An analytic, reflection method for time-domain florescence diffuse optical tomography based on a generalized pulse spectrum technique - art. no. 68500M

Abstract

An image reconstruction scheme for time-domain fluorescence diffuse optical tomography is proposed using a reflection-mode for a semi-infinite turbid geometry. The method is based on a generalized pulse spectrum technique that employs analytic expressions of the Laplace-transformed time-domain photon-diffusion model to construct a Born normalized inverse model, and a pair of real domain transform-factors to separate distributions of the fluorescent yield and lifetime. The methodology is validated with a specifically-developed fluorescent Monte-Carlo simulator or finite-element-based methods and its robustness to the background uncertainties is investigated.

Full text

An analytic, reflection method for time-domain florescence diffuse
optical tomography based on a generalized pulse spectrum technique
Huiyuan He, Limin Zhang, Feng Gao, Zheng Ma, Huijuan Zhao, and Jingying Jiang
College of Precision Instruments and Optoelectronics Engineering, Tianjin University, Tianjin,
300072, P. R. China
gaofeng@tju.edu.cn
ABSTRACT
An image reconstruction scheme for time-domain fluorescence diffuse optical tomography is proposed using a
reflection-mode for a semi-infinite turbid geometry. The method is based on a generalized pulse spectrum technique that
employs analytic expressions of the Laplace-transformed time-domain photon-diffusion model to construct a Born
normalized inverse model, and a pair of real domain transform-factors to separate distributions of the fluorescent yield
and lifetime. The methodology is validated with a specifically-developed fluorescent Monte-Carlo simulator or
finite-element-based methods and its robustness to the background uncertainties is investigated.
Keywords: Fluorescence molecular tomography, generalized pulse spectrum technique, finite-element-based methods,
analytic solution of the diffuse equation, Born approach
1. INTRODUCTION
Fluorescence molecular tomography (FMT) has become an important method to study fluorescent markers inside small
animal modes during the last decade. Disease and treatment progression can be monitored by imaging the fluorescence
emitted from fluorophores that are attached to specific molecules1. The simulative modes and the related detections of
FMT in steady-state and frequency-domain have been heavily explored, it is natural and imminent to extend the
techniques to time-domain2, 3, where not only the simultaneous reconstructions of fluorescent yield and lifetime but also
the analysis of multiple components could be achieved in a direct way.
One major challenge in FMT is to obtain a feasible reconstruction scheme resembling the correct result as well as
possible. In the past years, rapid progress has been made in the reconstruction of FMT, such as finite-element-based
methods, finite-difference methods and so on. But this paper presents a novel reconstruction scheme for a semi-infinite
turbid geometry. It is based on the real-domain Laplace-transformed diffusion model, referred to as the generalized pulse
spectrum technique (GPST), which has been shown to possess many advantages in flexibility, stability, robustness and
computational efficiency4, 5, that employs analytic expressions to construct a Born normalized inverse model, and a pair
of real domain transform-factors to separate distributions of the fluorescent yield and lifetime. The methodology is
validated with a specifically-developed fluorescent Monte-Carlo simulator or finite-element-based methods (FEM) and
its robustness to the background uncertainties is investigated.
2. METHODOLOGY
The task of image reconstruction in FMT is expressed as an inverse issue for a given photon migration model. In this
section, we describe the Monte Carlo fluorescence model and the frameworks for both the forward and inverse models of
time-domain FMT, aiming at simultaneous reconstruction of fluorescent yields and lifetimes of components.
2.1 Fluorescence Monte Carlo Model
Monte Carlo simulation is an important tool in the field of biomedical optics 6, 7. Although Monte Carlo modeling suffers
from poor efficiency, it has a realistically broad generality and can cope with arbitrary media complexity.
The purpose of fluorescence Monte Carlo simulations is to study the process of fluorescence emission from a turbid
medium containing a mixture of fluorophores of interest and to provide an effective tool for the validation of the
proposed inversion procedure. For the time-domain case, an ultra-short pulsed laser at the excitation wavelength
λ
x is
Multimodal Biomedical Imaging III, edited by Fred S. Azar, Xavier Intes,
Proc. of SPIE Vol. 6850, 68500M, (2008)
1605-7422/08/$18 · doi: 10.1117/12.756377
Proc. of SPIE Vol. 6850 68500M-1
2008 SPIE Digital Library -- Subscriber Archive Copy

