Uniqueness theorems in bioluminescence tomography
Ge Wang
a)
Bioluminescence Tomography Laboratory and CT/MicroCT Laboratory,Departments of Radiology,
Biomedical Engineering,and Mathematics,University of Iowa,Iowa City,Iowa 52242
Yi Li
b)
Hunan Normal University,Changsha 410081,Hunan,China and Department of Mathematics,
University of Iowa,Iowa City,Iowa 52242
Ming Jiang
c)
LMAM,School of Mathematical Sciences,Peking University,Beijing 100871,China
~Received 25 November 2003;revised 27 April 2004;accepted for publication 7 May 2004;
published 26 July 2004!
Motivated by bioluminescent imaging needs for studies on gene therapy and other applications in
the mouse models,a bioluminescence tomography ~BLT!system is being developed in the Univer
sity of Iowa.While the forward imaging model is described by the wellknown diffusion equation,
the inverse problem is to recover an internal bioluminescent source distribution subject to Cauchy
data.Our primary goal in this paper is to establish the solution uniqueness for BLT under practical
constraints despite the illposedness of the inverse problem in the general case.After a review on
the inverse source literature,we demonstrate that in the general case the BLT solution is not unique
by constructing the set of all the solutions to this inverse problem.Then,we show the uniqueness
of the solution in the case of impulse sources.Finally,we present our main theorem that solid/
hollow ball sources can be uniquely determined up to nonradiating sources.For better readability,
the exact conditions for and rigorous proofs of the theorems are given in the Appendices.Further
research directions are also discussed. 2004 American Association of Physicists in Medicine.
@DOI:10.1118/1.1766420#
Key words:Bioluminescence tomography ~BLT!,diffusion equation,inverse source problem,
solution uniqueness
I.INTRODUCTION
Small animals,particularly genetically engineered mice,are
of increasing importance for development of the modern
medicine.Small animal imaging offers a major opportunity
to understand pathophysiological and therapeutic processes
at anatomical,functional,cellular,and molecular levels.For
example,gene therapy is a recent breakthrough,which prom
ises to cure diseases by modifying gene expression.Akey for
the development of gene therapy is to monitor the gene
transfer and evaluate its ef®cacy in the living mouse model.
Traditional biopsy methods are insensitive,invasive,and
limited in the extent.To depict the distribution of the admin
istered gene,reporter genes such as those producing lu
ciferase are used to generate light signals within a mouse in
vivo.These signals can be externally measured by a highly
sensitive CCD camera.
1
Such a twodimensional 2D biolu
minescent view can be superimposed onto a photograph of
the mouse for localization of the reporter gene activity.In
addition to its application in gene therapy,this new imaging
tool has great potentials in various other biomedical applica
tions as well.
2±6
However,the single view based biolumines
cent imaging,like the traditional radiography,takes only a
2D image,and is incapable of tomographic reconstruction
of internal features of interest,that is,the 3D distribution of
the bioluminescent source inside the mouse.
Supported by the National Institutes of Biomedical Imag
ing and Bioengineering ~USA!,our team is developing bi
oluminescence tomography ~BLT!as a new modality for mo
lecular imaging,initially of living mice.
7,8
The novel concept
is to collect emitted photons from multiple 3D directions
with respect to a living mouse marked by bioluminescent
reporter luciferases,and reconstruct an internal biolumines
cent source distribution based on both the outgoing biolumi
nescent signals and a prescanned tomographic volume,such
as a CT/microCT volume,of the same mouse.
Traditionally,optical tomography utilizes incoming vis
ible or near infrared light to probe a scattering object,and
reconstructs the distribution of internal optical properties,
such as one or both of absorption and scattering coef®cients.
In contrast to this active imaging mode,BLT reconstructs an
internal bioluminescent source distribution,generated by lu
ciferase induced by reporter genes,from external optical
measures.In BLT,the complete knowledge on the optical
properties of anatomical structures of the mouse is estab
lished from an independent tomographic scan,such as a CT/
microCT scan,by image segmentation and optical property
mapping.That is,we can segment the CT/microCT image
volume into a number of structures,and assign optical prop
erties to each structure using a database of the optical prop
erties compiled for this purpose.
The outline of this paper is as follows.In Sec.II,we
present the basics for BLT,including the diffusion approxi
mation for the radiative transfer equation,or Boltzmann
equation,and formulate the BLT problem.In Sec.III,we
2289 2289Med.Phys.31 8,August 2004 00942405Õ2004Õ318Õ2289Õ11Õ$22.00 2004 Am.Assoc.Phys.Med.
review known theoretical results relevant to the solution
uniqueness of BLT.In Sec.IV,we present the main results on
the solution uniqueness of BLT.In Sec.V,we discuss related
issues and future work,and conclude the paper.Because an
accurate presentation of our results requires rather math
ematical terms,in the main text we only summarize our re
sults as three theorems in engineerfriendly terms;then we
give their complete conditions and proofs in the Appendices.
All the theorems in the main text are referenced by the ro
man numbers,while those in the Appendices are indexed by
the roman letters.
II.PROBLEM STATEMENT
Let V be a domain in the threedimensional Euclidean
space R
3
that contains the object to be imaged.Let u(x,u)
be the light ¯ux in direction uPS
2
at xPV,where S
2
is the
unit sphere.A general model for light migration in a random
medium is given by the radiative transfer equation,or
Boltzmann equation:
9±11
1
c
]u
]t
~
x,u,t
!
1u"
x
u
~
x,u,t
!
1m
~
x
!
u
~
x,u,t
!
5m
s
~
x
!
E
S
2
h
~
uu
8
!
u
~
x,u
8
,t
!
du
8
1q
~
x,u,t
!
,~1!
for t.0,and xPV,where c denotes the particle speed,m
5m
a
1m
s
with m
a
and m
s
being the absorption and scatter
ing coef®cients respectively,the scattering kernel his nor
malized such that
*
S
2
h(uu
8
)du
8
51,and q is the internal
light source.The initial condition for u is formulated as
u
~
x,u,0
!
50,for xPV and uPS
2
,~2!
while the boundary condition for u represents the incoming
¯ux g
2
:
u
~
x,u,t
!
5g
2
~
x,u,t
!
,
for t.0,and xPG,
uPS
2
,such that n
~
x
!
u<0,~3!
where n(x) is the exterior normal at x on the boundary G of
V.Although we have g
2
50 in a typical BLT case,we prefer
keeping g
2
here for generality of the formulation.For ex
ample,if we perform BLT of two mice simultaneously,the
outgoing ¯ux of one mouse would be partially intercepted by
the other mouse as its incoming ¯ux.Then,we want to re
construct the internal light source q from measurements of
the outgoing radiation,i.e.,the escaping energy through a
unit area at xPG perpendicular to the exterior normal n(x)
on G,
10,11
g
~
x,t
!
