level. Typically, in biological media, the tissues are composed of cells and organelles with size comparable to light wavelength making them an ideal dif-fuse media. Diffusion of light is also a common phenomenon seen in nature, propagation of light in milk, clouds, white paint, lipid, foam in tides and waves, tissues are few exam-ples.

In a clear medium, Lambert-Beer law relates absorption to the intensity of transmitted light.

µa= −

ln (I/I0)

L (1.1)

whereas I and I0 represent the transmitted and the incident

light intensities, respectively; L is the distance travelled by photons into the medium, which coincides with the thick-ness of the sample (Fig. 1.1). However, in a diffusive media, owing to scattering, photon undergoes complex path. The attenuation of light can be related to both absorption and scattering events. A complex light propagation model is needed to characterize the absorption (µa) and scattering

(µs) coefficients of a diffusive medium. "Photon Migration"

is a term used by the physicists to refer to the light

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1.2 Light-tissue Interaction

agation in diffuse media. A detailed description on Photon Migration models used in this thesis is provided in chapter 2.

1.2 Light-tissue Interaction

Photons interact with tissues in several ways, of these ab-sorption and scattering are two prominent processes. Ab-sorption attenuates photons as they propagate into the medium. Absorption contrast of light was used as early as 1929 [2] to identify tumors in breast tissues. Furthermore, the discov-ery of low photon attenuation window (therapeutic window, 600-1100 nm) by Jobsis [3] showed high penetration depth of photons deep into tissues, sowing seed to the idea of broad-band in vivo spectroscopy. Scattering can be classified into two kinds, elastic and inelastic scattering. The elastic scat-tering, also known as Rayleigh or Tyndall scattering sim-ply changes the path of light photons whereas the inelastic scattering alters also the energy(wavelength), around visi-ble range of electromagnetic spectrum, Raman scattering is an example. Studying scattering in detail can reveal many structural properties of tissues. In Raman scattering, the newly generated Raman lines holds highly specific informa-tion about chemical and structural properties of molecules in the media. Additionally, in a tissue composed of moving particles, the dynamics of these particles can be studied by using coherence property of light photons.

In last few decades, diffuse optical studies of biological media have been evolved particularly on the above men-tioned properties. Various techniques, methods and method-ologies have been formulated to understand the behavior of light photons in a biological diffusive media. Few of the im-portant techniques exploited by this thesis can be classified into three domains, namely:

• Diffuse Optical Spectroscopy (uses absorption and dif-fuse scattering (Rayleigh) properties of light)

• Diffuse Raman Spectroscopy (uses Raman scattering properties of diffusive media)

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• Diffuse Correlation Spectroscopy (uses Coherece na-ture of light to monitor dynamics of diffusive media) Depending on the kind of light source, detection meth-ods, and probe geometries, each of these domains are further classified multiple branches.

1.3 Diffuse Optical Spectroscopy (DOS)

Diffuse Optical Spectroscopy is a technique to extract opti-cal properties (absorption µa, reduced scattering µ0s) of a

dif-fusive media [4,5]. Most of the in vivo biological tissues are diffusive in nature. Especially, human tissue constituents namely oxy-,deoxy-hemoglobine, water, lipid, collagen have significantly different absorption spectra. Thus, a broad-band spectroscopy on in vivo tissue followed by comparison with individual spectra can quantify the key tissue chro-mophores in the measured location. Unfortunately, the DOS below 600 nm is limited by high absorption of hemoglobin, whereas above 1100 nm, the absorption of water, lack of affordable and efficient detectors are few reasons prevented the detection of deeply propagated photons inside tissue. The later is addressed in this work chapter 3, where a new detection strategy was used to extend the range to 500-1350 nm and 500-1700 nm.

Fig.1.3 shows an example of in vivo spectrum and the corresponding weights of tissue constituent spectra. The concentration of each constituents is extracted by fitting the measured spectrum with tissues constituents spectra as shown in Eq.1.2

µa(λ) =

X

i

cii(λ) (1.2)

where λ is the wavelength of photons, the concentration (free fit parameter) and the specific absorption of constituents are represented by ciand i, respectively. Using this

method-ology, various tissues have been characterized in the litera-ture for clinical diagnostics. Breast cancer assessment [6,7], heamodaynamics in brain tissues [8, 9], optical biopsy of bone tissue (calcaneus bone) [10] are few instances. An extensive work on characterizing human in vivo locations

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1.3 Diffuse Optical Spectroscopy (DOS)

Figure 1.3: Sample in vivo spectrum of human Manubrium with weight spectra of tissue constituents.

were performed as part of this thesis, chapter 4 presents var-ious results and conclusions obtained during clinical trials on humane superficial bone locations, thyroid, manubrium and abdomen fat tissue.

1.3.1 Techniques and Instrumentation

DOS has been realized by different methods by various groups around the world. These methods can be classified based on the spatial and temporal distribution of light source, and the properties of detection system. The famous among them are continuous wave (CW) [11], frequency domain (FD) [12], time-domain (TD) [13], also the hybrid variants like CW-FD [14], modulated imaging [15]. In the following sec-tions, it will be made clear that time-domain has unique na-ture to decouple naturally the optically properties (µa, µ0s),

which gives it an upper hand among the existing techniques. Though Diffuse Optics has been adopted in different fields like for example Diffuse Optical Tomography (DOT) [16], functional Near Infrared Spectroscopy (fNIR) [9], Diffuse Optical Imaging (DOI) [17] etc, i will be focusing only on

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the broadband diffuse optical spectroscopic systems which are of interest to this thesis.

Figure 1.4: Time-Domain technique, pulsed laser exci-tation and time resolved de-tector collection

Time-Domain (TD)

The TD method uses pico-second pulsed lasers and time resolved detectors. Upon propagation into diffuse media, owing to the absorption and scattering, the laser pulse gets broadened. To be more specific, the absorption (µa) affects

the slope of the pulse tail, whereas the scattering (µ0

s) shifts

the peak. Thus, TD naturally decouples the optical proper-ties (µa, µ0s). Fig.1.4 shows the effect on pulse shape upon

propagating in a diffusive medium. Time-Domain Diffuse Optical Spectroscopy (TD-DOS) has been realized in differ-ent configurations. With advancemdiffer-ent of technology in the field of source and detectors, the TD-DOS has been evolved from a laboratory based system to a hand-held compact de-vice [18]. The earlier systems are built on laboratory based high power laser facility and streak cameras which have lim-itation owing to bulkiness, low signal to noise ratio (SNR) and high cost [19]. The advent of new photonic crystal fiber based supercontinuum sources [20] enabled cheaper yet ef-ficient, portable Time-Correlated Single-Photon Counting (TCSPC) techniques [21].

