Numerical approximation of bioluminescence tomography ... - CiteSeerX

Numerical approximation of bioluminescence tomography based on a new formulation Xiaoliang Cheng ([email protected]) Department of Mathematics, Zhejiang University, Hangzhou 310027, P.R. China.

Rongfang Gong ([email protected]) Department of Mathematics, Zhejiang University, Hangzhou 310027, P.R. China.

Weimin Han ([email protected]) Department of Mathematics, University of Iowa, Iowa City, IA 52242, U.S.A. Abstract. Bioluminescence tomography (BLT) is a promising new method in biomedical imaging. The BLT problem is an ill-posed inverse source problem, usually studied through a regularization technique. In this paper, a new approach is proposed for solving the BLT problem based on the use of an adjoint equation. Numerical examples show that the new formulation provides accurate solutions. Keywords: Bioluminescence tomography, inverse problem, finite element methods

1. Introduction Molecular imaging is a rapidly developing biomedical imaging technique for studying physiological and pathological processes in vivo at the cellular and molecular levels; see e.g. [1–5] and references therein. The goal of molecular imaging is to depict non-invasively cellular and molecular processes in vivo sensitively; applications include the monitoring of multiple molecular events, cell trafficking and targeting etc., in tumorigenesis studies, cancer diagnosis, metastasis detection, drug discovery and development, gene therapies, and orthopedic research [3, 6–8]. In general, molecular imaging is mainly based on three technologies: nuclear imaging [9, 10], magnetic resonance imaging (MRI) [11, 12], and optical imaging [13, 14]. Based on the three technologies, there have been numerous instrumentations. For instance, nuclear imaging includes positron emission tomography (PET) [15–17] and single photon emission computed tomography (SPECT) [18], while optical imaging mainly involves fluorescence molecular tomography (FMT) [14, 19] and bioluminescent imaging (BLI) [20–22]. The difference between FMT and BLI can be found in [23]. Different technologies can also be used in a combined system [25]. BLT has received considerable attention in recent years because of its unique advantages regarding sensitivity and specificity. A major issue in BLT is the determination of the distribution of a bioluminesc 2008 Kluwer Academic Publishers. Printed in the Netherlands.

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2

X. Cheng, R. Gong, W. Han

cent source. With the introduction of BLT, a bioluminescent source distribution inside a living small animal can be localized and quantified in three dimensions. Without BLT, bioluminescence imaging is primarily qualitative. With BLT, quantitative and localized analysis of a bioluminescent source distribution becomes feasible in a living subject [24–28]. In BLT, an internal bioluminescent source is constructed from a measured bioluminescent signal on the external surface of a small animal. The problem of determining the photon density on the small animal surface from the bioluminescent source distribution inside the animal requires accurate representation of photon transport in biological tissue. Generally, the bioluminescent photon propagation in biological tissue can be well described by either a radiative transfer equation (RTE) or a Monte Carlo model. However, at present, because transmission of the bioluminescent photons through the biological tissue is subject to both scattering and absorption, neither description is computationally feasible in most practical applications. In practical performance, approximation of the diffusion equation of RTE is adopted if scattering is dominant over absorption in the process of propagation of light inside a small animal [29]. The remainder of the paper is organized as follows. In the following section, a mathematical statement of the BLT problem is given. In Section 3, we introduce a new formulation for BLT, based on the use of an adjoint equation. In Section 4, we discuss the discretization of the new formulation by FEMs. Several numerical examples are presented in Section 5 to demonstrate the feasibility of our method. Some concluding remarks are given in the last section.

2. Problem statement Let Ω ⊂ R3 with boundary Γ of class C 0,1 for the domain occupied by a biological medium. We denote by u0 = u0 (x, θ, t) the light flux in direction θ ∈ S 2 (S 2 : the unit sphere) at x ∈ Ω and t > 0, and by p0 = p0 (x, θ, t) the internal light source. Then light transportation in the biological tissue can be described by RTE [29, 30] as follows 1 ∂u0 + θ · ∇x u0 + µu = µs c ∂t

Z

S2

k(θ · θ ′ )u0 (x, θ ′ , t)dθ ′ + p0

in Ω, (1)

where c stands for the photon speed, µ = µa + µs , µa , µs are absorption and scattering coefficients. The scattering kernel function k ≥ 0 such R that S 2 k(θ · θ ′ )dθ ′ = 1. The initial boundary conditions for equation (1) are u0 (x, θ, 0) = 0, x ∈ Ω, θ ∈ S 2 ,

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Numerical approximation of BLT based on a new formulation

3

and t > 0, x ∈ Γ, θ ∈ S 2 such that ν(x) · θ ≤ 0,

u0 (x, θ, t) = g− (x, θ, t),

respectively, where ∂ν is the outward normal differentiation operator on the boundary Γ. To reconstruct the internal light source p0 , we impose measurements of the outgoing radiation on the boundary Γ: g(x, t) =

Z

S2

ν(x) · θu0 (x, θ, t) dθ,

t > 0, x ∈ Γ.

