ABSTRACT
The efflux protein P-glycoprotein (P-gp) is over expressed in many cancer cells and has a known capacity to confer multi-drug resistance to cytotoxic therapies. We provide a mathematical model for the direct cell-to-cell transfer of proteins between cells and the indirect transfer between cells and the surrounding liquid. After a mathematical analysis of the model, we construct an adapted numerical scheme and give some numerical simulations. We observe that we obtain a better fit with the experimental data when we take into account the indirect transfer of the protein released in a dish. This quantity, usually neglected by the experimenters, seems to influence the results.
Subject(s)
ATP Binding Cassette Transporter, Subfamily B, Member 1/metabolism , Breast Neoplasms/metabolism , Models, Biological , Biological Transport , Breast Neoplasms/pathology , Cell Communication , Drug Resistance, Neoplasm , Humans , MCF-7 Cells , Models, TheoreticalABSTRACT
A model for cell-cell adhesion, based on an equation originally proposed by Armstrong et al. [A continuum approach to modelling cell-cell adhesion, J. Theor. Biol. 243 (2006), pp. 98-113], is considered. The model consists of a nonlinear partial differential equation for the cell density in an N-dimensional infinite domain. It has a non-local flux term which models the component of cell motion attributable to cells having formed bonds with other nearby cells. Using the theory of fractional powers of analytic semigroup generators and working in spaces with bounded uniformly continuous derivatives, the local existence of classical solutions is proved. Positivity and boundedness of solutions is then established, leading to global existence of solutions. Finally, the asymptotic behaviour of solutions about the spatially uniform state is considered. The model is illustrated by simulations that can be applied to in vitro wound closure experiments.
Subject(s)
Cell Adhesion , Cell Communication , Cell Movement , Cell Proliferation , Computer Simulation , Developmental Biology , Diffusion , Humans , Linear Models , Models, Biological , Motion , Neoplasm Invasiveness , Neoplasms/metabolism , Wound HealingABSTRACT
We analyze the asymptotic behaviour of solutions of the abstract differential equation u'(t)=Au(t)-F(u(t))u(t)+f. Our results are applicable to models of structured population dynamics in which the state space consists of population densities with respect to the structure variables. In the equation the linear term A corresponds to internal processes independent of crowding, the nonlinear logistic term F corresponds to the influence of crowding, and the source term f corresponds to external effects. We analyze three separate cases and show that for each case the solutions stabilize in a way governed by the linear term. We illustrate the results with examples of models of structured population dynamics -- a model for the proliferation of cell lines with telomere shortening, a model of proliferating and quiescent cell populations, and a model for the growth of tumour cord cell populations.
Subject(s)
Cell Proliferation , Logistic Models , Models, Biological , Algorithms , Animals , Cell Count , Cell Survival/physiology , Cellular Senescence/physiology , Humans , Neoplasms/pathology , Neoplasms/physiopathology , Telomere/physiologyABSTRACT
We analyse an age-structured model of telomere loss in a proliferating cell population. The cell population is divided into telomere classes, which shorten each round of division. The model consists of a nonlinear system of partial differential equations for the telomere classes. We prove that if the highest telomere class is exempted from mortality, then all the classes stabilize to a nontrivial equilibrium dependent on the initial state of cells in the highest telomere class.
Subject(s)
Cell Line, Tumor/ultrastructure , Logistic Models , Telomere/ultrastructure , Animals , Cell Death , Cell Proliferation , Humans , Models, Biological , Nonlinear DynamicsABSTRACT
A model of a proliferating cell population is analyzed. The model distinguishes individual cells by cell age, which corresponds to phase of the cell cycle. The model also distinguishes individual cells by proliferating or quiescent status. The model allows cells to transit between these two states at any age, that is, any phase of the cell cycle. The model also allows newly divided cells to enter quiescence at cell birth, that is, cell age 0. Sufficient conditions are established to assure that the cell population has asynchronous exponential growth. As a consequence of this asynchronous exponential growth the population stabilizes in the sense that the proportion of the population in any age range, or the fraction in proliferating or quiescent state, converges to a limiting value as time evolves, independently of the age distribution and proliferating or quiescent fractions of the initial cell population. The asynchronous exponential growth is proved by demonstrating that the strongly continuous linear semi-group associated with the partial differential equations of the model is positive, irreducible, and eventually compact.
