Donaghey [12] |
Discrete |
CA |
Higher proportion of cells will enter the G0 absorbing cell state as the total cell population gets larger. |
Düchting and Vogelsaenger [13] |
Discrete |
CA |
After treatment, undamaged G0 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 10−6 to 10−6cm2/h. |
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. |