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Voronoi Polyhedra Analysis of Optimized Arterial Tree Models

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Abstract

Topological and metric properties of Voronoi polyhedra (VP) generated by the distal end points of terminal segments in arterial tree models grown by the method of constrained constructive optimization (CCO) are analyzed with the aim to characterize the spatial distribution of their supply sites relative to randomly distributed points as a reference model. The distributions of the number N f of Voronoi cell faces, cell volume V, surface area S, area A of individual cell faces, and asphericity parameter α of the CCO models are all significantly different from the ones of random points, whereas the distributions of V, S, and α are also significantly different among CCO models optimized for minimum intravascular volume and minimum segment length (p < 0.0001). The distributions of N f , V, and S of the CCO models are reasonably well approximated by two-parameter gamma distributions. We study scaling of intravascular blood volume and arterial cross-sectional area with the volume of supplied tissue, the latter being represented by the VP of the respective terminal segments. We observe scaling exponents from 1.20 ± 0.007 to 1.08 ± 0.005 for intravascular blood volume and 0.77 ± 0.01 for arterial cross-sectional area. Setting terminal flows proportional to the associated VP volumes during tree construction yields a relative dispersion of terminal flows of 37% and a coefficient of skewness of 1.12. © 2003 Biomedical Engineering Society.

PAC2003: 8719Uv, 8710+e, 4720Ky, 0260Pn, 0230Oz

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References

  1. Allen, M. P., and D. J. Tildesley. Computer Simulation of Liquids. Oxford: Oxford University Press, 1987, pp. 54–55.

    Google Scholar 

  2. Arts, T., R. T. I. Kruger, W. VanGerven, J. A. C. Lambregts, and R. S. Reneman. Propagation velocity and reflection of pressure waves in the canine coronary artery. Am. J. Physiol.237:H469–H474, 1979.

    Google Scholar 

  3. Aurenhammer, F., and R. Klein. Voronoi diagrams. In: Handbook of Computational Geometry, edited by J. R. Sack and J. Urrutia. Amsterdam: Elsevier Science B. V., 2000, pp. 201–290.

    Google Scholar 

  4. Barber, C. B., and H. Huhdanpaa. Qhull version 2.6. The Geometry Center, University of Minnesota, Minneapolis, 1998. http://www.geom.umn.edu/locate/qhull.

    Google Scholar 

  5. Barber, C. B., D. P. Dobkin, and H. Huhdanpaa. The quickhull algorithm for convex hulls. ACM Trans. Math. Softw.22:469–483, 1996.

    Google Scholar 

  6. Bassingthwaighte, J. B., R. B. King, and S. A. Roger. Fractal nature of regional myocardial blood flow heterogeneity. Circ. Res.65:578–590, 1989.

    Google Scholar 

  7. Beard, D. A., and J. B. Bassingthwaighte. The fractal nature of myocardial blood flow emerges from a whole-organ model of arterial network. J. Vasc. Res.37:282–296, 2000.

    Google Scholar 

  8. Beard, D. A., and J. B. Bassingthwaighte. Modeling advection and diffusion of oxygen in complex vascular networks. Ann. Biomed. Eng.29:298–310, 2001.

    Google Scholar 

  9. Brown, D. Voronoi diagrams from convex hulls. Inf. Process. Lett.9:223–228, 1979.

    Google Scholar 

  10. Chilian, W. M., S. M. Layne, E. C. Klausner, C. L. Eastham, and M. L. Marcus. Redistribution of coronary microvascular resistance produced by dipyridamole. Am. J. Physiol.256:H383–H390, 1989.

    Google Scholar 

  11. Chilian, W. M.Microvascular pressures and resistances in the left ventricular subepicardium and subendocardium. Circ. Res.69:561–570, 1991.

    Google Scholar 

  12. Dirichlet, G. L.über die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. J. Reine Angew. Math.40:209–227, 1850.

    Google Scholar 

  13. Fung, Y. C. Biomechanics: Motion, Flow, Stress, and Growth. New York: Springer, 1990, pp. 155–195.

    Google Scholar 

  14. Glenny, R. W., and H. T. Robertson. Fractal properties of pulmonary blood flow: Characterization of spatial heterogeneity. J. Appl. Physiol.69:532–545, 1990.

    Google Scholar 

  15. Glenny, R. W., and H. T. Robertson. A computer simulation of pulmonary perfusion in three dimensions. J. Appl. Physiol.79:357–369, 1995.

    Google Scholar 

  16. Gil Montoro, J. C., and J. L. F. Abascal. The Voronoi polyhedra as tools for structure determination in simple disordered systems. J. Phys. Chem.97:4211–4215, 1993.

    Google Scholar 

  17. Gil Montoro, J. C., F. Bresme, and J. L. F. Abascal. Ionic association in electrolyte solutions: A Voronoi polyhedra analysis. J. Chem. Phys.101:10892–10898, 1994.

