Practical reconstruction method for bioluminescence tomography

Wenxiang Cong; Ge Wang; Durairaj Kumar; Yi Liu; Ming Jiang; Lihong V. Wang; Eric A. Hoffman; Geoffrey McLennan; Paul B. McCray; Joseph Zabner; Alexander Cong

doi:10.1364/OPEX.13.006756

Optics Express
Vol. 13,
Issue 18,
pp. 6756-6771
(2005)
•https://doi.org/10.1364/OPEX.13.006756

Practical reconstruction method for bioluminescence tomography

Wenxiang Cong, Ge Wang, Durairaj Kumar, Yi Liu, Ming Jiang, Lihong V. Wang, Eric A. Hoffman, Geoffrey McLennan, Paul B. McCray, Joseph Zabner, and Alexander Cong

Open Access

Get PDF
Email
Share
Get Citation
Copy Citation Text
Wenxiang Cong, Ge Wang, Durairaj Kumar, Yi Liu, Ming Jiang, Lihong V. Wang, Eric A. Hoffman, Geoffrey McLennan, Paul B. McCray, Joseph Zabner, and Alexander Cong, "Practical reconstruction method for bioluminescence tomography," Opt. Express 13, 6756-6771 (2005)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Citation alert
Save article

Abstract

Bioluminescence tomography (BLT) is used to localize and quantify bioluminescent sources in a small living animal. By advancing bioluminescent imaging to a tomographic framework, it helps to diagnose diseases, monitor therapies and facilitate drug development. In this paper, we establish a direct linear relationship between measured surface photon density and an unknown bioluminescence source distribution by using a finite-element method based on the diffusion approximation to the photon propagation in biological tissue. We develop a novel reconstruction algorithm to recover the source distribution. This algorithm incorporates a priori knowledge to define the permissible source region in order to enhance numerical stability and efficiency. Simulations with a numerical mouse chest phantom demonstrate the feasibility of the proposed BLT algorithm and reveal its performance in terms of source location, density, and robustness against noise. Lastly, BLT experiments are performed to identify the location and power of two light sources in a physical mouse chest phantom.

Full Article | PDF Article

More Like This

In vivo mouse studies with bioluminescence tomography

Ge Wang, Wenxiang Cong, Kumar Durairaj, Xin Qian, Haiou Shen, Patrick Sinn, Eric Hoffman, Geoffrey McLennan, and Michael Henry
Opt. Express 14(17) 7801-7809 (2006)

Three-dimensional Bioluminescence Tomography based on Bayesian Approach

Jinchao Feng, Kebin Jia, Chenghu Qin, Guorui Yan, Shouping Zhu, Xing Zhang, Junting Liu, and Jie Tian
Opt. Express 17(19) 16834-16848 (2009)

An optimal permissible source region strategy for multispectral bioluminescence tomography

Jinchao Feng, Kebin Jia, Guorui Yan, Shouping Zhu, Chenghu Qin, Yujie Lv, and Jie Tian
Opt. Express 16(20) 15640-15654 (2008)

Previous Article Next Article

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.

Fig. 1. Numerical simulation for BLT reconstruction of one source. (a) The true source distribution in the left lung consisting of 6 volume elements and having a homogeneous density of 200.0pico-Watts/mm³, and (b) the counterpart reconstructed from the surface data corrupted by 10% Gaussian noise. The average density error is 0.5%.

Download Full Size | PDF

Fig. 2. Numerical simulation for BLT reconstruction of two sources in the left and right lungs, respectively. (a) The true source distribution with a density of 200.0pico-Watts/mm³ for each source, and (b) the counterparts reconstructed from the surface data corrupted by 10% Gaussian noise. The average density error is 3%.

Download Full Size | PDF

Fig. 3. Numerical simulation for BLT reconstruction of two sources in the left lung. (a) The true source distribution with density 200.0pico-Watts/mm³ for both the sources, and (b) the counterparts reconstructed from the surface data corrupted by 10% Gaussian noise. The average density error is 5%.

Download Full Size | PDF

Fig. 4. Numerical simulation for BLT reconstruction of three sources: two in the left lung and one in the right lung. (a) The true source distribution in which each source consists of several volume elements and has density 200.0pico-Watts/mm³, and (b) the counterparts reconstructed from the surface data corrupted by 10% Gaussian noise. The average density error is 3%.

