Principal component analysis of dynamic fluorescence tomography in measurement space.

Abstract

Challenges remain in resolving metabolic processes of drugs within small animals using a fluorescence tomographic image. In our previous work, using principal component analysis (PCA), we detected functional structures with different kinetic behaviors, where PCA was applied in fluorescence tomographic sequence (i.e. in the image space). As a result, all measurement data had to be reconstructed before performing PCA, which imposed a large computational burden. In this paper, we propose a new approach and apply PCA directly to fluorescence projection sequence (i.e. in the measurement space). Utilizing the compression property of PCA, it is possible to resolve regions with different kinetics by reconstructing only a few principal components. Hence, the computational cost can be significantly reduced. To evaluate the performance of the new method, numerical simulation and a phantom experiment are performed on a hybrid fluorescence and x-ray computed tomography imaging system. The results demonstrate that the proposed method greatly reduces the computational time compared with the previous method, while keeping a similar resolving capability.

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Principal component analysis of dynamic fluorescence tomography in measurement space
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IOP PUBLISHING
PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 57 (2012) 2727–2742
doi:10.1088/0031-9155/57/9/2727
Principal component analysis of dynamic fluorescence
tomography in measurement space
Xin Liu1, Bin Zhang1, Jianwen Luo1,2and Jing Bai1
1Department of Biomedical Engineering, School of Medicine, Tsinghua University,
Beijing 100084, People’s Republic of China
2Center for Biomedical Imaging Research, Tsinghua University, Beijing 100084,
People’s Republic of China
E-mail: deabj@tsinghua.edu.cn
Received 24 October 2011, in final form 23 February 2012
Published 17 April 2012
Online at stacks.iop.org/PMB/57/2727
Abstract
Challenges remain in resolving metabolic processes of drugs within small
animals using a fluorescence tomographic image. In our previous work,
using principal component analysis (PCA), we detected functional structures
with different kinetic behaviors, where PCA was applied in fluorescence
tomographic sequence (i.e. in the image space). As a result, all measurement
data had to be reconstructed before performing PCA, which imposed a large
computationalburden.Inthispaper,weproposeanewapproachandapplyPCA
directly to fluorescence projection sequence (i.e. in the measurement space).
UtilizingthecompressionpropertyofPCA,itispossibletoresolveregionswith
differentkineticsbyreconstructingonlyafewprincipalcomponents.Hence,the
computationalcostcanbesignificantlyreduced.Toevaluatetheperformanceof
thenewmethod,numericalsimulationandaphantomexperimentareperformed
onahybridfluorescenceandx-raycomputedtomographyimagingsystem.The
resultsdemonstratethattheproposedmethodgreatlyreducesthecomputational
time compared with the previous method, while keeping a similar resolving
capability.
(Some figures may appear in colour only in the online journal)
1. Introduction
Dynamicopticalimagingtechniques,suchasdynamicfluorescencediffuseopticaltomography
(FDOT), allow one to tomographically image the absorption, distribution and elimination of
fluorescent bio-markers in vivo (Liu et al 2010a, 2011, Patwardhan et al 2005, Vasquez
et al 2011). The technique is helpful in better understanding the complex mechanism in drug
deliveryanddiseaseprogression.However,thelowspatialresolutionofFDOTandthecomplex
dynamic behaviors of drugs together make it difficult to clearly resolve organs/functional
structures with different kinetic behaviors within small animals directly from the single
fluorescence tomographic image.
0031-9155/12/092727+16$33.00© 2012 Institute of Physics and Engineering in Medicine Printed in the UK & the USA2727

