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A Critical Review of Multiphase Modelling of Blood Flow in Human Cardiovascular System

  • Review Article
  • Published:
Journal of the Indian Institute of Science Aims and scope

Abstract

In the human body, blood acts as a transporter of oxygen and other nutrients as well as carbon dioxide and other waste materials to and from all the organs. Therefore, continuous supply of blood to all the organs is critical for proper functioning of the human body. Blood is a complex fluid and has more than 40% flexible particles which include red blood cells, white blood cells, platelets and other proteins suspended in a water-like fluid, plasma. The dynamics of blood flow, known as haemodynamics, is critical in the development, diagnosis and treatment planning of vascular diseases and design and development of cardiovascular devices. Whilst the most advanced flow measurement techniques such as X-ray imaging, magnetic resonance imaging and ultrasound imaging are used in the diagnosis and treatment of vascular diseases, it is not possible to obtain the complete information of pressure and velocity field experimentally via in vivo methods. Therefore, in silico methods or computational modelling techniques are being increasingly employed not only to understand the haemodynamics but also for use in the clinical setting. Whilst blood is treated as a homogeneous, single-phase fluid in several studies, it is possible to capture several features of the flow of blood only by modelling it as a multiphase fluid. A number of approaches have been adopted to model multiphase flow of blood. A broad categorisation can be based on whether the cell boundary is captured explicitly, e.g. immersed boundary method, or the phases are treated as interpenetrating and two or more phases can exist simultaneously at a point, e.g. Euler–Euler method. In the literature, both the approaches have been adopted to model the flow of blood. Particle-based methods, such as smoothed particle hydrodynamics and dissipative particle dynamics have also been employed by researchers to study the complex interactions associated with the flow of blood. In this article, we discuss different multiphase modelling approaches and their application in the haemodynamics modelling.

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Figure 1:

Adapted from Ottesen et al. 1.

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Notes

  1. Inner cross-section of the vessel.

  2. The space surrounding blood vessels and cells.

  3. Erythrocytes: Erythors: red, cyte: cell.

  4. Haematocrit: Hemato: blood; crit/krinein: to separate.

  5. Leukocytes: Leuk: white; cyte: cell.

  6. Plaque build-up in the arteries of legs or arms.

  7. Formation of blood clot in the vein.

  8. Circulating blood clot in the blood vessels (Thrombos: clot, emboli: foreign substance travelling through the blood stream).

  9. Bulging of a weakened vessel wall.

  10. By birth.

  11. Narrowing of blood vessels.

  12. Blood vessel rupture.

  13. Eulerian description in fluid mechanics refers to flow of fluid in a control volume and the flow properties such as velocity and pressure are a function of time and space and any specific fluid particle.

  14. Two phases can exist simultaneously at a location.

  15. Abnormally constricted.

  16. In the Lagrangian description of fluid flow, the motion of individual fluid particles is tracked.

  17. Connection between two blood vessels.

  18. Process of separation of one or more components of the blood.

  19. Intracellular fluid or fluid present in the cell and does not include organelles.

References

  1. Ottesen JT, Olufsen MS, Larsen JK (2004) Applied mathematical models in human physiology. Soc Ind Appl Math. https://doi.org/10.1137/1.9780898718287

    Article  Google Scholar 

  2. Brust M, Schaefer C, Doerr R, Pan L, Garcia M, Arratia PE, Wagner C (2013) Rheology of human blood plasma: viscoelastic versus Newtonian behavior. Phys Rev Lett. https://doi.org/10.1103/PhysRevLett.110.078305

    Article  PubMed  Google Scholar 

  3. Rodrigues T, Mota R, Gales L, Campo-Deaño L (2022) Understanding the complex rheology of human blood plasma. J Rheol 66(4):761–774. https://doi.org/10.1122/8.0000442

    Article  CAS  Google Scholar 

  4. Guyton A, Hall A (2020) Textbook of medical physiology, 3rd edn. Elsevier Health Science

    Google Scholar 

  5. Beris AN, Horner JS, Jariwala S, Armstrong MJ, Wagner NJ (2021) Recent advances in blood rheology: a review. Soft Matter 17(47):10591–10613. https://doi.org/10.1039/d1sm01212f

    Article  CAS  PubMed  Google Scholar 

  6. Fåhræus R, Lindqvist T (1931) The viscosity of the blood in narrow capillary tubes. Am J Physiol Legacy Content 96(3):562–568. https://doi.org/10.1152/ajplegacy.1931.96.3.562

    Article  Google Scholar 

  7. Secomb TW, Pries AR (2013) Blood viscosity in microvessels: experiment and theory. C R Phys 14(6):470–478. https://doi.org/10.1016/j.crhy.2013.04.002

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  8. Malek AM, Alper SL, Izumo S (1999) Hemodynamic shear stress and its role in atherosclerosis. JAMA 282(21):2035–2042. https://doi.org/10.1001/jama.282.21.2035

    Article  CAS  PubMed  Google Scholar 

  9. Baratchi S, Chen YC, Peter K (2020) Helical flow: a means to identify unstable plaques and a new direction for the design of vascular grafts and stents. Atherosclerosis 300:34–36. https://doi.org/10.1016/j.atherosclerosis.2020.03.002

    Article  CAS  PubMed  Google Scholar 

  10. Esmaily Moghadam M, Vignon-Clementel IE, Figliola R, Marsden AL (2013) A modular numerical method for implicit 0D/3D coupling in cardiovascular finite element simulations. J Comput Phys 244:63–79. https://doi.org/10.1016/j.jcp.2012.07.035

    Article  Google Scholar 

  11. Figueroa CA, Vignon-Clementel IE, Jansen KE, Hughes TJR, Taylor CA (2006) A coupled momentum method for modeling blood flow in three-dimensional deformable arteries. Comput Methods Appl Mech Eng 195(41–43):5685–5706. https://doi.org/10.1016/j.cma.2005.11.011

