Reconstructing initial pressure and speed of sound distributions simultaneously in photoacoustic tomography

TL;DR

This work tackles the ill-posed problem of reconstructing both the initial pressure $p_0(r)$ and the speed of sound $c(r)$ in photoacoustic tomography by using multiple datasets generated from different initial pressures $p_0^i$, thereby providing additional information to stabilize the inversion. A gradient-based framework with bound constraints and a multigrid strategy solves the joint inverse problem, leveraging adjoint-state gradients for $p_0^i$ and $c$ and updating via $x_{k+1}=\mathcal{P}(x_k-\alpha_k H_k \nabla\varepsilon(x_k))$. Numerical 2D simulations in simple and tissue-mimicking phantoms show that jointly estimating $p_0^i$ and $c$ improves reconstruction quality compared to a single-$p_0$ approach, with multi-direction illumination notably enhancing identifiability, while multi-wavelength data alone may be insufficient. The results highlight the potential to exploit variable optical excitations to extract tissue-structure information from $c(r)$, improving quantitative PAT, albeit with higher computational cost and the need for further validation in 3D and on real data.

Abstract

Image reconstruction in photoacoustic tomography relies on an accurate knowledge of the speed of sound in the target. However, the speed of sound distribution is not generally known, which may result in artefacts in the reconstructed distribution of initial pressure. Therefore, reconstructing the speed of sound simultaneously with the initial pressure would be valuable for accurate imaging in photoacoustic tomography. Furthermore, the speed of sound distribution could provide additional valuable information about the imaged target. In this work, simultaneous reconstruction of initial pressure and speed of sound in photoacoustic tomography is studied. This inverse problem is known to be highly ill-posed. To overcome this, we study an approach where the ill-posedness is alleviated by utilising multiple photoacoustic data sets that are generated by different initial pressure distributions within the same imaged target. Then, these initial pressure distributions are reconstructed simultaneously with the speed of sound distribution. A methodology for solving this minimisation problem is formulated using a gradient-based iterative approach equipped with bound constraints and a multigrid approach. The methodology was evaluated with numerical simulations. Different approaches for generating multiple initial pressure distributions and their effect on the solution of the image reconstruction problem were studied. The results show that initial pressure and speed of sound can be simultaneously reconstructed from photoacoustic data. Furthermore, utilising multiple initial pressure distributions improves the reconstructions such that the locations of initial pressure and speed of sound inhomogeneities can be better distinguished and image artifacts are reduced.

Figures (8)

• Figure 1: Reconstructions in a simple geometry phantom with a low speed of sound contrast. First column: True parameters. Second column: Reconstructed initial pressure $p_{0}^{1}$, $p_{0}^{2}$, $p_{0}^{3}$ and $p_{0}^{4}$, and speed of sound $c$ distributions using the proposed approach. Third column: Reconstructed initial pressure $p_0^{\mathrm{1}}$ and speed of sound $c$ distributions using the reference approach. Units are in $\mathrm{Pa}$ and $\mathrm{m/s}$ for initial pressures and speed of sound, respectively.
• Figure 2: Reconstructions in a simple geometry phantom with a low speed of sound contrast and a water bath mimicking layer. First column: True parameters. Second column: Reconstructed initial pressure $p_{0}^{1}$, $p_{0}^{2}$, $p_{0}^{3}$ and $p_{0}^{4}$, and speed of sound $c$ distributions using the proposed approach. Third column: Reconstructed initial pressure $p_0^{\mathrm{1}}$ and speed of sound $c$ distributions using the reference approach. Units are in $\mathrm{Pa}$ and $\mathrm{m/s}$ for initial pressures and speed of sound, respectively.
• Figure 3: Reconstructions in a simple geometry phantom with a high speed of sound contrast. First column: True parameters. Second column: Reconstructed initial pressure $p_{0}^{1}$, $p_{0}^{2}$, $p_{0}^{3}$ and $p_{0}^{4}$, and speed of sound $c$ distributions using the proposed approach. Third column: Reconstructed initial pressure $p_0^{\mathrm{1}}$ and speed of sound $c$ distributions using the reference approach. Units are in $\mathrm{Pa}$ and $\mathrm{m/s}$ for initial pressures and speed of sound, respectively.
• Figure 4: Optical absorption $\mu_{\mathrm{a}}$ and reduced scattering $\mu_{\mathrm{s}}'$ coefficients used to simulate initial pressure distributions when multiple illuminations were used. Units are in $\mathrm{mm^{-1}}$.
• Figure 5: Volume fractions of deoxygenated $\mathcal{V}_{\mathrm{HHb}}$, and oxygenated $\mathcal{V}_{\mathrm{HbO_2}}$ hemoglobin, water $\mathcal{V}_{\mathrm{water}}$ and fat $\mathcal{V}_{\mathrm{fat}}$, and reference scattering $\mu_{\mathrm{s,ref}}'$ at wavelength $\lambda_{\mathrm{ref}} = 800 \: \mathrm{nm}$ used to simulate initial pressure distributions when multiple wavelengths or exogenous absorbers were utilised. Locations of exogenous absorbers are highlighted with green color in the top left figure. Units are in $\mathrm{mm^{-1}}$ for reference scattering.