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2728 X Liu et al
To address the problem, several methods have been proposed (Hillman and Moore 2007,
Liu et al 2010b, Welsher et al 2011). Considering that in drug delivery each organ exhibits
a distinctive time course, it is possible to resolve and identify major internal organs of
small animals by characterizing the temporal signature of each organ. Principal component
analysis (PCA), as a common statistical processing method, is suitable for the examination
because this technique is able to group signals that vary similarly in time. For example, using
dynamic fluorescence reflectance images as the input data of PCA, Hillman and Moore have
successfully resolved anatomical information of various internal organs of mouse because
of their high, but distinct, variance in time (Hillman and Moore 2007). Recently, we have
extractedthefunctionalstructureswithdifferentkineticbehaviorsusingPCA,wherePCAwas
applied in the reconstructed fluorescence tomographic sequence, i.e. in the image space (Liu
et al 2010b). However, in our previous study, all measurement data (fluorescence projection
images) were required to be reconstructed before performing PCA and therefore imposed a
large computational burden.
Consideringtheapproximatelinearrelationshipbetweenthemeasuredfluorescencesignal
and the reconstructed fluorescence concentration (Alacam and Yazici 2009), we propose
an alternative approach in this paper and apply PCA directly to fluorescence projection
sequence, i.e. in the measurement space. Since PCA has a dimensionality reduction property,
after transforming all measurement data, the signal is generally compressed into the first
few principal components (PCs) with the higher order components containing little useful
information. Utilizing this property, it is possible to resolve the organs/functional structures
with different kinetic behaviors only by reconstructing the few PCs, which avoid the need
for reconstructing all fluorescence projection images. Hence, the computational cost can be
significantly reduced.
To evaluate the performance of the new method, numerical simulation and a phantom
experiment are performed. In the simulation study, the poly(DL-lactic-co-glycolic acid)
(PLGA) nanoparticle delivery of indocynaine green (ICG) (Saxena et al 2006) through the
heart and the lungs of a 3D digital mouse (Dogdas et al 2007) is simulated and imaged.
In the phantom experiment, with a hybrid FDOT and x-ray computed tomography (XCT)
imaging system, two tubes containing different concentrations of ICG at different time points
are imaged to imitate the above dynamic course. The results suggest that the proposed method
can greatly reduce the computational cost in contrast to the previous method (Liu et al 2010b),
while keeping a similar resolving capability.
The paper is organized as follows. In section 2, we present the methods used in
our experiments. In section 3, we describe the experimental materials and procedure. The
experimental results are summarized in section 4. Finally, we discuss relevant issues and
conclude the major findings in section 5.
2. Methods
2.1. Imaging model
In the near infrared spectral window, the photon propagation in biological tissues can be
modeled using the diffusion equation. For the continuous wave FDOT with point excitation
source, the fluorescence signal detected at a detector point rddue to an excitation source at rs
can be expressed based on the first-order Born approximation (O’Leary et al 1996)
?
?(rd,rs) =
V
Gλfl(rd,rp)n(rp)Gλexc(rp,rs)drp,
(1)

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Principal component analysis of dynamic fluorescence tomography in measurement space2729
where rp is the point inside the volume V considered for reconstruction; n(rp) is the
fluorescence yield at rp, which is proportional to fluorophore concentration (Landsman
et al 1976); Green’s function Gλfl(rd,rp) describes the light propagating from rpto rdand
Gλexc(rp,rs) describe the light propagating from rsto rp. With the known optical properties,
these Green’s functions can be obtained by solving the following diffusion equation using the
finite element method (Arridge et al 1993):
− ∇ · [D(r)∇G(r)] + μa(r)G(r) = δ(r − rs),
where μa(r) is the absorption coefficient and D(r) is the diffusion coefficient at position r.
(2)
2.2. PCA for fluorescence projection sequence
PCA is employed to resolve organs/functional structures with different kinetic behaviors.
In this paper, the input data of PCA are N frames fluorescence projection images denoted
as ?mea
1
,?mea
2
,...,?mea
t
,...,?mea
tth frame fluorescence projection image taken in scan t. If ?mea
detector pairs, we can further represent the frame data as a column vector ?mea
(?mea
at the frame t. By transforming all frames, the input data (fluorescence projection sequence)
are arranged as an M × N matrix ?.
Mathematically, PC transformation starts with computing the eigenvalues (sorted
according to decreasing magnitude) and the corresponding eigenvectors of the matrix,
S =
value from each column of the data matrix ? and T represents transposition operation. For
the N × N matrix S, N eigenvectors and corresponding eigenvalues can be obtained using the
following equation:
N, where N is the number of frame (scan) and ?mea
t
is the
t
has totally M source–
t
=
t,1,?mea
t,2,...,?mea
t,k,...,?mea
t,M)T, where ?mea
t,kis the density of the kth source–detector pair
1
M−1¯?T¯?, where¯? is a data set with zero mean obtained by subtracting the mean
SU = UL,
(3)
where L is the diagonal matrix of eigenvalues and the diagonal elements of L, l1,l2,...,lN,
are eigenvalues of the matrix S; U is the matrix of eigenvectors and the columns of U,
u1,u2,...,uNare eigenvectors of the matrix S. The eigenvector represents the direction of the
greatest variance for a given PC. Ordering the eigenvectors by eigenvalues from highest to
lowest further gives the PCs in order of significance. The first principal component (PC1) has
the highest variance in the input data. The second principal component (PC2) represents the
direction of next highest variance that is orthogonal to the PC1. The other PCs are deduced
likewise. By projecting the input data onto the PCs, the matrix of score is obtained as follows:
P = ?U,
(4)
where the first column of P is made up of the PC score with respect to the PC1, the next
column has the PC score with respect to the PC2 and so on. In this paper, the PC score is
generated by projecting original input data ? onto the PC, as shown in equation (4). It is
different from our previous study, where the PC score is generated using the data set with zero
mean¯? (Liu et al 2010b). Furthermore, the absolute values of positive and negative elements
of the reconstructed PC scores are respectively utilized as a weight factor for creating images,
termed as PC-FDOT images, which are used to illustrate the organs/functional structures with
different kinetic behaviors. Since PCA tends to compact signal information into the first few
PCs (the higher order PCs are dominated by noise), it is possible to resolve the regions with
different kinetics by reconstructing a few PC scores. Comparison of the proposed method and
the previous method is illustrated by the flowchart shown in figure 1.

