Modelling and analysis of time-lags in some basic patterns of cell proliferation (original) (raw)
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Modelling and analysis of time-lags in cell proliferation
1997
In this paper, we present a systematic approach for obtaining qualitatively and quantitatively correct mathematical models of some biological phenomena with time-lags. Features of our approach are the development of a hierarchy of related models and the estimation of parameter values, along with their non-linear biases and standard deviations, for sets of experimental data. We demonstrate our method of solving parameter estimation problems for neutral delay differential equations by analyzing some models of cell growth that incorporate a time-lag in the cell division phase. We show that these models are more consistent with certain reported data than the classic exponential growth model. Although the exponential growth model provides estimates of some of the growth characteristics, such as the population-doubling time, the time-lag growth models can additionally provide estimates of: (i) the fraction of cells that are dividing, (ii) the rate of commitment of cells to cell division, (iii) the initial distribution of cells in the cell cycle, and (iv) the degree of synchronization of cells in the (initial) cell population.
A short note on delay effects in cell proliferation
1995
We show that a growth model for cell proliferation that incorporates a time-lag in the cell division phase is more consistent with certain reported data than the classical exponential growth model. Although both models provide estimates of some of the growth characteristics, such as the population doubling-time, the time-lag growth model additionally provides estimates of: (i) the cell-doubling time, (ii) the fraction of the cells that are dividing, (iii) the rate of commitment to cell division and, (iv) the initial distribution of cells in the cell cycle.
Application of Stochastic Delay Differential Equations on Cell Growth
Modelling of biological system via ordinary differential equations (ODEs) and stochastic differential equations (SDEs) has become an intensive research over last few years. In both types of equations the unknown function and its derivatives are evaluated at the same instant time, t. However, many of the natural phenomena do not have an immediate effect from the moment of their occurrence. A patient, for example, shows symptoms of an illness days or even weeks after the day in which he was infected. The dynamics of the systems differ dramatically if the corresponding characteristic equations involve time delay. Therefore, ODEs and SDEs which are simply depending on the present state are insufficient to explain this process. Such phenomenon can be modelled via stochastic delay differential equations (SDDEs). Batch fermentation is one of the systems that subject to the presence of noise and delay effects. It is necessary to model this process via SDDEs. To the best of our knowledge, the mathematical model of this system takes the form of ODEs and SDEs.
Mathematical Modelling of Cell Growth and Proliferation
IFAC Proceedings Volumes, 1988
Models able to describe the events of cellular growth and divi sion and the dyn ami cs of cell populations are useful for the understanding of control mechanisms and for theoretical support for the automated analysis of flow cytometric data and of cell volume distributions. This paper reports on models that have been developed by the Authors with this aim, describing in a rather unitary frame the cell cycle of eukaryotic cells, like mammalian cells and yeast, and of prokaryotic cells. The model is based on the assumption that the progression of the nuclear divi sion cycle is regulated by a sequential attainment of two threshold protein levels.lt accounts for a number of features of cell growth and division in population of actively growing cells, it explains all the different patterns of cell cycle which are experimentally found and yields quantitative relations between timing of the cell cycle and macromolecular composition of the cells. The model is also used to study the effect of various sources of variability on the statistical properties of cell populations and the main source of variability appears to be an inaccuracy of the molecular mechanism that monitors the cell size. Besides in normal mammalian cells a second source of variability is apparent, which depends upon the interaction with growth factors which give competence. An extended version of the model, which comprises also this additional variability, is also considered and used to describe properties of normal and transformed cell growth. Keywords Mathematical models; cell growth and proliferation; cell populations; normal and transformed cells.
A delay differential equation model for tumor growth
Journal of Mathematical Biology, 2003
We present a competition model of tumor growth that includes the immune system response and a cycle-phase-specific drug. The model considers three populations: Immune system, population of tumor cells during interphase and population of tumor during mitosis. Delay differential equations are used to model the system to take into account the phases of the cell cycle. We analyze the stability of the system and prove a theorem based on the argument principle to determine the stability of a fixed point and show that the stability may depend on the delay. We show theoretically and through numerical simulations that periodic solutions may arise through Hopf Bifurcations.
Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2015
For the model of periodic chronic myelogenous leukemia considered by Pujo-Menjouet, Mackey et al., model consisting of two delay differential equations, the equation for the density of so-called “resting cells” was studied from numerical and qualitative point of view in several works. In this paper we focus on the equation for the density of proliferating cells and study it from a qualitative point of view.
Electronic Journal of Differential Equations Electronic Only, 2003
In this paper, we investigate a linear population model of cells that are capable of simultaneous proliferation and maturation. We consider the case when the time required for a cell to divide depends on its maturity. This model is described by first order partial differential system with a retardation of the maturation variable and a time delay depending on this maturity. Both delays are due to cell replication.
For the model of periodic chronic myelogenous leukemia considered by Pujo-Menjouet, Mackey et al., model consisting of two delay differential equations, the equation for the density of so-called "resting cells" was studied from numerical and qualitative point of view in several works. In this paper we focus on the equation for the density of proliferating cells and study it from a qualitative point of view.
Delay equation formulation of a cyclin-structured cell population model
The aim of this paper is to derive a system of two renewal equations from individual-level assumptions concerning a cyclin-structured cell population. Nonlinearity arises from the assumption that the rate at which quiescent cells become proliferating is determined by feedback. In fact we assume that this rate is a nonlinear function of a weighted population size. We characterize steady states and establish the validity of the principle of linearized stability.
Computational and Mathematical Methods in Medicine, 1999
Cell proliferation and differentiation phenomena are key issues in immunology, tumour growth and cell biology. We study the kinetics of cell growth in the immune system using mathematical models formulated in terms of ordinary and delay differential equations. We study how the suitability of the mathematical models depends on the nature of the cell growth data and the types of differential equations by minimizing an objective function to give a best-fit parameterized solution. We show that mathematical models that incorporate a time-lag in the cell division phase are more consistent with certain reported data. They also allow various cell proliferation characteristics to be estimated directly, such as the average cell-doubling time and the rate of commitment of cells to cell division. Specifically, we study the interleukin-2-dependent cell division of phytohemagglutinin stimulated T-cellsthe model of which can be considered to be a general model of cell growth. We also review the numerical techniques available for solving delay differential equations and calculating the least-squares best-fit parameterized solution.