Mathematical Models of Cancer Cell Plasticity

Table 1

The important models reviewed here are organised by modelling approach and a summary of their key finding(s). The list is nonexhaustive.

Reference	Model	Submodel	Key result
Donaghey [12]	Discrete	CA	Higher proportion of cells will enter the absorbing cell state as the total cell population gets larger.
Düchting and Vogelsaenger [13]	Discrete	CA	After treatment, undamaged cells are recruited into the cell cycle again and stimulate tumour growth.
Duchting and Dehl [14]	Discrete	CA	Critical initial number of tumour cells of a tumour nucleus is necessary for the growth of a tumour. Additional high-influence variables are the mean life span of a tumour cell and the amount of tumour cell loss.
Smolle and Stettner [23]	Discrete	CA	Histological tumour patterns depend complexly on the autocrine and paracrine factors.
Smolle et al. [24]	Discrete	CA	Relative degree of motility to proliferation decreases from benign to primary malignant and metastatic, but the absolute degree of motility is increasing.
Ferreira et al. [25]	Discrete	CA	Growth patterns of the tumour are compact with gyration radius, surface roughness, and number of peripheral cells.
Schmitz et al. [19]	Discrete	CA	A tumorous subpopulation is most highly favoured when the interfacial area among strains is maximized. Total volumetric fraction of nonlocalized strains is not important in tumour development.
Jiao and Torquato [21]	Discrete	CA	Quantitative properties of the host microenvironment can significantly affect tumour morphology and growth dynamics.
Jiao and Torquato [22]	Discrete	CA	Strong cell-cell adhesion can suppress the invasive behaviour of the tumours growing in soft microenvironments; cancer malignancy can be significantly enhanced by harsh microenvironmental conditions, such as exposure to high pressure levels.
Xie et al. [20]	Hybrid	CA, diffusion reaction	In chemotherapy, constant dosing is generally more effective in suppressing primary tumour growth than periodic dosing, due to the resulting continuous high drug concentration.
Hatzikirou et al. [27]	Discrete	LGCA	Width of the travelling front is proportional to the front speed.
Chopard et al. [28]	Discrete	LGCA	There is a positive effect of fibre track on glioma growth.
Graner and Glazier [29]	Discrete	CPM	Long-distance cell movement leads to sorting with a logarithmic increase in the length scale of homogeneous cell clusters.
Jiang et al. [30]	Discrete	CPM	The microenvironmental conditions required for tumour cell survival and growth promoters and inhibitors have diffusion coefficients in the range to .
Turner and Sherratt [31]	Discrete	CPM	Increased proliferation rate results in an increased depth of invasion into the extracellular matrix.
Shirinifard et al. [32]	Discrete	CPM	Simulated avascular tumours form cylinders following the blood vessels, leading to a differential distribution of hypoxic cells within the tumour.
Anderson and Chaplain [34]	Continuum and discrete (discrete model is the discretized form of the continuum model)	Diffusion-reaction equation
Random walk model	Both chemotaxis and haptotaxis are necessary for the formation of a capillary network at large scales.
Anderson et al. [35]	Continuum and discrete (discrete model is the discretized form of the continuum model)	Diffusion-reaction equation
Random walk model	ECM structures can aid or hinder the migration of individual cells that have the potential to metastasis. As time increases, small cell clusters can be observed.
Chaplain and Stuart [65]	Continuum	PDE	Possible explanation for anastomosis
Bazaliy and Friedman [66]	Continuum	PDE	Establish the existence and uniqueness of a solution for some time interval
Friedman [69]	Continuum	PDE	For the densities of three types of cells: proliferating, quiescent and necrotic, the nutrient concentration, fluid velocity, and pressure have a unique smooth solution, with a smooth free boundary for a small time interval
Chen et al. [68]	Continuum	Partial integro-differential equations	Stationary solution of the model is linearly asymptotically stable
Cui and Friedman [71]	Continuum	ODE	Initial value problem has a one-parameter family of solutions and there exists a unique solution to the free boundary problem.
Zhang and Tao [75]	Continuum	PDE	Prove the global solvability of the model
Xu and Wu [74]	Continuum	PDE	Prove the existence and stability of the steady-state solutions when the rate at which the tumour attracts blood vessels is constant.