Download Full Size | PDF

Fig. 5. Numerical simulation for BLT reconstruction of four sources. (a) Four bioluminescent source separations of 1mm between first and second, 2mm between second and third, and 3mm between third and fourth source with the density of 200.0 pico-Watts/mm³ for each source, and (b) the counterparts reconstructed from the surface data corrupted by 10% Gaussian noise. The source positions are accurately identified with their density being recovered to 196.1 pico-Watts/mm³, 184.7 pico-Watts/mm³, 175.1 pico-Watts/mm³ and 181.5 pico-Watts/mm³, respectively.

Download Full Size | PDF

Fig. 6. (a), (b) and (c) are the reconstructed source distributions (with unit pico-Watts/mm³) from the surface noise-free data subject to permissible source regions

Ω_{s}^{1}

Ω_{s}^{2}

and

Ω_{s}^{3}

, respectively. They are identical to the actual source in position and strength.

Download Full Size | PDF

Fig. 7. (a), (b) and (c) are the reconstructed source distribution (with unit pico-Watts/mm³) from the surface data corrupted by 5% Gaussian noise subject to permissible source regions

Ω_{s}^{1}

Ω_{s}^{2}

and

Ω_{s}^{3}

, respectively.

Download Full Size | PDF

Fig. 8. (a), (b) and (c) are the reconstructed source distributions (with unit pico-Watts/mm³) from the surface data corrupted by 10% Gaussian noise subject to permissible source regions

Ω_{s}^{1}

Ω_{s}^{2}

and

Ω_{s}^{3}

, respectively.

Download Full Size | PDF

Fig. 9. (a), (b) and (c) are the reconstructed source distribution (with unit pico-Watts/mm³), subject to permissible source regions

Ω_{s}^{1}

Ω_{s}^{2}

and

Ω_{s}^{3}

, respectively. The measured surface data are corrupted by 15% Gaussian noise.

Download Full Size | PDF

Fig. 10. Mouse Chest phantom. (a) A heterogeneous mouse phantom consisting of bone (B), heart (H), lungs (L), and muscle (M); (b) a middle cross-section through two hollow cylinders for hosting luminescent sources in the left lung. The four arrows show the direction of the CCD camera during data acquisition.

Download Full Size | PDF

Fig. 11. Comparison of experimental and computational photon density profiles for determination of the optical parameters of the phantom materials: (a) Muscle (M), (b) Lung (L), (c) Heart (H), and (d) Bone (B).

Download Full Size | PDF

Fig. 12. Luminescent views of the side surface covering cylindrical phantom taken using a CCD camera in four directions 90 degrees apart. (a) Front view, (b) Right view, (c) Back view, and (d) Left view.

Download Full Size | PDF

Fig. 13. (a) Finite element model for a middle portion of the mouse chest phantom. (b) Physical experiment on BLT reconstruction of two sources in the left lung of the mouse chest phantom. The difference between the reconstructed and real source centers was less than 1mm for both the sources at height 15.0mm. The maximum error of source power was about 18.5%.

Download Full Size | PDF

Fig. 14. Comparison between measured and computational photon density profiles along the detection circle on the phantom surface at heights (a) 10.6mm, (b) 15.9mm, and (c) 21.1mm, from the top surface of the model.

Download Full Size | PDF

Tables (3)

Table 1. Optical parameters for the numerical phantom

View Table | View all tables in this article

Table 2. Relative error (%) with BLT results of total source power

View Table | View all tables in this article

Table 3. Optical parameters of the mouse chest phantom.

View Table | View all tables in this article

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

{\begin{matrix} - \nabla \cdot (D (x) ∇Φ (x)) + μ_{a} (x) Φ (x) = S (x) (x ∊ Ω) \\ D (x) = {(3 (μ_{a} (x) + (1 - g) μ_{s} (x)))}^{- 1} \end{matrix}

Φ (x) + 2 A (x; n, n′) D (x) (v (x) \cdot ∇Φ (x)) = 0 (x ∊ \partial Ω)

A (x; n, n′) \approx \frac{(1 + R (x))}{(1 - R (x))}

Q (x) = - D (x) (v \cdot ∇Φ (x)) = \frac{Φ (x)}{(2 A (x; n, n′))} (x ∊ \partial Ω) .

