CO-REGISTRATION OF BIOLUMINESCENCE TOMOGRAPHY AND ANATOMICAL IMAGING MODALITIES FOR CELL

TRACKING AND SOURCE QUANTIFICATION

by

Moussa Chehade

A thesis submitted to Johns Hopkins University in conformity with the requirements for the degree of Master of Science in Engineering

Baltimore, Maryland

May, 2014

© 2014 Moussa Chehade All Rights Reserved

Abstract

Bioluminescence tomography (BLT) is a molecular imaging tool that provides

three-dimensional, quantitative reconstructions of bioluminescent sources in vivo.

A main limitation of BLT to date, however, has been a lack of validation and

demonstrated utility in preclinical research. An approach employing a fusion of

BLT with other, well-established imaging modalities was used in this work to

validate results obtained with BLT and improve the performance of source

quantification.

In the first chapter of this thesis, a method was developed to co-register BLT

to magnetic resonance (MR) and computed tomography (CT) anatomical data

for tracking cell transplants using a specialized animal holder. Using a luciferase-

expressing tumor model in mice, MRI was shown to be superior at locating cells

while BLT provided a more sensitive measure of cell proliferation. A multimodal

approach incorporating BLT can therefore provide a better understanding of cell

dynamics in vivo in preclinical research than with anatomical imaging alone.

In the second chapter of this thesis, anatomical MRI and CT images were

segmented to provide hard spatial priors to quantify the power of calibrated

luminescent sources implanted in mice. To do this, a finite element (FEM)

implementation of the diffusion approximation was used as a forward model for

light propagation and validated through a phantom experiment. Source powers

quantified using hard prior information showed a 65% reduction in average

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deviation compared to traditional BLT using four spectral bins and comparable

performance to eight bins. BLI imaging times using hard spatial priors were

reduced by 16-fold and 100-fold compared to the four- and eight-bin BLT

methods, respectively. Together with the results of the first chapter, these results

show value in incorporating data from other imaging modalities into BLT.

Thesis Committee: Jeff W. M. Bulte, Professor of Biomedical Engineering, Thesis Advisor Piotr Walczak, Associate Professor of Radiology Kevin J. Yarema, Associate Professor of Biomedical Engineering

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Acknowledgements

The numerous medical and pre-clinical imaging modalities available today are a

testament to the creativity of the individuals working in this research field, one

that I have been fortunate enough to be a part of over the past two years. I

would like to take this opportunity to thank a very talented group of individuals

for their support; this thesis would not have been possible without them.

First and foremost, I would like to thank my advisor, Jeff W. M. Bulte, for

his continuous guidance and support during my graduate studies. Dr. Bulte’s

emphasis on treating all members of the research group, regardless of experience,

as independent scientists was intimidating to me at first, but in retrospect a

tremendous help in developing my skills as a researcher. I am also thankful for

the support of Piotr Walczak, who provided valuable suggestions on my work

and for the opportunities he provided me with to improve my skills in image

analysis and visualization. I would also like to thank Kevin Yarema for his

valuable feedback on this thesis.

Very special thanks go to Amit Srivastava for his mentorship over the past

two years. Amit has taught me much of what I know in wet lab skills and was

always willing to sacrifice his time and effort to assist me with experimentation. I

am indebted to him for his help.

I would like to acknowledge the help of Irina Shats for sharing her wisdom in

and training me in cryosectioning and staining protocols. A thanks goes out to

Lisa Song for her guidance on the relevant literature and software for

bioluminescence tomography. I would also like to acknowledge Antje Arnold and

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Anna Jablonska for their advice in day-to-day issues that came up during the

course of my graduate work and their always appreciated sense of humor. To my

lab mates and others who I may have forgotten, I am grateful to you for your

willingness to help and your friendship. It has been a pleasure working with you

all.

Last but not least, I would like to extend a heartfelt thanks to my parents for

their unconditional support and words of encouragement during my time at

Johns Hopkins. Without them I would not have gotten this far, and for that I am

truly grateful.

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Table of Contents

Abstract ...................................................................................................... ii

Acknowledgements ..................................................................................... iv

Table of Contents ...................................................................................... vi

List of Tables ............................................................................................. ix

List of Figures ............................................................................................. x

Introduction ................................................................................................ 1

Chapter 1: Bioluminescence Tomography and MRI for Cell Tracking ........ 3

1.1 Background ................................................................................... 3

1.1.1 Bioluminescent Imaging ..................................................... 3

1.1.2 Limitations of Planar BLI .................................................. 4

1.1.3 Bioluminescence Tomography ............................................ 7

1.1.4 Alternative Cellular and Molecular Imaging Modalities ... 10

1.1.5 Co-registration and Image Fusion .................................... 12

1.1.6 Approach and Significance of this Work .......................... 13

1.2 Methodology and Animal Holder Design ..................................... 14

1.2.1 Equipment ........................................................................ 14

1.2.2 Imaging Workflow ............................................................ 14

1.2.3 Holder Requirements ........................................................ 15

1.2.4 Holder Design ................................................................... 16

1.3 Phantom Tests and Co-Registration Procedure .......................... 19

1.3.1 Phantom .......................................................................... 19

1.3.2 Imaging and Registration ................................................. 19

1.3.3 Results ............................................................................. 20

1.4 Imaging Protocols ....................................................................... 23

1.4.1 MR Imaging ..................................................................... 23

1.4.2 CT Imaging Protocol........................................................ 23

1.4.3 Bioluminescent Imaging ................................................... 24

1.4.4 BLT Reconstruction ......................................................... 24

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1.4.5 MR to BLT/CT Registration ........................................... 25

1.5 In Vivo Validation ...................................................................... 26

1.5.1 Overview and Approach ................................................... 26

1.5.2 Cell Culture and Transplantation .................................... 27

1.5.3 Animal Imaging Protocol ................................................. 27

1.5.4 Histological Analysis ........................................................ 28

1.5.5 Results ............................................................................. 28

1.6 Conclusions ................................................................................. 35

Chapter 2: Quantitative Bioluminescence Tomography using Prior Spatial Information ....................................................................................... 36

2.1 Motivation .................................................................................. 36

2.1.1 Spatial Prior Knowledge in BLT ...................................... 37

2.1.2 Approach in this Chapter ................................................. 38

2.2 Background Theory .................................................................... 40

2.2.1 Light Propagation in Biological Tissues ........................... 40

2.2.2 The Radiative Transfer Equation ..................................... 42

2.2.3 Solutions to the RTE ....................................................... 43

2.3 Finite Difference Method ............................................................ 46

2.3.1 Implementation ................................................................ 46

2.3.2 Boundary Conditions ....................................................... 47

2.3.3 Numerical Validation ....................................................... 48

2.3.4 Limitations of FDM Approach ......................................... 52

2.4 Finite Element Method ............................................................... 54

2.4.2 Implementation ................................................................ 54

2.4.3 Matrix Assembly .............................................................. 56

2.4.4 Conversion of Photon Density to Radiance ...................... 59

2.5 Implementation and Source Quantification ................................. 61

2.6 Method Validation ...................................................................... 67

2.6.1 Numerical Validation of Forward Model .......................... 67

2.6.2 Validation against Tissue Mimicking Phantom ................ 70

2.7 In Vivo Testing ........................................................................... 73

2.7.1 Procedure ......................................................................... 73

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2.7.2 Results ............................................................................. 75

2.7.3 Discussion......................................................................... 78

2.8 Conclusions and Future Work .................................................... 80

Summary and Conclusions ........................................................................ 81

Glossary of Terms and Notation ............................................................... 83

Appendix .................................................................................................. 84

Appendix A: Histological Analysis .................................................... 84

Appendix B: FDM Matrix Coefficients ............................................. 86

Appendix C: Scaling Coefficient From Cross-Correlation ................. 88

Bibliography ............................................................................................. 90

Curriculum Vitae ...................................................................................... 98

viii

List of Tables

Table 1.1 MRI to CT transformation repeatability errors along each axis ..21

Table 2.1 List of optical properties for uniform and non-uniform spherical cases used in FDM validation ......................................................50

Table 2.2 List of optical properties for uniform and non-uniform spherical cases used in FEM validation ......................................................67

Table 2.3 Comparison of source power quantification using multispectral BLT and the hard spatial prior approach ....................................73

Table 2.4 Standard and mean absolute deviations of the normalized datasets from the calibrated bead power for each of the quantification methods .......................................................................................76

Table 2.5 Comparison of time needed for source quantification using planar BLI, multispectral BLT, and the hard spatial prior method .......77

ix

List of Figures

Figure 1.1 Cytosolic luciferase acts on D-Luciferin in the presence of oxygen and ATP to produce light. ........................................................... 4

Figure 1.2 BLT imaging procedure. .............................................................10

Figure 1.3 CAD model of animal holder (left). Animal holder in use during BLI, showing restrained mouse (right) ........................................17

Figure 1.4 Surface coil insertion into animal holder during MRI .................18

Figure 1.5 Air-water tube phantom used to determine coordinate transformation between CT and MRI scanners ...........................19

Figure 1.6 Axial (left) and coronal (right) sections of phantom in CT (blue) and MRI (yellow), showing agreement between the co-registered volumes. .......................................................................................20

Figure 1.7 Co-registered MRI (orange) and CT (greyscale) in a live mouse, showing good agreement between the modalities .........................25

Figure 1.8 BLT (hot color scale) reconstructed Luc+ cell location superimposed over T2 MRI at day 27 for all three test subjects. 29

Figure 1.9 3D visualization of MR-segmented tumor volume (green) and BLT reconstruction (orange) .......................................................30

Figure 1.10 Ex vivo analysis of subject 1 brain after day 27 ...........................32

Figure 1.11 Plot of total bioluminescence signal and MR-segmented tumor volume over the duration of the study, normalized to day 1 values (n=3) ...........................................................................................33

Figure 1.12 Coronal MRI slices at 1, 2, and 4 weeks after transplantation. Increase in tumor size is only apparent by week 4. ......................33

Figure 1.13 Comparison of total bioluminescence (left) and BLT-reconstructed source power (right) over the duration of the study, showing a similar trend but increased variance with BLT (n=3) .................34

Figure 2.1 Schematic of light propagation in biological tissues. Light traveling through tissue may be absorbed or scattered several times before exiting through the surface of the tissue. .................40

x

Figure 2.2 Geometry for spherical volume with regions of varying optical properties. The light source is isotropic within a radius 𝑟𝑞 ..........49

Figure 2.3 Photon density vs. distance from center for the FDM compared against the ODE in the optically homogeneous case....................51

Figure 2.4 Photon density vs. distance from center for the FDM compared against the ODE in the optically inhomogeneous case, showing a discrepancy beyond a radius of 3 mm. .........................................51

Figure 2.5 The FDM approach in a voxelized mouse volume (left) causes banding artifacts in the photon density at the surface (right) due to attenuation over one voxel length ...........................................52

Figure 2.6 Tetrahedral mesh representation (right) of mouse CT volume (left) ............................................................................................54

Figure 2.7 Flowchart of source quantification process using prior spatial information ..................................................................................61

Figure 2.8 The CT volume is cropped and used to generate a mesh ...........62

Figure 2.9 Cross-correlation of simulated and measured BLI images shows a single peak near the center. .........................................................65

Figure 2.10 Output images showing simulated and measured images of radiance on the top surface of a tissue-mimicking phantom and an error image (right). ......................................................................66

Figure 2.11 Photon density vs. distance from center for the FEM model compared against the ODE in the optically homogeneous case. ..68

Figure 2.12 Photon density vs. distance from center for the FEM model compared against the ODE in the optically inhomogeneous case. ....................................................................................................68

Figure 2.13 Photon density vs. distance from center for the FEM model compared against TIM-OS in an optically homogeneous medium. ....................................................................................................69

Figure 2.14 XFM-2 tissue-mimicking phantom ...............................................70

Figure 2.15 Emission spectra of firefly luciferase and tritium bead ................71

Figure 2.16 Coronal CT sections of XFM-2 phantom, showing two possible locations for tritium bead placement ...........................................72

xi

Figure 2.17 Implanted tritium bead is visible in CT and shown segmented in red. ..............................................................................................74

Figure 2.18 Overall mouse volume (left) and segmented bone and brain tissue (right) obtained from CT images .................................................75

Figure 2.19 Comparison of reconstructed source powers using corrected total flux from BLI, BLT with differing number of bins used, and the hard prior method (n=9). Dashed line represents the actual power of the calibrated light source. Whiskers show the data minimum and maximum. .............................................................................76

Figure 2.20 Comparison of variances in reconstructed source powers after normalization of the dataset from Fig. 2.19 to give a mean source power of 1.15x1010 photons/s for each method. Dashed line represents the actual power of the calibrated light source. Whiskers show the data minimum and maximum. ......................77

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Introduction

Cell transplantation therapies, including stem cell transplants, are an active area

of research that has shown potential for treating a wide range of diseases such as

neurological disorders, cardiovascular diseases, and stroke [1,2,3]. A persistent

challenge, however, has been the need to monitor cell targeting, survival, and

tumorigenicity when evaluating potential treatments in vivo [2,3,4]. Today, this is

accomplished using a wide range of non-invasive molecular and cellular imaging

modalities, primarily: optical, magnetic resonance (MR), and nuclear imaging

[5,6,7]. As one of the most popular optical methods, bioluminescent imaging

(BLI) is a widely used molecular imaging tool for imaging of small animals, with

the main advantages of this modality being its relative low cost, high throughput,

and use of a robust reporter gene. A key limitation of BLI, however, has been its

inability to provide spatial information comparable to anatomical imaging

modalities such as MRI or CT.

Over the past two decades, optical tomography has emerged as a promising

solution to this limitation of BLI but has yet to be widely adopted for in vivo

use. The aim of this work was to investigate the utility and limitations of

tomographic BLI when paired with anatomical modalities such as MRI and CT,

and improve upon the capabilities of optical tomography using a multimodal

imaging approach.

