Improved bioluminescence and fluorescence reconstruction algorithms using diffuse optical tomography, normalized data, and optimized selection of the permissible source region

Mohamed A. Naser; Michael S. Patterson

doi:10.1364/BOE.2.000169

Improved bioluminescence and fluorescence reconstruction algorithms using diffuse optical tomography, normalized data, and optimized selection of the permissible source region

Mohamed A. Naser and Michael S. Patterson

Biomedical Optics Express
Vol. 2,
Issue 1,
pp. 169-184
(2011)
•https://doi.org/10.1364/BOE.2.000169

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Mohamed A. Naser and Michael S. Patterson, "Improved bioluminescence and fluorescence reconstruction algorithms using diffuse optical tomography, normalized data, and optimized selection of the permissible source region," Biomed. Opt. Express 2, 169-184 (2011)

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Related Topics
Table of Contents Category
- Image Reconstruction and Inverse Problems
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Medical and biological imaging (170.3880)
- Tomography (170.6960)

About this Article
History
- Original Manuscript: October 13, 2010
- Revised Manuscript: December 16, 2010
- Manuscript Accepted: December 17, 2010
- Published: December 20, 2010

Accessible

Open Access

Abstract

Reconstruction algorithms are presented for two-step solutions of the bioluminescence tomography (BLT) and the fluorescence tomography (FT) problems. In the first step, a continuous wave (cw) diffuse optical tomography (DOT) algorithm is used to reconstruct the tissue optical properties assuming known anatomical information provided by x-ray computed tomography or other methods. Minimization problems are formed based on L1 norm objective functions, where normalized values for the light fluence rates and the corresponding Green’s functions are used. Then an iterative minimization solution shrinks the permissible regions where the sources are allowed by selecting points with higher probability to contribute to the source distribution. Throughout this process the permissible region shrinks from the entire object to just a few points. The optimum reconstructed bioluminescence and fluorescence distributions are chosen to be the results of the iteration corresponding to the permissible region where the objective function has its global minimum This provides efficient BLT and FT reconstruction algorithms without the need for a priori information about the bioluminescence sources or the fluorophore concentration. Multiple small sources and large distributed sources can be reconstructed with good accuracy for the location and the total source power for BLT and the total number of fluorophore molecules for the FT. For non-uniform distributed sources, the size and magnitude become degenerate due to the degrees of freedom available for possible solutions. However, increasing the number of data points by increasing the number of excitation sources can improve the accuracy of reconstruction for non-uniform fluorophore distributions.

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References

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[Crossref] [PubMed]

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[Crossref]

L. Cao, M. Breithaupt, and J. Peter, “Geometrical co-calibration of a tomographic optical system with CT for intrinsically co-registered imaging,” Phys. Med. Biol. 55(6), 1591–1606 (2010).
[Crossref] [PubMed]

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[Crossref] [PubMed]

M. A. Naser and M. S. Patterson, “Algorithms for bioluminescence tomography incorporating anatomical information and reconstruction of tissue optical properties,” Biomed. Opt. Express 1(2), 512–526 (2010).
[Crossref]

2009 (2)

Y. Lin, H. Yan, O. Nalcioglu, and G. Gulsen, “Quantitative fluorescence tomography with functional and structural a priori information,” Appl. Opt. 48(7), 1328–1336 (2009).
[Crossref] [PubMed]

Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express 17(10), 8062–8080 (2009).
[Crossref] [PubMed]

2008 (6)

Y. Tan and H. Jiang, “Diffuse optical tomography guided quantitative fluorescence molecular tomography,” Appl. Opt. 47(12), 2011–2016 (2008).
[Crossref] [PubMed]

J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express 16(20), 15640–15654 (2008).
[Crossref] [PubMed]

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[Crossref] [PubMed]

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27(5), 48–57 (2008).
[Crossref] [PubMed]

S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol. 53(14), 3921–3942 (2008).
[Crossref] [PubMed]