    Article  Google Scholar 

  12. Kim HJ, Vignon-Clementel IE, Coogan JS, Figueroa CA, Jansen KE, Taylor CA (2010) Patient-specific modeling of blood flow and pressure in human coronary arteries. Ann Biomed Eng 38(10):3195–3209. https://doi.org/10.1007/s10439-010-0083-6

    Article  CAS  PubMed  Google Scholar 

  13. Marsden AL, Esmaily-Moghadam M (2015) Multiscale modeling of cardiovascular flows for clinical decision support. Appl Mech Rev. https://doi.org/10.1115/1.4029909

    Article  Google Scholar 

  14. Shang JK, Esmaily M, Verma A, Reinhartz O, Figliola RS, Hsia T-Y, Feinstein JA, Marsden AL (2019) Patient-specific multiscale modeling of the assisted bidirectional Glenn. Ann Thorac Surg 107(4):1232–1239. https://doi.org/10.1016/j.athoracsur.2018.10.024

    Article  PubMed  Google Scholar 

  15. Vignon-Clementel IE, Figueroa CA, Jansen KE, Taylor CA (2010) Outflow boundary conditions for 3D simulations of non-periodic blood flow and pressure fields in deformable arteries. Comput Methods Biomech Biomed Engin 13(5):625–640. https://doi.org/10.1080/10255840903413565

    Article  CAS  PubMed  Google Scholar 

  16. Apostolidis AJ, Beris AN (2014) Modeling of the blood rheology in steady-state shear flows. J Rheol 58(3):607–633. https://doi.org/10.1122/1.4866296

    Article  CAS  Google Scholar 

  17. Bird B, Stewart W, Lightfoot E (2006) Transport phenomena. Wiley

    Google Scholar 

  18. Anand M, Kwack J, Masud A (2013) A new generalized Oldroyd-B model for blood flow in complex geometries. Int J Eng Sci 72:78–88. https://doi.org/10.1016/j.ijengsci.2013.06.009

    Article  Google Scholar 

  19. Horner JS, Armstrong MJ, Wagner NJ, Beris AN (2018) Investigation of blood rheology under steady and unidirectional large amplitude oscillatory shear. J Rheol 62(2):577–591. https://doi.org/10.1122/1.5017623

    Article  CAS  Google Scholar 

  20. Johnston BM, Johnston PR, Corney S, Kilpatrick D (2004) Non-Newtonian blood flow in human right coronary arteries: steady state simulations. J Biomech 37(5):709–720. https://doi.org/10.1016/j.jbiomech.2003.09.016

    Article  PubMed  Google Scholar 

  21. Boyd J, Buick JM, Green S (2007) Analysis of the Casson and Carreau-Yasuda non-Newtonian blood models in steady and oscillatory flows using the lattice Boltzmann method. Phys Fluids. https://doi.org/10.1063/1.2772250

    Article  Google Scholar 

  22. Casson N (1959) Flow equation for pigment-oil suspensions of the printing ink-type. Rheol Disperse Syst, 84–104

  23. Chandran KB, Rittgers SE, Yoganathan AP (2012) Biofluid Mechanics. CRC Press. https://doi.org/10.1201/b11709

    Book  Google Scholar 

  24. Doost SN, Zhong L, Su B, Morsi YS (2016) The numerical analysis of non-Newtonian blood flow in human patient-specific left ventricle. Comput Methods Programs Biomed 127:232–247. https://doi.org/10.1016/j.cmpb.2015.12.020

    Article  PubMed  Google Scholar 

  25. Johnston BM, Johnston PR, Corney S, Kilpatrick D (2006) Non-Newtonian blood flow in human right coronary arteries: transient simulations. J Biomech 39(6):1116–1128. https://doi.org/10.1016/j.jbiomech.2005.01.034

    Article  PubMed  Google Scholar 

  26. Morbiducci U, Gallo D, Massai D, Ponzini R, Deriu MA, Antiga L, Redaelli A, Montevecchi FM (2011) On the importance of blood rheology for bulk flow in hemodynamic models of the carotid bifurcation. J Biomech 44(13):2427–2438. https://doi.org/10.1016/j.jbiomech.2011.06.028

    Article  PubMed  Google Scholar 

  27. Quemada D (1981) A rheological model for studying the hematocrit dependence of red cell-red cell and red cell-protein interactions in blood1. Biorheology 18(3–6):501–516. https://doi.org/10.3233/BIR-1981-183-615

    Article  CAS  PubMed  Google Scholar 

  28. Fry BC, Roy TK, Secomb TW (2013) Capillary recruitment in a theoretical model for blood flow regulation in heterogeneous microvessel networks. Physiol Rep. https://doi.org/10.1002/phy2.50

    Article  PubMed  PubMed Central  Google Scholar 

  29. Pries AR, Reglin B, Secomb TW (2001) Structural adaptation of microvascular networks: functional roles of adaptive responses. www.ajpheart.org

  30. Arciero JC, Carlson BE, Secomb TW (2008) Theoretical model of metabolic blood flow regulation: roles of ATP release by red blood cells and conducted responses. Am J Physiol Heart Circ Phys-Iol 295:1562–1571. https://doi.org/10.1152/ajpheart.00261.2008.-A

    Article  Google Scholar 

  31. Carlson BE, Arciero JC, Secomb TW (2008) Theoretical model of blood flow autoregulation: roles of myogenic, shear-dependent, and metabolic responses. Am J Physiol Heart Circul Physiol. https://doi.org/10.1152/ajpheart.00262.2008

    Article  Google Scholar 

  32. Cornelissen AJM, Dankelman J, Vanbavel ED, Spaan JAE, Vanbavel E, Spaan JAE (2002) Balance between myogenic, flow-dependent, and metabolic flow control in coronary arterial tree: a model study. Am J Physiol Heart Circ Physiol 282:2224–2237. https://doi.org/10.1152/ajpheart.00491.2001.-Myogenic

    Article  Google Scholar 

  33. Ursino, M., Colantuoni, A., & Bertuglia, S. (1998). Vasomotion and Blood Flow Regulation in Hamster Skeletal Muscle Microcirculation: A Theoretical and Experimental Study.