5
E
S
2
n
~
x
!
uu
~
x,u,t
!
du,t.0 and xPG.~4!
Reconstruction of the light source q is quite complex based
on the measurement g and above initialboundary conditions
with the radiative transfer equation ~1!as the governing
equation,closely related to the dif®culty in computing the
¯ux u as the forward problem ~1!,~2!,and ~3!.Then,we seek
an approximation to simplify the radiative transfer equation
~1!.Because the meanfree path of the particle is between
0.005 and 0.01 mm in biological tissues,which is very small
compared to a typical object in this context,the predominant
phenomenon is scattering instead of transport.
11
Hence,we
can approximate the radiative transfer equation ~1!with a
much simpler equation,the diffusion equation,which has
already been widely used in optical tomography.
10,11
Let u
0
be the average photon ¯ux in all directions,i.e.,the diffusion
approximation,
u
0
~
x,t
!
5
1
4p
E
S
2
u
~
x,u,t
!
du,~5!
and q
0
be de®ned similarly,
q
0
~
x,t
!
5
1
4p
E
S
2
q
~
x,u,t
!
du.~6!
It can be shown that u
0
satis®es the following initial
boundary value problem ~omitting the refraction at the
boundary without loss of generality!,
10,11
1
c
]u
0
]t
2"
~
Du
0
!
1m
a
u
0
5q
0
,t.0 and xPV,
~7!
u
0
~
x,t
!
12D
~
x
!
]u
0
]n
~
x,t
!
5g
2
~
x,t
!
,
t.0 and xPG,~8!
u
0
~
x,t50
!
50,xPV,~9!
where
D
~
x
!
5
1
3m
a
~
x
!
1m
s
8
~
x
!
.~10!
The measurement equation ~4!after the diffusion approxima
tion reads
10,11
as
g
~
x,t
!
52D
~
x
!
]u
0
]n
~
x,t
!
,t.0 and xPG.~11!
The above diffusion approximation procedure is also called
the P
1
approximation.
10,11
Because the internal bioluminescence distribution induced
by reporter genes is relatively stable,we can use the station
ary version of Eqs.~7!±~9!as the forward model for BLT.By
discarding all the time dependent terms and Eq.~9!,the sta
tionary forward model is
2"
~
D¹u
0
!
1m
a
u
0
5q
0
,xPV,~12!
u
0
~
x
!
12D
~
x
!
]u
0
]t
~
x
!
5g
2
~
x
!
,xPG,~13!
and measurement equation ~11!reads as
g
~
x
!
52D
~
x
!
]u
0
]n
~
x
!
,xPG.~14!
Given the measurement ~14!,it follows that the boundary
value of u
0
(x) can be obtained according to ~13!as follows:
2290 Wang,Li,and Jiang:Uniqueness theorems in bioluminescence tomography 2290
Medical Physics,Vol.31,No.8,August 2004
u
0
~
x
!
5g
2
~
x
!
12g
~
x
!
,xPG.~15!
Hence,u
0
satis®es the following Cauchy condition on the
boundary G:
12
u
0
~
x
!
5g
2
~
x
!
12g
~
x
!
,xPG,~16!
D
~
x
!
]u
0
]
v
~
x
!
52g
~
x
!
,xPG.~17!
Therefore,BLT is equivalent to reconstruct the source q
0
of
Eq.~12!from given u
0
(x) and (]u
0
/]n)(x) for xPG,under
the governing diffusion equation ~12!.
In summary,the BLT problem can be stated as follows:
Given the incoming ¯ux g
2
(x) and outgoing ¯ux g (x) for
xPG,®nd a source q
0
with one corresponding photon ¯ux u
to satisfy
~
BLT
!
5
2"
~
Du
0
!
1m
a
u
0
5q
0
,xPV,
u
0
~
x
!
12D
~
x
!
]u
0
]n
~
x
!
5g
2
~
x
!
,xPG,
D
~
x
!
]u
0
]n
~
x
!
52g
~
x
!
,xPG,
~18!
or,equivalently,
~
BLT
!
H
2"
~
Du
0
!
1m
a
u
0
5q
0
,xPV,
u
0
~
x
!
5g
2
~
x
!
12g
~
x
!
,xPG,
D
~
x
!
]u
0
]n
~
x
!
52g
~
x
!
,xPG.
~19!
The optical parameters D and m
a
can be established point
wise from a prerequisite tomographic scan,such as a CT/
microCT scan.
7,8
In this paper,we assume that V is a
bounded smooth domain of R
N
,although the case of our
main interest is N53.We always assume that the parameters
D.D
0
.0 for some positive constant D
0
and that m
a
>0 are
bounded functions in this work.We further assume that D is
suf®ciently regular near G,e.g.,D is equal to a constant near
G.
In practice,it is dif®cult to obtain all the measurements
along the boundary G.We consider the case in which the
measurement can only be taken on a portion P
0
,G.The
BLT problem then becomes
BLT
~
P
0
!
5
2"
~
Du
0
!
1m
a
u
0
5q
0
,xPV,
u
0
~
x
!
12D
~
x
!
]u
0
]n
~
x
!
5g
2
~
x
!
,xPP
0
,
D
~
x
!
]u
0
]n
~
x
!
52g
~
x
!
,xPP
0
.
~20!
III.LITERATURE REVIEW
BLT as formulated above is for the reconstruction of an
internal source from Cauchy data,which is called the inverse
source problem of partial differential equations.
13
There are
several theoretical studies relevant to the uniqueness of the
solution to this type of problems.Although they do not pro
vide a satisfactory answer to the solution uniqueness of BLT,
these results do form a background for us to establish the
uniqueness theorems under practical constraints for BLT.For
a detailed historical survey,please refer to Ref.13 and the
references therein.
In Ref.13,when the domain V is a bounded Lipschitz
domain in the ndimensional Euclidean space R
n
,the source
q
0
5aq
1
1q
2
with ]q
i
/]x
n
50 for i51,2 and ]a/]x
n
>0,
where ais given,and the coef®cient D does not depend on
x
n
and m
a
>0,then q
0
is uniquely determined by the Cauchy
data ~16!and ~17!.
In Ref.14,V is a cylindrical domain V5V
8
3V
9
,
V
8
,R
n
8
,V
9
,R
n
9
.The governing equation is the Poisson
equation,
2Du
0
5q
0
,~21!
i.e.,D51 and m
a
50.The source is assumed to be cylindri
cal,
q
0
~
x
!
5b
~
x
8
!
h
~
x
9
!
,x5
~
x
8
,x
9
!
.~22!
If q
0
is with one known factor and a positive height part,
then it is uniquely determined by Cauchy data ~16!and ~17!.
