  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.
