[10] W. Han, W.X. Cong, and G. Wang, Mathematical study and numerical ... Shen, W.X. Cong, S. Zhao, and G.W. Wei, Temperature-modulated bioluminescence.
To appear in a special issue of Applicable Analysis, 2009
Studies of a Mathematical Model for Temperature-modulated Bioluminescence Tomography Weimin Han1 , Haiou Shen2 , Kamran Kazmi3 , Wenxiang Cong2 , and Ge Wang2
Abstract. We introduce and study a mathematical model for temperature-modulated bioluminescence tomography (TBT). The model is capable of self-adjusting values of experimental parameters that are used in the formulation. Major theoretical results of the paper include: solution existence of the model, convergence of numerical solutions, an iterative scheme based on linearization, studies of the solution limiting behaviors when normalized total energy function and/or some or all the energy percentages in individual spectral bands are known exactly. Several numerical examples are included to illustrate the improvement of the accuracy of the reconstructed bioluminescent source distribution due to the employment of measurements from multiple temperature distributions.
1
Introduction
In the past several years, molecular imaging has been rapidly developed to study physiological and pathological processes in vivo at the cellular and molecular levels ([20, 18]). Among molecular imaging modalities, optical imaging has attracted major attention for its unique advantages, especially performance and cost-effectiveness ([6, 21, 14]). Among various optical molecular imaging techniques, the fluorescence molecular tomography (FMT) ([15]) and the bioluminescence tomography (BLT) ([16, 17, 5]) are two emerging and complementary modes. In contrast to fluorescent imaging, bioluminescent imaging has unique capabilities in probing molecular and cellular processes. Furthermore, there is no or little background auto-fluorescence with bioluminescent imaging. With BLT, quantitative and localized analyses on a bioluminescent source distribution become feasible in a mouse, which reveal information important for numerous biomedical studies. Some uniqueness results for the BLT problem are given in [17]. A comprehensive theoretical and numerical analysis of the BLT problem is performed in [9]. More theoretical investigations and numerical solutions of the BLT problem are reported in [11, 8, 2, 3]. A further step along the direction of BLT development is multi-spectral bioluminescence tomography (MBT) that uses multi-spectral measurement data in reconstructing the bioluminescent source distribution. Some theoretical studies of MBT, including results from numerical simulations, can be found in [10] for a particular formulation. A general framework for MBT is introduced and analyzed in [12]. See [13] for a summary account. Inspired by the experimental results reported in [22] that bioluminescent spectra can be significantly affected by temperature, the temperature-modulated bioluminescence tomography (TBT) was proposed in [19] for more accurate reconstruction of the bioluminescent source distribution. The purpose of the current paper is to establish a general mathematical framework for the study of TBT and to provide 1
Department of Mathematics, University of Iowa, Iowa City, IA 52242, U.S.A. Division of Biomedical Imaging, Virginia Tech–Wake Forest University School of Biomedical Engineering and Sciences, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, U.S.A. 3 Department of Mathematics, University of Wisconsin, Oshkosh, WI 54901, U.S.A. 2
1
a thorough theoretical investigation. The rest of the paper is organized as follows. We describe the TBT problem in Section 2. In Section 3, we introduce the mathematical framework for the study of the TBT problem through minimizing the mismatch between predicted boundary values by boundary value problems and boundary measurements, coupled with regularization on the source distribution function and penalization for the inaccuracy in the normalized total energy function and energy percentage functions. Solution existence issue is addressed in Section 4. Note that the TBT problem can be solved only approximately; so in Section 5, we consider discretization of the TBT problem using the finite element method. The main result there is on convergence of the numerical solutions. Then in Section 7, we explore the limiting behavior of the solution of our framework as some of the penalization parameters tend to infinity. We report some numerical results on simulation of the TBT problem in Section 8. The numerical results illustrate that using measurements from multiple temperature distributions leads to accuracy improvement in the reconstructed bioluminescent source distribution.
2
Problem description
We denote by Ω ⊂ Rd the domain occupied by a biological medium and by Γ = ∂Ω for its boundary. The integer d refers to the space dimension. In applications, d = 3. However, the mathematical theory can be developed for any dimension. The diffusion boundary value problem at the normal body temperature is −div(D∇u) + µu = pχΩ0
in Ω,
u + 2AD∂ν u = 0 on Γ = ∂Ω.
(2.1) (2.2)
Here, D(x) = 1/[3 (µ(x)+µ′ (x))], µ(x) and µ′ (x) are the absorption coefficient and the reduced scattering coefficient, ∂/∂ν stands for the outward normal derivative, A(x) =
1 + R(x) , 1 − R(x)
R(x) ≈ −1.4399 γ −2 + 0.7099 γ −1 + 0.6681 + 0.0636 γ
with γ being the refractive index of the medium, Ω0 is a subset of Ω, known as the permissible region, and χΩ0 is the characteristic function of Ω0 . The four coefficients 1.4399, . . . , 0.0636 are dimensionless and are emperically determined. The homogeneous boundary condition (2.2) corresponds to a dark environment for the experiments. The purpose is to reconstruct the source function p from multispectral measurements of outgoing photon densities on portions of the boundary Γ, resulted from several sets of experiments with modulated temperatures. Our discussion in this paper will be given in the more general context of the multispectral bioluminescence tomography (MBT). In MBT, the optical imaging study is performed with multiple reporters of different spectra for multiple molecular and cellular targets. Multispectral data are measured in spectral bands on the surface of the biological body, which are then employed for the reconstruction of the distributions of multiple biomarkers. For a detailed theoretical analysis of MBT, see [12]. We now introduce additional symbols necessary for a description of the problem. In multispectral bioluminescence tomography, the spectrum is divided into certain numbers of bands, say i0 bands Λ1 , · · · , Λi0 , with Λi = [λi−1 , λi ), 1 ≤ i ≤ i0 − 1,
Λi0 = [λi0 −1 , λi0 ].
