Global stability in a delayed partial differential equation describing cellular replication

Mackey, Michael C.; Rudnicki, Ryszard

doi:10.1007/bf00160175

Cited by 89 publications

(98 citation statements)

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“…A survey of many applications is given in the book edited by Metz and Diekmann [20]. Our model includes models of a population of organisms reproducing by binary fission with equal [8,10,12,16,17] and unequal division [2,11,13,15]. Both types of binary fission models were considered in [29].…”

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confidence: 99%

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Asynchronous Exponential Growth of a General Structured Population Model

2011

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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.

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confidence: 99%

“…It is then interesting to note that even a nonlinear model of cell population (cf. [17]) can be reduced to a Markov semigroup.…”

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Asynchronous Exponential Growth of a General Structured Population Model

2011

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“…Other models structured in age and maturity have been proposed [1,184,185], with comparable transport equations. It is also possible to discretise the maturity variable to yield a proliferationquiescence model with communicating compartments for differentiation, with distinction between self-renewal and differentiation rates, as in [3].…”

Section: (M Q(t M))q(t M)mentioning

confidence: 99%

Modelling Physiological and Pharmacological Control on Cell Proliferation to Optimise Cancer Treatments

Clairambault

2009

Math. Model. Nat. Phenom.

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Abstract. This review aims at presenting a synoptic, if not exhaustive, point of view on some of the problems encountered by biologists and physicians who deal with natural cell proliferation and disruptions of its physiological control in cancer disease. It also aims at suggesting how mathematicians are naturally challenged by these questions and how they might help, not only biologists to deal theoretically with biological complexity, but also physicians to optimise therapeutics, on which last point the focus will be set here. To this purpose, mathematical modelling should represent proliferating cell population dynamics with natural built-in control targets (which implies modelling the cell division cycle), together with the distribution of drugs in the organism and their molecular actions on different targets at the cell level on proliferation, i.e., molecular pharmacokinetics-pharmacodynamics of antiproliferative drugs. This should make possible optimal control of drug delivery with constraints to be determined according to the main pharmacological issues encountered in the clinic: unwanted toxic side-effects, occurrence of drug resistance. Mathematical modelling should also take into account physiological determinants of cell and tissue proliferation, such as intervention of the immune system, circadian control on cell cycle checkpoint proteins, and activity of intracellular drug processing enzymes together with individual variations in the activities of these proteins (genetic polymorphism). Taking these points into account will add to the rich scenery of normal or disrupted cell and tissue regulations, and their corrections by drugs, a natural environmental, whole body physiological, frame. It is necessary indeed to consider such a framework if one wants to eventually be actually helpful to clinicians who routinely treat by combinations of drugs living Humans with their complex whole body regulations, often dependent on genotypic variations, and not isolated cells or tissues.Key words: cell proliferation, population dynamics, physiologically structured partial differential equations, pharmacological control, pharmacokinetics-pharmacodynamics, physiological modelling, cancer treatments, therapeutic optimisation, individualised medicine AMS subject classification: 92-02, 92B05 Biological common knowledge on cancerThis section shortly presents the general scenery of cancer evolution and features that must be included in models designed to describe and control it, bearing in mind a more descriptive (physicist's) than a therapeutic (physician's) point of view, only to reverse this perspective in the following sections, in which will be stressed the interest of modelling from the point of view of control, at least if one wants to design models for therapeutic applications. This biological section will be only a short sketch, if one considers that on this subject many books have already been written, e.g. [24,78,96,97,223]. Cancer: a challenging public health problemCardiovascular diseases and cancer are by fa...

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“…Since then it has been improved and analyzed by many authors, including Mackey and co-authors [17,18,20,21] and Adimy et al [1,2,3,4]. All these works considered that the nonlinear term, which describes the rate of introduction of nonproliferating cells in the proliferating compartment, depends only on the nonproliferating cell number.…”

Section: Introductionmentioning

confidence: 99%

Delay Model of Hematopoietic Stem Cell Dynamics: Asymptotic Stability and Stability Switch

Crauste

2009

Math. Model. Nat. Phenom.

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Abstract. A nonlinear system of two delay differential equations is proposed to model hematopoietic stem cell dynamics. Each equation describes the evolution of a sub-population, either proliferating or nonproliferating. The nonlinearity accounting for introduction of nonproliferating cells in the proliferating phase is assumed to depend upon the total number of cells. Existence and stability of steady states are investigated. A Lyapunov functional is built to obtain the global asymptotic stability of the trivial steady state. The study of eigenvalues of a second degree exponential polynomial characteristic equation allows to conclude to the existence of stability switches for the unique positive steady state. A numerical analysis of the role of each parameter on the appearance of stability switches completes this analysis.

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Global stability in a delayed partial differential equation describing cellular replication

Cited by 89 publications

References 19 publications

Asynchronous Exponential Growth of a General Structured Population Model

Asynchronous Exponential Growth of a General Structured Population Model

Modelling Physiological and Pharmacological Control on Cell Proliferation to Optimise Cancer Treatments

Delay Model of Hematopoietic Stem Cell Dynamics: Asymptotic Stability and Stability Switch

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