An optimal control problem in photoacoustic tomography

Abstract

International audienceThis article is devoted to the introduction and study of a photoacoustic tomography model, an imaging technique based on the reconstruction of an internal photoacoustic source distribution from measurements acquired by scanning ultrasound detectors over a surface that encloses the body containing the source under study. In a nutshell, the inverse problem consists in determining absorption and diffusion coefficients in a system coupling a hyperbolic equation (acoustic pressure wave) with a parabolic equation (diffusion of the fluence rate), from boundary measurements of the photoacoustic pressure. Since such kinds of inverse problems are known to be generically ill-posed, we propose here an optimal control approach, introducing a penalized functional with a regularizing term in order to deal with such difficulties. The coefficients we want to recover stand for the control variable. We provide a mathematical analysis of this problem, showing that this approach makes sense. We finally write necessary first order optimality conditions and give preliminary numerical results

On the uniqueness and stability of an inverse problem in photo-acoustic tomography

“…We have proved the uniqueness of the solution of the optimal control problem studied in [12] and given a stability result. We also provide a characterization of the derivative of the optimal control with respect both to the source and the observation.…”

“…This energy is converted into heat creating a thermally induced pressure jump that propagates as a sound wave, which can be detected. The fluence rate I μ , that is the average of the luminous intensity in all the directions, satisfies the diffusion equation (see [1,5,12])…”

“…We refer [12] to get a complete description of the model. Let us briefly mention that we deal with two coupled partial differential equations that describes the light intensity (fluence) behavior inside a body that is excited by a laser (pulsed) source and the acoustic pressure wave which is generated by this excitation.…”

“…Let us briefly mention that we deal with two coupled partial differential equations that describes the light intensity (fluence) behavior inside a body that is excited by a laser (pulsed) source and the acoustic pressure wave which is generated by this excitation. The authors of [12] have investigated the model and obtained an optimal control formulation to recover some parameters of interest, namely the absorption and diffusion coefficients (μ, D).…”

How to position sensors in thermo-acoustic tomography

“…• The first system of equations describes the generation of the heating process inside the body. In the PAT, this system involves the fluence equation (the fluence rate is the average of the light intensity in all directions) which is a diffusion equation [6]; in the TAT case, this equation is replaced by Maxwell equations [1]. Then the temperature is driven by the classical heat equation, than can be neglected in the PAT case because the high speed of light implies that the thermal effect is quasi instantaneous.…”