Here, λ0 < λ1 < · · · < λi0 is a partition of the whole spectrum range. The case of TBT based on the ordinary single spectrum BLT is recovered by setting i0 = 1. 2
0.5 energy percentages in band 1
normalized total energy
1.1 1 0.9 0.8 0.7
25
30 35 temperature
0.4 0.35 0.3 0.25 25
40
30 35 temperature
40
30 35 temperature
40
0.45 energy percentages in band 3
0.34 energy percentages in band 2
0.45
0.33 0.32 0.31 0.3 0.29 0.28 0.27 25
30 35 temperature
0.4 0.35 0.3 0.25 0.2 25
40
Figure 1: Approximate normalized total energy function and energy percentages in three spectral bands Several experiments are to be performed at different temperature distributions. Denote by j0 the number of experiments, and for j th (1 ≤ j ≤ j0 ) experiment, let Tj = Tj (x) be the corresponding temperature distribution in the medium. Experimental results indicate that the photon density satisfies a diffusion differential equation of the form (2.1) with the right hand side modified by a temperature dependent scaling parameter. Specifically, in the j th experiment, the portion of the photon density in the spectral band Λi is described by the following boundary value problem, adapted from (2.1)–(2.2): −div(D∇uij ) + µuij = ω(Tj )wi (Tj )pχΩ0 uij + 2AD∂ν uij = 0 on Γ.
in Ω,
(2.3) (2.4)
Here, ω is the normalized total energy function, wi is the energy percentage function for the spectral band Λi , 1 ≤ i ≤ i0 . The value of ω at the normal body temperature 37◦ C is set to be 1: ω(37) = 1. At any temperature T , we have i0 X wi (T ) = 1. i=1
The corresponding multispectral boundary measurements are Qij = −D∂ν uij =
1 uij 2A
on Γi .
(2.5)
Here Γi , 1 ≤ i ≤ i0 , are subsets of Γ, where we measure the multispectral outgoing photon densities. These subsets can be the same, and they can equal to the entire boundary Γ. Based on prior experimental results, we suppose we know approximate values of ω and w = (w1 , · · · , wi0 )T ; (0) (0) the known approximate values are denoted as ω (0) and w(0) = (w1 , · · · , wi0 )T ([22, 19]). Figure 1 shows (0)
graphs of the functions ω (0) and wi , 1 ≤ i ≤ 3, for the temperature range [25◦ C, 39◦ C]. The three spectral bands are Λ1 = [500, 590) nm, Λ2 = [590, 625) nm, Λ3 = [625, 750] nm. Then the TBT problem is to determine p, ω and w from (2.3), (2.4) and (2.5) for 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 . As is well-known, this pointwise formulation is ill-posed, see, e.g., [9]. So we study the TBT problem through minimizing the mismatch between predicted boundary values by boundary value problems and 3
boundary measurements, coupled with regularization on the source distribution function and penalization for the inaccuracy in the normalized total energy function and energy percentage functions.
3
A general framework for studying TBT problem
We first make assumptions on the data. Assume Ω ⊂ Rd is a Lipschitz domain, A(x) ∈ [Al , Au ] for some constants Al , Au ∈ (0, ∞), D ∈ L∞ (Ω) with D(x) ≥ D0 for some constant D0 > 0, and µ ∈ L∞ (Ω) with µ(x) ≥ 0. Let fij = 2AQij and assume fij ∈ L2 (Γi ) for 1 ≤ j ≤ j0 , 1 ≤ i ≤ i0 . Let Tl and Tu be the lower and upper bounds of temperature distributions involved in the experiments. Denote IT = (Tl , Tu ). We will use the following function spaces: V = H 1 (Ω), Q0 = L2 (Ω0 ), QΓ = L2 (Γ), QΓi = L2 (Γi ) for 1 ≤ i ≤ i0 , and VT = H 1 (IT ). In our formulation, unknowns are the source function p, the normalized total energy function ω, and the multispectral energy percentage vector function w = (w1 , · · · , wi0 )T . We search for p in a set Qp ⊂ Q0 , the function ω in the set Qω = {ω ∈ VT | 0 ≤ ω ≤ 2}, and w in the set Qw = {w ∈ (VT )i0 | 0 ≤ wi ≤ 1, 1 ≤ i ≤ i0 ,
X
wi = 1}.
i
We assume Qp is a non-empty, closed, convex set in Q0 . Define the admissible set Qad = Qp × Qω × Qw . Then Qad is a non-empty, closed, convex set in the space Q0 × VT × (VT )i0 . For any (p, ω, w) ∈ Q0 × VT × (VT )i0 , we define uij = uj (p, ω, wi ) ∈ V by Z Z Z 1 (D∇uij · ∇v + µ uij v) dx + ω(Tj ) wi (Tj ) p v dx uij v ds = Γ 2A Ω Ω0
∀v ∈ V
(3.1)
for 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 . Note that (3.1) is a weak form of the boundary value problem (2.3)–(2.4). By the assumptions made on the data D, µ and A, it follows from an application of the Lax-Milgram lemma ([1, 7]) that the solution uij is uniquely defined by (3.1). For positive numbers ε, Mω and Mw , we introduce the objective functional J(p, ω, w) =
ε 1X kuj (p, ω, wi ) − fij k2QΓ + kpk2Q0 i 2 2 i,j
Mω Mw X (0) + kwi − wi k2VT . kω − ω (0) k2VT + 2 2
(3.2)
i
Here ε is meant to be small, whereas Mω and Mw are meant to be large. Throughout the paper, the indices i and j range between 1 and i0 , and 1 and j0 , respectively. We then introduce the following for the study of the reconstruction problem described in Section 2. Problem 3.1 Find (p, ω, w) ∈ Qad such that the objective functional J(p, ω, w) is minimal over Qad .
4
4
Solution existence
We first present and prove a solution existence result. Theorem 4.1 Under the assumptions stated in the beginning of Section 3 on the data, Problem 3.1 has a solution. Proof. By definition of the infimum, we have a sequence {(pn , ωn , w n )}n ⊂ Qad , wn = (w1,n , · · · , wi0 ,n )T , such that J(pn , ωn , w n ) → inf J(p, ω, w) as n → ∞. (p,ω,w)∈Qad
We note that kpn kQ0 , kωn kVT and kwi,n kVT , 1 ≤ i ≤ i0 , are all uniformly bounded, independent of n. Denote uij,n = uj (pn , ωn , wi,n ). From the definition (3.1), we deduce that kuij,n kV is also uniformly bounded, independent of n. Thus, there is a subsequence {n′ } of the sequence {n}, and functions (p∞ , ω∞ , w∞ ) ∈ Qad , w∞ = (w1,∞ , · · · , wi0 ,∞ )T , and uij,∞ ∈ V such that as n′ → ∞, pn′ ⇀ p∞ in Q0 ,
(4.1)
ωn′ ⇀ ω∞ in VT , ωn′ → ω∞ in C(IT ),
(4.2)
uij,n′ ⇀ uij,∞ in V, uij,n′ → uij,∞ in QΓ , 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 .
