Energetic and spatial constraints of arterial networks

Rossitti, Sandro

doi:10.1590/S0004-282X1995000200028

Abstracts

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.

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.

artérias cerebrais; artérias retinianas; fluxo sangüíneo; geometria fractal; hemodinâmica; mecanismos de controle; vasodilatação; vasos sangüíneos

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

SUMMARY

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.

RESUMO

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 P_f para propulsão do sangue através dos vasos (que aumenta com a redução do calibre das artérias) e do custo metabólico P_m do tecido sangüíneo e dos vasos (que diminui com a redução do calibre das artérias). O equilíbrio entre P_f e P_m 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+r₂³assume 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.

1. Batjer H, Suss RA, Samson D. Intracranial arteriovenous malformations associated with aneurysms. Neurosurgery 1986, 18: 29-35.
2. Bevan JA, Bevan R (eds). The human brain circulation. London: Humana Press (Chapman & Hall Medical), 1994.
3. Bevan JA, Joyce EH. Flow-induced resistance artery tone: balance between contrictor and dilator mechanisms. Am J Physiol 1990, 258: H663-H668.
4. Bevan JA, Laher I. Pressure and flow-dependent vascular tone. FASEB J 1991, 5: 2267-2273.
5. Bevan JA, Wellman GC. Intraluminal flow-initiated hyperpolarization and depolarization shift the membrane potential of arterial smooth muscle toward an intermendiate level. Circ Res 1993, 73: 1188-1192.
6. Brown N. A mathematical model for the formation of cerebral aneurysms. Stroke 1991, 22: 619-625.
7. Cvitanovic P (ed). Universality in Chaos. Ed 2. Bristol: Adam Higer, 1989.
8. Damiani G. Evolutionary meaning, functions and morphogenesis of branching structures in biology. In: Nonnenmacher TF, Losa, GA Weibel ER (eds). Fractals in biology and medicine. Basel: Birkhäuser, 1994, p 104-115.
9. Fung YC, Liu SQ. Elementary mechanics of the endothelium of blood vessels. ASME J Biomech Eng 1993, 115: 1-12.
10. Griffith TM. Chaos and fractals in vascular biology. Vasc Med Rev 1994, 5: 161-182.
11. Griffith TM, Edwards DH. Basal EDRF activity helps to keep the geometrical configuration of arterial bifurcations close to the Murray optimum. J Theor Biol 1990, 146: 545-573.
12. Hao Bai-Lin (ed). Chaos II. Singapore: World Scientific, 1990.
13. Hassler W. Hemodynamic aspects of cerebral angiomas. Acta Neurochir (Wien) 1986, 37 (Suppl): 1-136.
14. Hutcheson, IR, Griffith TM. Release of endothelium-derived relaxing factor is modulated both by frequency and amplitude of pulsatile flow. Am J Physiol 1991, 261: H257-H262.
15. Kim C, Cervós-Navarro J, Kikuchi H. Alterations in cerebral vessels in experimental animals and their possible relationship to the development of aneurysms. Surg Neurol 1992, 38: 331-337.
16. La Barbera M. Principles of design of fluid transport systems in zoology. Science 1990, 249:992-1000.
17. Lasjaunias P, Piske R, Terbrugge K, Wilinsky R. Cerebral arteriovenous malformations (CAVM) and associated arterial aneurysms (AA): analysis of 101 CAVM cases with 37 AA in 23 patients. Acta Neurochir (Wien) 1988, 91: 29-36.
18. Levčfre J. Teleonomical optimization of a fractal model of the pulmonary arterial bed. J Theor Biol 1983, 102: 225-248.
19. Liu YH, Ritman EL. Branching pattern of pulmonary arterial tree in anesthetized dogs. ASME J Biomech Eng 1986, 108: 289-293.
20. Mandelbrot BB. Trees and diameter exponent. In: The Fractal Geometry of Nature. New York: Freeman, 1983, pl56-165.
21. Mayrovitz HN. An optimal flow-radius equation for microvessel non-newtonian blood flow. Microvasc Res 1987, 34: 380-384.
