The mathematical modelling of tumour angiogenesis and invasion

In order to accomplish the transition from avascular to vascular growth, solid tumours secrete a diffusible substance known as tumour angiogenesis factor (TAF) into the surrounding tissue. Endothelial cells which form the lining of neighbouring blood vessels respond to this chemotactic stimulus in a well-ordered sequence of events comprising, at minimum, of a degradation of their basement membrane, migration and proliferation. Capillary sprouts are formed which migrate towards the tumour eventually penetrating it and permitting vascular growth to take place. It is during this stage of growth that the insidious process of invasion of surrounding tissues can and does take place. A model mechanism for angiogenesis is presented which includes the diffusion of the TAF into the surrounding host tissue and the response of the endothelial cells to the chemotactic stimulus. Numerical simulations of the model are shown to compare very well with experimental observations. The subsequent vascular growth of the tumour is discussed with regard to a classical reaction-diffusion pre-pattern model

Mathematical modeling of tumor-induced angiogenesis

Angiogenesis, the growth of a network of blood vessels, is a crucial component of solid tumor growth, linking the relatively harmless avascular and the potentially fatal vascular growth phases of the tumor. As a process, angiogenesis is a well-orchestrated sequence of events involving endothelial cell migration and proliferation; degradation of tissue; new capillary vessel formation; loop formation (anastomosis) and, crucially, blood flow through the network. Once there is flow associated with the nascent network, subsequent growth evolves both temporally and spatially in response to the combined effects of angiogenic factors, migratory cues via the extracellular matrix, and perfusion-related hemodynamic forces in a manner that may be described as both adaptive and dynamic. In this article, we first present a review of previous theoretical and computational models of angiogenesis and then indicate how recent developments in flow models are providing insight into antiangiogenic and chemotherapeutic drug treatment of solid tumors

Mathematical modelling of flow in 2D and 3D vascular networks: applications to anti-angiogenic and chemotherapeutic drug strategies

The aim of this paper is to investigate the conditions required to optimize the amount of chemotherapeutic drug absorbed by a solid tumour through a network of blood vessels. This work is based on a study of vascular networks generated from a discrete mathematical model of tumour angiogenesis, which describes the formation of a capillary network in response to chemical stimuli released by a solid tumour. Simulations of blood flow in the vasculature connecting the parent vessel to the solid tumour are then performed by adapting modelling techniques from the field of petroleum engineering to this biomedical application We begin with a qualitative, comparative study relating to the efficiency of drug delivery in 2D and 3D tumour-induced vasculatures and evaluate the influence of key parameters (mean capillary radius, blood viscosity and delivery regime) upon uptake by the tumour. We then go on to examine the impact of the vascular architecture upon nutrient (e.g., oxygen) and drug delivery by comparing the efficiency of three vasculatures characterized by different spatial distributions of branching order and anastomosis density (i.e., the number of fused loops or connections). We identify the main criteria required of a tumour-induced vascular network for optimized delivery of nutrients and/or cytotoxic agents. We conclude by focusing on a particular vascular network and investigate how ?capillary pruning? (i.e., network re-modelling) modifies the network connectivity and associated blood flow distribution. We demonstrate how random removal of vessels may lead to a significant increase in the amount of drug delivered to the tumour. Selective removal of vessels characterized by low flow is seen to speed up delivery, whilst the targeting of high flow capillaries leads to a rapid shut down of the entire capillary bed. These results allow us to propose the possibility of optimized cancer treatment therapies, based upon a coupled anti-angiogenic/chemotherapy strategy?the anti-angiogenesis treatment would be used to optimize network efficiency, thereby maximizing drug uptake during subsequent chemotherapy treatments

Avascular growth, angiogenesis and vascular growth in solid tumours: the mathematical modelling of the stages of tumour development

The growth and development of solid tumours occurs in two distinct stages?the avascular growth phase and the vascular growth phase. During the former growth phase the tumour remains in a diffusion-limited, dormant state of a few millimetres in diameter (cf. multicell spheroids, carcinoma in situ) while during the latter growth phase, invasion and metastasis may take place. In order to accomplish the transition from avascular to vascular growth, solid tumours may secrete diffusible substances known as tumour angiogenesis factors (TAF) into the surrounding tissue. Endothelial cells which form the lining of neighbouring blood vessels respond to this chemotactic stimulus in a well-ordered sequence of events. Capillary sprouts are formed which migrate towards the tumour, eventually penetrating it and permitting vascular growth to take place. This paper will present several mathematical models which deal with the various stages of growth and development of solid tumours