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 ≥ 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. The growth of the two kinds of perfusions appears in the same tissue environment; there is no reason why, in the same tissue, for an identical physiologic function, there should be a difference in the growing pattern of these vessels. Only the origin of these blood vessels is different, and this is taken into account by the model in having different initiators; this explains the difference in macroscopic aspects. Finally, several variables of this mathematical model are discussed.SCOPUS: ar.j