(4.4)
wi,n′ ⇀ wi,∞ in VT , wi,n′ → wi,∞ in C(IT ), 1 ≤ i ≤ i0 ,
(4.3)
Let us show that uij,∞ = uj (p∞ , ω∞ , wi,∞ ). For any v ∈ V , consider the difference Z Z ω∞ (Tj ) wi,∞ (Tj ) p∞ v dx. ωn′ (Tj ) wi,n′ (Tj ) pn′ v dx − Ω0
Ω0
It can be rewritten as Z Z ωn′ (Tj ) wi,n′ (Tj ) − ω∞ (Tj ) wi,∞ (Tj ) pn′ v dx +
Ω0
Ω0
ω∞ (Tj ) wi,∞ (Tj ) (pn′ − p∞ ) v dx.
As n′ → ∞, the first term tends to zero because of (4.2) and (4.3), and the second term tends to zero because of (4.1). Therefore, Z Z ω∞ (Tj ) wi,∞ (Tj ) p∞ v dx. lim ωn′ (Tj ) wi,n′ (Tj ) pn′ v dx = ′ n →∞ Ω0
Ω0
In the defining relation Z Z D∇uij,n′ · ∇v + µ uij,n′ v dx + Ω
Γ
1 uij,n′ v ds = 2A
Z
ωn′ (Tj ) wi,n′ (Tj ) pn′ v dx Ω0
let n′ → ∞, use the convergence of the right side and the property (4.4) to obtain Z Z Z 1 ω∞ (Tj ) wi,∞ (Tj ) p∞ v dx uij,∞ v ds = (D∇uij,∞ · ∇v + µ uij,∞v) dx + Ω0 Γ 2A Ω
∀ v ∈ V,
∀ v ∈ V.
In other words, uij,∞ = uj (p∞ , ω∞ , wi,∞ ). Using this relation and the properties (4.1)–(4.4), we obtain J(p∞ , ω∞ , w ∞ ) ≤ lim inf J(pn′ , ωn′ , wn′ ). ′ n →∞
5
Therefore, J(p∞ , ω∞ , w ∞ ) =
inf
(p,ω,w)∈Qad
J(p, ω, w),
and (p∞ , ω∞ , w∞ ) is a solution of Problem 3.1. Next we present a necessary condition for a solution of Problem 3.1. Associated with an element (p, ω, w) ∈ Qad , we introduce the set Qad (p, ω, w) = (∆p , ∆ω , ∆w ) ∈ Q0 × VT × (VT )i0 | (p + ∆p , ω + ∆ω , w + ∆w ) ∈ Qad .
Observe that Qad (p, ω, w) is a non-empty, closed, convex set in the space Q0 × VT × (VT )i0 . For any (p, ω, w) ∈ Qad , any (∆p , ∆ω , ∆w ) ∈ Qad (p, ω, w) with ∆w = (∆w1 , . . . , ∆wi0 )T , and 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 , we define eij ≡ ej (p, ω, wi ; ∆p , ∆ω , ∆wi ) ∈ V by the relation Z Z 1 eij v ds (D ∇eij · ∇v + µ eij v) dx + 2A Γ Ω Z [ω(Tj ) ∆wi (Tj ) ∆p + wi (Tj ) ∆ω (Tj ) ∆p + p ∆ω (Tj ) ∆wi (Tj )] v dx ∀ v ∈ V. (4.5) = Ω0
We can show that if (p, ω, w) ∈ Qad is a solution of Problem 3.1, then X (uj (p, ω, wi ) − fij , eij )QΓ + ε (p, ∆p )Q0 + Mω (ω − ω (0) , ∆ω )VT i
i,j
X (0) (wi − wi , ∆wi )VT ≥ 0 + Mw i
∀ (∆p , ∆ω , ∆w ) ∈ Qad (p, ω, w).
(4.6)
This relation can be proved similar to that of (2.15) in [11].
5
Numerical approximation
Let {ThΩ }hΩ be a regular family of finite element partitions of Ω, {ThΩ0 }hΩ0 a regular family of finite element partitions of Ω0 , and {ThT }hT a regular family of finite element partitions of IT . In the following, we will use the symbol h for the triplet (hΩ , hΩ0 , hT ). By h → 0, we mean max{hΩ , hΩ0 , hT } → 0. Corresponding to the partitions ThΩ , ThΩ0 and ThT , we construct the related linear finite element hΩ0
hΩ
space V hΩ , piecewise constant function space Q0 0 , and linear finite element space VThT . Define Qp Qp ∩
hΩ Q0 0 ,
QhωT = Qω ∩
VThT ,
QhwT
=
Qw ∩ (VThT )i0 , Qhad =
hΩ Qp 0
=
and the discrete admissible set
× QhωT × QhwT .
We assume Qhad is non-empty. Then Qhad is a closed and convex subset of Qad . For any (ph , ω h , w h ) ∈ Qhad , we define uhij = uhj (ph , ω h , wih ) ∈ V hΩ by Z Z Z 1 h h D∇uhij · ∇v h + µ uhij v h dx + ω h (Tj ) wih (Tj ) ph v h dx uij v ds = 2A Γ Ω Ω0
∀ v h ∈ V hΩ
(5.1)
for 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 . Applying the Lax-Milgram lemma, we deduce that uhij ∈ V hΩ is uniquely defined by (5.1). We then introduce a discrete analog of the functional (3.2): ε 1X h h h h kuj (p , ω , wi ) − fij k2QΓ + kph k2Q0 J h (ph , ω h , w h ) = i 2 2 i,j
+
Mw X h Mω h (0) kwi − wi k2VT , kω − ω (0) k2VT + 2 2 i
and that of Problem 3.1: 6
(5.2)
Problem 5.1 Find (ph , ω h , w h ) ∈ Qhad such that the functional J h (ph , ω h , wh ) is minimal over Qhad . Similar to Problem 3.1, we can show that Problem 5.1 has a solution. Our emphasis here is a convergence result. h , w h )} be a sequence of discrete solutions, h → 0. Then there exist a Theorem 5.2 Let {(ph∞ , ω∞ ∞ h 0 , w 0 ) ∈ Qh such that as h′ → 0, subsequence {h′ = (h′Ω , h′Ω0 , h′T )} of {h} and an element (p0∞ , ω∞ ∞ ad ′
′
ph∞ → p0∞ in Q0 ,
′
h 0 ω∞ → ω∞ in VT ,
h′
h 0 wi,∞ → wi,∞ in VT , 1 ≤ i ≤ i0 ,
uij,∞ → u0ij,∞ in V, 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 , ′
′
′
′
(5.3) (5.4)
′
0 0 0 0 0 0 h , wh ), u0 where uhij,∞ = uhj (ph∞ , ω∞ i,∞ ij,∞ = uj (p∞ , ω∞ , wi,∞ ). Moreover, (p∞ , ω∞ , w ∞ ) is a solution of Problem 3.1. h k , kwh k h Proof. We first notice that kph∞ kQ0 , kω∞ VT i,∞ VT for 1 ≤ i ≤ i0 , and kuij,∞ kVT for 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 , are all uniformly bounded, independent of h. Thus for a subsequence {h′ = (h′Ω , h′Ω0 , h′T )} 0 , w 0 ) ∈ Qh and functions u0 of {h}, an element (p0∞ , ω∞ ∞ ad ij,∞ ∈ V , 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 , such that as ′ h → 0, ′
ph∞ ⇀ p0∞ in Q0 ,
(5.5)
h′
h′
0 0 ω∞ ⇀ ω∞ in VT , ω∞ → ω∞ in C(IT ),
h′ 0 wi,∞ ⇀ wi,∞ in VT , ′ uhij,∞ ⇀ u0ij,∞ in V,
h′ wi,∞ ′ uhij,∞
→ →
0 wi,∞ u0ij,∞
(5.6)
in C(IT ), 1 ≤ i ≤ i0 ,
(5.7)
in QΓ , 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 .