Because the RTE is highly dimensional, it presents a serious challenge for accurate numerical simulations given the current level of development of computer software and hardware. However, because the mean-free path of the photon is between 500nm and 1000nm in biological tissues, which is very small compared to a typical object in this context, the scattering dominates transport [29]. Moreover, by noticing that the internal bioluminescence distribution induced by reporter genes is relatively stable, we neglect the time dependence. The diffusion approximation of the RTE (1) is (see [31, 32] for detail) −div(D∇u) + µa u = p

in Ω.

(2)

Here u = u(x) and p = p(x), defined as u(x) =

1 4π

Z

S2

u0 (x, θ) dθ,

p(x) =

1 4π

Z

S2

p0 (x, θ) dθ,

are the average light flux and average light source distribution in any ′ ′ direction respectively, D = [3(µa +µs )]−1 , µs = (1−k)µs is the reduced scattering coefficient with k=

1 4π

Z

S2

θ · θ ′ k(θ · θ ′ )dθ ′ .

Accordingly, the boundary condition reduces to u + 2D∂ν u = g− ≡ g2

on Γ

(3)

while the boundary measurement reduces to g = −D∂ν u

on Γ.

(4)

From (3) and (4), we obtain a third boundary condition for equation (2): u = g− + 2g ≡ g1 on Γ. (5) Note that only two of the three boundary conditions (3), (4) and (5) are independent. As pointed out in [26], to determine the source

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function p, we may associate one of the above three boundary conditions (3), (4) and (5) with the differential equation (2) to form a boundary-value problem, while choosing one of the remaining boundary conditions to form an inverse problem for p. In particular, in [26], the inverse problem was discussed for the boundary value problem (2) and (3) with the measurement matching for (5): Problem 1. Given D > 0, µa ≥ 0, g1 and g2 , suitably smooth, find a source function p such that the solution of the boundary-value problem

−div(D∇u) + µa u = pχΩ0 u + 2D∂ν u = g2 on Γ

in Ω

(6)

satisfies u = g1

on Γ.

(7)

Here Ω0 is a measurable subset of Ω, known as the permissible region, χΩ0 is the characteristic function of Ω0 , i.e., its value is 1 in Ω0 and is 0 outside Ω0 . We note that, even with this simplification, it still remains to develop efficient ways of simulating diffusion based BLT. In this regard, we mention a few references [26–28, 33–36]. Next, we discuss BLT problem based on Problem 1. The purpose of this paper is to introduce a new formulation as the basis of a potentially more effective numerical method to solve the BLT problem. Different from conventional ways which often involve minimizing a cost functional subject to a system of differential equations, we convert the determination of a source function p to solving a system of equations. In this way, the BLT problem is solved through linear systems of elliptic partial differential equations without the need of an optimization process. As will be seen later, our new method provides good numerical results. In what follows, we let Ω be a domain in Rd (d = 3 for applications) with the boundary Γ. Moreover, for our future needs, we introduce a few symbols. For G = Ω, Ω0 or Γ, and s ≥ 0, we denote by H s (G) the standard Sobolev space associated with inner product (·, ·)s,G and norm k · ks,G , H 0 (G) = L2 (G). Denote V = H 1 (Ω), V0 = H01 (Ω) and Q = L2 (Ω0 ). For any q ∈ Q, we denote by u = u(q) ∈ V the solution of the problem Z

Ω

(D∇u·∇v+µa u v)dx+

1 2

Z

Γ

u vds =

Z

q vdx+ Ω0

1 2

Z

Γ

g2 vds

∀ v ∈ V.

(8) Note that this is a weak formulation of the boundary value problem defined by (6) with p replaced by q. Suppose the admissible source function p belongs to a closed convex subset Qad of the space Q.

