DEPARTMENT OF MECHANICS AND APPLIED MATHEMATICS
THE FRACTAL MODEL OF THE BLOOD-VESSEL SYSTEM
HEAD OF THE PROJECT Jaan KALDA, Ph.D., senior researcher
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)
the length of the largest vessels (aorta)
and the total length of capillaries,
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 = (
Using the definition of the similarity dimension we can easily find
Ds = -1/ log2a
The fractal model of the blood-vessel system with Ds
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
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
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
and then projected into the space of dimensionality
D < Ds.
As a result of such a projection, the dimensions
Ds become equal to the new value of
D, whereas the similarity dimension will evidently remain unchanged.
- 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"