Cellular potts modeling of tumor growth, tumor invasion, and tumor evolution

Abstract

Despite a growing wealth of available molecular data, the growth of tumors, invasion of tumors into healthy tissue, and response of tumors to therapies are still poorly understood. Although genetic mutations are in general the first step in the development of a cancer, for the mutated cell to persist in a tissue, it must compete against the other, healthy or diseased cells, for example by becoming more motile, adhesive, or multiplying faster. Thus, the cellular phenotype determines the success of a cancer cell in competition with its neighbors, irrespective of the genetic mutations or physiological alterations that gave rise to the altered phenotype. What phenotypes can make a cell "successful" in an environment of healthy and cancerous cells, and how? A widely used tool for getting more insight into that question is cell-based modeling. Cell-based models constitute a class of computational, agent-based models that mimic biophysical and molecular interactions between cells. One of the most widely used cell-based modeling formalisms is the cellular Potts model (CPM), a lattice-based, multi particle cell-based modeling approach. The CPM has become a popular and accessible method for modeling mechanisms of multicellular processes including cell sorting, gastrulation, or angiogenesis. The CPM accounts for biophysical cellular properties, including cell proliferation, cell motility, and cell adhesion, which play a key role in cancer. Multiscale models are constructed by extending the agents with intracellular processes including metabolism, growth, and signaling. Here we review the use of the CPM for modeling tumor growth, tumor invasion, and tumor progression. We argue that the accessibility and flexibility of the CPM, and its accurate, yet coarse-grained and computationally efficient representation of cell and tissue biophysics, make the CPM the method of choice for modeling cellular processes in tumor development.

Keywords: cell-based modeling; cellular Potts model; evolutionary tumor model; multiscale modeling; tumor growth model; tumor invasion model; tumor metastasis model.

Figure 1

Tumor growth models. (A) Cross section of the 3D avascular tumor model of Stott et al. (1999). Black cells in the middle of the tumor are necrotic, surrounded by quiescent cells (light gray). The outer layer of the tumor consists of proliferating cells (dark gray). The tumor is embedded in stroma, represented by stromal (white) cells. Image reproduced from Stott et al. (1999) with permission. (B) Cross section of the 3D avascular tumor model of Jiang et al. (2005), with a numerical simulation of nutrient and waste diffusion, cell cycle regulation, and cell metabolism. The figure shows the three layers of avascular tumors. The stroma is modeled as a continuum, depicted in blue. Image reproduced from Jiang et al. (2005) with permission. (C) Avascular tumors with a homogeneous population of tumor cells and mixed cancer stem cells and transient amplifying cancer cells (Sottoriva et al., 2011). Homogeneous tumors produce spherical aggregates, whereas a heterogeneous population gives rise to a rugged surface, enhancing metastasis. The lower images show the distribution of a couple of clones that illustrates the growth dynamics within the aggregates. Image reproduced from Sottoriva et al. (2011) with permission.

Figure 2

Vascular tumor growth of Shirinifard et al. (2009). (A) Number of normal proliferative tumor cells in the non-angiogenic (red curve) and angiogenic (black curve) model, showing different stages of development. Black arrows: (1) the exponential growth phase of the spherical tumor; (2) no growth; (3) the linear-spherical phase; (4) slow growth; (5) the linear-cylindrical phase; (6) the linear-sheet phase. Red Arrows: (1) the exponential growth phase of the spherical tumor; (2) slow growth; (3) cylindrical growth phase. (B) Cylindrical shaped non-angiogenic tumor. Tumor cells are shown in green, the vasculature is red. (C) Paddle-shaped angiogenic tumor. Neovascular endothelial cells are shown in purple. Images reproduced from Shirinifard et al. (2009) with permission.

Figure 3

Tumor invasion with homogeneous and heterogeneous ECM. (A) Invasion front of tumor penetrating the stroma via “viscous fingers”. Image reproduced from Turner and Sherratt (2002) with permission. (B) Avascular tumor model of Rubenstein and Kaufman (2008), exploring invasion along ECM fibers. Image reproduced from Rubenstein and Kaufman (2008) with permission. (C) Invasion front of persistently moving cancer cells penetrate in “fingers” along ECM fibers describe penetration dynamics even without cell division. Image produced based on the model of Szabó et al. (2012).