COL 10(2), 021701(2012) CHINESE OPTICS LETTERS February 10, 2012
Improved AFEM algorithm for bioluminescence
tomography based on dual-mesh alternation strategy
Wei Li (
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Life Sciences Research Center, School of Life Sciences and Technology, Xidian University, Xi’an 710071, China
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Corresponding author: jimleung@mail.xidian.edu.cn
Received April 19, 2011; accepted August 3, 2011; posted online September 30, 2011
Adaptive finite element method (AFEM) is broadly adopted to recover the internal source in biological
tissues. In this letter, a novel dual-mesh alternation strategy (dual-mesh AFEM) is developed for biolumi-
nescence tomography. By comprehensively considering the error estimation of the finite element method
solution on each mesh, two different adaptive strategies based on the error indicator of the reconstructed
source and the photon flux density are used alternately in the process. Combined with the constantly
adjusted permissible region in t he adaptive process, the n ew algorithm can achieve a more accurate source
location compared with the AFEM in the previous experiments.
OCIS codes: 170.6960, 170.3010.
doi: 10.3788/COL201210.021701.
Bioluminescence tomography (BLT), as a molecular
imaging modality, has attracted considerable attention
for the study of biological processes in vivo at the cel-
lular and molecular levels. Moreover, it is a potential
technique for cancer detection, drug discovery, and gene
expression visualiza tio n
[1,2]
. The major advantages of
BLT are sensitivity, low cost, ease of operation, and low
noise (in contrast to fluorescence imaging). BLT also
allows localization and quantification of internal biolog-
ical sources generated by the luminescent enzyme and
luciferase, which may reveal va rious molecular and cellu-
lar activities in three dimensions
[3]
.
The source reconstruction algorithm has remained a
challenging task due to the ill-posedness of the inverse
problem in theo ry. Hence, sufficient a priori informa-
tion including optical parameters, a natomical structure,
and permissible source region have tremendous effect on
the algorithm
[3−6]
. In addressing the irregular heteroge-
neous region, the finite element method (FEM), a clas-
sical technique has been widely used, particularly in the
inverse problem
[7−9]
. Accurate numerical solutions re-
quire fine discre tizations of the tissue volume and large
computational reso urces. Adaptive FEM (AFEM) can
achieve a relatively accurate result with low computa-
tion cost. In the adaptive process, the strategies of mesh
refinement indicated by the error estimation are the core
issue. Thes e strategies have s ignificant effect on the pre-
cision of the rec onstruction result.
In previous studies, Lv et al. proposed a multilevel
AFEM method which employed two different a posteriori
error es timations in the forbidden and permissible source
regions on the same mesh
[10]
. Han et al. developed an
hp-FEM for BLT using linear or quadratic interpolation
basis functions for selected elements
[11]
. In this letter, a
novel AFEM for BLT was extended. Both the discrete
error e stimation of the so urce and the flux density on
the whole mesh were co ns idered comprehensively. Conse-
quently, both the direct maximum selection method and
Kelly’s err or estimation of flux density were utilized as in-
dicators. Subsequently, the mesh refinement was carrie d
out alter nately according to the two indicators on the two
meshes to avoid overly detailed refinement which could
aggravate the ill-posedness of the problem
[12]
. Further-
more, in the adaptive process, hp-refinement was chosen
for the selected elements on a certain mesh, incorporat-
ing the constantly adjusted source permissible region in
the adaptive process. As such, more accurate and stable
results were obtained. Actual mouse experiments demon-
strated the advantage of our new algorithm.
In bioluminescence imaging experiments, the biolumi-
nescent photon scattering predominates over a bs orption
in biological tissues, and the photon transport c an be
described by the diffusion equation
[5,10]
. The goal of
BLT is to obtain the reconstr uction of the internal source
from the measurement of the emission of light on the sur-
face. Using FEM and the classical Tikhonov regulariza-
tion method
[6]
, we can express the solution of BLT as
min
S
inf
6S
per
6S
sup
Θ(S
per
) = ||AS
per
− Φ
b
||
L
2
(∂Ω)
+ λ||S
per
||
L
2
(Ω)
, (1)
where Θ(S
per
) is the objective function, S
inf
and S
sup
are the lower and upper bounds of the source density
S
per
(W/mm
3
), Ω is the solving domain, A is the system
matrix, Φ
b
is the measured flux density on the boundary
∂Ω, and λ denotes the regularization para meter which is
manually optimized in this letter.
Considering the approximation Φ
h
(S
h
) and S
h
to the
exact solution Φ(S) and the uniqueness solution of the
BLT problem S, the error bo und is derived as
[13]
||Φ(S) − Φ
h
(S
h
)||
2
L
2
(∂Ω)
+
√
λ||S − S
h
||
2
L
2
(Ω)
6 ch
3/4
.
(2)
In the AFEM framework, hp-refinement is used for the
mesh because hp-refinement can achieve a higher con-
vergence rate compared with h-refinement. Thus , for a
tetrahedral element in three dimensions, we have
[14]
||Φ(S) − Φ
h
(S
h
)||
L
2
(∂Ω)
6 c
′
h
p
p
−(t−1/2)
||Φ(S)||
L
2
(Ω)
.
(3)
Incorporating Eq . (2) with Eq. (3), the convergence
1671-7694/2012/021701(4) 021701-1
c
2012 Chinese Optics Lett ers