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Problem 1 is usually studied via a least-squares optimization approach. Problem 2. Find p ∈ Qad such that J(p) = inf J(q), q∈Qad

where J(q) =

1 ku(q) − g1 k20,Γ . 2

As noted in [26], Problem 1 or 2 is ill-posed. In general, there are infinitely many solutions. When the form of the source function is prespecified, there is no solution if the data are inconsistent. Also, the source function does not depend continuously on the data. To circumvent these difficulties, a Tikhonov-type regularization version of the problem is introduced and numerically solved. Define a functional 1 ε ku(q) − g1 k20,Γ + kqk20,Ω0 , ε ≥ 0. 2 2 Then for BLT reconstruction, we study the following constrained optimization problem. Jε (q) =

Problem 3. Find pε ∈ Qad such that Jε (pε ) = inf Jε (q). q∈Qad

By Theorems 3.2 and 3.3 in [26], this reformulation is well-posed for ε > 0. Moreover, it is shown there that the reformulated problem with ε > 0 leads to stable and convergent numerical solutions. 3. A new formulation for BLT Our main objective in this section is to introduce a new formulation for studying the BLT problem. Again, let Ω ⊂ Rd be an open and bounded convex set with boundary Γ. It is shown in [37, Section 1.2] that the boundary of an open, bounded and convex set is Lipschitz continuous; consequently, the unit outward normal vector exists a.e. on Γ. Assume the coefficients D, µa and measurements g1 , g2 belongs to space L∞ (Ω) such that D ≥ D0 a.e. in Ω for some constant D0 > 0, µa ≥ 0 a.e. in Ω, and µa > 0 a.e. in a subset of Ω with positive measure. For the derivation of our new formulation for BLT, let u = u(q) be the solution of problem

−div(D∇u) + µa u = qχΩ0 in Ω u + 2D∂ν u = g2 on Γ.

(9)

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e the solutions of problems Denote by u∗ = u(pε ) and u

and

−div(D∇u∗ ) + µa u∗ = pε χΩ0 in Ω u∗ + 2D∂ν u∗ = g2 on Γ

(10)

e ) + µa u e = qχΩ0 in Ω −div(D∇u e + 2D∂ν u e=0 u on Γ

(11)

respectively. Together with the problem (9), we have, for any q ∈ Qad , e. Then, for any t ∈ R and q ∈ Qad , u(pε + t q) = u∗ + t u 1 Jε (pε + t q) − Jε (pε ) = 2

Z

Z

ε [u(pε + t q) − g1 ] ds + (pε + t q)2 dx 2 Ω0 Γ Z Z ε 1 ∗ 2 (u − g1 ) ds − p2 dx − 2 Γ 2 Ω0 ε

= t

2

Z

+

Z

eds + ε (u∗ − g1 ) u

Γ t2

2

Z

Γ

2

e ds + ε u

Z

Ω0

pε qdx

2

q dx

Ω0

and the Gateaux derivative of Jε at pε in direction q ∈ Qad is Jε (pε + t q) − Jε (pε ) t→0 t Z Z

Jε′ (pε ) q = lim =

Γ

eds + ε (u∗ − g1 ) u

pε qdx.

Ω0

(12)

Let w be the solution of the adjoint problem

−div(D∇w) + µa w = 0 in Ω w + 2D∂ν w = u∗ − g1 on Γ.

(13)

e, integrate over Ω and integrate by parts to get Multiply (13) by u Z

Ω

1 2

e + µa w u e) dx + (D∇w · ∇u

Z

Similarly, from (11), we have Z

Ω

Γ

eds = wu

e + µa w u e) dx + (D∇w · ∇u

1 2

Z

Γ

Combining (14), (15) and (12), we obtain Jε′ (pε ) q

=

Z

Ω0

1 2

Z

Γ

eds. (u∗ − g1 ) u

e wds = u

(2w + εpε ) qdx.

Z

q wdx.

(14)

(15)

Ω0

(16)

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Because Qad is a convex and closed subset of space Q, the following first order necessary and sufficient condition [38, 39] of the solution pε ∈ Qad of Problem 3 holds: Jε′ (pε ) (q − pε ) ≥ 0 or

Z

Ω0

∀ q ∈ Qad

(2w + εpε ) (q − pε )dx ≥ 0 ∀ q ∈ Qad .

Therefore, pε is the projection of −2w/ε on Qad with respect to the L2 (Ω0 ) inner product ([38, Section 5.3]). In the case where Qad consists of the non-negatively valued L2 (Ω0 ) functions, we have 2 pε = max{− w, 0} in Ω0 . ε If Qad = Q, then

2 pε = − w in Ω0 . (17) ε Also, (17) holds if it happens that −2w/ε ∈ Qad . Consequently, we consider the following equations and boundary conditions:   −div(D∇u∗ ) + µa u∗ + 2ε wχΩ0 = 0 in Ω     in Ω  −div(D∇w) + µa w = 0

p = −2w

(18)

in Ω0 on Γ on Γ.