22. Mayrovitz HN, Roy J. Microvascular blood flow: evidence indicating a cubic dependence on arteriolar diameter. Am J Physiol 1983, 245: H1031-H1038.
23. Murray CD. The physiological principle of minimum work: I. The vascular system and cost of blood volume. Proc Nat Acad Sci USA 1926,12:207-214.
24. Murray CD. The physiological principle of minimum work applied to angle of branching of arteries. J Gen Physiol 1926, 9: 835-841.
25. Nerem RM.Hemodynamics and the vascular endothelium. ASME J Biomech Eng 1993,115:510-514.
26. Nonnenmacher TF, Losa, GA, Weibel ER (eds). Fractals in biology and medicine. Basel: Birkhäuser, 1994.
27. Norlén G. The cerebral circulation in supratentorial angiomas as studied by angiography before and after removal. In Proceedings of the 1 st International Congress of Neurosurgery. Bruxelles: Les Editions "Acta Medica Bélgica", 1957, p 217-222.
28. Okamoto S, Handa H, Hashimoto N. Location of intracranial aneurysms asssociated with cerebral arteriovenous malformations: statistical analysis. Surg Neurol 1984, 22: 335-340.
29. Ostergaard JR. Risk factors in intracranial saccular aneurysms: aspects on the formation and rupture of aneurysms, and development of cerebral vasospasm. Acta Neurol Scand 1989, 80: 81-98.
30. Pile-Spellman JMD, Baker KF, Liszczak TM, Sandrew BB, Oot RF, Debrun G, Zervas NT, Taveras JM. High-flow angiopathy: cerebral blood vessel changes in experimental chronic arteriovenous fistula. Am J Neuroradiol 1986, 7: 811-815.
31. Rossitti S. Optimality principles and regulation of arterial caliber. Can J Physiol Pharmacol 1994, 72 (Suppl 4): 22.
32. Rossitti S, Frangos J, Girardi PR, Bevan J. Regulation of vascular tone. Can J Physiol Pharmacol (in press).
33. Rossitti S, Frisén L. Remodelling of the retinal arterioles in descending optic atrophy follows the principle of minimum work. Acta Physiol Scand 1994, 152: 333-340.
34. Rossitti S, Löfgren J. Vascular dimensions of the cerebral arteries follow the principle of minimum work. Stroke 1993, 24: 371-377.
35. Rossitti S, Löfgren J. Optmality principles and flow orderliness at the branching points of cerebral arteries. Stroke 1993, 24: 1029-1032.
36. Rossitti S, Löfgren J. Optimality principles and the geometry of the cerebral arterial network. Biomech Sem 1993, 7: 21-26.
37. Rossitti S, Löfgren J. Why do the cerebral arteries develop saccular aneurysms? Upsala J Med Sci 1993, 52 (Suppl): 42.
38. Rossitti S, Löfgren J beskriven, Stephensen H. Blodflödeshastighetens tidsmăssiga heterogenicitet i artéria cerebri media hos människa med fraktalanalys. Hygiea 1993, 102: 241.
39. Rossitti S, Stephensen H. Temporal heterogeneity of the blood flow velocity at the middle cerebral artery in the normal human characterized by fractal analysis. Acta Physiol Scand 1994, 151: 191-198.
40. Sherman TF. On connecting large vessels to small: the meaning of Murrray's law. J Gen Physiol 1981,78: 431-453.
41. Smiesko V, Johnson PC. The arterial lumen is controlled by flow-related shear stress. News Physiol Sci 1993,8: 34-38.
42. Stehbens WE. Pathology of the cerebral blood vessels. Saint Louis: C V Mosby 1972.
43. Stehbens WE. Etiology of intracranial berry aneurysms. J Neurosurg 1989, 70:823-831.
44. Steiger HJ. Pathophysiology of development and rupture of cerebral aneurysms. Acta Neurochir (Wien) 1990, 48 (Suppl): 1-57.
45. Uylings HBM. Optimization of diameters and bifurcation angles in lung and vascular tree structures. Bull Math Biol 1977, 39: 509-520.
46. Vane JR. The endothelium: maestro of the blood circulation. Phi Trans R Soc Lond B 1994, 343: 225-246.
47. West BJ. Fractal physiology and chaos in medicine. Singapore: World Scientific, 1990.
48. Zamir M. Cost analysis of arterial branching in the cardiovascular systems of man and animals. J Theor Biol 1986, 120: 111-123.