IJnIJCJJJJJ xcou bpoou
LUC1(
GUGLLJ LuqoJubp 1JJUJ
JJJLOIJ oLbpo
IJ
12 p? UnoLobIJ
GUGL!U UflOLCGLJC bpoou
coujbnu rnuujoitc)LCLJCG bpoou
1JTHJC1J!U ipq UflOLCUCG bpoou
fin JJOLJ oi UUOLG2CGUC. bpoou
oi UflJ 2CJCG bpooi bp jup
m1LrO1J 04 UflOLCUCG bpoou
NG!IJ nbq
pLG2pojq AJflG
LOflJ 211LMA
jr bpoou
uq
CJflJG UJLfl!0U
qnciou LLJOLJJJ
rccoqrn o
nJi qLJpnpou
A
LGUC1(
normally incidenting to the surface of a semi-infinite turbid medium, as illustrated in Fig. 1. The excitation photon
propagates in the medium with the given simulated path until it escapes the boundary or is absorbed. When it is absorbed
by a fluorophore, a new fluorescence photon is launched at emission wavelength
λ
m after a certain delay determined by
the fluorescence lifetime of the fluorophore. The fluorescence photon propagates through the medium and then either is
absorbed or emitted. Secondary emission of the fluorescence photons by the fluorophore is neglected here. The whole
process is showed in Fig. 2.
Figure 1. The process of fluorescence emission from a semi-infinite turbid medium is illustrated.
Figure 2. Flow chart of Monte Carlo fluorescence simulation
scattering
absorption
emission
internal reflection excitation
λ
x
excitation photon
absorbed, emission
photon launched
λ
m
Proc. of SPIE Vol. 6850 68500M-2

