  Photo-acoustic Tomography (PAT) and Thermo-acoustic Tomography (TAT) are
medical imaging modalities that combine the high contrast of radiative
properties of tissues with the high resolution of ultrasound. In both
modalities, a first step concerns the reconstruction of the radiation-induced
source of ultrasound. Transient Elastography (TE) and Magnetic Resonance
Elastography (MRE) combine the high elastic contrast of tissues with the high
resolution of ultrasound and magnetic resonance, respectively. In both
modalities, a first step concerns the reconstruction of the elastic
displacement.
  The result of this first step, which is not considered in this paper, is the
availability of internal functionals of the unknown tissue properties. All
imaging modalities are recast as the reconstruction of parameters in elliptic
equations from knowledge of solutions to such equations. This paper provides a
characterization of the parameters that may or may not be reconstructed from
such internal functionals. We provide explicit reconstruction procedures and
indicate how stable they are with respect to errors in the available
measurements.
  The modalities PAT, TAT, TE, and MRE allow us to reconstruct high-contrast
optical and elastic properties of tissues with the high resolution of
ultrasound or magnetic resonance imaging. They provide a means to reconstruct
second-order tensors modeling tissue anisotropy as well as complex-valued
coefficients modeling absorbing and dissipative effects.