In the standard case of n
9
51,b and h are referred to as the
base and height of the source,respectively.
In Ref.15,V is a bounded Lipschitz domain in a two
dimensional Euclidean space R
2
.The governing equation is
the Helmholtz equation,
Du
0
1k
2
u
0
5q
0
,~23!
i.e.,D51 and m
a
52k
2
.The source is assumed of either the
form
q
0
~
x
!
5r
~
x
!
x
B
~
x
!
,~24!
where B is an open subset of V,x
B
is the characteristic
function of B,or the form
q
0
~
x
!
5div
@
r
~
x
!
x
B
~
x
!
a
#
,~25!
where a is a nonzero constant vector.Under some additional
technical conditions,the convex hull of the source support B
can be uniquely reconstructed given Cauchy data ~16!and
~17!.
In Ref.16,V is a bounded domain of R
n
with a suf®
ciently regular boundary and partitioned into connected sub
domains coated in layers ~see Ref.16 for a precise presenta
tion!.The governing equation is
2"
~
Du
0
!
5q
0
,~26!
i.e.,m
a
50.The coef®cient D is constant in each subdomain.
The source distribution is assumed to be of the form
q
0
5
(
k51
m
x
v
k
,~27!
where x
v
k
is the characteristic function of a ball v
k
with
center S
k
and radius r
k
.The centers must be distinct but the
balls may overlap each other.It was proved that the number
m of balls v
k
and their parameters S
k
and r
k
can be uniquely
determined by Cauchy data.Note that these sources are as
sumed to have identical intensity values;otherwise,the
2291 Wang,Li,and Jiang:Uniqueness theorems in bioluminescence tomography 2291
Medical Physics,Vol.31,No.8,August 2004
uniqueness does not hold.There is a counterexample in Ref.
16 that q
0
5l
i
x
v
i
with different l
i
and v
i
for i51,2 such
that
2Du5l
i
x
v
i
,~28!
u
i
5f,on G,~29!
]u
i
]n
5g,on G,~30!
with the same f and g.To that effect,it suf®ces to set the
parameters for both v
i
such that
S
1
5S
2
,and l
1
r
1
2
5l
2
r
2
2
.~31!
It is interesting that the solution uniqueness holds for the
equation ~26!,assuming a combination of mono and dipolar
sources of the following form:
16
q
0
~
x
!
5
(
k51
m
1
l
k
d
S
k
1
(
j 51
m
2
p
j
d
C
j
,~32!
where m
1
and m
2
are positive integers,S
k
and C
j
are points
in V
0
,l
k
and p
j
are,respectively,scalar and vector quanti
ties,d
S
k
and d
C
j
are a dfunction,and the gradient of a d
function at S
k
and C
j
,respectively.
The counterexample given in ~28!±~31!shows the nonu
niqueness of the solution to inverse source problems.Recon
structing sources of the form q
0
5lx
v
with the Poisson
equation ~21!as the governing equation is related to the in
verse gravimetry problem in geophysics,where the unique
ness does not hold unless the source support is starshaped or
convex.
13
For the Helmholtz equation,the inverse source
problem does not admit a unique solution because of the
possible nonradiating sources within the source support
V.
17±19
In Ref.14,with the Poisson equation as the govern
ing equation it was proved that all the solutions to the inverse
source problem with Cauchy data can be expressed as
q
0
5q
1
1q
2
;~33!
where q
1
is the minimal L
2
norm solution of the inverse
source problemsatisfying 2Dq
1
50 and q
2
52Dh for some
h with zero Cauchy data.Hence,q
1
is unique.The q
2
part
corresponds to the nonradiating source.
For clarity,our literature overview is summarized in Table
I.As shown in Table I,the solution uniqueness results are not
available for BLT,in which the diffusion equation assumes
spatially variable optical properties m
a
and D.
IV.RESULTS
Given its physical meaning,BLT must have at least one
solution.Therefore,in this section we will not discuss the
existence of the BLT solution,and primarily focus on the
solution uniqueness of BLT.To convey our main points
clearly,we will just present our three theorems in a manner
easily accessible to physicists and engineers while giving
rigorous statements and proofs in the Appendices.
The ®rst result is about the solution structure of the BLT
problem ~18!,which is a generalization of ~33!in Ref.14.
Let L be the following differential operator:
L
@v#
52"
~
D
v
!
1m
a
v
,~34!
we have the following.
Theorem IV.1:Assume that the BLT problem is solvable.
There is one special solution q
H
for the BLT problem (18),
which is of the minimal L
2
norm among all the solutions.All
the solutions can be expressed as q
0
5q
H
1L
@
m
#
,for any
mPH
0
2
(V),which is the closure of all smooth functions in V
Å
vanishing on G up to order one.(cf.Theorem B.2.)
Given the dif®culty that there is no unique solution to
BLT in the general case by Theorem IV.1 ~as a matter of fact,
solvable problems always have many distinct solutions!,we
must restrict the solution space to a subspace of biolumines
cent source distributions so that the solution uniqueness may
be established in that speci®c case.For example,we can
study source distributions in a certain parametrized form to
remove the ambiguity in the BLT solution.
In the following,we ®rst consider the case of a linear
combination of bioluminescent impulses,
T
ABLE
I.Summary of known inverse source results.
Reference Domain Equation m
a
D Source Uniqueness of q
0
13 general diffusion ~12!arbitrary
]D
]x
n
50 q
0
5aq
1
1q
2
,known a;
]q
i
]x
n
50,
]a
]x
n
>0;
yes
14 cylindrical Poisson ~21!0 D51 b(x
8
)h(x
9
),one known factor yes
15 general Helmholtz ~23!0 negative constant r(x)x
B
(x) convex hull of B
16 general diffusion ~26!0 piecewise constant
(
k51
m
x
v
k
yes
16 general diffusion ~26!0 piecewise constant
q
0
~
x
!
5
(
k51
m
1
l
k
d
S
k
1
(
j51
m
2
p
j
d
C
j
yes
16 general Poisson ~21!0 D51 q
0
(x)5lx
S
no
14 general Poisson ~21!0 D51 arbitrary no ( q
0
5q
1
1q
2
)
2292 Wang,Li,and Jiang:Uniqueness theorems in bioluminescence tomography 2292
Medical Physics,Vol.31,No.8,August 2004
q
0
~
y
!
5
(
i51
m
a
i
d
~
y2y
i
!
,~35!
where each a
i
is a constant coef®cient,and y
i
the location of
a point source inside V,for i51,...,m.We have the follow
ing theorem.