ε ε   ∗ + 2D∂ u∗ = g  u ν 2    ∗

w + 2D∂ν w = u − g1

A new formulation in studying the BLT problem is the following. Problem 4. Find (u∗ , w) ∈ V × V such that  ∗  a(u , v) + b1 (w, v) = f1 (v)

∀ v ∈ V, b2 (u∗ , v) + a(w, v) = f2 (v) ∀ v ∈ V,  pε = − 2ε w in Ω0 ,

where Z

1 a(u, v) = (D∇u·∇v+µa u v)dx+ 2 Ω 1 b2 (u, v) = − 2

Z

Γ

u vds,

1 f1 (v) = 2

Z

u vds,

Γ

Z

Γ

g2 vds,

b1 (u, v) =

(19)

Z

1 f2 (v) = − 2

Ω0

Z

Γ

2 u vdx, ε g1 vds.

In this paper, we focus on the numerical performance of the new formulation in solving the BLT problem.

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4. Finite element approximation In this section, we consider the problem of approximating the solution of Problem 4 with FEMs. For simplicity of statement, assume both Ω ⊂ Rd and Ω0 ⊂ Ω are polyhedral convex set. We note that Ω and Ω0 can be any bounded open convex domains. Let {Th }h and {T0,H }H be regular families of finite element partitions of Ω and Ω0 , respectively. For any triangulation Th = {K}, T0,H = {T }, define the finite element spaces k V h = {v ∈ C(Ω) | v |K ∈ PK or QkK , ∀ K ∈ Th }, QH = {q ∈ Q | q|T ∈ PTt or QtT ∀ T ∈ T0,H }, s and Qs denote spaces of polynomials over the set G of where PG G total degree and individual degree less than or equal to s, respectively. Denote by ΠH the orthogonal projection operator from space Q onto space QH :

(ΠH p, q H )0,Ω0 = (p, q H )0,Ω0

∀ p ∈ Q, q H ∈ QH .

(20)

Then we have the approximation to continuous Problem 4 as follows. Problem 5. Find (u∗h , wh ) ∈ V h × V h such that  ∗   a(uh , vh ) + b1 (wh , vh ) = f1 (vh )  

∀ v ∈ V h, ∈ V h,

b2 (u∗h , vh ) + a(wh , vh ) = f2 (vh ) ∀ v 2 H 2 H pH ε,h = − ε wh = − ε Π wh in Ω0 .

(21)

Let ϕi (x) ∈ V h , i = 1, 2, · · ·, n (n is the number of the nodes of triangulation Th ), be the node basis functions of finite element space V h associated with grid nodes xi . Then the solution u∗h ∈ V h can be Pn ∗ written as uh = i=1 u∗i ϕi , where u∗i denotes the value of function u∗ at gridPnode xi , i.e. u∗i = u∗ (xi ). Similarly, wh can be expanded by wh = ni=1 wi ϕi with wi = w(xi ). Moreover, let φs (x) ∈ QH , s = 1, 2, · · ·, N (N is the number of the nodes of triangulation T0,H ), be the node basis functions of finite element space QH associated with grid nodes xs , and denote ϕ0,l , l = 1, 2, · · ·, N0 , for those basis functions associated with nodes x0,l contained in Ω0 . Then Problem 5 is equivalent to the following. Problem 6. Solve linear system K Y = F,

(22)

and compute the approximation light source function p by N

pH ε,h = −

N X 0 2X ( wl ϕ0,l (xs ))φs in Ω0 , ε s=1 l=1

(23)

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where K,

A B1 B2 A

9

, Y , (U t , W t )t , F , (F1t , F2t )t ,

U = (u∗1 , u∗2 , · · ·, u∗n )t , W = (w1 , w2 , · · ·, wn )t , Fk = (fk1 , fk2 , · · ·, fkn )t , fkj = fk (ϕj ), A = (aji ), aji = a(ϕi , ϕj ), B1 = (bji 1 ), B2 = (bji 2 ),

bji 1 = b1 (ϕi , ϕj ),

bji 2 = b2 (ϕi , ϕj ), i, j = 1, 2, · · ·, n, k = 1, 2,

and (·)t is the transposition of (·). We note that if Th = {K} and T0,H = {T } are consistent, then H = h, N = N0 and (23) reduces to phε,h

