Skip to main content

Advertisement

SpringerLink
Log in

Mathematical Modeling of Blood Flow Through Human Femoral Arteries and the Analysis of Model Parameters

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

In this work, we propose mathematical models for blood flow through human femoral arteries, fit statistical models to the data generated by simulations and identify model parameters that are significant on the blood flow. It is known that blood is a complex fluid whose viscosity varies with the shear rate, and this biophysical parameter, i.e., the blood viscosity is a critical determinant of friction against the vessel walls. Thus, we built two mathematical models; one describes blood as a Newtonian fluid (model-1) that defines viscosity as a constant and the other as a non-Newtonian fluid (model-2) that describes viscosity as a function of shear rate. The models developed incorporating the salient features of arterial blood flow consisted of three sets of parameters called the fluid parameters (to describe blood), the flow parameters (to describe the nature of blood flow), and the material and geometric parameters (to describe the nature of arteries). With the help of publicly available data on the human femoral artery for both male and female populations with different ages ranging from 19 years to 60+ , we carried out simulations using the developed models. Later, we performed statistical analysis of the data generated through simulations and identified the potential predictor variables on the two physical quantities computed: the wall shear stress and the volumetric flow rate. An interesting idea in this study is that we projected these models to describe the different states of health of human arteries: model-1 for “healthy,” model-2 for “transient,” and that with large values of a non-Newtonian parameter for the “unhealthy.”

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data Availability Statement

The data that has been generated and analysed in this study can be shared on request.

Abbreviations

\(\overrightarrow{q}\) :

Velocity vector

\(\rho \) :

Density (kg/m3)

\({\delta }_{ij}\) :

Kronecker symbol

\({e}_{ij}\) :

Rate of deformation tensor

\({m}_{ij}\) :

Couple stress tensor

\({t}_{ij}^{\left(A\right)}\) :

Skew-symmetric part of\({t}_{ij}\)

\({c}_{k}\) :

k-component of body moment vector

\(m\) :

Trace of \({m}_{ij}\)

\({\eta }_{1},{\eta }_{1}^{^{\prime}}\) :

Couple stress viscosity coefficients

\(\overrightarrow{c}\) :

Body moment per unit mass

\({a}_{0}\) :

Average blood pressure (Pa m−1)

\(\Omega \) :

Frequency of oscillations \(\left(2\uppi \cdot \mathrm{HR}\right)\)

\(F\) :

Taper fraction

\({R}_{0}\) :

Inlet radius (at \(t=0\))

\(\Delta \) :

Aspect ratio \(\left(\frac{{R}_{0}}{L}\right)\)

\(C\) :

Dimensionless couple stress parameter

\(p\) :

Thermodynamic pressure

\({\tau }_{ij}\) :

Deviatoric stress tensor

\(\mu \) :

Viscosity (Pa  s)

\({t}_{ij}\) :

Non-symmetric stress tensor

\({t}_{ij}^{(s)}\) :

Symmetric part of \({t}_{ij}\)

\({d}_{ij}\) :

Rate of deformation tensor

\({\varepsilon }_{ijk}\) :

Levi–Civita symbol

\({\mu }_{1}, {\lambda }_{1}\) :

Viscosity coefficients

\(\overrightarrow{f}\) :

Body force per unit mass

\(\sigma \) :

Couple stress parameter

\({a}_{1}\) :

Pulse difference (Pa m−1)

\(HR\) :

Heart rate

\(L\) :

Length of the artery

\(\beta \) :

Contracting/Expanding parameter

\(\alpha \) :

Womersley number \(\left(\sqrt{\frac{\rho\Omega {R}_{0}^{2}}{\mu }}\right)\)

\(h\),\(k\),\(l,n\) :

Auxiliary parameters

References:

  1. Maas, A.H.E.M., Appelman, Y.E.A.: Gender differences in coronary heart disease. Netherlands Hear. J. (2010). https://doi.org/10.1007/s12471-010-0841-y

    Article  Google Scholar 

  2. Dhingra, R., Vasan, R.S.: Age as a cardiovascular risk factor. Med. Clin. North Am. 96(1), 87–91 (2012)

    Article  Google Scholar 

  3. Steenman, M., Lande, G.: Cardiac aging and heart disease in humans. Biophys. Rev. (2017). https://doi.org/10.1007/s12551-017-0255-9

    Article  Google Scholar 

  4. Marchione, V.: Resting heart rate chart: factors that influence heart rate in elderly (2018). https://www.belmarrahealth.com/resting-heart-rate-chart-factors-influence-heart-rate-elderly/.

