Journal of Oncology / 2019 / Article / Tab 1

Review Article

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.

ReferenceModelSubmodelKey result

Donaghey [12]DiscreteCAHigher proportion of cells will enter the absorbing cell state as the total cell population gets larger.
Düchting and Vogelsaenger [13]DiscreteCAAfter treatment, undamaged cells are recruited into the cell cycle again and stimulate tumour growth.
Duchting and Dehl [14]DiscreteCACritical 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]DiscreteCAHistological tumour patterns depend complexly on the autocrine and paracrine factors.
Smolle et al. [24]DiscreteCARelative 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]DiscreteCAGrowth patterns of the tumour are compact with gyration radius, surface roughness, and number of peripheral cells.
Schmitz et al. [19]DiscreteCAA 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]DiscreteCAQuantitative properties of the host microenvironment can significantly affect tumour morphology and growth dynamics.
Jiao and Torquato [22]DiscreteCAStrong 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]HybridCA, diffusion reactionIn 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]DiscreteLGCAWidth of the travelling front is proportional to the front speed.
Chopard et al. [28]DiscreteLGCAThere is a positive effect of fibre track on glioma growth.
Graner and Glazier [29]DiscreteCPMLong-distance cell movement leads to sorting with a logarithmic increase in the length scale of homogeneous cell clusters.
Jiang et al. [30]DiscreteCPMThe microenvironmental conditions required for tumour cell survival and growth promoters and inhibitors have diffusion coefficients in the range to .
Turner and Sherratt [31]DiscreteCPMIncreased proliferation rate results in an increased depth of invasion into the extracellular matrix.
Shirinifard et al. [32]DiscreteCPMSimulated 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 modelBoth 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 modelECM 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]ContinuumPDEPossible explanation for anastomosis
Bazaliy and Friedman [66]ContinuumPDEEstablish the existence and uniqueness of a solution for some time interval
Friedman [69]ContinuumPDEFor 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]ContinuumPartial integro-differential equationsStationary solution of the model is linearly asymptotically stable
Cui and Friedman [71]ContinuumODEInitial value problem has a one-parameter family of solutions and there exists a unique solution to the free boundary problem.
Zhang and Tao [75]ContinuumPDEProve the global solvability of the model
Xu and Wu [74]ContinuumPDEProve the existence and stability of the steady-state solutions when the rate at which the tumour attracts blood vessels is constant.