A Maturity-Structured Mathematical Model of Mutation, Acquisition in the Absence of Homeostatic Regulation

Abstract

Abstract. Most mammalian tissues are organized into a hierarchical structure of stem, progenitor, and differentiated cells. Tumors exhibit similar hierarchy, even if it is abnormal in comparison with healthy tissue. In particular, it is believed that a small population of cancer stem cells drives tumorigenesis in certain malignancies. These cancer stem cells are derived from transformed stem cells or mutated progenitors that have acquired stem-cell qualities, specifically the ability to selfrenew. Similar to their normal counterparts, cancer stem cells are long-lived, can self-renew and differentiate, albeit aberrantly, and are capable of generating tissue, resulting in tumor formation. Although identified and characterized in several forms of malignancy, the specific multi-step process that causes the formation of cancer stem cells is uncertain. Here, a maturity-structured mathematical model is developed to investigate the sequential order of mutations that causes the fastest emergence of cancer stem cells. Using model predictions, we discuss conditions for which genetic instability significantly speeds cancer onset and suggest that unbalanced stem-cell selfrenewal and inhibition of progenitor differentiation contribute to aggressive forms of cancer. To our knowledge, this is the first continuous maturity-structured mathematical model used to investigate mutation acquisition within hierarchical tissue in order to address implications of cancer stem cells in tumorigenesis.

A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics with Mutation Accumulation and Telomere Length Hierarchies

“…The numerical simulation of the nonlinear model shows growth of the cancer cell population similar to the Gompertzian curve, which is achieved in many models [24][41] [14]. This result supports the usefulness of the model framework for modeling both in-vivo and in-vitro experiments by showing its consistency with previous work.…”

“…The resulting vector of the cancer cell populations is (2690.99, 244.64, 40.7741, 9.40954, 2.68848, 0.896172, 0.336069, 0.261392) T , where the first number refers to the number, in millions, of differentiated cancer cells (j = 1 telomere, i = 5 mutations) and the last -the number, in millions, of cancer stem cells (j = 8 telomeres, i = 5 mutations) in the region. The curve that shows the growth of the cancer cells is similar to the Gompertzian curve, which is achieved in many models [24][41] [14]. The results, depending on the interpretation of the scenario, is that the cancer cells overtake either the region in the organ or the cell culture dish.…”

“…0 (linear simulation) Mutation probability, q j,k,i [24] 0.00001 0.00001 The main purpose of this section is to show numerical simulations of the mathematical results and demonstrate them in a visual setting. In Table 1 one can see the parameters used, ranges for the values of the parameters, based on evidence or previous models, and the values used in the simulations.…”

“…The mathematical model proposed in the present paper differs from theirs in several ways. In [24], the model considers cell maturity as a continuous variable correlated to differentiation, while in the present work differentiation is discrete and determines cell classes. The present model tracks cell lineages through multiple generations, not unlike [7], and focuses on the role of the stem cells as generators of this lineage.…”

“…Their hypothesis, that a cell goes through several mutations to become cancerous, is a very important assumption for our model. In [24], Gentry, Ashkenazi, and Jackson developed a model that combined cell hierarchy with sequential accumulation of mutations. They focused on the effect different types of mutations had on cancer growth.…”