A Survey of Cell Population Dynamics (original) (raw)
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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.
Mathematical and Computer Modelling, 2008
We analyse both theoretically and numerically a nonlinear model of the dynamics of a cell population divided into proliferative and quiescent compartments that is described in [F. Bekkal Brikci, J. Clairambault, B. Ribba, B. Perthame, An age-and-cyclinstructured cell population model with proliferation and quiescence, INRIA Research Report No 5941, 2006]. It is a physiological age and molecule-structured population model for the cell division cycle, which aims at representing both healthy and tumoral tissues. A noticeable feature of this model is to exhibit tissue homeostasis for healthy tissue and unlimited growth for tumoral tissue. In particular, the present paper analyses model parameters for which a tumoral tissue exhibits polynomial growth and not mere exponential growth. Polynomial tumour growth has been recently advocated by several authors, on the basis either of experimental observations or of individual cell-based simulations which take space limitations into account. This model is able to take such polynomial growth behaviour into account without considerations of space, by proposing exchange functions between the proliferative and quiescent compartments.
Cell growth and division: a deterministic/probabilistic model of the cell cycle
Journal of Mathematical Biology, 1986
A model of the cell cycle, incorporating a deterministic cell-size monitor and a probabilistic component, is investigated. Steady-state distributions for cell size and generation time are calculated and shown to be globally asymptotically stable. These distributions are used to calculate various statistical quantities, which are then compared to known experimental data. Finally, the results are compared to distributions calculated from a Monte-Carlo simulation of the model.
Model of Cellular Proliferation
2013
We analyse the asymptotic behaviour of a nonlinear mathematical model of cellular proliferation which describes the production of blood cells in the bone marrow. This model takes the form of a system of two maturity structured partial differential equations, with a retardation of the maturation variable and a time delay depending on this maturity. We show that the stability of this system depends strongly on the behaviour of the immature cells population. We obtain conditions for the global stability and the instability of the trivial solution.
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.
Simulation of mammalian cell population dynamics
Applied is a systems view to modelling of mammalian cell cultivation in a bioreactor for biotechnological applications. Proposed is a model with a hierarchical structure with three levels: the macroscopic level of a reactor, microscopic level of a cell population, and molecular level of protein interactions with cyclin dependent kineases. The macroscopic model provides basis for production process control by optimal feeding of nutrients and growth factors during cultivation. The microscopic model simplifies a cell population into three pools of cells: P proliferating, Q quiescent, and D dead cells. Dynamics of cell population is determined by the least square estimation of the specific rates of the pool transitions. The molecular model enables theoretical basis for prediction of G1/S transition and the rate of transition from proliferating cells into quiescent state. Conceptual application of the molecular model in conjunction with the mammalian cell production system is discussed
Modeling of cell population dynamics
Oscillatory yeast cell dynamics are observed in glucose-limited growth environments. Under such conditions, both glucose and the excreted product ethanol can serve as substrates for cell growth. The cell dynamics is described by a PDE (partial differential equation) system containing one PDE for the cell population and 8 ODEs for 8 substrates variations (extracellular glucose, extracellular ethanol, intracellular glucose, intracellular ethanol, exhausted oxygen, exhausted carbon dioxide, dissolved oxygen and dissolved carbon dioxide). The system is solved by the numerical method of characteristics (MOC) modified . Here, mesh points are added at the smallest cell size and also deleted at the largest cell size at the same time level. The modified MOC provides a solution free of numerical dissipation error caused by the cell growth term (i.e., convection term), owing to the mass axis moving along a cell growth pathline. The oscillatory behavior of the cell number and the cell mass is predicted from the simulation. The cell number variation affects the extracellular glucose/ethanol and oxygen/carbon-dioxide concentrations in the gas exhaust stream. As ethanol is excreted primarily by budded cells, it is shown for the extracellular ethanol concentration to slowly reach a regular oscillatory state. Dynamics of the evolved oxygen concentration ratio and carbon-dioxide concentration ratio are also shown.
A Structured Population Model of Cell Differentiation
SIAM Journal on Applied Mathematics, 2011
We introduce and analyze several aspects of a new model for cell differentiation. It assumes that differentiation of progenitor cells is a continuous process. From the mathematical point of view, it is based on partial differential equations of transport type. Specifically, it consists of a structured population equation with a nonlinear feedback loop. This models the signaling process due to cytokines, which regulate the differentiation and proliferation process. We compare the continuous model to its discrete counterpart, a multi-compartmental model of a discrete collection of cell subpopulations recently proposed by to investigate the dynamics of the hematopoietic system. We obtain uniform bounds for the solutions, characterize steady state solutions, and analyze their linearized stability. We show how persistence or extinction might occur according to values of parameters that characterize the stem cells self-renewal. We also perform numerical simulations and discuss the qualitative behavior of the continuous model vis a vis the discrete one.
Comptes Rendus Biologies, 2004
In this paper we consider cell cycle models for which the transition operator for the evolution of birth mass density is a simple, linear dynamical system with a stochastic perturbation. The convolution model for a birth mass distribution is presented. Density functions of birth mass and tail probabilities in n-th generation are calculated by a saddle-point approximation method. With these probabilities, representing the probability of exceeding an acceptable mass value, we have more control over pathological growth. A computer simulation is presented for cell proliferation in the age-dependent cell cycle model. The simulation takes into account the fact that the age-dependent model with a linear growth is a simple linear dynamical system with an additive stochastic perturbation. The simulated data as well as the experimental data (generation times for mouse L) are fitted by the proposed convolution model. To cite this article: