ABSTRACT
We present a model that describes the growth, division and death of a cell population structured by size. The model is an extension of that studied by Hall and Wake (1989) and incorporates the asymmetric division of cells. We consider the case of binary asymmetrical splitting in which a cell of size ε divides into two daughter cells of different sizes and find the steady size distribution (SSD) solution to the non-local differential equation. We then discuss the shape of the SSD solution. The existence of higher eigenfunctions is also discussed.
Subject(s)
Cell Division/physiology , Cell Enlargement , Cell Proliferation/physiology , Models, Biological , Models, Statistical , Animals , Cell Count , Cell Size , Computer Simulation , HumansABSTRACT
Most anti-cancer drugs in use today exert their effects by inducing a programmed cell death mechanism. This process, termed apoptosis, is accompanied by degradation of the DNA and produces cells with a range of DNA contents. We have previously developed a phase transition mathematical model to describe the mammalian cell division cycle in terms of cell cycle phases and the transition rates between these phases. We now extend this model here to incorporate a transition to a programmed cell death phase whereby cellular DNA is progressively degraded with time. We have utilised the technique of flow cytometry to analyse the behaviour of a melanoma cell line (NZM13) that was exposed to paclitaxel, a drug used frequently in the treatment of cancer. The flow cytometry profiles included a complex mixture of living cells whose DNA content was increasing with time and dying cells whose DNA content was decreasing with time. Application of the mathematical model enabled estimation of the rate constant for entry of mitotic cells into apoptosis (0.035 per hour) and the duration of the period of DNA degradation (51 hours). These results provide a dynamic model of the action of an anticancer drug that can be extended to improve the clinical outcome in individual cancer patients.
Subject(s)
Antineoplastic Agents, Phytogenic/pharmacology , Apoptosis/drug effects , Mathematics , Models, Biological , Paclitaxel/pharmacology , Cell Cycle , Cell Line, Tumor , DNA, Neoplasm/metabolism , Flow Cytometry , Humans , Melanoma/drug therapy , Melanoma/metabolism , Melanoma/pathologyABSTRACT
The growth of human cancers is characterised by long and variable cell cycle times that are controlled by stochastic events prior to DNA replication and cell division. Treatment with radiotherapy or chemotherapy induces a complex chain of events involving reversible cell cycle arrest and cell death. In this paper we have developed a mathematical model that has the potential to describe the growth of human tumour cells and their responses to therapy. We have used the model to predict the response of cells to mitotic arrest, and have compared the results to experimental data using a human melanoma cell line exposed to the anticancer drug paclitaxel. Cells were analysed for DNA content at multiple time points by flow cytometry. An excellent correspondence was obtained between predicted and experimental data. We discuss possible extensions to the model to describe the behaviour of cell populations in vivo.