(5.8)
0 , w0 ) and Similar to the proof of Theorem 4.1, we can verify that u0ij,∞ = uj (p0∞ , ω∞ i,∞ ′
′
′
′
0 h J(p0∞ , ω∞ , w 0∞ ) ≤ lim′ inf J h (ph∞ , ω∞ , wh∞ ) h →0
= lim′ inf h →0
inf
′ (ph′ ,ω h′ ,wh′ )∈Qh ad
′
′
′
′
J h (ph , ω h , wh ).
(5.9)
Now let (p, ω, w) ∈ Qad be a solution of Problem 3.1 and let uij = uj (p, ω, wi ) be defined by (3.1). Let hΩ
Π p ∈ Q0 0 be the piecewise average of p, I hT ω ∈ VThT and I hT wi ∈ VThT be the continuous piecewise linear interpolants of ω and wi . Then by the finite element theory ([4]), hΩ0
kΠhΩ0 p − pkQ0 → 0, kI hT ω − ωkVT → 0, kI hT wi − wi kVT → 0, as h → 0.
(5.10)
Define uhij ∈ V hΩ by Z Z 1 h h h h h h D∇uij · ∇v + µ uij v dx + uij v ds Γ 2A Ω Z I hT ω(Tj ) I hT wi (Tj ) ΠhΩ0 p v h dx ∀ v h ∈ V hΩ . =
(5.11)
Ω0
Then from the definitions (3.1) and (5.11), we can derive the following inequality: kuij − uhij kV ≤ c inf kuij − v hΩ kV + kω(Tj ) wi (Tj ) p − I hT ω(Tj ) I hT wi (Tj ) ΠhΩ0 pkQ0 . vhΩ ∈V hΩ
7
(5.12)
Because of the uniform boundedness of kI hT ωkC(IT ) , kI hT wi kC(IT ) and kΠhΩ0 pkQ0 , we deduce from (5.12) that h h kuij − uij kV ≤ c inf kuij − v hΩ kV + kω − I hT ωkC(IT ) h v Ω ∈V hΩ i + kwi − I hT wi kC(IT ) + kp − ΠhΩ0 pkQ0 . In particular, this implies kuij − uhij kV → 0 as h → 0. Thus, inf
(ph ,ω h ,wh )∈Qh ad
J h (ph , ω h , w h ) ≤ J h (ΠhΩ0 p, I hT ω, I hT w) → J(p, ω, w) as h → 0.
Therefore, lim sup h′ →0
inf
′ (ph′ ,ω h′ ,wh′ )∈Qh ad
′
′
′
′
′
′
′
′
J h (ph , ω h , w h ) ≤ J(p, ω, w).
(5.13)
Combining (5.9) and (5.13), we deduce that h 0 , wh∞ ) = J(p, ω, w). J(p0∞ , ω∞ , w 0∞ ) = lim J h (ph∞ , ω∞ ′ h →0
0 , w 0 ) is a solution of Problem 3.1, and as h′ → 0, So (p0∞ , ω∞ ∞ ′
kph∞ kQ0 → kp0∞ kQ0 , ′
(0)
′
h 0 kω∞ − ω (0) kVT → kω∞ − ω (0) kVT , (0)
h 0 kwi,∞ − wi kVT , 1 ≤ i ≤ i0 . − wi kVT → kwi,∞
These relations and the weak convergence of (5.5)–(5.7) together imply the strong convergence in (5.3) ([1]). To show the strong convergence (5.4), we use the definitions (3.1) and (5.1) and write Z Z h i 1 ′ ′ ′ D |∇(uhij,∞ − u0ij,∞ )|2 + µ |uhij,∞ − u0ij,∞|2 dx + |uhij,∞ − u0ij,∞ |2 ds 2A Γ Ω Z h i ′ ′ ′ ′ h h 0 0 ω∞ (Tj ) wi,∞ (Tj ) ph∞ − ω∞ (Tj ) wi,∞ (Tj ) p0∞ uhij,∞ dx = Ω0 Z Z ′ 1 0 ′ h 0 0 0 0 ω∞ (Tj ) wi,∞ (Tj ) p∞ uij,∞ − uij,∞ dx + + uij,∞ (u0ij,∞ − uhij,∞ ) ds Γ A Ω0 Z h i ′ ′ D ∇u0ij,∞ · ∇(u0ij,∞ − uhij,∞ ) + µ u0ij,∞ (u0ij,∞ − uhij,∞ ) dx. +2 Ω
′
As h′ → 0, the first term on the right side approaches zero because of the boundedness of {kuhij,∞ kL2 (Ω0 ) }h′ h′ 0 h′ → ω 0 and wh′ → w0 and the strong convergence ω∞ ∞ i,∞ i,∞ in VT , p∞ → p∞ in Q0 . The other terms approach zero because of the convergence property (5.8). Thus, Z Z h i ′ 1 ′ ′ |uhij,∞ − u0ij,∞ |2 ds → 0 as h′ → 0. D |∇(uhij,∞ − u0ij,∞ )|2 + µ |uhij,∞ − u0ij,∞ |2 dx + 2A Γ Ω So we have the strong convergence (5.4).