Theorem IV.2:Assume that the conditions in Theorem
D.4 (Appendix D) hold.If q
0
(y)5
(
i51
m
a
i
d(y2y
i
) and
Q
0
(y)5
(
j 51
M
A
j
d(y2Y
j
) are two solutions to the BLT prob
lem (18),then m5M and there is a permutation tof
@
1,m
#
such that a
i
5A
t(i)
and y
i
5Y
t(i)
.
Then,let us consider a linear combination of solid/hollow
ball sources,
q
0
~
y
!
5
(
i51
m
l
i
x
B
r
0
i
,r
1
i
~
x
i
!
,~36!
for the more general BLT( P
0
) problem ~20!,which covers
the BLT problem as a special case.To present our ®nding in
this case,we need the following notations.For each 0 <r
0
,r
1
,`,x
0
PR
N
,let B
r
0
,r
1
(x
0
) denote a hollow ball speci
®ed by r
0
,
u
x2x
0
u
,r
1
for r
0
.0 and a solid ball speci®ed
by
u
x2x
0
u
,r
1
for r
0
50.To study the solution uniqueness,
we assume that the domain V is partitioned into I sub
domains.Another assumption is that the coef®cients D and
m
a
must be piecewise constants,which is also reasonable in
practice.Please see Theorem D.4 ~Appendix D!to ®nd the
exact conditions for the following theorem.
Theorem IV.3:Assume that the conditions in Theorem
D.4 (Appendix D) hold.If q
1
(y)5
(
i51
m
l
i
x
B
r
0
i
,r
1
i
(x
i
) and
q
2
(y)5
(
j 51
M
L
j
x
B
R
0
i
,R
1
i
(X
i
) are two solutions to the
BLT(P
0
) problem (20),then m5M,and there exists a per
mutation tof
@
1,m
#
and a map C:
@
1,m
#
!
@
1,I
#
such that
x
i
5X
t(i)
PV
C(i)
and
l
i
E
r
0
i
r
1
i
r
N21
w
C
~
i
!
~
r
!
dr5L
t
~
i
!
E
R
0
t
~
i
!
R
1
t
~
i
!
r
N21
w
C
~
i
!
~
r
!
dr,
for i51,...,I,~37!
where w
j
is the unique solution of
2D
j
S
w
j
9
1
N21
r
w
j
8
D
1m
j
w
j
50,~38!
w
j
~
0
!
51,w
j
8
~
0
!
50.~39!
V.DISCUSSIONS AND CONCLUSION
Theorem IV.1 reveals a fundamental feature of BLT.That
is,without the incorporation of effective priori knowledge
on the source distribution,there would be no hope to deter
mine a unique solution.Actually,no matter how many
higherorder derivatives are measured,the uniqueness of the
solution cannot be claimed without the use of additional con
straints on the source.For example,if z
0
is a solution and
m(x) is any smooth function with compact support in V and
D
a
m
u
G
50 for all a,then it is straightforward to prove that
w5z
0
1aL
@
m
#
is also a solution to the BLT problem.Physi
cally speaking,no matter how many orders of measures are
taken in an open band around the boundary of the domain V,
we will not be able to ®nd the solution uniquely without
utilization of adequate priori knowledge.In other words,
Theorem IV.1 suggests that one must utilize all possible in
formation on the source distribution to achieve the best pos
sible reconstruction for BLT.
Theorem IV.2 is not only theoretically inspiring but also
practically useful.As a modality for molecular imaging,BLT
is often intended for the detection of small pathological
events and changes such as for cancer screening.In this con
text,a combination of bioluminescent impulses may model
the early stage of tumor development very well.With in
creasingly more imaging probes and smart drugs available,
the solution uniqueness in that case would de®nitely facili
tate an early diagnosis and better treatment of the cancer in
general.
Theorem IV.3 is our main result in this paper.Interest
ingly,if we only consider solid ball sources and assume that
their intensities are known,it can be readily shown that the
solution to the BLT problem is unique.Practically,the source
intensity is closely related to the strength of the molecular/
cellular activity,such as gene expression.Hence,it is often
reasonable to take the intensity or its parametric form as
known to ®nd the unique solution.
Our uniqueness results are instrumental for the recon
struction of a bioluminescent source distribution.For sources
as parametrized in Theorem IV.3,once a solution is found,
any other solution can be easily constructed by adjusting a
limited number of source related parameters ~intensity l
i
and
so on!according to the relationships given in Theorem IV.3,
subject to any other available anatomical and physiological
constraints.Note that since a practical source function can be
approximated by a linear combination of solid/hollow ball
sources as parametrized in Theorem IV.3,our uniqueness
results cover a quite general class of source distributions,
spanned by those solid/hollow ball sources.
We emphasize that BLT as de®ned in this paper is a new
area,and there remain many theoretical,numerical and ex
perimental issues to be resolved.Theoretically,we would
like to relax the assumptions on the properties of the scatter
ing media and enrich the family of parametric source distri
butions.The solution uniqueness with some additional inter
nal measurement,such as endoscopic measurement,may
improve the wellposedness of BLT.The stablity of the BLT
solution is also an important problem to be addressed.The
perspective for multispectral and dynamic BLT should be
even more challenging.While the continuous domain formu
lation is important,various digital algorithms must be de
signed for practical BLT.However,the development and
evaluation of these algorithms are beyond the scope of this
theoretical paper.Currently,we are developing our BLT pro
totype with an initial emphasis on mouse models of various
lung diseases.
8
While we were in the stage of ®nalizing this paper,it
came to our attention that some similar work was performed
at Xenogen,as reported in an SPIE paper
20
and the company
website ~http://www.xenogen.com/!.Some 3D imaging sys
tems have been recently released to a few test sites,which
2293 Wang,Li,and Jiang:Uniqueness theorems in bioluminescence tomography 2293
Medical Physics,Vol.31,No.8,August 2004
take multiple views around a mouse or rat.A diffuse lumi
nescent imaging tomography algorithm is used to reconstruct
an internal source,coupled with a homogeneous scattering
media assumption.Clearly,this approach may reveal subcu
taneous depth information,but satisfactory reconstruction of
a bioluminescent source distribution ~both geometric and
power!cannot be achieved in general without compensation
for the heterogeneous anatomy of the mouse.
In conclusion,we have determined the set of the solutions
to BLT in the general case to demonstrate that the generic
BLT problem is not uniquely solvable.Then,we have estab
lished the solution uniqueness in the cases of ~i!impulse
sources,and ~ii!solid/hollow ball sources ~up to nonradiating
sources!,assuming that the scattering media are piecewise
constant in terms of D and m
a
.It has been emphasized that
by introducing the priori knowledge on the bioluminescent
source structure,the BLT problem becomes wellde®ned.
Therefore,the BLT is feasible for the localization and quan
ti®cation of the bioluminescent source distribution.We be
lieve that BLT will grow into an important molecular imag
ing modality,and play a signi®cant role in development of
molecular medicine.
