Narrow Results

The aim of the present work is twofold: to develop numerical procedures for a priori determining whether a given cell population, having a distributed cell-cycle duration, will grow or decay when subjected to prescribed chemotherapy; to evaluate the cumulative error in the long-term predictions for such populations. We show that cell population dynamics under drug treatment can be modelled by iterative application of a compact operator on the initial cell age-distribution. We further show that this model can be approximated by iterative application of matrices on some finite-dimensional vector, containing initial conditions. Moreover, we develop a method for estimating the growth rate of cell population and show that in fully periodic treatments the estimated error does not grow as time tends to infinity. From the biomedical viewpoint this means that only fully periodic (strictly periodic) schedules can be considered for successfully predicting the long-term effect of chemotherapy. Thus, cyclic drug treatment is shown to be advantageous, not only in increasing selectivity of chemotherapy, as has been previously demonstrated, but also in increasing long-term predictability of specific treatment schedules.

The generally accepted Moolgavkar's theory of carcinogenesis assumes that all cancers are clonal, i.e. that they arise from progressive genetic deregulation in a cell pedigree originating from a single ancestral cell.¹⁸ However, recently the clonal theory has been challenged by the field theory of carcinogenesis, which admits the possibility of simultaneous changes in tissue subject to carcinogenic agents, such as tobacco smoke in lung cancer. Axelrod et al.¹ formulated a more detailed framework, in which partially transformed cells depend in a mutualistic way on growth factors they produce, in this way enabling these cells to proliferate and undergo further transformations. On the other hand, the field theory assumes spatial distribution of precancerous cells and indeed there exists evidence that early-stage precancerous lesions in lung cancer progress along linear, tubular, or irregular surface structures. This seems to be the case for the atypical adenomatous hyperplasia (AAH),¹⁰ a likely precursor of adenocarcinoma of the lung. In this paper we explore the consequences of linking the model of spatial growth of precancerous cells,¹² with the mutualistic hypothesis. We investigate the solutions of the model using analytical and computational techniques. The picture emerging from our modelling indicates that production of growth factors by cells considered may lead to diffusion-driven instability, which in turn may lead either to decay of both population, or to emergence of local growth foci, represented by spike-like solutions. Mutualism may, in some situations, increase the stability of solutions. One important conclusion is that models of field carcinogenesis, which include spatial effects, generally have very different behaviour compared to ODE models.

This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, some of the novel mathematical problems in the field. This review attempts to capture, from the appropriate literature, the main issues involved in the modelling of phenomena related to cancer dynamics at all scales which characterise this highly complex system: from the molecular scale up to that of tissue. The last part of the paper discusses the challenge of developing a mathematical biological theory of tumour onset and evolution.

In this paper, we investigate the scaling law relating the size of the boundary of a solid tumor and the rate at which it is lysed by a cell population of non-infiltrating cytotoxic lymphocytes. We do it in the context of enzyme kinetics through geometrical, analytical and numerical arguments. Following the Koch island fractal model, a scale-dependent function that describes the constant rate of the decay process and the fractal dimension is obtained. Then, in silico experiments are accomplished by means of a stochastic hybrid cellular automaton model. This model is used to grow several tumors with varying morphology and to test the power decay law when the cell-mediated immune response is effective, confirming its validity.