8
6
An iterative scheme based on linearization
The discussion in the section is made at the continuous level, i.e., for Problem 3.1. All the results can be extended to the discrete Problem 5.1 in a straightforward fashion. Note that Problem 3.1 has three unknowns p, ω and w, but approximate values ω (0) and w(0) of ω and w are available from prior experiments. We may first determine an approximate value p(0) for p, and then develop an iterative scheme based on the idea of linearization. First we compute p(0) . Consider Problem 3.1 with freezed ω = ω (0) and w = w(0) . In other words, we consider the following problem. Problem 6.1 Find p ∈ Qp such that the objective functional J(p, ω (0) , w(0) ) is minimal over Qp . This problem is similar to the standard BLT/MBT problems studied in the literature, [9, 12]. There is a unique solution p(0) ∈ Qp , which is characterized by an inequality X (0) (0) (0) uij (p(0) ) − fij , uij (p) − uij (p(0) ) + ε p(0) , p − p(0) ≥ 0 ∀ p ∈ Qp . QΓi
i,j
Q0
(0)
Here uij (p) ∈ V is defined by a boundary value problem: Z Z 1 (0) (0) (0) D∇uij (p) · ∇v + µ uij (p) v dx + u (p) v ds 2A ij Γ Ω Z (0) ω (0) (Tj ) wi (Tj ) p v dx ∀ v ∈ V. = Ω0
Let us discuss the idea of linearization. Write p = p(0) + δp ,
ω = ω (0) + δω ,
w = w(0) + δ w .
Here δ w = (δw1 , · · · , δwi0 )T . These δ terms are expected to be small, and we drop terms of second or higher orders of these to obtain (0)
(0)
ω(Tj ) wi (Tj ) p ≈ ω (0) (Tj ) wi (Tj ) p(0) + ω (0) (Tj ) wi (Tj ) δp (0)
+ wi (Tj ) p(0) δω (Tj ) + ω (0) (Tj ) p(0) δwi (Tj ).
Then the solution uj (p, ω, wi ) of the problem (3.1) satisfies (0)
uj (p, ω, wi ) ≈ uij (p(0) ) + δu,ij , where we define δu,ij ∈ V by Z Z 1 δu,ij v ds (D∇δu,ij · ∇v + µ δu,ij v) dx + Γ 2A Ω Z h i (0) (0) ω (0) (Tj ) wi (Tj ) δp + wi (Tj ) p(0) δω (Tj ) + ω (0) (Tj ) p(0) δwi (Tj ) v dx = Ω0
∀ v ∈ V.
Based on the above discussion, we propose the following iterative scheme. (k) For k = 0, 1, · · · , with (p(k) , ω (k) , w (k) ) ∈ Qad known, define uij ∈ V by the problem Z Z Z 1 (k) (k) (k) (k) D∇uij · ∇v + µ uij v dx + ω (k) (Tj ) wi (Tj ) p(k) v dx ∀ v ∈ V. uij v ds = 2A Γ Ω Ω0 9
(6.1)
Define
o n (k) Qad = (δp , δω , δ w ) ∈ Q0 × VT × (VT )i0 | (p(k) + δp , ω (k) + δω , w(k) + δ w ) ∈ Qad . (k)
It is easy to show that Qad is a non-empty, closed, convex set in the space Q0 × VT × (VT )i0 . For (k) (δp , δω , δ w ) ∈ Qad , define δu,ij ∈ V by Z Z 1 (D∇δu,ij · ∇v + µ δu,ij v) dx + δu,ij v ds 2A Γ Ω Z h i (k) (k) ω (k) (Tj ) wi (Tj ) δp + wi (Tj ) p(k) δω (Tj ) + ω (k) (Tj ) p(k) δwi (Tj ) v dx ∀ v ∈ V, (6.2) = Ω0
and introduce the functional (k)
Jδ (δp , δω , δ w ) =
1 X (k) ε kuij + δu,ij − fij k2QΓ + kδp k2Q0 i 2 2 i,j
+
Mw X Mω kδwi k2VT . kδω k2VT + 2 2
(6.3)
i
(k)
(k)
(k)
We determine (δp , δω , δ w ) from the following problem: (k)
(k)
(k)
(k)
(k)
Problem 6.2 Find (δp , δω , δ w ) ∈ Qad that minimizes the functional J (k) (δp , δω , δ w ) over Qad . As output of the iteration step, we let p(k+1) = p(k) + δp(k) ,
ω (k+1) = ω (k) + δω(k) ,
w (k+1) = w (k) + δ (k) w .
(6.4)
Regarding Problem 6.2, we have the following result. (k)
(k)
(k)
(k)
Theorem 6.3 Problem 6.2 has a unique solution (δp , δω , δ w ) ∈ Qad , which is characterized by an inequality X (k) (k) (k) + ε δp(k) , δp − δp(k) uij + δu,ij − fij , δu,ij − δu,ij Q0
QΓi
i,j
+ Mω δω(k) , δω − δω(k)
VT
X (k) (k) , δ − δ δw + Mw wi wi i
VT
i
(k)
≥0
(k)
∀ (δp , δω , δ w ) ∈ Qad ,
(6.5)
(k)
where uij ∈ V is defined by (6.1), δu,ij ∈ V is defined by (6.2), and δu,ij ∈ V is defined by Z Z 1 (k) (k) D∇δu,ij · ∇v + µ δu,ij v dx + δu,ij v ds Γ 2A Ω Z h i (k) (k) (k) ω (k) (Tj ) wi (Tj ) δp(k) + wi (Tj ) p(k) δω(k) (Tj ) + ω (k) (Tj ) p(k) δw (T ) v dx = j i Ω0
∀ v ∈ V.
Numerical methods for Problem 6.2 can be similarly developed, and convergence of the numerical solutions can be shown.
