Cell kinetics in tumour cords studied by a model with variable cell cycle length.
ABSTRACT A mathematical model is developed that describes the proliferative behaviour at the stationary state of the cell population within a tumour cord, i.e. in a cylindrical arrangement of tumour cells growing around a blood vessel and surrounded by necrosis. The model, that represents the tumour cord as a continuum, accounts for the migration of cells from the inner to the outer zone of the cord and describes the cell cycle by a sequence of maturity compartments plus a possible quiescent compartment. Cell-to-cell variability of cycle phase transit times and changes in the cell kinetic parameters within the cord, related to changes of the microenvironment, can be represented in the model. The theoretical predictions are compared against literature data of the time course of the labelling index and of the fraction of labelled mitoses in an experimental tumour after pulse labelling with 3H-thymidine. It is shown that the presence of cell migration within the cord can lead to a marked underestimation of the actual changes along cord radius of the kinetics of cell cycle progression.
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ABSTRACT: The proliferative behaviour at the stationary state of a cell population within a tumour cord has been described by a non-classical boundary value problem for a hyperbolic flrst order integro-partial difierential equation (2). The model was theoretically ana- lyzed in (5), where su-cient conditions are given on the fraction of cells which enter proliferation to assure the existence of a unique steady state. Also, a more complex model, where cells are distinguished by maturity, had been studied in (4). In later works, models with variable cell-cycle length (1) or with the efiects of drugs and ra- diation (3) has been developed. From a numerical point of view, there is only an algorithm proposed in (3) in order to make extensive simulations. In this work, we are going to develop a numerical method for this kind of problem. The numerical scheme we will introduce take into account that the straightforward discretization of the integro-partial difierential equation by a flnite difierences method needs of additional data on the characteristic curve representing the wall of the blood vessel and also that the coupling of the boundary conditions involves the solution of a nonlinear system of equations. The numerical simulations with the method are used to make a quantitative study of the best functional form (within several classes of functions) for the radial depen- dency of the function that describes the fraction of newborn cells which become quiescent in the model (3), when compared with experimental fleld data. The goal of this study is to validate the model under study. ReferencesMathematical and Computer Modelling. 01/2010; 52:992-998.
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ABSTRACT: The model proposed here links together two approaches to describe tumours: a continuous medium to describe the movement and the mechanical properties of the tissue, and a population dynamics approach to represent internal genetic inhomogeneity and instability of the tumour. In this way one can build models which cover several stages of tumour progression. In this paper we focus on describing transition from aerobic to purely glycolytic metabolism (the Warburg effect) in tumour cords. From the mathematical point of view this model leads to a free boundary problem where domains in contact are characterized by different sets of equations. Accurate stitching of the solution was possible with a modified ghost fluid method. Growth and death of the cells and uptake of the nutrients are related through ATP production and energy costs of the cellular processes. In the framework of the bi-population model this allowed to keep the number of model parameters relatively small.Journal of Theoretical Biology 03/2009; 258(4):578-90. · 2.35 Impact Factor
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ABSTRACT: Two major mechanisms are involved in the formation of blood vasculature: vasculogenesis and angiogenesis. The former term describes the formation of a capillary-like network from either a dispersed or a monolayered population of endothelial cells, reproducible also in vitro by specific experimental assays. The latter term describes the sprouting of new vessels from an existing capillary or post-capillary venule. Similar mechanisms are also involved in the formation of the lymphatic system through a process generally called lymphangiogenesis. A number of mathematical approaches have been used to analyse these phenomena. In this article, we review the different types of models, with special emphasis on their ability to reproduce different biological systems and to predict measurable quantities which describe the overall processes. Finally, we highlight the advantages specific to each of the different modelling approaches.Journal of Theoretical Biology 05/2013; · 2.35 Impact Factor