A unified Bayesian inversion approach for a class of tumor growth models with different pressure laws

TL;DR

This work develops a unified Bayesian inversion framework for a family of tumor growth models governed by porous-medium type equations with pressure–density relations p = \\frac{m}{m-1}\\rho^{m-1} and m in [2,\\infty). It proves well-posedness and stability of the posterior for each m and demonstrates convergence of the posterior to the incompressible limit as m \\to \\infty, ensuring uniform inference across the index. The numerical method combines a plain Metropolis–Hastings sampler with an asymptotic-preserving forward solver to efficiently handle forward models for all m, validated by extensive 2D experiments that show uniform accuracy with respect to noise level and model index. The results provide a robust framework for data assimilation in tumor-growth models with varying pressure laws, enabling reliable parameter and function recovery under model uncertainty. These theoretical and computational advances offer a scalable approach for integrating heterogeneous tumor-growth models with observational data in practice.

Abstract

In this paper, we use the Bayesian inversion approach to study the data assimilation problem for a family of tumor growth models described by porous-medium type equations. The models contain uncertain parameters and are indexed by a physical parameter $m$, which characterizes the constitutive relation between density and pressure. Based on these models, we employ the Bayesian inversion framework to infer parametric and nonparametric unknowns that affect tumor growth from noisy observations of tumor cell density. We establish the well-posedness and the stability theories for the Bayesian inversion problem and further prove the convergence of the posterior distribution in the so-called incompressible limit, $m \rightarrow \infty$. Since the posterior distribution across the index regime $m\in[2,\infty)$ can thus be treated in a unified manner, such theoretical results also guide the design of the numerical inference for the unknown. We propose a generic computational framework for such inverse problems, which consists of a typical sampling algorithm and an asymptotic preserving solver for the forward problem. With extensive numerical tests, we demonstrate that the proposed method achieves satisfactory accuracy in the Bayesian inference of the tumor growth models, which is uniform with respect to the constitutive relation.

Figures (6)

• Figure 1: Test 1 (a) with $\sigma=0.1$, by using $M=800$. Left: histogram for the posterior samples. Right: prior and posterior distributions for $h$.
• Figure 2: Test 1 (b). Posterior distributions of $h$ for different $\sigma$ (fix $m=40$) and different $m$ (fix $\sigma=0.25$).
• Figure 3: Test 2 with $\sigma = 0.5$ (top) and $\sigma=0.02$ (bottom). Histogram for posterior samples of $h$ and $c_1$.
• Figure 4: Test 3 with $\sigma=0.25$ and $m=40$. True $h(x,y)$ function, the prior mean (which is $h_0=2$ at all points), and the estimator of the posterior mean of $h(x,y)$ computed pointwisely at each $(x,y)$ point.
• Figure 5: Test 3 with $\sigma=0.5$. Density is computed by using the posterior mean of $h(x,y)$, where $m=5$ (left) and $m=50$ (right). The second row shows the corresponding difference between the posterior mean of $h(x,y)$ and the true function.

Theorems & Definitions (20)

• Theorem 3.2: Lemma 2,benilan1996singular
• Theorem 3.3: Theorem 2,benilan1996singular
• Remark 3.4
• Theorem 3.5: Theorem 1.1,igbida2021a
• Theorem 3.6: Theorem 2.1,igbida2021b
• Proposition 3.8
• Remark 3.9
• Lemma 3.10: Lemma 3.3,dashti2017bayesian
• Lemma 3.11
• proof