10
7
Limiting behaviors (1)
We first consider the limiting behavior of the solution of Problem 3.1 as Mω → ∞. Let Qad = Qp × Qw . (1) (1) Then Qad is a non-empty, closed, convex set in the space Q0 × (VT )i0 . For any (p, w) ∈ Qad , we define (1) (1) uij = uj (p, wi ) ∈ V by Z Z Z 1 (1) (1) (1) D∇uij · ∇v + µ uij v dx + ω (0) (Tj ) wi (Tj ) p v dx ∀ v ∈ V (7.1) uij v ds = Γ 2A Ω Ω0 for 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 . By the assumptions made on the data D, µ and A, it follows from an (1) application of the Lax-Milgram lemma that the solution uij is uniquely defined by (7.1). For positive numbers ε and Mw , we introduce the objective functional ε 1 X (1) Mw X (0) kuj (p, wi ) − fij k2QΓ + kpk2Q0 + kwi − wi k2VT (7.2) J (1) (p, w) = i 2 2 2 i,j
i
and the following problem: (1)
(1)
Problem 7.1 Find (p, w) ∈ Qad such that the objective functional J (1) (p, w) is minimal over Qad . Notice that Problem 7.1 is a degenerate case of Problem 3.1 when Qω is replaced by a one element set {ω (0) }. In other words, we study Problem 7.1 when ω (0) is known to be the exact normalized total energy function. Similar to Theorem 4.1, we can show Problem 7.1 has a solution. Our focus here is on relationship between Problem 3.1 and Problem 7.1. Theorem 7.2 Let (pn , ωn , w n ) ∈ Qad be a solution of Problem 3.1 for Mω = Mn with Mn → ∞ as n → ∞. Then ωn → ω (0) in VT as n → ∞, (7.3) (1)
(1)
(1)
and there exist a subsequence {n′ } of the sequence {n} and a solution (p∞ , w ∞ ) ∈ Qad of Problem 7.1 such that as n′ → ∞, pn′ → p(1) ∞ in Q0 , (1)
(1)
wi,n′ → wi,∞ in VT , 1 ≤ i ≤ i0 ,
uij,n′ → uij,∞ in V, 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 , (1)
(1)
(1)
(7.4) (7.5)
(1)
where uij,n′ = uj (pn′ , ωn′ , wi,n′ ), uij,∞ = uj (p∞ , wi,∞ ). Proof. To stress the dependence of the functional J(·, ·, ·) of (3.2) on Mω = Mn , we write it as Jn (·, ·, ·) (1) in this proof. Let (p∞ , w ∞ ) ∈ Qad be a solution of Problem 7.1. Then Jn (pn , ωn , wn ) ≤ Jn (p∞ , ω (0) , w ∞ ) = J (1) (p∞ , w∞ ).
(7.6)
In particular, this implies ωn → ω (0) in VT . So (7.3) holds. Notice that kpn kQ0 , kwi,n kVT and kuij,n kV , 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 , are uniformly bounded. Hence, there exists a subsequence {(pn′ , wn′ )}, an element (1) (1) (1) (1) (p∞ , w∞ ) ∈ Qad and uij,∞ ∈ V , 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 , such that pn′ ⇀ p(1) ∞ in Q0 ,
(7.7)
(1)
(1)
(7.8)
(1)
(1)
(7.9)
wi,n′ ⇀ wi,∞ in VT , wi,n′ → wi,∞ in C(IT ), 1 ≤ i ≤ i0 , uij,n′ ⇀ uij,∞ in V, uij,n′ → uij,∞ in QΓ , 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 , 11
We need to show (1) (1) (1) (1) (a) uij,∞ = uj (p∞ , wi,∞ ); (1)
(1)
(b) (p∞ , w ∞ ) is a solution of Problem 7.1; (c) strong convergence (7.4) and (7.5). It is easy to see that (a) is valid. For (b) and (c), we first deduce the following inequality from (7.6): lim sup Jn (pn , ωn , w n ) ≤ J (1) (p∞ , w ∞ ).
(7.10)
n→∞
Then we have the following sequence of inequalities: (1) inf J (1) (pn′ , w n′ ) J (1) (p(1) ∞ , w ∞ ) ≤ lim ′ n →∞
≤ lim sup J (1) (pn′ , w n′ ) n′ →∞ Mn′ (0) 2 = lim sup Jn′ (pn′ , ωn′ , wn′ ) − kωn′ − ω kVT 2 n′ →∞ ≤ lim sup Jn′ (pn′ , ωn′ , w n′ ) ≤
n′ →∞ J (1) (p∞ , w ∞ ) (1)
(1)
where in the last step, we used (7.10). So (p∞ , w∞ ) is a solution of Problem 7.1 and (1) lim J (1) (pn′ , wn′ ) = J (1) (p(1) ∞ , w ∞ ).
n′ →∞
Taking (7.7), (7.8) and (7.9) into consideration, the equality implies that as n′ → ∞, kpn′ kQ0 → kp(1) ∞ kQ0 ,
(1)
(1)
kwi,n′ kVT → kwi,∞ kVT , 1 ≤ i ≤ i0 .
The above norm sequence convergence and the weak convergence property (7.7) and (7.8) together imply the strong convergence (7.4). Proof of the strong convergence (7.5) is similar to that of (5.4). From the proof of the theorem, we find that lim Mn′ kωn′ − ω (0) k2VT = 0.
n′ →∞
This relation indicates that the speed with which ωn′ converges to ω (0) in VT is faster than that with √ which 1/ Mn′ converges to zero. (2)
We then consider the limiting behavior of the solution of Problem 3.1 as Mw → ∞. Let Qad = (2) (2) Qp × Qω . Then Qad is a non-empty, closed, convex set in the space Q0 × VT . For any (p, ω) ∈ Qad , we (2) (2) define uij = uij (p, ω) ∈ V by Z Z Z 1 (2) (2) (2) (0) D∇uij · ∇v + µ uij v dx + (7.11) ω(Tj ) wi (Tj ) p v dx ∀ v ∈ V uij v ds = 2A Γ Ω Ω0 for 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 . By the assumptions made on the data D, µ and A, it follows from an (2) application of the Lax-Milgram lemma that the solution uij is uniquely defined by (7.11). For positive numbers ε and Mω , we introduce the objective functional ε Mω 1 X (2) kuij (p, ω) − fij k2QΓ + kpk2Q0 + kω − ω (0) k2VT (7.12) J (2) (p, w) = i 2 2 2 i,j
and the following problem: 12
(2)
(2)
Problem 7.3 Find (p, ω) ∈ Qad such that the objective functional J (2) (p, ω) is minimal over Qad . Notice that Problem 7.3 is a degenerate case of Problem 3.1 when Qw is replaced by a one element set {w (0) }. In other words, we study Problem 7.3 when w(0) is known to be the exact multispectral energy percentage vector function. We can show that Problem 7.3 has a solution. Similar to Theorem 7.2, we have the next result regarding Problem 7.3. Theorem 7.4 Let (pn , ωn , wn ) ∈ Qad be a solution of Problem 3.1 for Mw = Mn with Mn → ∞ as n → ∞. Then (0) wi,n → wi in VT as n → ∞, 1 ≤ i ≤ i0 , (2)
(2)
(2)
and there exist a subsequence {n′ } of {n} and a solution (p∞ , ω∞ ) ∈ Qad of Problem 7.3 such that as n′ → ∞, pn′ → p(2) ∞ in Q0 , (2)
(2) in VT , ω n′ → ω ∞
uij,n′ → uij,∞ in V, 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 , (2)
(2)
(2)
(2)
