The multifractal structure of arterial trees.

Fractal properties of arterial trees are analysed using the cascade model of turbulence theory. It is shown that the branching process leads to a non-uniform structure at the micro-level meaning that blood supply to the tissue varies in space. From the model it is concluded that, depending on the branching parameter, vessels of a specific size contribute dominantly to the blood supply of tissue. The corresponding tissue elements form a dense set in the tissue. Furthermore, if blood flow in vessels can get obstructed with some probability, the above set of tissue elements may not be dense anymore. Then there is the risk that, spread out over the tissue, nutrient and gas exchange fall short.

[1]  Michael F. Barnsley,et al.  Fractals everywhere , 1988 .

[2]  Daniel A. Lidar,et al.  The Limited Scaling Range of Empirical Fractals , 1998 .

[3]  Jaap A. Kaandorp,et al.  Morphological analysis of growth forms of branching marine sessile organisms along environmental gradients , 1999 .

[4]  A. R. Imre.  Ideas in Theoretical Biology - Comment About the Fractality of the Lung , 1999 .

[5]  Shaun Lovejoy,et al.  Universal Multifractals: Theory and Observations for Rain and Clouds , 1993 .

[6]  Radomír Mech,et al.  Visual Models of Plant Development , 1997, Handbook of Formal Languages.

[7]  Stanley J. Wiegand,et al.  Vascular-specific growth factors and blood vessel formation , 2000, Nature.

[8]  B. Mandelbrot.  Intermittent turbulence in self-similar cascades : divergence of high moments and dimension of the carrier , 2004 .

[9]  Roberto Benzi,et al.  On the multifractal nature of fully developed turbulence and chaotic systems , 1984 .

[10]  Grzegorz Rozenberg,et al.  Handbook of Formal Languages , 1997, Springer Berlin Heidelberg.

[11]  A. Kolmogorov.  A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number , 1962, Journal of Fluid Mechanics.

[12]  P. Dodds,et al.  Unified view of scaling laws for river networks. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  D. Turcotte,et al.  Networks with side branching in biology. , 1998, Journal of theoretical biology.

[14]  V V Gafiychuk,et al.  On the principles of the vascular network branching. , 2001, Journal of theoretical biology.

[15]  M. Zamir.  On fractal properties of arterial trees. , 1999, Journal of theoretical biology.

[16]  M Zamir,et al.  Fractal dimensions and multifractility in vascular branching. , 2001, Journal of theoretical biology.

[17]  Daniel A. Lidar,et al.  Is the Geometry of Nature Fractal? , 1998, Science.