ACKNOWLEDGMENTS
The authors thank Dr.Mike Cable,Dr.Eric Hoffman,Dr.
Ruo Li,Dr.Paul McCray,Dr.Geoffrey McLennan,Dr.
Lihong Wang,Dr.Joseph Zabner for discussions;anonymous
referees for constructive comments.This work is partially
supported by the National Institutes of Health ~EB001685
and EB002667!.Dr.Ming Jiang was supported in part by the
National Basic Research Program of China ~2003CB716101!
and the National Science Foundation of China ~60325101,
60272018,and 60372024!.Yi Li was supported in part by
the National Science Foundation of China ~10171036!.
APPENDIX A:MATHEMATICAL PRELIMINARIES
1.Notations
The following function spaces
21±23
are used in the proofs
below:
L
2
~
V
!
5
H
u:
E
V
u
f
~
x
!
u
2
dx,`
J
~A1!
with the inner product de®ned by
^
u,
v
&
L
2
~
V
!
5
E
V
u
~
x
!
v~
x
!
dx.~A2!
We need the Sobolev spaces,
H
1
~
V
!
5
$
uPL
2
~
V
!
:uPL
2
~
V
!
%
,~A3!
where u is the derivative in the sense of distribution,with
the inner product de®ned by
^
u,
v
&
H
1
~
V
!
5
^
u,
v
&
L
2
~
V
!
1
^
u,
v
&
L
2
~
V
!
,~A4!
and
H
2
~
V
!
5
$
uPH
1
~
V
!
:¹
2
uPL
2
~
V
!
%
,~A5!
with the inner product de®ned by
^
u,
v
&
H
2
~
V
!
5
^
u,
v
&
H
1
~
V
!
1
^
¹
2
u,¹
2
v
&
L
2
~
V
!
.~A6!
The subspaces H
0
1
(V) and H
0
2
(V) of H
1
(V) and H
2
(V) are
the closure of smooth functions with compact support inside
V in H
1
(V) and H
2
(V) with the associated norms,respec
tively.In fact,there is a family of Sobolev spaces,denoted
by H
s
(V),for sPR.Similarly,we can de®ne H
0
s
(V).
To solve the boundary value problems of partial differen
tial equations,we need functions on G.We can de®ne the
space L
2
(G) on G similarly.De®nitions for the Sobolev
spaces H
t
(G) on G involve tedious speci®cs,
21±23
and are
skipped here.For a smooth function u,its boundary value is
de®ned by restriction of u to G:u
u
G
(x)5u(x),for xPG.For
a Sobolev space,it can be established that there is a unique
map tfrom H
s
(V) to H
s21/2
(G) such that ~1!t
@
u
#
5u
u
G
for
a smooth u;~2!tis continuous and onto.tis called the trace
operator.Hence,for example,the space for characterizing
the boundary values of functions in H
1
(V) is naturally
H
1/2
(G).It can be proved that uPH
0
s
(V) if and only if
t
@
u
#
50.It is wellknown that L
2
(V),H
s
(V) and H
t
(G) are
Hilbert spaces with the norms induced from the correspond
ing inner products.
We need the following notations from functional
analysis.
24
Let A be a linear operator from a Banach space X
to a Banach space Y.The kernel or null space of A is de®ned
as N
@
A
#
5
$
xPX:A
@
x
#
50
%
,and the range of A is R
@
A
#
5
$
yPY:y5A
@
x
#
for some xPX
%
.For a subspace M of a
Hilbert space H,M
'
is the set of all yPH,such that
^
y,x
&
50 for all xPM.
In the following,we will make the mathematical presen
tations as precise as possible except for some rather technical
and tedious assumptions ~on the coef®cients and so on!not
perfectly stated.For details,please see Ref.22.
2.DirichlettoNeumann map
To make the presentation concise,we introduce the fol
lowing notations.Let g
0
and g
1
be the boundary value maps,
g
0
@
u
#
5u
u
G
,~A7!
and
g
1
@
u
#
5D
]u
]n
U
G
.~A8!
Let L
@
u
#
be the differential operator,
L
@
u
#
52"
~
Du
!
1m
a
u.~A9!
Then,the forward model can be written as
L
@
u
#
5q
0
,in V,~A10!
g
0
@
u
#
12g
1
@
u
#
5g
2
,on G.~A11!
Given f PH
1/2
(G),let w
1
PH
1
(V) be the solution of the
following boundary value problem:
22,25
L
@
w
1
#
50,in V,~A12!
g
0
@
w
1
#
5f,on G.~A13!
2294 Wang,Li,and Jiang:Uniqueness theorems in bioluminescence tomography 2294
Medical Physics,Vol.31,No.8,August 2004
We de®ne a linear operator N from H
1/2
(G) to H
21/2
(G) by
N
@
f
#
5g
1
@
w
1
#
.~A14!
N is the wellknown DirichlettoNeumann ~or Steklov±
Poincare
Â
!map.
13
On the other hand,for q
0
PL
2
(V),we consider the prob
lem
L
@
w
2
#
5q
0
,in V,~A15!
g
0
@
w
2
#
50,on G,~A16!
and de®ne another linear operator L by
L
@
q
0
#
5g
1
@
w
2
#
.~A17!
From the regularity theory for secondorder elliptic partial
differential equations,w
2
PH
2
(V)ùH
0
1
(V) and g
1
@
w
2
#
PH
1/2
(G).
22,25
In terms of g
0
and g
1
,the BLT problem is to ®ndq
0
such
that
L
@
u
#
5q
0
,in V,~A18!
g
0
@
u
#
12g
1
@
u
#
5g
2
,on G,~A19!
g
1
@
u
#
52g,on G,~A20!
given the observed g and assumed g
2
,where u is unknown.
Assume that such a source q
0
exists.Then,we can ®nd u by
solving the following boundary value problem:
L
@
u
#
5q
0
,in V,~A21!
g
0
@
u
#
5g
2
12g,on G.~A22!
Let w
1
be de®ned as in ~A12!±~A13!with f 5g
2
12g,and
w
2
be de®ned as in ~A15!±~A16!.It follows that u5w
1
1w
2
.The measurement equation implies that
2g5g
1
@
u
#
5g
1
@
w
1
#
1g
1
@
w
2
#
5N
@
g
2
12g
#
1L
@
q
0
#
,
~A23!
i.e.,
L
@
q
0
#
52N
@
g
2
12g
#
2g.~A24!
Conversely,if there exists a q
0
satisfying ~A24!,we can
construct u as indicated above.It follows easily that u satis
®es the forward model and the measurement equation.In
summary,we have the following proposition.
Proposition A.1:q
0
is a solution for the inverse problem
(A18)±(A20) if and only it is a solution to (A24).
