Numerical simulation of blood flow and pressure drop in the pulmonary arterial and venous circulation

Abstract

A novel multiscale mathematical and computational model of the pulmonary circulation is presented and used to analyse both arterial and venous pressure and flow. This work is a major advance over previous studies by Olufsen et al. (Ann Biomed Eng 28:1281–1299, 2012) which only considered the arterial circulation. For the first three generations of vessels within the pulmonary circulation, geometry is specified from patient-specific measurements obtained using magnetic resonance imaging (MRI). Blood flow and pressure in the larger arteries and veins are predicted using a nonlinear, cross-sectional-area-averaged system of equations for a Newtonian fluid in an elastic tube. Inflow into the main pulmonary artery is obtained from MRI measurements, while pressure entering the left atrium from the main pulmonary vein is kept constant at the normal mean value of 2 mmHg. Each terminal vessel in the network of ‘large’ arteries is connected to its corresponding terminal vein via a network of vessels representing the vascular bed of smaller arteries and veins. We develop and implement an algorithm to calculate the admittance of each vascular bed, using bifurcating structured trees and recursion. The structured-tree models take into account the geometry and material properties of the ‘smaller’ arteries and veins of radii \(\ge \)50 \(\upmu \)m. We study the effects on flow and pressure associated with three classes of pulmonary hypertension expressed via stiffening of larger and smaller vessels, and vascular rarefaction. The results of simulating these pathological conditions are in agreement with clinical observations, showing that the model has potential for assisting with diagnosis and treatment for circulatory diseases within the lung.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Notes

  1. 1.

    As is common for studies of cardiovascular dynamics, all pressures are given in mmHg. The conversion to SI units is 1 mmHg \(=\) 133.3 Pa.

References

  1. Alastruey J, Parker KH, Peiro J, Byrd SM, Sherwin SJ (2007) Modelling the circle of Willis to assess the effects of anatomical variations and occlusions on cerebral flows. J Biomech 40:1794–1805

    Article  Google Scholar 

  2. Alastruey J, (2011) Numerical assessment of time-domain methods for estimation of local arterial pulse wave speed. J Biomech 44:885–891

    Google Scholar 

  3. Attinger EO (1963) Pressure transmission in pulmonary arteries related to frequency and geometry. Circ Res 12(6):623–641

    Article  Google Scholar 

  4. Azer K, Peskin CS (2007) A one-dimensional model of blood flow in arteries with friction and convection based on the womersley velocity profile. Cardiovasc Eng 7(2):51–73

    Article  Google Scholar 

  5. Barnes PJ, Liu SF (1995) Regulation of pulmonary vascular tone. Pharmacol Rev 47:87–131

    Google Scholar 

  6. Barst RJ, McGoon M, Torbicki A, Sitbon O, Krowka MJ, Olschewski H, Gaine S (2004) Diagnosis and differential assessment of pulmonary arterial hypertension. J Am Coll Cardiol 43:40S–47S

    Article  Google Scholar 

  7. Bovendeerd PH, Borsje P, Arts T, van de Vosse FN (2006) Dependence of intramyocardial pressure and coronary flow on ventricular loading and contractility: a model study. Ann Biomed Eng 34:1833–1845

    Article  Google Scholar 

  8. Burton AC (1972) Physiology and biophysics of the circulation. Year Book Medical Publishers, Chicago, IL, pp 86–94

    Google Scholar 

  9. Castelain V, Herve P, Lecarpentier Y, Duroux P, Simonneau G, Chemla D (2001) Pulmonary artery pulse pressure and wave reflection in chronic pulmonary thromboembolism and primary pulmonary hypertension. J Am Coll Cardiol 7:1085–1092

    Article  Google Scholar 

  10. Clipp RB, Steele BN (2009) Impedance boundary conditions for the pulmonary vasculature including the effects of geometry, compliance, and respiration. IEEE Trans Biomed Eng 56:862–870

    Article  Google Scholar 

  11. Clipp RB, Steele BN (2012) An evaluation of dynamic outlet boundary conditions in a 1D fluid dynamics model. Math Biosci Eng 9:61–74

    Article  MATH  MathSciNet  Google Scholar 

  12. Cousins W, Gremaud PA (2012) Boundary conditions for hemodynamics: The structured tree revisited. J Comp Phys 231:6086–6096

    Google Scholar 

  13. Cousins W, Gremaud PA, Tartakovsky DM (2013) A new physiological boundary condition for hemodynamics. SIAM J Appl Math 3(73):1203–1233

    Article  MathSciNet  Google Scholar 

  14. Dartevelle P, Fadell E, Mussot S, Chapelier A, Hervel P, de Perrot M, Cerrinal J, Laduriel FL, Lehouerou D, Humbert M, Sitbon O, Simonneau G (2004) Chronic thromboembolic pulmonary hypertension. Eur Respir J 23:637–648