Chapter 1 of this thesis examines the co-registration of tomographic BLI

(BLT) with CT and MRI for in vivo cell tracking in small animals. A specialized

1

animal holder was designed to simplify the registration of BLT to MRI and

applied to an in vivo tumor model in mice to examine the utility of BLT paired

with MRI.

In Chapter 2 of this thesis, the use of prior spatial information from co-

registered anatomical images in BLT source quantification was examined and

compared to current BLT methods with implications for in vivo applications such

as quantifying cell number.

2

Chapter 1: Bioluminescence Tomography and MRI for Cell Tracking

1.1 BACKGROUND

1.1.1 Bioluminescent Imaging

Bioluminescent imaging (BLI) is a molecular imaging technique that captures

light emitted from biochemical processes within a cell. By using a light-emitting

probe, BLI allows for non-invasive tracking of cells expressing the probe as well

as measuring the relative expression of a target molecular process in these cells.

Currently used BLI reporters fall within the class of luciferase enzymes; several

variants of luciferases occur naturally in certain organisms and differ in the

substrate and co-factors required in the chemical reaction, reaction kinetics, and

wavelength of light produced. By far the most commonly employed variant is

Firefly luciferase (fLuc), which acts upon a luciferin substrate in the presence of

ATP and emits with a peak of 560 nm at room temperature [8], shown in Fig.

1.1. The fLuc emission spectrum is red-shifted in vivo to a peak emission at 612

nm [9]. Other variants include Red and Green Click Beetle luciferase (544 nm

and 611 nm, respectively), and Renilla luciferase which acts instead on

coelenterazine in the absence of ATP to produce light at 480 nm [8]. In current

studies the most widely used variant is firefly luciferase.

3

Figure 1.1 Cytosolic luciferase acts on D-Luciferin in the presence of oxygen and ATP to produce light.

While BLI is more complex than other optical methods, such as fluorescence

imaging, in that it requires the injection of a chemical substrate to produce a

signal, BLI benefits from a high signal-to-noise ratio (SNR) with minimal

background. As described by Sadikot and Blackwell, “bioluminescence is

appealing as an approach for in vivo optical imaging in mammalian tissues

because these tissues have low intrinsic bioluminescence.” [10]

1.1.2 Limitations of Planar BLI

Despite being one of the most widely used small animal imaging modalities, BLI

nonetheless has a few key limitations:

4

1) Dependence of Luciferase Activity on External Factors

The bioluminescence reaction is dependent on the action of the intracellular

luciferase enzyme on D-luciferin, ATP, and oxygen, as well as the presence of

Mg2+ as a cofactor. Ideally, the rate of this reaction is dependent only on the level

of expression of luciferase so that the intensity of the BLI signal correlates

directly with the level of gene expression or the number of viable cells, either in

vitro or in vivo. To do this, enough luciferin must be administered so that the

enzyme is saturated. A widely used dose in BLI today is a standard

intraperitoneal injection of 150 mg/kg of luciferin per animal body weight.

While earlier works have presented evidence that the standard dose is

sufficient to saturate the available luciferase [11,12], more recent and

comprehensive studies contradict this result. Lee et al. used a radioassay to study

the biodistribution of luciferin following intraperitoneal injection and found that

the lowest concentrations were found in the brain, bone, muscle, and

myocardium, with luciferin concentrations in the brain 20 times lower than in the

systemic circulation [13]. They concluded that given the relatively poor transport

of luciferin across the cell membrane, intracellular concentrations are low enough

to limit luciferase activity. In another study of the brain, Aswendt et al. found

that BLI signal intensity increased beyond doses of 700 mg/kg [14]. Zhang et al.

noted that for in vitro BLI of luc-expressing HEK-293T cells, activity did not

saturate at the standard dose nor at an equivalent dose of 1000 mg/kg [15].

5

In addition to substrate dependence, Moriyama et al. have documented a

decrease in BLI output in response to reduced oxygenation in vitro, which they

attribute to reduced ATP [16]. This may be relevant to studies involving tumors,

which often contain hypoxic regions. Inhibition of the BLI signal was also found

in vivo with the use of inhalant anesthetics such as isoflurane, which the authors

attributed to changes in hemodynamics [17,18]. Finally, work by Bollinger

demonstrated a dependence of BLI intensity on animal temperature, concluding

that it is necessary to control body temperature between imaging sessions [18].

2) Attenuation of Signal due to Tissue Optical Properties

Light produced in vivo by bioluminescence must travel through tissue and exit

the surface of the specimen before being detected by a camera or detector. All

other variables unchanged, the BLI signal decreases with increasing depth in the

tissue as a result of scattering and attenuation. An estimate for the effective

attenuation coefficient, 𝜇𝑒𝑓𝑓 , using mouse tissue properties at the peak emission

of luciferase (600 nm) [19] is 1 mm-1, meaning that the BLI signal will decrease

by 99% for roughly every 5 mm that it must travel through tissue. While this

may be less of an issue for subcutaneous transplants, it results in a significant

loss of signal in deeper transplants. The topic of light propagation through tissue

in vivo is explored in detail in Chapter 2.

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3) Limited Spatial Information

In traditional BLI, where a camera is used to image the specimen from a given

direction, the captured images are planar and do not provide explicit depth

information. In addition, the highly scattering nature of biological tissue causes

light to spread as it travels within the specimen, causing a depth-dependent loss

of spatial resolution. A rule of thumb is that the spatial resolution of BLI is

roughly equal to the depth of the light source in the tissue [9], or on the order of

several millimeters for typical in vivo applications.

The dependence of luciferase activity on external factors can be somewhat

mitigated by standardizing the imaging procedure to use consistent luciferin

doses, anesthetic conditions, and animal body temperature. In addition, imaging

at a fixed time after luciferin administration is necessary to account for in vivo

bioluminescence kinetics resulting from the distribution, metabolism, and

clearance of luciferin. To allow for quantification of the bioluminescence signal

using an in vitro assay, where the light output per cell per second is measured,

cells in the assay should be subjected to physiologically relevant oxygen and

luciferin concentrations that correspond to the tissue of interest.

1.1.3 Bioluminescence Tomography

A relatively recent extension of BLI, bioluminescence tomography (BLT),

compensates for the lack of spatial information in BLI by providing a three

7

dimensional reconstruction of the light source in vivo. The BLT procedure

typically follows the following four steps [20]:

1) Acquisition of 2D BLI images. Most commonly, a multispectral approach

is used where a series of spectrally-limited BLI images are acquired at

various wavelengths. This method provides depth-related information

based on the fact that the bioluminescent emission spectrum is typically

red-shifted as it travels through increasing distance in biological tissues.

2) The geometry of the sample-environment boundary is obtained.

3) A forward model is established that maps from the light source inside the

tissue to the light measurements on the sample boundary. Forward models

for light propagation are covered in depth in Chapter 2.

4) An inverse problem that solves for the light source distribution given the

BLI measurements along the boundary.

In this manner, BLT provides an estimate of the light source distribution

producing the acquired BLI images. In addition, since the forward model

incorporates the attenuation of light as it travels through the sample, BLT

provides a quantitative measure of the in vivo light source power. In comparison,

2D BLI can provide only semi-quantitative measurements.

An outline of the procedure for generating a BLT reconstruction in a small

animal using a commercially available BLI/CT imager is shown in Fig. 1.2 below.

While the capabilities of BLT go beyond planar BLI, it is still under development

with two key areas currently under investigation:

8

1) Improvements to Reconstruction Accuracy

Unlike other tomographic modalities such as MRI and CT, which provide a

faithful reconstruction of the signal or geometry being imaged, the reconstruction

obtained by BLT is dependent on the model used for light propagation in the

tissue. Consequently, BLT reconstruction accuracy is dependent on a model that

is physically accurate and a reconstruction algorithm that is robust to noise in

measurements. As of this time, there are ongoing efforts to improve the accuracy

of BLT reconstructions [21,22].

2) Demonstrated Utility and in vivo Applications

A survey of existing literature shows a limited number of studies demonstrating

the utility of BLT in research applications [23,24] or validating the quantification

of source power through BLT [22]. For BLT to gain popularity over traditional

BLI, it needs to be validated in its ability to provide a quantitative, spatially

accurate reconstruction in a demonstrated small animal application.

9

Figure 1.2 BLT imaging procedure. (A) Luciferase-expressing cells are transplanted into the animal. (B) A solution of luciferin is administered 10-15 minutes prior to imaging in the BLI/CT scanner (C). A series of spectrally filtered BL images and CT volume (D) are used to reconstruct the bioluminescence source using a BLT algorithm (E).

1.1.4 Alternative Cellular and Molecular Imaging Modalities

In addition to BLI, other modalities are commonly used for in vivo cellular

imaging and are summarized below:

10

Magnetic Resonance Imaging (MRI)

A popular method of in vivo cellular imaging is through the use of MRI labeling

agents [25,26,27] and, currently in development, MR reporter genes [28]. For

example, labeling of cells with superparamagnetic iron oxide (SPIO) prior to

transplantation allows MRI to locate single cells [29]. The high spatial resolution

of MRI and flexibility in choice of pulse sequences allows it to localize cells in

space with high precision and relative to anatomical details, something that other

modalities such as BLI, BLT, or PET cannot do. MRI may also be used for

quantitative assessment of transplanted cell count using 19-F labeling or MR

reporter genes, however these methods are limited to a minimum number of

roughly 104 detectable cells [30]. Conversely, BLI is capable of detecting and

quantifying down to 102-103 cells in vivo [14,31].

Positron Emission Tomography (PET)

PET uses targeted radiotracers, such as 18F-FDG, that emit gamma radiation

when decaying and allows for a tomographic reconstruction of the tracer

distribution. Unlike BLT which uses low energy photons, the gamma emissions in

PET travel with minimal interaction with biological tissue, allowing for a faithful

reconstruction of the tracer location. Compared to MRI, PET benefits from high

signal specificity but suffers from limited anatomical information [24] and poor

spatial resolution (> 1 mm) due to fundamental limits such as positron range

[32]. Compared to PET, BLI (and by extension BLT) provide lower limits of

11

detection, higher throughput, and avoids the cost and safety concerns with the

use of radiotracers [24].

1.1.5 Co-registration and Image Fusion

A multimodal imaging approach, where different imaging modalities are co-

registered, has the potential to compensate for the shortcomings of the individual

modalities. One approach that has been explored in literature has been to use co-

registered images in an attempt to improve BLT reconstruction accuracy. In an

early example of this approach, Allard et al. registered BLI to MRI to provide a

reference for light propagation simulations to investigate the accuracy of various

BLT reconstructions [33]. Similarly, Beattie et al. have investigated the

registration of planar BLI to CT and MRI using a specialized animal holder,

providing anatomical information which they suggest may improve BLT

reconstruction accuracy [34,35]. Phantom studies by Yan et al. also suggest that

prior knowledge of structural information obtained from co-registration may

improve the accuracy of BLT reconstructions [36].

While a growing body of work has examined the co-registration of BLI and

MRI in feasibility studies, a currently underdeveloped area is the fusion of BLT

with other modalities in pre-clinical or discovery research [22]. Among the few

examples of this approach in literature, Virostko et al. co-registered BLT and

PET images to evaluate three new PET radiotracers for imaging human

pancreatic beta cells [37]. Similarly, Deroose et al. used a multimodal BLI-

fluorescence-PET reporter gene to provide planar BLI and co-registered PET-CT

12

imaging of tumors [24]. A more common approach thus far has been the use of

BLI in multimodal imaging without the use of image fusion. In one example,

Zhang et al. used independently acquired MRI and planar BLI in order to look at

stem cell survival in rat models of myocardial infarct [38]. In a similar vein, a

protocol by Tennstaedt et al. outlined multimodal imaging of neural stem cell

transplants into the mouse brain using MRI and BLI [39], but without the use of

BLT or co-registration. A remaining line of work in this area is the co-

registration of tomographic BLI to MRI, or other modalities, in an in vivo

application.

1.1.6 Approach and Significance of this Work

The remainder of this chapter demonstrates a method for the co-registration of

reconstructed BLT volumes to MR and CT anatomical data for tracking cell

transplants in a small animal model. Furthermore, this work investigates whether

this particular combination of imaging modalities provides a better understanding

of transplanted cell dynamics than can be obtained with a single modality, due to

the superior resolution of MRI and lower limits of detection of BLI/BLT. More

importantly, it would help validate BLT as a method for cell tracking by

providing a reference modality for comparison. Finally, some researchers have

suggested that, given that the BLT algorithm is typically an underdetermined

problem [40], the incorporation of a priori structural information from co-

registered CT/MR images into BLT may improve its performance [36,41,42].

13

1.2 METHODOLOGY AND ANIMAL HOLDER DESIGN

1.2.1 Equipment

In this work, an IVIS Spectrum CT (PerkinElmer Inc.) was used to perform BLI

and CT imaging for subsequent BLT reconstruction. The scanner contains a

2048x2048 pixel, cooled CCD camera to capture BLI with a filter wheel

containing 18 bandpass emission filters. The IVIS also includes a rotating

platform to perform microCT with a detector size of 3072x864 pixels, 50 kV x-ray

energy at 1 mA, and a field of view of 126x126x31 mm with an effective

resolution of 0.15 mm. A quad-core 2.8 GHz computer workstation with a 256-

core CUDA GPU is used to perform CT and BLT reconstructions. The IVIS-

generated BLT volumes are by default co-registered with CT volumes. The IVIS

Spectrum CT was acquired June 2012 through NIH #S10 OD010744.

MR imaging was done in a Bruker Biospec 117/16 (Bruker Corporation)

11.7T horizontal bore MRI scanner, which is specifically designed for pre-clinical,

small animal imaging. The Bruker 117/16 is actively shielded with a bore

diameter of 160 mm.

