MATHEMATICAL MODELS OF THE VASCULAR BED STRUCTURE OF THE HUMAN BODY
(LITERATURE REVIEW) 

Dmitriev Andrey Viktorovich, Candidate of medical sciences, head of x-ray endovascular surgery unit, Institute of Emergency Surgery (47 Leninsky avenue, Donetsk, DPR), E-mail: dmitriev72@list.ru
Lysykh Yaroslav Arturovich, Applicant, head of laboratory, Faculty of Physics and Engineering, Donetsk National University (24 Universitetskaya street, Donetsk, DPR), E-mail: shadowintheflames@gmail.com
Zenin Oleg Konstantinovich, Doctor of medical sciences, professor, sub-department of human anatomy, Penza State University (40 Krasnaya street, Penza, Russia), E-mail: zen.olegz@gmail.com 

611.11 

10.21685/2072-3032-2019-2-8 

Background. The aim of the work is to compare and systematize information available in the literature on mathematical modeling of the structure of the human vascular bed. 51 literary sources of the last decade were analyzed.
Along with the hydrodynamic modeling of the cardiovascular system, there is also structural modeling, which explores both the structural structure of various t№ and organs and the principle of the construction of the vascular bed. Structure models differ primarily in the accuracy and performance associated with the presentation of materials, while approaches to constructing the hierarchical structure of the vascular bed may differ at a fundamental level. The described approaches of structural modeling are constantly evolving both by improving the old models and developing new ones, and by increasing the available computing power. The results of the analysis indicate the need for further development in this area and the creation of a unified universal theory. 

