Rongfang Gong

Bioluminescence tomography for media with spatially varying refractive index

Biomedical imaging has developed into the level of molecular imaging. Bioluminescence tomography (BLT), as an optical imaging modality, is a rapidly developing new and promising field. So far, much of the theoretical analysis of BLT is based on a diffusion approximation equation for media with constant refractive index. In this article, we study the BLT problem for media with spatially varying refractive index. We introduce a general framework with Tikhonov regularization for this purpose, present its well-posedness and establish the error bounds for its numerical solution by the finite element method. Numerical results are reported on simulations of the BLT problem for media with spatially varying refractive index.

A novel coupled complex boundary method for solving inverse source problems

In this paper, we consider an inverse source problem for elliptic partial differential equations with Dirichlet and Neumann boundary data. The unknown source term is to be determined from additional boundary conditions. Unlike the existing methods found in the literature, which usually use some of the boundary conditions to form a boundary value problem for the elliptic partial differential equation and the remaining boundary conditions in the objective functional for optimization to determine the source term, the novel method that we propose here has coupled complex boundary conditions. We use a complex elliptic partial differential equation with a Robin boundary condition coupling the Dirichlet and Neumann boundary data, and optimize with respect to the imaginary part of the solution in the domain to determine the source term. Then, on the basis of the complex boundary value problem, Tikhonov regularization is used to obtain a stable approximate source function and the finite element method is used for discretization. Theoretical analysis is given for both the continuous model and the discrete model. Several numerical examples are provided to show the usefulness of the proposed coupled complex boundary method.

A novel approach for studies of multispectral bioluminescence tomography

Bioluminescence tomography (BLT) is a promising new area in biomedical imaging. The goal of BLT is to provide quantitative reconstruction of bioluminescent source distribution within a small animal from optical signals on the animal’s body surface. The multispectral version of BLT takes advantage of the measurement information in different spectrum bands. In this paper, we propose a novel approach for the multispectral BLT. The new feature of the mathematical framework is to use numerical prediction results based on two related but distinct boundary value problems. This mathematical framework includes the conventional framework in the study of multispectral BLT. For the new framework introduced here, we establish the solution existence, uniqueness and continuous dependence on data, and characterize the limiting behaviors when the regularization parameter approaches zero or when the penalty parameter approaches infinity. We study two kinds of numerical schemes for multispectral BLT and derive error estimates for the numerical solutions. We also present numerical examples to show the performance of the numerical methods.

A fast solver for an inverse problem arising in bioluminescence tomography

Bioluminescence tomography (BLT) is a new method in biomedical imaging, with a promising potential in monitoring non-invasively physiological and pathological processes in vivo at the cellular and molecular levels. The goal of BLT is to quantitatively reconstruct a three dimensional bioluminescent source distribution within a small animal from two dimensional optical signals on the surface of the animal body. Mathematically, BLT is an under-determined inverse source problem and is severely ill-posed, making its numerical treatments very challenging. In this paper, we provide a new Tikhonov regularization framework for the BLT problem. Compared with the existing reconstruction methods about BLT, our new method uses an energy functional defined over the whole problem domain for measuring the data fitting, associated with two related but different boundary value problems. Based on the new formulation, a fast solver is introduced by transforming the proposed optimization model into a system of partial differential equations. Moreover, a finite element method is used to obtain a regularized discrete solution. Finally, numerical results show that the fast solver for BLT is feasible and effective.

A general framework for integration of bioluminescence tomography and diffuse optical tomography

Diffuse optical tomography (DOT) and bioluminescence tomography (BLT) are different but complementary modalities in optical molecular imaging. The goal of DOT is to reconstruct optical parameters of a medium from surface measurements induced by external sources, whereas that of BLT is to reconstruct running a bioluminescent source distribution in the medium from surface measurements induced by internal bioluminescent sources. A pre-requisite for BLT reconstruction is knowledge on the distribution of optical parameters within the medium, which is the output of DOT. In this paper, we propose a mathematical model integrating BLT and DOT at the fundamental level; that is, performing the two types of reconstructions simultaneously instead of doing them sequentially. The model is introduced through minimizing the difference between predicted quantities and boundary measurements, as well as incorporating regularization terms. In practice, the optical parameters are assumed to be piecewise constants or piecewise smooth functions. Hence, we seek the optical parameters in the space of functions with bounded variation (BV). For the BV-regularized data-fitting functional and its discretized approximations, we demonstrate the existence of minima and show convergence of the minima of the discretized functionals to that of the non-discretized one. We also present numerical results to illustrate the strength of our proposed model.

A coupled complex boundary method for an inverse conductivity problem with one measurement

We recently proposed in [Cheng, XL et al. A novel coupled complex boundary method for inverse source problems Inverse Problem 2014 30 055002] a coupled complex boundary method (CCBM) for inverse source problems. In this paper, we apply the CCBM to inverse conductivity problems (ICPs) with one measurement. In the ICP, the diffusion coefficient q is to be determined from both Dirichlet and Neumann boundary data. With the CCBM, q is sought such that the imaginary part of the solution of a forward Robin boundary value problem vanishes in the problem domain. This brings in advantages on robustness and computation in reconstruction. Based on the complex forward problem, the Tikhonov regularization is used for a stable reconstruction. Some theoretical analysis is given on the optimization models. Several numerical examples are provided to show the feasibility and usefulness of the CCBM for the ICP. It is illustrated that as long as all the subdomains share some portion of the boundary, our CCBM-based Tikhonov regularization method can reconstruct the diffusion parameters stably and effectively.

A coupled complex boundary method for the Cauchy problem

Considered in this paper is a Cauchy problem governed by an elliptic partial differential equation. In the Cauchy problem, one wants to recover the unknown Neumann and Dirichlet data on a part of the boundary from the measured Neumann and Dirichlet data, usually contaminated with noise, on the remaining part of the boundary. The Cauchy problem is an inverse problem with severe ill-posedness. In this paper, a coupled complex boundary method (CCBM), originally proposed in [Cheng XL, Gong RF, Han W, et al. A novel coupled complex boundary method for solving inverse source problems. Inverse Prob. 2014;30:055002], is applied to solve the Cauchy problem stably. With the CCBM, all the data, including the known and unknown ones on the boundary are used in a complex Robin boundary on the whole boundary. As a result, the Cauchy problem is transferred into a complex Robin boundary problem of finding the unknown data such that the imaginary part of the solution equals zero in the domain. Then the Tikhonov regularization is applied to the resulting new formulation. Some theoretical analysis is performed on the CCBM-based Tikhonov regularization framework. Moreover, through the adjoint technique, a simple solver is proposed to compute the regularized solution. The finite-element method is used for the discretization. Numerical results are given to show the feasibility and effectiveness of the proposed method.

A homotopy method for bioluminescence tomography

Abstract Bioluminescence tomography (BLT) aims at the determination of the distribution of a bioluminescent source quantitatively. The mathematical problem involved is an inverse source problem and is ill-posed. With the Tikhonov regularization, an optimization problem is formed for the light source reconstruction and it is usually solved by gradient-type methods. However, such iterative methods are often locally convergent and thus the solution accuracy depends largely on initial guesses. In this paper, we reformulate the reduced regularized optimal problem as a nonlinear equation and apply a homotopy method, which is a powerful tool for solving nonlinear problem due to its globally convergent property, to it. Numerical experiments show that the application of the homotopy technique is feasible and can produce satisfactory approximate solutions for a very large range of initial guesses.