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)
l_{0}
0.5 m
and the total length of capillaries,
L
_{0}N
100,000 km.
The total number of capillaries N can be expressed via the effective number of generations n_{eff} as
N = 2^{neff}.
Being guided by the assumption of self-similarity, we can express the similarity factor a as a = (
_{0}/
l_{0})^{1/neff}.
Using the definition of the similarity dimension we can easily find

D_{s} = -1/ log_{2}a
3.4(1)

The fractal model of the blood-vessel system with D_{s}
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
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

- 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"

07/04/1998 webmaster