We present SIBIA (Scalable Integrated Biophysics-based Image Analysis), a framework for coupling biophysical models with medical image analysis. It provides solvers for an image-driven inverse brain tumor growth model and an image registration problem, the combination of which can eventually help in diagnosis and prognosis of brain tumors. The two main computational kernels of SIBIA are a Fast Fourier Transformation (FFT) implemented in the library AccFFT to discretize differential operators, and a cubic interpolation kernel for semi-Lagrangian based advection. We present efficiency and scalability results for the computational kernels, the inverse tumor solver and image registration on two x86 systems, Lonestar 5 at the Texas Advanced Computing Center and Hazel Hen at the Stuttgart High Performance Computing Center. We showcase results that demonstrate that our solver can be used to solve registration problems of unprecedented scale, 40963 resulting in ∼ 200 billion unknowns - a problem size that is 64x larger than the state-of-the-art. For problem sizes of clinical interest, SIBIA is about 8x faster than the state-of-the-art.