Cell cycle length and long‐time behavior of an age‐size model (original) (raw)
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Cell cycle length and long-time behaviour of an age-size model
arXiv: Analysis of PDEs, 2021
We consider an age-size structured cell population model based on the cell cycle length. The model is described by a first order partial differential equation with initial-boundary conditions. Using the theory of semigroups of positive operators we establish new criteria for an asynchronous exponential growth of solutions to such equations. We discuss the question of exponential size growth of cells. We show how to incorporate into our description models with constant increase of size and with target size division. We also present versions of the model when the population is heterogeneous.
A model of proliferating cell populations with inherited cycle length
Journal of Mathematical Biology, 1986
A mathematical model of cell population growth introduced by J. L. Le~owitz and S. I. Rubinow is analyzed. Individual cells are distinguished by age and cell cycle length. The cell cycle length is viewed as an inherited property determined at birth. The density of the population satisfies a first order linear partial differential equation with initial and boundary conditions. The boundary condition models the 15rocess of cell division of mother cells and the inheritance of cycle length by daughter cells. The mathematical analysis of the model employs the theory of operator semigroups and the spectral theory of linear operators. It is proved that the solutions exhibit the property of asynchronous exponential growth.
Asynchronous Exponential Growth of a General Structured Population Model
Acta Applicandae Mathematicae, 2011
We consider a class of structured cell population models described by a first order partial differential equation perturbed by a general birth operator which describes in a unified way a wide class of birth phenomena ranging from cell division to the McKendrick model. Using the theory of positive stochastic semigroups we establish new criteria for an asynchronous exponential growth of solutions to such equations.
Asymptotic behavior of nonlinear semigroup describing a model of selective cell growth regulation
Journal of Mathematical Biology, 1991
A new scheme of regulation of cell population growth is considered, called the selective growth regulation. The principle is that cells are withdrawn from proliferation depending on their contents of certain biochemical species. The dynamics of the cell population structured by the contents of this species is described by the functional integral equation model, previously introduced by the authors. The solutions of the model equations generate a semigroup of nonlinear positive operators. The main problem solved in this paper concerns stability of the equilibria of the model. This requires stating and proving of an original abstract result on the spectral radius of a perturbation of a semigroup of positive linear operators. Biological applications are discussed.
Journal of Mathematical Analysis and Applications, 2005
A continuous cell population model, which represents both the cell cycle phase structure and the kinetic heterogeneity of the population following Shackney's ideas [J. Theor. Biol. 38 (1973) 305-333], is studied. The asynchronous exponential growth property is proved in the framework of the theory of strongly continuous semigroups of bounded linear operators. 2004 Elsevier Inc. All rights reserved.
Asynchronous exponential growth in an age structured population of proliferating and quiescent cells
Mathematical biosciences
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 stron...
The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 1990
A population of cells growing and dividing often goes through a phase of exponential growth of numbers, during which the size distribution remains steady. In this paper we study the function differential equation governing this steady size distribution in the particular case where the individual cells themselves are growing exponentially in size. A series solution is obtained for the case where the probability of cell division is proportional to any positive power of the cell size, and a method for finding closed-form solutions for a more general class of cell division functions is developed.
Functional differential equations arising in cell-growth
Pamm, 2007
Non-local differential equations are notoriously difficult to solve. Cell- growth models for population growth of a cohort structured by size, simultaneously growing and dividing, give rise to a class of non-local eigenvalue problems, whose "principal" eigenvalue is the time-constant for growth/decay. These and other novel non-local problems are described and solved in special cases in this paper.