Skip to main page content
Access keys NCBI Homepage MyNCBI Homepage Main Content Main Navigation
Review
, 18 (12), 1271-1286

Mathematical Models of Tumor Cell Proliferation: A Review of the Literature

Affiliations
Review

Mathematical Models of Tumor Cell Proliferation: A Review of the Literature

Angela M Jarrett et al. Expert Rev Anticancer Ther.

Abstract

A defining hallmark of cancer is aberrant cell proliferation. Efforts to understand the generative properties of cancer cells span all biological scales: from genetic deviations and alterations of metabolic pathways to physical stresses due to overcrowding, as well as the effects of therapeutics and the immune system. While these factors have long been studied in the laboratory, mathematical and computational techniques are being increasingly applied to help understand and forecast tumor growth and treatment response. Advantages of mathematical modeling of proliferation include the ability to simulate and predict the spatiotemporal development of tumors across multiple experimental scales. Central to proliferation modeling is the incorporation of available biological data and validation with experimental data. Areas covered: We present an overview of past and current mathematical strategies directed at understanding tumor cell proliferation. We identify areas for mathematical development as motivated by available experimental and clinical evidence, with a particular emphasis on emerging, non-invasive imaging technologies. Expert commentary: The data required to legitimize mathematical models are often difficult or (currently) impossible to obtain. We suggest areas for further investigation to establish mathematical models that more effectively utilize available data to make informed predictions on tumor cell proliferation.

Keywords: Computational; biophysical; cancer; cell growth; oncology.

Conflict of interest statement

Financial and competing interests disclosure

The authors report no conflicts of interest.

Figures

Figure 1:
Figure 1:
Panel (a) displays example population curves for exponential, logistic, Gompertz, and Allee type growth models. Observe that exponential growth is constant and therefore the population will grow without bound as opposed to the logistic, Gompertz, and Allee growth models, which are all bounded by the cell population size, but with differing growth phases (i.e., different steepness of growth). Panel (b) presents Michaelis-Menten type growth with examples of different concentration curves representing (for example) nutrient concentrations for cellular growth. The blue curve results from a constant source of nutrient, where the food source is constantly replenished. For the red curve, the nutrient decays with time due to (for example) wash out of nutrient degradation. The yellow curve results from the nutrient decreasing with cell number representing a sustainable, but limited source of food. Finally, the purple curve has nutrient that is periodic in time, where the nutrient decays and is replenished. Notice that using a constant nutrient source results in characteristically exponential growth of the population, and population dependent nutrient is similar to logistic.
Figure 2:
Figure 2:
Control (red) and dose-response (blue) model fits for different subtypes of triple negative breast cancer cell lines: MDA-MB-468 (basal-like 1), SUM-149PT (basal-like 2), MDA-MB-231 (mesenchymal), and MDA-MB-453 (luminal expressing androgen receptor). Each cell line was plated and serially imaged via fluorescence microscopy for 30 days following a six hour doxorubicin treatment (156 nM). Cells were grown for at least three days to allow for a pre-treatment proliferation rate to be estimated. Nuclear counts are displayed in black with error bars representing the 95% confidence interval from six experimental replicates. These counts are used to fit Eqs. (i-iii). Eq. (iv) is a weighted average approach being used to incorporate both Eqs. (ii) and (iii) in the treatment response model (i), with NTC,A and NTC,B being the solutions of Eq. (i) using the term kd(t,D) as described in Eqs. (ii) and (iii), respectively. The A and B subscripts refer to the two different model formulations for death (kd) defined in Eqs. (ii) and (iii), respectively. Model weights were calculated from the Akaike Information Criterion for models (ii) and (iii). For this concentration and exposure time, we can see that doxorubicin acts reducing proliferation, but growth is resumed for greater times.
Figure 3:
Figure 3:
A comparison of a solid growth theory model and a reaction-diffusion model is presented for a rat with a C6 glioma. Panel (a) shows a T2-weighted MRI of the central slice of the rat head while panels (b) and (c) show the estimated tumor cell number, N(x), on days 10 and 14 in the region indicated by the white dashed box in (a). Panels (d-f) show an example of the linearized growth model calibrated over days 10 to 14. Panel (d) shows a schematic of volume change represented by Jg(x), the ratio between the growth-induced volume Vg and the initial volume V0 of the tumor. Panel (e) shows the spatial distribution of Jg(x) used to grow the tumor from day 10 to 14. The model simulated N(x) is shown in panel (f). Panel (g) shows a 1D example of a reaction (or proliferation) diffusion model of tumor growth. Outward expansion is governed by tumor cell diffusion (or motility) while increase in cell number is governed by tumor cell proliferation. Panel (h) shows the spatial distribution of k(x) used to grow the tumor from day 10 to 14. The model simulated N(x) is shown for the reaction-diffusion model in panel (i). While representing different phenomena, Jg(x) and k(x) generally have increased values in areas of rapid tumor expansion, and decreased values in areas of slow tumor expansion.
Figure 4:
Figure 4:
Simulation of the hybrid model described in section 3.3 with the tumor cells modeled by an agent-based model (left column), and the nutrient diffusion by a reaction-diffusion equation (right column) at three different time points (corresponding to each row). The nutrient is consumed by the tumor cells, and if the nutrient concentration drops below a threshold, the tumor cells become hypoxic. As the nutrient is depleted, the hypoxic cells transition to necrotic cells. Panels (a) and (b) displays the simulation at 10 days—the tumor is heterogeneous mixture of proliferative (green), quiescent (grey), and apoptotic cells (purple). Panels (c) and (d) displays the simulation at 12.5 days—(c) the tumor now presents hypoxic cells (yellow) at the center due to nutrient depletion (d). Panels (e) and (f) displays the simulation at 15 days—(e) the tumor becomes necrotic (blue) at the center due to continuous nutrient consumption (f).

Similar articles

See all similar articles

Cited by 3 articles

Publication types

LinkOut - more resources

Feedback