  34. Leighton D, Acrivos A (1987) The shear-induced migration of particles in concentrated suspensions. J Fluid Mech 181(1):415. https://doi.org/10.1017/S0022112087002155

    Article  CAS  Google Scholar 

  35. Phillips RJ, Armstrong RC, Brown RA, Graham AL, Abbott JR (1992) A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration. Phys Fluids A 4(1):30–40. https://doi.org/10.1063/1.858498

    Article  CAS  Google Scholar 

  36. Mansour MH, Bressloff NW, Shearman CP (2010) Red blood cell migration in microvessels. Biorheology 47(1):73–93. https://doi.org/10.3233/BIR-2010-0560

    Article  PubMed  Google Scholar 

  37. Biasetti J, Spazzini PG, Hedin U, Gasser TC (2014) Synergy between shear-induced migration and secondary flows on red blood cells transport in arteries: considerations on oxygen transport. J R Soc Interface 11(97):20140403. https://doi.org/10.1098/rsif.2014.0403

    Article  PubMed  PubMed Central  Google Scholar 

  38. Chandran K, Dalal IS, Tatsumi K, Muralidhar K (2020) Numerical simulation of blood flow modeled as a fluid- particulate mixture. J Nonnewton Fluid Mech 285:104383. https://doi.org/10.1016/j.jnnfm.2020.104383

    Article  CAS  Google Scholar 

  39. Ranade VV, Utikar RP (eds) (2022) Multiphase flows for process industries. Wiley. https://doi.org/10.1002/9783527812066

    Book  Google Scholar 

  40. White FM (2009) Fluid Mechanics, 7th edn. Mc-Graw Hill

    Google Scholar 

  41. Schiller L (1933) A drag coefficient correlation. Zeit Ver Deutsch Ing 77:318–320

    Google Scholar 

  42. Jung J, Hassanein A, Lyczkowski RW (2006) Hemodynamic computation using multiphase flow dynamics in a right coronary artery. Ann Biomed Eng 34(3):393–407. https://doi.org/10.1007/s10439-005-9017-0

    Article  PubMed  Google Scholar 

  43. Jung J, Lyczkowski RW, Panchal CB, Hassanein A (2006) Multiphase hemodynamic simulation of pulsatile flow in a coronary artery. J Biomech 39(11):2064–2073. https://doi.org/10.1016/j.jbiomech.2005.06.023

    Article  PubMed  Google Scholar 

  44. Berthier B, Bouzerar R, Legallais C (2002) Blood flow patterns in an anatomically realistic coronary vessel: influence of three different reconstruction methods. J Biomech 35(10):1347–1356. https://doi.org/10.1016/S0021-9290(02)00179-3

    Article  CAS  PubMed  Google Scholar 

  45. Jung J, Hassanein A (2008) Three-phase CFD analytical modeling of blood flow. Med Eng Phys 30(1):91–103. https://doi.org/10.1016/j.medengphy.2006.12.004

    Article  PubMed  Google Scholar 

  46. Qiao Y, Zeng Y, Ding Y, Fan J, Luo K, Zhu T (2019) Numerical simulation of two-phase non-Newtonian blood flow with fluid-structure interaction in aortic dissection. Comput Methods Biomech Biomed Engin 22(6):620–630. https://doi.org/10.1080/10255842.2019.1577398

    Article  PubMed  Google Scholar 

  47. Ling Y, Tang J, Liu H (2021) Numerical investigation of two-phase non-Newtonian blood flow in bifurcate pulmonary arteries with a flow resistant using Eulerian multiphase model. Chem Eng Sci 233:116426. https://doi.org/10.1016/J.CES.2020.116426

    Article  CAS  Google Scholar 

  48. Fontan F, Baudet E (1971) Surgical repair of tricuspid atresia. Thorax 26(3):240–248. https://doi.org/10.1136/thx.26.3.240

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  49. Rao MS, Bhan A, Talwar S, Sharma R, Choudhary SK, Airan B, Saxena A, Kothari SS, Juneja R, Venugopal P (2000) Modified blalock-taussig shunt in neonates: determinants of immediate outcome. Asian Cardiovasc Thorac Ann 8(4):339–343. https://doi.org/10.1177/021849230000800410

    Article  Google Scholar 

  50. Lyras KG, Lee J (2022) A finite volume coupled level set and volume of fluid method with a mass conservation step for simulating two-phase flows. Int J Numer Meth Fluids 94(8):1027–1047. https://doi.org/10.1002/fld.5082

    Article  CAS  Google Scholar 

  51. Gidaspow D, Huang J (2009) Kinetic theory based model for blood flow and its viscosity. Ann Biomed Eng 37(8):1534–1545. https://doi.org/10.1007/s10439-009-9720-3

    Article  PubMed  Google Scholar 

  52. Gidaspow D, Bacelos MS (2018) Kinetic theory based multiphase flow with experimental verification. Rev Chem Eng 34(3):299–318. https://doi.org/10.1515/revce-2016-0044

    Article  CAS  Google Scholar 

  53. Huang J, Lyczkowski RW, Gidaspow D (2009) Pulsatile flow in a coronary artery using multiphase kinetic theory. J Biomech 42(6):743–754. https://doi.org/10.1016/j.jbiomech.2009.01.038

    Article  PubMed  Google Scholar 

  54. Chen T, Liu X, Si B, Feng Y, Zhang H, Jia B, Wang S (2021) Comparison between single-phase flow simulation and multiphase flow simulation of patient-specific total cavopulmonary connection structures assisted by a rotationally symmetric blood pump. Symmetry 13(5):912. https://doi.org/10.3390/sym13050912