  5. Marchione, V.: Understanding blood pressure readings key to overall well-being (2017)

  6. Zhou, D., Xi, B., Zhao, M., Wang, L., Veeranki, S.P.: Uncontrolled hypertension increases risk of all-cause and cardiovascular disease mortality in US adults: The NHANES III linked mortality study. Sci. Rep. (2018). https://doi.org/10.1038/s41598-018-27377-2

    Article  Google Scholar 

  7. Bozkurt, S.: Mathematical modeling of cardiac function to evaluate clinical cases in adults and children. PLoS ONE (2019). https://doi.org/10.1371/journal.pone.0224663

    Article  Google Scholar 

  8. Li, X., Bai, J., Cui, S., Wang, S.: Simulation study of the cardiovascular functional status in hypertensive situation. Comput. Biol. Med. (2002). https://doi.org/10.1016/S0010-4825(02)00020-3

    Article  Google Scholar 

  9. Majonga, E.D., Rehman, A.M., McHugh, G., Mujuru, H.A., Nathoo, K., Patel, M.S., Munyati, S., Odland, J.O., Kranzer, K., Kaski, J.P., Ferrand, R.A.: Echocardiographic reference ranges in older children and adolescents in sub-Saharan Africa. Int. J. Cardiol. (2017). https://doi.org/10.1016/j.ijcard.2017.06.109

    Article  Google Scholar 

  10. Quarteroni, A.: Modeling the Cardiovascular system-a mathematical adventure: part-II. SIAM News. 34 (2001)

  11. Peskin, C.S., McQueen, D.M.: A three-dimensional computational method for blood flow in the heart I. Immersed elastic fibers in a viscous incompressible fluid. J. Comput. Phys. (1989). https://doi.org/10.1016/0021-9991(89)90213-1

    Article  MathSciNet  MATH  Google Scholar 

  12. Verma, N., Parihar, R.S.: Mathematical model of blood flow through a tapered artery with mild stenosis and hematocrit. J. Mod. Math. Stat. 4, 38–43 (2010)

    Article  MathSciNet  Google Scholar 

  13. Karimipour, A., Mokhtari, H., Akbari, M., Toghraie, D., Karimipour, A.: Simulation of blood flow into the popliteal artery to explain the effect of peripheral arterial disease: Investigation the conditions and effects of different foot states during the daily activity of the patient. Comput. Methods Programs Biomed. 195, 105638 (2020). https://doi.org/10.1016/j.cmpb.2020.105638

    Article  Google Scholar 

  14. Awrejcewicz, J., Zafar, A.A., Kudra, G., Riaz, M.B.: Theoretical study of the blood flow in arteries in the presence of magnetic particles and under periodic body acceleration. Chaos, Solitons Fractals 140, 110204 (2020). https://doi.org/10.1016/j.chaos.2020.110204

    Article  MathSciNet  Google Scholar 

  15. Udupa, M., Shankar Narayan, S., Saha, S.: A study of the blood flow using newtonian and non-newtonian approach in a stenosed artery. In: Proc. Int. Conf. Trends Comput. Cogn. Eng., Springer, pp. 257–269 (2021). https://doi.org/10.1007/978-981-15-5414-8_21

  16. Vasu, B., Dubey, A., Bég, O.A., Gorla, R.S.R.: Micropolar pulsatile blood flow conveying nanoparticles in a stenotic tapered artery: non-Newtonian pharmacodynamic simulation. Comput. Biol. Med. (2020). https://doi.org/10.1016/j.compbiomed.2020.104025

    Article  Google Scholar 

  17. Majekodunmi Joshua, T., Anwar, K., Abdullah, N. : Numerical simulation of non-newtonian blood flow through a tapered stenosed artery using the cross model. In: IOP Conf. Ser. Mater. Sci. Eng. (2020). https://doi.org/10.1088/1757-899X/864/1/012200

  18. Liu, Y., Liu, W.: Blood flow analysis in tapered stenosed arteries with the influence of heat and mass transfer. J. Appl. Math. Comput. 63, 523–541 (2020). https://doi.org/10.1007/s12190-020-01328-5

    Article  MathSciNet  MATH  Google Scholar 

  19. Ponalagusamy, R., Manchi, R.: Mathematical modeling of electro-magnetohydrodynamic pulsatile flow of an elastico-viscous fluid through an inclined porous tapered arterial stenosis. Math. Eng. Sci. Aerosp. (2020)

  20. Fatih, S., Öztop, H.F., Chamkha, A.J.: Role of magnetic field on forced convection of nanofluid in a branching channel. Int. J. Numer. Methods Heat Fluid Flow. 30, 1755–1772 (2019). https://doi.org/10.1108/HFF-10-2018-0568

    Article  Google Scholar 

  21. Chamkha, A.J.: Unsteady laminar hydromagnetic fluid-particle flow and heat transfer in channels and circular pipes. Int. J. Heat Fluid Flow. 21, 740–746 (2000). https://doi.org/10.1016/S0142-727X(00)00031-X

    Article  Google Scholar 

  22. Tu, J., Inthavong, K., Wong, K.K.L.: Computational Hemodynamics-Theory, Modelling and Applications. Springer, Berlin (2015)

    Book  Google Scholar 

  23. Mazumdar, J.N.: Biofluid Mechanics. World Scientific (1992). https://doi.org/10.1142/1623

    Article  MATH  Google Scholar 

  24. Stokes, V.K.: Theories of Fluids with Microstructure. Springer, Berlin Heidelberg (1984)

    Book  Google Scholar 

  25. Uchida, S., Aoki, H.: Unsteady flows in a semi-infinite contracting or expanding pipe. J. Fluid Mech. 82, 371–387 (1977). https://doi.org/10.1017/S0022112077000718

