We present a numerical scheme for solving an inverse problem for parameter estimation in tumor growth models for glioblastomas, a form of aggressive primary brain tumor. The growth model is a reaction-diffusion partial differential equation (PDE) for the tumor concentration. We use a PDE-constrained optimization formulation for the inverse problem. The unknown parameters are the reaction coefficient (proliferation), the diffusion coefficient (infiltration), and the initial condition field for the tumor PDE. Segmentation of Magnetic Resonance Imaging (MRI) scans drive the inverse problem where segmented tumor regions serve as partial observations of the tumor concentration. Like most cases in clinical practice, we use data from a single time snapshot. Moreover, the precise time relative to the initiation of the tumor is unknown, which poses an additional difficulty for inversion. We perform a frozen-coefficient spectral analysis and show that the inverse problem is severely ill-posed. We introduce a biophysically motivated regularization on the structure and magnitude of the tumor initial condition. In particular, we assume that the tumor starts at a few locations (enforced with a sparsity constraint on the initial condition of the tumor) and that the initial condition magnitude in the maximum norm is equal to one. We solve the resulting optimization problem using an inexact quasi-Newton method combined with a compressive sampling algorithm for the sparsity constraint. Our implementation uses PETSc and AccFFT libraries. We conduct numerical experiments on synthetic and clinical images to highlight the improved performance of our solver over a previously existing solver that uses standard two-norm regularization for the calibration parameters. The existing solver is unable to localize the initial condition. Our new solver can localize the initial condition and recover infiltration and proliferation. In clinical datasets (for which the ground truth is unknown), our solver results in qualitatively different solutions compared to the two-norm regularized solver.
Related work.Although there has been a lot of work on forward problems for tumor growth, there has been less work on inverse problems. The latter has different aspects. The first is the underlying biophysical model. The second is the inverse problem setup, observation operators and the existence of scans at multiple points, the noise models, inversion parameters and constraints. And the third one is the solution algorithm.Regarding the underlying model, like us, most researchers focus on parameter calibration of a handful of model parameters using single-species reaction-diffusion equations [7,22,24,28,33,39,51,58,59]. While more complex models describing processes like mass effect, angiogenesis and chemotaxis [23,26,48,54,60] exist, they have not been considered for calibration due to theoretical and computational challenges. However, several groups, including ours, are working to address these challenges.Regarding the inverse problem setup, in most studies t...