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2730X Liu et al
(a)(b)
Figure 1. The flowchart of the proposed method and the previous method (Liu et al 2010b).
(a) The proposed method, i.e. PCA is applied in fluorescence projection sequence (in the
measurement space). (b) The previous method, i.e. PCA is applied in fluorescence tomographic
sequence (in the image space).
2.3. Reconstruction of principal component
To reconstruct each generated PC score, the integral equation (1) should be discretized. After
that, a linear system is generated as follows:
P = Wn,
(5)
where W is the weight matrix and each element wijpresents the contribution of the jth voxel
of the discrete field of interest (FOI) to the measurement due to the ith source–detector pair.
In this paper, the minimization of equation (5) is performed using a least-squares algorithm
(LSQR) (Paige and Saunders 1982). This algorithm relaxes the positivity constraint, which is
necessary because some of the PC scores can be negative.
2.4. Dice coefficient
To further evaluate the similarity between actual organs/structures and the organs/structures
resolved by PCA, the Dice coefficient is calculated as (Dice 1945)
Dice =2|X ∩Y|
where|·|denotesthenumberofvoxels;Xdescribesactualorgan/structure;Ydescribesresolved
organ/structure, which is obtained by the proposed method or the previous method (Liu et al
2010b).
|X| + |Y|,
(6)
2.5. Summary of approach
The implementation procedure of the proposed method is summarized as follows.
Step1.ConstructtheM×N inputmatrixofPCA?fromfluorescenceprojectionsequence.
Step 2. Generate the N × N matrix S =
Step 3. Find the eigenvectors and corresponding eigenvalues of the matrix S.
1
M−1¯?T¯?.

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Principal component analysis of dynamic fluorescence tomography in measurement space2731
Figure 2. The hybrid fluorescence and micro-XCT imaging system.
Step 4. Compute the PC scores P by projecting the original input data ? onto the PCs.
Step5.ReconstructthePCscorestoobtainnbysolvingP = WnusingtheLSQRmethod.
Step 6. Generate the positive and negative PC-FDOT images based on the reconstructed
PC scores n.
3. Materials
3.1. Experimental setup
The numerical simulation and phantom experiment were performed on a hybrid FDOT/XCT
imaging system, similar to that described in Guo et al (2010). Briefly, the imaged object was
placed on a custom-built rotating stage. Around the rotational stage, a free-space FDOT and a
micro-XCT imaging system were constructed to acquire the optical and structural data of the
imaged object, respectively. The sketch of the system is shown in figure 2.
3.2. Numerical simulation
3.2.1. Setup for numerical simulation.
was used as the imaged object and suspended on the rotating stage for collecting measurement
data from multiple projection angles. The mouse torso from the neck to the base of the lungs
was selected as the investigated region, totally 1.6 cm in length (see figure 3(a)). To simulate
photon propagation in heterogeneous tissues, the absorption and the scattering coefficients
fromAlexandrakisetal(2005)wereassignedtotheheart,thelungsandthebones,.Theoptical
properties outside the organs were regarded as homogeneous. For fluorescence imaging, the
collimated light beam was modeled as an isotropic point source, located at one mean free
path of photon transportation (1/μ?
an optimal strategy described by Wang et al (2009), 360◦full angle fluorescence tomography
was performed with 24 projections in 15◦step.
A digital mouse model from Dogdas et al (2007)
s) beneath the surface, at the height of 0.8 cm. Based on
3.2.2. Simulated metabolic processes of PLGA/ICG through heart and lungs of the mouse.
Based on the above imaging model, we simulated the metabolic processes of PLGA