2.2 Forward Problem
The mathematical formulation of fluorescent light detection relies on a coupled diffusion equation between the excitation
light and the emitted fluorescent light8. To apply the GPST, the time-dependent forward model is converted into the
complex frequency domain by using Laplace transform
ss
ss
[()](,,)(-)
[ () ] (, , ) (, , ) ()/(1 ())
xax x
mam m x af
cq q
cq q c q s
κµ δ
κµ ηµ τ
∇⋅ ∇− − Φ =−
⎧
⎪
⎨∇⋅ ∇− − Φ =− Φ +
⎪
⎩
rrrrr
rrr rrrr
, (1)
where x
κ
and m
κ
are diffusion coefficients of excitation and emission wavelengths respectively; ax
µ
and am
µ
are the
absorption coefficients of excitation and emission wavelengths respectively; s
(, , )
xq
Φ
rr and s
(, , )
mqΦrr are the Laplace
transforms of the time-dependent photon densities with a complex transform-factor s; the fluorescence parameters are the
fluorescent yield ()
af
ηµ
r and lifetime ()
τ
r; c is the speed of light in medium; '
/3( )
as
c
κ
µµ
=+
. These quantities, in
general, are the functions of the position vector r.
2.3 Inverse Problem
According to Eq. (1), the fluorescence density at a detector position rd, due to an excitation source in rs, is a contribution
of all fluorescent volume fractions dV through the volume. Hence the fluorescence flux can be described by Eqs. (2)9
ds d s
(r , r , ) G(r , r, ) (r,r , ) (r, )
(r, ) /(1 (r))
mx
V
af
s
cs sxsdV
xs s
ηµ τ
⎧Φ= Φ
⎪
⎨=+
⎪
⎩
∫, (2)
where ds
(,,)
mqΦrr is the Laplace transform of the transient emission density measured at boundary site rd and or
excitation site rs, and d
G( ,,)
mqrr the density at rd for a source at r.
To reconstruct the fluorescence parameters, the normalized Born approach is applied10. Utilizing this approach and
by applying the Fick’s Law at emission wavelength, the fluorescence flux normalized with the excitation wavelength
flux at a detector position rd with the excitation source placed rs is given by
ds
sd sd
sd
sd sd sd
G( , , ) ( , , ) ( , )
(,,) (,,)
(,,) (,,) (,,) (,,)
x
mmV
nb
xx x
cq qxqdV
qq
qqq q
Φ
ΓΦ
Γ= = =
ΓΦ Φ
∫rr rr r
rr rr
rr rr rr rr . (3)
Applying Monte Carlo fluorescence simulation, we can also obtain
sd
sd
sd
(,,)
(,,) (,,)
MC
m
nb MC
x
q
qq
Γ
Γ=
Γ
rr
rr rr . (4)
So, from Eq. (3) and Eq. (4), the fluorescence density is given by
sd
sd sd d s
sd
(,,)
(, ,) (, ,) G(,,) (, ,)(,)
(,,)
MC
m
mxx
MC V
x
q
qqcqqxqdV
q
Γ
Φ= ⋅Φ= Φ
Γ∫
rr
rr rr r r rr r
rr . (5)
To solve for the unknown (, )
x
qr, the volume integral in Eq. (5) is discredited into Nvoxels where each cube has a
volume of V∆[mm3].Let 1
(, ) ( ) () ()u()
voxel
NT
nn
n
xq xqu xq
=
≈=
∑
rrr
, where, T
11
( )=[ ( ), ( ), , ( )]
voxel
N
qxqxq x qxL, Eq. (5) can be
described as a matrix equation
() ()()qqq
=
ΦWx , (6)
where
T
11 21 D S
11 21 D S
11 21 D S
(r ,r , ) (r ,r , ) (r ,r , )
( ) [ (r ,r , ), (r ,r , ), , (r ,r , )]
(r ,r , ) (r ,r , ) (r ,r , )
MC MC MC
mm m
xx x
MC MC MC
xx x
qq q
qqq q
qq q
ΓΓ Γ
Φ = ⋅Φ ⋅Φ ⋅Φ
ΓΓ Γ
L (7)
and
11 11 11
21 21 21
(,,,1),(,,,2),,(,,, )
(,,,1),(,,,2),,(,,, )
()
(,,,1),(,,,2), ,(,,, )
voexl
voexl
DS DS DS voexl
Wrrq Wrrq WrrqN
Wr rq Wr rq Wr r qN
q
Wr r q Wr r q Wr r qN
⎡⎤
⎢⎥
⎢⎥
=⎢⎥
⎢⎥
⎣⎦
W
L
L
M
L
(8)
with the element given by
Proc. of SPIE Vol. 6850 68500M-3

