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Mathematical and Computational Modelling of Blood Pressure and Flow

  • Carole LeguyEmail author
Chapter
Part of the Series in BioEngineering book series (SERBIOENG)

Abstract

Since William Harvey discovered in 1628 that blood circulates in a closed loop in the body, and that the contraction of the heart delivered the driving force to move the blood [1], cardiovascular mechanics has gained a lot of attention and is still subject to research. During the last decades, mathematical models have been developed to grasp the diversity of blood flow patterns and pressure propagation phenomena within the cardiovascular system [2]. The geometry of the arterial and venous system is difficult to describe; blood, a complex non-Newtonian fluid, circulates in vessels with non-linear viscoelastic walls. It is therefore very difficult to take into account the complexity of the cardiovascular system within mathematical or numerical models in a comprehensive manner. Thus, mathematical and numerical models generally focus on particular aspects of the cardiovascular circulation. Two approaches can be used to simulate blood flow: either a phenomenon is simulated locally (and in detail) using 3D Computational Fluid Dynamics (CFD) or fluid structure interaction (FSI) models, or with lumped or wave propagation models to simulate the entire systemic circulation considering a simplified geometry. These models have proven their value to understand normal physiology better [3], to simulate the effects of pathophysiological symptoms, or to predict the effect of medical interventions [4]. In the following sections, physiological considerations will be reviewed and the basic equations that govern blood flow and pressure dynamics within arteries and veins will be presented. Later, lumped, 1D and 3D models will be introduced and finally, clinical relevance and applications will be introduced