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2732 X Liu et al
(a)(b)
Figure 3. Setup for the simulation study. (a) The 3D geometrical model of the digital mouse used
in the numerical simulation, with a length of 1.6 cm from the neck to the base of the lungs. The red
pointsin(a)depictthepositionsofthediscreteisotropicpointsources.Theanatomicalinformation
of different organs are depicted using different colors (heart: cyan; lungs: magenta; bones: yellow;
surface: gray). (b) The PLGA/ICG concentration used in the simulation study, which is obtained
by interpolating the original concentration data (Saxena et al 2006) using the piecewise cubic
Hermite method. The circles in (b) depict the PLGA/ICG concentrations at corresponding time
points in the heart (cyan) and the lungs (magenta).
nanoparticles entrapping ICG (PLGA/ICG) through the heart and the lungs of the digital
mouse. The PLGA/ICG concentration curves in the heart and the lungs (see figure 3(b)) were
acquired by interpolating the original concentration data (Saxena et al 2006) using a piecewise
cubic Hermite method. In detail, firstly, according to figure 3(b), a series of PLGA/ICG
concentrations (fluorescent yields) were assigned to the heart and the lungs at different time
points (from 5 to 120 min). Considering that the imaging time of one circle/frame took
approximately 1 min in our in vivo experiment (Liu et al 2010a, 2011), here, the interval of
each frame was set to be 1 min. Secondly, with the forward model described in section 2.1,
116 frames of fluorescence projection images were generated (COMSOL Mulitphysics 3.3,
COMSOL, Inc., Burlington, MA). Then these generated data (116 frames in total, with an
interval of 1 min) were assembled to form fluorescence image sequence, which was used to
depict the PLGA/ICG metabolic processes in the heart and the lungs of the mouse.
3.2.3. PCA for fluorescence projection images.
heart and the lungs, PCA was performed. Here, the input data were 116 frame fluorescence
projectionimagesandeachframecontained34696source–detectorpairs.Basedonthemethod
described in section 2.2, each frame data was further transformed as a column vector with
34696 elements. By transforming all frames, the input data were arranged as a matrix with
34696 rows and 116 columns. After performing PCA, 116 PC scores were generated, which
were then reconstructed to obtain the corresponding PC-FDOT images.
To resolve the anatomical information of the
3.2.4. Reconstruction of principal components.
located on the boundary finite element nodes between 1.4 cm height range and inside 150◦
FOV. The reconstruction region was 1.9 cm×2.7 cm×1.6 cm with 0.1 cm mesh spacing, and
only the mesh inside the imaged object was used for reconstruction. The number of LSQR
iterations was fixed to 30, selected empirically based on the simulation results.
For the reconstruction, the detectors were

