  Magnetic resonance elastography is a motion-sensitive image modality that
allows to measure in vivo tissue displacement fields in response to mechanical
excitations. This paper investigates a data assimilation approach for
reconstructing tissue displacement and pressure fields in an in silico brain
model from partial elastography data. The data assimilation is based on a
parametrized-background data weak methodology, in which the state of the
physical system -- tissue displacements and pressure fields -- is reconstructed
from the available data assuming an underlying poroelastic biomechanics model.
For this purpose, a physics-informed manifold is built by sampling the space of
parameters describing the tissue model close to their physiological ranges to
simulate the corresponding poroelastic problem, and computing a reduced basis
via Proper Orthogonal Decomposition. Displacements and pressure reconstruction
is sought in a reduced space after solving a minimization problem that
encompasses both the structure of the reduced-order model and the available
measurements. The proposed pipeline is validated using synthetic data obtained
after simulating the poroelastic mechanics of a physiological brain. The
numerical experiments demonstrate that the framework can exhibit accurate joint
reconstructions of both displacement and pressure fields. The methodology can
be formulated for an arbitrary resolution of available displacement data from
pertinent images.