N 2X =− ws φs in Ω0 . ε s=1

(24)

5. Numerical examples In this section, we present numerical results on two model problems. The main purpose is to demonstrate the feasibility of our new formulation for BLT problem. Let Ω be the problem domain and Ω0 the permissible region. Assume the absorption coefficient µa ≡ 0.02 and the reduced scattering coefficient µ′s ≡ 1.00 in Ω. We use uniform square partitions of the regions Ω and Ω0 with mesh parameters h = H, where h and H are the maximal diameters of the elements in the partitions Th and T0,H , respectively. For finite element spaces V h and Qh , we use continuous piecewise bilinear functions and piecewise constant functions corresponding to the partitions Th and T0,h . From the boundary value of a finite element solution of BVP (2)–(3) for a small enough meshsize we compute the function g1 ; in our experiment, we choose 1/512 for this small meshsize. In the following text, we compute the approximate source function phε,h for different grid parameter h and different regularization parameter ε. 5.1. Model 1 In this model, we set Ω = (0, 1) × (0, 1), Ω0 = (0.5, 0.75) × (0.5, 0.75). We take p ≡ 1 pW for the true light source in Ω0 and set g2 ≡ 0 on the boundary Γ = ∂Ω. Denote the error E = phε,h − p. We first show the performance of the solution phε,h of BLT which is obtained from our new formulation when the data g1 and g2 are

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10


1.003 1.01

1.002

1.0075

1.001

1.005

1

pε,h

1.0025

0.999

h

1 0.998 0.9975 0.997 0.995 0.996 0.9925 0.995

0.99 0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.45 0.8

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.994 0.993

y

x

Figure 1. Model 1: Reconstructed light source function phε,h when meshsize h = 1/64 and regularization parameter ε=1e-5.

noise-free. For different meshsize and regularization parameter, we reconstruct light source functions phε,h and plot one of them in Figure 1 for h = 1/64 and ε=1e-5. Tables I–II exhibit the dependence of the error in the approximate light source function phε,h on the meshsize and the regularization parameter, respectively. Note that k · k∞ and k · k0 are the L∞ and L2 norms, respectively. We conclude from them that the accuracy of discrete solution improves as h or ε gets smaller. More behaviors of the discrete solution can be seen from Figures 2–3. Figure 2 gives the error of the approximate solution in the norm L∞ versus (vs.) the meshsize. The red, blue, black, green and yellow curves correspond to parameter ε=1e-1, 1e-3, 1e-5, 1e-7, and 1e-9, respectively. It appears ε = 1e − 5 is a best choice (the black curve in Figure 2). Figure 3 plots the error vs. regularization parameter. These figures suggest values of h and ε should match in order to achieve optimal numerical performance. Table I. Model 1: Error in approximate light source function phε,h when regularization parameter ε=1e-5 h

1/4

1/8

1/16

1/32

1/64

1/128

kEk∞ kEk0

4.1211e+0 1.0303e+0

3.4971e-1 4.7054e-2

5.9186e-2 5.2836e-3

2.1386e-2 1.2056e-3

7.1987e-3 3.8184e-4

2.8881e-3 2.0781e-4

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5 ε=1e−1 ε=1e−3 ε=1e−5 ε=1e−7 ε=1e−9

4 3

log10||E||∞

2 1 0 −1 −2 −3

2

3

4

5

6

7

−log2 h

Figure 2. Model 1: Error in approximate light source function phε,h vs. meshsize h in scale −log2 and log10 respectively for different values of regularization parameter ε. Table II. Model 1: Error in approximate light source function phε,h when meshsize h = 1/64 ε

1e-1

1e-2

1e-3

1e-4

1e-5

kEk∞ kEk0

6.2449e-1 1.5149e-1

1.7197e-1 3.3547e-2

4.5229e-2 4.9663e-3

1.0708e-2 9.1736e-4

7.1987e-3 3.8184e-4

To examine the stability of our modality, we introduce random noise on observation data g1 with three noise level δ = 1%, 10%, and 20%. For each noise level δ, we compute the discrete solution phε,h ten times and use the average error of the corresponding solutions as the error of the approximate solution associated with δ. Figure 4 plots the L∞ – norm of the error of the approximate solution vs. meshsize h for four noise levels 0, 1%, 10%, and 20%. We observe that higher noise level leads to bigger solution error. The same conclusion can be drawn from Figure 5 for the error of the approximate solution in L∞ –norm vs. regularization parameter ε. We comment that these numerical results demonstrate clearly that the numerical solution is rather stable with respect to the perturbation in the measurement data.

