Review

Mathematical methods for modeling the microcirculation

  • Received: 02 March 2017 Accepted: 09 May 2017 Published: 07 June 2017
  • The microcirculation plays a major role in maintaining homeostasis in the body. Alterations or dysfunctions of the microcirculation can lead to several types of serious diseases. It is not surprising, then, that the microcirculation has been an object of intense theoretical and experimental study over the past few decades. Mathematical approaches offer a valuable method for quantifying the relationships between various mechanical, hemodynamic, and regulatory factors of the microcirculation and the pathophysiology of numerous diseases. This work provides an overview of several mathematical models that describe and investigate the many different aspects of the microcirculation, including geometry of the vascular bed, blood flow in the vascular networks, solute transport and delivery to the surrounding tissue, and vessel wall mechanics under passive and active stimuli. Representing relevant phenomena across multiple spatial scales remains a major challenge in modeling the microcirculation. Nevertheless, the depth and breadth of mathematical modeling with applications in the microcirculation is demonstrated in this work. A special emphasis is placed on models of the retinal circulation, including models that predict the influence of ocular hemodynamic alterations with the progression of ocular diseases such as glaucoma.

    Citation: Julia C. Arciero, Paola Causin, Francesca Malgaroli. Mathematical methods for modeling the microcirculation[J]. AIMS Biophysics, 2017, 4(3): 362-399. doi: 10.3934/biophy.2017.3.362

    Related Papers:

  • The microcirculation plays a major role in maintaining homeostasis in the body. Alterations or dysfunctions of the microcirculation can lead to several types of serious diseases. It is not surprising, then, that the microcirculation has been an object of intense theoretical and experimental study over the past few decades. Mathematical approaches offer a valuable method for quantifying the relationships between various mechanical, hemodynamic, and regulatory factors of the microcirculation and the pathophysiology of numerous diseases. This work provides an overview of several mathematical models that describe and investigate the many different aspects of the microcirculation, including geometry of the vascular bed, blood flow in the vascular networks, solute transport and delivery to the surrounding tissue, and vessel wall mechanics under passive and active stimuli. Representing relevant phenomena across multiple spatial scales remains a major challenge in modeling the microcirculation. Nevertheless, the depth and breadth of mathematical modeling with applications in the microcirculation is demonstrated in this work. A special emphasis is placed on models of the retinal circulation, including models that predict the influence of ocular hemodynamic alterations with the progression of ocular diseases such as glaucoma.


