Solvability for Photoacoustic Imaging With Idealized Piezoelectric Sensors

Most reconstruction algorithms for photoacoustic imaging assume that the pressure field is measured by the ultrasound sensors placed on a detection surface. However, such sensors do not measure pressure exactly due to their nonuniform directional and frequency responses, and resolution limitations. This is the case for piezoelectric sensors that are commonly employed for photoacoustic imaging. In this article, using the method of matched asymptotic expansions and the basic constitutive relations for piezoelectricity, we propose a simple mathematical model for piezoelectric transducers. The approach simultaneously models how the pressure waves induce the piezoelectric measurements and how the presence of the sensors affects the pressure waves. Using this model, we analyze whether the data gathered by the piezoelectric sensors lead to the mathematical solvability of the photoacoustic imaging problem. We conclude that this imaging problem is well posed in certain normed spaces and under a geometric assumption. We also propose an iterative reconstruction algorithm that incorporates the model for piezoelectric measurements. A numerical implementation of the reconstruction algorithm is presented.