A mathematical model for the production and secretion of tumour angiogenesis factor in tumours

Solid tumour growth was hypothesized by Folkman (1976) to take place in two phases: the avascular phase and the vascular phase. In the first (avascular) phase, the tumour obtains its nutrients and disposes of its metabolic wastes by diffusion transport processes alone. Since the mechanism for growth is diffusion-limited, these tumours cannot expand indefinitely, but grow to a dormant state in which they have ceased expanding. The second (vascular) phase involves the eliciting of new blood vessels from the surrounding tissue and there is now firm evidence that tumour cells produce a chemical compound which triggers this process. The compound has been termed tumour angiogenesis factor (TAF) and considerable research has been carried out to try and isolate it and identify its biological structure as well as to elucidate its effects on the endothelial cells which form the lining of the blood vessels. In this second phase, the tumour grows rapidly and can spread to other parts of the body via blood-borne metastases. In this paper, the authors present a theoretical model for the production of the TAF within the tumour while in its diffusion-limited state and prior to its release into the surrounding host tissue. Using experimental results on vascularized tumours in conjunction with the findings of Oosaki et al. (1987), it is assumed that the profile of the TAF concentration within the tumour prior to secretion is qualitatively the same as that of the blood vessels found in neovascularized tumours. The TAF concentration c(x, t) is taken to satisfy the diffusion equation and the TAF production is accounted for either by the inclusion of a production term phi(c) in the diffusion equation itself or via inclusion in the boundary conditions. Taking phi(c) to be of the form 1/(1-c) produces the desired TAF profile and also leads to the possibility of a critical level being reached. It is shown that if the tumour is small enough this critical level can never be attained. However, if the tumour exceeds a certain size, then the critical level is attained and the TAF is subsequently secreted into the external tissue. This is described mathematically by the phenomenon of quenching, that is, the solution c(x, t) remains finite while some derivative becomes unbounded in finite time.