Publication Dates

Publication in this collection
20 Jan 2011
Date of issue
June 1995

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

[1] 1. Batjer H, Suss RA, Samson D. Intracranial arteriovenous malformations associated with aneurysms. Neurosurgery 1986, 18: 29-35.

[2] 2. Bevan JA, Bevan R (eds). The human brain circulation. London: Humana Press (Chapman & Hall Medical), 1994.

[3] 3. Bevan JA, Joyce EH. Flow-induced resistance artery tone: balance between contrictor and dilator mechanisms. Am J Physiol 1990, 258: H663-H668.

[4] 4. Bevan JA, Laher I. Pressure and flow-dependent vascular tone. FASEB J 1991, 5: 2267-2273.

[5] 5. Bevan JA, Wellman GC. Intraluminal flow-initiated hyperpolarization and depolarization shift the membrane potential of arterial smooth muscle toward an intermendiate level. Circ Res 1993, 73: 1188-1192.

[6] 6. Brown N. A mathematical model for the formation of cerebral aneurysms. Stroke 1991, 22: 619-625.

[7] 7. Cvitanovic P (ed). Universality in Chaos. Ed 2. Bristol: Adam Higer, 1989.

[8] 8. Damiani G. Evolutionary meaning, functions and morphogenesis of branching structures in biology. In: Nonnenmacher TF, Losa, GA Weibel ER (eds). Fractals in biology and medicine. Basel: Birkhäuser, 1994, p 104-115.

[9] 9. Fung YC, Liu SQ. Elementary mechanics of the endothelium of blood vessels. ASME J Biomech Eng 1993, 115: 1-12.

[10] 10. Griffith TM. Chaos and fractals in vascular biology. Vasc Med Rev 1994, 5: 161-182.

[11] 11. Griffith TM, Edwards DH. Basal EDRF activity helps to keep the geometrical configuration of arterial bifurcations close to the Murray optimum. J Theor Biol 1990, 146: 545-573.

[12] 12. Hao Bai-Lin (ed). Chaos II. Singapore: World Scientific, 1990.

[13] 13. Hassler W. Hemodynamic aspects of cerebral angiomas. Acta Neurochir (Wien) 1986, 37 (Suppl): 1-136.

[14] 14. Hutcheson, IR, Griffith TM. Release of endothelium-derived relaxing factor is modulated both by frequency and amplitude of pulsatile flow. Am J Physiol 1991, 261: H257-H262.

[15] 15. Kim C, Cervós-Navarro J, Kikuchi H. Alterations in cerebral vessels in experimental animals and their possible relationship to the development of aneurysms. Surg Neurol 1992, 38: 331-337.

[16] 16. La Barbera M. Principles of design of fluid transport systems in zoology. Science 1990, 249:992-1000.

[17] 17. Lasjaunias P, Piske R, Terbrugge K, Wilinsky R. Cerebral arteriovenous malformations (CAVM) and associated arterial aneurysms (AA): analysis of 101 CAVM cases with 37 AA in 23 patients. Acta Neurochir (Wien) 1988, 91: 29-36.

[18] 18. Levčfre J. Teleonomical optimization of a fractal model of the pulmonary arterial bed. J Theor Biol 1983, 102: 225-248.

[19] 19. Liu YH, Ritman EL. Branching pattern of pulmonary arterial tree in anesthetized dogs. ASME J Biomech Eng 1986, 108: 289-293.

[20] 20. Mandelbrot BB. Trees and diameter exponent. In: The Fractal Geometry of Nature. New York: Freeman, 1983, pl56-165.

[21] 21. Mayrovitz HN. An optimal flow-radius equation for microvessel non-newtonian blood flow. Microvasc Res 1987, 34: 380-384.

[22] 22. Mayrovitz HN, Roy J. Microvascular blood flow: evidence indicating a cubic dependence on arteriolar diameter. Am J Physiol 1983, 245: H1031-H1038.

[23] 23. Murray CD. The physiological principle of minimum work: I. The vascular system and cost of blood volume. Proc Nat Acad Sci USA 1926,12:207-214.

[24] 24. Murray CD. The physiological principle of minimum work applied to angle of branching of arteries. J Gen Physiol 1926, 9: 835-841.

[25] 25. Nerem RM.Hemodynamics and the vascular endothelium. ASME J Biomech Eng 1993,115:510-514.