    Google Scholar 

  18. Gödde, R., and H. Kurz. Structural and biophysical simulation of angiogenesis and vascular remodeling. Dev. Dyn.220:387–401, 2001.

    Google Scholar 

  19. Hinde, A. L., and R. E. Miles. Monte Carlo estimates of the distributions of the random polygons of the Voronoi tessellation with respect to a Poisson process. J. Stat. Comput. Simul.10:205–223, 1980.

    Google Scholar 

  20. Honda, H.Description of cellular patterns by Dirichlet domains: The two-dimensional case. J. Theor. Biol.72:523–543, 1978.

    Google Scholar 

  21. Hoofd, L., Z. Turek, K. Kubat, B. E. M. Ringnalda, and S. Kazda. Variability of intercapillary distance estimated on histological sections of rat heart. Adv. Exp. Med. Biol.191:239–247, 1985.

    Google Scholar 

  22. Hudlicka, O., A. J. Wright, and A. M. Ziada. Angiogenesis in the heart and skeletal muscle. Can. J. Cardiol.2:120–123, 1986.

    Google Scholar 

  23. Jedlovszky, P.Voronoi polyhedra analysis of the local structure of water from ambient to supercritical conditions. J. Chem. Phys.111:5975–5985, 1999.

    Google Scholar 

  24. Kamiya, A., and T. Togawa. Optimal branching structure of the vascular tree. Bull. Math. Biophys.34:431–438, 1972.

    Google Scholar 

  25. Karch, R., F. Neumann, M. Neumann, and W. Schreiner. A three-dimensional model for arterial tree representation, generated by constrained constructive optimization. Comput. Biol. Med.29:19–38, 1999.

    Google Scholar 

  26. Karch, R., F. Neumann, M. Neumann, and W. Schreiner. Staged growth of optimized arterial model trees. Ann. Biomed. Eng.28:495–511, 2000.

    Google Scholar 

  27. Kassab, G. S., C. A. Rider, N. J. Tang, and Y.-C. B. Fung. Morphometry of pig coronary arterial trees. Am. J. Physiol.265:H350–H365, 1993.

    Google Scholar 

  28. Kassab, G. S., J. Berkley, and Y. C. B. Fung. Analysis of pig's coronary arterial blood flow with detailed anatomical data. Ann. Biomed. Eng.25:204–217, 1997.

    Google Scholar 

  29. King, R. B., L. J. Weissman, and J. B. Bassingthwaighte. Fractal descriptions for spatial statistics. Ann. Biomed. Eng.18:111–121, 1980.

    Google Scholar 

  30. Kumar, S., S. K. Kurtz, J. R. Banavar, and M. G. Sharma. Properties of a three-dimensional Poisson–Voronoi tesselation: A Monte Carlo study. J. Stat. Phys.67:523–551, 1992.

    Google Scholar 

  31. Kumar, S., and S. K. Kurtz. Properties of a two-dimensional Poisson–Voronoi tesselation: A Monte-Carlo study. Mater. Charact.31:55–68, 1993.

    Google Scholar 

  32. Kurz, H., and K. Sandau. Modelling of blood vessel development. Bifurcation pattern and hemodynamics, optimality and allometry. Comments Theor. Biol.4:261–291, 1997.

    Google Scholar 

  33. Levy, S., T. Munzer, M. Phillips, C. Fowler, N. Thurston, D. Krech, S. Wisdom, D. Meyer, and T. Rowley. Geomview version 1.6.1. The Geometry Center, University of Minnesota, Minneapolis, 1998. http://www.geomview.org.

    Google Scholar 

  34. Lipowsky, H. H., and B. W. Zweifach. Methods for the simultaneous measurement of pressure differentials and flows in single unbranched vessels of the microcirculation for rheological studies. Microvasc. Res.14:345–361, 1977.

    Google Scholar 

  35. Meijering, J. L.Interface area, edge length, and number of vertices in crystal aggregates with random nucleation. Philips Res. Rep.8:270–290, 1953.

    Google Scholar 

  36. Murray, C. D.The physiological principle of minimum work. I. The vascular system and the cost of blood volume. Proc. Natl. Acad. Sci. U.S.A.12:207–214, 1926.

    Google Scholar 

  37. Neumann, M., F. J. Vesely, O. Steinhauser, and P. Schuster. Solvation of large dipoles. I. A molecular dynamics study. Mol. Phys.35:841–855, 1978.

    Google Scholar 

  38. Neumann, F., M. Neumann, R. Karch, and W. Schreiner. Visualization of computer-generated arterial model trees. In: Simulation Modelling in Bioengineering, edited by M. Cerrolaza, D. Jugo, and C. A. Brebbia. Southampton: Computational Mechanics, 1996, pp. 259–268.

    Google Scholar 

  39. Okabe, A., B. Boots, K. Sugihara, and S. N. Chiu. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, Second Edition. Chichester: Wiley, 1999.