structural modeling, cardiovascular system, hemodynamics 

1. Owen B. et al. Biomechanics and Modeling in Mechanobiology. 2018, vol. 17, no. 5, pp. 1217–1242.
2. Fedosov D. A., Caswell B., Karniadakis G. E. Biophysical Journal. 2010, vol. 98, no. 10, pp. 2215–2225.
3. Li J. et al. Biophysical Journal. 2005, vol. 88, pp. 3707–3719.
4. Nakamura M., Bessho S., Wada S. International Journal for Numerical Methods in Biomedical Engineering. 2013, vol. 29, no. 1, pp. 114–128.
5. Zhao A. R. et al. European Journal of Cardio-Thoracic Surgery. 2008, vol. 34, no. 3, pp. 623–629.
6. Figueroa C. A. et al. Computer Methods in Applied Mechanics and Engineering. 2006, vol. 195, no. 41–43, pp. 5685–5706.
7. Crosetto P. et al. Computers & Fluids. 2011, vol. 43, no. 1, pp. 46–57.
8. Gent A. N. Engineering with rubber: how to design rubber components. 3rd edition. Munich: Hanser Publishers, 2012, 451 p.
9. Gerhard A. et al. American Journal of Physiology: Hearth and Circulatory Physiology. 2005, vol. 289, no. 5, pp. 2048–2058.
10. Gerhard A. H., Thomas C. G., Ray W. O. Journal of elasticity and the physical science of solids. 2000, vol. 61, no. 1–3, pp. 1–48.
11. Fung Y. C. American Journal of Physiology. 1967, vol. 213, no. 6, pp. 1532–1544.
12. Skalak R. et al. Biophysical Journal. 1973, vol. 13, no. 3, pp. 245–264.
13. Taylor C. A., Hughes T. J., Zarins C. K. Annals of Biomedical Engineering. 1998, vol. 26, no. 6, pp. 975–987.
14. Oshima M., Torii R., Kobayashi T. Computer Methods in Applied Mechanics and Engineering. 2001, vol. 191, no. 6-7, pp. 661–671.
15. Kim H. J., Jansen K. E., Taylor C. A. Annals of Biomedical Engineering. 2010, vol. 38, no. 7, pp. 2314–2330.
16. Abdelaziz Y., Hamouine A. Computers & Structures. 2008, vol 86, no. 11–12, pp. 1141–1151.
17. Liu W. K. et al. Computer Methods in Applied Mechanics and Engineering. 2006, vol. 195, no. 13–16, pp. 1722–1749.
18. Abraham F. F. Goulian M. Europhysics Letters. 1992, vol. 19, no. 4, pp. 293–296.
19. Bejan A. Shape and Structure: from Engineering to Nature. Cambridge: Univ. Press, 2000, 210 p.
20. Chernous'ko F. L. Prikladnaya matematika i mekhanika [Applied mathematics and mechanics]. 1977, vol. 41, no. 2, pp. 376–383. [In Russian]
21. La Barbera M. Science. 1990, vol. 249, pp. 992–1000.
22. Balabanov V. A., Kizilova N. N. Vіsnik Kharkіvs'kogo natsіonal'nogo unіversitetu іmenі V. N. Karazіna [Bulletin of Kharkov National University named after V.N. Karazin]. 2017, pp. 5–17.
23. Balabanov V. O., Kіzіlova N. M. Vіsnik Kiїvs'kogo natsіonal'nogo unіversitetu іmenі Tarasa Shevchenka. Ser.: Fіziko-matematichnі nauki. Spetsvipusk [Bulletin of Kyev National University named after Taras Shevchenko. Series: Physical and mathematical sciences. Special issue]. 2015, pp. 27–32.
24. Balabanov V. A., Kizilova N. N. Mekhanika. Issledovaniya i innovatsii [Mechanics. Research and innovations]. 2016, iss. 9, pp. 18–26. [In Russian]
25. Jeroen P. H. M. et al. Journal of Biomechanics. 2013, vol. 46, no. 2, pp. 229–239.
26. Olufsen M. et al. Annals of Biomedical Engineering. 2000, vol. 28, no. 11, pp. 1281–1299
27. Schreiner W., Neumann F., Neumann M., Karch R., End A., Roedler S. M. Journal of general physiology. 1997, vol. 109, no. 2, pp. 129–140.
28. Karch R. et al. Computers in Biology and Medicine. 1999, vol. 29, pp. 19–38.
29. Dovgyallo Yu. V. et al. Pitannya eksperimental'noї ta klіnіchnoї meditsini, Zbіrnik statey [Issues of Experimental and Clinical Medicine, Collected Articles]. 2013, vol. 2, iss. 17, pp. 164–169.
30. Perdikaris P., Grinberg L., Karniadakis G. E. Annals of Biomedical Engineering. 2015, vol. 43, no. 6, pp. 1432–1442.
31. Cassot F. et al. Brain Research. 2010, vol. 1313, pp. 62–78.
32. Comerford A., Förster C., Wall W. A. ASME. Journal of Biomechanical Engineering. 2010, vol. 132, no. 8. Available at: https://biomechanical.asmedigitalcollection.asme.org/article.aspx?articleid=1426383 (accessed Dec. 20, 2018).
33. Du T., Hu D., Cai D. PLoS One. 2015, vol. 10, no. 5. Available at: https://doi.org/10.1371/journal.pone.0128597 (accessed Dec. 20, 2018).
34. Rojas A. M. Torres et al. PLoS One. 2015, vol. 10, no. 6. Available at: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0128111 (accessed Dec. 20, 2018)
35. Aydin B. et al. eprint arXiv. 2008. Available at: https://arxiv.org/abs/0810.0944 (accessed Dec. 21, 2018).
36. Reymond P. et al. American Journal of Physiology| Hearth and Circulatory Physiology. 2009, vol. 297, no. 1, pp. 208–222.
37. Grasman J. et al. J Theor Biol. 2003, vol. 220, pp. 75–82.
38. Cassot F. et al. Microcirculation. 2009, vol. 16, pp. 331–344.
39. Kizilova N. N. Lecture Notes in Computer Sci. 2004, vol. 3044, pp. 476–485.
40. Rozen R. Printsip optimal'nosti v biologii [The principle of optimality in biology]. Moscow: Mir, 1968, 212 p. [In Russian]
41. Dawson C. A., Krenz G. S., Karau K. L. et al. J. Appl. Physiol. 1999, vol. 86, pp. 569–583.
42. Zenin O. K. et al. Arterial'naya sistema cheloveka v tsifrakh i formulakh [Human arterial system in figures and formulas]. Donetsk: Donbass, 2002, 196 p. [In Russian]
43. Hassani K., Navidbakhsh M., Rostami M. Biomedical papers. 2001, vol. 150, no. 1, pp. 105–112.
44. Mossa H. A. L. The 1st Regional Conference of Eng Sci NUCEJ Spatial Issue. 2008, vol. 11, no. 2, pp. 307–314.
45. Mirzaee M., Ghasemalizadeh O., Firoozabadi B. The Internet Journal of Bioengineering. 2008, vol. 3, no. 2. Available at: http://ispub.com/IJBE/3/2/6880 (accessed Dec. 24, 2018).
46. Naik K., Bhathawala P. H. International Journal of Mathematical and Computational Sciences. 2017, vol. 11, no. 1, pp. 73–84.
47. Tsalikakis D. G., Fotiadis D. I., Sideris D. Computers in Cardiology. 2003, vol. 30, pp. 445−448.
48. Arathi S., Ananda B. International journal of advances in scientific research and engineering. 2016, vol. 2, no. 11, pp. 8–15.
49. Frolov S. V. et al. Journal of Mechanics in Medicine and Biology. 2017, vol. 17, no. 03. Available at: https://www.worldscientific.com/doi/abs/10.1142/S0219519417500567 (accessed Dec. 24, 2018).
50. Lim E. et al. PLoS ONE. 2013, vol. 8, no. 10. Available at: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0077357 (accessed Dec. 24, 2018).
51. Duanmu Z. et al. Scientific Reports. 2018, vol. 8. Available at: https://www.nature.com/articles/s41598-018-19164-w (accessed Dec. 25, 2018).