()
(
)
(
)
,,, , ,
ds mdn x ns i
WqncG V
=
Φ∆rr rr rr (9)
where D and S are the numbers of the detectors and sources, respectively;
(
)
ndm
Grr , and
()
snx rr ,Φ are the values of
d
(,,)
m
Gqrr and s
(, , )
xqΦrr of the volume center
(
)
voxel
N1,...,n
=
n
r. In a reflection mode for a semi-infinite geometry,
()
ndm
Grr ,,
()
snx rr ,Φ and ds
(,,)
xqΦrr are all the analytic solutions to the Laplace-transformed diffusion equation with
extrapolated-boundary condition 11.
For solving Eq. (6) that is in general of large-scale, under-determined and ill-posed. A Kacmarcz method, commonly
known as the Algebraic Reconstruction Technique (ART) might be very efficient and robust 12,13,14. The ART strongly
relies on the initial guess of the unknowns. Sometimes, the prior information on the background as well as the targets is
necessarily required to attain an image reconstruction of high-quality.
To solve both the fluorescent yield (r)
af
ηµ
and lifetime (r)
τ
, we uniquely employ a pair of real transform-factors:
1,2 0.1qQ=m, where () () ()
1/[1/( ) 1/( ) ]
BBB
ax am
Qcc
µµτ
=++
15. According to Eq. (2), the fluorescent parameters are given by
12 1 2 1 1 2 2
121122
() ( ) (, ) (, )/[ (, ) (, )]
() [(, ) (, )]/[ (, ) (, )]
af qqxqxq qxq qxq
xq xq qxq qxq
ηµ
τ
=− −
⎧
⎪
⎨=− − −
⎪
⎩
rrrrr
rr r r r . (10)
3. VALIDATIONS
The principle of methodology is applicable to three-dimensional (3-D) models of the geometry and the results represent
an ideal situation. The domain is divided into 60×60×60 cube. Sources and detectors are placed as Fig. 3, of which the
detectors collect the exiting photons as the sources illuminate the surface. This leads to a total of 64 time-resolved
measurements.
Figure 3． Optode configuration employed in the study
3.1 Monte Carlo Case
The validation is firstly enforced by the data generated from Monte Carlo simulation. In inverse model, a circulation
number 25 and a relaxation parameter 0.5 for ART are used. In Phantom 1, a fluorescent cylinder target (the radius is 5
mm and the height is 10) is embedded 2 mm below the surface to investigate the ability of the algorithm to reconstruct
the parameters. The optical properties of the target are the same as the background, which are set to
1
,0.035
ax m mm
µ
−
=and 1
,
'1.0
sx m mm
µ
−
=for both the excitation and emission wavelengths. These values are in the range
of the optical properties for in vivo muscle16. The fluorescent parameters of the background are 1
0.0023
af mm
ηµ
−
= and
1000ps
τ
=, while the fluorescent parameters of the target are 1
0.0115
af mm
ηµ
−
= and 1000ps
τ
=. The reconstructed
image is shown in Fig. 4(a) and (b).
3.2 FEM Case
source and detector
detector
30mm
30mm
60mm
60mm
60mm
Proc. of SPIE Vol. 6850 68500M-4

Lifetime
60 1100 60 1100 60
40 Ib00 40 IOH 401
20 20 I 20I
900 1900
20 40 60 20 40 60 20 40 60
x(mm) z4mm x(mm) z8mm x(mm) zl2mm I
1100
1000
20 40 60 900
x(mm) z4Omm
Yield
60 )aoo
!A I4OI]
5.2oI 52O] 5
20 40 60 20 40 60 20 40 60 20 40 60
x(mm) z4mm x(mm) z8mm x(mm) zl2mm x(mm) z4Omm
— —
— —
y(rrn,) 0.
— —
Lifetime
60 60
40
1800
1800 1800
11000 •11000 601
20 H 20
H600600 600
20 40 60 20 40 60
x(mm) zgmm x(mm) zlOmm :
1000
800
600
20 40 60
x(mm) z4Omm
20 40 60
x(mm) z4mm
In this case, the validation is enforced by the data that are obtained from FEM. We use Phantom 2 to probe the
performance of the algorithm in different cases. The fluorescent lifetime of the background is 560ps and other
parameters are the same as Phantom 1. The reconstructed images are shown in Fig. 5(a) and (b), where all the differences
in the yield, lifetime and size are disclosed. Meanwhile, it is worthy to note that the quantitative accuracy of the yield is
slightly worse than the lifetime.
Figure 4． Reconstructed images of yield and lifetime in MC case for Phantom 1.
(a)
010 20 30 40 50 60
0
0.002
0.004
0.006
0.008
0.01
0.012
x(mm)
Yield(1/mm)
reco nstructi on
ideal
010 20 30 40 50 60
0
0.002
0.004
0.006
0.008
0.01
0.012
y[mm]
Yield[1/mm]
reconstruction
ideal
010 20 30 40 50 60
2
4
6
8
10
12x 10
-3
z[mm]
Yield[1/mm]
recons truct ion
ideal
010 20 30 40 50 60
400
500
600
700
800
900
1000
1100
x[mm]
lifetime[p s]
recons tructi on
ideal
010 20 30 40 50 60
500
600
700
800
900
1000
1100
y[mm]
lifetim e[ ps ]
reco nstruct ion
ideal
010 20 30 40 50 60
500
600
700
800
900
1000
1100
z[mm]
lifet im e [p s ]
recons truc tion
ideal
(b)
Figure 5． (a) Reconstructed images of fluorescent yield and lifetime for Phantom 1,
and (b) their profiles along X-,Y- and Z-axes in FEM case
Proc. of SPIE Vol. 6850 68500M-5

