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The Diffusive Ultrasound Modulated Bioluminescence Tomography with Partial Data and Uncertain Optical Parameters

Tianyu Yang, Yang Yang

TL;DR

This work formulates diffusive ultrasound-modulated bioluminescence tomography with partial boundary data and uncertain optical parameters, casting the imaging problem into a PDE-based inverse problem with internal data derived from plane-wave ultrasound modulation. It develops a reconstruction framework that uses a positive adjoint solution to convert internal data into a solvable elliptic PDE for the unmodulated photon density, from which the bioluminescent source $S$ is recovered, and extends it to partial boundary measurements. The authors then quantify robustness through continuous and discretized uncertainty quantification bounds that bound the source error in terms of perturbations in the diffusion and absorption coefficients, and they discretize the model on a staggered grid to enable finite-dimensional analysis and numerics. Numerical experiments with uncertainty generation via generalized polynomial chaos validate the theory, showing that uncertainties in the diffusion coefficient $D$ more strongly affect the reconstructed source than uncertainties in the absorption coefficient $\sigma_a$, while averaging over many realizations mitigates the effect of noise.

Abstract

The paper studies an imaging problem in the diffusive ultrasound-modulated bioluminescence tomography with partial boundary measurement in an anisotropic medium. Assuming plane-wave modulation, we transform the imaging problem to an inverse problem with internal data, and derive a reconstruction procedure to recover the bioluminescent source. Subsequently, an uncertainty quantification estimate is established to assess the robustness of the reconstruction. To facilitate practical implementation, we discretize the diffusive model using the staggered grid scheme, resulting in a discrete formulation of the UMBLT inverse problem. A discrete reconstruction procedure is then presented along with a discrete uncertainty quantification estimate. Finally, the reconstruction procedure is quantitatively validated through numerical examples to demonstrate the efficacy and reliability of the proposed approach and estimates.

The Diffusive Ultrasound Modulated Bioluminescence Tomography with Partial Data and Uncertain Optical Parameters

TL;DR

This work formulates diffusive ultrasound-modulated bioluminescence tomography with partial boundary data and uncertain optical parameters, casting the imaging problem into a PDE-based inverse problem with internal data derived from plane-wave ultrasound modulation. It develops a reconstruction framework that uses a positive adjoint solution to convert internal data into a solvable elliptic PDE for the unmodulated photon density, from which the bioluminescent source is recovered, and extends it to partial boundary measurements. The authors then quantify robustness through continuous and discretized uncertainty quantification bounds that bound the source error in terms of perturbations in the diffusion and absorption coefficients, and they discretize the model on a staggered grid to enable finite-dimensional analysis and numerics. Numerical experiments with uncertainty generation via generalized polynomial chaos validate the theory, showing that uncertainties in the diffusion coefficient more strongly affect the reconstructed source than uncertainties in the absorption coefficient , while averaging over many realizations mitigates the effect of noise.

Abstract

The paper studies an imaging problem in the diffusive ultrasound-modulated bioluminescence tomography with partial boundary measurement in an anisotropic medium. Assuming plane-wave modulation, we transform the imaging problem to an inverse problem with internal data, and derive a reconstruction procedure to recover the bioluminescent source. Subsequently, an uncertainty quantification estimate is established to assess the robustness of the reconstruction. To facilitate practical implementation, we discretize the diffusive model using the staggered grid scheme, resulting in a discrete formulation of the UMBLT inverse problem. A discrete reconstruction procedure is then presented along with a discrete uncertainty quantification estimate. Finally, the reconstruction procedure is quantitatively validated through numerical examples to demonstrate the efficacy and reliability of the proposed approach and estimates.
Paper Structure (15 sections, 7 theorems, 58 equations, 8 figures)

This paper contains 15 sections, 7 theorems, 58 equations, 8 figures.

Table of Contents

  1. Introduction and Problem Formulation
  2. Reconstruction with Full Data
  3. Reconstruction with Partial Data
  4. Uncertainty Quantification with Continuous Diffusive Model
  5. Uncertainty Quantification with Discretized Diffusive Model
  6. Discretization with Isotropic Diffusion Coefficients.
  7. Discretization of the Forward Problem.
  8. Discretization of the Adjoint Problem.
  9. Discretization of the Internal Data Problem.
  10. Discrete Uncertainty Quantification Estimate.
  11. Discretization with Anisotropic Diffusion Coefficients.
  12. Numerical Experiment
  13. Uncertainty Generation
  14. Numerical Implementation.
  15. Experiment 2:

Key Result

Proposition 1

Assume that then eq:mixDE-eq:mixDEbc2 has a unique solution $\psi\in C^2(\overline{\Omega}\setminus\overline{\Gamma})\cap C^0(\overline{\Omega})$

Figures (8)

  • Figure 1: The illustration of staggered grid scheme. The zero and second order terms are defined on the grid points (black dots), the first order terms and $D$ are defined on the edges (while dots).
  • Figure 2: The outgoing vector at the corner
  • Figure 3: Left: Diffusion coefficient $D$. Middle: Absorption coefficient $\sigma_a$. Right: Shepp-Logan Source $S$.
  • Figure 4: The distribution of the error with respect to the inaccuracies in optical coefficients
  • Figure 5: Reconstructed source $\tilde{S}$ and its error under $10\%$ Gaussian random noise.
  • ...and 3 more figures

Theorems & Definitions (16)

  • Proposition 1: lieberman1986mixed
  • Theorem 2
  • proof
  • Remark 3
  • Proposition 4: dong2022w
  • Theorem 5
  • proof
  • Remark 6
  • Definition 7
  • Proposition 8
  • ...and 6 more