1.2.2 Imaging Workflow

Co-registration between modalities may be accomplished in one of two ways,

namely:

• Maintaining subject position between imaging modalities and using a

coordinate transformation between scanners (rigid transformation)

14

• Freely positioning the subject in each scanner and using deformable/non-

linear registration to account for changes in subject position

Intermodality non-linear registration is a non-trivial task, therefore a rigid-

transformation approach was chosen despite requiring additional hardware to

maintain the animal position between the two scanners. This approach provides a

quick workflow since the coordinate transformation between the two scanners

may be determined a priori. In a pre-clinical setting, where high imaging

throughput is desirable, this feature is important. This method of co-registering

BLT/CT to MRI entails the following steps:

1. Animal is imaged in BLI/CT scanner

2. Transportation to MRI scanner without disturbing position

3. Animal is imaged in MRI scanner

4. BLT/CT volumes aligned to MRI using prior transformation

Due to the need to administer luciferin via intraperitoneal (IP) injection, BLI

must be performed before MRI to maintain the animal position. Step 2 entails

the use of a specialized animal holder.

1.2.3 Holder Requirements

For successful use in both BLI and MRI, it was decided that the animal holder

should meet the following criteria at minimum:

• Immobilization of the animal during BLI and MRI to reduce motion

artifacts

15

• Allow transfer of the animal between the BLI/CT and MR imagers

without altering the position of the animal

• Allow repeatable positioning in each imager to simplify co-registration by

use of an a priori determined transformation

• Made of MR- and CT-compatible material, which should be: non-ferrous,

low radiodensity, low MR signal

• Compatible with BLI: made of non-reflective material with low

autofluorescence and autoluminescence. Should minimize interference with

light transport from the animal to the CCD camera

1.2.4 Holder Design

Main Features

The animal holder consists of a custom-modified design of the Mouse Imaging

Shuttle (Perkin Elmer) provided with the IVIS Spectrum and is shown in Fig.

1.3. The original shuttle consisted of a rectangular bed with a nosecone for

anesthetic delivery and a clear plastic lid for use in either BLI or fluorescent

imaging, which immobilizes the mouse by applying pressure from above when the

lid is attached. In this work, the lid was omitted and the shuttle modified to fit

an RF read coil on top of the mouse during MRI. The shuttle is machined from

non-fluorescent, MR-compatible Delrin plastic and is colored black to minimize

light scattering during BLI. A set of clear, elastic polyurethane straps were added

to gently restrain the mouse during BLI (Fig. 1.3) without contributing

detectable autofluorescence or autoluminescence in the wavelengths of interest

16

(580-680 nm). During imaging, gas anesthesia is provided from the IVIS

Spectrum and MR scanners through an entry port in the front of the holder. The

subject and holder are kept at 37°C by the heating beds in the IVIS and MR

scanners.

Figure 1.3 CAD model of animal holder (left). Animal holder in use during BLI, showing restrained mouse (right)

Respiration Monitoring

Tubing running through the back of the holder allows for an MR-compatible

pressure transducer (Biopac Systems Inc.) to be placed under the mouse to

monitor respiration during MR imaging. This feature is used for respiration

gating to reduce the effect of motion artifacts during MRI. Rotation of the stage

for CT in the IVIS Spectrum precludes the use of the sensor during BLI/CT;

however this is less of an issue since respiration artifacts, which mostly introduce

motion in the dorsal-ventral axis, are less significant in BLI where the animal is

imaged from above.

17

Removable MR Coil

During MRI, a phased-array surface read coil, suitable for brain or cervical

imaging, is inserted into the holder as shown in Fig. 1.4. The coil may be omitted

and the volume coil in the 11.7T scanner used instead, allowing for whole-body

imaging at the cost of reduced SNR. The use of a removable MRI coil allows the

shuttle to maintain an open top during BLI and avoids optical distortions or

attenuation of light traveling to camera.

Figure 1.4 Surface coil insertion into animal holder during MRI

Shuttle Positioning

Repeatable positioning of the holder is accomplished by a snap-fit mechanism

that locks it into the stage of the IVIS and MR scanners. The imaging stage is

fixed in position in the IVIS spectrum and movable in the MR scanner. In the

Bruker 11.7T scanner, a motorized positioning stage with 0.1 mm precision is

used to position the shuttle along the axis of the bore. The imaging stage in the

MR scanner is freely adjustable along the two axes in the sagittal plane using

manual adjustment knobs in order to accommodate stage inserts of varying size.

18

1.3 PHANTOM TESTS AND CO-REGISTRATION PROCEDURE

1.3.1 Phantom

To determine the transformation that maps between the coordinate systems of

the BLI/CT and MRI scanners, an air-water phantom visible to both CT and

MR was made out of a 15mL polypropylene tube filled with deionized water and

smaller air-filled tubes (Fig. 1.5). The phantom was imaged using the MR and

CT protocols described below, repeating the procedure three times with removal

of the shuttle, reinsertion, and re-adjustment of the stage position knobs in the

MR scanner to measure repeatability under typical use.

Figure 1.5 Air-water tube phantom used to determine coordinate transformation between CT and MRI scanners

1.3.2 Imaging and Registration

Phantom imaging utilized the scanner’s volume coil and a T2-weighted RARE

sequence with parameters: repetition time (TR) = 3400 ms, echo time (TE) = 30

ms, FOV = 6x2x1.6 cm, number of slices (Nslice) = 32 with 0.5mm spacing,

matrix = 360x128, number of averages (Navg) = 1.

19

To determine the transformation between the MR and CT coordinates, the

phantom datasets were imported into Amira 5.3 (Visualization Sciences Group)

and co-registered by manual positioning followed by automatic registration using

a rigid transformation and normalized mutual information metric. Since

automatic registration methods may converge to local minima depending on the

start position, registration accuracy was examined visually slice-by-slice. The

transformation between the CT and MR coordinate systems was taken as the

mean of the transformations obtained from the three trials.

1.3.3 Results

The co-registered CT and MR volumes of the phantom are shown in Fig. 1.6

Figure 1.6 Axial (left) and coronal (right) sections of phantom in CT (blue) and MRI (yellow), showing agreement between the co-registered volumes.

20

Two measures of positioning repeatability were taken: 1) the RMS error between

the individual transformation components along each axis against the averaged

transformation above 2) the 95% confidence interval of the transformations. The

results are given in Table 1.1 below.

Table 1.1 MRI to CT transformation repeatability errors along each axis

Mediolateral (ML)

Dorsoventral (DV)

Anterioposterior (AP)

Mean Translation (mm) -55.18 -13.99 -50.49 + x

RMS Error (mm) 7.59x10-3 0.93 0.77

95% C.I. (mm) 8.59x10-3 1.05 0.87

Where x is the MR stage position readout, measured towards the bore and relative to the typical imaging position of 870 mm (𝑥 = 𝑠𝑡𝑎𝑔𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 − 870)

The repeatability test showed that errors were greatest in the DV and AP axes,

with RMS errors of 7.6x10-3 mm, 0.93 mm, and 0.78 mm along the ML, DV, and

AP axes respectively. Rotation errors were negligible. For reference, the MR and

CT voxel resolutions were roughly 0.15 mm. Ideally, the rigid transformation that

was obtained between the BLT/CT and MR coordinates eliminates the need for

subsequent software registration. However, the results obtained with the

phantom indicate that this method still requires software co-registration along

the DV and AP axes to correct for the <1 mm deviations observed. This may be

attributed to the design of the MR scanner stage, which includes manual fine-

positioning knobs that translate the stage along the sagittal plane and are

21

necessary to allow the scanner to accommodate stages and inserts of different

geometries. Nonetheless, the current animal holder design is valuable in that it

maintains the animal in a fixed position, eliminating the need for non-rigid

deformation-based registration, and greatly simplifies the registration procedure

from a six-degree-of-freedom problem to a simple translation along two axes.

22

1.4 IMAGING PROTOCOLS

For imaging small animals under co-registered MR, BLI, and CT, the following

protocols were developed:

1.4.1 MR Imaging

A 2x2 phased array surface coil (Bruker Corporation) was placed into the shuttle.

The coil has a cylindrical profile and rests closely over the mouse head. Each

mouse was imaged using two sequences to generate different contrast images:

1) a T1-weighted FLASH sequence with sequence parameters: TR = 480 ms, TE =

6.3 ms, FOV = 1.6x1.6 cm, Nslice = 40 with 0.35mm spacing, matrix = 196x196,

Navg = 1, for a total scan time of 3 min.

2) a T2-weighted RARE sequence with identical FOV geometry and parameters:

TR = 4000 ms, TE = 31.9ms, flip angle (FA) = 180°, matrix = 256x256, Navg =

3, and a total scan time of 10:25 min. Respiration gating was used for both

animal imaging sequences to suppress motion artifacts.

1.4.2 CT Imaging Protocol

All CT images were acquired in the IVIS Spectrum CT with the following

parameters: 50 kVp at 1mA current, 50 ms exposure time, aluminum filter. A

total of 720 projections spaced 0.5° apart were acquired and the CT volume

reconstructed using the IVIS’s Living Image software (PerkinElmer Inc.), giving a

field of view (FOV) of 12.0 x 12.0 x 3.0 cm at an isotropic resolution of 0.15 mm.

23

1.4.3 Bioluminescent Imaging

Bioluminescent images of the animals were acquired using a cooled CCD camera

in the IVIS Spectrum CT (Perkin Elmer). For each animal, anesthesia was

induced using 2% isoflurane gas in oxygen and 150 mg/kg body weight of D-

luciferin were injected intraperitoneally. Images were acquired 10 min after

injection to maximize the bioluminescence signal. During imaging, four spectrally-

binned images were acquired using emission filters at 600, 620, 640, and 660 nm

with a bandwidth of 20 nm each. Imaging parameters were: exposure time = 180

s, aperture = f/1, FOV = 13x13 cm, 2048x2048 pixel resolution. Pixel binning

was set to 8x8 for an effective image resolution of 256x256.

1.4.4 BLT Reconstruction

Reconstruction of the bioluminescent source and superposition over the CT

volume was done using the DLIT algorithm in Living Image 4.3 (Perkin Elmer),

derived from work described by Kuo et al. in 2007 [43]. Briefly, the algorithm

uses single-view, multispectral BLI images to constrain the reconstruction with

segmentation of the CT images providing the mouse volume boundary.

Bioluminescent source and tissue absorption spectra for the luciferase reporter

and mouse tissue were predefined in the software. The source distribution was

visualized in Living Image using a voxel size of 0.31 mm and no smoothing.

24

1.4.5 MR to BLT/CT Registration

MR volumes were co-registered with BLT/CT using the same procedure

developed for the tube phantom. As mentioned before, the datasets were loaded

into Amira using the coordinate transformation determined for the phantom, and

translation was adjusted along the AP and DV axes until the MR and CT

volumes were aligned. The animal holder was effective in restraining the live

mouse between modalities as seen in Fig. 1.7 below.

Figure 1.7 Co-registered MRI (orange) and CT (greyscale) in a live mouse, showing good agreement between the modalities

25

1.5 IN VIVO VALIDATION

1.5.1 Overview and Approach

To examine the application of BLT in tracking the progress and safety of cell

transplant therapy, the approach followed in this work was to graft rapidly

growing, luminescent in an animal model. The cells, tagged to be MR-visible,

were then imaged using the method of co-registering BLT and MRI developed

previously, allowing a comparison of how both modalities track the growth and

location of the transplant. A mouse embryonic stem cell line was used because of

its relevance to preclinical stem cell therapy research, as well as the propensity to

proliferate rapidly and form tumors in vivo. The brain was chosen as a relevant

site of transplantation for two main reasons:

1) Brain tissue provides a relatively uniform background in MRI, making it

easier to locate the cells

2) A large research effort has focused on stem cell transplants in the brain for

the treatment of neurodegenerative diseases [1,44,45]

The choice of the brain as the site of transplantation introduces some

complexities into BLT, including the limited transport of luciferin into the brain

from the bloodstream [13] and an optically heterogeneous environment including

brain, bone, and skin that may pose a challenge for the BLT reconstruction.

Nonetheless, it provided a convenient location for comparison against MRI that is

relevant to currently researched therapies.

26

1.5.2 Cell Culture and Transplantation

Luciferase-expressing HBG3 mouse embryonic cells (mESC) were engineered by

transducing them with a lentivirus carrying the fLuc reporter gene under control

of the ubiquitin promoter. For MR-visible labeling, fLuc-mESCs were incubated

overnight with Molday ION-Rhodamine SPIO nanoparticles (BioPal, Inc.) prior

to transplantation. Fluorescence imaging of the cells prior to resuspension was

used to verify that the SPIO particles were taken up by the cells.

Three male BALB/c mice (3 weeks old, Harlan Laboratories) were

anesthetized using 2% isoflurane, shaved, then immobilized in a stereotactic

frame (Harvard Apparatus). 2 μL of the iron-tagged fLuc-mES cell suspension

containing 5x104 cells in each volume were loaded into a 31G needle and injected

into the brain (coordinates AP: 0 mm, ML: 2 mm, DV: 1.5 mm) using a

motorized injector (Stoelting Co.) at a rate of 0.5 µL/min. The needle was

carefully withdrawn 2 min after the end of the injection to minimize backflow.

Animals were kept on a heated blanket during surgery to maintain body

temperature. All animal procedures were approved and conducted in accordance

with the institutional guidelines for the care of laboratory animals. Mice were

immunosuppressed using FK506/rapamycin (1 mg/kg, daily IP injection)

1.5.3 Animal Imaging Protocol

To monitor the growth of the transplant, mice were imaged the next day after

transplantation, then weekly for a total of four weeks. In each imaging session,

anesthesia was induced using 3% isoflurane in oxygen and maintained using 1

27

L/min of 1-2% isoflurane throughout the session. Fur was trimmed in proximity

to the transplantation site to improve BLI signal strength and avoid introducing

errors in the BLT reconstruction. The mice were gently restrained in the shuttle

using the elastic straps in the prone position with the nose fully inserted into the

anesthetic nosecone. BLI and CT images were then acquired in the IVIS imager.

