New concepts in the study of tissue vascularization : a mathematical model of skin vascularization

A preliminary study demonstrated the existence of a fractal structure for perforator arterial vessels of the skin and proved to be a useful tool to compare vascular trees on the basis of their complexity. Fractal analysis of axial-perforator arteriovenous vascular trees was performed on the skin of mice after injection of the arterial network by india ink. Fractal analysis was performed by box counting. Fractal dimension D was determined for 35 venous and 31 arterial perforator vessels (D = 1.302 and 1.264, respectively) and 5 venous and 3 arterial axial vessels (D = 1.374 and 1.328, respectively) (r2 > or = 0.985). All vascular networks show a fractal structure, characterized by a specific D. These values are relatively constant; D is a function of the anatomic and physiologic characteristics. There was no significant difference between venous and arterial networks, nor was there between axial and perforator networks (p < 0.05); this demonstrates a similar efficacy in terms of perfusion of the skin. A computer simulation based on fractal theory has been developed to reproduce the two kinds of vascular networks. Fractals are the result of a construction procedure that is repeated and repeated so that the iteration of a very simple rule can produce seemingly complex shapes, such as vascular networks. The basic module that is repeated in the whole structure is Y-shaped and is termed the generator; this generator is applied to a basic structure, called the initiator. After a few iterations, a vascular network is obtained. The difference between axial and perforator vascular networks is the choice of the initiator, whereas the generator is identical.