Streak camera based system: one of the early

sys-tem used streak camera coupled to a spectrometer for de-tection [19]. A supercontinuum source producing broadband pulses is used as source. Though this system had an advan-tage of single shot complete spectral acquisition of temporal photon profile, low signal to noise ratio due to small numeri-cal aperture and inherent noise of detection system increased the acquisition time to few minutes. The operational range of the system is also limited to 600-850 nm. The delicate-ness and huge size of the system limited its application to laboratory purposes.

TCSPC based system: Most of the TD systems in

lit-erature are built on TCSPC technology. Cost effectiveness,

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1.3 Diffuse Optical Spectroscopy (DOS)

portability coupled with recent advancement in single pho-ton counting technology have enabled TCSPC as a versatile tool in TD-DOS. One such system, a first of its kind was built by Politecnico di Milano [21]. The system is based on supercontiniuum source and spectrometer based 16-channel time resolved detectors acquisition, this system can acquire 16 wavelengths in one go, but had limitations of SNR owing due to low numerical aperture (NA), and challenged in vivo studies owing to maximum admissible power on subjects.

Alternatively, sequential systems are proposed [13, 22, 23]

Figure 1.5: Optical layout of one of the earlier sequential acquisition based TCSPC system [13]

. Essentially, sequential approach is better than parallel acquisition systems in many ways, it provides more flexi-bility in detector selection with specific temporal and spec-tral properties, optimization of power and acquisition time for each wavelength, even fluorescence can be eliminated by proper use of filter or monochromator. Importantly, it lim-its the power on the in vivo sample to single wavelength instead of broadband illumination, preventing its heating or damage. However, long acquisition time is a key drawback to this approach, nevertheless, this can be tackled with im-proved detection efficiency. Fig.1.5 depicts an optical chain of a typical TCSPC system [13], the operational range is between 550 to 1050 nm.

Continuous Wave (CW)

As the name suggests, the CW technique uses CW light sources and detectors, thus, its a cost effective and widely

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applied technique. However, the key limitation is that op-tical properties (µa, µ0s) are entangled in CW technique.

Figure 1.6: Pictorial depiction of two variants of Continuous Wave (CW) technique

They can be disentangled to some extent by performing spa-tially resolved (multiple source detector separation) or an-gle resolved measurements, which is depicted pictorially in Fig.1.6. To be more specific, light collected at short source detector separation is affected more by scattering than at large separation. The analytical solution to extract the opti-cal properties can easily be obtained by integrating TD solu-tion [24]. Even implementasolu-tion of broadband CW system is straight forward as it only needs CW light like incandescent sources and CCD camera coupled to spectrometer. In liter-ature, successful CW systems have been developed [25, 26] and employed in various in vivo studies [27].

Frequency Domain (FD) and hybrid FD-CW

Figure 1.7: Principle of Frequency Domain (FD) technique

In Frequency domain, the source injected into the sample is modulated at radio wave frequencies, the optical prop-erties are extracted from amplitude and phase changes at sample output for different modulated frequencies. In brief, diffusive medium acts as a low pass filter, lets amplitude to decrease and phase to increase with increasing frequency. A new concept of diffuse photon density wave (DPDW) [28,29] is introduced. According to DPDW, the photons in the ran-dom walk collectively contribute to the spherical wave of photon density. DPDW obeys all propagation law like elec-tromagnetic waves. The analytical solution for FD can also be obtained from Fourier transform of TD solution. Like TD, FD also naturally decouples optical properties. How-ever, in literature, owing to difficulties of broadband mod-ulators, FD systems are limited to only few wavelengths [30,31].

In recent years, a hybrid FD-CW has been devised to exploit the best of both FD and CW technologies. In this tech-nique, broadband measurements are performed with CW light and FD measurements are performed at few points to extracted mie scattering coefficients of diffusive media. The extracted scattering values are used to decouple absorption from broadband CW measurements, thus obtaining broad-band optical properties. FD-CW has been developed [14]

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1.4 Diffuse Correlation Spectroscopy (DCS)

and applied for breast related in vivo diagnosis [32].

From the above discussions, it is clear that each tech-nique has there own set of pros and cons. Certainly, TD holds more information about diffused photons and natu-rally disentangles optical properties. However, expensive single photon detectors, pulsed lasers make TD less cost ef-fective. CW and FD techniques are cheap and cost effective, but come at the price of uncertainty in predicted optical val-ues. A hybrid FD-CW approach can be a viable solution, still, the assumption (scattering follows Mie law) made may fail under certain conditions. The DOS part of this the-sis uses TD for characterizing new tissue chromophores and various in vivo locations which is presented in upcoming chapters 4 5.

1.4 Diffuse Correlation Spectroscopy (DCS)

Diffuse Correlation spectroscopy (DCS) quantifies the dy-namic property of particles in a diffusing medium [33, 34]. In biological media, assessing dynamic property of tissue like blood flow is a typical example [35]. DCS is a powerful technique for monitoring dynamic changes and has been val-idated against various existing standard like perfusion mag-netic resonance imaging [36,37], laser Doppler [38], Doppler ultrasound [39]. However, knowledge of optical properties (µa, µ0s) is necessary for DCS to make absolute assessment of

blood flow rate in tissues. Therefore, often the DCS systems are combined with DOS system that provide optical prop-erties (µa, µ0s) for the absolute quantification of the blood

flow of tissues [40]. In our thesis, the in vivo studies are performed using both TD-DOS and DCS techniques. The optical properties input needed by DCS is provided by clin-ical TD-DOS system, thus enabling the combined system to assesses both static (µa, µ0s) and dynamic (blood flow)

