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      Probabilistic approach to a cell growth model

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          Abstract

          We consider the time evolution of the supercritical Galton-Watson model of branching particles with extra parameter (mass). In the moment of the division the mass of the particle (which is growing linearly after the birth) is divided in random proportion between two offsprings (mitosis). Using the technique of moment equations we study asymptotic of the mass distribution of the particles. Mass distribution of the particles is the solution of the equation with linearly transformed argument: functional, functional-differential or integral. We derive several limit theorems describing the fluctuations of the density of the particles, first two moments of the total masses etc.

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          A mathematical model for analysis of the cell cycle in cell lines derived from human tumors.

          Marianne E. Marshall, Bruce Van Every, Bruce C. Baguley … (2003)
          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.
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            Mathematical determination of cell population doubling times for multiple cell lines.

            Liene Daukste, Britta Basse, Bruce C. Baguley … (2012)
            Cell cycle times are vital parameters in cancer research, and short cell cycle times are often related to poor survival of cancer patients. A method for experimental estimation of cell cycle times, or doubling times of cultured cancer cell populations, based on addition of paclitaxel (an inhibitor of cell division) has been proposed in literature. We use a mathematical model to investigate relationships between essential parameters of the cell division cycle following inhibition of cell division. The reduction in the number of cells engaged in DNA replication reaches a plateau as the concentration of paclitaxel is increased; this can be determined experimentally. From our model we have derived a plateau log reduction formula for proliferating cells and established that there are linear relationships between the plateau log reduction values and the reciprocal of doubling times (i.e. growth rates of the populations). We have therefore provided theoretical justification of an important experimental technique to determine cell doubling times. Furthermore, we have applied Monte Carlo experiments to justify the suggested linear relationships used to estimate doubling time from 5-day cell culture assays. We show that our results are applicable to cancer cell populations with cell loss present.
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              Structure of Population Inside Propagating Front

              L. Koralov, S. Molchanov (2013)
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                Author and article information

                Journal
                Publication date Created: 16 August 2018
                Article
                ArXiV ID: 1808.05677
                SO-VID: 06bfb8b1-3582-410e-b47f-0d236f46bfd0
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                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Comments 18 pages
                Categories math.PR

                ScienceOpen disciplines: Probability
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                ScienceOpen disciplines: Probability

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