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 that 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.

The influence of optimization target selection on the structure of arterial tree models generated by constrained constructive optimization.

The computational method of constrained constructive optimization was used to generate complex arterial model trees by optimization with respect to a target function. Changing the target function also changes the tree structure obtained. For a parameterized family of target functions a series of trees was created, showing visually striking differences in structure that can also be quantified by appropriately chosen numerical indexes. Blood transport path length, pressure profile, and an index for relative segment orientation show clear dependencies on the optimization target, and the nature of changes can be explained on theoretical grounds. The main goal was to display, quantify, and explain the structural changes induced by different optimization target functions.

The branching angles in computer-generated optimized models of arterial trees.

The structure of a complex arterial tree model is generated on the computer using the newly developed method of "constrained constructive optimization." The model tree is grown step by step, at each stage of development fulfilling invariant boundary conditions for pressures and flows. The development of structure is governed by adopting minimum volume inside the vessels as target function. The resulting model tree is analyzed regarding the relations between branching angles and segment radii. Results show good agreement with morphometric measurements on corrosion casts of human coronary arteries reported in the literature.

Anatomical variability and functional ability of vascular trees modeled by constrained constructive optimization.

The aim of this study was to investigate the extent that functional capability of vascular trees is related to anatomical variability. To these ends we used the method of Constrained Constructive Optimization (CCO) to generate optimized computer models of coronary arterial trees. All these model trees were optimized according to the same principle under equal boundary conditions of pressures and flows. However, by stochastically casting the locations of the terminal segments, different tree structures were generated, each of which conformed to the same boundary conditions. The structural variability of these models was interpreted as the correlate of the anatomical variability found in real arterial trees. The advantage that computer model trees are known in numerical detail was exploited to perform comprehensive and exact classifications of all segments into bifurcation levels, STRAHLER orders and composite vessels, and to compute the area expansion ratio. The unexpected result was that, despite striking visual differences in anatomical structure, the model trees were almost identical with regard to functional performance. We conclude that models optimized on the computer for a given perfusion task show little differences in their morphometric parameters even if they differ considerably regarding the course of the large vessels.

Shear stress distribution in arterial tree models, generated by constrained constructive optimization.

Models of arterial trees are generated by the algorithm of Constrained Constructive Optimization (CCO). Straight cylindrical, binary branching tubes are arranged in an optimized fashion so as to convey blood to the terminal sites of the tree, which are distributed over a predefined area, representing the tissue to be perfused. All terminal segments supply equal flows at a unique terminal pressure, and the radii of parent and daughter segments are related via a bifurcation law. The connective structure and geometry of the model are optimized according to a target function such as total intravascular volume. The shear rate between blood and the vessel walls is computed in each segment and a new method is presented for rescaling a given CCO tree to a desired value of shear rate in the root segment. The effect of viscosity varying with shear rate is evaluated and a new method is presented for rescaling a CCO-tree segment by segment to consistent values of radii and variable viscosity. Shear stress is evaluated for its deviation from being proportional to shear rate and then subjected to various types of analyses. Usually both, shear stress and its variability, are found to be larger in the smaller than in the larger segments of the CCO-model trees. However, it is shown how the shear-stress distribution can be reshuffled between small and large segments when rescaling a CCO tree to obey a different bifurcation law, while its whole geometry remains unchanged and all boundary conditions remain fulfilled. The selection of optimization target is found to drastically affect shear-stress variability within bifurcations, which reaches a distinct minimum if the model is optimized according to intravascular volume. Finally, a rank-analysis of shear stress within each bifurcation shows that only two out of six possible rank patterns actually occur: the parent segment always experiences medium shear stress while minimum shear stress resides mostly in the larger, less frequently in the smaller daughter.