[26] 26. Nonnenmacher TF, Losa, GA, Weibel ER (eds). Fractals in biology and medicine. Basel: Birkhäuser, 1994.

[27] 27. Norlén G. The cerebral circulation in supratentorial angiomas as studied by angiography before and after removal. In Proceedings of the 1 st International Congress of Neurosurgery. Bruxelles: Les Editions "Acta Medica Bélgica", 1957, p 217-222.

[28] 28. Okamoto S, Handa H, Hashimoto N. Location of intracranial aneurysms asssociated with cerebral arteriovenous malformations: statistical analysis. Surg Neurol 1984, 22: 335-340.

[29] 29. Ostergaard JR. Risk factors in intracranial saccular aneurysms: aspects on the formation and rupture of aneurysms, and development of cerebral vasospasm. Acta Neurol Scand 1989, 80: 81-98.

[30] 30. Pile-Spellman JMD, Baker KF, Liszczak TM, Sandrew BB, Oot RF, Debrun G, Zervas NT, Taveras JM. High-flow angiopathy: cerebral blood vessel changes in experimental chronic arteriovenous fistula. Am J Neuroradiol 1986, 7: 811-815.

[31] 31. Rossitti S. Optimality principles and regulation of arterial caliber. Can J Physiol Pharmacol 1994, 72 (Suppl 4): 22.

[32] 32. Rossitti S, Frangos J, Girardi PR, Bevan J. Regulation of vascular tone. Can J Physiol Pharmacol (in press).

[33] 33. Rossitti S, Frisén L. Remodelling of the retinal arterioles in descending optic atrophy follows the principle of minimum work. Acta Physiol Scand 1994, 152: 333-340.

[34] 34. Rossitti S, Löfgren J. Vascular dimensions of the cerebral arteries follow the principle of minimum work. Stroke 1993, 24: 371-377.

[35] 35. Rossitti S, Löfgren J. Optmality principles and flow orderliness at the branching points of cerebral arteries. Stroke 1993, 24: 1029-1032.

[36] 36. Rossitti S, Löfgren J. Optimality principles and the geometry of the cerebral arterial network. Biomech Sem 1993, 7: 21-26.

[37] 37. Rossitti S, Löfgren J. Why do the cerebral arteries develop saccular aneurysms? Upsala J Med Sci 1993, 52 (Suppl): 42.

[38] 38. Rossitti S, Löfgren J beskriven, Stephensen H. Blodflödeshastighetens tidsmăssiga heterogenicitet i artéria cerebri media hos människa med fraktalanalys. Hygiea 1993, 102: 241.

[39] 39. Rossitti S, Stephensen H. Temporal heterogeneity of the blood flow velocity at the middle cerebral artery in the normal human characterized by fractal analysis. Acta Physiol Scand 1994, 151: 191-198.

[40] 40. Sherman TF. On connecting large vessels to small: the meaning of Murrray's law. J Gen Physiol 1981,78: 431-453.

[41] 41. Smiesko V, Johnson PC. The arterial lumen is controlled by flow-related shear stress. News Physiol Sci 1993,8: 34-38.

[42] 42. Stehbens WE. Pathology of the cerebral blood vessels. Saint Louis: C V Mosby 1972.

[43] 43. Stehbens WE. Etiology of intracranial berry aneurysms. J Neurosurg 1989, 70:823-831.

[44] 44. Steiger HJ. Pathophysiology of development and rupture of cerebral aneurysms. Acta Neurochir (Wien) 1990, 48 (Suppl): 1-57.

[45] 45. Uylings HBM. Optimization of diameters and bifurcation angles in lung and vascular tree structures. Bull Math Biol 1977, 39: 509-520.

[46] 46. Vane JR. The endothelium: maestro of the blood circulation. Phi Trans R Soc Lond B 1994, 343: 225-246.

[47] 47. West BJ. Fractal physiology and chaos in medicine. Singapore: World Scientific, 1990.

[48] 48. Zamir M. Cost analysis of arterial branching in the cardiovascular systems of man and animals. J Theor Biol 1986, 120: 111-123.

Brasil

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Energetic and spatial constraints of arterial networks

Optimação energética e espacial de redes arteriais

Abstracts

Publication Dates