    Article  Google Scholar 

  55. Melka B, Gracka M, Adamczyk W, Rojczyk M, Golda A, Nowak AJ, Białecki RA, Ostrowski Z (2018) Multiphase simulation of blood flow within main thoracic arteries of 8-year-old child with coarctation of the aorta. Heat Mass Transf 54(8):2405–2413. https://doi.org/10.1007/s00231-017-2136-y

    Article  Google Scholar 

  56. Wen CY, Yu YH (1966) A generalized method for predicting the minimum fluidization velocity. AIChE J 12(3):610–612. https://doi.org/10.1002/aic.690120343

    Article  CAS  Google Scholar 

  57. Syamlal M (1987) The particle-particle drag term in a multiparticle model of fluidization. topical report DOE/MC 21353 2373 NTIS/DE 87006500

  58. Bouchnita A, Belyaev AV, Volpert V (2021) Multiphase continuum modeling of thrombosis in aneurysms and recirculation zones. Phys Fluids. https://doi.org/10.1063/5.0057393

    Article  Google Scholar 

  59. Longest PW, Kleinstreuer C, Buchanan JR (2004) Efficient computation of micro-particle dynamics including wall effects. Comput Fluids 33(4):577–601. https://doi.org/10.1016/j.compfluid.2003.06.002

    Article  Google Scholar 

  60. Biglarian M, Firoozabadi B, Saidi MS (2021) Atheroprone sites of coronary artery bifurcation: Effect of heart motion on hemodynamics-dependent monocytes deposition. Comput Biol Med 133:104411. https://doi.org/10.1016/j.compbiomed.2021.104411

    Article  PubMed  Google Scholar 

  61. De Gruttola S, Boomsma K, Poulikakos D (2005) Computational simulation of a non-Newtonian model of the blood separation process. Artif Organs 29(12):949–959. https://doi.org/10.1111/j.1525-1594.2005.00164.x

    Article  PubMed  Google Scholar 

  62. Childress EM, Kleinstreuer C, Kennedy AS (2012) A new catheter for tumor-targeting with radioactive microspheres in representative hepatic artery systems—part ii: solid tumor-targeting in a patient-inspired hepatic artery system. J Biomech Eng. https://doi.org/10.1115/1.4006685

    Article  PubMed  Google Scholar 

  63. Kleinstreuer C, Basciano CA, Childress EM, Kennedy AS (2012) A new catheter for tumor targeting with radioactive microspheres in representative hepatic artery systems. Part I: impact of catheter presence on local blood flow and microsphere delivery. J Biomech Eng. https://doi.org/10.1115/1.4006684

    Article  PubMed  Google Scholar 

  64. Hirt CW, Nichols BD (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39(1):201–225. https://doi.org/10.1016/0021-9991(81)90145-5

    Article  Google Scholar 

  65. Osher S, Fedkiw RP (2001) Level set methods: an overview and some recent results. J Comput Phys 169(2):463–502. https://doi.org/10.1006/jcph.2000.6636

    Article  CAS  Google Scholar 

  66. Jacqmin D (1999) Calculation of two-phase Navier-Stokes flows using phase-field modeling. J Comput Phys 155(1):96–127. https://doi.org/10.1006/jcph.1999.6332

    Article  Google Scholar 

  67. Tryggvason G, Bunner B, Esmaeeli A, Juric D, Al-Rawahi N, Tauber W, Han J, Nas S, Jan Y-J (2001) A front-tracking method for the computations of multiphase flow. J Comput Phys 169(2):708–759. https://doi.org/10.1006/jcph.2001.6726

    Article  CAS  Google Scholar 

  68. Peskin CS (2002) The immersed boundary method. Acta Numer 11:479–517. https://doi.org/10.1017/S0962492902000077

    Article  Google Scholar 

  69. Doddi SK, Bagchi P (2008) Lateral migration of a capsule in a plane Poiseuille flow in a channel. Int J Multiph Flow 34(10):966–986. https://doi.org/10.1016/j.ijmultiphaseflow.2008.03.002

    Article  CAS  Google Scholar 

  70. Eggleton CD, Popel AS (1998) Large deformation of red blood cell ghosts in a simple shear flow. Phys Fluids 10(8):1834–1845. https://doi.org/10.1063/1.869703

    Article  CAS  Google Scholar 

  71. Balogh P, Bagchi P (2017) A computational approach to modeling cellular-scale blood flow in complex geometry. J Comput Phys 334:280–307. https://doi.org/10.1016/j.jcp.2017.01.007

    Article  Google Scholar 

  72. Krüger T, Kusumaatmaja H, Kuzmin A, Shardt O, Silva G, Viggen EM (2017) The lattice Boltzmann method. Springer International Publishing. https://doi.org/10.1007/978-3-319-44649-3

    Book  Google Scholar 

  73. Hernández-Rodríguez M, C Rosales-Hernández M, E Mendieta-Wejebe J, Martínez-Archundia M, Correa Basurto J (2016) Current tools and methods in molecular dynamics (MD) simulations for drug design. Curr Med Chem 23(34):3909–3924. https://doi.org/10.2174/0929867323666160530144742

    Article  CAS  PubMed  Google Scholar 

  74. Bhatnagar PL, Gross EP, Krook M (1954) A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys Rev 94(3):511–525. https://doi.org/10.1103/PhysRev.94.511

    Article  CAS  Google Scholar 

  75. Kumar K (1967) The chapman-enskog solution of the boltzmann equation: a reformulation in terms of irreducible tensors and matrices. Aust J Phys 20(3):205–252

    Article  Google Scholar 

  76. Peskin CS (1977) Numerical analysis of blood flow in the heart. J Comput Phys 25(3):220–252. https://doi.org/10.1016/0021-9991(77)90100-0

    Article  Google Scholar 

  77. Sun C, Munn LL (2008) Lattice-Boltzmann simulation of blood flow in digitized vessel networks. Comput Math Appl 55(7):1594–1600. https://doi.org/10.1016/j.camwa.2007.08.019

