A Simple Mathematical Model Based on the Cancer Stem Cell Hypothesis Suggests Kinetic Commonalities in Solid Tumor Growth

Abstract

BackgroundThe Cancer Stem Cell (CSC) hypothesis has gained credibility within the cancer research community. According to this hypothesis, a small subpopulation of cells within cancerous tissues exhibits stem-cell-like characteristics and is responsible for the maintenance and proliferation of cancer.Methodologies/Principal FindingsWe present a simple compartmental pseudo-chemical mathematical model for tumor growth, based on the CSC hypothesis, and derived using a “chemical reaction” approach. We defined three cell subpopulations: CSCs, transit progenitor cells, and differentiated cells. Each event related to cell division, differentiation, or death is then modeled as a chemical reaction. The resulting set of ordinary differential equations was numerically integrated to describe the time evolution of each cell subpopulation and the overall tumor growth. The parameter space was explored to identify combinations of parameter values that produce biologically feasible and consistent scenarios.Conclusions/SignificanceCertain kinetic relationships apparently must be satisfied to sustain solid tumor growth and to maintain an approximate constant fraction of CSCs in the tumor lower than 0.01 (as experimentally observed): (a) the rate of symmetrical and asymmetrical CSC renewal must be in the same order of magnitude; (b) the intrinsic rate of renewal and differentiation of progenitor cells must be half an order of magnitude higher than the corresponding intrinsic rates for cancer stem cells; (c) the rates of apoptosis of the CSC, transit amplifying progenitor (P) cells, and terminally differentiated (D) cells must be progressively higher by approximately one order of magnitude. Simulation results were consistent with reports that have suggested that encouraging CSC differentiation could be an effective therapeutic strategy for fighting cancer in addition to selective killing or inhibition of symmetric division of CSCs.

A Multicompartment Mathematical Model of Cancer Stem Cell-Driven Tumor Growth Dynamics

“…Focusing on the exponential tumor growth phase, Johnston et al (2010) and Molina-Peña and Álvarez (2012) developed ordinary differential equation models of stem, transit-amplifying (or progenitor), and fully differentiated cell populations to study constant cancer stem cells fractions in tumors. Depending on the tumor growth and differentiation rates balance, Johnston et al (2010) showed that stem cells can comprise any proportion of the tumor, and that higher stem cell proportions likely yield more aggressive tumors.…”

A Branching Process to Characterize the Dynamics of Stem Cell Differentiation

“…Previous theoretical approaches to study stem cell dynamics focused in the general aspects of stem cell biology [15,33], applied to colon crypts [13,14], hematopoietic [16,34], or cancer stem cells [35]. In the context of motor neuron generation [2], we recently developed a mathematical model based on chemical kinetics formalism that predicted a switch in the division mode in synchrony with changes in Shh activity.…”

Less Utilized Prognostic Markers in Acute Myeloid Leukemia

“…Other mathematical models has been established to characterize stem cells based on signaling interaction with the environment [61] and determine how stem cells coordinate homeostasis through cross-talk between genetic and epigenetic regulation [62] . Mathematical and computational techniques were also used to gain insights into the biology of cancer stem cells in initiation, progression and response to treatment in chronic myeloid leukaemia (reviewed in Michor, 2008 [63] ), to determine how mutations deregulate stem cell homeostasis and induces cancer [64] , how kinetics of cancer stem cell sustain growth [65] and to strategise how to eliminate cancer stem cells [66] . The lesser requirement for resources and lower complexity compared to clinical trials and laboratory procedures may make mathematical modeling a niche to guide in drug selection and prediction of treatment outcome in personalized medicine.…”

“…Complete remission is achievable in 65% de novo AML and 41% secondary AML. Median overall survival for these patients was 1119 days for age 16-55 years (N=554), 350 days for age [56][57][58][59][60][61][62][63][64][65] years (N = 437) and 80 days for patients aged 76-89 year old (N = 968) [1] . The French-American-British (FAB) classification of acute myeloid leukemias, the first classification established more than 40 years ago, identifies eight subtypes (M0-M7) based on morphology and cytochemistry staining.…”