S. C. Davis, B. W. Pogue, R. Springett, C. Leussler, P. Mazurkewitz, S. B. Tuttle, S. L. Gibbs-Strauss, S. S. Jiang, H. Dehghani, and K. D. Paulsen, “Magnetic resonance-coupled fluorescence tomography scanner for molecular imaging of tissue,” Rev. Sci. Instrum. 79(6), 064302 (2008).
[Crossref] [PubMed]

2007 (8)

M. Guven, B. Yazici, and V. Ntziachristos, “Fluorescence diffuse optical image reconstruction with a priori information,” Proc. SPIE 6431, 643107, 643107-15 (2007).
[Crossref]

C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt. 12(2), 024007 (2007).
[Crossref] [PubMed]

P. Mohajerani, A. A. Eftekhar, J. Huang, and A. Adibi, “Optimal sparse solution for fluorescent diffuse optical tomography: theory and phantom experimental results,” Appl. Opt. 46(10), 1679–1685 (2007).
[Crossref] [PubMed]

S. C. Davis, H. Dehghani, J. Wang, S. Jiang, B. W. Pogue, and K. D. Paulsen, “Image-guided diffuse optical fluorescence tomography implemented with Laplacian-type regularization,” Opt. Express 15(7), 4066–4082 (2007).
[Crossref] [PubMed]

A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescence sources,” J. Opt. Soc. Am. A 24(6), 1601–1608 (2007).
[Crossref]

L. Hervé, A. Koenig, A. Da Silva, M. Berger, J. Boutet, J. M. Dinten, P. Peltié, and P. Rizo, “Noncontact fluorescence diffuse optical tomography of heterogeneous media,” Appl. Opt. 46(22), 4896–4906 (2007).
[Crossref] [PubMed]

N. Cao, A. Nehorai, and M. Jacobs, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express 15(21), 13695–13708 (2007).
[Crossref] [PubMed]

X. Song, D. Wang, N. Chen, J. Bai, and H. Wang, “Reconstruction for free-space fluorescence tomography using a novel hybrid adaptive finite element algorithm,” Opt. Express 15(26), 18300–18317 (2007).
[Crossref] [PubMed]

2006 (4)

H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, “Spectrally resolved bioluminescence optical tomography,” Opt. Lett. 31(3), 365–367 (2006).
[Crossref] [PubMed]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Effect of optical property estimation accuracy on tomographic bioluminescence imaging: simulation of a combined optical-PET (OPET) system,” Phys. Med. Biol. 51(8), 2045–2053 (2006).
[Crossref] [PubMed]

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys. 33(1), 61–68 (2006).
[Crossref] [PubMed]

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantification of bioluminescence images of point source objects using diffusion theory models,” Phys. Med. Biol. 51(15), 3733–3746 (2006).
[Crossref] [PubMed]

2005 (4)

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005).
[Crossref] [PubMed]

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50(23), 5421–5441 (2005).
[Crossref] [PubMed]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50(17), 4225–4241 (2005).
[Crossref] [PubMed]

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13(18), 6756–6771 (2005).
[Crossref] [PubMed]

2004 (3)

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9(3), 488–496 (2004).
[Crossref] [PubMed]

X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express 12(17), 3996–4000 (2004).
[Crossref] [PubMed]

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. 31(8), 2289–2299 (2004).
[Crossref] [PubMed]

2003 (1)

A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. 42(16), 3081–3094 (2003).
[Crossref] [PubMed]

2002 (1)

C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4(1), 235–260 (2002).
[Crossref] [PubMed]

2001 (1)

R. Weissleder and U. Mahmood, “Molecular imaging,” Radiology 219(2), 316–333 (2001).
[PubMed]

1999 (2)

A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging 18(3), 262–271 (1999).
[Crossref] [PubMed]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), 022 (1999).
[Crossref]

1998 (1)

H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. 37(22), 5337–5343 (1998).
[Crossref] [PubMed]

1995 (1)

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[Crossref] [PubMed]

1993 (1)

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[Crossref] [PubMed]

Adibi, A.

Ahn, S.

