Application Of The Finite Element Method And Regularization Strategy For The Optical Molecular Imaging

Molecular imaging is a promising and rapidly developing biomedical research field, which enables people to non-invasively reveal the molecular activities of living body in vivo. Compared with the traditional imaging modalities, molecular imaging can investigate the same objects continually, thus making the corresponding research more convincing. Currently, molecular imaging technique has been widely applied to early disease diagnosis, tumor cell detection and drug improvement. Among the various molecular imaging modalities, optical molecular imaging has become a research focus over the past years due to the advantages of non-radiativity, high sensitivity, rapid measurement and lowcost.The transmission of photon in the biological tissues is complex, and the image reconstruction is a severely ill-posed problem due to the underdetermined linear equations in mathematics. How to establish the appropriate forward model and get accurate reconstruction result still remain a problem. The main contents of this dissertation can be summarized as follows:An extended finite element method (XFEM) for the forward model of 3-D optical molecular imaging is developed with simplified spherical harmonics approximation (SPN). In order to improve the accuracy of the forward model, we use the SPN equations instead of the diffusion equation, which is not suitable for tissues with high absorption. Owing to the complex and curvilinear geometries associated with the biological tissues, FEM is necessary, but it needs the fine mesh to conform the complex internal boundary of the tissue, which leads high computation cost. In the XFEM scheme of SPN equations, the signed distance function is employed to accurately represent the internal tissue boundary, and then it is used to construct the enriched basis function of the finite element scheme. Therefore, the finite element calculation can be carried out without the time-consuming internal boundary mesh generation. Moreover, the required overly fine mesh conforming to the complex tissue boundary which leads to excess time cost can be avoided. XFEM conveniences its application to tissues with complex internal structure and improves the computational efficiency. Phantom and digital mouse experiments were carried out to validate the efficiency of the proposed method. Compared with standard finite element method and classical Monte Carlo (MC) method, the validation results show the merits and potential of the XFEM for optical imaging.Bioluminescence Tomography(BLT) has attracted more attention in the area of optical whole body molecular imaging and it has been studied for about several years. The adaptive finite element method (FEM) is broadly adopted to predict light propagation or recover the internal source in biological tissues. In the adaptive progress, the error estimation indicates which region needs to be refined, to make full use of the error estimation and get the banlace between the accuracy and efficency. A novel dual-mesh alternation strategy (dual-mesh A-FEM) is developed for BLT. Considering the error estimation of the FEM solution on each mesh comprehensively, two different adaptive strategies according to the error indicator of the reconstructed source and the photon flux density are used alternately in the process. Combined with the intelligent permissible region selected in the adaptive process, the new algorithm can achieve more accurate source location compared with the previous adaptive FEM. Both numerical and mouse experiments were conducted to demonstrate the potential of the algorithm for BLT applications.As an emerging optical imaging technology, Fluorescence Molecular Tomography can efficiently reconstruct optical parameters of the fluorophore and diagnose early cancer. For the purpose of improving the performance of the reconstructed image, simplified spherical harmonics approximation based on radiative transport equation (RTE) is proposed to perform the forward simulation. To overcome the influence of the ill-posedness and receive a more accurate reconstructed image, Laplace regularization method is introduced to reconstruct the inverse problem. Compared to the classical Tikhonov method, the numerical experiments reveal that the proposed method can reconstruct the small object and multi-object and improve the image resolution.In order to reduce the severely ill-posedness of the BLT inverse problem, l2 regularization methods are widely used in the BLT, but they may lead to the overly smooth of the result. Based on the compressed sensing theory and considering the sparsity of bioluminescent source distribution, a comparison of five l1 regularization methods is conducted to get the sparse result. Numerical simulations of a 3D mouse atlas demonstrate the robustness of the five methods. In addition, Homotopy and Iterative Shrinkage Thresholding method (IST) are more effective than the other three methods.