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.

翻译：肿瘤侵袭和转移动力学的研究对于癌症生物学和治疗的进展至关重要。为研究肿瘤细胞对宿主组织的侵袭，已发展出多种数学模型。本文提出了一种新颖的随机相互作用粒子-场（SIPF）算法，该算法能够精确模拟趋触性平流扩散（HAD）系统中的癌细胞侵袭过程。我们的方法通过粒子相互作用的经验测度结合谱方法计算的更光滑场变量——细胞外基质浓度（ECM）——来逼近解。基于以拉普拉斯算子减正常数为特征的椭圆算子的显式格林函数，我们采用隐式欧拉离散化推导了随机粒子位置与场变量的单步时间递推关系。数值实验表明，所提算法尤其在计算三维（3D）空间中具有薄自由边界的癌细胞生长方面表现出优越性能。数值结果显示，SIPF算法具有无网格、自适应性及低计算成本的特点。此外，相较于有限差分法（FDM）和谱方法等传统数值技术，该算法更为精确高效。