    Google Scholar 

  40. Preparata, F. P., and M. I. Shamos. Computational Geometry. An Introduction. New York: Springer, 1985.

    Google Scholar 

  41. Prothero, J. W.Scaling of blood parameters in mammals. Comparative Biochem. Physiol.67A:649–657, 1980.

    Google Scholar 

  42. Qian, H., and J. B. Bassingthwaighte. A class of flow bifurcation models with lognormal distribution and fractal dispersion. J. Theor. Biol.205:261–268, 2000.

    Google Scholar 

  43. Rahman, A. Liquid structure and self-diffusion. J. Chem. Phys.45:2584–2592, 1966.

    Google Scholar 

  44. Ripley, B. D.Test of ‘randomness’ for spatial point patterns. J. R. Statist. Soc. B41:368–374, 1979.

    Google Scholar 

  45. Rosen, R. Optimality Principles in Biology. London: Butterworth, 1967, pp. 40–60.

    Google Scholar 

  46. Ruocco, G., M. Sampoli, and R. Vallauri. Analysis of the network topology in liquid water and hydrogen sulphide by computer simulation. J. Chem. Phys.96:6167–6176, 1992.

    Google Scholar 

  47. Sandau, K., and H. Kurz. Modelling of vascular growth processes: A stochastic biophysical approach to embryonic angiogenesis. J. Microsc.175:205–213, 1994.

    Google Scholar 

  48. Schreiner, W.Computer generation of complex arterial tree models. J. Biomed. Eng.15:148–149, 1993.

    Google Scholar 

  49. Schreiner, W., and P. Buxbaum. Computer-optimization of vascular trees. IEEE Trans. Biomed. Eng.40:482–491, 1993.

    Google Scholar 

  50. Schreiner, W., M. Neumann, F. Neumann, S. M. Roedler, A. End, P. Buxbaum, M. R. Müller, and P. Spieckermann. The branching angles in computer-generated optimized models of arterial trees. J. Gen. Physiol.103:975–989, 1994.

    Google Scholar 

  51. Seiler, C., R. L. Kirkeeide, and K. L. Gould. Basic structure-function relations of the epicardial coronary vascular tree. Circulation85:1987–2003, 1992.

    Google Scholar 

  52. Stoyan, D., and H. Stoyan. Fractals, Random Shapes and Point Fields. Chichester: Wiley, 1994.

    Google Scholar 

  53. Strahler, A. N.Quantitative analysis of watershed geomorphology. Trans. Am. Geophys. Union38:913–920, 1957.

    Google Scholar 

  54. Thompson, D. W. On Growth and Form. Cambridge, MA: Cambridge University Press, 1917.

    Google Scholar 

  55. Van Bavel, E., and J. A. E. Spaan. Branching patterns in the porcine coronary arterial tree. Estimation of flow heterogeneity. Circ. Res.71:1200–1212, 1992.

    Google Scholar 

  56. Van Beek, J. H. G. M., S. A. Roger, and J. B. Bassingthwaighte. Regional myocardial flow heterogeneity explained with fractal networks. Am. J. Physiol.257:H1670–H1680, 1989.

    Google Scholar 

  57. Vaz, M. F., and M. A. Fortes. Grain size distribution: The lognormal and the gamma distribution functions. Scr. Metall.22:35–40, 1988.

    Google Scholar 

  58. Voronoi, G.Nouvelles applications des parametres continus a la theorie des formes quadratiques. J. Reine Angew. Math.134:198–287, 1908.

    Google Scholar 

  59. Wang, C. Y., and J. B. Bassingthwaighte. Capillary supply regions. Math. Biosci.173:103–114, 2001.

    Google Scholar 

  60. West, G. B., J. H. Brown, and B. J. Enquist. A general model for the origin of allometric scaling laws in biology. Science276:122–126, 1997.

    Google Scholar 

  61. Woldenberg, M. J., and K. Horsfield. Relation of branching angles to optimality for four cost principles. J. Theor. Biol.122:187–204, 1986.

    Google Scholar 

  62. Zamir, M.Optimality principles in arterial branching. J. Theor. Biol.62:227–251, 1976.

    Google Scholar 

  63. Zamir, M., J. A. Medeiros, and T. K. Cunningham. Arterial bifurcations in the human retina. J. Gen. Physiol.74:537–548, 1979.

    Google Scholar 

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Authors and Affiliations

  1. Department of Medical Computer Sciences, University of Vienna, Spitalgasse 23, A-1090, Wien, Austria

    Rudolf Karch, Friederike Neumann, Paul Szawlowski & Wolfgang Schreiner

  2. Institute of Experimental Physics, Section for Computational Physics, University of Vienna, Strudlhofgasse 4, A-1090, Wien, Austria

    Martin Neumann

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Karch, R., Neumann, F., Neumann, M. et al. Voronoi Polyhedra Analysis of Optimized Arterial Tree Models. Annals of Biomedical Engineering 31, 548–563 (2003). https://doi.org/10.1114/1.1566444

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