  We present the SIBIA (Scalable Integrated Biophysics-based Image Analysis)
framework for joint image registration and biophysical inversion and we apply
it to analyse MR images of glioblastomas (primary brain tumors). In particular,
we consider the following problem. Given the segmentation of a normal brain MRI
and the segmentation of a cancer patient MRI, we wish to determine tumor growth
parameters and a registration map so that if we "grow a tumor" (using our tumor
model) in the normal segmented image and then register it to the patient
segmented image, then the registration mismatch is as small as possible. We
call this "the coupled problem" because it two-way couples the biophysical
inversion and registration problems. In the image registration step we solve a
large-deformation diffeomorphic registration problem parameterized by an
Eulerian velocity field. In the biophysical inversion step we estimate
parameters in a reaction-diffusion tumor growth model that is formulated as a
partial differential equation.
  In this paper, our contributions are the introduction of a PDE-constrained
optimization formulation of the coupled problem, the derivation of the
optimality conditions, and the derivation of a Picard iterative scheme for the
solution of the coupled problem. In addition, we perform several tests to
experimentally assess the performance of our method on synthetic and clinical
datasets. We demonstrate the convergence of the SIBIA optimization solver in
different usage scenarios. We demonstrate that we can accurately solve the
coupled problem in three dimensions ($256^3$ resolution) in a few minutes using
11 dual-x86 nodes. Also, we demonstrate that, with our coupled approach, we can
successfully register normal MRI to tumor-bearing MRI while obtaining Dice
coefficients that match those achieved when registering of normal-to-normal
MRI.
