Theoretical Analysis And Numerical Simulation For An Inverse Problem In Bioluminescence Tomography

With the development of science and technology, these years have witnessed the rapid progress in biomedical imaging. Molecular imaging is a rapidly developing biomedical imaging technique for studying physiological and pathological processes in vivo at the cellular and molecular levels. The goal of molecular imaging is to depict non-invasively cellular and molecular processes in vivo sensitively. Molecular imaging is broadly based on three technologies: nuclear imaging,magnetic resonance imaging and optical imaging. Bioluminescent imaging (BLI), as one of the optical imaging modalities, has its own advantages over traditional imaging methods such as computed tomography (CT), positron emission tomography (PET), magnetic resonance imaging (MRI) as well as their combinations. For in vivo studies in humans, optical imaging is largely limited to superficial sites owing to the absorption and scattering properties of tissue, and MRI or PET are preferred modalities. However, in small animals, due to shorter path lenghts, a large fraction of photons reaches the surface of the animal and detection of bioluminescent and fluorescent signals is possible at significantly lower cost compared to MRI and PET. Imaging of bioluminescent sources is particularly attractive for in vivo applications because no external excitation source is needed, and, in turn, background noise is low and sensitivity is high.Bioluminescence tomography (BLT) is a promising BLI because of the possibility of revealing molecular and cellular activities in real time. Over the past several years, numerous work has been devoted to theoretical analysis and numerical simulations of BLT. The main object of BLT is to determine the photon density distribution within small animals or on the superficial of some big organs from the light measurement on the boundary. With BLT, a bioluminescent source distribution inside a living small animal can be localized and quantified in 3D. The first step of BLT is to determine the optical properties of tissue and this is the issue of a diffuse optical tomography (DOT) problem. Because the transport of light in any entity is subject to both absorption and scattering, accurate representation of the photon transport in biological tissue is required. In general, the bioluminescent photon propagation in a tissue can be described accurately by either the radiative transfer equation (RTE) or the Monte Carlo model (MCM). However, at the moment, neither of them is computationally feasible. Usually a diffusion approximation equation of the RTE is employed when the wavelength of light is in the range of around or bigger than 600 nm. Mathematically, BLT is an ill-posed inverse source problem, usually studied through a regularization technique. In this thesis, we discuss theoretical analysis and numerical solution of BLT and its multispectral version. This thesis includes five chapters.In Chapter 1, we at first give a brief introduction of biomedical background of BLT. Then in Section 1.2, its mathematical formulation is presented. At last, in Section 1.3, we give some principle symbols which will be used repeatedly in the following chapters.Chapter 2 is devoted to some theoretical study and numerical experiments for the classic BLT. Section 2.1 gives a presentation of the classic BLT. Then in Section 2.2, under some assumptions, we convert a regularized minimal problem into a couple of partial differential equations (PDEs) by using an adjoint strategy. This makes it possible for reconstructing light source function fast. In Section 2.3, we study the BLT problem through a general framework together with a Tikhonov regularization. For the proposed formulation, we establish a well-posedness result and explore its relation to the formulation studied previously in other papers. In Section 2.4, a new variational formulation for the BLT problem is developed. It is used to explain rigorously the reason behind the loss of the continuous dependence of the light source function solution on the measurements. We also prove an optimal error order in finite element solution. Additionally, by using adjoint equations, a simple but efficient iterative scheme is also explored.In the third chapter, we focus on multispectral BLT. In Section 3.2, based on a penalization strategy, 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. Then in Section 3.3, we first improve finite element error order compared to [1]. In Subsection 3.3.2, a linear finite element is used for the discretization of the space of light source. Using active set strategy, a higher finite element error order is obtained.We continue to discuss BLT extensively in Chapter 4. We study the BLT problem for media with spatially varying refractive index in Section 4.2. We introduce a general framework with Tikhonov regularization for this purpose, present its well-posedness and establish error bounds for its numerical solutions by the finite element methods (FEMs). Numerical results are reported on simulations of the BLT problem for media with spatially varying refractive index. Then in Section 4.3, 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. In practice, the optical parameters are assumed to be piecewise constants or piecewise smooth functions. We seek the optical parameters in the space of functions with bounded variation. We show the solution existence, prove convergence of the numerical solution and introduce numerical schemes.At last, some conclusions are given in Chapter 5. We also list some problems which may deserve future effort.