A Nonlinear Age and Maturity Structured Model of Population Dynamics

Dyson, Janet; Villella-Bressan, Rosanna; Webb, Glenn

doi:10.1006/jmaa.1999.6656

Cited by 49 publications

(27 citation statements)

References 14 publications

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“…In [8] and [9], Dyson et al considered a time-age-maturity structured equation in which the age for a cell to divide is not identical between cells. They presented the basic theory of existence and uniqueness and properties of the solution operator.…”

Section: Introductionmentioning

confidence: 99%

Global stability of a partial differential equation with distributed delay due to cellular replication

Adimy

Crauste

2003

Nonlinear Analysis: Theory, Methods & Applications

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In this paper, we investigate a nonlinear partial differential equation, arising from a model of cellular proliferation. This model describes the production of blood cells in the bone marrow. It is represented by a partial differential equation with a retardation of the maturation variable and a distributed temporal delay. Our aim is to prove that the behaviour of primitive cells influences the global behaviour of the population.

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Section: Introductionmentioning

confidence: 99%

Global stability of a partial differential equation with distributed delay due to cellular replication

Adimy

Crauste

2003

Nonlinear Analysis: Theory, Methods & Applications

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“…Assuming that they are locally Lipschitz continuous (which is always the case for smooth functions, say continuously differentiable; in particular, for β and k given by (17) in Section 4), one can easily check, from Hale and Verduyn Lunel [20], that system (2) has a unique continuous solution (Q(t), W (t)), which is well-defined for all t ≥ 0 and for a continuous initial condition. Moreover, we easily see that, for nonnegative initial conditions, the solutions of (2) remain nonnegative for t ≥ 0.…”

Section: The Modelmentioning

confidence: 99%

“…Many authors studied properties of the model introduced by Mackey [26] and other models related to this one, in order to understand the role played by hematopoietic stem cells in the stability of hematopoiesis. We refer to Mackey and Rey [28,29,30], Mackey and Rudnicki [31,32], Dyson et al [15,16,17,18], Pujo-Menjouet and Rudnicki [37], Adimy and Pujo-Menjouet [7], Adimy and Crauste [1,2] and Adimy et al [4] for studies of structured models of hematopoiesis, the structure being either age, maturity or age-maturity. We want to point out that we do not consider any structure in the model analyzed in this paper, but these references could be used to improve the present work.…”

Section: Introductionmentioning

confidence: 99%

Periodic oscillations in leukopoiesis models with two delays

Adimy

Crauste

Ruan

2006

Journal of Theoretical Biology

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The term leukopoiesis describes processes leading to the production and regulation of white blood cells. It is based on stem cells differentiation and may exhibit abnormalities resulting in severe diseases, such as cyclical neutropenia and leukemias. We consider a nonlinear system of two equations, describing the evolution of a stem cell population and the resulting white blood cell population. Two delays appear in this model to describe the cell cycle duration of the stem cell population and the time required to produce white blood cells. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analyzing roots of a second degree exponential polynomial characteristic equation with delay-dependent coefficients. We also prove the existence of a Hopf bifurcation which leads to periodic solutions. Numerical simulations of the model with parameter values reported in the literature demonstrate that periodic oscillations (with short and long periods) agree with observations of cyclical neutropenia in patients.

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“…Stage (i) can be achieved by maintaining a constant culture environment and is characterized by timeindependent transition rates. Balanced growth is achieved upon completion of stage (ii) so the age densities are also time independent (Dyson et al, 2000;Fredrickson et al, 1967). This feature of balanced growth implies that the system has reached a state such that the population densities of the phases will maintain constant ratio to one another (Dyson et al, 2002).…”

Section: Introductionmentioning

confidence: 99%

Identification of age‐structured models: Cell cycle phase transitions

Sherer

Tocce

Hannemann

et al. 2008

Biotech & Bioengineering

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A methodology is developed that determines age-specific transition rates between cell cycle phases during balanced growth by utilizing age-structured population balance equations. Age-distributed models are the simplest way to account for varied behavior of individual cells. However, this simplicity is offset by difficulties in making observations of age distributions, so age-distributed models are difficult to fit to experimental data. Herein, the proposed methodology is implemented to identify an age-structured model for human leukemia cells (Jurkat) based only on measurements of the total number density after the addition of bromodeoxyuridine partitions the total cell population into two subpopulations. Each of the subpopulations will temporarily undergo a period of unbalanced growth, which provides sufficient information to extract age-dependent transition rates, while the total cell population remains in balanced growth. The stipulation of initial balanced growth permits the derivation of age densities based on only age-dependent transition rates. In fitting the experimental data, a flexible transition rate representation, utilizing a series of cubic spline nodes, finds a bimodal G(0)/G(1) transition age probability distribution best fits the experimental data. This resolution may be unnecessary as convex combinations of more restricted transition rates derived from normalized Gaussian, lognormal, or skewed lognormal transition-age probability distributions corroborate the spline predictions, but require fewer parameters. The fit of data with a single log normal distribution is somewhat inferior suggesting the bimodal result as more likely. Regardless of the choice of basis functions, this methodology can identify age distributions, age-specific transition rates, and transition-age distributions during balanced growth conditions.

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A Nonlinear Age and Maturity Structured Model of Population Dynamics

Cited by 49 publications

References 14 publications

Global stability of a partial differential equation with distributed delay due to cellular replication

Global stability of a partial differential equation with distributed delay due to cellular replication

Periodic oscillations in leukopoiesis models with two delays

Identification of age‐structured models: Cell cycle phase transitions

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