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1.5 Diffuse Raman Spectroscopy (DRS)

In recent times, Raman spectroscopy has emerged as an important tool to probe biological diffuse media [41–43]. High specificity to chemical and structural properties of the molecule makes it an unique tool in biomedical diagnosis. Unlike DOS, DRS needs only a single excitation wavelength and generated broadband Raman lines, providing molecular information of a diffusive medium. Raman lines being 106

times weaker than excitation photons is the key limitation of DRS, however, increasing technological advancement in detection systems has improved collection efficiency. Typi-cally, biological tissues are fluorescing media. But, a cafully chosen red shifted Raman exciting wavelength can re-duce the fluorescence that overwhelms the Raman signal. Similar to DOS, DRS has been realized in both time

do-Figure 1.8: Principle of Spatial Offset Raman Spec-troscopy

main [44] and continuous wave [45]. A brief overview on recent technological advancement in DRS that are interest to this thesis work is given in the following subsections.

Spatial Offset Raman Spectroscopy (SORS)

SORS is the most widely used current tool in DRS domain. It work on the simple, yet powerful principle that large source detector separation d carries the depth information of the diffusive medium [45]. Fig.1.8 depicts the principle of SORS. Its invention in 2005 can be considered as one of the important turning point in the deep layer Raman spectroscopy of diffusive media. This technique has found wide range of applications; clinical diagnostics of breast can-cer [46], bone diagnosis [47], pharmaceutical quality con-trol [48], explosive detection in airport [49] are few key ar-eas where it has shown significant advancement. However SORS suffers from low spatial resolution due to measure-ments at multiple source detector separations and low sig-nal to noise (SNR) ratio at large d. Importantly, during this thesis period we have developed a new technique called Frequency Offset Raman Spectroscopy (FORS) which can extract deep layer information with relatively high spatial resolution and SNR as compared to SORS. More detailed

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1.5 Diffuse Raman Spectroscopy (DRS)

description of FORS can be found in chapter 6.

Time-Domain DRS

Figure 1.9: Principle of Time-Domain Dif-fuse Raman Spectroscopy (TD-DRS)

Time domain diffuse Raman spectroscopy (TD-DRS) is an emerging tool in biomedical diffuse optics [50]. A system based on temporal gating has shown to remove fluorescence by gating earlier photons [51]; as fluorescence is not an in-stantaneous process like Raman. Importantly, with tempo-ral gating of Raman photons emerging from diffusive medium, we can slice the Raman information emerging from various layers of the medium, this is the concept of late photons carrying deep information which is shown diagrammatically in Fig.1.9. Though literature witnessed few TD-DRS sys-tems based on Kerr gating [50], gated SPAD [52], gated intensified CCD [44, 53], all these works reported the lack of time resolved Raman signal compared to CW Raman. During this thesis period, for the first time in literature, a TCSPC camera based TD-DRS system has been setup using a novel time resolved micro-channel plate camera. In-terestingly, our preliminary studies showed that our system has similar count rate compared to CW systems thus mak-ing TD-DRS technically more feasible like CW Raman. An elaborate description on our system and various discoveries can be found in chapter 7. Along with instrumentation, few fundamental problems pertaining to quantification of TD-DRS signal are answered during this thesis period. To this end, a key part of this thesis was devoted to the development of new analytical tools and simulations to model migration of Raman photons in diffusive media which is discussed in detail in Chapter 7.

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CHAPTER

2

Photon Migration in Diffusive Media

-Theory

This chapter elaborates on analytical tools that were used in this thesis dissertation. Both TD-DOS and DRS are key as-pects of this thesis. Analytical solutions for different geometry of diffusion media are derived and stated, a brief account on derivation of diffusion equation from parent radiative trans-port equation is provided. A new methodology to estimate penetration depth of photons is elaborated in chapter 4, how-ever, an introduction to basic analytical solution of penetra-tion depth is provided here. Importantly, this chapter doesn’t emphasis the details of TD-DRS, as the work is part of this dissertation which is elaborated in chapter 6

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Ch 2. Photon Migration in Diffusive Media -Theory

As discussed in Chapter 1, owing to high scattering, the photons undergo random walk in a diffuse media. Model-ing light propagation in such medium is challengModel-ing, though, the Maxwell equation is considered as an ideal tool to model photon propagation. In practice, it turns out to be com-plex and cumbersome. As pointed out by Ishimaru [54,55], there are not yet any practical models to describe multiple-scattering problems. However, an analytical based empirical solution can be less complex and can provide useful practi-cal models for real time application. One solution is based on the historic radiative transport equation (RTE). Though this theory lacks a rigorous mathematical treatment, it has been demonstrated that under certain assumptions it can be derived from electromagnetic theory in discrete random me-dia considering multiple scattering [56]. RTE solution with certain assumptions and modeling can be implemented to diffusive media, the corresponding solution is called diffu-sion equation (DE), which will be discussed in detail in the following sections.

2.1 Modeling Photon Migration: RTE

to DE

Historically, Radiative transport equation was developed by astrophysics [57], and later on used to describe neutron en-ergy transport in nuclear reactor [58]. RTE has been suc-cessfully applied to a diffusive medium [59, 60]. In this model, the medium is considered to be distributed with ab-sorption and scattering centers; abab-sorption annihilates pho-ton while the scattering elastically deviates the initial tra-jectory.

The complex integro-differential nature of RTE equa-tion makes it computaequa-tionally intensive [61]. Recently, few analytical solutions have been proposed [59] for infinite ran-dom media, and also numerical methods [60, 62] or analyt-ical approximations are usually considered. However, the complexity of RTE can be further simplified with diffusion approximation (DA), which has a less complex solution for different configurations of system. DA assumes that occur-rence of scattering events are much higher than absorption

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2.1 Modeling Photon Migration: RTE to DE

events. RTE with DA has been the major tool in Diffuse optics to probe turbid media.

