A generalization of the optimal models of arterial branching

Roy, Alain; Woldenberg, Michael J.

doi:10.1016/s0092-8240(82)80016-5

Cited by 31 publications

(40 citation statements)

References 14 publications

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“…A classification between delivering and distributing paths in vascular trees was earlier made for the human heart, 19 although no branching angles were reported in that study. In addition, we used our data to test two optimality principles, 15 relating the branching angles at the bifurcations to the junction exponent and the bifurcation index. One principle minimizes lumen volume and pumping power, whereas the other principle minimizes lumen surface and total shear force on the endothelial tissue.…”

Section: Discussionmentioning

confidence: 99%

“…(3)), optimal values for the branching angles at the bifurcations can be calculated. 15 Minimization of lumen volume and pumping power results in:…”

Section: Branching Anglesmentioning

confidence: 99%

“…Given an inlet pressure, Eqs. (15) and (16) form a set of linear equations with the nodal pressures as unknowns:…”

Section: Relative Shear Stressmentioning

confidence: 99%

“…1,2,9,11,15,16 Most of these studies report theoretical analyses of the relationship between the anatomical structure of the tree and one or more hemodynamic variables, such as shear stress, 16 drag and power loss, 15 and perfusion volume. 9 Unfortunately, the bulk of these models are not well supported by empirical data.…”

Section: Introductionmentioning

confidence: 99%

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Branching Morphology of the Rat Hepatic Portal Vein Tree: A Micro-CT Study

2006

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In contrast to the lung and the myocardium, the liver is a relatively homogeneous organ with fewer anatomic constraints on vascular branching. Hence, we hypothesize that the hepatic vasculature could more closely follow optimization of branching geometry than is the case in other organs. The geometrical and fractal properties of the rat hepatic portal vein tree were investigated, with the aid of three-dimensional micro-computed tomography data. Frequency distributions of vessel radii were obtained at three different voxel resolutions and fitted to a theoretical model of dichotomous branching. The model predicted an average junction exponent of 3.09. Hemodynamic model calculations showed that with generation, relative shear stress decreases. Branching angles were found to oscillate between those predicted by two optimality principles of minimum power loss and volume, and of minimum shear stress and surface. The liver shows a variation in branching morphology similar to that of other organs. Therefore, we conclude that anatomic constraints do not have a major perturbing impact.

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Section: Discussionmentioning

confidence: 99%

“…(3)), optimal values for the branching angles at the bifurcations can be calculated. 15 Minimization of lumen volume and pumping power results in:…”

Section: Branching Anglesmentioning

confidence: 99%

“…Given an inlet pressure, Eqs. (15) and (16) form a set of linear equations with the nodal pressures as unknowns:…”

Section: Relative Shear Stressmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Branching Morphology of the Rat Hepatic Portal Vein Tree: A Micro-CT Study

2006

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“…Shoshenko et al (1982) and Mayrovitz and Roy (1983) have derived z from direct measurements of blood flow and radius. Some predict z from optimality equations (Murray, 1926a;Zamir, 1976;Uylings, 1977;Sherman, 1981;Roy and Woldenberg, 1982;Woldenberg and Horsfield, 1983). Others have derived z from overall system morphometry established by ordering (Leopold and Miller, 1956;Woldenberg et al, 1970;Woldenberg, 1972;Horsfield and Thurlbeck, 1981).…”

Section: Mathematical Conceptsmentioning

confidence: 99%

Diameters and cross‐sectional areas of branches in the human pulmonary arterial tree

Horsfield¹,

Woldenberg²

1989

Anat. Rec.

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Measurements were made of the diameters of the three branches meeting at each of 1,937 bifurcations in the pulmonary arterial tree, using resin casts from two fully inflated human lungs. Cross-sectional areas of the parent branch and of the daughter branches were calculated and plotted on a log-log plot, which showed that mean cross-sectional area increases in a constant proportion of 1.0879 at bifurcations. The mean value of the ratio of daughter branch diameters at bifurcations was 0.7849. The mean value of the exponent z in the equation flow = k (diameter(z)) was found to be 2.3 +/- 0.1, which is equal to the optimal value for minimizing power and metabolic costs for fully developed turbulent flow. Although Reynolds number may exceed 2,000 in the larger branches of the pulmonary artery, turbulent flow probably does not occur, and in the peripheral branches Reynolds number is always low, excluding turbulent flow in these branches. This finding seems to be incompatible with the observed value of z. A possible explanation may be that other factors may need to be taken into account when calculating the theoretical optimum value of z for minimum power dissipation, such as the relatively short branches and the disturbances of flow occurring at bifurcations. Alternatively, higher arterial diameters reduce acceleration of the blood during systole, reduce turbulent flow, and increase the reservoir function of the larger arteries. These higher diameters result in a lower value of z.

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2005

Complexity

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A generalization of the optimal models of arterial branching

Cited by 31 publications

References 14 publications

Branching Morphology of the Rat Hepatic Portal Vein Tree: A Micro-CT Study

Branching Morphology of the Rat Hepatic Portal Vein Tree: A Micro-CT Study

Diameters and cross‐sectional areas of branches in the human pulmonary arterial tree

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