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Determining Sources in the Bioluminescence Tomography Problem

Ming-Hui Ding, Rongfang Gong, Hongyu Liu, Catharine W. K. Lo

TL;DR

This paper addresses the bioluminescence tomography (BLT) inverse source problem under the diffusion approximation, aiming to reconstruct an internal light source from boundary measurements. It advances the theory by proving uniqueness for sources of the form $q=\varphi\chi_\omega$ when the domain $\omega$ is either a $C^2$-smooth region or a polyhedral/corona-shaped region, using a single boundary datum, and it supports these results with comprehensive numerical experiments. The authors develop a microlocal CGO-based analysis to handle corners and non-smooth geometries, and they implement a Levenberg-Marquardt inversion with Tikhonov regularization to numerically recover both the support and intensity of the sources. The work broadens the class of geometries for which BLT uniqueness can be guaranteed and demonstrates practical recoverability for both smooth and non-smooth source domains, with potential impact on in vivo imaging and related biomedical applications.

Abstract

In this paper, we revisit the bioluminescence tomography (BLT) problem, where one seeks to reconstruct bioluminescence signals (an internal light source) from external measurements of the Cauchy data. As one kind of optical imaging, the BLT has many merits such as high signal-to-noise ratio, non-destructivity and cost-effectiveness etc., and has potential applications such as cancer diagnosis, drug discovery and development as well as gene therapies and so on. In the literature, BLT is extensively studied based on diffusion approximation (DA) equation, where the distribution of peak sources is to be reconstructed and no solution uniqueness is guaranteed without adequate a priori information. Motivated by the solution uniqueness issue, several theoretical results are explored. The major contributions in this work that are new to the literature are two-fold: first, we show the theoretical uniqueness of the BLT problem where the light sources are in the shape of $C^2$ domains or polyhedral- or corona-shaped; second, we support our results with plenty of problem-orientated numerical experiments.

Determining Sources in the Bioluminescence Tomography Problem

TL;DR

This paper addresses the bioluminescence tomography (BLT) inverse source problem under the diffusion approximation, aiming to reconstruct an internal light source from boundary measurements. It advances the theory by proving uniqueness for sources of the form when the domain is either a -smooth region or a polyhedral/corona-shaped region, using a single boundary datum, and it supports these results with comprehensive numerical experiments. The authors develop a microlocal CGO-based analysis to handle corners and non-smooth geometries, and they implement a Levenberg-Marquardt inversion with Tikhonov regularization to numerically recover both the support and intensity of the sources. The work broadens the class of geometries for which BLT uniqueness can be guaranteed and demonstrates practical recoverability for both smooth and non-smooth source domains, with potential impact on in vivo imaging and related biomedical applications.

Abstract

In this paper, we revisit the bioluminescence tomography (BLT) problem, where one seeks to reconstruct bioluminescence signals (an internal light source) from external measurements of the Cauchy data. As one kind of optical imaging, the BLT has many merits such as high signal-to-noise ratio, non-destructivity and cost-effectiveness etc., and has potential applications such as cancer diagnosis, drug discovery and development as well as gene therapies and so on. In the literature, BLT is extensively studied based on diffusion approximation (DA) equation, where the distribution of peak sources is to be reconstructed and no solution uniqueness is guaranteed without adequate a priori information. Motivated by the solution uniqueness issue, several theoretical results are explored. The major contributions in this work that are new to the literature are two-fold: first, we show the theoretical uniqueness of the BLT problem where the light sources are in the shape of domains or polyhedral- or corona-shaped; second, we support our results with plenty of problem-orientated numerical experiments.
Paper Structure (12 sections, 12 theorems, 89 equations, 11 figures, 1 algorithm)

This paper contains 12 sections, 12 theorems, 89 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setup and Background
  3. Discussion and Organisation of this Paper
  4. Preliminaries
  5. Smooth Domains
  6. Polyhedral Domains
  7. Geometrical Setup
  8. Proof of Main Results
  9. Inversion algorithm
  10. Numerical results and discussion
  11. Smooth Domains
  12. Polyhedral Domains

Key Result

Theorem 2.1

For any $q\in L^2(\Omega)$, the weak solution $u\in H^1(\Omega)$ to eq:StatBLT is such that and for every open subset $U\Subset\Omega$, the estimate holds for some constant $C$ depending only on $U$, $\Omega$, $D$ and $\mu$. Suppose further that $\Omega$ is such that $\partial\Omega$ is $C^2$. Then the weak solution $u\in H^1(\Omega)$ is such that and satisfies the estimate for a different con

Figures (11)

  • Figure 3.1: An example of a nest partition
  • Figure 4.1: An example of a corona shape
  • Figure 4.2: (a) Admissible polygons, (b) Admissible corona shapes
  • Figure 4.3: (a) Admissible polygonal-nest or polyhedral-nest partition, (b) Admissible corona-nest partition
  • Figure 6.1: \newlabelcircle0 (a) (b) (c) (d) reconstructed solutions at different iteration steps, (e) exact solution, (f) $e_r$ for different iteration steps $i$.
  • ...and 6 more figures

Theorems & Definitions (39)

  • Theorem 2.1: see for instance LadyzhenskayaUraltsevaBook
  • Theorem 2.2: Theorem IV.1 of BLT2003
  • Theorem 2.3: Theorems IV.2 and IV.3 of BLT2003
  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Proof 1
  • Proof 2: Proof of Theorem \ref{['thm:mainSmooth']}
  • Corollary 3.4
  • Proof 3
  • ...and 29 more