where uij,n′ = uj (pn′ , ωn′ , wi,n′ ), uij,∞ = uij (p∞ , ω∞ ). It is easy to extend the above discussion to the situation where only some of the energy percentage functions are known exactly. Let I ⊂ {1, · · · , i0 } be the index set for those energy percentage functions ˜ ad = Qp × Qω × Q ˜ w,I where Q ˜ w,I = {w ∈ Qw | wi = w(0) ∀ i 6∈ I}. that are not known exactly. Define Q i ˜ ad is a non-empty, closed, convex set in the space Q0 × VT × (VT )i0 . Modify (3.2) to Then Q X X X 1 (0) kuj (p, ω, wi ) − fij k2QΓ kuj (p, ω, wi ) − fij k2QΓ + J˜(p, ω, w) = i i 2 j
+
i∈I
i6∈I
Mw X Mω ε (0) kwi − wi k2VT , kpk2Q0 + kω − ω (0) k2VT + 2 2 2 i∈I
˜ ω, w) over the set Q ˜ ad . Similar theoretical results hold for this problem. and then minimize J(p, We can also consider the limiting behavior of the solution of Problem 3.1 as both Mω → ∞ and (3) (3) Mw → ∞. Let Qad = Qp . Then Qad is a non-empty, closed, convex set in the space Q0 . For any (3) (3) (3) p ∈ Qad , we define uij = uij (p) ∈ V by Z Z (3) (3) D∇uij · ∇v + µ uij v dx + Ω
Γ
1 (3) u v ds = 2A ij
Z
Ω0
(0)
ω (0) (Tj ) wi (Tj ) p v dx
∀v ∈ V
(7.13)
for 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 . By the assumptions made on the data D, µ and A, it follows from an (3) application of the Lax-Milgram lemma that the solution uij is uniquely defined by (7.13). For a positive number ε, we introduce the objective functional J (3) (p) =
ε 1 X (3) kuij (p) − fij k2QΓ + kpk2Q0 i 2 2
(7.14)
i,j
and the following problem: (3)
(3)
Problem 7.5 Find p ∈ Qad such that the objective functional J (3) (p) is minimal over Qad . 13
Notice that Problem 7.5 is a degenerate case of Problem 3.1 when Qω is replaced by a one element set {ω (0) } and Qw is replaced by {w(0) }. In other words, Problem 7.5 is intended for the situation where both the normalized total energy function and multispectral energy percentage vector function are known exactly. We can show this problem has a solution. Similar to Theorems 7.2 and 7.4, we have the following result on Problem 7.5. Theorem 7.6 Let (pn , ωn , w n ) ∈ Qad be a solution of Problem 3.1 for Mω = M2,n and Mw = M3,n , with M2,n → ∞ and M3,n → ∞ as n → ∞. Then as n → ∞, ωn → ω (0) in VT ,
(0)
wi,n → wi
in VT , 1 ≤ i ≤ i0 . (3)
(3)
Moreover, there exist a subsequence {n′ } of {n} and a solution p∞ ∈ Qad of Problem 7.5 such that as n′ → ∞, pn′ → p(3) ∞ in Q0 , (3)
uij,n′ → uij,∞ in V, 1 ≤ i ≤ i0 , 1 ≤ j ≤ j0 , (3)
(3)
(3)
where uij,n′ = uj (pn′ , ωn′ , wi,n′ ), uij,∞ = uij (p∞ ).
8
Numerical examples
In this section we report some numerical results to illustrate the usefulness of measurements from multiple temperature distributions in improving the accuracy of the reconstructed bioluminescent source distribution. Example 8.1 We consider a two dimensional domain Ω = (0, 1) × (0, 1). The region Ω is divided into four subregions Ωm , 1 ≤ m ≤ 4, with Ω1 = (0, 0.5)×(0, 0.5), Ω2 = (0.5, 1)×(0, 0.5), Ω3 = (0, 0.5)×(0.5, 1) and Ω4 = (0.5, 1) × (0.5, 1). In each subregion Ωm , 1 ≤ m ≤ 4, the values of the optical parameters D and µ are constant. The values of D in the four regions are 0.3268, 0.1916, 0.5464, and 0.2415, and the values of µ are 0.02, 0.14, 0.01, and 0.08, respectively. We choose A = 1. The source p is placed in the region (0.25, 0.375) × (0.125, 0.375) where the function value is 6. The permissible region is taken to be Ω0 = (0.125, 0.375) × (0.125, 0.375). We divide Ω0 into eight congruent triangles and take the admissible hΩ set Qp 0 to consist of piecewise constant functions. The numbering of the eight triangles is shown in Figure 8.1. We use constant temperature distribution Tj in each experiment j, 1 ≤ j ≤ 4. The four temperatures distributions used are T1 = 37◦ C, T2 = 39◦ C, T3 = 35◦ C and T4 = 33◦ C. The corresponding exact values of normalized total energy function at these temperatures are ω = (1, 1.0509, 0.9414, 0.8771)T and the exact energy percentage function values in the three spectral bands are w1 = (0.2844, 0.2598, 0.3095, 0.3364)T , w2 = (0.3264, 0.3307, 0.3204, 0.3130)T , w3 = (0.3892, 0.4095, 0.3701, 0.3506)T . We use linear elements on uniform triangular partitions of the domain Ω. The uniform meshes are obtained by dividing the interval [0, 1] into 1/h equal parts in both x and y directions. We start with 14
(0.375, 0.375)
(0.125, 0.375)
6
8 7
5
4
2 1
3
(0.125, 0.125)
(0.375, 0.125)
Figure 2: Numbering of the eight triangles for the permissible region
Table element h p , h = 1/8 ph , h = 1/16 ph , h = 1/32 p
1: Simulation results 1 2 3 0.2107 0 6.9921 0.5458 0 7.3273 0.5939 0 6.8690 0 0 6
with one 4 4.5387 4.7643 4.6132 6
temperature distribution 5 6 7 1.7817 1.1067 5.3777 1.4998 0.5741 5.4138 1.5639 0.8177 5.2959 0 0 6
8 4.1501 4.1343 4.4156 6
an initial mesh with h = 1/8 and then successively halve h to obtain more refined meshes. We take the numerical solutions of the BVP (3.1) computed using the exact values of p, ω and w and on the mesh with mesh size h = 1/512 as the true solution and use it to obtain the measured quantities fij on Γi = Γ, 1 ≤ i ≤ 3, 1 ≤ j ≤ 4. The numerical solutions of the problem (5.1) are then computed using these values of fij on the meshes with mesh size h = 1/8, 1/16, 1/32 using Matlab and its Optimization Toolbox. We distinguish two cases. First, we assume the normalized total energy functions ω and the energy percentage function w to be known and reconstruct the unknown source function p from multi-spectral measurements of outgoing photon densities. Since the only unknown is p, we replace (5.2) by the following functional ε 1X h h kuj (p ) − fij k2QΓ + kph k2Q0 , J h (ph ) = i 2 2 i,j
hΩ
