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 luciferaseexpressing 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
ii
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
iii
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
iv
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.
v
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
vi
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
vii
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
xii
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 coregistered 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.
6
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 coregistered, has the potential to compensate for the shortcomings of the individual
modalities. One approach that has been explored in literature has been to use coregistered 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 BLIfluorescence-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 coregistration 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 coregistered 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 256core CUDA GPU is used to perform CT and BLT reconstructions. The IVISgenerated 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/nonlinear registration to account for changes in subject position
Intermodality non-linear registration is a non-trivial task, therefore a rigidtransformation 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 finepositioning 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 Dluciferin were injected intraperitoneally. Images were acquired 10 min after
injection to maximize the bioluminescence signal. During imaging, four spectrallybinned 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 nonunique [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−𝜇𝑎 𝑥
𝑃𝑠 (𝑥) = 𝑒−𝜇𝑠 𝑥
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]:
1 𝜕𝐿
+ 𝒔̂ ⋅ ∇𝐿(𝒓, 𝑡, 𝒔̂) + (𝜇𝑎 + 𝜇𝑠 )𝐿(𝒓, 𝑡, 𝒔̂) = 𝜇𝑠 � 𝑃 ( 𝒔̂, 𝒔̂′)𝐿(𝒓, 𝑡, 𝒔̂′)𝑑2 𝒔̂′ + 𝑞𝑠 (𝒓, 𝑡, 𝒔̂)
𝜈 𝜕𝑡
𝑆
(1)
Where L is the radiance at position r in direction 𝒔̂, 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, 𝒔̂), the volumetric density of photons traveling in direction 𝒔̂,
[66]:
�𝒔̂ ⋅ ∇ + 𝜇𝑠 (𝒓) + 𝜇𝑎 (𝒓)�𝜙(𝒓, 𝒔̂) = 𝜇𝑠 (𝒓) � 𝑃 (𝒔̂, 𝒔̂′ )𝜙(𝒓, 𝒔̂′ )𝑑𝒔̂′ + 𝑞𝑠 (𝒓, 𝒔̂)
(2)
𝑆
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]:
(3)
Φ(𝒓) = � 𝜙(𝒓, 𝒔̂)𝑑𝒔̂
𝑆
(4)
𝑞(𝒓) = � 𝑞𝑠 (𝒓, 𝒔̂)𝑑𝒔̂
𝑆
The resulting DA equation is shown below, with derivations available in the
relevant literature [65,68]:
(5)
𝜇𝑎 (𝒓)Φ(𝒓) − ∇ ⋅ 𝜅(𝒓)∇Φ(𝒓) = 𝑞(𝒓)
Where 𝜅(𝑟) = 3(𝜇 1+𝜇′ ) and 𝜇′𝑠 = 𝜇𝑠 (1 − 𝑔)
𝑎
𝑠
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:
(5)
𝜇𝑎 (𝒓)Φ(𝒓) − ∇ ⋅ 𝜅(𝒓)∇Φ(𝒓) = 𝑞(𝒓)
