The growth of solid tumors and metastases beyond a certain size limit of 1.4 to 2.5 mm in diameter (1) is enabled by the tumor-induced sprouting of blood vessels, a process termed angiogenesis. Folkman and colleagues (2) first identified a soluble growth-factor referred to as tumor-angiogenesis factor (TAF) produced by tumors that could induce the growth of such new capillaries. This in turn led to the concept of antiangiogenesis as tumor therapy (3). Experimental verification of this mechanism in a rabbit model (4), research discovering details of the cellular processes of tumor angiogenesis (5), and in vivo inhibition of tumor-induced angiogenesis (6) form the basis of most of the work that followed.

In parallel to the experimental work, attempts were being made early on to mathematically model the endothelial cell (EC) sprouting, the interconnection and looping of blood vessels, and their growth toward the tumor induced by TAFs (7). Meanwhile, a model of the extravascular transport of reporter molecules in tumor and normal tissue showed a good fit to one-dimensional diffusion. Furthermore, real tumors were found to provide fewer hindrances to macromolecular transport by diffusion than were surrounding tissue (8).Modeling work that compared experimentally measured TAF profiles predicted that this factor could be secreted outside the neoplasm only after the tumor exceeds a threshold size (9). Patterns of angiogenesis have been examined subsequently using reaction diffusion simulations that focused (for example) on TAF diffusion (10). Furthermore, work by Byrne and Chaplain (11) examined the role of the migration rate of newly sprouted capillaries and its consequences on brush borders and the parameters contributing to successful vascularization of tumors.