Modelling aspects of cancer dynamics: a review

Abstract

Cancer is a complex disease in which a variety of factors interact over a wide range of spatial and temporal scales with huge datasets relating to the different scales available. However, these data do not always reveal the mechanisms underpinning the observed phenomena. In this paper, we explain why mathematics is a powerful tool for interpreting such data by presenting case studies that illustrate the types of insight that realistic theoretical models of solid tumour growth may yield. These range from discriminating between competing hypotheses for the formation of collagenous capsules associated with benign tumours to predicting the most likely stimulus for protease production in early breast cancer. We will also illustrate the benefits that may result when experimentalists and theoreticians collaborate by considering a novel anti-cancer therapy.

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

“…Given an initialized level set function ϕ, only the points that fall within a fixed distance of the interface are updated during level set operations (e.g., velocity extensions and level set reinitialization). In the level set context, the narrow band can be identified by (9) where R > 0 is a fixed constant that is chosen to suit the problem. In our work with the tumor problem, we use R = 20Δx.…”

“…Cancer is a fundamental scientific and societal problem, and in the past several decades, intensive research has been focused on understanding the complexity of cancer progression, developing new therapies, and formulating optimal treatment protocols. While much work has been done in the mathematical community on tumor modeling (e.g., see the reviews [3,6,7,9,46,47]), to date there has been little work in modeling tumor growth in realistic, heterogeneous tissues on large spatial scales. The methods we present in this paper will provide the foundation for a biologically-detailed millimeter-to-centimeter-scale model of tumor growth in heterogeneous tissues with realistic features (e.g., mechanically soft and hard regions, bone, and inhomogeneous nutrient delivery) [33,35,39].…”

“…Frieboes et al [16,17] and Wise et al [53] have begun studying 3D tumor growth using a diffuse interface approach, while others have begun studying the tumor problem using multiphase mixture models (e.g., see [4,8], and [10]). Still others use discrete models, such as cellular automata and agents (e.g., see [1,5], and [9] for some recent examples).…”

A Mechano-Chemical Model of a Solid Tumor for Therapy Outcome Predictions

“…In the study we introduce angiostatin locally in the 4th month of tumor development and continue its administration till the end of month 10. This inhibitor neutralizes the growth factor directly as described by (2). We apply AST off-center in the lower left quadrant and not inside the tumor, to prevent the vessels from reaching the tumor at all.…”

“…Simulation of such complex systems are computationally most demanding, yet the rapid development of hardware, especially emerging distributed parallel computing concepts, enable more and more realistic modeling of physiological systems. Tumor development has been studied extensively over the last three decades, reviews and exhaustive survey of the approaches can be found elsewhere [1,2] or more recently in [3]. A further discussion of the relevant literature may be found in our previous works [4,5].…”