# Inverse Problem Regularization for 3D Multi‐Species Tumor Growth Models

## TL;DR

The paper introduces a multi-species tumor growth model for brain cancer and a method to calibrate it using MRI scans, improving tumor estimation accuracy.

## Contribution

A novel two-stage regularization approach for calibrating a multi-species tumor growth model from single-time MRI data.

## Key findings

- The proposed regularization improves tumor Dice score by 5%–10% compared to single-species reconstructions.
- Model parameter reconstruction errors are reduced by 4%–80% with regularization.
- The model can estimate infiltrative tumor cells using observable tumor species.

## Abstract

We present a multi‐species partial differential equation (PDE) model for tumor growth and an algorithm for calibrating the model from magnetic resonance imaging (MRI) scans. The model is designed for glioblastoma multiforme (GBM) a fast‐growing type of brain cancer. The modeled species correspond to proliferative, infiltrative, and necrotic tumor cells. The model calibration is formulated as an inverse problem and solved by a PDE‐constrained optimization method. The data that drives the calibration is derived by a single multi‐parametric MRI image. This is a typical clinical scenario for GBMs. The unknown parameters that need to be calibrated from data include 10 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. Our regularization improves the tumor Dice score by 5%–10% compared to single‐species model reconstruction. Also, our regularization reduces model parameter reconstruction errors by 4%–80% in cases with known initial condition and brain anatomy compared to cases without regularization. Importantly, our model can estimate infiltrative tumor cells using observable tumor species.

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