    Article  PubMed  PubMed Central  Google Scholar 

  78. Zhang J, Johnson PC, Popel AS (2007) An immersed boundary lattice Boltzmann approach to simulate deformable liquid capsules and its application to microscopic blood flows. Phys Biol 4(4):285–295. https://doi.org/10.1088/1478-3975/4/4/005

    Article  CAS  PubMed  Google Scholar 

  79. Zhang J, Johnson PC, Popel AS (2009) Effects of erythrocyte deformability and aggregation on the cell free layer and apparent viscosity of microscopic blood flows. Microvasc Res 77(3):265–272. https://doi.org/10.1016/j.mvr.2009.01.010

    Article  PubMed  PubMed Central  Google Scholar 

  80. Enjalbert R, Hardman D, Krüger T, Bernabeu MO (2021) Compressed vessels bias red blood cell partitioning at bifurcations in a hematocrit-dependent manner: Implications in tumor blood flow. Proc Natl Acad Sci. https://doi.org/10.1073/pnas.2025236118

    Article  PubMed  PubMed Central  Google Scholar 

  81. Zhou Q, Perovic T, Fechner I, Edgar LT, Hoskins PR, Gerhardt H, Krüger T, Bernabeu MO (2021) Association between erythrocyte dynamics and vessel remodelling in developmental vascular networks. J R Soc Interface 18(179):20210113. https://doi.org/10.1098/rsif.2021.0113

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  82. Rashidi Y, Simionato G, Zhou Q, John T, Kihm A, Bendaoud M, Krüger T, Bernabeu MO, Kaestner L, Laschke MW, Menger MD, Wagner C, Darras A (2023) Red blood cell lingering modulates hematocrit distribution in the microcirculation. Biophys J 122(8):1526–1537. https://doi.org/10.1016/j.bpj.2023.03.020

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  83. Karniadakis GE, Kevrekidis IG, Lu L, Perdikaris P, Wang S, Yang L (2021) Physics-informed machine learning. Nature Rev Phys 3(6):422–440. https://doi.org/10.1038/s42254-021-00314-5

    Article  Google Scholar 

  84. Wang L, Dong D, Tian F-B (2022) Fast prediction of blood flow in stenosed arteries using machine learning and immersed boundary-lattice Boltzmann method. Front Physiol. https://doi.org/10.3389/fphys.2022.953702

    Article  PubMed  PubMed Central  Google Scholar 

  85. Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Not R Astron Soc 181(3):375–389. https://doi.org/10.1093/mnras/181.3.375

    Article  Google Scholar 

  86. Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. Astron J 82:1013. https://doi.org/10.1086/112164

    Article  Google Scholar 

  87. Monaghan JJ (1992) Smoothed particle hydrodynamics. Ann Rev Astron Astrophys 30(1):543–574. https://doi.org/10.1146/annurev.aa.30.090192.002551

    Article  Google Scholar 

  88. Liu MB, Liu GR (2010) Smoothed particle hydrodynamics (SPH): an overview and recent developments. Arch Comput Methods Eng 17(1):25–76. https://doi.org/10.1007/s11831-010-9040-7

    Article  CAS  Google Scholar 

  89. Violeau D, Rogers BD (2016) Smoothed particle hydrodynamics (SPH) for free-surface flows: past, present and future. J Hydraul Res 54(1):1–26. https://doi.org/10.1080/00221686.2015.1119209

    Article  Google Scholar 

  90. Ye T, Pan D, Huang C, Liu M (2019) Smoothed particle hydrodynamics (SPH) for complex fluid flows: recent developments in methodology and applications. Phys Fluids. https://doi.org/10.1063/1.5068697

    Article  Google Scholar 

  91. Lind SJ, Rogers BD, Stansby PK (2020) Review of smoothed particle hydrodynamics: towards converged Lagrangian flow modelling. Proc R Soc A: Math Phys Eng Sci. https://doi.org/10.1098/rspa.2019.0801

    Article  Google Scholar 

  92. Pozorski J, Olejnik M (2023) Smoothed particle hydrodynamics modelling of multiphase flows: an overview. Acta Mech. https://doi.org/10.1007/s00707-023-03763-4

    Article  Google Scholar 

  93. Becker M, Teschner M (2007) Weakly compressible SPH for free surface flows, Proceedings of the 2007 ACM SIGGRAPH/Eurographics symposium on computer animation, San Diego, CA, USA (2007), pp 1–8

  94. Monaghan JJ (1994) Simulating free surface flows with SPH. J Comput Phys 110(2):399–406. https://doi.org/10.1006/jcph.1994.1034

    Article  Google Scholar 

  95. Antuono M, Colagrossi A, Marrone S, Molteni D (2010) Free-surface flows solved by means of SPH schemes with numerical diffusive terms. Comput Phys Commun 181(3):532–549. https://doi.org/10.1016/j.cpc.2009.11.002

    Article  CAS  Google Scholar 

  96. Cummins SJ, Rudman M (1999) An SPH projection method. J Comput Phys 152(2):584–607. https://doi.org/10.1006/jcph.1999.6246

    Article  CAS  Google Scholar 

  97. Hosseini SM, Feng JJ (2009) A particle-based model for the transport of erythrocytes in capillaries. Chem Eng Sci 64(22):4488–4497. https://doi.org/10.1016/j.ces.2008.11.028

    Article  CAS  Google Scholar 

  98. Chui Y-P, Heng P-A (2010) A meshless rheological model for blood-vessel interaction in endovascular simulation. Prog Biophys Mol Biol 103(2–3):252–261. https://doi.org/10.1016/j.pbiomolbio.2010.09.003

    Article  PubMed  Google Scholar 

  99. Gholami B, Comerford A, Ellero M (2014) A multiscale SPH particle model of the near-wall dynamics of leukocytes in flow. Int J Numer Methods Biomed Eng 30(1):83–102. https://doi.org/10.1002/cnm.2591

