Uniqueness theorems in bioluminescence tomography

Ge Wang

a)

Bioluminescence Tomography Laboratory and CT/Micro-CT 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 well-known 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 ill-posedness 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 two-dimensional 2-D 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

2-D image,and is incapable of tomographic reconstruction

of internal features of interest,that is,the 3-D 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 3-D 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/micro-CT 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/

micro-CT scan,by image segmentation and optical property

mapping.That is,we can segment the CT/micro-CT 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 0094-2405Õ2004Õ318Õ2289Õ11Õ$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 engineer-friendly 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 three-dimensional 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 initial-boundary 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 mean-free 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 pre-requisite tomographic scan,such as a CT/

micro-CT 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 n-dimensional 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 d-function,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 star-shaped 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

higher-order 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 well-posedness 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 3-D 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 well-de®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 well-known 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.Dirichlet-to-Neumann 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 well-known Dirichlet-to-Neumann ~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 second-order 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

well-known:

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 sub-domains 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 sub-domain 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 sub-domains.Let V

k

be any

adjacent sub-domain 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 sub-domains.

Now,we can proceed with the rest of the sub-domains and

show that the conclusion of the theorem holds for each of

those sub-domains.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:ge-wang@ieee.org

b!

Electronic mail:yi-li@uiowa.edu

c!

Electronic mail:jiangm@math.pku.edu.cn

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