  We propose a PDE-constrained shape registration algorithm that captures the
deformation and growth of biological tissue from imaging data. Shape
registration is the process of evaluating optimum alignment between pairs of
geometries through a spatial transformation function. We start from our
previously reported work, which uses 3D tensor product B-spline basis functions
to interpolate 3D space. Here, the movement of the B-spline control points,
composed with an implicit function describing the shape of the tissue, yields
the total deformation gradient field. The deformation gradient is then split
into growth and elastic contributions. The growth tensor captures addition of
mass, i.e. growth, and evolves according to a constitutive equation which is
usually a function of the elastic deformation. Stress is generated in the
material due to the elastic component of the deformation alone. The result of
the registration is obtained by minimizing a total energy functional which
includes: a distance measure reflecting similarity between the shapes, and the
total elastic energy accounting for the growth of the tissue. We apply the
proposed shape registration framework to study zebrafish embryo epiboly process
and tissue expansion during skin reconstruction surgery. We anticipate that our
PDE-constrained shape registration method will improve our understanding of
biological and medical problems in which tissues undergo extreme deformations
over time.