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5 h=1/4 h=1/8 h=1/16 h=1/32 h=1/64 h=1/128

4 3

log10||E||∞

2 1 0 −1 −2 −3

1

2

3

4

5 −log10ε

6

7

8

9

Figure 3. Model 1: Error in approximate light source function phε,h vs. regularization parameter ε in scale −log10 and log 10 respectively for different values of meshsize h.

5 h=1/4 h=1/8 h=1/16 h=1/32 h=1/64 h=1/128

4 3

log10||E||∞

2 1 0 −1 −2 −3

1

2

3

4

5 −log10ε

6

7

8

9

Figure 4. Model 1: Error in approximate light source function phε,h vs. meshsize h in scale −log2 and log10 respectively for different values of noise level δ when regularization parameter ε =1e-5.

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−0.2 δ=0 δ=1% δ=10% δ=20%

−0.4 −0.6

log10||E||∞

−0.8 −1 −1.2 −1.4 −1.6 −1.8 −2 −2.2

1

2

3

4

5 −log10ε

6

7

8

9

Figure 5. Model 1: Error in approximate light source function phε,h vs. regularization parameter ε in scale −log10 and log10 respectively for different values of noise level δ when meshsize h = 1/64.

5.2. Model 2 In the second model, we assume the permissible region Ω0 of light source function is the union of k disjoint subdomains Ω0i , i = 1, 2, · · · , k. In our simulation, we take k = 4, Ω01 = (0.25, 0.375) × (0.25, 0.375), Ω02 = (0.625, 0.75) × (0.25, 0.375), Ω03 = (0.25, 0.375) × (0.625, 0.75) and Ω04 = (0.625, 0.75) × (0.625, 0.75). The true light source function is piecewise constant: p(x) = pi in Ω0i , i = 1, 2, 3, 4; we take p1 = 1.0, p2 = 1.3, p3 = 1.0 and p4 = 1.3. Again we let H = h and g2 ≡ 0 on the boundary Γ = ∂Ω. Similar to Model 1, we consider boundary measurement g1 with or without noise, obtaining similar results. We show reconstructed light source phε,h in Figure 6. Tables III and IV give the dependence of the error in both L∞ and L2 –norms in the approximate light source function on the meshsize h and regularization parameter ε. Numerical results corresponding to noise-free data and noise data are reported in Figures 7–8 and in Figures 9–10 respectively. Finally, we conclude this section by emphasizing that the numerical values of − 2ε whH in Ω0 are always non-negative in our numerical experiments which justifies the use of (17) in these examples.

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Figure 6. Model 2: Reconstructed light source function phε,h when meshsize h = 1/128 and parameter ε=1e-7.

3.5 ε=1e−1 ε=1e−3 ε=1e−5 ε=1e−7 ε=1e−9

3 2.5

log10||E||∞

2 1.5 1 0.5 0 −0.5 −1

3

3.5

4

4.5

5 −log2 h

5.5

6

6.5

7

Figure 7. Model 2: Error in approximate light source function phε,h vs. meshsize h in scale −log2 and log10 respectively for different values of regularization parameter ε.

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3.5 h=1/4 h=1/8 h=1/16 h=1/32 h=1/64

3 2.5

log10||E||∞

2 1.5 1 0.5 0 −0.5 −1

1

2

3

4

5 −log10ε

6

7

8

9

Figure 8. Model 2: Error in approximate light source function phε,h vs. regularization parameter ε in scale −log10 and log 10 respectively for different values of meshsize h.

1.5 δ=0 δ=1% δ=10% δ=20%

log10||E||∞

1

0.5

0

−0.5

−1

3

3.5

4

4.5

5 −log2 h

5.5

6

6.5

7

Figure 9. Model 2: Error in approximate light source function phε,h vs. meshsize h in scale −log2 and log10 respectively for different values of noise level δ when regularization parameter ε=1e-7.