\int_{Ω} (D (x) (\nabla Φ (x)) \cdot (\nabla Ψ (x)) + μ_{a} (x) Φ (x) Ψ (x)) d x

+ \frac{\int_{\partial Ω} Φ (x) Ψ (x)}{(2 A (x; n, n')) d x} = \int_{Ω} S (x) Ψ (x) d x

Φ (x) \approx Φ^{h} (x) = \sum_{k = 1}^{T} ϕ_{k} φ_{k} (x) when x ∊ Ω,

S (x) \approx S^{h} (x) = \sum_{k = 1}^{N_{s}} S_{k} γ_{k} (x) when x ∊ Ω

([K] + [C] + [B]) {Φ} = [M] {Φ} = [F] {S},

{\begin{matrix} k_{ij} = \int_{Ω} (D (x) (\nabla φ_{i} (x)) \cdot (\nabla φ_{i} (x)) d x \\ c_{ij} = \int_{Ω} μ_{a} (x) φ_{i} (x) φ_{j} (x) d x \\ f_{ij} = \int_{Ω} φ_{i} (x) γ_{j} (x) d x \\ b_{ij} = \frac{\int_{\partial Ω} φ_{i} (x) φ_{j} (x)}{(2 A (x : n, n')) d x} \end{matrix}

[\begin{matrix} M_{11} & M_{12} \\ M_{12}^{T} & M_{22} \end{matrix}] {\begin{matrix} Φ^{m} \\ Φ^{*} \end{matrix}} = [\begin{matrix} F_{11} & F_{12} \\ F_{21} & F_{22} \end{matrix}] {\begin{matrix} S^{p} \\ S^{*} \end{matrix}},

(M_{11} - M_{12} M_{22}^{- 1} M_{12}^{T}) Φ^{m} = (F_{11} - M_{12} M_{22}^{- 1} F_{21}) S^{p} .

Φ^{m} = {(M_{11} - M_{12} M_{22}^{- 1} M_{12}^{T})}^{- 1} (F_{11} - M_{12} M_{22}^{- 1} F_{21}) S^{p},

p_{k} = {(\frac{1}{2 π Φ_{k}^{meas}})}^{\frac{1}{2}} \exp [- \frac{{(Φ_{k}^{m} - Φ_{k}^{meas})}^{2}}{2 Φ_{k}^{meas}}]

p (S^{p}) = {(\frac{\det (W)}{π^{M}})}^{\frac{1}{2}} \exp [- \sum_{k = 1}^{M} \frac{{(Φ_{k}^{m} - Φ_{k}^{meas})}^{2}}{2 Φ_{k}^{meas}}],

Θ (S^{p}) = {(Φ^{m} - Φ^{meas})}^{T} W (Φ^{m} - Φ^{meas}) .

min_{U_{i} \geq s_{i} \geq 0} Θ (S^{p}) .

Ω_{s}^{1} = \{(x, y, z)| x < 0, 5.6 < z < 7.0, (x, y, z) ∊ L\},

Ω_{s}^{2} = \{(x, y, z)| x < 0, 5.6 < z < 8.0, (x, y, z) ∊ L\}

Material	Muscle (M)	Lung (L)	Heart (H)	Bone (B)
μ_a [mm^-1]	0.10	0.22	0.21	0.16
μ_s ′ [mm^-1]	1.20	2.30	2.00	1.28

Permissible region	Source	Noise (0%)	Noise (5%)	Noise (10%)	Noise (15%)
$Ω_{s}^{1}$	Source-1	1.0	5.0	13.0	4.8
$Ω_{s}^{1}$	Source-2	0.5	0.5	1.2	9.5
$Ω_{s}^{2}$	Source-1	1.0	5.1	9.1	14.1
$Ω_{s}^{2}$	Source-2	0.5	4.5	9.3	14.4
$Ω_{s}^{3}$	Source-1	7.0	4.2	13.2	24.1
$Ω_{s}^{3}$	Source-2	8.5	23.0	33.0	12.6

Material	Muscle (M)	Lung (L)	Heart (H)	Bone (B)
μ_a [mm^-1]	0.007	0.023	0.011	0.001
μ_s ′ [mm^-1]	1.031	2.000	1.096	0.060

Material	Muscle (M)	Lung (L)	Heart (H)	Bone (B)
μ_a [mm^-1]	0.10	0.22	0.21	0.16
μ_s ′ [mm^-1]	1.20	2.30	2.00	1.28

Permissible region	Source	Noise (0%)	Noise (5%)	Noise (10%)	Noise (15%)
$Ω_{s}^{1}$	Source-1	1.0	5.0	13.0	4.8
$Ω_{s}^{1}$	Source-2	0.5	0.5	1.2	9.5
$Ω_{s}^{2}$	Source-1	1.0	5.1	9.1	14.1
$Ω_{s}^{2}$	Source-2	0.5	4.5	9.3	14.4
$Ω_{s}^{3}$	Source-1	7.0	4.2	13.2	24.1
$Ω_{s}^{3}$	Source-2	8.5	23.0	33.0	12.6

Abstract

Cited By

Figures (14)

Tables (3)

Equations (19)

Optics Express