Ale, A.

Alexandrakis, G.

Antich, P. P.

Arridge, S. R.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), 022 (1999).
[Crossref]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[Crossref] [PubMed]

Bachmann, M. H.

C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4(1), 235–260 (2002).
[Crossref] [PubMed]

Bading, J. R.

Bai, J.

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27(5), 48–57 (2008).
[Crossref] [PubMed]

Bao, S.

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27(5), 48–57 (2008).
[Crossref] [PubMed]

Berger, M.

Boas, D. A.

A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. 42(16), 3081–3094 (2003).
[Crossref] [PubMed]

Bouman, C. A.

A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. 42(16), 3081–3094 (2003).
[Crossref] [PubMed]

Boutet, J.

Breithaupt, M.

Cao, L.

Cao, N.

Chan, T. F.

Chatziioannou, A. F.

Chaudhari, A. J.

Chen, N.

Chen, X.

Cherry, S. R.

Comsa, D. C.

Cong, A.

Cong, W.

Contag, C. H.

C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4(1), 235–260 (2002).
[Crossref] [PubMed]

Conti, P. S.

Coquoz, O.

Da Silva, A.

Darvas, F.

Davis, S. C.

Dehghani, H.

Delpy, D. T.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[Crossref] [PubMed]

Dinkelborg, L. M.

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. 7(7), 591–607 (2008).
[Crossref] [PubMed]

Dinten, J. M.

Douraghy, A.

Eftekhar, A. A.

Eppstein, M. J.

Farrell, T. J.

Feng, J.

Gambhir, S. S.

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. 7(7), 591–607 (2008).
[Crossref] [PubMed]

Gibbs-Strauss, S. L.

Godavarty, A.

Gu, X.

X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express 12(17), 3996–4000 (2004).
[Crossref] [PubMed]

Gulsen, G.

Y. Lin, H. Yan, O. Nalcioglu, and G. Gulsen, “Quantitative fluorescence tomography with functional and structural a priori information,” Appl. Opt. 48(7), 1328–1336 (2009).
[Crossref] [PubMed]

Gurfinkel, M.

Guven, M.

M. Guven, B. Yazici, and V. Ntziachristos, “Fluorescence diffuse optical image reconstruction with a priori information,” Proc. SPIE 6431, 643107, 643107-15 (2007).
[Crossref]

Hanson, K. M.

He, X.

Hervé, L.

Hielscher, A. H.

Hiraoka, M.

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[Crossref] [PubMed]

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J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27(5), 48–57 (2008).
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R. Weissleder and U. Mahmood, “Molecular imaging,” Radiology 219(2), 316–333 (2001).
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Fig. 1 Schematic of the object used in the simulation illustrating 6 different regions: E1 is the skin, E2 is the bowel, E3 and E4 are the kidneys, E5 is bone and E6 is adipose tissue. The source-detectors setup is also shown in the figure; 5 sources indicated by the star uniformly distributed in a field of view of 90°. 121 detectors forming an approximately 270° of view; the total number of data readings is 5x121 = 605 for DOT and FT. The total number of reading used for BLT is 3 (wavelengths) x121 which is 363.

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Fig. 2 The objective function as a function of the number of reconstructed BL source points in the permissible region for the true source distribution of Fig. 3(a).

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Fig. 3 BLT reconstruction of two uniform bioluminescence sources localized in the left and right kidneys. The actual sources in (a) are 3mm in diameter with magnitude 1 nW/mm. The reconstructed sources are shown in (d). The actual sources in (b) are 3mm in diameter with magnitude 0.5 nW/mm² for the left source and 1 nW/mm² for the right source. The reconstructed sources shown in (e) have total powers within 20% of the true value but the size and magnitude are not accurately estimated. In (c) the left source is 2 mm diameter with magnitude 1 nW/mm² and the right source is the same magnitude with diameter 3 mm. The reconstructed sources in (f) have total powers within 10% and the size and magnitude are also recovered with comparable accuracy.