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Principal component analysis of dynamic fluorescence tomography in measurement space2733
(a) (b)
Figure 4. Setup for the phantom experiment. (a) The phantom used in the experiment, which is
made of a polymethyl methacrylate cylinder (diameter of 3.0 cm). The matching fluid (intralipid,
ink and water) is added to the phantom, with μa= 0.3 cm−1and μ?s= 10.0 cm−1. Two glass
tubes (outer diameter of 4.1 mm) filled with different concentrations of ICG are immersed inside
the phantom, as the fluorescence targets. (b) ICG concentration curves in tubes 1 and 2. The circles
in (b) depict the ICG concentrations at corresponding time points. Different colors correspond to
time courses of different tubes (tube 1: cyan; tube 2: magenta).
3.3. Phantom experiment
3.3.1. Setupforthephantomexperiment.
modality FDOT/XCT imaging system (see figure 2). Here, a cylinder made of polymethyl
methacrylate, with 3.0 cm diameter and 4.1 cm length, was employed as the phantom. The
optical properties in the phantom were assigned as μa= 0.3 cm−1and μ?
similar to the bulk optical properties of mouse tissues. Two transparent glass tubes (0.41 cm
outer diameter) (see figure 4(a)) filled with different concentrations of ICG were immersed in
the cylinder phantom, as the fluorescence targets.
Thephantomexperimentwasperformedonadual-
s= 10.0 cm−1,
3.3.2. Dynamicmodelinginphantomexperiment.
intheheartandthelungsofthemouse,theICGconcentrationsintubes1and2atdifferenttime
points were set as shown in figure 4(b). For each experiment at specific time point (frame), 24
emission and excitation images were collected at every 15◦, with the excitation source located
at the height of 2.0 cm. The total excitation light power delivered on the phantom surface was
about 3 mW.
TosimulatethetimecourseofPLGA/ICG
3.3.3. PCA for fluorescence projection images.
PCA were 16 frames fluorescence images with 9374 source–detector pairs contained in each
frame. After performing PCA, we obtained 16 PC scores, which were then reconstructed to
generate the corresponding PC-FDOT images.
In the phantom experiment, the input data of
3.3.4. Reconstruction of principal components.
located on the boundary finite element nodes which were between 2.5 cm height range and
inside150◦FOVcorrespondingtoeachpointsource.Thevolumeconsideredforreconstruction
was 3.0 cm × 3.0 cm × 4.1 cm and sampled to 31 × 31 × 42 voxels. The reconstruction was
terminated after 30 LSQR iterations.
In the reconstruction, the detectors were

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2734 X Liu et al
(a)(b)
Figure 5. The reconstructed fluorescence tomographic images and the generated PC-FDOT
images. (a) The fluorescence reconstruction results from a numerical simulation, which depicts
the PLGA/ICG metabolic processes throughout the heart and the lungs of a digital mouse.
The reconstruction results are obtained assuming that accurate optical properties are known.
Additionally, noise is not contained in the synthetic projection sequence. All images are displayed
in the same range. (b) The PC-FDOT images obtained by applying PCA to fluorescence projection
sequence and then reconstructing these generated PC scores using LSQR. The anatomical
information of the lungs and the heart are well illustrated using the positive and the negative
PC2-FDOTimages,respectively.Differentcolorsindicatetheactualboundariesofdifferentorgans
(heart: cyan; lungs: magenta; surface: red).
4. Results
4.1. Simulation study
Figure 5(a) depicts the reconstructed fluorescence tomographic images from the dynamic
simulation study. The slice was selected through the chest of the mouse, including the heart
and the lungs, at z = 0.6 cm. The results suggest that it is quite difficult to account for the
metabolism of PLGA/ICG in the heart and the lungs directly from these tomographic images
(seefigure5(a)).Thisisprobablycausedbythediffusivenatureofphotonmigrationintissues.
In comparison, when PCA was applied in the 116 frame fluorescence projection images, we
obtained 116 PC scores. After reconstructing these PC scores, we acquired the corresponding
PC-FDOT images (see figure 5(b)), which could be used to illustrate the organs/functional
structures with different kinetic behaviors.
Figure 5(b) shows the results obtained by applying PCA to the 116 frame fluorescence
projection images and then reconstructing these obtained PC scores using LSQR, termed
as PC1-, PC2- and PC116-FDOT images. Based on the anatomical information of the digital
mouse,thelungsandtheheartareclearlyindicatedinthepositiveandthenegativePC2-FDOT
images, respectively. Additionally, we also observe that the higher order PC-FDOT images
contain little available information.
In the actual measurement, noise was an important factor whose effect could not be
neglected. To evaluate the noise effect, zero-mean, Gaussian noise with different levels was
added to the synthetic projection sequence (Roy and Sevick-Muraca 1999)
ˆ?mea= ?mea(1 + δG(0,1)),
(7)

