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Arquivos de Neuro-Psiquiatria

Print version ISSN 0004-282X

Arq. Neuro-Psiquiatr. vol.53 no.2 São Paulo June 1995 

Energetic and spatial constraints of arterial networks


Optimação energética e espacial de redes arteriais



Sandro Rossitti

From the Department of Clinical Neurosciences, Section of Neurosurgery, Göteborg University, Göteborg, Sweden




The principle of minimum work (PMW) is a parametric optimization model for the growth and adaptation of arterial trees. A balance between energy dissipation due to frictional resistance of laminar flow (shear stress) and the minimum volume of the blood and vessel wall tissue is achieved when the vessel radii are adjusted to the cube root of the volumetric flow. The PMW is known to apply over several magnitudes of vessel calibers, and in many different organs, including the brain, in humans and in animals. Animal studies suggest that blood flow in arteries is approximately proportional to the cube of the vessel radius, and that arteries alter their caliber in response to sustained changes of blood flow according to PMW. Remodelling of the retinal arteriolar network to long-term changes in blood flow was observed in humans. Remodelling of whole arterial networks occurs in the form of increase or diminishing of vessel calibers. Shear stress induced endothelial mediation seems to be the regulating mechanism for the maintenance of this optimum blood flow/vessel diameter relation. Arterial trees are also expected to be nearly space filing. The vascular system is constructed in such a way that, while blood vessels occupy only a small percentage of the body volume leaving the bulk to tissue, they also crisscross organs so tightly that every point in the tissue lies on the boundary between an artery and a vein. This review describes how the energetic optimum principle for least energy cost for blood flow is also compatible with the spatial constraints of arterial networks according to concepts derived from fractal geometry.

Key Words: blood vessels, cerebral arteries, fractals, hemodynamics, optimality concepts, retinal arteries, shear stress, vasodilation.


Aneurismas intracranianos e vasculopatia de hiperperfusão em pacientes com malformações arteriovenosas cerebrais resultam do elevado stress hemodinâmico na rede arterial cerebral. O estabelecimento de uma norma para a geometria arterial cerebral deve resultar em melhores critérios preventivos e de planejamento terapêutico para essas patologias. Uma rede arterial deve distribuir-se no espaço para todo o órgão perfundido, e ao mesmo tempo possibilitar a perfusão tecidual com adequado custo energético e mínimo stress hemodinâmico. O custo total do fluxo sangüíneo é a soma do custo Pf para propulsão do sangue através dos vasos (que aumenta com a redução do calibre das artérias) e do custo metabólico Pm do tecido sangüíneo e dos vasos (que diminui com a redução do calibre das artérias). O equilíbrio entre Pf e Pm resulta no mínimo custo total quando artérias de grande e pequeno calibre organizam-se em uma hierarquia de ramificações tal que o raio interno do vaso é proporcional ao cubo do fluxo sangüíneo (princípio do trabalho mínimo), o que significa que em cada ramificação arterial o raio do tronco (r0) e dos ramos (r1 e r2) estão relacionados de acordo com a regra r03=r13+r23
O exponente de bifurcação "n", definido r0n=r13+r23 assume em condição ótima o valor 3. A regra n
3 é também compatível à optimação espacial de redes arteriais de acordo com princípios de geometria fractal. O exponente de bifurcação aproxima-se desse valor em redes arteriais de diversas espécies de mamíferos e nas artérias cerebrais humanas. Arteríolas retinianas reorganizam-se após atrofia óptica (quando o fluxo sangüíneo retiniano diminui) de acordo com essa regra. Estudos experimentais in vivo e in vitrocorroboram esse princípio. O controle do calibre arterial ocorre através de mediação endotelial, com a produção de vasomediadores e alterações da polaridade das membranas celulares, desse modo controlando o tônus vascular em curtos intervalos de tempo, e resultando em remodelação anatômica a longo prazo.

Palavras-chave: artérias cerebrais, artérias retinianas, fluxo sangüíneo, geometria fractal, hemodinâmica, mecanismos de controle, vasodilatação, vasos sangüíneos.



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Aceite: 3-outubro-1994.


This article is based on lecture notes of communications at the 7th Biomechanics Seminar (Göteborg, Sweden, April 22-23,1993), at the 2nd International Conference on Stroke of the World Federation of Neurology (Geneva, Switzerland, May 12-15, 1993) and at the 8th International Symposium on Vascular Neuroeffector Mechanisms (Alberta, Canada, August 1-4, 1994).
Dr. Sandro Rossitti - Nordenskiöldsgatan 6 - S-413 09 Göteborg, Sweden.

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