2.1.1 Radiative Transfer Equation

The basic principle of RTE is balance of energy (principle of conservation) in photons propagating in a volume ele-ment dV for a given absorption and scattering. It can be obtained by balancing the various mechanisms that are re-sponsible for radiance changes in volume element dV . The time dependent form of RTE can be written as

1 v ∂I(~r, ˆs, t) ∂t = − ∇·[ˆsI(~r, ˆs, t)] − (µa+ µs)I(~r, ˆs, t)+ + µs Z 4π p(ˆs, ˆs0_)I(~_{r, ˆ}_{s, t) dΩ}0_{+ (~}_{r, ˆ}_{s, t)} (2.1)

where (~r, ˆs, t) is the source term representing the pho-ton generated per unit volume, solid angle along ˆs and Ω0_,

The physical interpretation of various terms in Eq. (2.1) is as follows:

- ∇·[ˆsI(~r, ˆs, t)] is the propagation of net energy flux through the volume element dV along ˆs;

- I(~r, ˆs, t) [W m−₂_sr−₁_{] is the Radiance, represents the}

average power flowing through the unit area at time t in direction ˆs per unit solid angle;

- µa [cm−1] and µs [cm−1] are the probability of

pho-ton absorption and scattering per unit length, respec-tively;

- (µa+ µs)I(~r, ˆs, t)is the energy loss due to absorption

and deflection of photons along directions different from ˆs; the mean free path of light is represented by (µt)−1 = (µa+ µs)−1;

- p(ˆs, ˆs0₎_{is the scattering phase function defined as the}

probability of a photon changing its propagating di-rection from ˆs0 _{to ˆs within the unit solid angle dΩ;}

- µs

R

4πp(ˆs, ˆs

0_)I(~_{r, ˆ}_{s, t) dΩ}0_{represents the gain in energy}

due to photons arriving with a random initial direc-tion which are deflected in the ˆs direcdirec-tion;

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- v represents the speed of light in diffusive media. A Green’s function solution [63] can be obtained for Eq. (2.1), provided the source term (~r, ˆs, t) = δ3_(~_r−~_r0_)δ(ˆ_{s− ˆ}_s0_)δ(t−t0₎

is a Dirac delta function. Considering linear nature of RTE, a general source distribution can be written as a superposi-tion integral as follows:

I(~r, ˆs, t) = Z V Z 4π Z ∞ −∞ IGF(~r, ~r0, ˆs, ˆs0, t, t0)(~r0, ˆs0, t0) dV0dΩ0dt0 (2.2) where I(~r, ˆs, t) represents the complete solution whereas IGF(~r, ~r0, ˆs, ˆs0, t, t0) is the Green function solution to the

RTE for a delta source.

Treatment of Absorption: Interestingly, if the source term is a delta function in time, i.e. (~r, ˆs, t) = S(~r, ˆs)δ(t), the effect of the absorption can be treated separately: con-sider I(~r, ˆs, t) as a solution of a non-absorbing medium then the solution considering absorption can be written as

Ia(~r, ˆs, t) = exp(−µavt)I(~r, ˆs, t) (2.3)

where µa is the absorption coefficient of the medium.

2.1.2 Diffusion Approximation

As discussed before, DA is the widely preferred and less complex tool that provides wide range of solutions to model diffusive media. The most successful models of Diffusion Approximation [54,64–69] are built on the following assump-tions:

- Isotropic radiance: as a result of dominance of scattering events over absorption events, the radiance term can be written as

I(~r, ˆs, t) = 1

4πΦ(~r, t) + 3

4πJ (~~ r, t) · ˆs (2.4) where Φ(~r, t) [W m−2_]_{and ~}_{J (~}_{r, t)} _{[W m}−2_]_represent

the isotropic Fluence and a small directional Flux, respectively;

- Negligible flux variation, the change in flux of pho-tons in a time corresponding to a mean free path

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2.1 Modeling Photon Migration: RTE to DE

(t = (vµt)−1) is negligible as compared to the total

Flux;

- Radial symmetry, the modeled problem has radial symmetry: i.e. p(ˆs, ˆs0₎_{is a function of (ˆs · ˆs}0₎ _alone;

- Isotropic light source

On implementing these assumption, the simplified time-dependent diffusion equation (DE) from RTE for a homogeneous medium [70] can be written as 1 v ∂ ∂t− D∇ 2_{+ µ} a Φ(~r, t) = q0(~r, t) (2.5) where q0(~r, t) = R

4π(~r, ˆs, t), dΩ is the isotropic light

source term, and D = 1 3µ0

s

represents the diffusion coeffi-cient. µ0

s= µs(1−g)and g =< cos θ >= 2π R₀πcos θp(θ) sin θ dθ

are reduced scattering and anisotropy coefficient, respec-tively. Anisotropy coefficient is defined as the average co-sine of the scattering angle. The transport mean free path l0 = (1/µ0_s) is the inverse of reduced scattering coefficient, which represents the mean distance traveled by photons in a homogeneous medium before they forget their initial di-rection.

DA is independent of absorption µa [71,72], however, in

analogy to Eq.2.3, if Φ(~r, t) represents a solution to Eq. (2.5) for a non-absorbing medium with a delta function in time as source, then

Φ(~r, t, µa) = Φ(~r, t, µa= 0) exp(−µavt) (2.6)

is the extension of solution for the same medium by con-sidering absorption coefficient µa. The flux can be related

to fluence by Fick’s Law [73] ~

J (~r, t) = −D∇Φ(~r, t) (2.7) Thus the property in Eq.2.6 can be extended to flux and radiance.

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Figure 2.1: Positive (red) and negative (blue) sources for the method of imges in case of the Extrapolated Boundary Conditions and the Zero Boundary Conditions, respectively

2.1.3 Boundary Conditions

In practical scenario, while dealing with real samples certain boundary condition are imposed on Eq. (2.5) [74–76] to ac-count for their finite dimension. For example, at the sample interface, due to refractive index mismatch part of light is reflected back into sample which need to be addressed using boundary conditions.

The most commonly used boundary conditions are Zero Boundary Condition (ZBC) [77, 78] and the Extrapo-lated Boundary Condition (EBC) [74–76]. Simple as-sumption of fluence equal to zero at physical boundary is made by ZBC. Though this works most of the time, under certain conditions it fails leading to overestimation of ab-sorption. However, EBC sets the Fluence to zero on an ex-trapolated boundary surface situated at distance ze= 2AD

from its boundary, outside the medium; ze effectively

ac-counts to refractive index miss-match with A whereas the optical properties are taken into account by D.

