Optimizing Quantitative Photoacoustic Imaging Systems: The Bayesian Cramér-Rao Bound Approach

TL;DR

This work introduces a novel computational approach for the optimal experimental design of qPACT imaging systems based on the Bayesian Cramér–Rao bound (CRB), the first work to propose Bayesian CRB based design for systems governed by PDEs.

Abstract

Quantitative photoacoustic computed tomography (qPACT) is an emerging medical imaging modality that carries the promise of high-contrast, fine-resolution imaging of clinically relevant quantities like hemoglobin concentration and blood-oxygen saturation. However, qPACT image reconstruction is governed by a multiphysics, partial differential equation (PDE) based inverse problem that is highly non-linear and severely ill-posed. Compounding the difficulty of the problem is the lack of established design standards for qPACT imaging systems, as there is currently a proliferation of qPACT system designs for various applications and it is unknown which ones are optimal or how to best modify the systems under various design constraints. This work introduces a novel computational approach for the optimal experimental design (OED) of qPACT imaging systems based on the Bayesian Cramér-Rao bound (CRB). Our approach incorporates several techniques to address challenges associated with forming the bound in the infinite-dimensional function space setting of qPACT, including priors with trace-class covariance operators and the use of the variational adjoint method to compute derivatives of the log-likelihood function needed in the bound computation. The resulting Bayesian CRB based design metric is computationally efficient and independent of the choice of estimator used to solve the inverse problem. The efficacy of the bound in guiding experimental design was demonstrated in a numerical study of qPACT design schemes under a stylized two-dimensional imaging geometry. To the best of our knowledge, this is the first work to propose Bayesian CRB based design for systems governed by PDEs.

Figures (10)

• Figure 1: Illustration of the cone-beam illumination system model. The power of the cone-beam is non-isotropic and decays as the angle $\theta$ between the direction of the cone-beam and $\mathbf{u}$ increases, where $\mathbf{x}$ is a point on the domain's boundary and $\mathbf{s}$ is the location of the source. The flux density transmitted into the domain at $\mathbf{x}$ then depends on angle between $\mathbf{u}$ and the domain normal vector $\boldsymbol{\eta}(\mathbf{x})$ at $\mathbf{x}$.
• Figure 2: Study 1: Four i.i.d. samples of the absorption coefficient $\mu_a$ from the log-Gaussian prior (top) and the corresponding reconstruction results in the single-illumination setting (bottom). As can be seen, the samples have the general structure of patterned occlusions found in biological tissue and the range of $\mu_a$ values is physiologically plausible. The reconstructions capture the main features of the prior samples, with the best performance near the boundary of domain, where the signal-to-noise ratio is highest.
• Figure 3: Illustration of the (a) contiguous and (b) interlaced design schemes. In each design scheme, cone-beam illuminators are placed in a circle around the domain. After each illumination, the illuminators are rotated by an angle $\alpha$ along the circle, where $\alpha = \pi/2$ for the contiguous scheme and $\alpha = \pi/20$ for the interlaced scheme.
• Figure 4: (a) Fluence distribution from a single illumination and (b) the sum of the fluence distributions over the four illuminations from the contiguous and interlaced design schemes. Here the fluences were computed with the absorption coefficient and reduced scattering coefficient set as the prior mean. As can be seen, both design schemes can illuminate the entire domain.
• Figure 5: Study 1: Pointwise mean square error (MSE) and the corresponding Bayesian CRB bound for the parameter $m_1$ (top row) and the optical absorption coefficient $\mu_a$ (bottom row) in the single-illumination regime. The difference between the pointwise MSE and the bound is shown in the last column. As can be seen, the Bayesian CRB provides a tight lower bound on the error in $m_1$. The bound is less sharp for $\mu_a$ but still captures the general spatial structure of the error.

Theorems & Definitions (2)

• Theorem 1
• proof : Proof Sketch