Uniqueness theorems in bioluminescence tomography

Abstract

Motivated by bioluminescent imaging needs for studies on gene therapy and other applications in the mouse models, a bioluminescence tomography (BLT) system is being developed in the University of Iowa. While the forward imaging model is described by the well-known diffusion equation, the inverse problem is to recover an internal bioluminescent source distribution subject to Cauchy data. The primary goal of this paper is to establish the solution uniqueness for BLT under practical constraints despite the illposedness of the inverse problem in the general case.After a review on the inverse source literature, we demonstrate that in the general case the BLT solution is not unique by constructing the set of all the solutions to this inverse problem. Then, we show the uniqueness of the solution in the case of impulse sources. Finally, we present our main theorem that solid/hollow ball sources can be uniquely determined up to non-radiating sources. For better readability, the exact conditions for and rigorous proofs of the theorems are given in the appendices. Further research directions are also discussed. Index TermsBioluminescence tomography (BLT), diffusion equation, inverse source problem, solution uniqueness.

Identification of multipolar sources in a bioluminescent tomography problem

“…As proven in [17], in the general case where the source F ∈ L 2 (Ω), the source reconstruction is not unique. To overcome this difficulty, we assume that some a priori information on the source are available, depending on the underlying physical problem.…”

“…However, since in biological tissues [16,17], the mean-free path of the particle is between 0.005 and 0.01 mm, the scattering phenomenon dominates on the absorption phenomenon and thus the transport equation is approximated by a diffusion equation [4]. Moreover, since the internal bioluminescent distribution induced by the reporter genes is relatively stable, we consider the stationary diffusion equation where the photon fluence rate u = u(x) satisfies the system:…”

Methods in diffuse optical imaging

“…In the study of Gu et al [14], the bioluminescent source distribution was recovered from a monochromatic dataset and theoretical aspects of uniqueness of solutions were discussed in Wang et al [15] and Cong et al [16]. The idea of using multiple wavelengths in bioluminescence imaging was introduced by Chaudhari et al [17].…”

Theoretical and numerical analysis on multispectral bioluminescence tomography

“…Today, fluorescent and bioluminescent imaging modes are most widely applied in mouse studies, and to a limited extent in clinical research as well. Among various optical molecular imaging techniques, fluorescence molecular tomography (FMT) ( [17]) and bioluminescence tomography (BLT) ( [19,21,7]) are two emerging and complementary modes. In contrast to fluorescent imaging, bioluminescent imaging has unique capabilities in probing molecular and cellular processes.…”