  We present an optimization framework for photo-acoustic tomography of brain
based on a system of coupled equations that describe the propagation of sound
waves in linear isotropic inhomogeneous and lossy elastic media with the
absorption and physical dispersion following a frequency power law using
fractional Laplacian operators. The adjoint of the associated continuous
forward operator is derived, and a numerical framework for computing this
adjoint based on a k- space pseudospectral method is presented. We analytically
show that the derived continuous adjoint matches the adjoint of an associated
discretised operator. We include this adjoint in a first-order positivity
constrained optimization algorithm that is regularized by total variation
minimization, and show that the iterates monotonically converge to a minimizer
of an objective function, even in the presence of some error in estimating the
physical parameters of the medium.