lifetime
Secondly, the spatial resolution of the algorithm is probed by using a phantom with the same domain and
background optical/fluorescent properties as the above example. Fig. 6 illustrates the reconstruction results as the
center-to-center spacing (CCS) changing. It is seen from the results that the two targets can still be resolved as the CCS is
less than 8 mm for yield and 12 mm for lifetime, and the recovery of the fluorescent yield appears to have better
resolution.
(a)
010 20 30 40 50 60
-2
0
2
4
6
8
10
12x 10
-3
x[mm ]
yield[1/mm]
ccs=12mm
ccs=14mm
ccs=20mm
ccs=10mm
ccs=8mm
010 20 30 40 50 60
500
600
700
800
900
1000
1100
x[mm]
lifetime[ps]
ccs=20mm
ccs=14mm
ccs=12mm
ccs=10mm
(c)
Figure 6． (a) Schematic of the phantom used for evaluating spatial resolution of the algorithm; (b) profiles of the
reconstructed yield along the X-axis for CCS=8,10,12,14,20mm,respectively; (c) profiles of the
reconstructed lifetime along the X-axis for CCS=10,12,14,20mm,respectively.
Thirdly, we demonstrate the robustness of the algorithm to noise by using noisy data with different signal-to-noise
ratio (SNR). Same phantom is used, but fix the CCS of the two targets at 16 mm. Originally, the noise is embedded in the
measured time-resolved flux and is composed of many types. But here we model the noise directly in each of the
featured data-types, i.e., the Laplace transforms of the time-resolved data, as an additive Gaussian random variable with
a standard deviation proportional to the data-type: -/20
ds ds
(r ,r , ) Γ(r ,r , )10ss
χ
σ
=, where
χ
is the SNR in decibels. The
reconstructed images of varying SNR are shown as Fig.7. The results reveal that the noise-robustness of the algorithm is
moderate and the reconstruction of the yield is much more insensitive to the noise but less accurate in the
quantitativeness than that of the lifetime.
010 20 30 40 50 60
0
0.01
0.02
yield[1/mm ]
x[mm ]
010 20 30 40 50 60
500
1000
1500
lifetim e[p s]
x[mm ]
45dB
ideal
40dB
35dB
45dB
ideal
40dB
35dB
(a) (b)
Figure 7． Investigation on the noise robustness of the algorithm by imaging the same phantom as in Fig. 5, with the
CCS of the two targets equal to 16 mm. (a) Reconstructed yield and lifetime images for a varying SNR of
y
x
Proc. of SPIE Vol. 6850 68500M-6