    Article  Google Scholar 

  15. Evans RL, Pelley JW, Quenemoen L (1960) Some simple geometric and mechanical characteristics of mammalian blood vessels. Am J Physiol 199:1150–1152

    Google Scholar 

  16. Figueroa CA, Vignon-Clementel IE, Jansen KE, Hughes T, Taylor CA (2006) A coupled momentum method for modeling blood flow in three-dimensional deformable arteries. Comput Methods Appl Mech Eng 195:5685–5706

    Article  MATH  MathSciNet  Google Scholar 

  17. Fonseca GH, Souza R, Salemi VM, Jardim CV, Gualandro SF (2012) Pulmonary hypertension diagnosed by right heart catheterization in sickle cell disease. Eur Respir J 39(1):112–8

    Article  Google Scholar 

  18. Formaggia L, Lamponi D, Tuveri M, Veneziani A (2006) Numerical modelling of 1D networks coupled with a lumped parameters description of the heart. Comput Methods Biomech Biomed Eng 9:273–288

    Article  Google Scholar 

  19. Fullana J, Zaleski S (2009) A branched one-dimensional model of vessel networks. J Fluid Mech 621:183–204

    Article  MATH  MathSciNet  Google Scholar 

  20. Fung YC (1996) Biomechanics: circulation, 2nd edn. Springer, New York

    Google Scholar 

  21. Gao Y, Raj UJ (2005) Role of veins in regulation of pulmonary circulation. Am J Physiol Lung Cell Mol Physiol 288:L213–L226

    Article  Google Scholar 

  22. Greenfield JC, Douglas MG (1963) Relation between pressure and diameter in main pulmonary artery of man. J Appl Physiol 18:557–559

    Google Scholar 

  23. Hachulla E, Gressin V, Guillevin L et al (2005) Early detection of pulmonary arterial hypertension in systemic sclerosis: a French nationwide prospective multicenter study. Arthritis Rheum 52:3792–3800

    Article  Google Scholar 

  24. Hall JE (2011) Guyton and Hall textbook of medical physiology, 12th edn. Saunders Elsevier, Philadelphia

    Google Scholar 

  25. Herve P, Musset D, Simonneau G, Wagner W Jr, Duroux P (1989) Almitrine decreases the distensibility of the large pulmonary arteries in man. Chest 96:572–577

    Article  Google Scholar 

  26. Hollander EH, Wang JJ, Dobson GM, Parker KH, Tyberg JV (2001) Negative wave reflections in pulmonary arteries. Am J Physiol Heart Circ Physiol 281:H895–902

    Google Scholar 

  27. Huang W, Yen RT, McLaurine M, Bledsoe G (1996) Morphometry of the human pulmonary vasculature. J Appl Physiol 81:2123–2133

    Google Scholar 

  28. Huo Y, Kassab GS (2007) A hybrid one-dimensional/Womersley model of pulsatile blood flow in the entire coronary arterial tree. Am J Physiol Heart Circ Physiol 292:H2623–H2633

    Article  Google Scholar 

  29. Kato R, Lickfett L, Meininger G, Dickfeld T, Wu R, Juang G, Angkeow P, LaCorte J, Bluemke D, Berger R, Halperin HR, Calkins H (2003) Pulmonary vein anatomy in patients undergoing catheter ablation of atrial fibrillation. Lessons learned by use of magnetic resonance imaging. Circulation 107:2004–2010

    Article  Google Scholar 

  30. Kawahira Y, Kishimoto H, Kawata H, Ikawa S, Ueda H, Nakajima T, Kayatani F, Inamura N, Nakada T (1997) Diameters of the pulmonary arteries and veins as an indicator of bilateral and unilateral pulmonary blood flow in patients with congenital heart disease. J Card Surg 12:253–260

    Article  Google Scholar 

  31. Kim YH, Marom EM, Herndon JE, McAdams HP (2005) Pulmonary vein diameter, cross-sectional area, and shape: CT analysis. Radiology 235:43–50

    Article  Google Scholar 

  32. Krenz GS, Dawson CA (2003) Flow and pressure distributions in vascular networks consisting of distensible vessels. Am J Physiol 284:H2192–H2203

    Google Scholar 

  33. Lankhaar JW, Westerhof N, Faes TJC, Marques KMJ, Marcus JT, Post-mus PE, Vonk-Noordegraaf A (2006) Quantification of right ventricular afterload in patients with and without pulmonary hypertension. Am J Physiol Heart Circ Physiol 291:H1731–H1737

