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.
"These structures are observed in sections taken from in vivo primary and experimental tumors, and may be a key organizational structure for vascular tumors at the sub-tissue level  . The tumor cord has been investigated mathematically in many contexts as it lends itself to useful assumptions about the symmetry of tumor growth  . If one models growth in a thin crosswise section so that significant lengths of the microvessel/tumor extend in both directions, one can make the assumption that the section has axial symmetry. "
[Show abstract][Hide abstract] ABSTRACT: We investigate a new model of tumor growth in which cell motility is considered an explicitly separate process from growth. Bulk tumor expansion is modeled by individual cell motility in a density-dependent diffusion process. This model is implemented in the context of an in vivo system, the tumor cord. We investigate numerically microscale density distributions of different cell classes and macroscale whole tumor growth rates as functions of the strength of transitions between classes. Our results indicate that the total tumor growth follows a classical von Bertalanffy growth profile, as many in vivo tumors are observed to do. This provides a quick validation for the model hypotheses. The microscale and macroscale properties are both sensitive to fluctuations in the transition parameters, and grossly adopt one of two phenotypic profiles based on their parameter regime. We analyze these profiles and use the observations to classify parameter regimes by their phenotypes. This classification yields a novel hypothesis for the early evolutionary selection of the metastatic phenotype by selecting against less motile cells which grow to higher densities and may therefore induce local collapse of the vascular network.
"Diffusion and haptotaxis terms account for the spatial dynamics of the system in the models under study. Age, size and/or space structure has also been used in models of tumor cords    . Computational and software considerations often limit scientists from incorporating physiological structure directly into a model. "
[Show abstract][Hide abstract] ABSTRACT: We present multiscale models of cancer tumor invasion with components at the molecular, cellular, and tissue levels. We provide biological justifications for the model components, present computational results from the model, and discuss the scientific-computing methodology used to solve the model equations. The models and methodology presented in this paper form the basis for developing and treating increasingly complex, mechanistic models of tumor invasion that will be more predictive and less phenomenological. Because many of the features of the cancer models, such as taxis, aging and growth, are seen in other biological systems, the models and methods discussed here also provide a template for handling a broader range of biological problems.
SIAM Journal on Multiscale Modeling and Simulation 04/2005; 5(1). DOI:10.1137/050629215 · 1.63 Impact Factor
"The authors concluded that a correct analysis of radioactive labelling data from tumour microregions requires the possible cell migration through the regions to be taken into account. While both the model given by Bertuzzi and Gandolfi (2000) and that given by Bertuzzi et al. (2002) neglected the process of cell death within the tumour cord, a subsequent model given by Bertuzzi et al. (2003) considered the dynamics of tumour cords under the action of a cytotoxic agent. This model incorporated both a random cell death—either spontaneous or induced by the cytotoxic agent—and a cell death which results from insufficient nutrient availability. "
[Show abstract][Hide abstract] ABSTRACT: A miscellany of new strategies, experimental techniques and theoretical approaches are emerging in the ongoing battle against cancer. Nevertheless, as new, ground-breaking discoveries relating to many and diverse areas of cancer research are made, scientists often have recourse to mathematical modelling in order to elucidate and interpret these experimental findings. Indeed, experimentalists and clinicians alike are becoming increasingly aware of the possibilities afforded by mathematical modelling, recognising that current medical techniques and experimental approaches are often unable to distinguish between various possible mechanisms underlying important aspects of tumour development. This short treatise presents a concise history of the study of solid tumour growth, illustrating the development of mathematical approaches from the early decades of the twentieth century to the present time. Most importantly these mathematical investigations are interwoven with the associated experimental work, showing the crucial relationship between experimental and theoretical approaches, which together have moulded our understanding of tumour growth and contributed to current anti-cancer treatments. Thus, a selection of mathematical publications, including the influential theoretical studies by Burton, Greenspan, Liotta et al., McElwain and co-workers, Adam and Maggelakis, and Byrne and co-workers are juxtaposed with the seminal experimental findings of Gray et al. on oxygenation and radio-sensitivity, Folkman on angiogenesis, Dorie et al. on cell migration and a wide variety of other crucial discoveries. In this way the development of this field of research through the interactions of these different approaches is illuminated, demonstrating the origins of our current understanding of the disease.
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