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Principal component analysis of dynamic fluorescence tomography in measurement space 2735
Table 1. Calculations of total variance of the first six PCs in the simulation study, with different
noise levels (1%, 3% and 5%) used in the synthetic data sets.
Total variance (%)
PC1% Gaussian3% Gaussian 5% Gaussian
1
2
3
4
5
6
99.9008
0.0772
0.0005
0.0005
0.0004
0.0004
99.7221
0.0797
0.0050
0.0044
0.0038
0.0034
99.3664
0.0847
0.0136
0.0120
0.0106
0.0094
where G(0,1) was a Gaussian distribution with zero mean and unit variance, δ was the
noise level, with δ = 0.01,0.03,0.05 used in the paper, ?meawere the simulated, noise-free
measurement data. Then, the synthetic data sets are used as the input data of PCA.
The PCA has a dimensionality reduction property, wherein the signal can be compressed
into the first few components, with higher order components containing little useful signal
information. By calculating eigenvalues (the variance of each PC), we found that the first two
PCs accounted for most of the total variance defined as follows:
Total variance(%) =Eigenvalue of the interest PC
Total eigenvalues
with very little contained in the remaining PCs (see table 1). In addition, the PC1 image is
similar to a weighted mean over all the images, where the functional structures with different
kinetics are hardly resolved (Hillman and Moore 2007, Welsher et al 2011). The PC1-FDOT
image shown in figure 5(b) also validated this finding. Based on the above analysis, it is
possible to use PC2 only to resolve the anatomical structures of the heart and the lungs. Hence,
for each case above, only PC2 is used while applying the proposed method.
Additionally, to compare the resolving performance of the proposed method and the
previous method (Liu et al 2010b), we also generated the PC-FDOT images by applying PCA
to dynamic fluorescence tomographic images. In the implementation, the input matrix of PCA
had 9520 rows and 116 columns, which was constructed from 116 frames of 3D reconstructed
tomographic images with a spatial size of 20 × 28 × 17 pixels. The same weight matrix and
reconstruction method were used in the reconstruction processes.
Figure 6 compares the PC2-FDOT images obtained by the proposed method and the
previous method (Liu et al 2010b). The first row of figure 6 depicts the merged PC2-FDOT
images obtained by applying PCA to 116 frame fluorescence projection images and then
reconstructing PC2 score using LSQR. In comparison, the second row of figure 6 depicts the
merged PC2-FDOT images obtained when PCA is applied in the 116 frames fluorescence
tomographic images. The results suggest that even if 5% Gaussian noise is added to the
synthetic data set, the anatomical information of the heart and the lungs can still be resolved
using the proposed method. On the other hand, we also observe that no matter whether
projection sequence or tomographic sequence is used as the input data of PCA, the obtained
PC-FDOT images are similar.
Table 2 compares the Dice coefficients obtained from the proposed method, the previous
method and the reconstructed fluorescence tomographic image of frame 1. The Dice
coefficients were calculated from axis slices (from z = 0.4 to 1.1 cm, with an interval of
0.1 cm). Other slices did not contain the anatomical information of the heart and the lungs.
It can be seen that the Dice coefficients obtained from frame 1 are lower than that from the
proposed method and the previous method, indicating that the image quality can be improved
× 100,
(8)

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2736X Liu et al
Figure 6. Comparison of the PC2-FDOT images obtained with the proposed method (PCA is
applied in fluorescence projection sequence, i.e. in the measurement space) and the previous
method (PCA is applied in fluorescence tomographic images, i.e. in the image space). The first
row depicts the merged PC2-FDOT images obtained by applying PCA to 116 frame fluorescence
projection images and then reconstructing PC2 score using LSQR. The second row depicts the
merged PC2-FDOT images obtained by applying PCA to 116 frames fluorescence tomographic
images. In the reconstruction processes, the same weight matrix and the reconstruction method
are used. Red color depicts the generated positive PC2-FDOT images and green color depicts the
generated negative PC2-FDOT images.
Table 2. The Dice coefficients in the heart and the lungs acquired from the proposed method,
the previous method and the reconstructed fluorescence tomographic image of frame 1 in the
simulation study, with different noise levels (noise free, 1%, 3% and 5%) used in the synthetic data
sets.
Dice coefficients in heartDice coefficients in lungs
Noise
level
Measurement Image
space
Measurement Image
space
space Frame 1a
space Frame 1a
Noise free
1% Gaussian 0.75 ± 0.13
3% Gaussian 0.74 ± 0.13
5% Gaussian 0.73 ± 0.12
aFrame 1 means the reconstructed fluorescence tomographic image of the first frame.
0.76 ± 0.13 0.74 ± 0.16 0.34 ± 0.08 0.63 ± 0.15
0.74 ± 0.16 0.34 ± 0.08 0.63 ± 0.15
0.73 ± 0.16 0.33 ± 0.08 0.62 ± 0.15
0.72 ± 0.14 0.32 ± 0.08 0.60 ± 0.16
0.60 ± 0.16 0.48 ± 0.13
0.59 ± 0.16 0.48 ± 0.13
0.59 ± 0.16 0.48 ± 0.13
0.57 ± 0.16 0.47 ± 0.14
by performing PCA. In addition, the Dice coefficients obtained from the proposed method
and the previous method match closely to each other, indicating that the PC2-FDOT images
obtained via these two methods provide a similar image quality. However, by using the
proposed method, it is possible to only perform one reconstruction, i.e. reconstructing PC2