Anesthetic delivery was briefly interrupted at the end of BLI while the shuttle

was transported to the Bruker MR scanner for imaging, and resumed before the

animals could recover.

1.5.4 Histological Analysis

All animals were euthanized after the conclusion of the imaging session at week 4.

Ex vivo analysis of the brains was performed using the following staining

protocols:

1) Hematoxylin and Eosin (H&E) staining to visualize tumor morphology

2) Prussian Blue staining to visualize SPIO deposits

3) Immunostaining to locate fLuc+ cells derived from the initial transplant

Sample preparation and staining protocols are described in Appendix A.

1.5.5 Results

Reported Cell Location

At the conclusion of the study, MR coronal images showed two features of

interest: an original implantation site indicated by a hypointense region and

corresponding to a concentration of SPIO, and increased T2 signal intensity due

28

to tumor formation by migratory cells. The BLT reconstructions, shown

superimposed over MRI below in Fig. 1.8, indicated a single, diffuse region of

viable fLuc-mES cells.

Figure 1.8 BLT (hot color scale) reconstructed Luc+ cell location superimposed over T2 MRI at day 27 for all three test subjects.

In two out of the three subjects, BLT overestimated the depth of the Luc cells by

several mm, compared to a BLT voxel size of 0.4 mm. Looking at Fig. 1.8,

however, the diffuse reconstructions suggest that the effective resolution of BLT

in practice is lower than the voxel resolution. Subject 1 provided a more difficult

reconstruction task for BLT since MRI showed the formation of a secondary

tumor site by migratory cells from the original transplantation site. The BLT

reconstruction in this case shows a single cell location between the original and

secondary tumor sites.

The MRI tumor volume was manually segmented from the T2 images. A 3D

visualization of the tumor volume and BLT reconstruction is shown in Fig. 1.9

below, showing reasonable overlap between the modalities given the lower

resolution of BLT.

29

Figure 1.9 3D visualization of MR-segmented tumor volume (green) and BLT reconstruction (orange)

Histological analysis (Fig. 1.10) was used to validate the results obtained from in

vivo imaging. H&E staining confirmed the presence of a tumor mass both in the

hypointense region and area of increased signal intensity seen in MRI. Prussian

blue staining confirmed the presence of iron deposits seen as the hypointense

region in MRI. fLuc staining (green) indicated the presence of expressing cells

located at both the original transplantation and migrated cell sites. These results

suggest that BLT in the mouse brain is only able to provide reconstructions that

are accurate to within a few mm. The use of MRI is necessary where more

accuracy is required. There are several factors which may account for the

relatively low accuracy of BLT relative to MRI:

• The principle of operation of MRI provides a spatially faithful

reconstruction of the signal. The scattering nature of light in biological

tissue makes this difficult in optical imaging and necessitates accurate

models for light propagation.

30

• An accurate BLT reconstruction requires knowledge of the in vivo optical

properties in order to model light propagation correctly. The

heterogeneous nature of biological tissues makes this more challenging.

The BLT algorithm used by the IVIS Spectrum approximates the mouse

tissue as optically homogeneous.

• Luciferin kinetics cause the BLI signal to diminish over the course of

imaging. For multi-spectral imaging, where several images are acquired in

succession, if the imaging time per spectral bin is large enough (several

minutes) the BLI kinetic profile will cause a significant drop in signal

intensity during imaging. As a result BLT results may be inaccurate unless

corrections are made.

In this study only four spectral bins were used to limit imaging time and mitigate

the effect of the BLI kinetic profile. A tissue-mimicking phantom, detailed in

Chapter 2, was used to test the best-case performance of BLT using four spectral

bins and was able to locate the light source locations with a mean error of 3.0

mm (n=3). Furthermore, as was noted in later experiments in Chapter 2, the

accuracy of BLT was found to increase with the number of bins up to eight bins

tested. For future experiments where the BLI signal is sufficiently intense, the

imaging time per bin may be shortened and a greater number of spectral bins

used in the imaging window allowed by the kinetic profile. Alternatively, time

course corrections to the BLI signal could be used to compensate for BLI kinetics.

31

Figure 1.10 Ex vivo analysis of subject 1 brain after day 27 (a,b) H&E stained coronal section showing tumor near implantation site (c,d) Prussian blue stained section with Nuclear Fast Red counterstain. SPIO appears as blue deposits in the stain (e,f) Immunohistological stain for Luc (green) against DAPI nuclear counterstain, showing Luc-expressing cells at both the original transplantation site and superficial lesion

Cell Viability and Proliferation

A plot of BLI intensity compared to MR tumor volume (Fig 1.11) shows a

significant (p<0.05) increase in signal intensity within 1 week. Segmented tumor

volume from the MR datasets showed minimal change over first few weeks and it

was not until Week 4 that rapid tumor growth was seen (Fig 1.12).

32

Figure 1.11 Plot of total bioluminescence signal and MR-segmented tumor volume over the duration of the study, normalized to day 1 values (n=3)

Figure 1.12 Coronal MRI slices at 1, 2, and 4 weeks after transplantation. Increase in tumor size is only apparent by week 4.

Quantification of the BLT reconstructed light source showed a significant

increase in source intensity by Week 1 and followed a similar trend as BLI, as

expected (Fig 1.13). The larger variance in the BLT plot compared to BLI is

likely due to the errors in locating the BLI source. Since BLT accounts for

attenuation of the signal through the tissue, an incorrectly estimated depth will

result in an incorrectly quantified source.

33

Figure 1.13 Comparison of total bioluminescence (left) and BLT-reconstructed source power (right) over the duration of the study, showing a similar trend but increased variance with BLT (n=3)

34

1.6 CONCLUSIONS

As shown, BLT was a suitable modality for measuring cell proliferation and was

able to detect the onset of tumor formation earlier than could be seen from the

MR images. Conversely, MRI was superior in its ability to locate the

transplanted cells compared to BLT, which suffered from inaccuracy in the

depths of the reconstructed sources. The modalities used in this work provided

complementary information on cell fate after transplantation that would be

otherwise incomplete if a single modality were used.

It is notable that the plot of absolute light output in Fig. 1.13, quantified

using BLT, showed greater variability than the relative measurements made with

just planar BLI (Fig. 1.11), the cause of which is likely due to the depth errors

seen in the BLT-reconstructed light sources. Since this quantification is

dependent on the source being located correctly due to the attenuation of light as

it travels through the tissue, an area worth investigation would be on the use of

prior spatial information from the co-registered MR images to improve the

performance of BLT in this area. By incorporating accurate anatomical

information on the location of the light source, the performance of BLT in

quantifying the absolute output from the bioluminescent source might be

improved. This approach is investigated in the next chapter.

35

Chapter 2: Quantitative Bioluminescence Tomography using Prior Spatial Information

2.1 MOTIVATION

A fundamental challenge with bioluminescent tomography (BLT) is that it is an

ill-posed problem: namely, it attempts to solve for a light source distribution

inside an entire volume of tissue given a single measurement of light exiting from

the top surface of the tissue boundary. In the general case this solution is non-

unique [46] and additional information is required, which may include the

following:

• Assumptions on the geometry of the light source e.g. point source or

spherical [46,47]

• Multispectral imaging, based on the principle that the spectrum of a BLI

image is altered according to the depth of tissue through which it travels

[48,49,50]

• Multiview imaging, where BLI images are acquired from multiple

directions [48,51]

The performance of BLT is dependent on the use of a priori knowledge such as

the optical characteristics of the tissue being imaged [52]. There is ongoing

research as well on the use of spatial a priori information in BLT, which includes

anatomical or structural information on the tissue or bioluminescent source and is

summarized in the section below.

36

2.1.1 Spatial Prior Knowledge in BLT

The use of spatial information from other imaging modalities in optical

tomography has been most commonly studied in a closely related modality to

BLT, fluorescence molecular tomography (FMT). In one approach, Davis et al.

used MRI coupled with FMT to reconstruct fluorophore concentrations in tumors

in mice. Segmentation of the tumor and surrounding tissue in MRI provided hard

spatial priors on the distribution of the fluorophore [53,54,55]. Similarly, Zhou et

al. obtained a priori anatomical information from MRI on the expected

distribution of fluorophore in the lungs. By using a soft prior approach, where a

penalty is assigned for deviation of the FMT reconstruction from the prior

information, they demonstrated improved FMT spatial resolution [56]. Lin et al.

used co-registered CT and Diffuse Optical Tomography (DOT) to improve the

quantification of fluorophore concentration in a phantom study [57].

In the case of BLT, some groups have used DOT to obtain the spatial

distribution of optical properties prior to reconstruction [52,58]. Naser et al.

extended this approach further by using CT as prior spatial information for

DOT, which was then subsequently used to assist BLT [59]. In another

application, groups have used a mouse organ atlas or segmented the organ

locations directly from anatomical images in order to determine the true

distribution of optical properties within the tissue, improving the quantitative

performance of BLT [60,61,62].

37

2.1.2 Approach in this Chapter

Building upon the results of Chapter 1, in this chapter co-registered MRI and/or

CT will be used to provide prior spatial information on the location of the

bioluminescent source in BLT, an aspect that has not been thoroughly examined

in literature. Anatomical imaging modalities such as MRI are better suited for

localizing cells compared to BLT due to their superior accuracy and resolution.

Furthermore, the previous results suggest that errors in the reconstructed source

depth in BLT may be a factor in the accuracy of source power quantification,

which is supported by the work of Allard et al. [33]. It would be expected, then,

that the use of spatial priors on the location of the bioluminescent source would

improve the accuracy of source power estimation in BLT.

The approach in this chapter is as follows: coregistered anatomical images,

obtained either from MRI or CT, will be segmented to provide an estimate of the

true light source location. This information will be used as a hard spatial prior,

which assumes that the bioluminescent light source is distributed in the tissue

exactly as indicated by the anatomical images. A simulation of light propagation

through the tissue from the source, assumed to have unit power, will allow

quantification of source power from a single BLI image, in comparison to the

multispectral approach in traditional BLT.

This approach may be more accurately described as BLI attenuation

correction than BLT, but nonetheless uses the same framework in modeling light

propagation. In addition, it is expected that the accuracy of this technique is

38

dependent on the accuracy of prior spatial information. A more robust approach,

which is beyond the scope of this work, would be to use soft spatial priors which

instead penalize deviation of the reconstructed source from the prior information.

An expected benefit of the hard prior method over multispectral BLT, when used

in a multimodal imaging study, is that only a single BLI image is needed since

there is no need to use multispectral imaging to reconstruct the source location.

This single BLI image may either be integrated for a longer time period to

improve sensitivity when quantifying weak light sources, or used to reduce

imaging time, speeding up the workflow and reducing the influence of the

bioluminescence kinetic profile.

39

2.2 BACKGROUND THEORY

2.2.1 Light Propagation in Biological Tissues

A defining characteristic of light propagation in biological tissues is the high ratio

of scattering to absorption events. Unlike radiographic modalities such as PET

and SPECT, the low energy photons in optical imaging scatter several times

before exiting the tissue and traveling through the air to the camera. This is

illustrated in Fig. 2.1 below.

Figure 2.1 Schematic of light propagation in biological tissues. Light traveling through tissue may be absorbed or scattered several times before exiting through the surface of the tissue.

Light transport in scattering media, including biological tissues, is typically

described by the following set of parameters [63]:

40

𝜇𝑎 Absorption coefficient. Describes the probability of light absorption per unit path length and has units of [length]-1

𝜇𝑠 Scattering coefficient. Describes the probability of a scattering event per

unit path length and has units of [length]-1

g Scattering anisotropy factor, equal to the mean of the cosine of the

scattering angle. A small value of g indicates isotropic scattering. [64]

n Index of refraction, which influences interactions at the tissue-air boundary due to mismatch of the indices between air and the tissue.

For a total path length of x, the probabilities of absorption and scattering

occurring in a medium are, respectively [63]:

𝑃𝑎(𝑥) = 𝑒−𝜇𝑎𝑥 𝑃𝑠(𝑥) = 𝑒−𝜇𝑠𝑥

A meaningful measure is also the inverse of the two coefficients, giving the mean

path length that a photon travels in the medium before encountering and

absorption or scattering event [63]. It is important to note that these parameters

vary widely in biological systems and depend on:

• The type of tissue

• Physiological conditions such as oxygenation, in vivo vs. ex vivo

• Wavelength of light

The wavelength dependence of 𝜇𝑎 is the most apparent in optical imaging, where

red-shifted probes provide better penetration through tissue.

41

2.2.2 The Radiative Transfer Equation

Simulations and reconstructions of bioluminescent sources in biological tissue are

dependent on an accurate model for the propagation of light in scattering media.

By far the most commonly applied transport equation is the Radiative Transfer

Equation (RTE) [65]:

1𝜈

𝜕𝐿𝜕𝑡

+ 𝒔̂ ⋅ ∇𝐿(𝒓, 𝑡, 𝒔)̂ + (𝜇𝑎 + 𝜇𝑠)𝐿(𝒓, 𝑡, 𝒔)̂ = 𝜇𝑠 � 𝑃(𝑆

𝒔,̂ 𝒔′̂)𝐿(𝒓, 𝑡, 𝒔′̂)𝑑2𝒔′̂ + 𝑞𝑠(𝒓, 𝑡, 𝒔)̂

(1)

Where L is the radiance at position r in direction 𝒔,̂ 𝜈 is the speed of light in the

medium, P is the scattering phase function, and 𝑞𝑠 is the light source term. The

integral is taken over all solid angles, 4π. A glossary of mathematical notation

used in this work is provided at the end of this thesis.