Where 𝜇𝑎 , Φ, 𝜅, 𝑞 vary arbitrarily over space. Expanding the gradient and
divergence operators gives:
𝜇𝑎 Φ − �
∂
𝜕Φ
∂
𝜕Φ
∂
𝜕Φ
�𝜅 � +
�𝜅 � + �𝜅 �� = 𝑞
∂x
𝜕𝑥
∂y
𝜕𝑦
∂z
𝜕𝑧
(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:
(7)
(𝝁𝒂 𝐼 − 𝑀 )𝚽 = 𝒒
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:
(8)
𝚽 = (𝝁𝒂 𝐼 − 𝑀)−𝟏 𝒒
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) + 2𝜅𝐴𝒏̂ ⋅ ∇Φ(𝑟) = 0
𝐴=
(9)
𝑟 ∈ 𝜕Ω
(10)
1 + 𝑅𝑖
1 − 𝑅𝑖
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]:
(11)
𝑅𝑖 ≅ −1.4399𝑛−2 + 0.7099𝑛−1 + 0.6681 + 0.0636𝑛
Assuming an index of refraction n = 1 for the environment. In the FDM solution
to
the
diffusion
approximation,
the
entries
in
the
matrix
(𝝁𝒂 𝐼 − 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
−𝜅
𝑑2 Φ
2𝜅
𝑑𝜅 𝑑Φ
−
�
+
�
+ 𝜇𝑎 Φ = 𝑞(𝑟)
𝑑𝑟2
𝑟
𝑑𝑟 𝑑𝑟
(12)
With boundary conditions:
𝑑Φ
=0
𝑑𝑟
Φ + 2𝜅𝐴
𝑎𝑡 𝑟 = 0
𝑑Φ
=0
𝑑𝑟
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 nonuniform 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
𝜇𝑎 (mm-1)
𝜇𝑠 ′ (mm )
-1
Non-uniform
𝜇𝑎 (mm )
-1
𝜇𝑠 ′ (mm-1)
𝑟1= 7 mm
0 ≤ 𝑟 ≤ 𝑟1
𝑟2 = 7 mm
𝑟1 ≤ 𝑟 ≤ 𝑟2
𝑟3 = 7 mm
𝑟2 ≤ 𝑟 ≤ 𝑟3
2
2
2
𝑟1= 1 mm
0 ≤ 𝑟 ≤ 𝑟1
𝑟2 = 3 mm
𝑟1 ≤ 𝑟 ≤ 𝑟2
𝑟3 = 7 mm
𝑟2 ≤ 𝑟 ≤ 𝑟3
0.1
1
2
0.1
2
0.1
1
0.1
0.01
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 4core 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:
(5)
𝜇𝑎 (𝒓)Φ(𝒓) − ∇ ⋅ 𝜅(𝒓)∇Φ(𝒓) = 𝑞(𝒓)
Subject to the RBC on the boundary:
Φ(𝒓) + 2𝜅𝐴𝒏̂ ⋅ ∇Φ(𝒓) = 0
(9)
𝒓 ∈ 𝜕Ω
54
The weak formulation of this differential equation can be written as [65,66]:
� [∇φ𝑖 (𝒓) ⋅ 𝜅(𝒓)∇𝜑𝑗 (𝒓) + 𝜇𝑎 (𝒓)𝜑𝑖 (𝒓)𝜑𝑗 (𝒓)]Φ(𝒓)𝑑𝑟 + �
Ω
𝜕Ω
1
𝜑 (𝒓)𝜑𝑗 (𝒓)𝑑𝑟
2𝐴 𝑖
(13)
= � 𝑞(𝒓)𝜑𝑖 (𝒓)𝜑𝑗 (𝒓)𝑑𝑟
Ω
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
𝜑𝑖 which 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:
4
(14)
𝑓(𝒓) ≅ � 𝑁𝑖 𝜑𝑖 (𝒓)
𝑖=1
Where 𝑁𝑖 is the value of 𝑓(𝒓) at node i . Using this, Eq. (13) is written in matrix
form:
(15)
[𝐾 + 𝐶 + 𝐵]𝚽 = 𝒒
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 𝑑𝑟
(16)
𝐶𝑖,𝑗 = � 𝜇𝑎 𝜑𝑖 𝜑𝑗 𝑑𝑟
Ω
𝐵𝑖,𝑗 =
(17)
Ω
1
� 𝜑𝑖 𝜑𝑗 𝑑𝑟
2𝐴
(18)
𝒒𝑖 = � 𝑞𝜑𝑖 𝑑𝑟
∂Ω
(19)
Ω
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) =
1
𝐹𝒏
− (𝒓 − 𝒓𝒃 ) ⋅ 𝑖 𝒊
4
3𝑉𝑒
𝑖 ∈ [1,2,3,4]
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
(20)
Similarly, an expression for the gradient of the shape function [89]:
∇𝜑𝑖 (𝒓) = −
𝐹𝑖 𝒏𝒊
3𝑉𝑒
(21)
𝑖 ∈ [1,2,3,4]
Using Eq. (21), the local matrix K can be simplified:
𝑙𝑜𝑐𝑎𝑙
𝐾𝑖,𝑗
= � ∇𝜑𝑖 ⋅ 𝜅∇𝜑𝑗 𝑑𝑟 =
Ω
𝐹𝑖 𝐹𝑗 𝒏𝒊 ⋅ 𝒏𝒋
𝐹𝑖 𝐹𝑗 𝒏𝒊 ⋅ 𝒏𝒋 𝜅(𝒓𝒃 )
� 𝜅 𝑑𝑟 =
2
9𝑉𝑒
9𝑉𝑒
(22)
Ω
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:
𝐾 𝑙𝑜𝑐𝑎𝑙 =
𝜅(𝒓𝒃 ) 𝑘11
� ⋮
9𝑉𝑒 𝑘
41
⋯
⋱
⋯
𝑘14
⋮ �
𝑘44
(23)
Where 𝑘𝑖𝑗 = 𝐹𝑖 𝐹𝑗 𝒏𝒊 ⋅ 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]:
⎧
�
𝑚!
𝑛!
� (𝜑𝑖 )𝑚 (𝜑𝑗 )𝑛 𝑑𝑟 =
6𝑉𝑒 =
⎨
(𝑚 + 𝑛 + 3)!