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Principal component analysis of dynamic fluorescence tomography in measurement space2737
Figure 7. The PC-FDOT images obtained from the proposed method combined with different PC
scores. The first to fifth columns depict the merged PC-FDOT images obtained by applying PCA
to 116 frame fluorescence projection images and then reconstructing PC1 score, PC3 score, PC4
score, PC5 score and PC116 score, respectively, using LSQR. Red color depicts the generated
positive PC-FDOT images and green color depicts the generated negative PC-FDOT images.
Table 3. Comparison of the computational time cost of the proposed method and the previous
method in the simulation study.
MethodReconstruction time PCA time Total time
PCA in the image space
PCA in the measurement space
1061.2 s
9.2 s
0.7 s
2.5 s
1061.9 s
11.7 s
score, to resolve the anatomical structures of the heart and the lungs. As a result, we require
only 1.11% of the computational time in contrast to the previous method (see table 3). It
should also be noted that in this simulation study, the PCA computational time of the proposed
method, i.e. 2.5 s, is longer than that of the previous method (Liu et al 2010b), i.e. 0.7 s. The
main reason is that the input data of PCA used in the proposed method is a 34696 × 116
matrix, which is larger in size than the input matrix (9520×116) used in the previous method.
Nevertheless, the increased computational time of PCA is very short when compared to the
total time used in the previous method (see table 3). In this paper, the computation was
performed on a personal computer with 2.66 GHz Quad processor and 8 GB RAM.
Figure 7 shows the PC-FDOT images obtained by applying the proposed method to other
PC scores. As expected, the PC-FDOT images generated with the PC3 score, PC4 score,

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2738 X Liu et al
Figure 8. The fluorescence reconstruction results from a dynamic phantom experiment. The
reconstruction results (frame 1, frame 2, frame 3, frame 4, frame 5 and frame 13) are registered to
the corresponding coronal slice (x = −0.1 cm) from XCT volume. All images are displayed in the
same range.
Table 4. Total variance of the first six principal components in the phantom study.
PC Total variance (%)
1
2
3
4
5
6
99.4583
0.4275
0.0509
0.0259
0.0108
0.0055
PC5 score and PC116 score, do not contain useful structural information. Additionally, the
generated PC1-FDOT image is similar to a weighted mean over all the reconstruction images.
These results further indicate that the use of PC2 is reasonable in the cases.
4.2. Phantom experiment
Figure 8 shows the fluorescence reconstruction results of experimental data from the same
coronalslice(x=−0.1cm)butfromdifferentframes,whichareusedtodescribethedynamics
of ICG in tubes 1 and 2. Similar to the simulation results (see figure 5(a)), it is difficult to
account for ICG dynamics in two adjacent tubes only based on static tomographic images.
In contrast, PCA is used to resolve the spatial structures of tubes 1 and 2 with different
dynamics. Table 4 shows the total variance of the first six PCs in the phantom study. As
expected, the first two components accounted for most of the total variance. In addition, based
ontheaboveanalysis,thePC1-FDOTimageissimilartothemeanimage,wherethefunctional
structures with different kinetics are hardly resolved. Hence, in the phantom study, the PC2 is
used while applying the proposed method.
Figures 9(b) and (c) show the positive and the negative PC2-FDOT images, which are
obtained from the proposed method, i.e. by applying PCA to 16 frame fluorescence projection
images and then reconstructing PC2 score using LSQR. The results suggest that the spatial
structures of tubes 1 and 2 are well illustrated using the proposed method. Again, to evaluate

