Mathematical Modelling and Simulation of Blood Flow in the Human Ascending Aorta: An Analytical Approach
Keywords:
Aorta blood flow, Newtonian fluid, Reynolds Transport Theorem, Wall pressure.
Abstract
The pulsatile behaviour of blood flow through a healthy aorta was modelled using Navier-Stokes and continuity
equations, while the nature of the aorta wall was accounted for by Hooke’s law. The resulting balance equations
were transformed by using Reynolds transport theorem, and the variables expressed in Fourier modes. Substitution resulted in just one equation expressed in terms of Bessel functions, and which required one characteristic independent variable, k, to be determined. This was obtained by using Wolfram Mathematica to solve the equation along with aorta wall and blood properties (obtained from literature). The characteristic k value used for the prediction of the pulsatile nature of the flow was obtained as 0.139983 + 0.0590188i. Simulated results with
this k value showed variation in blood pressure, aorta expansion and elongation, for a healthy heart to be within typical ranges of 80 – 120 mmHg, and 0 – 4mm respectively. The wavelength and wave speed generated by the blood flow was determined as 14.0848m and 53.8626m/s respectively.
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human aorta with emphasis on outlet boundary conditions. Applied Mathematical Modelling, 35 (7): 3175-
3188.
Bessonov, N., Sequeira, A., Simakov, S., Vassilevskii, Yu, and Volpert, V. (2016). Methods of Blood Flow Modelling.
Math. Model. Nat. Phenom. 11(1): 1–25
Bird, B. R., Stewart, W. E., and Lightfoot, E. N. (2007). Transport Phenomena ( 2nd ed.). New York: John Wiley and
Sons Inc., 731-732.
Caballero, A.D. and Lain,S.(2015). Numerical simulation of non-Newtonian blood flow dynamics in human thoracic
aorta, Computer Methods in Biomechanics and Biomedical Engineering, 18 (11):200-1216.
Catanho M., Sinha M., and Vijayan V. (2012). Model of Aortic Blood Flow using the Windkessel Effect. BENG 221 -
Mathematical Methods in Bioengineering Report,
http://isn.ucsd.edu/classes/beng221/problems/2012/BENG221_Project-Catanho Sinha Vijayan.pdf,
accessed 13/10/17
Chandran, K.B. (1993). Flow dynamics in the human aorta. J. Biomech. Engng. 115: 611–616.
Educational Designers (2017) The human Heart, http://www.myschoolhouse.com/courses/O/1/94.asp, Accessed
1/11/17
Epps, R.F. (2012). The Human Aorta: Your Super Highway of Life, Gazelle Press, United States, 3-6
Fairchild, B. T., Krovetz, L. J., and Huckaba, C. E. (1966). Digital Computer Simulation of Arterial Blood Flow. In:
Hershey, D. (Ed) Chemical Engineering in Medicine and Biology. Proceedings of the 33rd Annual Chemical
Engineering Symposium of the Division of Industrial and Engineering Chemistry of the American Chemical
Society, Held at the University of Cincinnati, on October 20-21, 1966. Plenum Press, New York. 3 – 43.
Ferziger, J. H., and M. Peric (2002). Computational Methods for Fluid Dynamics (3rd ed.), New York; Springer, 1 - 20
Grinberg, L., Anor, T., Madsen, J.R. Yakhot, A. and G.E. Karniadakis (2009). Large-scale simulation of the human
arterial tree. Clinical and Experimental Pharmacology and Physiology, 36 (2):194 -205.
Hall, J.E. (2011). Guyton and Hall Textbook of Medical Physiology (12th Edition), Elsevier Inc., 177- 178
Kundu, P. K., Cohen, I. M., and Dowling, D. R. (2016). Fluid Mechanics (6th ed.). Oxford: Elsevier Inc., 779 -820, e11
- e20.
Leaning, M. S., Pullen, H. E., Carson, E. R., and Finkelstein, L. (1983). Modelling a Complex Biological System: The
Human Cardiovascular System - 1. Methodology and Model Description. Transactions on Institute of
Measurement and Control, 5 (2): 71–86.