�
Ω
⎩
1
𝑉
20 𝑒
1
𝑉
10 𝑒
𝑖≠𝑗
(24)
𝑖=𝑗
𝑖, 𝑗 ∈ [1,2,3,4]
Borrowing notation from Sharp where he defines 𝛿𝑖𝑗 = �
1
0
𝑖=𝑗
[81], the
𝑖≠𝑗
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:
𝑙𝑜𝑐𝑎𝑙
𝐶𝑖,𝑗
= � 𝜇𝑎 𝜑𝑖 𝜑𝑗 𝑑𝑟 =
Ω
1
𝑉 𝜇 (𝒓 )(1 + 𝛿𝑖𝑗 )
20 𝑒 𝑎 𝑏
(25)
𝑖, 𝑗 ∈ [1,2,3,4]
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]:
⎧1 𝐴
�
�12 𝑒
𝑚! 𝑛!
𝑚
𝑛
� (𝜑𝑖 ) (𝜑𝑗 ) 𝑑𝑟 =
2𝐴 = ⎨
(𝑚 + 𝑛 + 2)! 𝑒
�
Ω
� 1 𝐴𝑒
⎩6
𝑖≠𝑗
𝑖=𝑗
(26)
𝑖, 𝑗 ∈ [1,2,3]
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:
𝑙𝑜𝑐𝑎𝑙
𝐵𝑖,𝑗
=
1
𝐴 (1 + 𝛿𝑖𝑗 )
24𝐴 𝑒
(27)
𝑖, 𝑗 ∈ [1,2,3]
Finally, the load vector q from Eq. (19) is simplified by using Eq. (20):
1
𝒒𝑖 = � 𝑞𝜑𝑖 𝑑𝑟 ≅ 𝑞(𝒓𝒃 )𝜑𝑖 (𝒓𝒃 )𝑉𝑒 = 𝑞(𝒓𝒃 ) ⋅ 𝑉𝑒
4
Ω
58
(28)
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)� 𝒏̂ ⋅ ∇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) =
𝑐
Φ(𝒓)
𝑛
(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)
1 𝑐
=
Φ(𝒓)
𝜋
2𝐴 𝑛𝜋
(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 ) =
3�1 − 𝑅𝑒𝑓𝑓 �
𝑐
𝑇 (𝜃) �1 +
cos 𝜃� Φ(𝒓)
2
4𝜋𝑛
2�1 + 𝑅𝑒𝑓𝑓 �
(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:
𝑁
(33)
𝐿𝑡𝑜𝑡 = � 𝜂𝑖 𝐿(𝜆𝑖 )
𝑖=1
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:
𝑁
�
𝑖=1
(34)
𝜂𝑖 = 1
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
(36)
𝐴(𝑥 − 𝑚, 𝑦 − 𝑛) = 𝛼𝐵(𝑥, 𝑦)
Where (m,n) is any small offset between the images due to error in coregistration. The scale factor 𝛼 is obtained from the cross-correlation of the
images:
𝛼=
∑𝑥,𝑦 𝐴(𝑥 − 𝑚, 𝑦 − 𝑛)𝐵(𝑥, 𝑦)
∑𝑥,𝑦 𝐵(𝑥, 𝑦)𝐵(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
𝜇𝑎 (mm-1)
𝜇𝑠 ′ (mm-1)
Non-uniform
𝜇𝑎 (mm-1)
𝜇𝑠 ′ (mm )
-1
𝑟1= 7 mm
0 ≤ 𝑟 ≤ 𝑟1
0.2
𝑟2 = 7 mm
𝑟1 ≤ 𝑟 ≤ 𝑟2
0.2
𝑟3 = 7 mm
𝑟2 ≤ 𝑟 ≤ 𝑟3
0.2
1
1
1
𝑟1= 1 mm
0 ≤ 𝑟 ≤ 𝑟1
𝑟2 = 3 mm
𝑟1 ≤ 𝑟 ≤ 𝑟2
𝑟3 = 7 mm
𝑟2 ≤ 𝑟 ≤ 𝑟3
0.1
1
2
2
1
0.01
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
71
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
1
1
11.5
19.0
Living Image
FEM
MC
36%
-27%
7%
27%
1%
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
Std. Deviation
(109 photons/s)
Mean Absolute
Deviation
(109 photons/s)
2D BLI
Homogeneous,
BLT – 8 bins
Homogeneous,
BLT – 4 bins
Homogeneous,
DA
Heterogeneous,
MC
7.39
3.13
8.28
3.90
3.77
5.09
2.74
6.75
3.13
2.40
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.
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
2D BLI
Homogeneous BLT – 4 bins
Homogeneous BLT – 8 bins
Homogeneous DA-FEM
Heterogeneous MC
BLI Acquisition Time
(s)
Computation Time
(s)
0.5
8
51
0.5
0.5
N/A
≈1
≈1
50
186
77
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
78
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.