Method of Images: To implement the boundary condi-tions into diffusion equation at the physical boundary or at ze, a simple method of images tools are used [76,77,79]. In

this method, an infinite number of pairs of positive and neg-ative delta-sources are placed in addition to the real source in such a way that fluence goes to zero at proposed bound-ary. Fig. 2.1, depicts the placement of positive and negative sources to make fluence zero on sample boundary and ex-trapolated boundary for ZBC and EBC, respectively.

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2.2 Analytical Solutions for TD-DOS

One more possible boundary condition is known as the Partial Current Boundary Condition (PCBC) [54,74– 76,79]. The PCBC is expressed as h Φ(~r, t) − 2A ~J (~r, t) · ˆq i ~r∈Σ= 0 (2.8)

where ˆq is the unit vector normal to the surface Σ outwardly directed and A takes into account the refractive index mis-match at the boundary of the sample with environment.

Usually, the EBC is preferred over ZBC, EBC gives more accurate results and are in better agreement with the PCBC.

2.2 Analytical Solutions for TD-DOS

In this section, useful solutions of TD-DOS that are of prac-tical use to assess opprac-tical properties (µa, µ0s) are presented.

These include slab, semi-finite, parallelepiped models. Most of the biological media fall under the above mentioned mod-els. Method images solution and practical tips are pre-sented.

The solution to Eq. (2.5) for an isotropic delta source in an infinite homogeneous medium is given as

Φ(~r, t) = v (4πDvt)3/2exp − r 2 4Dvt− µavt (2.9)

where ~r represents the source-detector separation [70, 78]. Importantly, this solution is valid only for time events greater than t = r

v, which is the minimum time taken by ballistic photons to reach the detector.

As described in the previous section, the boundary con-ditions for different geometries can be manipulated. In prac-tice, neither the flux nor the fluence are measurable quan-tities. Hence, what we measure is the number of photons crossing the surface per unit area per unit time at a distance ρ from the z-axis. In general, this can be conceived in two geometries, Reflectance R(~r, t) and Transmittance T (~r, t):

R(ρ, t) = ~J (ρ, z = 0, t) · ˆq = D ∂

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T (ρ, t) = ~J (ρ, z = s, t) · ˆq = D ∂

∂zΦ(ρ, z = s, t) (2.11) where s is the thickness of the sample and ρ = px2_{+ y}2;

Fick law has been used to obtain the relation between R(~r, t) (T (~r, t)) and the fluence.

The laterally infinite and homogeneous Slab is a com-mon geometry widely used in Photon Migration studies. Most of the biological media is made of a diffusive medium bounded by parallel planes [80],e.g. biological tissues [76, 78,81–83], and also slab model is handy for finite thickness transmittance measurements. Analytical solutions for the Reflectance and the Transmittance of such medium can be retrieved by solving Eq. (2.5) with EBC and making use of Eq. (2.10) and (2.11): R(ρ, t) = −exp(− ρ2 4Dvt − µavt) 2(4πDv)3/2_t5/2 × +∞ X m=−∞ z3mexp −−z3m 2 4Dvt − z_4mexp −−z4m 2 4Dvt (2.12) T (ρ, t) = −exp(− ρ2 4Dvt − µavt) 2(4πDv)3/2_t5/2 × +∞ X m=−∞ z1mexp −−z1m 2 4Dvt − z_2mexp −−z2m 2 4Dvt (2.13) with _                         z1m= (1 − 2m)s − 4mze− zs, z2m= (1 − 2m)s − (4m − 2)ze+ zs, z3m= −2ms − 4mze− zs, z4m= −2ms − (4m − 2)ze+ zs, m = 0, ±1, ±2, · · · ± ∞, ze= 2AD.

Another important geometry to be considered is the Paral-lelepiped[84], this solution plays a key role especially when the physical dimension of sample is few hundred times the photon mean free path. Let us consider a homogeneous

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allelepiped of diffusive material with dimensions Lx, Ly, Lz.

To manipulate boundary conditions in 3D geometry, a lat-tice of positive and negative sources is applied at 6 planes to satisfy EBC located at x = −ze, y = −ze, z = −ze, x =

Lx+ ze, y = Ly+ ze, z = Lz+ ze. In general, transmittance

is the most common measurement configuration adopted for parallelepiped samples. For a delta source with unitary strength located at ~rs = (xs, ys, zs)the solution is

T (x, y, t) = 1 2(4πDv)3/2_t5/2exp(−µavt) × +∞ X l=−∞ exp −(x − x1l) 2 4Dvt − exp −(x − x2l) 2 4Dvt × +∞ X m=−∞ exp −(y − y1m) 2 4Dvt − exp −(y − y2m) 2 4Dvt × +∞ X n=−∞ (Lz− z1n) exp −(z − z1n) 2 4Dvt − (Lz− z2n) exp −(z − z2n) 2 4Dvt (2.14) where _                                     x1l = 2lLx+ 4lze+ xs, x2l = 2lLx+ (4l − 2)ze− xs, y1m= 2mLy+ 4mze+ ys, y2m= 2mLy+ (4m − 2)ze− ys, z1n= 2nLy+ 4nze+ zs, z2n= 2nLz+ (4n − 2)ze− zs, l, m, n = 0, ±1, ±2, · · · ± ∞, ze= 2AD.

2.2.1 Scope of Diffusion Approximation

The above solutions are valid and applicable only within the scope of diffusion approximation. Key assumption of DA is that scattering is far less than absorption which en-sures the validity of isotropic source, small change flux in a mean free path. Furthermore, isotropic radiance Eq. (2.4) holds true only if Φ(~r, t) 3| ~J | [85]. The source detector

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separation needs to be sufficiently large for the photons to come under DA model, thus, the large source detector sep-aration has better validity. The rise of absorption deviates the model from the diffusive regime, leading to coupling be-tween absorption and scattering under high absorption µa

condition [70].