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16


Table III. Model 2: Error in approximate light source function phε,h when regularization parameter ε=1e-10 h

1/8

1/16

1/32

1/64

1/128

kEk∞ kEk0

1.1511e+1 1.4389e+0

1.9453e+0 1.4762e-1

4.9522e-1 2.8379e-2

2.1376e-1 9.1141e-3

1.4795e-1 6.9120e-3

Table IV. Model 2: Error in approximate light source function phε,h when meshsize h = 1/128 ε

1e-1

1e-3

1e-5

1e-7

1e-9

kEk∞ kEk0

2.8833e+0 3.5897e-1

2.6776e-1 1.9108e-2

2.0564e-1 1.1957e-2

1.4795e-1 6.9120e-3

6.7984e-1 1.8558e-2

6. Conclusion Bioluminescence tomography (BLT) is a promising new method and is attracting more and more attention in the community of biomedical imaging. One major challenge in this area is to develop efficient and effective numerical methods for solving the BLT problem. The aim of this paper is to explore an efficient method for evaluating a good approximation to the light source of the BLT problem. Because of the ill-posedness of the BLT, we adopt a Tikhonov regularization. The conventional methods for the BLT problem are to treat it as a PDE constrained optimization problem, Problem 3. Consequently, their numerical simulations are usually handled by an expensive iterative procedure in which, at each iteration step, one needs to solve the BVP (8). In contrast, with our method, one only needs to solve the boundary value problem (19) just once. In this paper, FEMs are applied to discretize PDEs for obtaining a approximate light source function. Numerical experiments indicate that our method performs well. Problem 4 and Problem 3 are identical only if pε from Problem 4 is non-negatively valued, for the relevant BLT application. While currently there are no theoretical results that would guarantee the non-negativity of pε for Problem 4, numerical results on numerous simulations we performed provide non-negative discrete light source functions. In case the non-negativity condition is violated for a numerical solution of Problem 4, the problem may be fixed through the use of a projection of the numerical solution of pε onto the admissible set Qad , and if necessary, a few iterative resolutions of the system (18) where the

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17


0.5 δ=0 δ=1% δ=10% δ=20%

log10||E||∞

0

−0.5

−1

1

2

3

4

5 −log10ε

6

7

8

9

Figure 10. Model 2: Error in approximate light source function phε,h vs. regularization parameter ε in scale −log10 and log10 respectively for different values of noise level δ when meshsize h = 1/128. 2 εw

term on the left hand side of the first equation of (18) is replaced by the projected numerical values. We are exploring the usefulness of this idea. The method proposed in this paper can be discussed for source function reconstruction problems of more general PDEs if the admissible set Qad for the source function has a convexity structure such as nonnegativity of function values. In this paper, we focus our attention on solving the BLT problem because BLT is currently an active research topic in biomedical imaging; our proposed method provides good numerical results in solving the BLT problem. Acknowledgements. We thank the referees for their valuable comments that lead to an improvement on the presentation of the paper.

References 1. 2.

Cherry SR (2004) In vivo molecular and genomic imaging: new challenges for imaging physics. Phys Med Biol 49:13–48 Levin CS (2005) Primer on molecular imaging technology. European Journal of Nuclear Medicine and Molecular Imaging 32:325–345

CG3.tex; 3/08/2008; 15:06; p.17

18 3. 4.

5.

6. 7. 8. 9. 10. 11. 12. 13.

14. 15.

16.

17.

18. 19.

20.

21.

22.