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Fig. 4 BLT reconstruction of a large distributed source that fill the gut region. The magnitude of the actual sources in (a) is 1 nW/mm², and the reconstructed source is shown in (c). For the nonuniform source in (b), the reconstructed source in (d) shows almost uniform magnitude and smaller size than the true source.

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Fig. 5 FT reconstruction of two uniform fluorophore sources localized in the left and right kidneys. The actual sources in (a) are 3 mm in diameter with concentration 1 µmol/mm². The reconstructed sources are shown in (d) and they have total number of molecules within 10% of the true value. The actual sources in (b) are 3mm in diameter with concentration 0.5 µmol/mm² for the left source and 1 µmol/mm² for the right source. The reconstructed sources shown in (e) have total number of molecules within 15% of the true value but the size and magnitude are not accurately estimated. In (c) the left source is 2 mm diameter with concentration 1 µmol /mm² and the right source is the same concentration with diameter 3 mm. The reconstructed sources in (f) have total number of molecules within 10% and the size and magnitude are also recovered with comparable accuracy.

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Fig. 6 FT reconstruction of a large distributed fluorophore source that fills the gut region. The concentration of the actual sources in (a) is 1 µmol/mm², and the reconstructed source is shown in (c). For the nonuniform source in (b), the reconstructed source in (d) shows almost uniform magnitude and smaller size than the true source.

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Fig. 7 (a) The actual fluorophore concentration (same as Fig. 6(b)). (b) The reconstructed fluorophore concentration using 16 excitation sources uniformly distributed around the object; associated with every source are 121 detectors uniformly distributed in a field of view of 270 degree.

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Tables (4)

Table 1 Algorithm description for BLT

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Table 2 Algorithm description for FT

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Table 3 Scattering coefficients for each region in the heterogeneous object calculated at five different wavelengths. The values of the scattering coefficients are given in (mm⁻¹).

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Table 4 Absorption coefficients for each region in the heterogeneous object calculated at five different wavelengths. The values of the absorption coefficients are given in (mm⁻¹).

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Equations (15)

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[- (\frac{\partial}{\partial x} κ (x, y; λ) \frac{\partial}{\partial x} + \frac{\partial}{\partial y} κ (x, y; λ) \frac{\partial}{\partial y}) + μ_{a} (x, y; λ)] φ (x, y; λ) = s (x, y; λ),

κ (x, y; λ) = \frac{1}{3 (μ_{a} (x, y; λ) + {μ^{'}}_{s} (x, y; λ))},

φ (x, y; λ; ω) + 2 κ (x, y; λ) A \hat{n} (x, y) \cdot \nabla φ (x, y; λ; ω) = 0,

A = \frac{(2 / (1 - R_{0})) - 1 + {| \cos θ_{c} |}^{3}}{1 - {| \cos θ_{c} |}^{2}},

{[K]}_{N \times N} φ_{N \times 1} = {[A_{s s}]}_{N \times N} s_{N \times 1},

φ = {[K]}^{- 1} [A_{s s}] s = G s

G = {[K]}^{- 1} [A_{s s}]

φ_{d} = G_{d} s

\begin{array}{l} f =min \sum_{λ} {‖ \tilde{G} (d, R; λ) s (R) - \tilde{φ} (d; λ) ‖}^{1} \\ s . t . 0 \leq s \leq s_{\max} \\ s . t . R \in Permissible Region \end{array}

[- (\frac{\partial}{\partial x} κ_{x} (x, y) \frac{\partial}{\partial x} + \frac{\partial}{\partial y} κ_{x} (x, y) \frac{\partial}{\partial y}) + μ_{a x} (x, y)] φ_{x} (x, y) = s (x, y),

[- (\frac{\partial}{\partial x} κ_{m} (x, y) \frac{\partial}{\partial x} + \frac{\partial}{\partial y} κ_{m} (x, y) \frac{\partial}{\partial y}) + μ_{a m} (x, y)] φ_{m} (x, y) = φ_{x} (x, y) η μ_{a f} (x, y),

