A Stochastic Interacting Particle-Field Algorithm for a Haptotaxis Advection-Diffusion System Modeling Cancer Cell Invasion

TL;DR

This paper introduces a novel stochastic particle-field algorithm for simulating cancer cell invasion within a haptotaxis advection-diffusion system, demonstrating improved accuracy and efficiency over traditional methods in 3D models.

Contribution

The paper develops a new SIPF algorithm combining particle interactions with a spectral ECM field, offering a mesh-free, adaptive, and computationally efficient approach for tumor invasion modeling.

Findings

Superior accuracy in 3D cancer growth simulations

Mesh-free and self-adaptive algorithm

Outperforms finite difference and spectral methods

Abstract

The investigation of tumor invasion and metastasis dynamics is crucial for advancements in cancer biology and treatment. Many mathematical models have been developed to study the invasion of host tissue by tumor cells. In this paper, we develop a novel stochastic interacting particle-field (SIPF) algorithm that accurately simulates the cancer cell invasion process within the haptotaxis advection-diffusion (HAD) system. Our approach approximates solutions using empirical measures of particle interactions, combined with a smoother field variable - the extracellular matrix concentration (ECM) - computed by the spectral method. We derive a one-step time recursion for both the positions of stochastic particles and the field variable using the implicit Euler discretization, which is based on the explicit Green's function of an elliptic operator characterized by the Laplacian minus a positive constant. Our numerical experiments demonstrate the superior performance of the proposed algorithm, especially in computing cancer cell growth with thin free boundaries in three-dimensional (3D) space. Numerical results show that the SIPF algorithm is mesh-free, self-adaptive, and low-cost. Moreover, it is more accurate and efficient than traditional numerical techniques such as the finite difference method (FDM) and spectral methods.