    Article  Google Scholar 

  34. Levy BI, Ambrosio G, Pries AR, Struijker-Boudier HA (2001) Microcirculation in hypertension: a new target for treatment? Circulation 104(6):735–740

    Article  Google Scholar 

  35. Li CW, Cheng HD (1993) A nonlinear fluid model for pulmonary blood circulation. J Biomech 26:653–664

    Article  Google Scholar 

  36. Machado RF, Gladwin MT (2010) Pulmonary hypertension in hemolytic disorders: pulmonary vascular disease: the global perspective. Chest 137:30S–38S

    Article  Google Scholar 

  37. Matthys KS, Alastruey J, Peiro J, Khir AW, Segers P, Verdonck PR, Parker KH, Sherwin SJ (2007) Pulse wave propagation in a model human arterial network: assessment of 1-D numerical simulations against in vitro measurements. J Biomech 40:3476–3486

    Article  Google Scholar 

  38. Milnor WR (1989) Hemodynamics, 2nd edn. Williams and Wilkins, Baltimore

    Google Scholar 

  39. Mukerjee D, George D, St Coleiro B et al (2003) Prevalence and outcome in systemic sclerosis associated pulmonary arterial hypertension: application of a registry approach. Ann Rheum Dis 62:1088–1093

    Article  Google Scholar 

  40. Müller LO, Toro EF (2014) A global multi-scale mathematical model for the human circulation with emphasis on the venous system. Int J Num Methods Bio Med Eng. doi:10.1002/cnm.2622

  41. Nichols WW, O’Rourke MF (1998) MacDonald’s blood flow in arteries: theoretical, experimental and clinical principles, 4th edn. Edward Arnold, Philadelphia

    Google Scholar 

  42. Olufsen MS (1998) Modeling the arterial system with reference to an anesthesia simulator. PhD Thesis, Department of Mathematics, Roskilde University, Denmark

  43. Olufsen MS (1999) Structured tree outflow condition for blood flow in larger systemic arteries. Am J Physiol Heart Circ Physiol 276:H257–H268

    Google Scholar 

  44. Olufsen MS, Peskin CS, Kim WY, Pedersen EM, Nadim A (2000) Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann Biomed Eng 28:1281–1299

    Article  Google Scholar 

  45. Olufsen MS, Hill NA, Vaughan GDA, Sainsbury C, Johnson M (2012) Rarefaction and blood pressure in systemic and pulmonary arteries. J Fluid Mech 705:280–305

    Article  MATH  MathSciNet  Google Scholar 

  46. Patel DJ, Schilder DP, Mallos AJ (1960) Mechanical properties and dimensions of major pulmonary arteries. J Appl Physiol 15:92–106

    Google Scholar 

  47. Patel DJ, De Freitas FM, Mallos AJ (1962) Mechanical function of the main pulmonary artery. J Appl Physiol 17:205–208

    Google Scholar 

  48. Peacock AJ, Rubin LJ (2004) Pulmonary circulation: diseases and their treatment, 2nd edn. Hodder Arnold Publication, London

    Google Scholar 

  49. Peacock AJ, Murphy NF, McMurray JJV et al (2007) An epidemiological study of pulmonary arterial hypertension. Eur Respir J 30:104–109

    Article  Google Scholar 

  50. Peskin E (1961) Transient and steady-state analysis of electric networks. Van Nostrand Company, Princeton, NJ, pp 304–378

    Google Scholar 

  51. Pollanen MS (1992) Dimensional optimization at different levels at the arterial hierarchy. J Theor Biol 159:267–270

    Article  Google Scholar 

  52. Pries AR, Secomb TW, Gaehtgens P (1995) Design principles of vascular beds. Circ Res 77:1017–1023

    Article  Google Scholar 

  53. Reeves JT, Linehan JH, Stenmark KR (2005) Distensibility of the normal human lung circulation during exercise. Am J Physiol Lung Cell Mol Physiol 288:L419–L425

    Article  Google Scholar 

  54. 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–H222

    Article  Google Scholar 

  55. Sherwin SJ, Franke V, Perio J, Parker K (2003) One-dimensional modelling of a vascular network in space-time variables. J Eng Math 47:217–250

    Article  MATH  Google Scholar 

  56. Simonneau G, Galle N, Rubin LJ, Langleben D, Seeger W, Domenighetti G, Gibbs S, Lebrec D, Speich R, Beghetti M, Rich S, Fishman A (2004) Clinical classification of pulmonary hypertension. J Am Coll Cardiol 43:5–12

    Article  Google Scholar 

  57. Singhal S, Henderson R, Horsfield K, Harding K, Cumming G (1973) Morphometry of the human pulmonary arterial tree. Circ Res 33:190–197