References

  1. 1.
    Harvey W, Leake C (1928) Exercitatio anatomica de motu cordis et sanguinis in animalibus. Thomas, SpringfieldGoogle Scholar
  2. 2.
    van de Vosse F (2003) Mathematical modelling of the cardiovascular system. J Eng Math 47(3):175–183MathSciNetzbMATHGoogle Scholar
  3. 3.
    Pedley TJ (2003) Mathematical modelling of arterial fluid dynamics. J Eng Math 47(3):419–444MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bode AS, Huberts W, Bosboom EMH, Kroon W, van der Linden WPM, Planken RN, van de Vosse FN, Tordoir JHM (2012) Patient-specific computational modeling of upper extremity arteriovenous fistula creation: its feasibility to support clinical decision-making. PloS one 7(4):e34491Google Scholar
  5. 5.
    Fung Y (1996) Biomechanics: circulation. Springer, BerlinGoogle Scholar
  6. 6.
    Nichols W, O’Rourke M (2005) Mc Donald’s blood flow in arteries. Theoretic, experimental, and clinical principles, 5th edn. Oxford University Press, LondonGoogle Scholar
  7. 7.
    Segers P, Verdonck P (2000) Role of tapering in aortic wave reflection: hydraulic and mathematical model study. J Biomech 33(3):299–306Google Scholar
  8. 8.
    World Health Organization (2014) Global status report on noncommunicable diseases 2014Google Scholar
  9. 9.
    Laurent S, Cockcroft J, van Bortel L, Boutouyrie P, Giannattasio C, Hayoz D, Pannier B, Vlachopoulos C, Wilkinson I, Struijker-Boudier H (2006) Expert consensus document on arterial stiffness: methodological issues and clinical applications. Eur Heart J 27:2588–2605Google Scholar
  10. 10.
    Bussy C, Boutouyrie P, Lacolley P, Challande P, Laurent S (2000) Intrinsic stiffness of the carotid arterial wall material in essential hypertensives. Hypertension 35:1049–1054Google Scholar
  11. 11.
    Shibeshi SS, Collins WE (2005) The rheology of blood flow in a branched arterial system. Appl Rheol 15(6):398–405Google Scholar
  12. 12.
    Bessonov N, Sequeira A, Simakov S, Vassilevskii Y, Volpert V (2016) Methods of blood flow modelling. Math Model Nat Phenom 11(1):1–25MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gijsen F, Allanic E, van de Vosse F, Janssen J (1999) The influence of the non-newtonian properties of blood on the flow in large arteries: unsteady flow in a 90 degrees curved tube. J Biomech 32:705–713Google Scholar
  14. 14.
    Leguy CAD, Bosboom EMH, Gelderblom H, Hoeks APG, van de Vosse FN (2010) Estimation of distributed arterial mechanical properties using a wave propagation model in a reverse way. Med Eng Phys 32(9):957–67Google Scholar
  15. 15.
    Brands P, Hoeks A, Willigers J, Willekes C, Reneman R (1999) An integrated system for the non-invasive assessment of vessel wall and hemodynamic properties of large arteries by means of ultrasound. Eur J Ultrasound 9:257–266Google Scholar
  16. 16.
    Kwee RM, Truijman MTB, van Oostenbrugge RJ, Mess WH, Prins MH, Franke CL, Korten AGGC, Wildberger JE, Kooi ME (2012) Longitudinal MRI study on the natural history of carotid artery plaques in symptomatic patients. PLoS One 7(7)Google Scholar
  17. 17.
    Westerhof N (2010) Snapshots of hemodynamics: an aid for clinical research and graduate educationGoogle Scholar
  18. 18.
    Leguy C, Bosboom E, Hoeks A, van de Vosse F (2009) Model-based assessment of dynamic arterial blood volume flow from ultrasound measurements. Med Biol Eng Comput 47:641–648Google Scholar
  19. 19.
    Womersley J (1955) Mathematical theory of oscillating flow in an elastic tube. J Physiol 127:37–8PGoogle Scholar
  20. 20.
    Leguy CAD, Bosboom EMH, Hoeks APG, van de Vosse FN (2009) Assessment of blood volume flow in slightly curved arteries from a single velocity profile. J Biomech 42(11):1664–72Google Scholar
  21. 21.
    Schlichting H (1960) Boundary layer theory, 1st edn. Mc Graw-Hill, New YorkzbMATHGoogle Scholar
  22. 22.
    Frank O (1899) Die Grundform des arteriellen Pulses. Erste Abhandlung. Mathematische Analyse Z Biol 37:483–526Google Scholar
  23. 23.
    Westerhof N, Bosman F, de Vries C, Noordergraaf A (1969) Analog studies of the human systemic arterial tree. J Biomech 2:121–143Google Scholar
  24. 24.
    Stergiopulos N, Segers P, Westerhof N (1999) Use of pulse pressure method for estimating total arterial compliance in vivo. Am J Physiol 276:H424–H428Google Scholar
  25. 25.
    Stergiopulos N, Westerhof B, Westerhof N (1999) Total arterial inertance as the fourth element of the windkessel model. Am J Physiol 276:H81–H88Google Scholar
  26. 26.
    Westerhof N, Lankhaar J-W, Berend A, Westerhof E (2008) The arterial WindkesselGoogle Scholar
  27. 27.
    Huberts W, Bosboom E, van de Vosse F (2008) A lumped model for blood flow and pressure in the systemic arteries based on an approximate velocity profile function. Math Biosci Eng 6(1):27–40MathSciNetzbMATHGoogle Scholar
  28. 28.
    Liang F, Liu H (2005) A closed-loop lumped parameter computational model for human cardiovascular system. JSME Int Ser C 48:84–493Google Scholar
  29. 29.
    Olufsen M, Peskin C, Kim W, Pedersen E, Nadim A, Larsen J (2000) Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann Biomed Eng 28:1281–1299Google Scholar
  30. 30.
    Lanzarone E, Liani P, Baselli G, Costantino M (2007) Model of arterial tree and peripheral control for the study of physiological and assisted circulation. Med Eng Phys 29:542–555Google Scholar
  31. 31.
    Pietrabissa R, Mantero S, Marotta T, Menicanti L (1996) A lumped parameter model to evaluate the fluid dynamics of different coronary bypasses. Med Eng Phys 18:477–484Google Scholar