79
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
𝐿
Description
Radiance
Units
𝒓
Position vector
[m,m,m]
𝒔̂
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
photons/(m2·s·sr)
Scattering anisotropy
Φ(𝒓)
Scattering phase function, probability of
scattering from direction 𝒔̂′ to 𝒔̂
Directional source term, giving density of photons
at location 𝒓 traveling in direction 𝒔̂
Photon density at location 𝒓 traveling in direction
𝒔̂
Isotropic photon density at location 𝒓
𝜅(𝒓)
Diffusion coefficient
m
𝑞(𝒓)
Isotropic light source term at location 𝒓
photons/m4
𝑃 (𝒔̂, 𝒔̂′)
𝑞𝑠 (𝒓, 𝒔̂)
𝜙(𝒓, 𝒔̂)
𝒏̂
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
photons/(m4·sr)
photons/(m3·sr)
photons/m3
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 nonfluorescing 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:
𝜇𝑎 Φ − ∇ ⋅ 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) −
Φ(𝑥 + Δ, 𝑦, 𝑧) − 2Φ(𝑥, 𝑦, 𝑧) + Φ(𝑥 − Δ, 𝑦, 𝑧)
∂κ Φ(𝑥 + ∆, 𝑦, 𝑧) − Φ(𝑥 − Δ, y, z)
+𝜅
+⎤
⎡∂x
Δ2
2Δ
⎢
⎥
⎢
⎥
Φ(𝑥, 𝑦 + Δ, 𝑧) − 2Φ(𝑥, 𝑦, 𝑧) + Φ(𝑥, 𝑦 − Δ, 𝑧) ⎥
⎢∂κ Φ(𝑥, 𝑦 + ∆, 𝑧) − Φ(𝑥, y − Δ, z)
+𝜅
+⎥
⎢∂y
Δ2
2Δ
⎢
⎥
⎢
⎥
⎢ ∂κ Φ(𝑥, 𝑦, 𝑧 + ∆) − Φ(𝑥, y, z − Δ)
Φ(𝑥, 𝑦, 𝑧 + Δ) − 2Φ(𝑥, 𝑦, 𝑧) + Φ(𝑥, 𝑦, 𝑧 − Δ) ⎥
+𝜅
⎣ ∂z
⎦
Δ2
2Δ
= 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
(𝝁𝒂 𝐼 − 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 = (𝝁𝒂 𝐼 − 𝑀)−𝟏 𝒒
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
𝜅(𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 )
: −Δ
Φ(𝑥𝑖+1 , 𝑦𝑖 , 𝑧𝑖 )
:
1 𝜕𝜅
�
2Δ 𝜕𝑥 (𝑥𝑖 ,𝑦𝑖 ,𝑧𝑖 )
+ Δ12 𝜅(𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 )
Φ(𝑥𝑖−1 , 𝑦𝑖 , 𝑧𝑖 )
:
−1 𝜕𝜅
�
2Δ 𝜕𝑥 (𝑥𝑖 ,𝑦𝑖 ,𝑧𝑖 )
+ Δ12 𝜅(𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 )
Φ(𝑥𝑖 , 𝑦𝑖+1 , 𝑧𝑖 )
:
1 𝜕𝜅
�
2Δ 𝜕𝑦 (𝑥𝑖 ,𝑦𝑖 ,𝑧𝑖 )
+ Δ12 𝜅(𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 )
Φ(𝑥𝑖 , 𝑦𝑖−1 , 𝑧𝑖 )
:
−1 𝜕𝜅
�
2Δ 𝜕𝑦 (𝑥𝑖 ,𝑦𝑖 ,𝑧𝑖 )
+ Δ12 𝜅(𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 )
Φ(𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖+1 )
:
1 𝜕𝜅
�
2Δ 𝜕𝑧 (𝑥𝑖 ,𝑦𝑖 ,𝑧𝑖 )
+ Δ12 𝜅(𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 )
Φ(𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖−1 )
:
−1 𝜕𝜅
�
2Δ 𝜕𝑧 (𝑥𝑖 ,𝑦𝑖 ,𝑧𝑖 )
+ Δ12 𝜅(𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 )
87
APPENDIX C: SCALING COEFFICIENT FROM CROSSCORRELATION
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
=0
𝜕𝛼
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 − � 𝐵(𝑥, 𝑦)𝜖(𝑥, 𝑦)
𝑥,𝑦
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=
∑𝑥,𝑦 𝐴(𝑥 − 𝑚, 𝑦 − 𝑛)𝐵(𝑥, 𝑦)
∑𝑥,𝑦 𝐵(𝑥, 𝑦)𝐵(𝑥, 𝑦)
The numerator ∑𝑥,𝑦 𝐴(𝑥 − 𝑚, 𝑦 − 𝑛)𝐵(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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