Uncertainty-aware Bayesian inversion for tumor growth

The problem

Mechanistic models of tumor growth — reaction–diffusion equations such as the Fisher–Kolmogorov–Petrovsky–Piskunov (FKPP) model, or phase-field models of Allen–Cahn type — can forecast how a tumor evolves and how it responds to therapy. To be useful for an individual patient, their parameters (proliferation rate, diffusivity, response to treatment, …) have to be estimated from sparse, noisy imaging data, and the estimate must come with an honest measure of its uncertainty.

Bayesian inversion provides exactly this: given data $d$ and parameters $\theta$,

where $\mathcal{G}$ maps parameters to the model prediction. The catch is cost: sampling the posterior requires evaluating $\mathcal{G}$ — i.e. solving the PDE — many thousands of times.

Approach

We replace the full PDE solver with a Latent Dynamics Network (LDNet), a neural surrogate that evolves a low-dimensional latent state through a neural ODE and reconstructs the spatial field on demand. Once trained, the surrogate $\widetilde{\mathcal{G}}$ is cheap and differentiable, which enables a full uncertainty-quantification pipeline:

• MAP estimation by gradient-based optimisation through the surrogate;
• the Laplace approximation of the posterior around the MAP point;
• Markov chain Monte Carlo sampling for the full posterior.

Applications

The pipeline is validated on an FKPP model of breast cancer and on an Allen–Cahn phase-field model of prostate cancer. The work is carried out within the IN-DEEP doctoral network, in collaboration with Tecnalia Research & Innovation (Bilbao).

Results and figures will be added once the paper is public.