lifetime
U
35dB (top) ,40dB (middle) and 45dB (bottom), and (b) their profiles along the X-rays.
Finally, different sources and detectors numbers are experimented to see how it influent the reconstructed image.
Here we use 16 sources and 4 detectors, 16 sources and 16 detectors. The results are illustrated in Fig.8. Compared Fig.
5with Fig.8, we can see if let the source change into the detector, the results are the same, which fits the theory very well.
As sources and detectors numbers increasing, recovery of the fluorescent yield appears to be improved .
(a) (b)
Figure 8． Reconstructed images of fluorescent yield and lifetime for (a) 16 sources and 4 detectors,
and (b) 16 sources and 16 detectors.
4. CONCLUSIONS
We have presented an analytic, GPST-based methodology of time-domain FMT for simultaneous reconstruction of
fluorescent yield and lifetime of fluorescence makers in turbid medium, from which the algorithms for recovery of
Monte Carlo and FEM simulated data are exemplified, respectively. The simulative validations of the algorithm for a 3-D
domain have been performed for its abilities to discern differences in fluorescence parameters and for its
noise-robustness and spatial resolution. The results have proved the effectiveness of the methodology. We will explore
phantom and in vivo experimental validations of the methodology in the on-going work.
ACKNOWLEDGMENTS
The authors acknowledge the funding supports form the National Natural Science Foundation of China (60578008,
60678049), the National Basic Research Program of China (2006CB705700) and Tianjin municipal government of China
(07JCYBJC06600). Feng Gao and Huijuan Zhao also thank the support from the “111” project of the Ministry of
Education of China.
REFERENCES
1. V. Ntziachristos, J. Ripoll, L. Wang, and R. Weissleder, “Looking and listening to light: the evolution of
whole-body photonic imaging,” Nat. Biotechnol, 23,313-320(2005).
2. S. Lam, F. Lesage, and X. Intes, “Time domain fluorescent diffuse optical tomography: analytical expressions,”
Opt.Express 13, 2263-2275 (2005).
3. A.T.N. Kumar, J. Skoch, B.J. Bacskai, D.A. Boas, and A.K. Dunn, “Fluorescent-lifetime-based tomography for
turbid media,” Opt. Lett. 30, 3347-3349 (2005).
4. F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by
time-domain detection: methodology and phantom validation,” Phys. Med. Biol. 49, 1055-1078 (2004).
Proc. of SPIE Vol. 6850 68500M-7

5. F. Gao, Y. Tanikawa, H.J. Zhao and Y. Yamada, “Semi-three-dimensional algorithm for time-resolved diffuse
optical tomography by use of the generalized pulse spectrum technique,” Appl. Opt. 41, 7346-7358 (2002).
6. L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,”
Comput. Methods and Programs in Biomed. 47,131-146(1995).
7. S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo modeling of light propagation in highly
scattering tissue-1. Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36,
1162-1167(1989).
8. D. Paithankar, A. Chen, B. Pogue, M. Patterson, and E. Sevick-Muraca, “Imaging of fluorescent yield and lifetime
from multiply scattering media,” Appl. Opt. 36, 2260-2272 (1997).
9. M. O’Leary, D.Boas, X. Li, B. Chance, and A. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. 21,
158-160 (1996).
10. V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media
by use of a normalized Born approximation,” Opt. Lett. 26, 893-895 (2001).
11. A. Kienle and M. S. Patterson , “Improved solutions of the steady-state and the time-resolved diffusion equations
for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246-254 (1997).
12. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-93 (1999).
13. F. Gao, P. Poulet and Y. Yamada, “Simultaneous mapping of absorption and scattering coefficients from a
three-dimensional model of time-resolved optical tomography,” App. Opt 39, 5898-5910 (2001).
14. A.C. Kak and M. Slaney, Principle of Computerized Tomographic Imaging, (IEEE Press, New York, 1988).
15. F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “A linear, featured-data scheme for image reconstruction in
time-domain fluorescence molecular tomography,” Opt.Express 14, 7109-7124 (2006).
16. A.T.N. Kumar, J. Skoch, B.J. Bacskai, D.A. Boas, and A.K. Dunn, “Fluorescent-lifetime-based tomography for
turbid media,” Opt. Lett. 30, 3347-3349 (2005).
Proc. of SPIE Vol. 6850 68500M-8