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      Functional Characteristics of Optimized Arterial Tree Models Perfusing Volumes of Different Thickness and Shape

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          The relationship between the ‘shape of an organ’ and the ‘cost of blood transport’ to perfuse its tissue was evaluated on the basis of optimized arterial model trees simulated to perfuse square-based 100-cm<sup>3</sup> volumes of different shape (‘flat’ versus ‘thick’ as defined by the ratio of thickness to side-length h/s ≤1). Specifically, the effects of ‘shape’ on tree structure, blood transport, and on hemodynamic characteristics were investigated. Branching models of arterial trees were generated by constrained constructive optimization (CCO), based on an identical set of model parameters. All model trees were geometrically and topologically optimized for intravascular volume. Tree structures achieved tremendous savings of blood (transport medium) in comparison to a system of separate tubes. Thickening the perfusion volume (increasing h/s) resulted in a significant decrease of mean transport length, deposition time, and intravascular total volume in the tree. ‘Thick’ perfusion volumes induced CCO trees to branch more symmetrically into a number of equivalent subtrees repetitiously splitting into smaller ones; ‘flat’ structures were dominated throughout by a few asymmetrically branching major vessels. In summary, we conclude from systematic variation of shape that thicker perfusion volumes ( h/s >0.1) facilitate efficient delivery of blood in comparison to large amounts of ‘dead volume’ to be carried over long distances in very thin pieces of tissue.

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          Computer-optimization of vascular trees.

          Arterial branchings closely fulfill several "bifurcation rules" which are deemed to optimize blood flow. The question is whether these local criteria in conjunction with a general optimization principle can explain the overall structure of an arterial tree. We present a model of an arterial vascular tree which is grown on the computer by successively adding terminal vessel segments. Each new terminal segment is connected to the optimum site within the preexisting tree, and the new bifurcation is optimized geometrically. After each step of adding and optimizing, the whole tree is rescaled to meet invariant boundary conditions of pressure and flow at each terminal site. Thus, local geometric optimization is used to induce simultaneously an optimized global structure. The comparison between the model and real coronary arterial trees shows good agreement regarding structural appearance, morphometric parameters, and pressure profiles.
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            Limited Bifurcation Asymmetry in Coronary Arterial Tree Models Generated by Constrained Constructive Optimization

            Models of coronary arterial trees are generated by the algorithm of constrained constructive optimization (CCO). In a given perfusion area a binary branching network of straight cylindrical tubes is generated by successively adding terminal segments to the growing structure. In each step the site of connection is chosen according to an optimization target function (total intravascular volume), and in any stage of development the tree fulfills physiologic boundary conditions (constraints involving pressures, flows and bifurcation rules). CCO generates structures which in many aspects resemble real coronary arterial trees, except for very asymmetric bifurcations, occurring when a large branch gives off a tiny terminal segment. In the present work we evaluate an additional constraint within CCO, namely imposing a limit on the asymmetry of bifurcations during the construction process. Model trees are grown with different limits imposed, and the effects on structure are studied both phenomenologically and via statistical descriptors. As the limit to asymmetry is tightened, blood is conveyed to the perfusion sites via detours rather than directly and the comparison with measured data shows the structure to change from a conveying to a delivering type of function. Simultaneously total intravascular volume, surface and sum of segments' lengths increase. It is shown why and how local bifurcation asymmetry is able to determine the global structure of the optimized arterial tree model. Surprisingly, the pressure profile from inlet to terminals, being a functional characteristic, remains unaffected.

              Author and article information

              J Vasc Res
              Journal of Vascular Research
              S. Karger AG
              August 2000
              14 August 2000
              : 37
              : 4
              : 250-264
              aDepartment of Medical Computer Sciences, and bInstitute of Experimental Physics, University of Vienna, Vienna, Austria
              25739 J Vasc Res 2000;37:250–264
              © 2000 S. Karger AG, Basel

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              Page count
              Figures: 12, Tables: 2, References: 31, Pages: 15
              Research Paper


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