2.2.2 Numerical Methods: Monte Carlo

Approach

One of the ways to overcome the limitation of DA and im-prove the accuracy is to use Monte Carlo (MC) method [81, 86–92]. In this framework, photon migration in diffu-sive media can be formulated as a stochastic process and a random number generator is used to simulate the physical process of light propagation. In this way, there is no need to simplify the assumptions, and the accuracy is limited only by the intrinsic stochastic nature of the method.

In MC approach, every physical quantity governing pho-ton migration can be treated probabilistically:

- µa is the absorption probability that a photon is

ab-sorbed per unit-length;

- µs is the scattering probability that a photon is

scat-tered per unit-length;

- µa+µsis the probability per unit-length that a photon

is either scattered or absorbed;

- the phase function p(ˆs, ˆs0₎ _{is the probability that a}

photon coming along the direction ˆs is scattered along the new direction ˆs0_.

In MC framework, three random variables are defined to simulate the photons trajectories inside the medium:

* the photon step size l;

* the deflection angle θ of the photon after each scat-tering event;

* the azimuthal angle φ of the photon after each scat-tering event.

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Modeling MC variables: As mentioned above each phys-ical variable has to be modeled accordingly with random variables based on probabilities.The cumulative probability function step size of photon propagation in the medium fol-lows an exponential trend:

P(l) = 1 − exp(−µsl) (2.15)

P(l)is mapped onto a uniform distribution U[0, 1]. From the above relation a pseudo-random number ξ can be generated between 0 and 1, this can be associated to step-size l as follows

1 − exp[−µsl(ξ)] = ξ (2.16)

on rewriting the above equation

l(ξ) = −ln(1 − ξ) µs

= −ln(ξ) µs

. (2.17) where l(ξ) is sampled step-size. Similarly, the probability distribution function for the random variable θ can be writ-ten as

P(θ) = 2π Z θ(ξ)

0

p(θ0) sin(θ0) dθ0 = ξ (2.18)

here p(θ) is the phase function which is mapped on to uniform distribution function U[0, 1]. Eq. (2.17) is a random number uniformly distributed between 0 and 1, thus, 1 − ξ will also be a random number with the same distribution and thus we can replace (1 − ξ) with ξ. For simplified analytical models, Eq. (2.18) can be analytically inverted. Most often in biological tissues, Henyey-Greenstein (HG) function for the phase function which is defined as

p(θ) = 1 4π

1 − g2

(1 + g2_{− 2g cos θ)}3/2 (2.19)

The HG function is completely characterized by the pa-rameter g, which is an asymmetry factor for the function: hcosθi = g. For g = 0 the function is completely isotropic. If p(θ) follows the Mie Theory instead of HG function, the relation (2.18) has to be evaluated numerically to obtain θ = θ(ξ).

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uniform distribution in the range [0, 2π].

Running simulation: Multiple photons are launched inside the medium, the trajectories of photons are evaluated using the three random variable, which in turn are gener-ated by above mentioned methods using random number ξ. When the photon reaches the boundary, a pseudo-random number is mapped onto Fresnel coefficient to decide whether to back reflect or transmit it outside the medium. Snell’s law is applied in case of back-reflection. The photons are killed when the total travel length ltotof the photon crosses lM AX.

When photons arrive out of detector boundary, the travel length of photons are saved. The temporal point spread function (TSPF) is then reconstructed by converting pho-ton arrival histogram to their time-of-flight (tof) t = ltot/v,

where v speed of light in the diffusive medium.The tem-poral resolution is determined by the time bin size of the histogram.

MC technique yields more reliable results, as it is a out-come obtained by considering strictly the stochastic nature of the medium. However, the main setback comes from the central limit theorem which is proportional to number of photons received in the histogram. In brief, higher the number of received photons the better the outcome of MC simulation, making it a computational intense and time con-suming process. In the routines used in this dissertation, the launched photons are considered with zero absorption, later the absorption is added separately by obeying the Eq.2.3.

2.3 Optical Properties (µ

a

, µ

0_s

)

Estima-tion

The time-resolved measurements provide us the Reflectance/-Transmittance photons temporal distribution. To extract optical properties of the sample we need to employ either an inversion algorithm to the analytical formulae provided in the previous section (2.2) or to use a numerical method like MC simulations.

To apply inversion algorithm to an analytical model, the ex-perimental data are fitted to the analytical solutions of the DA. In practice, the real source is neither an isotropic nor a

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2.3 Optical Properties (µa, µ0s) Estimation

delta-function thus the convolution of theoretical curve with the Instrumental Response Function (IRF) is carried out to take into account the temporal width of the light source in-jected, the response time of the detector and the dispersion in optical fibers. Since the DA does not account for the early photons, the fitting range is usually set to exclude the points of the time-resolved curves with a number of counts lower than 80% of the peak value on the rising edge. Furthermore, to avoid distortions due to poor counts the tail of the curve is cut at 1% of the peak value. The free fit parameters are the optical properties (µa, µ0s). The Levenberg-Marquardt

algorithm for non-linear minimizations provides best fitting results [93]. Fig. 2.2 shows a typical example of fit per-formed to extract optical properties (µa, µ0s).

To retrieve optical properties (µa, µ0s) with a numerical

Figure 2.2: A typical example of fit. Green and red curves represent IRF and measurement, respectively. The overlap-ping yellow curve on red curve represents the theoretically fitted curve and the corresponding residuals at each point is shown below the curves

method, the first step is to create a library of MC sim-ulations at different scattering coefficients of interest and zero absorption; a CUDA (Compute Unified Device Archi-tecture) accelerated Monte Carlo code [94] is used for this

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purpose. In particular, pivot simulations are obtained sim-ulating curves at different scattering values according to a geometric progression, in order to have a denser sampling for low scattering values and a less dense sampling for higher scattering values. To include the effect of the absorption and thus to obtain a simulation at an arbitrary reduced scattering and absorption coefficients, a linear interpolation between pivot simulations is performed, followed by the mul-tiplication for the factor exp(−µavt), where v is the speed

of light. The simulated curve is then convoluted to the IRF and a Levenberg-Marquardt optimization procedure is per-formed [93]. This analysis method is usually applied when-ever the Diffusion Approximation fails, e.g. when a high absorption of the sample is involved or for small sample di-mensions [95].

