We present SIBIA (Scalable Integrated Biophysics-based Image Analysis), a framework for joint image registration and biophysical inversion and we apply it to analyse MR images of glioblastomas (primary brain tumors). We have two applications in mind. The first one is normal-to-abnormal image registration in the presence of tumor-induced topology differences. The second one is biophysical inversion based on single-time patient data. The underlying optimization problem is highly non-linear and non-convex and has not been solved before with a gradient-based approach. Given the segmentation of a normal brain MRI and the segmentation of a cancer patient MRI, we determine tumor growth parameters and a registration map so that if we ``grow a tumor'' (using our tumor model) in the normal brain and then register it to the patient image, then the registration mismatch is as small as possible. This ``\emph{coupled problem}'' two-way couples the biophysical inversion and the registration problem. 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 (PDE). In SIBIA, we couple these two sub-components in an iterative manner. We first presented the components of SIBIA in ``Gholami et al, Framework for Scalable Biophysics-based Image Analysis, IEEE/ACM Proceedings of the SC2017'', in which we derived parallel distributed memory algorithms and software modules for the decoupled registration and biophysical inverse problems.

In this paper, our contributions are the introduction of a PDE-constrained optimization formulation of the coupled problem, and the derivation of a Picard iterative solution scheme. We perform extensive 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 using SIBIA, we can accurately solve the coupled problem in three dimensions (256^3 resolution) in a few minutes using 11 dual-x86 nodes.