3.Green's formula
For
v
and pPH
2
(V) the following Green's formula is
wellknown:
22,25
E
V
v
L
@
p
#
2pL
@v#
dx52
E
G
v
g
1
@
p
#
2pg
1
@v#
dG.
~A25!
Let F(x,y) be the fundamental solution of L on R
N
with
coef®cients smoothly extended fromV to R
N
with the same
properties,which tends to zero at`for each ®xedxPR
N
,
26
i.e.,
L
y
F
~
x,y
!
5d
~
y2x
!
,lim
y!`
F
~
x,y
!
50,yPR
N
.~A26!
Then,we can apply Green's formula ~A25!to obtain a for
mula for the solution of the inverse problem ~A18!±~A20!.
Let u be the solution satisfying ~A18!±~A20!.For any x
PV,by Green's formula ~A25!with
v
5F(x,y) and p5u,
we have
E
V
@
F
~
x,y
!
L
@
u
#~
y
!
2u
~
y
!
d
~
x2y
!
#
dy
52
E
G
F
~
x,y
!
g
1
@
u
#~
y
!
2u
~
y
!
g
1
@
F
~
x,y
!
#
dG
y
.~A27!
Hence,
u
~
x
!
5
E
V
@
F
~
x,y
!
q
0
~
y
!
#
dy2
E
G
F
~
x,y
!
g
~
y
!
1g
2
~
y
!
12g
~
y
!
g
1
@
F
~
x,y
!
#
dG
y
,;xPV.~A28!
Note that L
@
F(x,)
#
50 if xPR
N
\V
Å
.We obtain,by Green's
formula again,
05
E
V
@
F
~
x,y
!
q
0
~
y
!
#
dy2
E
G
F
~
x,y
!
g
~
y
!
1g
2
~
y
!
12g
~
y
!
g
1
@
F
~
x,y
!
#
dG
y
,;xPR
N
\V
Å
.~A29!
APPENDIX B:PROOF OF THEOREM IV.1
By Proposition A.1,to study the uniqueness property of
the BLT solution we should characterize the kernel N@L#of
the operator L:L
2
(V)!H
1/2
,L
2
(G).We begin with deter
mining the adjoint L
*
of L,because N
@
L
#
5R
@
L
*
#
'
.
24
Let
cPH
1/2
(G) and f5T
@
c
#
as the unique solution in
H
1
(V),L
2
(V) of the boundary problem
L
@
f
#
50,in V,~B1!
g
0
@
f
#
52c,on G.~B2!
Then,by Green's formula ~A25!,~A16!,~A17!,~B1!,and
~B2!,
E
V
q
0
fdx5
E
V
L
@
w
2
#
fdx
52
E
G
2cL
@
q
0
#
2w
2
g
1
@
f
#
dG
1
E
V
w
2
L
@
f
#
dx5
E
G
cL
@
q
0
#
dG.
Thus,for the operators L:L
2
(V)!H
1/2
(G),L
2
(G) and
T:H
1/2
(G),L
2
(G)!L
2
(V),
^
q
0
,T
@
c
#
&
L
2
~
V
!
5
^
L
@
q
0
#
,c
&
L
2
~
G
!
,~B3!
i.e.,
L
*
5T.~B4!
Then,the kernel of L is
2295 Wang,Li,and Jiang:Uniqueness theorems in bioluminescence tomography 2295
Medical Physics,Vol.31,No.8,August 2004
N
@
L
#
5R
@
L
*
#
'
5R
@
T
#
'
.~B5!
We have the following proposition characterizing R
@
T
#
'
.
Proposition B.1:
R
@
T
#
'
5L
@
H
0
2
~
V
!
#
.~B6!
Proof:If qPL
@
H
0
2
(V)
#
with q5L
@
p
#
for some p
PH
0
2
(V),then for
v
5T
@
c
#
PR
@
T
#
,by Green's formula
~A25!,
^
q,
v
&
L
2
~
V
!
5
E
V
q
v
dx5
E
V
v
L
@
p
#
dx
5
E
G
pg
1
@v#
2
v
g
1
@
p
#
1
E
V
L
@v#
p dx
50,
because g
0
@
p
#
50,g
1
@
p
#
50 and L
@v#
50.Hence,
q'R
@
T
#
.Therefore,L
@
H
0
2
(V)
#
,R
@
T
#
'
.
Conversely,assume that qPR
@
T
#
'
5N
@
L
#
.We have,by
~A15!±~A17!,there exists w
2
such that
L
@
w
2
#
5q,in V,g
0
@
w
2
#
50,on G,
g
1
@
w
2
#
50,on G.
We have w
2
PH
2
(V) by the regularity theory for second
order elliptic partial differential equations.
22,25
The above
boundary conditions imply that w
2
PH
0
2
(V).Hence,q
5L
@
w
2
#
PL
@
H
0
2
(V)
#
.The conclusion follows immedi
ately.h
By Proposition A.1,all the solutions to the BLT problem
form a closed convex set in L
2
(V).There exists one unique
solution of the minimal L
2
norm among those solutions,
24
denoted as q
H
.Then,all the solutions can be expressed as
q
H
1N
@
L
#
.We summarize the above results into the follow
ing theorem.
Theorem B.2:Assume that the BLT problem is solvable.
For any couple (g
2
,g) such that
2N
@
g
2
12g
#
2gPH
1/2
~
G
!
;~B7!
there is one special solution q
H
for the BLT problem (18),
which is of the minimal L
2
norm among all the solutions.
Then,any solution can be expressed as q
0
5q
H
1L
@
m
#
,for
some mPH
0
2
(V).
Remark B.3:Naturally,the condition (B7) for (g
2
,g) is
automatically satis®ed when g is a normal trace g
1
@
u
#
,
where u is a solution of the forward model (A10) and (A11)
for q
0
PL
2
(V).
APPENDIX C:PROOF OF THEOREM IV.2
We present the exact conditions on V,D,m
a
,and q
0
for
Theorem IV.2,which are also part of conditions for Theorem
IV.3.
C1:V is a bounded C
2
domain of R
N
and partitioned into
nonoverlapping subdomains V
i
,i51,2,...,I.
C2:Each V
i
is connected with piecewise smooth bound
ary P
i
.
C3:D and m
a
are C
2
near the boundary of each subdo
main.
C4:D.D
0
.0 for some positive constant D
0
is Lipschitz
on each subdomain;m
a
>0 and m
a
PL
p
(V) for some
p.N/2.
Theorem C.1:Assume the conditions C1±C4 hold.If
q
0
(y)5
(
i51
m
a
i
d(y2y
i
) and Q
0
(y)5
(
j 51
M
A
j
d(y2Y
j
) are
two solutions to the BLT problem (18),then m5M,and
there is a permutation tof
@
1,m
#
such that a
i
5A
t(i)
and
y
i
5Y
t(i)
.