A reasonable simplification in bioluminescent imaging is to consider the RTE

at steady state, since the time scale of the measurement is far greater than that

of any time-dependent behavior. In this case, Arridge et al. give a steady-state

expression for 𝜙(𝒓, 𝒔)̂, the volumetric density of photons traveling in direction 𝒔,̂

[66]:

�𝒔̂ ⋅ ∇ + 𝜇𝑠(𝒓) + 𝜇𝑎(𝒓)�𝜙(𝒓, 𝒔)̂ = 𝜇𝑠(𝒓)� 𝑃(𝒔,̂ 𝒔′̂)𝜙(𝒓, 𝒔′̂)𝑑𝒔′̂ + 𝑞𝑠(𝒓, 𝒔)̂𝑆

(2)

With the exception of simple, semi-infinite geometries [67], the RTE must be

solved numerically. This is especially the case in biological applications, which

involve arbitrary geometries and distributions of optical parameters.

42

2.2.3 Solutions to the RTE

Diffusion Approximation

The diffusion approximation (DA) is a widely used approximation to the RTE in

biomedical optics due to its simplicity, and hence, low computational cost, and

forms the basis of many tomographic reconstruction methods. It is based on the

following assumptions [65]:

1) Scattering dominates absorption in light transport (𝜇𝑠 ≫ 𝜇𝑎)

2) Consequently, light propagation is mostly isotropic

These are generally valid deep within biological tissue where the scattering

coefficients are typically an order of magnitude larger than the absorption

coefficients. The assumptions tend to break down near light sources and tissue

boundaries, where light propagation is anisotropic, but nonetheless the DA has

been widely applied in practice. The photon density and light source terms may

be replaced with their isotropic counterparts [66]:

Φ(𝒓) = � 𝜙(𝒓, 𝒔)̂𝑑𝒔̂𝑆

(3)

𝑞(𝒓) = � 𝑞𝑠(𝒓, 𝒔)̂𝑑𝒔̂𝑆

(4)

The resulting DA equation is shown below, with derivations available in the

relevant literature [65,68]:

𝜇𝑎(𝒓)Φ(𝒓) − ∇ ⋅ 𝜅(𝒓)∇Φ(𝒓) = 𝑞(𝒓) (5)

Where 𝜅(𝑟) = 13(𝜇𝑎+𝜇𝑠

′ ) and 𝜇𝑠′ = 𝜇𝑠(1 − 𝑔)

43

Simplified Spherical Harmonics

The DA is a first-order approximation to the RTE that is in the class of spherical

harmonics equations [69]. A more accurate solution to the RTE may be obtained

using higher-order approximations known as the simplified spherical harmonics

(𝑆𝑃𝑁) equations, which gives a set of (N+1)/2 coupled diffusion equations, where

N is the order of the approximation [69]. In studies of bioluminescence

tomography, the third-order 𝑆𝑃3 approximation is gaining popularity and

provides more accurate reconstructions [70].

Monte Carlo

Monte Carlo (MC) simulations approximate a solution to light transport in a

different manner, namely by simulating the paths of randomly sampled photons

through the optical medium. As a result, MC methods are highly accurate but

computationally expensive due to the need to simulate a large number of photons

(105-106 in studies or greater) to ensure stable results [71]. This makes the use of

MC more difficult in BLT reconstructions since it is both time intensive and

statistical, lacking a closed-form expression unlike the DA or SPN methods. A few

groups, however, have studied iterative methods using MC to estimate source

distributions [72,73]. The issue of computational cost will likely become less

significant in the future given the steady increase in computing power available

to researchers. There currently exist several dedicated software packages for MC

simulations of photon transport in scattering media, notable examples being

MCML [71], MOSE [74], and TIM-OS [75].

44

Choice of Forward Model

BLT, despite being an inverse problem in that it attempts to reconstruct a light

source given surface measurements of exiting light, nonetheless requires a forward

model for the propagation of light that maps from the light source to the surface.

In this work, the DA will be examined first as a forward model since it is

efficiently solved and simple to implement. In addition, it provides a fair basis for

comparison with results obtained using the methods in Chapter 1 since the Living

Image software used to perform BLT uses the DA as its forward model [76,77].

MC simulations will be used as a gold standard for comparison using one of the

existing software packages. Solutions to the DA are implemented using numerical

methods, outlined in the next sections.

45

2.3 FINITE DIFFERENCE METHOD

The finite difference method (FDM) is a numerical method of approximating

solutions to partial differential equations, such as the DA, by discretizing the

problem on a grid and approximating the derivatives in the PDE at each grid

point by finite differences [78]. FDM solutions to light transport have been

previously used in literature [69,79,80,81] and are investigated below:

2.3.1 Implementation

The FDM implementation used here computes photon density on a uniform

rectangular grid. Starting from the DA expression for light propagation:

𝜇𝑎(𝒓)Φ(𝒓) − ∇ ⋅ 𝜅(𝒓)∇Φ(𝒓) = 𝑞(𝒓) (5)

Where 𝜇𝑎, Φ, 𝜅, 𝑞 vary arbitrarily over space. Expanding the gradient and

divergence operators gives:

𝜇𝑎Φ − � ∂∂x

�𝜅 𝜕Φ𝜕𝑥

� + ∂∂y

�𝜅𝜕Φ𝜕𝑦

� + ∂∂z

�𝜅𝜕Φ𝜕𝑧

�� = 𝑞 (6)

Working with three-dimensional tissue volumes, 𝜇𝑎, Φ, 𝜅, 𝑞 are represented as

voxels on a regular grid with spacing Δ between each voxel along each axis.

Approximation of the derivatives in Eq. (6) with central differences gives a

system of equations that may be represented in matrix form:

(𝝁𝒂𝐼 − 𝑀)𝚽 = 𝒒 (7)

46

The tissue volume is comprised of a total of m voxels. 𝝁𝒂, 𝚽, 𝒒 are vectors of

size m, I is the [m x m] identity matrix, M is an [m x m] matrix of coefficients.

The photon density is computed given the light source:

𝚽 = (𝝁𝒂𝐼 − 𝑀)−𝟏𝒒 (8)

With the necessary boundary conditions applied. M is a sparse matrix of

coefficients and is detailed in Appendix B.

2.3.2 Boundary Conditions

Solutions to the DA have typically used one of two boundary conditions in

previous work:

1) Dirichlet boundary condition

This simple boundary condition forces Φ to zero at the physical boundary, an

approach which is physically unrealistic [82]. A more physically accurate

extension of the Dirichlet boundary condition is the Extrapolated-Boundary

condition (EBC), where Φ is set to zero at a boundary extended a distance 𝑑𝑒𝑥𝑡

from the original boundary. More detail on this approach can be found in the

work of Haskell et al. [65] and Schweiger et al. [82]. While mathematically simple,

implementing the EBC may pose problems when dealing with complex geometries

such as biological tissues, where concavities may be present.

47

2) Robin boundary condition

The most commonly applied, and more physically accurate, boundary condition

in the diffusion approximation is the Robin-type boundary condition (RBC) [65]:

Φ(𝑟) + 2𝜅𝐴�̂� ⋅ ∇Φ(𝑟) = 0 𝑟 ∈ 𝜕Ω (9)

𝐴 = 1 + 𝑅𝑖1 − 𝑅𝑖

(10)

Where 𝑅𝑖 is a coefficient that accounts for internal reflection at the boundary due

to the refractive index mismatch, approximated by using a curve fit [65]:

𝑅𝑖 ≅ −1.4399𝑛−2 + 0.7099𝑛−1 + 0.6681 + 0.0636𝑛 (11)

Assuming an index of refraction n = 1 for the environment. In the FDM solution

to the diffusion approximation, the entries in the matrix (𝝁𝒂𝐼 − 𝑀)

corresponding to boundary voxels are replaced with a finite difference expression

for the RBC above.

2.3.3 Numerical Validation

The FDM implementation was validated against a numerical solution to a

simplified case of the diffusion approximation. For the case of a volume with

spherical symmetry and boundary at 𝑟 = 𝑟𝑜, as shown in Fig. 2.2, the DA

reduces to a 1-D differential equation:

48

−𝜅𝑑2Φ𝑑𝑟2 − �2𝜅

𝑟+ 𝑑𝜅

𝑑𝑟�𝑑Φ

𝑑𝑟+ 𝜇𝑎Φ = 𝑞(𝑟) (12)

With boundary conditions:

𝑑Φ𝑑𝑟

= 0 𝑎𝑡 𝑟 = 0

Φ + 2𝜅𝐴 𝑑Φ𝑑𝑟

= 0 𝑎𝑡 𝑟 = 𝑟𝑜

The ordinary differential equation (ODE) in Eq. (12) is solved easily using a

numerical software package. A comparison was done between the computed

photon densities by FDM and the ODE above for cases with uniform and non-

uniform optical properties. The optical parameters used are listed in Table 2.1.

Figure 2.2 Geometry for spherical volume with regions of varying optical properties. The light source is isotropic within a radius 𝑟𝑞

49

Table 2.1 List of optical properties for uniform and non-uniform spherical cases used in FDM validation

Uniform 𝑟1= 7 mm 𝑟2= 7 mm 𝑟3= 7 mm 0 ≤ 𝑟 ≤ 𝑟1 𝑟1 ≤ 𝑟 ≤ 𝑟2 𝑟2 ≤ 𝑟 ≤ 𝑟3

𝜇𝑎 (mm-1) 0.1 0.1 0.1 𝜇𝑠′ (mm-1) 2 2 2

Non-uniform 𝑟1= 1 mm 𝑟2= 3 mm 𝑟3= 7 mm 0 ≤ 𝑟 ≤ 𝑟1 𝑟1 ≤ 𝑟 ≤ 𝑟2 𝑟2 ≤ 𝑟 ≤ 𝑟3

𝜇𝑎 (mm-1) 2 1 0.01 𝜇𝑠′ (mm-1) 0.1 1 2

In both cases, 𝑟𝑞= 2 mm, 𝑞𝑜 = 100 photons/mm4 inside 𝑟𝑞, n = 1.4, Δ = 0.25

mm. Plots of photon density along r are shown in Fig. 2.3 for uniform and Fig.

2.4 for non-uniform optical properties and show good agreement between the

FDM and expected profile. The larger error in the non-uniform case (Fig. 2.4) is

likely in part due to discretization of the optical properties on the grid. On a grid

size of 0.25 mm, which was chosen to keep memory and computation costs

reasonable, errors in assigning optical properties are noticeable even over a single

voxel, with attentuations of 22% for 𝜇𝑎 = 1 mm-1 and 0.25% for 𝜇𝑎 = 0.01 mm-1

over one grid spacing (0.25 mm).

50

Figure 2.3 Photon density vs. distance from center for the FDM compared against the ODE in the optically homogeneous case.

Figure 2.4 Photon density vs. distance from center for the FDM compared against the ODE in the optically inhomogeneous case, showing a discrepancy beyond a radius of 3 mm.

51

2.3.4 Limitations of FDM Approach

The above implementation of FDM revealed a few limitations when dealing with

a voxelized representation of biological tissue where complex geometries are

involved:

Computation of Surface Normal

Calculating surface normals of a volume, needed for boundary conditions, is

typically taken from the gradient of the volume [83] but has added computational

overhead to avoid “staircase” artifacts [84,85] (Fig. 2.5).

Banding Artifacts

Due to a significant drop in photon density over one voxel length, the photon

density at the tissue boundary exhibits banding as seen in Fig. 2.5, which used an

optimal voxel spacing of 0.6 mm to keep computation times reasonable.

Figure 2.5 The FDM approach in a voxelized mouse volume (left) causes banding artifacts in the photon density at the surface (right) due to attenuation over one voxel length

52

Inefficiencies with Regular Grid

Uniformly spaced voxels are non-optimal in both memory usage and

computational cost for two main reasons: 1) changes in photon density are

greatest near the light source and boundary and require a higher grid density

there, 2) a voxelized representation of geometry needs a higher grid density near

small features and lower in smooth regions. For reference, a voxelized

representation of a mouse with a resolution of 200x298x398 would occupy 190 Mb

in memory and take an estimated 3h of computation using the FDM

implantation in this work. Downsampling to a low resolution of 50x74x100

resulted in a matrix M with size [106x106] elements (sparse) and a more

reasonable 2.3s of computation, at a cost of reduced detail in the mouse volume

and more pronounced banding artifacts. Computation was performed on a dual 4-

core Xeon E5 workstation running at 2GHz.

While it is possible to improve upon the issues above with more advanced

FDM methods using non-uniform grids [86,87] and surface normal interpolation

[84,85], it is more efficient to use Finite Element Methods (FEM), which are

better suited for dealing with non-uniform spacing and irregular geometry.

53

2.4 FINITE ELEMENT METHOD

The Finite Element Method (FEM) is a widely used numerical method that

provides an approximate solution to, typically a differential equation, over a

domain Ω by dividing that domain into small elements [88]. For the purposes of

modeling light propagation in three dimensions in a small animal such as a

mouse, the problem domain, which is the volume containing the tissue, can be

represented as a mesh of tetrahedral elements as shown in Fig. 2.6. The solution

to the DA is approximated by discrete values of the photon density Φ at the

vertices of these elements, called nodes.

Figure 2.6 Tetrahedral mesh representation (right) of mouse CT volume (left)

2.4.2 Implementation

Starting with the Diffusion Approximation:

𝜇𝑎(𝒓)Φ(𝒓) − ∇ ⋅ 𝜅(𝒓)∇Φ(𝒓) = 𝑞(𝒓) (5)

Subject to the RBC on the boundary:

Φ(𝒓) + 2𝜅𝐴�̂� ⋅ ∇Φ(𝒓) = 0 𝒓 ∈ 𝜕Ω (9)

54

The weak formulation of this differential equation can be written as [65,66]:

�[∇φ𝑖(𝒓) ⋅ 𝜅(𝒓)∇𝜑𝑗(𝒓) +Ω

𝜇𝑎(𝒓)𝜑𝑖(𝒓)𝜑𝑗(𝒓)]Φ(𝒓)𝑑𝑟 + � 12𝐴

𝜑𝑖(𝒓)𝜑𝑗(𝒓)𝑑𝑟𝜕Ω

= � 𝑞(𝒓)𝜑𝑖(𝒓)𝜑𝑗(𝒓)𝑑𝑟Ω

(13)

To approach this problem, the region Ω is divided into 𝑛𝑒 tetrahedral elements

and the boundary 𝜕Ω into 𝑛𝑏 triangles, although other element geometries are

possible. The functions Φ(𝒓) and 𝑞(𝒓) are represented using a piecewise linear

approximation. Consequently, the integrals can be taken element-by-element and,

due to the use of linear functions in each element, reduce to simple expressions,

shown later.

