Asynchronous exponential growth in an age structured population of proliferating and quiescent cells

Abstract

A model of a proliferating cell population is analyzed. The model distinguishes individual cells by cell age, which corresponds to phase of the cell cycle. The model also distinguishes individual cells by proliferating or quiescent status. The model allows cells to transit between these two states at any age, that is, any phase of the cell cycle. The model also allows newly divided cells to enter quiescence at cell birth, that is, cell age 0. Sufficient conditions are established to assure that the cell population has asynchronous exponential growth. As a consequence of this asynchronous exponential growth the population stabilizes in the sense that the proportion of the population in any age range, or the fraction in proliferating or quiescent state, converges to a limiting value as time evolves, independently of the age distribution and proliferating or quiescent fractions of the initial cell population. The asynchronous exponential growth is proved by demonstrating that the strongly continuous linear semigroup associated with the partial differential equations of the model is positive, irreducible, and eventually compact.

Introduction

In proliferating cell populations an individual cell can be quantitated by its age since cell birth, that is, since the division of its mother cell. In natural and experimental cell populations it is recognized that individual cells transit through the cell cycle with variability in intermitotic times. It is thus important to structure such populations with a continuous variable corresponding to cell age. The connection of cell age to the cell cycle arises from a specification of successive age intervals corresponding to the phases G₁ (first gap), S (synthesis phase), G₂ (second gap), and M (mitotic phase) of the cell cycle. It is also recognized that in most proliferating cell populations some cells are in a quiescent state in which progress through the cell cycle is suspended. Quiescent cells are especially important in tumor cell populations, in which it is observed that the fraction of cycling cells (the so-called growth fraction) is strongly dependent on such factors as tumor stage, tumor geometry, and therapeutic intervention.

In this paper we analyze a linear model of an age structured cell population with proliferating and quiescent compartments. The linearity of the model makes it applicable to experimental cell cultures, early stage tumors, and other populations considered before resource limitations produce the non-linear effects of crowding. In such cell populations a typical behavior is observed known as asynchronous exponential growth or balanced exponential growth. The general concept of asynchronous exponential growth was recognized in the early theoretical population investigations of Lotka and coworkers [1], [2] McKendrick [3]. A characteristic of asynchronous exponential growth in an age structured population is that the total population grows exponentially, but the age structure stabilizes in the sense that the fraction of cells in any age range converges to a limiting value independent of the initial age distribution. In our model asynchronous exponential growth also yields that the fraction of proliferating or quiescent cells stabilizes in the same sense as time evolves. Another feature of asynchronous exponential growth in our model is that both the proliferating and quiescent subpopulations ultimately disperse through all ages independently of any initial age or proliferating–quiescent synchronization.

Populations with proliferating and quiescent compartments have been investigated by many authors including [4], [5], [6], [7], [8], [9]. The mathematical formulation of asynchronous exponential growth has been developed by many authors including [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31]. Complete mathematical treatments of asynchronous exponential growth can be found in [11] (Part III, Sections 8–10) and [24] (Sections C-III and C-IV). The mathematical explanation of asynchronous exponential growth is tied to the domination of the spectrum of the linear operator associated with the solutions of the model by a simple eigenvalue. The phenomenological explanation of asynchronous exponential growth is tied to a mechanism which forces dispersion of the individual structure through successive generations. In an age structured population the mechanism driving asynchronous exponential growth is the variability of the age of division. In this paper we will establish sufficient conditions on the functions controlling division and transition between proliferating and quiescent compartments to assure asynchronization of the evolving population.

In an earlier study [4] sufficient conditions were established for asynchronous exponential growth in an age structured cell population with quiescence in the case that all newly divided cells entered the proliferating state. In this paper we investigate the case that only a fraction of newly divided cells enter the proliferating state and the remainder enter the quiescent state (the so-called G₀ phase of the cell cycle). For the case in [4] it was proved that asynchronous exponential growth occurs if the youngest proliferating cells have the possibility to transit to the quiescent state and the oldest quiescent cells have the possibility to transit to the proliferating state. For the case considered in this paper we will prove that if some fraction of newly divided cells can enter the quiescent state and the oldest quiescent cells have the possibility to transit to the proliferating state, then asynchronous exponential growth will occur.