where uhj (ph ) is determined by (5.1) with ω h = ω and wh = w, and look for ph ∈ Qp 0 which minimizes this functional. Using ε = 10−5 , we computed ph using one (T = T1 ) and four temperature distributions (T = T1 , T2 , T3 , T4 ), respectively. The values of the reconstructed source function are respectively given in Tables 1 and 2. The last row in each table gives the values of the exact source function p. In these and later tables, an entry 0 for a numerical result represents the value 0 or a value of size O(10−16 ). Note that the total source power, i.e., the L1 norm of the exact source p, is 0.1875. In Table 3, we list the reconstructed source powers for one and four temperature disctributions. Improvements gained through the use of measurements from four temperature distributions are evident. Next, we only assume the approximate values of ω and w and reconstruct the source function p as well as ω and w. We again perform the simulations using the one and four temperature distributions. 15
Table 2: Simulation results with four temperature distributions element 1 2 3 4 5 6 7 8 ph , h = 1/8 0 0 7.4592 4.0390 0.9639 0.6044 6.1966 4.7526 ph , h = 1/16 0 0 7.2504 4.3181 1.0064 0.1030 6.2447 5.1395 ph , h = 1/32 0 0 6.8316 4.4831 1.2203 0.0437 6.2296 5.2080 p 0 0 6 6 0 0 6 6
Table 3: Reconstructed source powers h 1/8 1/16 1/32 One temperatures 0.1887 0.1895 0.1888 Four temperatures 0.1876 0.1880 0.1876
The corresponding values of ω (0) and w(0) used are ω (0) = (1.1, 1.2, 0.85, 0.80)T , (0)
w1 = (0.22, 0.3, 0.35, 0.4)T , (0)
w2 = (0.46, 0.4, 0.25, 0.38)T , (0)
w3 = (0.32, 0.3, 0.4, 0.22)T . We choose ε = 10−4 , Mω = 10−2 , and Mw = 10−2 . In the first simulation we used only T = T1 , and in the second simulation we used all the four distributions in the reconstruction of the solution. Since we are using constant temperature distributions, P P (0) (0) the terms kω h − ω (0) k2VT and i kwih − wi k2VT in functional (5.2) are replaced by i |ωih − ωi |2 and P (0) 2 h i,j |wij − wij | , respectively, for computing the numerical solutions. We used the iterative scheme described in Section 6 to compute the solution performing only one iteration in all cases. Also we replace P the terms kδω k2VT and i kδwi k2VT in functional (6.3) by the corresponding sums to take account of the discrete temperature distributions used. Tables 4 and 5 give the computed values of ph using one and four temperatures distributions, respectively. Recall that the total source power of the exact source p is 0.1875. In Table 6, we list the reconstructed source powers for one and four temperature disctributions. Again, we observe improvement due to the use of measurements for the four temperature distributions. Example 8.2 In this example, we compare the TBT algorithm with the ordinary BLT algorithm using an anatomically realistic digital mouse chest phantom built from an segmented MRI mouse image volume.
Table 4: Simulation results with one temperature distribution element 1 2 3 4 5 6 7 ph , h = 1/8 1.9775 1.4441 3.8885 3.1770 2.2500 1.9762 3.4490 ph , h = 1/16 1.8622 1.1119 4.3459 3.4051 2.0469 1.5867 3.7594 ph , h = 1/32 1.6394 0.7047 4.5747 3.5151 1.9471 1.4257 4.0405 p 0 0 6 6 0 0 6
16
8 3.0554 3.2026 3.4385 6
Table 5: Simulation results with four temperature distributions element 1 2 3 4 5 6 7 ph , h = 1/8 1.3349 0.2514 5.3724 3.9005 2.0002 1.4610 4.6298 ph , h = 1/16 1.0537 0 5.7299 4.0914 1.7936 1.0421 5.0370 h p , h = 1/32 0.7388 0 5.8434 4.1310 1.6299 0.7995 5.3400 p 0 0 6 6 0 0 6
8 3.8130 4.0996 4.3535 6
Table 6: Reconstructed source powers h 1/8 1/16 1/32 One temperatures 0.1658 0.1666 0.1663 Four temperatures 0.1778 0.1785 0.1784
We obtain a finite-element model, as shown in Figure 3 (a), of the digital mouse chest phantom. This phantom consists of lungs, a heart, a liver, and muscle. Appropriate absorption µa and reduced scattering µ′s coefficients are assigned to these anatomical structures as shown in Table 7. The finite element model of the digital mouse phantom contains 14757 nodes and 80670 tetrahedral elements. In the simulation, we randomly put a source in a tetrahedral element in a spherical region of r = 3 mm, which contains 1111 tetrahedral elements, and center at (2, 2, 10) mm. In this region, there are 539, 284, and 288 elements belong to muscle, lung, and heart, respectively. We then set the permissible region as a spherical region of r = 6 mm, which contains 8792 tetrahedral elements, and center at (2, 2, 10) mm. To reduce the computational cost, we will only use one spectral. The power of the source will be adjust to create a moderate surface signal strength (maximum = 50). Fig. 3 (b) shows a typical surface bioluminescence signal distribution. Then Poisson noise and 20 dB white Gaussian noise will be added to the signal and rounded to the nearest integer. We then apply the TBT algorithm to the one-temperature and three-temperature data sets; the simulation result is shown in Table 8. In this simulation, we do 1000 random runs and the results shown in the table are the average over 1000 random runs. The results show that temperature data can reduce the reconstruction error. Here, the location error is defined to be the distance between the center of the true source location and the center of reconstructed source location, the latter being defined as the averaged center among the centers of all the elements in the permissible region weighted by the constant values of the reconstructed light source function on the elements. The power error is the relative error of the reconstructed source power with respect to the true source power. Table 7: Optical parameters Material Muscle −1 µa (mm ) 0.23 µ′s (mm−1 ) 1.00
for the Lung 0.35 2.30
17
mouse Heart 0.11 1.10
organ regions Liver 0.45 2.00
Figure 3: Finite element mouse chest model and associated bioluminescent measurement. (a) A geometrical model of the mouse chest consisting of muscle, a heart, lungs, and a liver, and (b) measured bioluminescent data mapped onto the surface of the finite element mouse chest model.
Table 8: Simulated TBT results in terms of source localization and power estimation errors Three-Temperature One-Temperature Location Error 0.7067mm 0.8124mm Power Error 2.52% 5.14%
18
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