This approximation relies on the use of “shape functions” or basis functions

𝜑𝑖which interpolate the nodal values inside the elements. For a given tetrahedral

element T containing four nodes, the value of an arbitrary function 𝑓(𝒓) inside

that element is approximated by:

𝑓(𝒓) ≅ � 𝑁𝑖𝜑𝑖(𝒓)4

𝑖=1 (14)

Where 𝑁𝑖 is the value of 𝑓(𝒓) at node i. Using this, Eq. (13) is written in matrix

form:

[𝐾 + 𝐶 + 𝐵]𝚽 = 𝒒 (15)

55

Where K, C, B are sparse matrices with size [n x n] and 𝚽,𝒒 are vectors of size

[n x 1], where n is the total number of nodes in the tetrahedral mesh. The entries

(i, j) of the matrices and entries i in the load vector q are given by:

𝐾𝑖,𝑗 = � ∇𝜑𝑖 ⋅ 𝜅∇𝜑𝑗 𝑑𝑟Ω

(16) 𝐶𝑖,𝑗 = � 𝜇𝑎𝜑𝑖𝜑𝑗 𝑑𝑟Ω

(17)

𝐵𝑖,𝑗 = 12𝐴

� 𝜑𝑖𝜑𝑗 𝑑𝑟∂Ω

(18) 𝒒𝑖 = � 𝑞𝜑𝑖 𝑑𝑟Ω

(19)

The integrals are evaluated only over elements containing nodes i and j since the

shape functions vanish outside this region.

2.4.3 Matrix Assembly

For each element T in the mesh, local 4x4 matrices are computed for K and C,

and local and 3x3 matrices for B since each tetrahedron T consists of 4 nodes and

each boundary triangle 3 nodes. For each T, a simple expression for the shape

functions is obtained by using a Barycentric coordinate system [89]:

𝜑𝑖(𝒓) = 14

− (𝒓 − 𝒓𝒃) ⋅ 𝐹𝑖𝒏𝒊3𝑉𝑒

𝑖 ∈ [1,2,3,4] (20)

Where 𝐹𝑖 is the area of the face opposite node i

𝒏𝒊 is the outward-facing unit normal to the face opposite node i

𝑉𝑒 is the volume of the element

𝒓𝒃 is the centroid of the tetrahedron

56

Similarly, an expression for the gradient of the shape function [89]:

∇𝜑𝑖(𝒓) = −𝐹𝑖𝒏𝒊3𝑉𝑒

𝑖 ∈ [1,2,3,4] (21)

Using Eq. (21), the local matrix K can be simplified:

𝐾𝑖,𝑗𝑙𝑜𝑐𝑎𝑙 = � ∇𝜑𝑖 ⋅ 𝜅∇𝜑𝑗 𝑑𝑟

Ω

= 𝐹𝑖𝐹𝑗𝒏𝒊 ⋅ 𝒏𝒋

9𝑉𝑒2 � 𝜅 𝑑𝑟

Ω

=𝐹𝑖𝐹𝑗𝒏𝒊 ⋅ 𝒏𝒋𝜅(𝒓𝒃)

9𝑉𝑒 (22)

Where 𝜅(𝒓𝒃) is the value of 𝜅(𝑟) at the centroid of the element. The local matrix

K for an element T is therefore:

𝐾𝑙𝑜𝑐𝑎𝑙 = 𝜅(𝒓𝒃)9𝑉𝑒

�𝑘11 ⋯ 𝑘14⋮ ⋱ ⋮

𝑘41 ⋯ 𝑘44

� (23)

Where 𝑘𝑖𝑗 = 𝐹𝑖𝐹𝑗𝒏𝒊 ⋅ 𝒏𝒋

A closed-form expression for the elements of matrix C is obtained by using a

result derived by Sharp for integrating the product of linear shape functions over

a tetrahedral element, which is rewritten here in consistent notation [81]:

�(𝜑𝑖)𝑚

Ω

(𝜑𝑗)𝑛 𝑑𝑟 = 𝑚!𝑛!(𝑚 + 𝑛 + 3)!

6𝑉𝑒 =

⎩�⎨�⎧

120

𝑉𝑒 𝑖 ≠ 𝑗110

𝑉𝑒 𝑖 = 𝑗

𝑖, 𝑗 ∈ [1,2,3,4]

(24)

Borrowing notation from Sharp where he defines 𝛿𝑖𝑗 = �1 𝑖 = 𝑗0 𝑖 ≠ 𝑗 [81], the

elements of matrix C may be written as follows, with the assumption that the

57

mesh is appropriately generated so that 𝜇𝑎 does not vary significantly over an

individual element:

𝐶𝑖,𝑗𝑙𝑜𝑐𝑎𝑙 = � 𝜇𝑎𝜑𝑖𝜑𝑗 𝑑𝑟

Ω

= 120

𝑉𝑒𝜇𝑎(𝒓𝑏)(1 + 𝛿𝑖𝑗)

𝑖, 𝑗 ∈ [1,2,3,4]

(25)

The matrix B enforcing boundary conditions (Eq. (18)) may also be simplified.

Noting that the boundary elements are triangles and that the shape functions

here are two dimensional. Again, a formula derived by Sharp is used here [81]:

�(𝜑𝑖)𝑚

Ω

(𝜑𝑗)𝑛 𝑑𝑟 = 𝑚!𝑛!(𝑚 + 𝑛 + 2)!

2𝐴𝑒 =

⎩��⎨��⎧ 1

12𝐴𝑒 𝑖 ≠ 𝑗

16

𝐴𝑒 𝑖 = 𝑗

𝑖, 𝑗 ∈ [1,2,3]

(26)

Where 𝐴𝑒 is the area of the triangular element Ω. This gives the following result,

noting that A is the reflection coefficient from Eq. (10) and not an area:

𝐵𝑖,𝑗𝑙𝑜𝑐𝑎𝑙 = 1

24𝐴𝐴𝑒(1 + 𝛿𝑖𝑗)

𝑖, 𝑗 ∈ [1,2,3] (27)

Finally, the load vector q from Eq. (19) is simplified by using Eq. (20):

𝒒𝑖 = � 𝑞𝜑𝑖 𝑑𝑟Ω

≅ 𝑞(𝒓𝒃)𝜑𝑖(𝒓𝒃)𝑉𝑒 = 𝑞(𝒓𝒃) ⋅ 14

𝑉𝑒 (28)

58

2.4.4 Conversion of Photon Density to Radiance

Eq. (15) provides a solution to the photon density, Φ, that satisfies the DA and

boundary conditions. Since the IVIS Spectrum CT bioluminescence imager used

in this work provides absolutely calibrated images of radiance, in units of

photons/(s·cm2·sr), it is necessary to convert between photon density at the

tissue boundary and surface radiance.

At the boundary, the outward photon flux density or exitance is given by the

following expression [90]:

𝐽(𝒓) = −𝜅(𝒓)� �̂� ⋅ ∇𝛷(𝒓)� = 𝛷(𝒓)2𝐴

(29)

Where 𝛷(𝒓) denotes “photon flux density” in Lu’s paper (units of photons/mm2·s)

rather than the photon density Φ used in this work (units of photons/mm3) [90].

Noting that the two are related [91]:

𝛷(𝒓) = 𝑐𝑛

Φ(𝒓) (30)

One approximation is to consider the tissue surface to be a Lambertian source

[92,93], in which case the radiance is angle-independent and given by the

following [94]:

𝐿(𝒓) = 𝐽(𝒓)𝜋

= 12𝐴

𝑐𝑛𝜋

Φ(𝒓) (31)

59

Alternatively, Kuo et al. provide an expression for surface radiance in terms of

photon density for the IVIS 200 Imaging system, similar to the IVIS Spectrum

CT used in this work [76]:

𝐿(𝒓, 𝜃𝑒) = 𝑐4𝜋𝑛2 𝑇 (𝜃) �1 +

3�1 − 𝑅𝑒𝑓𝑓�2�1 + 𝑅𝑒𝑓𝑓�

cos 𝜃� Φ(𝒓) (32)

Where Φ is the photon density at the tissue boundary and 𝜃𝑒 is the angle

between the tissue surface normal and camera lens. The geometry of the emission

angles 𝜃𝑒 and 𝜃, and coefficients 𝑇 (𝜃), 𝑅𝑒𝑓𝑓 are defined in the original paper by

Kuo et al. [76]. In subsequent real-world tests (Section 2.6.2), Eq. (31)

underestimated surface radiance by 30-60% while Eq. (32) provided accurate

results and was used in the remainder of this work.

60

2.5 IMPLEMENTATION AND SOURCE QUANTIFICATION

Software was written in MATLAB (MathWorks Inc.) to quantify the power of a

bioluminescent source using the hard spatial prior approach. The program flow is

outlined in Fig. 2.7

Figure 2.7 Flowchart of source quantification process using prior spatial information

Preprocessing and Segmentation

The CT volume (0.15mm isotropic resolution) is first filtered using a median

filter in the XY plane with a 3x3 kernel size and segmented to obtain the overall

tissue boundary. The segmented volume is cropped and filtered to remove

isolated segments and fill holes in preparation for mesh generation. Segmentation

61

of internal organ boundaries is done using CT to locate bones and co-registered

MRI to locate the brain. The current version of the software does not segment

other organs but may be extended to do so in the future. All other tissue is

assumed to have the optical properties of muscle.

The location of the bioluminescent source is segmented from either CT, when

radio-visible labeling is used, or from co-registered MRI. Source segmentation

from MRI may rely on intrinsic contrast or using cell labeling as previously

shown in Chapter 1: Figure 1.9, for example.

Mesh Generation

The iso2mesh toolbox [95] was used to generate a tetrahedral mesh from the

segmented anatomical volumes, as shown in Fig. 2.8. In this work, mesh

generation density was adjusted to give 500,000 elements on average and took

approximately 10s to generate, occupying 19 Mb in memory per mesh.

Figure 2.8 The CT volume is cropped and used to generate a mesh

62

Forward Simulation

The matrices in Eq. (15) are sparse and are assembled by iterating through each

mesh element and computing the local matrices according to Eq. (23), (25), and

(27). After previously segmenting the anatomical images to obtain a hard spatial

prior on the light source, the value of each non-zero voxel in the source prior

multiplied by its volume is added to the nearest mesh element according to Eq.

(28) to form the load vector q

The preconditioned conjugate gradient (PCG) method was used to solve for

the photon density 𝚽. The MATLAB 2013 implementation was used with the

following parameters: max. iterations = 750, tolerance (relative residual): 10-10.

While the entries of 𝚽 should all be greater than or equal to zero, the PCG

method was chosen over a non-negative least squares (NNLS) solver for increased

speed and provided reasonably accurate results. For a mesh density of 5x105

elements and n = 75,000 nodes, matrix assembly took 40s (sparse [n x n] matrix

with 106 non-zero elements). Solutions using PCG took an average of 1.1s for this

mesh size.

Calculation of radiance was done through Eq. 32 and an orthographic

projection from above of the photon density at the tissue-environment boundary.

Multispectral Calculations

The emission spectrum of firefly luciferase was divided into N = 8 bins (bin

centers: 540-680 nm, bin width: 20 nm). Forward simulations using the optical

63

properties at each bin wavelength gave N images of radiance, which were then

summed into an image of total radiance:

𝐿𝑡𝑜𝑡 = � 𝜂𝑖𝐿(𝜆𝑖)𝑁

𝑖=1 (33)

Where 𝜆𝑖 is the bin center wavelength, 𝐿(𝜆𝑖) is the radiance at that wavelength,

and 𝜂𝑖 is the fraction of the total emission spectrum power contained in that bin,

satisfying:

� 𝜂𝑖𝑁

𝑖=1= 1 (34)

Eq. (33) holds approximately true for the spectrum of firefly luciferase for the

choice of bins used, where the range of 530-690 holds 90% of the total spectral

power. For the remainder of the spectrum outside the bins used, a zero-order

approximation was used to add its contribution to the total radiance:

𝐿(𝜆𝑖) = 𝐿(𝜆1) 𝑖 < 1

𝐿(𝜆𝑖) = 𝐿(𝜆𝑁) 𝑖 > 𝑁 (35)

Source Quantification

The total radiance image computed was compared against radiance measured

from BLI without the use of an emission filter. If 𝐴(𝑥, 𝑦) is the simulated

radiance image using a unit power light source and 𝐵(𝑥, 𝑦) is the measured

image, then an accurate forward simulation will ensure that they are linearly

related:

64

𝐴(𝑥 − 𝑚, 𝑦 − 𝑛) = 𝛼𝐵(𝑥, 𝑦) (36)

Where (m,n) is any small offset between the images due to error in co-

registration. The scale factor 𝛼 is obtained from the cross-correlation of the

images:

𝛼 =∑ 𝐴(𝑥 − 𝑚, 𝑦 − 𝑛)𝐵(𝑥, 𝑦)𝑥,𝑦

∑ 𝐵(𝑥, 𝑦)𝐵(𝑥, 𝑦)𝑥,𝑦= 𝜌(𝐴, 𝐵; 𝑥 = 𝑚, 𝑦 = 𝑛)

𝜌(𝐵,𝐵; 𝑥 = 0, 𝑦 = 0) (37)

Where 𝜌(𝐴, 𝐵; 𝑥 = 𝑚, 𝑦 = 𝑛) is the value of the cross-correlation matrix of the

two images at location (m,n). A derivation is given in Appendix C. The offset in

practice was a few pixels at most and was accounted for by taking the peak value

of the cross-correlation (Fig. 2.9).