Proof:For xPR
N
\V
Å
,let
b
~
x
!
5
E
G
@
F
~
x,y
!
g
~
y
!
1
~
g
2
~
y
!
12g
~
y
!!
g
1
@
F
~
x,y
!
##
dG
y
.
~C1!
If q
0
(y)5
(
i51
m
a
i
d(y2y
i
) and Q
0
(y)5
(
j 51
M
A
j
d(y2Y
j
) are
both solutions to the BLT problem ~18!,then we have,by
~A29!,
E
V
F
~
x,y
!
F
(
i51
m
a
i
d
~
y2y
i
!
G
dy5b
~
x
!
,;xPR
N
\V
Å
,
~C2!
or
(
i51
m
a
i
F
~
x,y
i
!
5b
~
x
!
,;xPR
N
\V
Å
.~C3!
Similarly,we have
(
j 51
M
A
j
F
~
x,Y
j
!
5b
~
x
!
,;xPR
N
\V
Å
.~C4!
Now,let us de®ne two functions w and W on R
N
as follows:
w
~
x
!
5
(
i51
m
a
i
F
~
x,y
i
!
,~C5!
and
W
~
x
!
5
(
j 51
M
A
j
F
~
x,Y
j
!
.~C6!
Since F(x,y)5F(y,x),we have
2"
~
D¹w
!
1m
a
w50,in R
N
\
$
y
1
,...,y
m
%
,~C7!
2"
~
D¹W
!
1m
a
W50,in R
N
\
$
Y
1
,...,Y
M
%
,~C8!
and w(x)[W(x) in R
N
\V
Å
by ~C3!and ~C4!.Then,by the
unique continuation theory,
27
we have
w
~
x
!
[W
~
x
!
,in R
N
\
$
y
1
,...,y
m
,Y
1
,...,Y
M
%
.~C9!
Now,since
2"
~
Dw
!
1m
a
w5
(
i51
m
a
i
d
~
x2y
i
!
,in R
N
,~C10!
2"
~
DW
!
1m
a
W5
(
j 51
M
A
j
d
~
x2y
i
!
,in R
N
,~C11!
2296 Wang,Li,and Jiang:Uniqueness theorems in bioluminescence tomography 2296
Medical Physics,Vol.31,No.8,August 2004
and from ~C9!,
E
V
W
~
y
!
L
@
u
#~
y
!
dy5
E
V
w
~
y
!
L
@
u
#~
y
!
dy,~C12!
which implies that
(
i51
m
a
i
u
~
y
i
!
5
(
j 51
M
A
j
u
~
y
i
!
,~C13!
for all rapidly decaying C
2
functions,it follows that w and W
must possess the same singular point set,i.e.,
$
y
1
,...,y
m
%
5
$
Y
1
,...,Y
M
%
and their weights at each singular point be the
same,which ®nishes this proof.h
APPENDIX D:PROOF OF THEOREM IV.3
Lemma D.1:For any given source q
0
PL
2
(V) and any
nontrivial C
2
patch P,G,the solution u
0
of the forward
model is uniquely determined by the boundary values u
0
u
P
of
u
0
and ]u
0
/]n
u
P
of ]u
0
/]non P.
Proof:Because D and m
a
can be smoothly extended
across P by our assumption,the conclusion follows easily
from the unique continuation theory.
27
h
Lemma D.2:For any constant D.0,m
a
>0,and any so
lution u
0
of 2"(Du
0
)1m
a
u
0
50 in B
R
(x
0
),we have
E
r
0
,
u
x2x
0
u
,r
1
u
0
~
x
!
dx5
S
E
r
0
r
1
v
N
r
N21
w
~
r
!
dr
D
u
0
~
x
0
!
,
~D1!
where 0<r
0
,r
1
,R,v
N
is the surface area of the unit
sphere in R
N
,and w(r) is the unique positive radial solution
of
2DDw1m
a
w52D
S
w
9
1
N21
r
w
8
D
1m
a
w50,~D2!
with w~0!51 and w
8
~0!50.
Proof:De®ne
u
Å
~
r
!
[
1
v
N
r
N21
E
]B
r
~
x
0
!
u
0
~
x
!
ds
x
;
we have
u
Å
~
0
!
5 lim
r!01
u
Å
~
r
!
5u
~
x
0
!
,~D3!
and
2D
S
u
Å
9
1
N21
r
u
Å
8
D
1m
a
u
Å
50,~D4!
with u
Å
(0)5u(x
0
) and u
Å
8
(0)50.Hence,by the uniqueness
of the initial value problem,
u
Å
~
r
!
5u
0
~
x
0
!
w
~
r
!
.~D5!
Now,
E
r
0
,
u
x2x
0
u
,r
1
u
0
~
x
!
dx5
E
r
0
r
1
dr
E
u
x2x
0
u
5r
u
0
~
x
!
dS
x
~D6!
5
E
r
0
r
1
v
N
r
N21
u
Å
~
r
!
dr
5u
~
x
0
!
E
r
0
r
1
v
N
r
N21
w
~
r
!
dr.~D7!
h
Remark D.3:We have,for m
a
50,
w
~
r
!
51,~D8!
and for m
a
.0,
w
~
r
!
5
5
BesselI
S
0,
A
m
a
D
r
D
,N52,
sinh
S
A
m
a
D
r
D
A
m
a
D
r
,N53,
~D9!
where BesselI is a Bessel function of the ®rst kind.
Note that w(r)[1 for m
a
50 is equivalent to the mean
value theorem for harmonic functions.
Now,we present the additional conditions on V,D,m
a
and q
0
for Theorem IV.3.
C4
*
:D and m
a
are piecewise constant in the sense that
there exist constants D
1
,...,D
I
.0 and m
1
,...,m
I
>0 such that D(x)[D
i
and m
a
(x)[m
i
,;xPV
i
.
Note that condition C4
*
is a special case of condition C4.
C5:There exists a C
2
patch P
0
of G;
C6:For each subdomain V
m
,there exists a sequence of
indices i
1
,i
2
,...,i
k
P
@
1,I
#
with the following con
nectivity property:the intersection P
0
ùG
i
1
contains
a smooth C
2
open patch and P
i
j
ùP
i
j 11
contains a
smooth C
2
open patch,for j 51,...,k21,and V
i
k
5V
m
;
C7:q
0
is of the following form:
q
0
~
y
!
5
(
i51
m
l
i
x
B
r
0
i
,r
1
i
~
x
i
!
,~D10!
where each l
i
,i51,...,I,is constant,and each source
support B
r
0
i
,r
1
i
(x
i
),,V
k
for some kP
@
1,I
#
.This
means that B
r
0
i
,r
1
i
(x
i
) is compactly included in V
k
;
that is,there is a positive distance from B
r
0
i
,r
1
i
(x
i
) to
the boundary G
k
of V
k
.