    Article  Google Scholar 

  100. Polwaththe-Gallage H-N, Saha SC, Sauret E, Flower R, Senadeera W, Gu Y (2016) SPH-DEM approach to numerically simulate the deformation of three-dimensional RBCs in non-uniform capillaries. Biomed Eng Online 15(S2):161. https://doi.org/10.1186/s12938-016-0256-0

    Article  PubMed  PubMed Central  Google Scholar 

  101. Soleimani M, Sahraee S, Wriggers P (2019) Red blood cell simulation using a coupled shell–fluid analysis purely based on the SPH method. Biomech Model Mechanobiol 18(2):347–359. https://doi.org/10.1007/s10237-018-1085-9

    Article  PubMed  Google Scholar 

  102. Topalovic M, Nikolic A, Milovanovic V, Vulovic S, Ivanovic M (2022) Smoothed particle hydrodynamics for blood flow analysis: development of particle lifecycle algorithm. Comput Part Mech 9(6):1119–1135. https://doi.org/10.1007/s40571-021-00454-6

    Article  Google Scholar 

  103. Monteleone A, Viola A, Napoli E, Burriesci G (2023) Modelling of thrombus formation using smoothed particle hydrodynamics method. PLoS ONE 18(2):e0281424. https://doi.org/10.1371/journal.pone.0281424

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  104. Wang F, Xu S, Jiang D, Zhao B, Dong X, Zhou T, Luo X (2021) Particle hydrodynamic simulation of thrombus formation using velocity decay factor. Comput Methods Progr Biomed 207:106173. https://doi.org/10.1016/j.cmpb.2021.106173

    Article  Google Scholar 

  105. Hosseini SM, Feng JJ (2012) How Malaria parasites reduce the deformability of infected red blood cells. Biophys J 103(1):1–10. https://doi.org/10.1016/j.bpj.2012.05.026

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  106. Wu T, Feng JJ (2013) Simulation of malaria-infected red blood cells in microfluidic channels: passage and blockage. Biomicrofluidics. https://doi.org/10.1063/1.4817959

    Article  PubMed  PubMed Central  Google Scholar 

  107. Español P, Revenga M (2003) Smoothed dissipative particle dynamics. Phys Rev E 67(2):026705. https://doi.org/10.1103/PhysRevE.67.026705

    Article  CAS  Google Scholar 

  108. Müller K, Fedosov DA, Gompper G (2015) Smoothed dissipative particle dynamics with angular momentum conservation. J Comput Phys 281:301–315. https://doi.org/10.1016/j.jcp.2014.10.017

    Article  Google Scholar 

  109. Fedosov DA, Gompper G (2014) White blood cell margination in microcirculation. Soft Matter 10(17):2961–2970. https://doi.org/10.1039/C3SM52860J

    Article  CAS  PubMed  Google Scholar 

  110. Müller K, Fedosov DA, Gompper G (2014) Margination of micro- and nano-particles in blood flow and its effect on drug delivery. Sci Rep 4(1):4871. https://doi.org/10.1038/srep04871

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  111. Müller K, Fedosov DA, Gompper G (2016) Understanding particle margination in blood flow: a step toward optimized drug delivery systems. Med Eng Phys 38(1):2–10. https://doi.org/10.1016/j.medengphy.2015.08.009

    Article  PubMed  Google Scholar 

  112. Caballero A, Mao W, Liang L, Oshinski J, Primiano C, McKay R, Kodali S, Sun W (2017) Modeling left ventricular blood flow using smoothed particle hydrodynamics. Cardiovasc Eng Technol 8(4):465–479. https://doi.org/10.1007/s13239-017-0324-z

    Article  PubMed  PubMed Central  Google Scholar 

  113. Shahriari S, Kadem L, Rogers BD, Hassan I (2012) Smoothed particle hydrodynamics method applied to pulsatile flow inside a rigid two-dimensional model of left heart cavity. Int J Numer Methods Biomed Eng 28(11):1121–1143. https://doi.org/10.1002/cnm.2482

    Article  CAS  Google Scholar 

  114. Shahriari S, Maleki H, Hassan I, Kadem L (2012) Evaluation of shear stress accumulation on blood components in normal and dysfunctional bileaflet mechanical heart valves using smoothed particle hydrodynamics. J Biomech 45(15):2637–2644. https://doi.org/10.1016/j.jbiomech.2012.08.009

    Article  CAS  PubMed  Google Scholar 

  115. Biffi B, Gritti M, Grasso A, Milano EG, Fontana M, Alkareef H, Davar J, Jeetley P, Whelan C, Anderson S, Lorusso D, Sauvage E, Maria Bosi G, Schievano S, Capelli C (2019) A workflow for patient-specific fluid–structure interaction analysis of the mitral valve: a proof of concept on a mitral regurgitation case. Med Eng Phys 74:153–161. https://doi.org/10.1016/j.medengphy.2019.09.020

    Article  PubMed  Google Scholar 

  116. Caballero A, Mao W, McKay R, Primiano C, Hashim S, Sun W (2018) New insights into mitral heart valve prolapse after chordae rupture through fluid–structure interaction computational modeling. Sci Rep 8(1):17306. https://doi.org/10.1038/s41598-018-35555-5

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  117. Mao W, Caballero A, McKay R, Primiano C, Sun W (2017) Fully-coupled fluid-structure interaction simulation of the aortic and mitral valves in a realistic 3D left ventricle model. PLoS ONE 12(9):e0184729. https://doi.org/10.1371/journal.pone.0184729

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  118. Mao W, Li K, Sun W (2016) Fluid-structure interaction study of transcatheter aortic valve dynamics using smoothed particle hydrodynamics. Cardiovasc Eng Technol 7(4):374–388. https://doi.org/10.1007/s13239-016-0285-7