φ_{x} = {[K_{x}]}^{- 1} [A_{s s}] s

G_{m} = {[K_{m}]}^{- 1} [A_{s s}]

φ_{m} = G_{m} φ_{x} η μ_{a f} = G_{m} φ_{x} η \ln (10) ε C

\begin{array}{l} f =min \sum_{i = 1}^{N_{s}} {‖ (\ln (10) ε η) {\tilde{G}}_{m} (d_{i}, R) φ_{x i} (R) C (R) - {\tilde{φ}}_{m} (d_{i}) ‖}^{1} \\ s . t . 0 \leq C \leq C_{\max} \\ s . t . R \in Permissible Region \end{array}

Algorithm for BLT reconstruction
1. Calculate the Green’s function from Eq. (7)
2. Choose the rows corresponding to the detector points $G (d, :)$
3. Normalize the light fluence rate at every wavelength and get the corresponding Green’s function, $\tilde{φ} (d; λ) = φ (d; λ) / \max (φ (d; λ))$ and $\tilde{G} (d, R; λ) = G (d, R; λ) / \max (φ (d; λ))$
4. Initialize the permissible region R: points corresponding to the whole object domain except points of one transport length $1 / {μ^{'}}_{s}$ from the object boundary. $s (R) = zeros (length (R), 1)$
5. Initialize permissible region reduction factor $β = (N_{i} / N_{f})^(1 / N_{I T})$ , where $N_{i}$ is the initial number of points in the permissible region R, $N_{f}$ is the final number of points which can be chosen to be one, and $N_{I T}$ is the number of iterations.
6. Solve the iterative minimization problem:
	6.1. for i = 1: $N_{I T}$ solve the minimization problem in Eq. (9) and store the result $f (i) =min \sum_{λ} {‖ \tilde{G} (d, R; λ) s (R) - \tilde{φ} (d; λ) ‖}^{1}$
	6.2. Sort the source values and choose number of points equals $length (R) / β$ corresponding to the highest source values
	6.3. Update the permissible region R to be the new chosen points
	6.4. If the value of f(i) is smaller than the previous iteration, then update the best source estimation to be s: $s_{best} = s$
	6.5. End the for loop
7. End

Algorithm for FT reconstruction
1. Calculate the excitation light fluence rate due to every excitation source i, $φ_{x i}$ using Eq. (12)
2. Calculate the Green function at the emission wavelength, $G_{m}$ using Eq. (13)
3. Choose the rows corresponding to the detector points associated with every source i, $G_{m} (d_{i}, :)$
4. Normalize the emission light fluence rate at detectors corresponding to excitation source i and get the corresponding Green’s function, ${\tilde{φ}}_{m} (d_{i}) = φ_{m} (d_{i}) / \max (φ_{m} (d_{i}))$ and ${\tilde{G}}_{m} (d_{i}, R) = G_{m} (d_{i}, R) / \max (φ_{m} (d_{i}))$
5. Initialize the permissible region R: points corresponding to the whole object domain except points of one transport length $1 / {μ^{'}}_{s}$ from the object boundary. $C (R) = zeros (length (R), 1)$
6. Initialize permissible region reduction factor $β = (N_{i} / N_{f})^(1 / N_{I T})$ , where $N_{i}$ is the initial number of points in the permissible region R, $N_{f}$ is the final number of points which can be chosen to be one, and $N_{I T}$ is the number of iterations.
7. Solve the iterative minimization problem:
	7.1. for j = 1: $N_{I T}$ solve the minimization problem in Eq. (15) and store the result $f (j) =min \sum_{i = 1}^{N_{s}} {‖ (\ln (10) ε η) {\tilde{G}}_{m} (d_{i}, R) φ_{x i} (R) C (R) - {\tilde{φ}}_{m} (d_{i}) ‖}^{1}$
	7.2. Sort the fluorophore concentration values and choose number of points equals $length (R) / β$ corresponding to the highest concentration values.
	7.3. Update the permissible region R to be the new chosen points.
	7.4. If the value of f(j) is smaller than the previous iteration, then update the best fluorophore concentration estimation to be C: $C_{best} = C$
	7.5. End the for loop
8. End

