Inverse Problem Regularization for 3D Multi-Species Tumor Growth Models

TL;DR

A multi‐species partial differential equation (PDE) model for tumor growth and an algorithm for calibrating the model from magnetic resonance imaging (MRI) scans designed for glioblastoma multiforme a fast‐growing type of brain cancer are presented.

Abstract

We present a multi-species partial differential equation (PDE) model for tumor growth and a an algorithm for calibrating the model from magnetic resonance imaging (MRI) scans. The model is designed for glioblastoma (GBM) brain tumors. The modeled species correspond to proliferative, infiltrative, and necrotic tumor cells. The model calibration is formulated as an inverse problem and solved a PDE-constrained optimization method. The data that drives the calibration is derived by a single multi-parametric MRI image. This a typical clinical scenario for GBMs. The unknown parameters that need to be calibrated from data include ten scalar parameters and the infinite dimensional initial condition (IC) for proliferative tumor cells. This inverse problem is highly ill-posed as we try to calibrate a nonlinear dynamical system from data taken at a single time. To address this ill-posedness, we split the inversion into two stages. First we regularize the IC reconstruction by solving a single-species compressed sensing problem. Then, using the IC reconstruction, we invert for model parameters using a weighted regularization term. We construct the regularization term by using auxiliary 1D inverse problems. We apply our proposed scheme to clinical data. We compare our algorithm with single-species reconstruction and unregularized reconstructions. Our scheme enables the stable estimation of non-observable species and quantification of infiltrative tumor cells.

Figures (6)

• Figure 1: A schematic representation of the multi-species model dynamics in 1D. The tumor species growth process is represented, with vascularization indicated by oxygen.
• Figure 2: Ill-posedness analysis of multi-species model in 1D. The spectrum of Hessian eigenvalues for 3500 parameter combinations is presented in Figure \ref{['fig:eig_spec']}, with each eigenvalue normalized such that the max eigenvalue is 1. The singular values calculated in Algorithm \ref{['alg:reg']} are shown in Figure \ref{['fig:sing_values']}. We depict the absolute value of last four directions of the computed regularization term ($\mathbf{U}_{R}$) in \ref{['fig:directions']} corresponding to the four smallest singular values. The exponential decrease of eigenvalues in Figure \ref{['fig:eig_spec']} indicates high ill-posedness of the problem. Additionally, the wide range of small eigenvalues suggests varying degrees of ill-posedness among samples, which is accounted for in Algorithm \ref{['alg:reg']}. Figure \ref{['fig:sing_values']} demonstrates how different directions impact the ill-posedness, with smaller singular values indicating greater ill-posedness. Thus, using the inverse of the singular values can lead to improved inversion results. The results in \ref{['fig:directions']} clearly indicates ill-posedness for high values $\gamma_0$ and a combination of high values of $\rho$ and low values of $\delta_c$ contributes to ill-posedness.
• Figure 3: 1D inversion results of synthetic data with additive Gaussian noise ($5\%,10\%,20\%,30\%$). Here we show the reconstructed species concentration at $T=1$. First to fourth rows show relative noise levels. Plots in first to third columns show proliferative, necrotic, and infiltrative species, respectively. The green, gray, magenta, and blue lines represent the underlying ground truth species (before observation denoted as "Ground Truth"), ground truth species with noise ("Noisy Ground Truth"), reconstructed species without regularization ("Non-Reg. Inv."), and reconstructed species with regularization ("Reg. Inv."), respectively. Regularized inversion is observed to better match the underlying species even when the data is perturbed by noise, particularly in the case of infiltrative cells where deviations are greatly reduced with regularization.
• Figure 4: Evaluating 3D inversion using synthetic data with known IC and brain anatomy. We start by generating a tumor test-case and showing its species and segmentation in the first row (a). We then add $5\%$, $10\%$, $20\%$ and $30\%$ gaussian noise to the species and apply an observation operator to get noisy data. We perform inversion with and without regularization on this data. The inversion results are displayed in rows 2-4, with the enhancing, necrotic, and infiltrative concentrations in columns 1-3, and the segmentation in the last column. The regularized inversion yields acceptable segmentation for all noise levels, whereas the non-regularized case shows shows more variability for different noise levels.
• Figure 5: 3D inversion result with synthetic test-cases generated with an unknown IC and brain anatomy. Different normal brains with no mass deformation are used to generate the synthetic cases. In each row, we present a single test-case, with tumor species (Proliferative (EN), necrotic (NEC), and infiltrative cells) from the synthetic cases and inversion displayed in the first, second, and third columns, respectively. The given data and segmentation computed by the inversion algorithm are depicted in the fourth column. Moreover, we depicted the projected ground truth IC and estimated IC in the segmentation for data and inversion, respectively, Although the IC is an estimate, the reconstruction from the inversion is good as evidenced by the qualitative observation. The challenge of reconstructing multi-species arises due to the unknown initial brain anatomy and isotropic migration of tumor in all directions due to assigned white matter label.