    Article  PubMed  PubMed Central  Google Scholar 

  119. Toma M, Jensen MØ, Einstein DR, Yoganathan AP, Cochran RP, Kunzelman KS (2016) Fluid-structure interaction analysis of papillary muscle forces using a comprehensive mitral valve model with 3D chordal structure. Ann Biomed Eng 44(4):942–953. https://doi.org/10.1007/s10439-015-1385-5

    Article  PubMed  Google Scholar 

  120. Hoogerbrugge PJ, Koelman JMVA (1992) Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Europhys Lett (EPL) 19(3):155–160. https://doi.org/10.1209/0295-5075/19/3/001

    Article  Google Scholar 

  121. Español P, Warren P (1995) Statistical mechanics of dissipative particle dynamics. Europhys Lett (EPL) 30(4):191–196. https://doi.org/10.1209/0295-5075/30/4/001

    Article  Google Scholar 

  122. Español P, Warren PB (2017) Perspective: dissipative particle dynamics. J Chem Phys. https://doi.org/10.1063/1.4979514

    Article  PubMed  PubMed Central  Google Scholar 

  123. Groot RD, Warren PB (1997) Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation. J Chem Phys 107(11):4423–4435. https://doi.org/10.1063/1.474784

    Article  CAS  Google Scholar 

  124. Liu MB, Liu GR, Zhou LW, Chang JZ (2015) Dissipative particle dynamics (DPD): an overview and recent developments. Arch Comput Methods Eng 22(4):529–556. https://doi.org/10.1007/s11831-014-9124-x

    Article  Google Scholar 

  125. Moeendarbary E, Ng TY, Zangeneh M (2009) Dissipative particle dynamics: introduction, methodology and complex fluid applications: a review. Int J Appl Mecha 01(04):737–763. https://doi.org/10.1142/S1758825109000381

    Article  Google Scholar 

  126. Santo KP, Neimark AV (2021) Dissipative particle dynamics simulations in colloid and Interface science: a review. Adv Coll Interface Sci 298:102545. https://doi.org/10.1016/j.cis.2021.102545

    Article  CAS  Google Scholar 

  127. Pivkin IV, Caswell B, Karniadakis GE (2010) Dissipative particle dynamics. In: Lipkowitz KB (ed) Reviews in computational chemistry, vol 27. https://doi.org/10.1002/9780470890905.ch2

  128. Allen MP, Tildesley DJ (2017) Computer simulation of liquids. Oxford University Press, Oxford. https://doi.org/10.1093/oso/9780198803195.001.0001

    Book  Google Scholar 

  129. Dzwinel W, Boryczko K, Yuen DA (2003) A discrete-particle model of blood dynamics in capillary vessels. J Colloid Interface Sci 258(1):163–173. https://doi.org/10.1016/S0021-9797(02)00075-9

    Article  CAS  PubMed  Google Scholar 

  130. Español P (1998) Fluid particle model. Phys Rev E 57(3):2930–2948. https://doi.org/10.1103/PhysRevE.57.2930

    Article  Google Scholar 

  131. Boryczko K, Dzwinel W, YuenD A (2003) Dynamical clustering of red blood cells in capillary vessels. J Mol Model 9(1):16–33. https://doi.org/10.1007/s00894-002-0105-x

    Article  PubMed  Google Scholar 

  132. Boryczko K, Dzwinel W, Yuen DA (2004) Modeling fibrin aggregation in blood flow with discrete-particles. Comput Methods Progr Biomed 75(3):181–194. https://doi.org/10.1016/j.cmpb.2004.02.001

    Article  Google Scholar 

  133. Pivkin IV, Karniadakis GE (2008) Accurate coarse-grained modeling of red blood cells. Phys Rev Lett 101(11):118105. https://doi.org/10.1103/PhysRevLett.101.118105

    Article  CAS  PubMed  Google Scholar 

  134. Fedosov DA, Caswell B, Karniadakis GE (2010) A multiscale red blood cell model with accurate mechanics, rheology, and dynamics. Biophys J 98(10):2215–2225. https://doi.org/10.1016/j.bpj.2010.02.002

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  135. Pan W, Caswell B, Karniadakis GE (2010) A low-dimensional model for the red blood cell. Soft Matter 6(18):4366. https://doi.org/10.1039/c0sm00183j

    Article  CAS  Google Scholar 

  136. Fedosov DA, Caswell B, Suresh S, Karniadakis GE (2011) Quantifying the biophysical characteristics of Plasmodium-falciparum -parasitized red blood cells in microcirculation. Proc Natl Acad Sci 108(1):35–39. https://doi.org/10.1073/pnas.1009492108

    Article  PubMed  Google Scholar 

  137. Fedosov DA, Lei H, Caswell B, Suresh S, Karniadakis GE (2011) Multiscale modeling of red blood cell mechanics and blood flow in malaria. PLoS Comput Biol 7(12):e1002270. https://doi.org/10.1371/journal.pcbi.1002270

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  138. Fedosov DA, Pan W, Caswell B, Gompper G, Karniadakis GE (2011) Predicting human blood viscosity in silico. Proc Natl Acad Sci 108(29):11772–11777. https://doi.org/10.1073/pnas.1101210108

    Article  PubMed  PubMed Central  Google Scholar 

  139. Peng Z, Li X, Pivkin IV, Dao M, Karniadakis GE, Suresh S (2013) Lipid bilayer and cytoskeletal interactions in a red blood cell. Proc Natl Acad Sci 110(33):13356–13361. https://doi.org/10.1073/pnas.1311827110

    Article  PubMed  PubMed Central  Google Scholar 

  140. Gao C, Zhang P, Marom G, Deng Y, Bluestein D (2017) Reducing the effects of compressibility in DPD-based blood flow simulations through severe stenotic microchannels. J Comput Phys 335:812–827. https://doi.org/10.1016/j.jcp.2017.01.062