  32. 32.
    Wolters B, Emmer M, Rutten M, Schurink G, van de Vosse F (2007) Assessment of endoleak significance after endovascular repair of abdominal aortic aneurysms: a lumped parameter model. Med Eng Phys 29:1106–1118Google Scholar
  33. 33.
    Bessems D, Rutten M, van de Vosse F (2007) A wave propagation model of blood flow in large vessels using an approximate velocity profile function. J Fluid Mech 580:145–168MathSciNetzbMATHGoogle Scholar
  34. 34.
    Alastruey J, Moore S, Parker K, David T, Peiro J, Sherwin S (2008) Reduced modelling of blood flow in the cerebral circulation: coupling 1-d, 0-d and cerebral auto-regulation models. Int J Numer Methods Fluids 56(8):1061–1067MathSciNetzbMATHGoogle Scholar
  35. 35.
    van de Vosse FN, Stergiopulos N (2011) Pulse wave propagation in the arterial tree. Ann Rev Fluid Mech 43(1):467–499MathSciNetzbMATHGoogle Scholar
  36. 36.
    Bessems D, Giannopapa CG, Rutten MCM, van de Vosse FN, Anliker M, Rockwell L, Anliker M, Moritz W, Ogden E, Bessems D, Cox R, Fung Y, Giannopapa C, Hughes T, Lubliner J, Learoyd B, Taylor M (2008) Experimental validation of a time-domain-based wave propagation model of blood flow in viscoelastic vessels. J Biomech 41(2):284–91Google Scholar
  37. 37.
    Fullana J-M, Zaleski S (2009) A branched one-dimensional model of vessel networks. J Fluid Mech 621:183MathSciNetzbMATHGoogle Scholar
  38. 38.
    Keijsers JMT, Leguy CAD, Huberts W, Narracott AJ, Rittweger J, van de Vosse FN (2015) A 1D pulse wave propagation model of the hemodynamics of calf muscle pump function. Int J Numer Methods Biomed Eng 31(7):e02716MathSciNetGoogle Scholar
  39. 39.
    Marchandise E, Flaud P (2010) Accurate modelling of unsteady flows in collapsible tubes. Comput Methods Biomech Biomed Eng 13(2):279–290Google Scholar
  40. 40.
    Shapiro AH (1977) Steady flow in collapsible tubes. J Biomech Eng 99(3):126Google Scholar
  41. 41.
    Alastruey J, Khir AW, Matthys KS, Segers P, Sherwin SJ, Verdonck PR, Parker KH, Peiró J (2011) Pulse wave propagation in a model human arterial network: assessment of 1-D visco-elastic simulations against in vitro measurements. J Biomech 44(12):2250–2258Google Scholar
  42. 42.
    Reymond P, Merenda F, Perren F, Rüfenacht D, Stergiopulos N (2009) Validation of a one-dimensional model of the systemic arterial tree. Am J Physiol Heart Circ Physiol 297:H208–H222Google Scholar
  43. 43.
    Liu B, Tang D (2011) Influence of non-Newtonian properties of blood on the wall shear stress in human atherosclerotic right coronary arteries. Mol Cell Biomech MCB 8(1):3–90Google Scholar
  44. 44.
    Li M, Beech-Brandt J, John L, Hoskins P, Easson W (2007) Numerical analysis of pulsatile blood flow and vessel wall mechanics in different degrees of stenoses. J Biomech 40:715–3724Google Scholar
  45. 45.
    Morris PD, Narracott A, von Tengg-Kobligk H, Silva Soto DA, Hsiao S, Lungu A, Evans P, Bressloff NW, Lawford PV, Hose DR, Gunn JP (2016) Computational fluid dynamics modelling in cardiovascular medicine. Heart (Br Card Soc) 102(1):18–28Google Scholar
  46. 46.
    Steinman DA, Milner JS, Norley CJ, Lownie SP, Holdsworth DW (2003) Image-based computational simulation of flow dynamics in a giant intracranial aneurysm. AJNR. Am J Neuroradiol 24(4):559–66Google Scholar
  47. 47.
    Kayser-Herolda O, Matthies H (2007) A unified least-squares formulation for fluid-structure interaction problems. Comput Struct 85:998–1011MathSciNetGoogle Scholar
  48. 48.
    Moireau P, Xiao N, Astorino M, Figueroa CA, Chapelle D, Taylor CA, Gerbeau J-F (2012) External tissue support and fluid-structure simulation in blood flows. Biomech Model Mechanobiol 1(1–2):1–18Google Scholar
  49. 49.
    Taylor CA, Figueroa CA (2009) Patient-specific modeling of cardiovascular mechanics. Ann Rev Biomed Eng 11:109–34Google Scholar
  50. 50.
    Mynard JP, Wasserman BA, Steinman DA (2013) Errors in the estimation of wall shear stress by maximum Doppler velocity. Atherosclerosis 227(2):259–66Google Scholar
  51. 51.
    Neidlin M, Corsini C, Sonntag SJ, Schulte-Eistrup S, Schmitz-Rode T, Steinseifer U, Pennati G, Kaufmann TA (2016) Hemodynamic analysis of outflow grafting positions of a ventricular assist device using closed-loop multiscale CFD simulations: Preliminary results. J Biomech 49(13):2718–2725Google Scholar
  52. 52.
    Speelman L, Bosboom EMH, Schurink GWH, Buth J, Breeuwer M, Jacobs MJ, van de Vosse FN (2009) Initial stress and nonlinear material behavior in patient-specific AAA wall stress analysis. J Biomech 42(11):1713–9Google Scholar
  53. 53.
    Merkx MAG, Bode AS, Huberts W, Oliván Bescós J, Tordoir JHM, Breeuwer M, van de Vosse FN, Bosboom EMH (2013) Assisting vascular access surgery planning for hemodialysis by using MR, image segmentation techniques, and computer simulations. Med Biol Eng Comput 51(8):79–889Google Scholar
  54. 54.
    Leguy CAD, Bosboom EMH, Belloum ASZ, Hoeks APG, Van De Vosse FN (2011) Global sensitivity analysis of a wave propagation model for arm arteries. Med Eng Phys 33(8):1008–1016Google Scholar
  55. 55.
    Reymond P, Vardoulis O, Stergiopulos N (2012) Generic and patient-specific models of the arterial tree. J Clin Monit Comput 26(5):375–382Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of Cardiovascular Engineering CVEInstitute of Applied Medical Engineering AME, RWTH Aachen UniversityAachenGermany

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