Where did the tumor start? An inverse solver with sparse localization for tumor growth models

Abstract

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...

Multiatlas Calibration of Biophysical Brain Tumor Growth Models with Mass Effect

“…(1) requires initial conditions for the tumor c 0 and the precancer brain anatomy m 0 . Following [5], we parameterize c…”

“…Here, x i are voxels that are segmented as tumor and σ is one voxel, meaning m can be quite large (∼1000). This parameterization alleviates some of the ill-posedness associated with the inverse problem [5].…”

“…While there have been many studies to calibrate these models using inverse problems [9], [10], [12], [15], most do not invert for all unknown parameters (tumor initial condition and model parameters) or they assume the presence of multiple imaging scans (both these scenarios make the inverse problem more tractable). In our previous work [5], we presented a methodology to invert for tumor initial condition (IC) and cell proliferation and migration from a single scan; but we did not account for mass effect. We demonstrated the importance of using a sparse tumor IC (see §II) to correctly reconstruct for other tumor parameters.…”

“…O is an observation operator that defines the clearly observable tumor margin (see §II-D for its definition; for additional details see [16]). Following [5], we introduce two additional constraints to our optimization problem-Eq. 2c parameterizes the tumor initial condition using at most s Gaussians and with Eq.…”

“…We use them to deal with the several ill-posedness of the backward tumor growth PDE. In [5], we discuss this in detail.…”