    Article  CAS  Google Scholar 

  141. Hoque SZ, Anand DV, Patnaik BSV (2022) A dissipative particle dynamics simulation of a pair of red blood cells in flow through a symmetric and an asymmetric bifurcated microchannel. Comput Part Mech 9(6):1219–1231. https://doi.org/10.1007/s40571-021-00453-7

    Article  Google Scholar 

  142. Ye T, Phan-Thien N, Khoo BC, Lim CT (2014) Dissipative particle dynamics simulations of deformation and aggregation of healthy and diseased red blood cells in a tube flow. Phys Fluids. https://doi.org/10.1063/1.4900952

    Article  Google Scholar 

  143. Chang H-Y, Li X, Li H, Karniadakis GE (2016) MD/DPD multiscale framework for predicting morphology and stresses of red blood cells in health and disease. PLoS Comput Biol 12(10):e1005173. https://doi.org/10.1371/journal.pcbi.1005173

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  144. Hoque SZ, Anand DV, Patnaik BSV (2018) The dynamics of a healthy and infected red blood cell in flow through constricted channels: a DPD simulation. Int J Numer Methods Biomed Eng. https://doi.org/10.1002/cnm.3105

    Article  Google Scholar 

  145. Lei H, Karniadakis GE (2012) Quantifying the rheological and hemodynamic characteristics of sickle cell anemia. Biophys J 102(2):185–194. https://doi.org/10.1016/j.bpj.2011.12.006

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  146. Lei H, Karniadakis GE (2013) Probing vasoocclusion phenomena in sickle cell anemia via mesoscopic simulations. Proc Natl Acad Sci 110(28):11326–11330. https://doi.org/10.1073/pnas.1221297110

    Article  PubMed  PubMed Central  Google Scholar 

  147. Li X, Du E, Lei H, Tang Y-H, Dao M, Suresh S, Karniadakis GE (2016) Patient-specific blood rheology in sickle-cell anaemia. Interface Focus 6(1):20150065. https://doi.org/10.1098/rsfs.2015.0065

    Article  PubMed  PubMed Central  Google Scholar 

  148. Chang H-Y, Li X, Karniadakis GE (2017) Modeling of biomechanics and biorheology of red blood cells in type 2 diabetes mellitus. Biophys J 113(2):481–490. https://doi.org/10.1016/j.bpj.2017.06.015

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  149. Chang H-Y, Yazdani A, Li X, Douglas KAA, Mantzoros CS, Karniadakis GE (2018) Quantifying platelet margination in diabetic blood flow. Biophys J 115(7):1371–1382. https://doi.org/10.1016/j.bpj.2018.08.031

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  150. Deng Y, Papageorgiou DP, Li X, Perakakis N, Mantzoros CS, Dao M, Karniadakis GE (2020) Quantifying fibrinogen-dependent aggregation of red blood cells in type 2 diabetes mellitus. Biophys J 119(5):900–912. https://doi.org/10.1016/j.bpj.2020.07.026

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  151. Han K, Ma S, Sun J, Xu M, Qi X, Wang S, Li L, Li X (2023) In silico modeling of patient-specific blood rheology in type 2 diabetes mellitus. Biophys J 122(8):1445–1458. https://doi.org/10.1016/j.bpj.2023.03.010

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  152. Hareendranath S, Sathian SP (2023) Dynamic response of red blood cells in health and disease. Soft Matter 19(6):1219–1230. https://doi.org/10.1039/D2SM01090A

    Article  CAS  PubMed  Google Scholar 

  153. Filipovic N, Kojic M, Tsuda A (2008) Modelling thrombosis using dissipative particle dynamics method. Philos Trans R Soci A Math Phys Eng Sci 366(1879):3265–3279. https://doi.org/10.1098/rsta.2008.0097

    Article  CAS  Google Scholar 

  154. Zhang P, Gao C, Zhang N, Slepian MJ, Deng Y, Bluestein D (2014) Multiscale particle-based modeling of flowing platelets in blood plasma using dissipative particle dynamics and coarse grained molecular dynamics. Cell Mol Bioeng 7(4):552–574. https://doi.org/10.1007/s12195-014-0356-5

    Article  CAS  PubMed  Google Scholar 

  155. Yazdani A, Karniadakis GE (2016) Sub-cellular modeling of platelet transport in blood flow through microchannels with constriction. Soft Matter 12(19):4339–4351. https://doi.org/10.1039/C6SM00154H

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  156. Han C, Zhang P, Zhu Y, Cong G, Kozloski JR, Yang CC, Zhang L, Deng Y (2022) Scalable multiscale modeling of platelets with 100 million particles. J Supercomput 78(18):19707–19724. https://doi.org/10.1007/s11227-022-04648-4

    Article  Google Scholar 

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Acknowledgements

RG acknowledges the financial support provided by Indian Council of Medical Research (ICMR) vide Grant No. 2021-13210. MS acknowledges the Prime Minister’s Research Fellowship (ID: 1901276) from Ministry of Education.

Funding

The funding information was funded by the Indian Council of Medical Research, 2021-13210 and Prime Ministers' Research Fellowship (PMRF) India, 1901276.

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Authors and Affiliations

  1. Department of Chemical Engineering, Indian Institute of Technology Guwahati, Assam, 781039, India

    Raghvendra Gupta, Amit Kumar & Mudrika Singhal

  2. Centre for Sustainable Polymers, Indian Institute of Technology Guwahati, Assam, 781039, India

    Raghvendra Gupta & Amit Kumar

  3. Jyoti and Bhupat Mehta School of Health Sciences and Technology, Indian Institute of Technology Guwahati, Assam, 781039, India

    Raghvendra Gupta

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  1. Raghvendra Gupta
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Appendix

Appendix

See Table 2.

Table 2: List (not exhaustive) of software packages offering different modelling capabilities.
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Gupta, R., Kumar, A. & Singhal, M. A Critical Review of Multiphase Modelling of Blood Flow in Human Cardiovascular System. J Indian Inst Sci (2024). https://doi.org/10.1007/s41745-024-00430-y

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