Theorem D.4:Assume that the conditions C1±C4
*
,
C5±C7 hold.If q
1
(y)5
(
i51
m
l
i
x
B
r
0
i
,r
1
i
(x
i
) and q
2
(y)
5
(
j 51
M
L
j
x
B
R
0
i
,R
1
i
(X
i
) are two solutions to the BLT(P
0
)
problem (20);then m5M and there exists a permutation tof
@
1,m
#
and a map C:
@
1,m
#
!
@
1,I
#
such that x
i
5X
t(i)
PV
C(i)
and
2297 Wang,Li,and Jiang:Uniqueness theorems in bioluminescence tomography 2297
Medical Physics,Vol.31,No.8,August 2004
l
i
E
r
0
i
r
1
i
r
N21
w
C
~
i
!
~
r
!
dr5L
t
~
i
!
E
R
0
t
~
i
!
R
1
t
~
i
!
r
N21
w
C
~
i
!
~
r
!
dr,
for i51,...,I,~D11!
where w
j
is the unique solution of
2D
j
S
w
j
9
1
N21
r
w
j
8
D
1m
j
w
j
50,~D12!
w
j
~
0
!
51,w
j
8
~
0
!
50.~D13!
Proof:Let u
1
and u
2
be the solutions to ~20!correspond
ing to q
1
and q
2
,respectively.Let w5u
1
2u
2
;then w is a
solution of
2"
~
Dw
!
1m
a
w5q
1
2q
2
,in V,~D14!
w
u
P
0
5D
]w
]
v
U
P
0
50.~D15!
Based on the fact that the support G of q
1
øq
2
does not
touch any part of G or G
i
,for i51,...,I,in the following we
will show that w
u
G
i
5D(]w/]n)
u
G
i
50,i51,...,I.
First,let V
j
be any subdomain such that P
0
ùG
j
contains
a C
2
open patch,we have
2"
~
D
j
w
!
1m
j
w50,in V
j
\G,~D16!
w
u
P
0
ùG
j
5D
j
]w
]n
U
P
0
ùG
j
50.~D17!
Then,there exists an open peripheral narrow band B
j
of G
j
:
B
j
5
$
xPV
j
\G:dist(x,]V
j
),e
%
for a suf®ciently small
e.0.Here,dist~,!denotes a distance function.Clearly,B
j
can be covered from P
0
ùV
j
by overlapped open balls in
V
j
\G.Then,our Lemma D.1 implies that w
u
B
j
[0.Hence,
w
u
G
j
5D
j
(]w/]n)
u
G
j
50.
Next,let us deal with other subdomains.Let V
k
be any
adjacent subdomain such that G
j
ùG
k
contains a C
2
open
patch P
jk
.Then,we have
25
w
u
P
jk
5w
u
P
jk
and D
k
]w
]n
k
U
P
jk
1D
j
]w
]n
j
U
P
jk
50,~D18!
where n
k
and n
j
are the exterior normals of G
k
and G
j
,
respectively.That is,w satis®es
2"
~
D
k
w
!
1m
k
w50,in V
k
\G,~D19!
w
u
P
jk
5D
k
]w
]n
U
P
jk
50.~D20!
Similarly,we can conclude that there is an open band B
k
around G
k
in V
k
\G such that w
u
B
k
[0.Our connectivity as
sumption C6 guarantees that the above propagation proce
dure works for all the subdomains.
Now,we can proceed with the rest of the subdomains and
show that the conclusion of the theorem holds for each of
those subdomains.Therefore,without loss of generality we
may now assume that G,,V
1
.Let F
1
(x,y) be the funda
mental solution of 2"(D
1
u
0
)1m
1
u
0
with the Dirichlet
condition at`,that is,
2"D
1
F
1
~
x,y
!
1m
1
F
1
~
x,y
!
5d
~
x2y
!
,yPR
N
.
~D21!
Then,according to ~A29!,we have
E
V
1
F
1
~
x,y
!
q
1
~
y
!
2q
2
~
y
!
dy50,;xPR
N
\V
Å
1
.
~D22!
Also,we have
2D
1
D
y
F
1
~
x,y
!
1m
1
F
1
~
x,y
!
50,;xPR
N
\V
Å
1
.~D23!
For xPR
N
,let us de®ne
W
~
x
!
5
E
V
1
F
1
~
x,y
!
q
1
~
y
!
2q
2
~
y
!
dy.~D24!
Lemma D.2 implies that,for xPR
N
\V
Å
1
,
W
~
x
!
5
E
V
1
F
1
~
x,y
!
F
(
i51
m
l
i
x
B
r
0
i
,r
1
i
~
x
i
!
2
(
J51
M
Lx
B
R
0
j
,R
1
j
~
X
j
!
G
dy
5
(
i51
m
l
i
E
r
0
i
<
u
y2x
i
u
<r
1
i
F
1
~
x,y
!
dy
2
(
j 51
M
L
j
E
R
0
j
<
u
y2X
j
u
<R
1
j
F
1
~
x,y
!
dy
5
(
i51
m
l
i
S
E
r
0
i
r
1
i
w
n
r
N21
w
1
~
r
!
dr
D
F
1
~
x,x
i
!
2
(
j 51
M
L
j
S
E
R
0
j
R
1
j
w
n
r
N21
w
1
~
r
!
dr
D
F
1
~
x,X
j
!
50.
~D25!
Since
2D
1
DW1m
1
W50,on R
N
\
H
ø
i51
m
$
x
i
%
ø ø
j 51
M
$
X
j
%
J
,
~D26!
the unique continuation theory
27
implies that W[0 in
R
N
\
$
ø
i51
m
$
x
i
%
øø
j 51
M
$
X
j
%%
,which immediately leads to our
theorem.h
Remark D.5:Actually,the solid/hollow ball sources as
sumed in Theorem D.4 can be generalized to any radial
weight functions with radial supports,such as
2298 Wang,Li,and Jiang:Uniqueness theorems in bioluminescence tomography 2298
Medical Physics,Vol.31,No.8,August 2004
q
0
~
y
!
5
(
i51
m
r
i
~
y2x
!
x
B
r
0
i
,r
1
i
~
x
i
!
~
y
!
,~D27!
where B
r
0
i
,r
1
i
(x
i
),,V
k
for some kP
@
1,I
#
,and r
i
denotes
any radial distribution.The conclusion of Theorem D.4 can
be similarly derived but the proof is omitted here for brevity.
a!
Electronic mail:gewang@ieee.org
b!
Electronic mail:yili@uiowa.edu
c!
Electronic mail:jiangm@math.pku.edu.cn
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