MedFriendly. (2012). Aorta. http://medfriendly.com/aorta.html, Accessed 9/12/17
Menut M, Boussel L, Escriva X, Bou-Saïd B, Walter-Le Berre H, Marchesse Y, Millon A, Della Schiava N, Lermusiaux
P, Tichy J. (2018). Comparison between a generalized Newtonian model and a network-type multi-scale
model for hemodynamic behavior in the aortic arch: Validation with 4D MRI data for a case study. Journal
of Biomechanics 73, 119–126
Morris, L., Delassus, P., Callanan, A., Walsh, M., Wallis, F., Grace, P. and T. McGloughlin (2005). 3-D numerical
simulation of blood flow through models of the human. J Biomech Eng. Oct., 127 (5): 767-775.
Nico, W., Jan-Willem, K., and Berend, E. W. (2009). The Arterial Windkessel. Med. Biol. Eng. Comput. 47: 131-141
Olufsen, M. S., and Nadim, A. (2004). On Deriving Lumped Models for Blood Flow and Pressure in The Systemic
Arteries. Mathematical Biosciences and Engineering, 1(1), 61-80.
Quarteroni A. and Formaggia L., (2004). Computational Models for the Human Body. In Ayache N and P.G Ciarlet.
(Eds) Handbook of Numerical Analysis, Volume 12- Mathematical Modelling and Numerical Simulation of
the Cardiovascular System, 7 – 15
Pedley. T. J. (2008). The Fluid Mechanics of Large Blood Vessels Cambridge Monographs on Mechanics, Cambridge
University Press.
Rahman, M. S. and Haque, M. A. (2012). Mathematical Modelling of Blood Flow. In IEEE, International Conference
on Informatics, Electronics and Vision, ICIEV, Bangladesh, 672 - 676
San, O. and Staples A. E. (2012). An Improved Model for Reduced-Order Physiological Fluid Flows J. Mech. Med.
Biol. 12, (3): 1-28
Seta B., Torlak M. and Vila A. (2017) Numerical Simulation of Blood Flow through the Aortic Arch. In: Badnjevic A.
(Ed) IFMBE Proceedings of the International Conference on Medical and Biological Engineering. Springer,
Singapore, 62: 259-268
Silbernagl S. and Despopoulos A. (2009). Color Atlas of Physiology, 6th edition, Thieme, Stuggart. 88, 186, 190
Steinman D.A. (2012). Assumptions in modelling of large artery hemodynamics. In: Ambrosi D., Quarteroni A.,
Rozza G. (Eds) Modeling of Physiological Flows— Modeling, Simulation and Applications, Vol 5. Springer,
Milano, 1 - 18
Thomas B. and K. S. Sumam (2016). Blood Flow in Human Arterial System - a Review. Procedia Technology 24, 339
- 346.
Ursino, M. (1998). Interaction between Carotid Baroregulation and the Pulsating Heart: a Mathematical Model .
Am. J. Physiol., 275, 1733 - 1747.
Valentinuzzi M. E and Arini P.D. (2017). Mathematical Models in bioengineering. In Valentinuzzi M. E (Ed) Further
Understanding of the Human Machine - The Road to Bioengineering, Series on Bioengineering and
Biomedical Engineering: Volume 7, World Scientific, Singapore, pp 501 - 521
Vinoth, R., Kumar, D., Raviraj, A., and Vijay S. (2017). Non-Newtonian and Newtonian blood flow in human aorta: A
transient Analysis. Biomedical Research. 28 (7): 3194-3203
Wikipedia. (2017). Divergence theorem, https://en.wikipedia.org/wiki/Divergence_theorem Acessed 3/7/17
World Health Organization. (2017). Cardiovascular Diseases.
http://www.who.int/mediacentre/factsheets/fs317/en/, Accessed 14/12/17
Published
2020-02-22
How to Cite
Ajayi, T., Oludare, O., & Ogunkoya, E. W. (2020). Mathematical Modelling and Simulation of Blood Flow in the Human Ascending Aorta: An Analytical Approach. Journal of Engineering Research, 23(1), 43-54. Retrieved from http://jer.unilag.edu.ng/article/view/594
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