Apart from optical properties (µa, µ0s) as free fit

param-eters we can also include a third variable, that is the time shift between the reference and the data. This factor is usu-ally not set as a free fitting parameter, but rather it is kept fixed to an optimum value: the optimum is generally found fitting the shift together with the absorption and scattering coefficients for the data curve with the lowest absorption. This procedure helps to compensate for the inadequacy of standard diffusion at early times and to account for eventual temporal drifts due to misalignments or thermal effects [96].

2.4 Broadband Diffuse Spectroscopy

Extending the measurements on a broad spectral range, we are able to reconstruct both the absorption and the reduced scattering spectra: from these we can extract valuable infor-mation both on the chemical composition and on the micro-structure of the medium under investigation.

2.4.1 Second Level Fit

The absorption spectrum of a sample is related to the ab-sorption of its chemical constituents through the Beer’s law:

µa(λ) =

X

i

cii(λ) (2.20)

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2.4 Broadband Diffuse Spectroscopy

where ciand i are the concentration and the extinction

co-efficient of the i-th constituent, respectively. An important implication of Eq. (2.20) is that it is possible to fit the exper-imental absorption spectrum with a linear combination of the key constituent spectra to obtain their concentrations. A typical example of secondary fit is shown in Fig. 1.3.

Also the scattering spectrum can be analyzed taking into account its trend over a broadband spectrum. The phe-nomenon of scattering is due to the index mismatch among different structures: for example, in biological tissues it is caused by the light interaction with different cellular or-ganelles and membranes. Usually, in the wavelength range 600-1100 nm the reduced scattering is a decreasing function of the wavelength. A heuristic model, based on experimental results, has been developed expressing the reduced scatter-ing coefficients as a function of the wavelength:

µ0_s= a λ λ0

b

(2.21) where λ0 is a fixed reference wavelength, a is the scatter

amplitude and b is the scatter power [97]. Equation (2.21) can be interpreted in the light of simulations performed ap-plying the Mie Theory [97] and the two parameters a and b can be related to the properties of the scattering particles, assumed in a first approximation as homogeneous sphere with equal radius r: in particular, a stems from the par-ticles concentration and b depends on the equivalent Mie radius.

Hence, performing a second level fit after the optical parameters µa and µ0s have been obtained allows for the

re-covery of the main constituents concentrations and of two parameters linked to the micro-structure of the sample. Al-though these information are derived with the assumption of a homogeneous medium and there is no direct correlation of a and b with the physical reality of the sample structure, the information yielded by the constituents concentrations and the scatter amplitude and power are extremely valuable and can be used for example for classifying different types of tissues and for monitoring physiological or pathological alterations.

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Instead of applying a second level fit, it is possible to use the information coming from different wavelengths di-rectly during the inversion procedure, with the advantage of a better robustness and an increased stability [98–101]. The time-resolved Transmittance can be re-written uncou-pling the absorption term from the others which depend only on the reduced scattering:

T (ρ, d, t) = S(t, µ0_S) exp(−µavt) (2.22)

and making use of Eq. (2.20) and Eq. (2.21) we can write

T (t, λ, a, b, ci) = S(t, a λ λ0 b ) exp −vtX i cii(λ) ! (2.23) Then, for each wavelength the theoretical curve is convo-luted with the IRF of the system and a Levenberg-Marquardt algorithm is applied for minimizing the difference between the experimental and theoretical vectors of all the time-resolved curves acquired at the different wavelengths.

It is also possible to use a subset of wavelengths, with-out performing a real broadband measurement. Anyway, since there is always the possibility that the total absorp-tion spectrum is due also to unknown or disregarded ab-sorbers present at low concentrations (e.g. for biological tissues some proteins or biomolecules), it is preferable to adopt a broadband approach, exploiting the additional spec-tral information to better recover the concentration of the constituents. Moreover, time-resolved curves are affected by noise and the samples are not really homogeneous: any-way, with a broader wavelength range of investigation it is possible to overcome these limitations.

2.5 Penetration Depth in TD-DOS (DCS)

The knowledge of penetration depth of photons can help in understanding the contribution of different layers in a heterogeneous medium. Especially, while measuring multi-layer medium like superficial in vivo bone location, pene-tration depth calculations can quantify the contribution of deep layer over superficial layer. Different empirical

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tions based on analytical models have been proposed to mea-sure the mean maximum penetration depth of the medium [102, 103], recently, a completely analytical based solution has also been published [104]. In a slab of thickness s0, the

photons having maximum penetration depth zmaxbetween

z and d + dz for a given time t at a distance ρ is given by

Figure 2.3: hz_maxi varia-tion as a funcvaria-tion of ρ and µa

Figure 2.4: hzmaxi

varia-tion as a funcvaria-tion of ρ and µ0_s

R(s = z + dz, ρ, t) − R(s = z, ρ, t)

R(s = z, ρ, t) , (2.24) where R(s = s0, ρ, t) is TD-DOS reflectance of slab

cal-culated using 2.12 of thickness s0, the probability function

for the maximum penetration depth to be between z and z + dz can be formulated as f (z|t) = 1 R(s0, ρ, t) R(s = z + dz, ρ, t) − R(s = z, ρ, t) R(s = z, ρ, t) (2.25) Now, the mean maximum penetration depth is simple mean value of Eq.2.25, which can be written as

hz_max|ρ, ti = Z s0

0

zf (z|ρ, t)dz (2.26) Eq:2.26 provides the mean maximum penetration depth of photons, Fig.2.3 and Fig.2.4 show the effect of optical prop-erties and ρ on penetration depth. There is also an an-other parameter called mean penetration depth which can be heuristically defined as hzi = hzmax/2i, gives the mean

penetration depth reached by photons in a diffusive medium.

