Stochastic Proximal Gradient Algorithms for Multi-Source Quantitative Photoacoustic Tomography

Single-stage approach for estimating optical parameters in spectral quantitative photoacoustic tomography

In quantitative photoacoustic tomography, the optical parameters of a target, most importantly the concentrations of chromophores such as deoxygenated and oxygenated hemoglobin, are estimated from photoacoustic data measured on the boundary of the target. In this work, a numerical approximation of a forward model for spectral quantitative photoacoustic tomography is constructed by utilizing the diffusion approximation for light propagation, the acoustic wave equation for ultrasound propagation, and spectral models of optical absorption and scattering to describe the wavelength dependence of the optical parameters. The related inverse problem is approached in the framework of Bayesian inverse problems. Concentrations of four chromophores (deoxygenated and oxygenated hemoglobin, water, and lipid), two scattering parameters (reference scattering and scattering power), and the Grüneisen parameter are estimated in a single-stage from photoacoustic data. The methodology is evaluated using numerical simulations in different full-view and limited-view imaging settings. The results show that, utilizing spectral data and models, the spectral optical parameters and the Grüneisen parameter can be simultaneously estimated. Furthermore, the approach can also be utilized in limited-view imaging situations.

Image reconstruction for endoscopic photoacoustic tomography including effects of detector responses

In photoacoustic tomography (PAT), conventional image reconstruction methods are generally based on the assumption of an ideal point-like ultrasonic detector. This assumption is appropriate when the receiving surface of the detector is sufficiently small and/or the distance between the imaged object and the detector is large enough. However, it does not hold in endoscopic applications of PAT. In this study, we propose a model-based image reconstruction method for endoscopic photoacoustic tomography (EPAT), considering the effect of detector responses on image quality. We construct a forward model to physically describe the imaging process of EPAT, including the generation of the initial pressure due to optical absorption and thermoelastic expansion, the propagation of photoacoustic waves in tissues, and the acoustic measurement. The model outputs the theoretical sampling voltage signal, which is the response of the ultrasonic detector to the acoustic pressure reaching its receiving surface. The images representing the distribution map of the optical absorption energy density on cross-sections of the imaged luminal structures are reconstructed from the sampling voltage signals output by the detector through iterative inversion of the forward model. Compared with the conventional approaches based on back-projection and other imaging models, our method improved the quality and spatial resolution of the resulting images.

Bayesian Instability of Optical Imaging: Ill Conditioning of Inverse Linear and Nonlinear Radiative Transfer Equation in the Fluid Regime

For the inverse problem in physical models, one measures the solution and infers the model parameters using information from the collected data. Oftentimes, these data are inadequate and render the inverse problem ill-posed. We study the ill-posedness in the context of optical imaging, which is a medical imaging technique that uses light to probe (bio-)tissue structure. Depending on the intensity of the light, the forward problem can be described by different types of equations. High-energy light scatters very little, and one uses the radiative transfer equation (RTE) as the model; low-energy light scatters frequently, so the diffusion equation (DE) suffices to be a good approximation. A multiscale approximation links the hyperbolic-type RTE with the parabolic-type DE. The inverse problems for the two equations have a multiscale passage as well, so one expects that as the energy of the photons diminishes, the inverse problem changes from well- to ill-posed. We study this stability deterioration using the Bayesian inference. In particular, we use the Kullback–Leibler divergence between the prior distribution and the posterior distribution based on the RTE to prove that the information gain from the measurement vanishes as the energy of the photons decreases, so that the inverse problem is ill-posed in the diffusive regime. In the linearized setting, we also show that the mean square error of the posterior distribution increases as we approach the diffusive regime.

One-step fluorescence photoacoustic tomography with the optical radiative transfer model

We present adjoint-based Jacobian as well as gradient evaluations and corresponding reconstruction schemes to solve the fully nonlinear, optical radiative transfer modeled one-step fluorescence photoacoustic tomographic (FPAT) problem, which aims to reconstruct the map of absorption coefficient of the exogenous fluorophore from boundary photoacoustic data. The radiative transport equation (RTE) and frequency-domain photoacoustic equation have been employed to model light and photoacoustic wave propagation, respectively. Levenberg-Marquardt and Broyden-Fletcher-Goldfarb-Shanno reconstruction schemes have been used corresponding to the evaluated Jacobians and gradients, respectively. Numerical reconstructions obtained from the two schemes have been validated for scattering-dominant as well as nonscattering-dominant media in 2D. To the best of our knowledge, these are the first one-step FPAT reconstruction results in literature based on the optical RTE model.

Gradient-based one-step fluorescence photoacoustic tomography

We propose a gradient-based scheme to solve the fluorescence photoacoustic tomographic (FPAT) problem in a fully nonlinear one-step setting, which aims to reconstruct the map of the absorption coefficient of an exogenous fluorophore from boundary photoacoustic pressure data. Adjoint-based gradient evaluation is presented for the FPAT problem in a frequency-domain photoacoustic equation setting. Numerical validations of the resulting Broyden-Fletcher-Goldfarb-Shanno (BFGS) reconstruction scheme are carried out in two dimensions for full- as well as limited-data test cases, and the results are compared with existing Jacobian-based one-step FPAT reconstructions. The reasonably comparable results of the one-step gradient- and Jacobian-based FPAT reconstruction schemes, coupled with the significant computational savings of the former, potentially set up the one-step gradient-based schemes as an advantageous method of choice for FPAT reconstructions. Further reconstruction studies carried out using quantitative photoacoustic tomography (QPAT)-based chromophore reconstructions as inputs to the FPAT inversions show a robustness of fluorophore absorption coefficient reconstructions to the QPAT-obtained inputs.

Three-dimensional quantitative photoacoustic tomography using an adjoint radiance Monte Carlo model and gradient descent

Quantitative photoacoustic tomography aims to recover maps of the local concentrations of tissue chromophores from multispectral images. While model-based inversion schemes are promising approaches, major challenges to their practical implementation include the unknown fluence distribution and the scale of the inverse problem. We describe an inversion scheme based on a radiance Monte Carlo model and an adjoint-assisted gradient optimization that incorporates fluence-dependent step sizes and adaptive moment estimation. The inversion is shown to recover absolute chromophore concentrations, blood oxygen saturation, and the Grüneisen parameter from in silico three-dimensional phantom images for different radiance approximations. The scattering coefficient is assumed to be homogeneous and known a priori.