The term "angiogenesis" defines the fundamental process of the development and growth of new blood vessels from the pre-existing vasculature, and is essential for reproduction, development and wound repair [1]. Under these conditions, it is highly regulated: i.e. "turned on" for brief periods of time (days) and then completely inhibited.
The cyclic nature of the microvascular bed in the corpus luteum provides a unique experimental model for examining the discrete physiological steps of angiogenesis in the life cycle of endothelial cells which, together with pericytes (supportive vascular smooth muscle cells), carry all of the genetic information necessary to form tubes, branches and entire capillary networks.
However, many human diseases (including solid tumors) are driven by persistently up-regulated angiogenesis [1]. In some non-malignant processes, such as pyogenic granuloma or keloid formation [2], angiogenesis is prolonged but still self-limited; however, this is not true of tumor angiogenesis which, once begun, continues indefinitely until the entire tumor is eradicated or the host dies. Without blood vessels, tumors cannot grow beyond a critical size (1–2 mm) or metastasize to another organ.
Angiogenesis is one of the most complex dynamic processes in biology, and is highly regulated by a balance of pro- and anti-angiogenic molecules. It is now widely accepted that the "angiogenic switch" is "off" when the effects of pro-angiogenic molecules is balanced by that of anti-angiogenic molecules, and "on" when the net balance is tipped in favor of angiogenesis [1, 3]. Pro- and anti-angiogenic molecules can be secreted from cancer cells, endothelial cells, stromal cells, blood, and the extra-cellular matrix [4, 5], the relative contributions of which are likely to change with tumor type and site, as well as with tumor growth, regression and relapse [1].
Although considerable advances have been made in our molecular and cellular knowledge of the promotion, mediation and inhibition of angiogenesis, very little is known about its underlying complex dynamics. Vasculature and more generally tubular organs develop in a wide variety of ways involving many cell processes [6–8].
In mathematical terms, angiogenesis is a non-linear dynamic system that is discontinuous in space and time, but advances through qualitatively different states. The word state defines the configuration pattern of the system at any given moment, and a dynamic system can be represented as a set of different states and a number of transitions from one state to another over a certain time interval [9, 10].
At least seven critical steps have so far been identified in the sequence of angiogenic events on the basis of sprout formation: a) endothelial cells are activated by an angiogenic stimulus; b) the endothelial cells secrete proteases to degrade the basement membrane and extra-cellular matrix; c) a capillary sprout is formed as a result of directed endothelial cell migration, d) grows by means of cell mitoses and migration, and e) forms a lumen and a new basement membrane; f) two sprouts come together to form a capillary loop; and g) second-generation capillary sprouts begin to form [1, 11, 12] (Fig. 1).
The progression of these states generates a complex ramified structure that irregularly fills the surrounding environment (Fig. 2). The main feature of the newly generated vasculature is the structural diversity of the vessel sizes, shapes and connecting patterns.
Tumor vessels are structurally and functionally abnormal [1, 3]: unlike normal vessels, they are highly disorganized, tortuous and dilated, and have uneven diameters, and excessive branching and shunts. This may be mainly due to the heterogeneous distribution of angiogenic regulators, such as vascular-endothelial growth factor (VEGF), basic fibroblastic growth factor (bFGF) and angiopoietin [5, 13], leading to chaotic tumor blood flow, and hypoxic and acidic tumoral regions [5, 14–16]. Moreover, although it is commonly believed that the endothelial cells making-up tumor vessels are genetically stable, diploid cells (and thus different from genetically unstable neoplastic cells), tumor vasculature seems to be much more unpredictable [17].
These conditions all reduce the effectiveness of treatments, modulate the production of pro- and anti-angiogenic molecules, and select a subset of more aggressive cancer cells with higher metastatic potential [1].
A large number of clinical trials of anti-angiogenic therapies are being conducted throughout the world, but investigators are still concerned about how to achieve the maximum benefit from them and how to monitor patient response. There are currently no markers of the net angiogenic activity of a tumor that can help investigators to design specific anti-angiogenic treatment strategies [5, 18], but it is reasonable to resume that the quantification of various aspects of tumor vasculature may provide an indication of angiogenic activity.
One often-quantified element of tumor vasculature is microvessel density (MVD), which is used to allow a histological assessment of tumor angiogenesis. The results of studies carried out over the last decade have suggested the value of using tumor MVD as a prognostic index in a wide variety of solid cancers, and it has also recently been assumed that MVD may reveal the degree of angiogenic activity in a tumor. On the basis of these assumptions, the quantification of MVD is thought to be a surrogate marker of the efficacy of anti-angiogenic agents as well as a means of assessing which patients are good candidates for anti-angiogenic therapy. However, MVD has a number of substantial limitations, mainly due to the complex biology characterizing tumor vasculature [17], and the highly irregular geometry that the vascular system assumes in real space, which cannot be measured using the principles of Euclidean geometry because it is only capable of interpreting regular and smooth objects that are almost impossible to find in Nature.
However, quantitative descriptors of its geometrical complexity can be usefully abstracted from the fractal geometry introduced by Benoit Mandelbrot in 1975 [20, 21]. We here discuss the surface fractal dimension (D
S
) as a quantitative index of the 2-D geometrical complexity of vascular networks and their behavior during computer-simulated changes in vessel density and distribution.