2.6 Diffuse Correlation Spectroscopy (DCS)

DCS estimates the fluctuation of scattering centers in the diffusing medium. In biological tissues, DCS can monitor fluctuations of scattering centers like red blood cells, thereby monitoring the micro vasculature of tissues [35, 105]. Ini-tially, DCS was formulated as Diffusing Wave Spectroscopy (DWS) [33,106]. In general, DCS employs correlation trans-port theory to deduce diffusion equation of field auto-correlation function. The correlation fluence rate from correlation

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dif-Ch 2. Photon Migration in Diffusive Media -Theory

fusion equation [34] is written as h ∇.(D(r)∇) − vµa(r) − α 3vµ 0 sκ2h∇2(τ )i i G1(r, τ ) = −vS(r). (2.27) where, α denotes the fraction of photon scattering events among the moving particles in the medium. G1is correlation

fluence rate,

G1(r, τ ) =

Z

4π

GT₁(r, Ω, τ )dΩ = hE∗(r, t).E(r, t+τ )i (2.28) -E(r, t) denote the electric field at (r,t), and the isotropic source term S(r) is given by

S(r) = Z

4π

Q(r, Ω)dΩ (2.29) -h∇2_{(τ )i}_{represents the mean square displacement in time τ}

of the scattering centers. κ0 = 2π/λ is the wavenumber of

CW light. The Green’s function solution for the correlation diffusion equation2.27, in case of semi-infinite homogeneous medium is given as G1(ρ, z, τ ) = v 4πD exp(−K(τ )r1) r1 −exp(−K(τ )rb) rb (2.30) where K(τ) = p(µa+ αµ0sκ20h∇2(τ )i/3)v/D). It is

impor-tant to notice that the diffusion equation is applicable to the electric field auto correlation function whereas the DCS instrumentation measures intensity auto correlation func-tion. However, these two can be related to each other by the Siegert relation:

g1(τ ) =

s

g2(τ ) − 1

β (2.31) where β is a constant and is intrinsic to the instrumentation collection optics, approximately equal to inverse of number of detected speckles. The fitting process is begins by fit-ting experimentally obtained correlation curves with theo-retical intensity autocorrelation curves g2 predicted through

Siegert relation. Importantly, as we can see from Eq:2.30, g2 is also a function of absorption and scattering. Thus to yield

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accurate results we need to feed values of optical properties. Normally, DCS systems are accompanied by DOS system to access optical properties [107]. The major scatterers at 785 nm are organelles and blood cells, for this reason DCS is often exploited at 785 nm wavelength [40, 108]. In our work we used DCS at 785 nm and the optical properties were provided by our broadband TD-DOS system (chapter 3) for accurate quantification of blood flow.

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CHAPTER

3

Broadband (500-1350 nm) TD-DOS Clinical

Instrumentation: Development,

Characterization and Application

The key focus of this chapter is the TD-DOS instrumentation, an unique broadband clinical DOS system that was built, char-acterized and successfully implemented into clinical in vivo environment during this thesis period. Along with important functional aspects of the system, this chapter also elaborates on key parameters and critical decisions considered in the building process. Towards the end of the chapter, a spectral extension of the system till 1700 nm is also presented.

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Ch 3. Broadband (500-1350 nm) TD-DOS Clinical Instrumentation: Development, Characterization and Application

3.1 Introduction

Accompanied by the evolution of detector and source tech-nologies [18], diffuse optics has gained a strong foot hold in the field of tissue diagnostics; optical mammography [109– 111], in vivo characterization of tissues [112], bone diagno-sis [10], non-destructive tests on food quality [113], quality control of pharmaceutical medicine [107] are few examples. The common goal of all the above mentioned applications is to measure non-invasively the optical properties of diffusive media (i.e: absorption µa, reduced scattering µ0s). The

esti-mation of broadband µaand µ0svalues provides us with the

key information of tissue chromophore concentrations and structure, respectively.

In literature, most of the in vivo studies due to limited num-ber of wavelengths assume oxy- and deoxy-hemoglobin as the only absorbers [114]. With a broadband system like the one we designed, we can effectively quantify the other tissue constituents that are of clinical interest. The key tis-sue constituents like lipid, water, collagen have significant changes in 900-1350 nm, however it’s not possible to exploit this with silicon based detectors which have low quantum yield beyond 1100 nm thus poses severe challenges for the detection in this region.

Importantly, carrying out in vivo measurements in a clinical environment are quite challenging. Along with portability and compactness, stability and robustness of the system, self-adaptation, ease of use, automation are the prerequi-sites. Especially, time resolved systems in the absence of controlled lab environment are prone to stability challenges, leading to unwanted errors in measured optical properties.

All the above discussed parameters were carefully con-sidered while building our system. First of its kind, a broad-band (500-1350 nm) system was designed, developed, val-idated and successfully employed for preliminary trials in clinical environment. Importantly, a novel detection strat-egy is adopted to address the efficient light collection over the broadband range. Acquisition of the Instrument re-sponse function (IRF) in real time provides a self-stabilization

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3.2 Design and Build Process of TD-DOS

to the system from unwanted distortions and drift. Porta-bility, risk management and compactness are addressed with clinical applicability in view. Extensive characterization tests on system were carried out using MEDPHOT proto-col [115] which has been increasingly used for quantitative and objective assessment of diffuse optical system perfor-mance and agreed as an assessment protocol by several in-stitutions. Finally, in vivo trial were performed for retrieval of tissue constituents from measured broadband spectra at different in vivo locations.

3.2 Design and Build Process of

TD-DOS

This section highlights the unique features and strategies adopted to address broadband in vivo measurement of tis-sue locations in a clinical environment.

3.2.1 Optical Layout of System

Fig.3.1 depicts the schematic of the system’s optical chain. Both delivery and collection of light are made through fibers, this ensures patient safety also facilitates compactness and easy applicability. A supercontinuum laser source (SC450, Fianium, UK) based on photonic crystal provides broad-band source. The source is dispersed using F2 glass Pellin Broca prism and the minimum deviation wavelength is cou-pled onto a optical fiber (core diameter of 50 µm) by a lens (f=150 mm). By rotating the prism the wavelength tuning is achieved. The fiber laser runs at 40 MHz repetition rate. The spectral bandwidth of the source ranges from 3 nm at 600 nm to 7 nm at 1320 nm which is a result of nonlinear dispersion of the prism. The source is further split into two arms with 95% and 5%, the former is used as the source for measurements on the sample whereas the latter is the ref-erence arm that acquires IRF simultaneously for drift and distortion compensation. More details on this strategy is elaborated in the following sections. To avoid saturation of