Massoud TF, Gambhir SS (2003) Molecular imaging in living subjects: seeing fundamental biological processes in a new light. Genes Dev 17:545–580 Ray P, Wu AM, Gambhir SS (2003) Optical bioluminescence and bositron emission tomography imaging of a novel fusion reporter gene in tumor xenografts of living mice. Cancer Res 63:1160–1165 Troy T, Mcmullen DJ, Sambucetti L, Rice B (2004) Quantitative comparison of the sensitivity of detection of fluorescent and bioluminscent reporters in animal models. Molecular Imaging 3:9–23 Bhaumik S, Gambhir SS (2002) Optical imaging of Renilla luciferase reporter gene expression in living mice. Proc Natl Acad Sci USA 99:377–382 Contag C, Bachmann MH (2002) Advances in bioluminescence imaging of gene expression. Annu Rev Biomed Eng 4:235–260 Kuckuk PM, Boskey AL (2006) Molecular imaging promotes progress in orthopedic research. Bone 39:965–977 Blasberg R (2002) Imaging gene expression and endogenous molecular processes: molecular imaging. J Cereb Blood Flow Metab 22:1157–1164 Gambhir SS (2002) Molecular imaging of cancer with positron emission tomography. Nat Rev Cancer 2:683–693 Genove G, DeMarco U, Xu H, Goins WF, Ahrens ET (2005) A new transgene reporter for in vivo magnetic resonance imaging. Nat Med 11:450–454 Weissleder R, Ntziachristos V (2003) Shedding light onto live molecular targets. Nat Med 9:123–128 Ntziachristos V, Ripoll J, Wang LV, Weissleder R (2005) Looking and listening to light: the evolution of whole-body photonic imaging. Nat Biotechnol 23:313– 320 Ntziachristos V, Tung CH, Bremer C, Weissleder R (2002) Fluorescence molecular tomography resolves protease activity in vivo. Nat Med 8:757–760 Douraghy A, Prout DL, Silverman RW, Chatziioannou AF (2006) Evaluation of scintillator afterglow for use in a combined optical and PET imaging tomograph. Nuclear Instruments and Methods in Physics Research A 569:557–562 Loo WTY, Tong JMK, Cheung MNB, Chow LWC (2006) A new predictive and prognostic marker (ATP bioluminescence and positron emission tomography) in vivo and in vitro for delivering adjuvant treatment plan to invasive breast tumor patients. Biomedicine & Pharmacotherapy 60:285–288 Serganova I, Moroz E, Moroz M, Pillarsetty N, BlasbergR (2006) Non-invasive molecular imaging and reporter genes. Central European Journal of Biology 1:88–123 Baumjohann D, Lutz MB (2006) Non-invasive imaging of dendritic cell migration in vivo. Immunobiology 211: 587–597 Cong AX, Wang G (2006) Multispectral bioluminescence tomography: methodolgy and simulation. International Journal of Biomedical Imaging 2006:Article ID 57614, 7 pages Li SH, Driessen W, Sullivan S, Jiang HB (2006) Bioluminescence tomography based on phantoms with different concentrations of bioluminescent cancer cells. J Opt A: Pure Appl Opt 8: 743–746 Chaudhari AJ, Darvas F, Bading JR, Moats RA, Conti PS, Smith DJ, Cherry SR, Leahy RM (2005) Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging. Phys Med Biol 50:5421—5441 Doyle TC, Burns SM, Contag CH (2004)In vivo bioluminescence imaging for integrated studies of infection. Cellular Microbiology 6:303–317

CG3.tex; 3/08/2008; 15:06; p.18


23. 24. 25.

26. 27.

28. 29. 30. 31. 32. 33. 34. 35. 36.

37. 38. 39.

19

Hielscher AH (2005) Optical tomographic imaging of small animals. Current Opinion in Biotechnology 16:79–88 Wang G, Hoffman EA, et al. (2003) Development of the first bioluminescent CT scanner. Radiology 229(P):566 Alexandrakis G, Rannou FR, Chatziioannou AF (2005) Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study. Physics in Medicine and Biology 50:4225–4241 Han W, Cong WX, Wang G (2006) Mathematical theory and numerical analysis of bioluminescence tomography. Inverse Problems 22:1659–1675 Han W, Cong WX, Wang G (2006) Mathematical study and numerical simulation of multispectral bioluminescence tomography. International Journal of Biomedical Imaging 2006:Article ID 54390, 10 pages Han W, Wang G (2007) Theoretial and numerical analysis on multispectral boluminescence tomography. IMA Journal of Applied Mathematics 72:67–85 Natterer F, Wabbeling F (2001) Mathematical Methods in Image Reconstruction. SIAM, Philadephia Arridge S R (1999) Optical tomography in medical imaging. Inverse Problems 15:R41–R93 Wang G, Li Y, Jiang M (2004) Uniqueness theorems in bioluminescence tomography. Med Phys 31:2289–2299 Han W, Wang G (2008) Bioluminescence tomography: biomedical background, mathematical theorey, and numerical approximation. J Comp Math 26:259–270 Cheng XL, Gong RF, Han W (2008) A new general mathematical framework for bioluminescence tomography. Comput Methods Appl Mech Engrg 197:524–535 Cong WX, Durairaj K, Wang LV, Wang G (2006) A Born-type approximation method for bioluminescence tomography. Med Phys 33:679–686 Cong WX, Wang G, Kumar D, etc. (2005) A practical reconstruction method for bioluminescence tomography. Optics Express 13:6756–6771 Lv YJ, Tian J, Cong WX, Wang G, Luo J, Yang W, Li H (2006) A multilevel adaptive finite element algorithm for bioluminescence tomography. Optics Express 14:8211–8223 Grisvard p (1985) Elliptic Problems in Nonsmooth Domains. Pitman, Boston Atkinson K, Han W (2005) Theoretical Numerical Analysis: A Functional Analysis Framework, 2nd edn. Springer-Verlag, New York Lions J.-L. (1971) Optimal Control of Systems Governed by Partial Differential Equation. Springer, Berlin

CG3.tex; 3/08/2008; 15:06; p.19

CG3.tex; 3/08/2008; 15:06; p.20