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 n_eff as N = 2^n_eff. Being guided by the assumption of self-similarity, we can express the similarity factor a as a = ( ₀/ l₀)^1/n_eff. Using the definition of the similarity dimension we can easily find

D_s = -1/ log₂a 3.4(1)

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 D_HB and D_b are always equal to the topological dimension of the embedding space D. As for the similarity dimension, it is generally accepted that D_s coincides with the Hausdorff-Besicovitch dimension D_HB. Thus it may seem that always D_s D. However, the equality D_s = D_HB = D_b 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 D_in> D_s and then projected into the space of dimensionality D < D_s. As a result of such a projection, the dimensions D_HB and D_s 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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07/04/1998 webmaster