Filtered maximum likelihood expectation maximization based global reconstruction for bioluminescence tomography

Abstract

The reconstruction of bioluminescence tomography (BLT) is severely ill-posed due to the insufficient measurements and diffuses nature of the light propagation. Predefined permissible source region (PSR) combined with regularization terms is one common strategy to reduce such ill-posedness. However, the region of PSR is usually hard to determine and can be easily affected by subjective consciousness. Hence, we theoretically developed a filtered maximum likelihood expectation maximization (fMLEM) method for BLT. Our method can avoid predefining the PSR and provide a robust and accurate result for global reconstruction. In the method, the simplified spherical harmonics approximation (SP_N) was applied to characterize diffuse light propagation in medium, and the statistical estimation-based MLEM algorithm combined with a filter function was used to solve the inverse problem. We systematically demonstrated the performance of our method by the regular geometry- and digital mouse-based simulations and a liver cancer-based in vivo experiment.

Appendix

The detailed expressions of Mand F are as follows:

$$ \left\{\begin{array}{l}{M}_{m,n}^{1,1}={\int}_{\Omega}\left(\frac{1}{3{\mu}_{a1}}\nabla {\psi}_m(r)\nabla {\psi}_n(r)+{\mu}_a{\psi}_m(r){\psi}_n(r)\right) dr-{\int}_{\partial \varOmega}\left(\frac{\xi_{11}}{3{\mu}_{a1}}{\psi}_m(r){\psi}_n(r)\right) dr\\ {}{M}_{m,n}^{1,2}=-{\int}_{\partial \varOmega}\left(\frac{\xi_{12}}{3{\mu}_{a1}}{\psi}_m(r){\psi}_n(r)\right) dr\\ {}{M}_{m,n}^{2,1}=-{\int}_{\varOmega}\left(\frac{2{\mu}_a}{3}{\psi}_m(r){\psi}_n(r)\right) dr-{\int}_{\partial \varOmega}\left(\frac{\xi_{21}}{7{\mu}_{a3}}{\psi}_m(r){\psi}_n(r)\right) dr\\ {}{M}_{m,n}^{2,2}=-{\int}_{\Omega}\left(\frac{1}{7{\mu}_{a3}}\nabla {\psi}_m(r)\nabla {\psi}_n(r)+\left(\frac{4}{9}{\mu}_a+\frac{5}{9}{\mu}_{a2}\right){\psi}_m(r){\psi}_n(r)\right) dr-{\int}_{\partial \varOmega}\left(\frac{\xi_{22}}{7{\mu}_{a3}}{\psi}_m(r){\psi}_n(r)\right) dr\\ {}{F}_{m,n}^{11}={\int}_{\Omega}{\psi}_m(r){\psi}_n(r) dr\\ {}{F}_{m,n}^{22}=\hbox{-} \frac{2}{3}{\int}_{\Omega}{\psi}_m(r){\psi}_n(r) dr\end{array}\right. $$

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where ξ₁₁, ξ₁₂, ξ₂₁, ξ₂₂ could be deduced according to Eq. (1) and could be found in reference [8]. And the coefficients β_i (i = 1, 2) in Eq. (3) could be deduced as follows:

$$ \left\{\begin{array}{l}{\beta}_1=\frac{1}{4}+{J}_0-\left(\frac{0.5+{J}_1}{3{\mu}_{a1}}\right){\xi}_{11}-\left(\frac{J_3}{7{\mu}_{a3}}\right){\xi}_{21}\\ {}{\beta}_2=-\frac{1}{16}-\frac{2}{3}{J}_0+\frac{1}{3}{J}_2-\left(\frac{0.5+{J}_1}{3{\mu}_{a1}}\right){\xi}_{12}-\left(\frac{J_3}{7{\mu}_{a3}}\right){\xi}_{22}\end{array}\right. $$

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