The asynchronous exponential growth property in a model for the kinetic heterogeneity of tumour cell populations

Abstract

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.

Asymptotically exponential solutions in nonlinear integral and differential equations

“…(i) For all t 0 0 and ϕ ∈ C ([t 0 − τ , t 0 ], R n ) there exist d 0 (t 0 , ϕ), e 0 (t 0 , ϕ) ∈ R n such that the solution y(·; t 0 , ϕ) of Eq. (3.11) through (t 0 , ϕ) satisfies y(t; t 0 , ϕ) = e α 0 t t k d 0 (t 0 , ϕ) cos β 0 t + e 0 (t 0 , ϕ) sin β 0 t + o (1) ,…”

“…(3.22) is the so-called sunflower equation, which was investigated extensively (see, e.g., [20,21] and [27] and the references therein). Consider an associated linear equation y(t) + Aẏ(t) + B y(t − r) = 0, t 0, (3.24) and its characteristic equation λ 2 + Aλ + Be −λr = 0. x(t; 0, ϕ) = e α 0 t d 1 (ϕ) cos β 0 t + γ 1 (ϕ) + o (1) , t → +∞, (3.27) where d 1 (ϕ), γ 1 (ϕ) ∈ R and d 1 (ϕ 0 ) = 0 for some ϕ 0 ∈ C ([−r, 0], R).…”

“…x(t; 0, ϕ) = e α 0 t d α 0 (ϕ) cos β 0 t + e α 0 (ϕ) sin β 0 t + o (1) , t → +∞, (3.29) where d α 0 , e α 0 : C ([−r, 0], R) → R are such that |d α 0 (ϕ 0 )| + |e α 0 (ϕ 0 )| = 0, for some ϕ 0 ∈ C ([−r, 0], R).…”

“…Structured population models have been studied at least from the sixties [3], and it is still an intensively studied area [1,2,12,13,16,23,29]. One important property of age-structured population models is the so-called asynchronous exponential growth/decay property, i.e., when the age distribution tends to a limit independently of the initial age distribution (see, e.g., [10][11][12][13]15,23,[30][31][32]).…”

Estimating the Total Rate of DNA Replication Using Branching Processes

“…Several of them are deterministic (Bertuzzi et al 1983(Bertuzzi et al , 1984Basse et al, 2005), but there are also stochastic and semi-stochastic approaches (Arino et al, 2005;Bertuzzi et al, 2002;Chiorino et al, 2001;Larsson et al, 2005). Among the stochastic models, branching processes have been proven to be particularly suitable for modeling the growth of populations (Macdonald 1970(Macdonald , 1981Jagers 1983Jagers , 1991Kimmel and Axelrod, 2000;Haccou et al, 2005).…”