Figure 2.9 Cross-correlation of simulated and measured BLI images shows a single peak near the center.

65

Since the measured surface radiance is linear with respect to source intensity, the

source power is obtained by scaling of the source power used in the forward

simulation by 𝛼. A sample of the software output is shown in Fig. 2.10.

Figure 2.10 Output images showing simulated and measured images of radiance on the top surface of a tissue-mimicking phantom and an error image (right).

66

2.6 METHOD VALIDATION

2.6.1 Numerical Validation of Forward Model

1) Against ODE

The FEM implementation of the DA was tested in spherical geometry as was

done in Section 2.3.3. A table of optical properties used is given below:

Table 2.2 List of optical properties for uniform and non-uniform spherical cases used in FEM validation

Uniform 𝑟1= 7 mm 𝑟2= 7 mm 𝑟3= 7 mm 0 ≤ 𝑟 ≤ 𝑟1 𝑟1 ≤ 𝑟 ≤ 𝑟2 𝑟2 ≤ 𝑟 ≤ 𝑟3

𝜇𝑎 (mm-1) 0.2 0.2 0.2 𝜇𝑠′ (mm-1) 1 1 1

Non-uniform 𝑟1= 1 mm 𝑟2= 3 mm 𝑟3= 7 mm 0 ≤ 𝑟 ≤ 𝑟1 𝑟1 ≤ 𝑟 ≤ 𝑟2 𝑟2 ≤ 𝑟 ≤ 𝑟3

𝜇𝑎 (mm-1) 2 1 0.01 𝜇𝑠′ (mm-1) 0.1 1 2

In both cases, 𝑟𝑞= 2 mm, 𝑞𝑜 = 100 photons/mm4 inside 𝑟𝑞, n = 1.4. The results

indicate good agreement between the FEM and ODE, with the constant error

seen beyond 3 mm in the inhomogeneous case likely due to the method of

assigning optical properties to discrete elements in this test. A more sophisticated

mesh generation procedure, which avoids creating elements that span two

separate optical regions, would mitigate this error.

67

Figure 2.11 Photon density vs. distance from center for the FEM model compared against the ODE in the optically homogeneous case.

Figure 2.12 Photon density vs. distance from center for the FEM model compared against the ODE in the optically inhomogeneous case.

68

2) Comparison of DA with Monte Carlo

The validity of the DA was compared to a Monte Carlo (MC) simulation as a

gold standard using the TIM-OS software package by Shen et al. [75]. This case

tested a uniform spherical mesh of radius 7 mm with a point source emitting

from the center. 106 photons were simulated, taking 21s to simulate. Optical

properties were chosen to be representative of typical values seen in biological

tissues at around 600 nm: 𝜇𝑎 = 0.05 mm-1, 𝜇𝑠 = 10mm-1, g = 0.9, n = 1.4

Figure 2.13 Photon density vs. distance from center for the FEM model compared against TIM-OS in an optically homogeneous medium.

The largest deviations are seen near the light source (r = 0 mm) and near the

boundary where the assumptions of the DA are violated, as expected.

Nonetheless, the DA is reasonably accurate near the boundary (r = 7 mm)

compared to the MC simulation, with a maximum relative error of 10%.

69

2.6.2 Validation against Tissue Mimicking Phantom

Having verified the DA forward model, the accuracy of the overall source

quantification software was tested using calibrated light sources placed inside a

tissue mimicking, mouse-shaped phantom (XFM-2 Phantom, PerkinElmer Inc.),

shown in Fig. 2.14. Phantom optical properties and spectra were provided by

PerkinElmer Inc. and are roughly equal to the following at 600 nm: 𝜇𝑎 = 0.03

mm-1, 𝜇𝑠′ = 1.8 mm-1, n = 1.5

Figure 2.14 XFM-2 tissue-mimicking phantom

The calibrated light sources used were cylindrical, radioluminescent beads

(Traser, mb-microtec). Each bead measures 2.5 mm x 0.9 mm (length x

diameter) and is filled with tritium gas that excites a phosphor coating, giving a

steady light output.

70

Output power from the beads was measured using the IVIS Spectrum CT, which

provides 2D images of radiance, by integrating over the visible surface of the

beads. The bead emission spectrum was verified against manufacturer

specifications using the emission filters on the IVIS machine and is shown in Fig.

2.15. The spectrum closely matches that of firefly luciferase, making the beads a

good model for a bioluminescent light source.

Bead light output power was measured to be 1.15x1010 ± 0.14 x1010 photons/s

(95% CI), which is on the order of the bioluminescent source powers measured in

Chapter 1.

Figure 2.15 Emission spectra of firefly luciferase and tritium bead

The beads were easily visible in CT images of the phantom (Fig. 2.16). Source

power was quantified using three methods for comparison:

1) Multispectral BLT using Living Image software, similarly to Section 1.4.

Phantom optical properties and tritium bead spectra were predefined in

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software. Six spectral bins were used in the reconstruction, from 560 nm to

660 nm.

2) The hard spatial prior method (Section 2.5). Bead location was segmented

from the CT volume (Fig 2.16).

3) The hard spatial prior method with the TIM-OS Monte Carlo software

used in place of the DA forward model

Figure 2.16 Coronal CT sections of XFM-2 phantom, showing two possible locations for tritium bead placement The errors in the reconstructed source powers are given in Table 2.3 below. In

the case of the phantom, which is optically homogeneous, prior knowledge of the

light source location provides more accurate quantification of the source power.

The performance of the DA forward model is comparable to that of the MC

simulation. Interestingly, there is a bias in the errors of the FEM and MC

methods, with an increasing tendency to overestimate the source power with

increasing depth, suggesting that the optical properties of the phantom used in

the simulation slightly overestimate attenuation.

72

Table 2.3 Comparison of source power quantification using multispectral BLT and the hard spatial prior approach

Number of Sources

Source Depth(s) (mm)

No Spatial Prior Hard Spatial Prior

Living Image FEM MC 1 11.5 36% 7% 1% 1 19.0 -27% 27% 28% 2 11.5, 19.0 45% 18% 15% Mean Absolute

Error 36% 17% 15%

2.7 IN VIVO TESTING

2.7.1 Procedure

To measure real-world performance of the hard spatial prior approach, the

tritium beads, which provide a stable, calibrated light output mimicking the

spectrum of firefly luciferase, were implanted into mice to simulate

bioluminescent imaging. Nine BALB/c mice were euthanized and an incision was

made through the skin to allow a small hole to be drilled into the skull. A tritium

bead was inserted into the brain in varying locations and depths and the skin and

fur flap was restored and held in place until it adhered to the skull. The animals

were immediately imaged afterwards in the IVIS Spectrum CT. The procedure

was done on one animal at a time to minimize the time between the procedure

and imaging. While previous studies have inserted calibrated beads into the

abdomen of mice [33,35], in this work the brain was chosen as the site of

transplantation for consistency with the tests in Chapter 1 of this thesis. In

addition, the hard spatial prior approach is particularly valid in the brain where

73

MRI may be used to locate and segment tagged cells but is otherwise more

difficult to implement in other areas in the body.

Reconstruction of bead source power was done similarly to Section 2.6.2 using

eight bins using emission filters from 560-700 nm with a bandwidth of 20 nm

each. Imaging parameters were: aperture = f/1, FOV = 13x13 cm, 2048x2048

pixel resolution, 8x8 pixel binning. Exposure time was set to automatic, and

ranged from 30 s at 560 nm to 1 s at 640 nm. Bead location was segmented from

the CT volumes (Fig. 2.17).

Figure 2.17 Implanted tritium bead is visible in CT and shown segmented in red.

The Living Image BLT reconstruction treats the mouse as a homogeneous volume

and uses overall tissue properties. Likewise, the same uniform optical properties

were used in the hard spatial method for comparison. The optical properties are

proprietary to PerkinElmer Inc. and are approximately equal to the following at

600 nm: 𝜇𝑎 = 0.2 mm-1, 𝜇𝑠′ = 1 mm-1, n = 1.4. For reference, a survey of in vivo

mouse optical properties can be found in the literature [49,96].

74

To give an upper bound on the accuracy of the hard prior method, an MC

simulation was performed in place of the DA-FEM method, using a

heterogeneous model for the mouse tissue. Bone and brain tissue were segmented

from the CT volume captured by the IVIS imager in Amira; all other tissue was

assigned as muscle (Fig. 2.18). Optical properties for the bone and brain were

obtained from the Living Image software. The same spectral bins were used in

the MC simulation as the Living Image BLT and DA-FEM methods.

Figure 2.18 Overall mouse volume (left) and segmented bone and brain tissue (right) obtained from CT images

2.7.2 Results

The source powers obtained with each method are shown below in Fig. 2.19. For

comparison purposes, source power was also computed by integrating the total

surface flux (photons/s) from unfiltered 2D BLI images and using a

retrospectively determined attenuation correction factor of 0.24, chosen so that

the mean source power using this method matched the actual bead power of

1.15x1010 photons/s.

75

Figure 2.19 Comparison of reconstructed source powers using corrected total flux from BLI, BLT with differing number of bins used, and the hard prior method (n=9). Dashed line represents the actual power of the calibrated light source. Whiskers show the data minimum and maximum.

To give a meaningful comparison of the variances in the results in Fig. 2.19, the

datasets were normalized so that each method gave a mean source power equal to

the actual bead power (Fig. 2.20). In practice, this corresponds to using an

empirically determined correction factor. The standard deviations and mean

absolute deviations of the normalized source powers from the actual bead power

for each method are reported below:

Table 2.4 Standard and mean absolute deviations of the normalized datasets from the calibrated bead power for each of the quantification methods 2D BLI Homogeneous,

BLT – 8 bins Homogeneous, BLT – 4 bins

Homogeneous, DA

Heterogeneous, MC

Std. Deviation (109 photons/s) 7.39 3.13 8.28 3.90 3.77

Mean Absolute Deviation (109 photons/s)

5.09 2.74 6.75 3.13 2.40

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Figure 2.20 Comparison of variances in reconstructed source powers after normalization of the dataset from Fig. 2.19 to give a mean source power of 1.15x1010 photons/s for each method. Dashed line represents the actual power of the calibrated light source. Whiskers show the data minimum and maximum.

A comparison between the methods is given below, examining the time needed to

acquire the BLI images and perform the simulations/reconstructions.

Computation time includes both the time needed to assemble any applicable

matrices and solve for the photon density, but excludes image preprocessing and

mesh generation since it is common to all methods (excluding 2D BLI):

Table 2.5 Comparison of time needed for source quantification using planar BLI, multispectral BLT, and the hard spatial prior method

Method BLI Acquisition Time (s)

Computation Time (s)

2D BLI 0.5 N/A Homogeneous BLT – 4 bins 8 ≈1 Homogeneous BLT – 8 bins 51 ≈1 Homogeneous DA-FEM 0.5 50 Heterogeneous MC 0.5 186

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2.7.3 Discussion

A strong bias was observed in the source powers obtained using the hard prior

method as seen in Fig. 2.19. A possible cause may be the loss of tissue

oxygenation after euthanasia, causing an increase in the attenuation coefficient in

the tissues of the brain due to the increased absorption of light by

deoxyhemoglobin. As a result, the simulations underestimate the attenuation of

light and consequently the source power needed to produce the measured BLI

image. To test this, a subsequent test was done where a bead was implanted deep

into the leg muscle of one of the euthanized animals since skeletal muscle has

only 30% the blood perfusion of the brain in mice [97]. The hard prior method in

this case gave a source power of 1.01x1010 photons/s, closer to the true source

power compared to a mean of 4.15x109 photons/s in the brain (Fig. 2.19). This

suggests that correcting the optical properties used for the brain tissue, either

through empirical fitting or an in situ measurement, may correct the bias in the

errors from the hard prior method when testing in euthanized animals.

After normalizing the results in Fig. 2.20, both hard prior methods

outperformed multispectral BLT using four bins. Source quantification using the

homogeneous DA implementation was inferior to BLT using eight bins in,

however, with a greater spread in the reconstructed source powers. It is possible

that adjustment of the optical properties used, to correct the error seen in Fig.

2.19, would improve the performance of the hard prior DA-FEM method. Even

with the possible error in optical properties, the hard prior method using a

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heterogeneous tissue model and a MC simulation of light performed similarly to

multispectral BLT using eight bins. The comparison of the time needed by the

multispectral BLT and hard prior methods in Table 2.5 shows that while the

MATLAB code implementation of the DA in this thesis is not as efficient as the

calculations performed by the Living Image software, the hard prior method

saves significant time during imaging by eliminating the need for spectrally

binned images completely.

In terms of the applicability of the bead procedure to tests in live mice with

luciferase-expressing cells, it is apparent that the optical properties in the

euthanized animals used in this work would be different due to the decrease in

blood and tissue oxygenation. However, it is important to note that in testing the

performance of the hard prior and multispectral BLT methods in the brain, it

would not be ethical to implant a large object such as a luminescent bead into

live mice. It is likewise difficult to measure the performance of source

quantification by, for instance, injecting a known number of luminescent cells

with a calibrated light output, since changes in cell viability, the successfully

transplanted population, and proliferation after injection could impact the

results. The use of a calibrated light source in recently sacrificed animals provides

a heterogeneous optical environment with similar, but not identical, optical

properties to a live mouse. In comparison, current studies in BLT algorithms

have largely used tissue mimicking phantoms or numerical simulations to in order

to validate their results.