    Article  Google Scholar 

  58. Sitbon O, Lascoux-Combe C, Delfraissy JF et al (2008) Prevalence of HIV-related pulmonary arterial hypertension in the current antiretroviral therapy era. Am J Respir Crit Care Med 177:108–111

    Article  Google Scholar 

  59. Steele BN, Olufsen MS, Taylor CA (2007) Fractal network model for simulating abdominal and lower extremity blood flow during resting and exercise conditions. Comput Methods Biomech Biomed Eng 10:39–51

    Article  Google Scholar 

  60. Suwa N, Niwa T, Fukasawa H, Sasaki Y (1963) Estimation of intravascular blood pressure gradients by mathematical analysis of arterial casts. Tohoku J Exp Med 79:168–198

    Article  Google Scholar 

  61. Taylor CA, Draney MT, Ku JP, Parker D, Steele BN et al (1999) Predictive medicine: computational techniques in therapeutic decision-making. Comput Aided Surg 4:231–247

    Article  Google Scholar 

  62. Thurlbeck WM, Churg AM (1995) Pathology of the lungs, 2nd edn. Thieme Medical Publishers, New York

    Google Scholar 

  63. Tuder RM, Yun JH, Bhunia A, Fijalkowska I (2007) Hypoxia and chronic lung disease. J Mol Med 85:1317–1324

    Article  Google Scholar 

  64. Uylings HBM (1977) Optimization of diameters and bifurcation angles in lung and vascular tree structures. Bull Math Biol 39:509–520

    Article  MATH  Google Scholar 

  65. Valdez-Jasso D, Haider MA, Campbell AL, Bia D, Zocalo Y, Armentano RL, Olufsen MS (2009) Modeling viscoelastic wall properties of ovine arteries. In: Proceedings of ASME 2009, summer bioengineering conference SBC2009-205640

  66. Vaughan GDA (2010) Pulse propagation in the pulmonary and systemic arteries. PhD Thesis, Faculty of Information and Mathematical Sciences, University of Glasgow, UK

  67. Vignon-Clementel IE, Figueroa CA, Jansen KE, Taylor CA (2006) Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries. Comput Methods Appl Mech Eng 195:3776–3796

    Article  MATH  MathSciNet  Google Scholar 

  68. van de Vosse FN, Stergiopulos N (2011) Pulse wave propagation in the arterial tree. Annu Rev Fluid Mech 43:467–499

    Article  Google Scholar 

  69. Weibel ER (2009) What makes a good lung? The morphometric basis of lung function. Swiss Med Wkly 139:375–386

    Google Scholar 

  70. Xiao N, Humphrey JD, Figueroa CA (2013) Multi-scale computational model of three-dimensional hemodynamics within a deformable full-body arterial network. J Comput Phys 244:22–40

    Article  MathSciNet  Google Scholar 

  71. Yen RT, Rong Z, Zhang B (1990) Elasticity of pulmonary blood vessels in human lungs. In: Farrell Epstein MA, Ligas JR (eds) Respiratory biomechanics: engineering analysis of structure and function. Springer, New York, pp 109–116

  72. Yen RT, Sobin SS (1988) Elasticity of arterioles and venules in postmortem human lungs. J Appl Physiol 64(2):611–619

    Google Scholar 

  73. Zhuang FY, Fung YC, Yen RT (1983) Analysis of blood flow in cats lung with detailed anatomical and elasticity data. J Appl Physiol 55(4):1341–1348

    Google Scholar 

  74. Zhuang FY, Fung YC, Yen RT (1983) Analysis of blood flow in cats lung with detailed anatomical and elasticity data. J Appl Physiol 55(4):1341–1348

    Google Scholar 

Download references

Acknowledgments

Olufsen was funded in part by the National Science Foundation, Award No. NSF-DMS-1122424 and by the National Institute of Health Virtual Physiological Rat Center under Award No. 5-P50-GM094503-02. Vaughan was funded by a studentship from the UK EPSRC. Qureshi was funded by a scholarship from IIU and HEC Pakistan and by PG mobility scholarship from the College of Science and Engineering, University of Glasgow, to visit Olufsen’s group.

Author information

Affiliations

Authors

Corresponding author

Correspondence to N. A. Hill.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Qureshi, M.U., Vaughan, G.D.A., Sainsbury, C. et al. Numerical simulation of blood flow and pressure drop in the pulmonary arterial and venous circulation. Biomech Model Mechanobiol 13, 1137–1154 (2014). https://doi.org/10.1007/s10237-014-0563-y

Download citation

Keywords

  • Pulmonary circulation
  • Pulmonary hypertension
  • Resistance arteries
  • Structured tree
  • Multiscale mathematical model