	580 nm		600 nm		620 nm		785 nm		830 nm
	${μ^{'}}_{s}$	${μ^{'}}_{s r}$	${μ^{'}}_{s}$	${μ^{'}}_{s r}$	${μ^{'}}_{s}$	${μ^{'}}_{s r}$	${μ^{'}}_{s}$	${μ^{'}}_{s r}$	${μ^{'}}_{s}$	${μ^{'}}_{s r}$
Skin (1)	2.60	2.51	2.51	2.32	2.42	2.52	1.86	2.13	1.75	1.78
Bowel (2)	1.37	1.34	1.32	1.24	1.27	1.28	0.94	1.05	0.88	0.89
Kidney (3)	2.80	2.33	2.66	2.69	2.53	2.82	1.77	1.94	1.63	1.62
Kidney (4)	2.80	2.37	2.66	2.28	2.53	2.33	1.77	1.77	1.63	1.56
Bone (5)	3.08	2.92	2.93	2.90	2.80	2.81	1.98	1.99	1.82	1.59
Adipose (6)	1.30	1.29	1.28	1.24	1.26	1.26	1.11	1.20	1.08	1.06

	580 nm		600 nm		620 nm		785 nm		830 nm
	$μ_{a}$	$μ_{a r}$	$μ_{a}$	$μ_{a r}$	$μ_{a}$	$μ_{a r}$	$μ_{a}$	$μ_{a r}$	$μ_{a}$	$μ_{a r}$
Skin (1)	0.1299	0.1314	0.0391	0.0423	0.0178	0.0180	0.0305	0.0259	0.0261	0.0246
Bowel (2)	0.0202	0.0206	0.0073	0.0079	0.0041	0.0040	0.0057	0.0051	0.0053	0.0050
Kidney (3)	0.1216	0.1292	0.0372	0.0404	0.0176	0.0167	0.0292	0.0240	0.0254	0.0237
Kidney (4)	0.1216	0.1284	0.0372	0.0422	0.0176	0.0195	0.0292	0.0266	0.0254	0.0255
Bone (5)	0.1042	0.1083	0.0328	0.0338	0.0146	0.0135	0.0247	0.0241	0.0201	0.0220
Adipose (6)	0.0077	0.0076	0.0031	0.0028	0.0022	0.0023	0.0027	0.0028	0.0029	0.0034

Algorithm for BLT reconstruction
1. Calculate the Green’s function from Eq. (7)
2. Choose the rows corresponding to the detector points $G (d, :)$
3. Normalize the light fluence rate at every wavelength and get the corresponding Green’s function, $\tilde{φ} (d; λ) = φ (d; λ) / \max (φ (d; λ))$ and $\tilde{G} (d, R; λ) = G (d, R; λ) / \max (φ (d; λ))$
4. Initialize the permissible region R: points corresponding to the whole object domain except points of one transport length $1 / {μ^{'}}_{s}$ from the object boundary. $s (R) = zeros (length (R), 1)$
5. Initialize permissible region reduction factor $β = (N_{i} / N_{f})^(1 / N_{I T})$ , where $N_{i}$ is the initial number of points in the permissible region R, $N_{f}$ is the final number of points which can be chosen to be one, and $N_{I T}$ is the number of iterations.
6. Solve the iterative minimization problem:
	6.1. for i = 1: $N_{I T}$ solve the minimization problem in Eq. (9) and store the result $f (i) =min \sum_{λ} {‖ \tilde{G} (d, R; λ) s (R) - \tilde{φ} (d; λ) ‖}^{1}$
	6.2. Sort the source values and choose number of points equals $length (R) / β$ corresponding to the highest source values
	6.3. Update the permissible region R to be the new chosen points
	6.4. If the value of f(i) is smaller than the previous iteration, then update the best source estimation to be s: $s_{best} = s$
	6.5. End the for loop
7. End