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Principal component analysis of dynamic fluorescence tomography in measurement space 2739
(a)
(b)
(c)
(e)
(d)
(f)
Figure 9. Comparison of the PC2-FDOT images obtained from the proposed method and
the previous method. (a) The fluorescence reconstruction results of frame 6 at coronal view
(x = −0.1 cm). (b) and (c) depict the positive and the negative PC2-FDOT images obtained from
the proposed method. (e) and (f) depict the positive and the negative PC2-FDOT images obtained
from the previous method (Liu et al 2010b). (d) The 3D visualization results of the positive and
the negative PC2-FDOT images obtained from the proposed method.
Table 5. The Dice coefficients in tubes 1 and 2 obtained from the proposed method, the previous
method and the reconstructed fluorescence tomographic image of frame 1 in the phantom study.
Methods Dice coefficient in tube 1 Dice coefficient in tube 2
PCA in image space
PCA in measurement space
Frame 1a
0.23 ± 0.03
0.23 ± 0.03
0.11 ± 0.01
0.21 ± 0.05
0.20 ± 0.04
0.11 ± 0.01
aFrame 1 means the reconstructed fluorescence tomographic image from the first frame.
the resolving performance of the two methods, we also generated the PC-FDOT images using
the previous method (Liu et al 2010b), i.e. by applying PCA to 16 frames of 3D reconstructed
tomographic images with a spatial size of 31 × 31 × 42 pixels. The results are shown in
figures 9(e) and (f).
Table 5 further shows the Dice coefficients in tubes 1 and 2 obtained from the proposed
method,thepreviousmethodandthereconstructedfluorescencetomographicimageofframe1.
The Dice coefficients were calculated from axis slices (from z = 1.8 cm to z = 2.3 cm, with an
interval of 0.1 cm). Similar to the simulation results, the Dice coefficients obtained from frame
1arelowerthan thoseobtained fromtheproposed method andtheprevious method,indicating

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2740X Liu et al
Figure 10. The PC-FDOT images obtained from the proposed method combined with different
PC scores. Red color depicts the generated positive PC-FDOT images and green color depicts the
generated negative PC-FDOT images.
Table 6. Comparison of the computational time cost of the proposed method and the previous
method in the phantom study.
Method Reconstruction timePCA timeTotal time
PCA in the image space
PCA in the measurement space
307.6 s
19.2 s
0.12 s
0.02 s
307.72 s
19.22 s
that the image quality can be improved by performing PCA. In addition, we again observe that
the Dice coefficients obtained from the proposed and the previous methods are close to each
other, indicating that these two methods provide a similar resolving capability. On the other
hand, as expected, by using the proposed method, the computational time is greatly reduced
compared to the previous method. Comparison of the computational time obtained from the
two methods is summarized in table 6.
Figure 10 shows the PC-FDOT images obtained by applying the proposed method to
other PC scores. By comparing the structure information from XCT with the generated
PC-FDOT images, we observe that it is difficult to exactly resolve the spatial structures
of tubes 1 and 2 by other PC-FDOT images (e.g., PC1-FDOT, PC3-FDOT, PC4-FDOT,
PC5-FDOT and PC16-FDOT).
5. Discussion and conclusion
The analysis of dynamic fluorescence diffuse optical tomography (FDOT) is important for
basic drug deliver research. However, the low spatial resolution of FDOT and the complex
dynamic behaviors of drugs often make it difficult to resolve organs/functional structures
with different kinetics within small animals. In our previous study (Liu et al 2010b), by
using principal component analysis (PCA), we extracted the organs/functional structures with
differentkineticbehaviors,whichwereappliedinthereconstructiontomographicsequence.As
aresult,allmeasurementdatahadtobereconstructedbeforeperformingPCA,thusimposinga
large computational burden. In contrast, the main aim of this paper is to improve the resolving
efficiency by directly applying PCA to fluorescence projection sequence.
It was observed from the numerical simulation and the phantom experimental results
that the PC-FDOT images obtained from the two methods provided a similar image quality
(see figures 6 and 9). The quantitative analysis results in tables 2 and 5 further validated
this finding. In addition, it could be observed that in the simulation study, by the use of the
proposed method, the anatomical information of the heart and the lungs can be resolved by