Institute of Cybernetics at Tallinn University of Technology            RESEARCH OVERVIEW 1998

DEPARTMENT OF MECHANICS AND APPLIED MATHEMATICS

THE FRACTAL MODEL OF THE BLOOD-VESSEL SYSTEM

HEAD OF THE PROJECT Jaan KALDA, Ph.D., senior researcher

DESCRIPTION

The aim of the project is to elaborate a fractal model of the blood-vessel network as a whole, which would be in agreement with the current understanding of the processes governing the growth of the vascular network and with the empirical data.

Let us make a rough estimate of the similarity dimension of the blood-vessel tree. We can use the following empirical data: the length of the capillaries (i.e. the vessels of the last generation) 0 0.5, the length of the largest vessels (aorta) l0 0.5 m and the total length of capillaries, L 0N 100,000 km. The total number of capillaries N can be expressed via the effective number of generations neff as N = 2neff. Being guided by the assumption of self-similarity, we can express the similarity factor a as a = ( 0/ l0)1/neff. Using the definition of the similarity dimension we can easily find

Ds = -1/ log2a 3.4(1)

The fractal model of the blood-vessel system with Ds 3.4

The seemingly curious fact that the similarity dimension exceeds the topological dimension can be explained as follows. The Hausdorff-Besicovitch and box-counting dimensions of a space-filling fractal set DHB and Db are always equal to the topological dimension of the embedding space D. As for the similarity dimension, it is generally accepted that Ds coincides with the Hausdorff-Besicovitch dimension DHB. Thus it may seem that always Ds D. However, the equality Ds = DHB = Db can be applied only if all the dimensions are less than the dimension of the embedding space. Indeed, one can imagine that the fractal tree was originally embedded into a space of dimensionality Din> Ds and then projected into the space of dimensionality D < Ds. As a result of such a projection, the dimensions DHB and Ds become equal to the new value of D, whereas the similarity dimension will evidently remain unchanged.

ESF GRANTS

1. No 902 (1996-1998)

PUBLICATIONS and TALKS at CONFERENCES

• Kalda J. On the optimization of Monte-Carlo simulations. - Physica A, 1997, 246 - 646-658.

• International Conference on the Unity of the Sciences, Washington, 24.-29. November 1997.
Jaan Kalda: "On the Modelling of Fractal Tree-like Structures in Biology".

• 5th Int. Conf. Fractal 98, (forthcoming)
J. Kalda, "Fractality of the blood-vessel system: the model and its applications"

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