A Learnable Prior Improves Inverse Tumor Growth Modeling

TL;DR

This work showcases the effectiveness of integrating a rapid deep-learning algorithm with a high-precision evolution strategy in estimating brain tumor cell concentrations from magnetic resonance images, and proposes a novel framework that leverages the unique strengths of both approaches in a synergistic manner.

Abstract

Biophysical modeling, particularly involving partial differential equations (PDEs), offers significant potential for tailoring disease treatment protocols to individual patients. However, the inverse problem-solving aspect of these models presents a substantial challenge, either due to the high computational requirements of model-based approaches or the limited robustness of deep learning (DL) methods. We propose a novel framework that leverages the unique strengths of both approaches in a synergistic manner. Our method incorporates a DL ensemble for initial parameter estimation, facilitating efficient downstream evolutionary sampling initialized with this DL-based prior. We showcase the effectiveness of integrating a rapid deep-learning algorithm with a high-precision evolution strategy in estimating brain tumor cell concentrations from magnetic resonance images. The DL-Prior plays a pivotal role, significantly constraining the effective sampling-parameter space. This reduction results in a fivefold convergence acceleration and a Dice-score of 95%.

Figures (7)

• Figure 1: Fitting a tumor model with a combination of DL and evolution strategy. (a) T1c and FLAIR MR images are segmented into white matter (WM), gray matter (GM), and cerebrospinal fluid (CSF). The respective tumor segmentations are combined into one channel. (b) DL provides a fast estimation of the tumor model parameters $\theta = \{x, y, z, \mu_D, \mu_\rho\}$. A Gaussian prior is constructed based on an encoder followed by an ensemble of linear layers. (c) CMA-ES incorporates this DL-Prior and a Dice-score-based likelihood to further optimize the tumor model parameters.
• Figure 2: Example tumors created by different models are compared. The ground truth tumor is shown in black (a). Its outline is displayed in the other images as well. The atlas-based LMI approach (b), our DL-Prior (c), the ensemble mean (d), and the combination of the DL-Prior and sampling (e) are compared.
• Figure 3: Example simulations with parameters created by the ensemble of the networks. The difference (cyan, positive; pink, negative) to the ensemble mean (gray, Figure \ref{['fig:example']} d) is shown.
• Figure 4: Mean Dice-score results between prediction and ground truth at different threshold concentrations. The single network approach is compared to the ensemble mean, LMI and the final combined version with CMA-ES sampling + DL-Prior.
• Figure 5: (a) The mean likelihood over time for Naive Sampling and DL-Prior + Sampling is shown. Shaded areas represent standard error over different patients. DL-Prior + Sampling consistently yields higher likelihoods, with a significant difference around the clinically relevant 2-hour mark, resulting in a likelihood of 0.85 $\pm$ 0.03, while Naive Sampling only reached 0.41 $\pm$ 0.05. This translates to a p-value of $1.2 \times 10^{-5}$ and an effect size of 2.0. (b) The mean convergence time of our DL-inference (orange) and our combination of classical sampling with DL-Prior (blue) is compared to sampling without prior (purple, dashed). The paired t-test results in an effect size of 1.0 and a p-value of 0.003