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2.8 CONCLUSIONS AND FUTURE WORK

The hard spatial prior method for source quantification demonstrated in this

work adds the ability to quantify a bioluminescent source in vivo to studies

incorporating anatomical imaging modalities such as MRI or CT. The overhead

in this method is a single BLI image and relatively quick computations that can

be done after imaging. The benefit of this approach, in addition to reducing the

contribution of depth-related errors from BLT on quantification, as seen in

Chapter 1 for example, is that it mitigates the issue of BLI kinetics by reducing

the imaging time needed. While this work has examined the hard spatial prior

method in the brain, in other sites of transplantation where it is more difficult to

segment the location of light-producing cells from anatomical imaging, such as

near the viscera, the performance of multispectral BLT on its own may be

sufficient. For instance, previous work by Allard et al. indicates that the use of a

homogeneous tissue model for light sources implanted into in the abdomen of

mice introduces only marginal errors compared to a heterogeneous model [33].

A future extension of this work should examine the use of a soft prior

approach instead, which penalizes the deviation of the reconstructed light source

from prior spatial information. Since prior anatomical images may not always

reflect the true location of viable, luciferase-expressing cells, the soft prior

approach may be more robust to incorrect prior information than hard priors.

80

Summary and Conclusions

The co-registration of the relatively new optical imaging modality of BLT with

well-established anatomical imaging methods was examined in two applications in

this thesis. In Chapter 1, the benefits of a multimodal imaging approach

incorporating BLT and MRI for in vivo cell tracking were investigated. BLT

provided a low-resolution, low accuracy reconstruction of the light-producing cells

that was highly sensitive to changes in viable cell number. Conversely, MRI was

superior at localizing the transplanted cells both in accuracy and resolution, but

did not directly provide information on the viability of the transplanted cells. In

research applications, the complementary data obtained from multimodal imaging

may be used where transplanted cell fate needs to be examined in detail.

Alternatively, co-registered BLT may be used as a low-cost, high throughput

method to pinpoint areas of interest in an animal model, to be followed up with

more accurate, time-expensive MRI methods in the highlighted regions.

In Chapter 2 of this thesis, co-registered anatomical information was used in

an alternate approach by providing prior information on the light source location

in an attempt to improve the performance of BLT as a quantitative in vivo

imaging modality. While the hard spatial prior method did not show a clear

improvement in quantification accuracy over multispectral BLT, likely due to

inaccuracy in the optical properties used in this work, it showed significant

improvement in imaging times, making the incorporation of BLT into multimodal

imaging studies a more viable option.

81

To this date BLT remains a highly researched topic, due to the promise of a

low-cost, safe, and high-throughput quantitative imaging technique. While it is

unlikely that BLT will displace the better established molecular or cellular

imaging techniques in MRI, in the future it is hoped that multimodal approaches

to imaging incorporating BLT will be used to draw more valuable conclusions

from studies while still in their pre-clinical stages.

82

Glossary of Terms and Notation

Symbol Description Units 𝐿 Radiance photons/(m2·s·sr) 𝒓 Position vector [m,m,m] 𝒔 ̂ Unit direction vector 𝑐 Speed of light, in vacuum m/s 𝜈 Speed of light, in medium m/s 𝑛 Medium refractive index 𝜇𝑎 Absorption coefficient m-1 𝜇𝑠 Scattering coefficient m-1 𝜇′𝑠 Reduced scattering coefficient m-1 𝑔 Scattering anisotropy

𝑃(𝒔,̂ 𝒔′̂) Scattering phase function, probability of scattering from direction 𝒔′̂ to 𝒔 ̂

𝑞𝑠(𝒓, 𝒔)̂ Directional source term, giving density of photons at location 𝒓 traveling in direction 𝒔 ̂

photons/(m4·sr)

𝜙(𝒓, 𝒔)̂ Photon density at location 𝒓 traveling in direction 𝒔 ̂

photons/(m3·sr)

Φ(𝒓) Isotropic photon density at location 𝒓 photons/m3 𝜅(𝒓) Diffusion coefficient m 𝑞(𝒓) Isotropic light source term at location 𝒓 photons/m4 �̂� Unit normal vector to boundary 𝐴 Reflection coefficient Ω Interior region of mesh 𝜕Ω Boundary region of mesh

𝜌(𝐴, 𝐵) Cross-correlation matrix of images A and B

83

Appendix

APPENDIX A: HISTOLOGICAL ANALYSIS

Hematoxylin and Eosin (H&E) Staining

H&E staining was performed as described in previous work by our group [98].

Briefly, the mice were transcardially perfused with 1X phosphate-buffered saline

(PBS), followed by 4% paraformaldehyde in PBS (PFA). The brains were

removed, postfixed in PFA overnight at 4°C, cryopreserved in 30% sucrose, and

then snap frozen on dry ice. Serial coronal sections 30 µm thick were cut using a

Thermo Scientific HM 550 cryostat and transferred to electrostatically-charged

glass slides. Sections were then stained using H&E stains and examined under a

light microscope.

Prussian Blue Staining

A Prussian blue staining protocol was used to visualize SPIO deposits in the

sectioned tissue. Sections were dried for 2 h at 50°C then rehydrated overnight in

1X PBS at 4°C. Sections were incubated for 1 h in the dark in freshly prepared

Perls reagent (Prussian Blue Reagent, BioPAL Inc.), rinsed in PBS, then

counterstained with Nuclear Fast Red stain for 5 min. Sections were rinsed in

distilled water, dehydrated, and mounted on coverslips using a toluene-based

medium (SHUR/Mount, Triangle Biomedical Sciences) prior to being examined

under a light microscope.

84

Immunohistochemistry

Immunohistochemical analysis was done similar to previous work [99]. In brief,

coronal sections were dried for 2 h at 50°C then rehydrated in PBS for 15 min at

room temperature. Nonspecific binding was blocked using a blocking solution

consisting of 10% horse serum, 0.1% Triton X-100 in PBS for 2 h at room

temperature. Sections were then incubated overnight at 4°C with anti-firefly

luciferase (1:1000 dilution) (GeneTex) primary antibody in 0.1% Triton X-100

solution. The corresponding secondary antibodies were added 1:500 in 10% horse

serum for 2 h at room temperature. Sections were then rinsed with 0.1 M PBS,

counterstained with DAPI, and mounted on coverslips with aqueous non-

fluorescing medium (Fluoro-gel with Tris Buffer, Electron Microscopy Sciences).

Images were obtained with a Zeiss AX10 fluorescence microscope.

85

APPENDIX B: FDM MATRIX COEFFICIENTS

Starting with the diffusion approximation:

𝜇𝑎Φ − ∇ ⋅ 𝜅∇Φ = 𝑞

Expansion using the product rule and approximation of the derivatives of Φ with

central differences gives:

𝜇𝑎(x, y, z)Φ(x, y, z) −

⎣⎢⎢⎢⎢⎢⎢⎢⎡∂κ∂x

Φ(𝑥 + ∆, 𝑦, 𝑧) − Φ(𝑥 − Δ, y, z)2Δ

+ 𝜅Φ(𝑥 + Δ, 𝑦, 𝑧) − 2Φ(𝑥, 𝑦, 𝑧) + Φ(𝑥 − Δ, 𝑦, 𝑧)Δ2 +

∂κ∂y

Φ(𝑥, 𝑦 + ∆, 𝑧) − Φ(𝑥, y − Δ, z)2Δ

+ 𝜅Φ(𝑥, 𝑦 + Δ, 𝑧) − 2Φ(𝑥, 𝑦, 𝑧) + Φ(𝑥, 𝑦 − Δ, 𝑧)Δ2 +

∂κ∂z

Φ(𝑥, 𝑦, 𝑧 + ∆) − Φ(𝑥, y, z − Δ)2Δ

+ 𝜅Φ(𝑥, 𝑦, 𝑧 + Δ) − 2Φ(𝑥, 𝑦, 𝑧) + Φ(𝑥, 𝑦, 𝑧 − Δ)Δ2 ⎦

⎥⎥⎥⎥⎥⎥⎥⎤

= 𝑞(𝑥, 𝑦, 𝑧)

Where Δ is the grid spacing. Working with a voxelized representation of the

tissue with a total of m voxels, the equation relating the photon density Φ at

each voxel to the light source 𝑞 is

(𝝁𝒂𝐼 − 𝑀)𝚽 = 𝒒

𝝁𝒂, 𝚽, 𝒒 are vectors of size m, I is the [m x m] identity matrix, M is an [m x m]

matrix of coefficients. The photon density is computed given the light source:

𝚽 = (𝝁𝒂𝐼 − 𝑀)−𝟏𝒒

86

M is a sparse matrix with entries in each row chosen as follows: for a given voxel

i with photon density Φ(𝑥𝑖, 𝑦𝑖, 𝑧𝑖), row i of M is populated to multiply

Φ(𝑥𝑖, 𝑦𝑖, 𝑧𝑖) and neighboring voxels by the following coefficients:

Φ(𝑥𝑖, 𝑦𝑖, 𝑧𝑖) : − 6

Δ 𝜅(𝑥𝑖, 𝑦𝑖, 𝑧𝑖)

Φ(𝑥𝑖+1, 𝑦𝑖, 𝑧𝑖) : 12Δ

𝜕𝜅𝜕𝑥�(𝑥𝑖,𝑦𝑖,𝑧𝑖) + 1

Δ2 𝜅(𝑥𝑖, 𝑦𝑖, 𝑧𝑖)

Φ(𝑥𝑖−1, 𝑦𝑖, 𝑧𝑖) : −12Δ

𝜕𝜅𝜕𝑥�(𝑥𝑖,𝑦𝑖,𝑧𝑖) + 1

Δ2 𝜅(𝑥𝑖, 𝑦𝑖, 𝑧𝑖)

Φ(𝑥𝑖, 𝑦𝑖+1, 𝑧𝑖) : 12Δ

𝜕𝜅𝜕𝑦�(𝑥𝑖,𝑦𝑖,𝑧𝑖) + 1

Δ2 𝜅(𝑥𝑖, 𝑦𝑖, 𝑧𝑖)

Φ(𝑥𝑖, 𝑦𝑖−1, 𝑧𝑖) : −12Δ

𝜕𝜅𝜕𝑦�(𝑥𝑖,𝑦𝑖,𝑧𝑖) + 1

Δ2 𝜅(𝑥𝑖, 𝑦𝑖, 𝑧𝑖)

Φ(𝑥𝑖, 𝑦𝑖, 𝑧𝑖+1) : 12Δ

𝜕𝜅𝜕𝑧�(𝑥𝑖,𝑦𝑖,𝑧𝑖) + 1

Δ2 𝜅(𝑥𝑖, 𝑦𝑖, 𝑧𝑖)

Φ(𝑥𝑖, 𝑦𝑖, 𝑧𝑖−1) : −12Δ

𝜕𝜅𝜕𝑧�(𝑥𝑖,𝑦𝑖,𝑧𝑖) + 1

Δ2 𝜅(𝑥𝑖, 𝑦𝑖, 𝑧𝑖)

87

APPENDIX C: SCALING COEFFICIENT FROM CROSS-CORRELATION

Given two images 𝐴(𝑥, 𝑦) and 𝐵(𝑥, 𝑦) that are related by a scaling factor in

addition to some deviation 𝜖(𝑥, 𝑦) that is not linearly related:

𝐴(𝑥, 𝑦) = 𝛼𝐵(𝑥, 𝑦) + 𝜖(𝑥, 𝑦)

Additionally, image 𝐴(𝑥, 𝑦) is displaced by (𝑚, 𝑛) due to a slight registration

error:

𝐴(𝑥 − 𝑚, 𝑦 − 𝑛) = 𝛼𝐵(𝑥, 𝑦) + 𝜖(𝑥, 𝑦)

The sum of squared error between the images is:

𝐸 = �(𝐴(𝑥 − 𝑚, 𝑦 − 𝑛) − 𝛼𝐵(𝑥, 𝑦) − 𝜖(𝑥, 𝑦))2 𝑥,𝑦

The error as a function of the scaling factor is convex. Minimization with respect

to the scaling factor gives:

𝜕𝐸𝜕𝛼

= 0

0 = �−2𝐵(𝑥, 𝑦)(𝐴(𝑥 − 𝑚, 𝑦 − 𝑛) − 𝛼𝐵(𝑥, 𝑦) − 𝜖(𝑥, 𝑦)) 𝑥,𝑦

0 = �𝐴(𝑥 − 𝑚, 𝑦 − 𝑛)𝐵(𝑥, 𝑦) − �𝛼𝐵(𝑥, 𝑦)2

𝑥,𝑦−

𝑥,𝑦�𝐵(𝑥, 𝑦)𝜖(𝑥, 𝑦)𝑥,𝑦

If the non-linear term 𝜖(𝑥, 𝑦) is small, noting that 𝛼 is independent of x and y the

optimal scaling factor is:

88

𝛼 =∑ 𝐴(𝑥 − 𝑚, 𝑦 − 𝑛)𝐵(𝑥, 𝑦)𝑥,𝑦

∑ 𝐵(𝑥, 𝑦)𝐵(𝑥, 𝑦)𝑥,𝑦

The numerator ∑ 𝐴(𝑥 − 𝑚, 𝑦 − 𝑛)𝐵(𝑥, 𝑦)𝑥,𝑦 is simply the value at (𝑚, 𝑛) of the

cross-correlation matrix of images 𝐴(𝑥, 𝑦) and 𝐵(𝑥, 𝑦). Likewise, the denominator

is the value at the center of auto-correlation matrix of image 𝐵(𝑥, 𝑦).

89

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Curriculum Vitae

Moussa Chehade graduated from the University of

Toronto in 2012 with a bachelor’s degree in Engineering

Science, specializing in Biomedical Engineering. In

September of 2012, he enrolled in the Masters of Science

in Engineering (M.S.E) program at the Johns Hopkins

University in the Department of Biomedical Engineering. His graduate work was

done under the guidance of Dr. Jeff W. M. Bulte in the Cellular Imaging Section

at the Institute for Cell Engineering. Moussa was the recipient of an Alexander

Graham Bell Canada Graduate Scholarship in 2012. His research focus includes

medical image processing, co-registration, and visualization.

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