    Article  MathSciNet  MATH  Google Scholar 

  26. Mandal, P.K.: An unsteady analysis of non-Newtonian blood flow through tapered arteries with a stenosis. Int. J. Non. Linear. Mech. 40, 151–164 (2005). https://doi.org/10.1016/j.ijnonlinmec.2004.07.007

    Article  MATH  Google Scholar 

  27. Liao, S.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC, Boca Raton (2004)

    MATH  Google Scholar 

  28. Radhika, T.S.L., Iyengar, T.K.V., Rani, T.R.: Approximate analytical methods for solving ordinary differential equations, CRC Press, Hoboken, NJ (2015). https://cds.cern.ch/record/1999768

  29. Benetos, A., et al.: Arterial alterations with aging and high blood pressure. A noninvasive study of carotid and femoral arteries. Arterioscl. Throm. J. Vasc. Bio. 13, 90–97 (1993)

    Article  Google Scholar 

  30. San, O., Staples, A.E.: An improved model for reduced-order physiological fluid flows. J. Mech. Med. Bio. 12, 1–28 (2012)

    Article  Google Scholar 

  31. Tudnowski, R.J., Rico, R.C.: Specific gravity of blood and plasma at 4 and 37 ºC. Clin. Chem. 20, 615–616 (1974)

    Article  Google Scholar 

  32. Carallo, C., et al.: The effect of aging on blood and plasma viscosity. An 116 years follow-up study. Clin. Hemorheol. Microcirc. 47, 67–74 (2011)

    Article  Google Scholar 

  33. Chen, Y., Shen, F., Liu, J., Yang, G.Y.: Arterial stiffness and stroke: de-stiffening strategy, a therapeutic target for stroke, stroke. Vasc. Neurol. (2017). https://doi.org/10.1136/svn-2016-000045

    Article  Google Scholar 

  34. Thompson, M.M., Fitridge, R., Boyle, J., Thompson, M., Brohi, K., Hinchliffe, R.J., Cheshire, N., Naylor, A.R., Loftus, I., Davies, A.H.: Oxford Textbook of Vascular Surgery. Oxford University Press, Oxford (2016). https://doi.org/10.1093/med/9780199658220.001.0001.

    Book  Google Scholar 

  35. Stoewen, D.L.: Dimensions of wellness: change your habits, change your life. Can. Vet. J. 58, 861–862 (2017)

    Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the DST – FIST lab, Department of Mathematics, BITS Pilani-Hyderabad Campus for providing the computational facility.

Funding

Self-funded

Author information

Authors and Affiliations

  1. BITS Pilani, Hyderabad Campus, Hyderabad, 500078, India

    A. Karthik, T. S. L. Radhika & P. T. V. Praveen Kumar

Authors
  1. A. Karthik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. S. L. Radhika
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. T. V. Praveen Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. S. L. Radhika.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

See Tables 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30.

Table 14 Descriptive statistics for female population in model-1
Full size table
Table 15 Stepwise regression analysis for vol. flow rate in female population modeled using model-1
Full size table
Table 16 Descriptive statistics for male population in model-1
Full size table
Table 17 Stepwise regression analysis for WSS in male population modeled using model-1
Full size table
Table 18 Stepwise regression analysis for vol. flow rate in male population modeled using model-1
Full size table
Table 19 Descriptive statistics for female population in model-2
Full size table
Table 20 Stepwise regression analysis for WSS in female population modeled using model-2
Full size table
Table 21 Stepwise regression analysis for vol. flow rate in female population modeled using model-2
Full size table
Table 22 Descriptive statistics for male population in model-2
Full size table
Table 23 Stepwise regression analysis for WSS in male population modeled using model-2
Full size table
Table 24 Stepwise regression analysis for vol. flow rate in male population modeled using model-2
Full size table
Table 25 Descriptive statistics for female population in model-2 with small values of \(\sigma \)
Full size table
Table 26 Stepwise regression analysis for WSS in female population modeled using model-2 with small values of \(\sigma \)
Full size table
Table 27 Stepwise regression analysis for vol. flow rate in female population modeled model-2 with small values of \(\sigma \)
Full size table
Table 28 Descriptive statistics for male population in model-2 with small values of \(\sigma \)
Full size table
Table 29 Stepwise regression analysis for WSS in male population modeled using model-2 with small values of \(\sigma \)
Full size table
Table 30 Stepwise regression analysis for vol. flow rate in male population modeled using model-2 with small values of \(\sigma \)
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karthik, A., Radhika, T.S.L. & Praveen Kumar, P.T.V. Mathematical Modeling of Blood Flow Through Human Femoral Arteries and the Analysis of Model Parameters. Int. J. Appl. Comput. Math 8, 30 (2022). https://doi.org/10.1007/s40819-021-01228-7

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-021-01228-7

Keywords

Search

Navigation