Application of inverse source concepts to photoacoustic tomography

Abstract

Photoacoustic tomography (PAT), also known as optoacoustic or thermoacoustic tomography, is a hybrid imaging technique that possesses great potential for a wide range of biomedical imaging applications. Image reconstruction in PAT is tantamount to solving an inverse source problem, where the source represents the optical energy absorption distribution in the object that is induced by an interrogating pulsed optical waveform. In this work, we re-examine the PAT image reconstruction problem from a Fourier domain perspective by use of established time-harmonic inverse source concepts. A mathematical relationship between the photoacoustic pressure wavefield data on an aperture that encloses the object and the three-dimensional Fourier transform of the optical absorption distribution evaluated on a collection of concentric spheres is investigated. In addition to providing a framework for deriving both exact and approximate analytic reconstruction formulae, we demonstrate that this mapping provides an intuitive means of understanding certain spatial resolution characteristics of PAT.

Spectral power density of the random excitation for the photoacoustic wave equation

“…A. Anastasio et al studied the photoacoustic tomography (PAT) reconstruction problem in the frequency domain with inverse source concepts. 43 They also obtained a mathematical expression between the photoacoustic data and the optical absorption distribution in the frequency domain. In this work, the inverse problem dealt with is obtaining the pressure waves at the sources' locations from the measurements of the photoacoustic pressure waves on a spherical surface surrounding sources different from Anastasio's work.…”

Mathematics of Photoacoustic and Thermoacoustic Tomography

“…Since for all our purposes PAT results in the same mathematical model (although the biological features that TAT and PAT detect are different; see details in the chapter [15]), we will refer to TAT only.…”

“…We also denote by c(x) the speed of sound at a location x. Then, it has been argued, that the following model describes correctly the propagating pressure wave p(x, t) generated during the TAT procedure (e.g., [15,33,127,132,135]): p tt = c 2 (x)∆ x p, t ≥ 0, x ∈ R 3 p(x, 0) = f (x), p t (x, 0) = 0.…”

“…(Versions in all dimensions are known, see (16) and (15).) One can derive from here that knowledge of the function g(x, t) for x ∈ S and all t ≥ 0 is equivalent to knowing the spherical mean Rf (x, t) of the function f for any points x ∈ S and any t ≥ 0.…”

“…These means can be easily recovered from the measurements of the acoustic pressure using formulas (15) and (16) (see the discussion in [7]). In this case, image reconstruction becomes equivalent to inverting the spherical mean transform M. Thus, in what follows, we consider the problem of reconstructing a function f (x) supported within the region bounded by a closed surface S from known values of its spherical integrals g(y, r) with centers on S: g(y, r) =…”