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    [1] Tuma R, Duran WN, Ley K (2008) Microcirculation. 2Eds., Elsevier.
    [2] Gutterman DD, Chabowski DS, Kadlec AO, et al. (2016) The human microcirculation: regulation of flow and beyond. Circ Res 118: 157–172. doi: 10.1161/CIRCRESAHA.115.305364
    [3] Mortillaro N (1983) The physiology and pharmacology of the microcirculation. Elsevier.
    [4] Dalkara T (2015) Cerebral Microcirculation: An Introduction, In: Lanzer P, Editor, Springer Berlin Heidelberg, 655–680.
    [5] Navar LG, Arendshorst WJ, Pallone TL, et al. (2011) The renal microcirculation. Compr Physiol 9: 550–683.
    [6] Kvietys PR (2010) The gastrointestinal circulation. Morgan & Claypool Life Sciences.
    [7] Ivanov KP (2013) Circulation in the lungs and microcirculation in the alveoli. Respir Physiol Neurobiol 187: 26–30. doi: 10.1016/j.resp.2013.02.022
    [8] Murray A, Dinsdale G (2016) Imaging the microcirculation. Microcirculation 23: 335–336. doi: 10.1111/micc.12282
    [9] Leahy M (2012) Microcirculation Imaging. Wiley-Blackwell.
    [10] Eriksson S, Nilsson J, Sturesson C (2014) Non-invasive imaging of microcirculation: a technology review. Med Devices 7: 445–452.
    [11] Secomb TW, Beard DA, Frisbee JC, et al. (2008) The role of theoretical modeling in microcirculation research. Microcirculation 15: 693–698. doi: 10.1080/10739680802349734
    [12] Lee J, Smith NP (2008) Theoretical modeling in hemodynamics of microcirculation. Microcirculation 15: 699–714. doi: 10.1080/10739680802229589
    [13] Popel AS, Johnson PC (2005) Microcirculation and hemorheology. Annu Rev Fluid Mech 37: 43–69. doi: 10.1146/annurev.fluid.37.042604.133933
    [14] Gompper G, Fedosov DA (2016) Modeling microcirculatory blood flow: current state and future perspectives. Wiley Interdiscip Rev Syst Biol Med 8: 157–168. doi: 10.1002/wsbm.1326
    [15] Secomb TW (2017) Blood flow in the microcirculation. Annu Rev Fluid Mech 49: 443–461. doi: 10.1146/annurev-fluid-010816-060302
    [16] Liu D, Wood NB, Witt N, et al. (2009) Computational analysis of oxygen transport in the retinal arterial network. Curr Eye Res 34: 945–956. doi: 10.3109/02713680903230079
    [17] Gould IG, Linninger AA (2015) Hematocrit distribution and tissue oxygenation in large microcirculatory networks. Microcirculation 22: 1–18. doi: 10.1111/micc.12156
    [18] Reichold J, Stampanoni M, Keller AL, et al. (2009) Vascular graph model to simulate the cerebral blood flow in realistic vascular networks. J Cereb Blood Flow Metab 29: 1429–1443. doi: 10.1038/jcbfm.2009.58
    [19] Guibert R, Fonta C, Plouraboué F (2010) A new approach to model confined suspensions flows in complex networks: application to blood flow. Transport Porous Med 83: 171–194. doi: 10.1007/s11242-009-9492-0
    [20] Muller LO, Toro EF (2014) Enhanced global mathematical model for studying cerebral venous blood flow. J Biomech 47: 3361–3372. doi: 10.1016/j.jbiomech.2014.08.005
    [21] Ursino M, Lodi CA (1997) A simple mathematical model of the interaction between intracranial pressure and cerebral hemodynamics. J Appl Physiol 82: 1256–1269.
    [22] 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 Physiol 295: H1562–H1571. doi: 10.1152/ajpheart.00261.2008
    [23] Takahashi T, Nagaoka T, Yanagida H, et al. (2009) A mathematical model for the distribution of hemodynamic parameters in the human retinal microvascular network. J Biorheol 23: 77–86. doi: 10.1007/s12573-009-0012-1
    [24] Gabryś E, Rybaczuk M, Kędzia A (2005) Fractal models of circulatory system. Symmetrical and asymmetrical approach comparison. Chaos, Solitons & Fractals 24: 707–715.
    [25] Olufsen MS, Peskin CS, Kim WY, et al. (2000) Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann Biomed Eng 28: 1281–1299. doi: 10.1114/1.1326031
    [26] David T, Alzaidi S, Farr H (2009) Coupled autoregulation models in the cerebro-vasculature. J Eng Math 64: 403–415. doi: 10.1007/s10665-009-9274-2
    [27] Ganesan P, He S, Xu H (2011) Development of an image-based model for capillary vasculature of retina. Comput Meth Prog Bio 102: 35–46. doi: 10.1016/j.cmpb.2010.12.009
    [28] Schröder S, Schmid-Schönbein GW, Schmid-Schönbein H, et al. (1990) A method for recording the network topology of human retinal vessels. Klin Monbl Augenheilkd 197: 33–39. doi: 10.1055/s-2008-1046240
    [29] Causin P, Guidoboni G, Malgaroli F, et al. (2016) Blood flow mechanics and oxygen transport and delivery in the retinal microcirculation: multiscale mathematical modeling and numerical simulation. Biomech Model Mechanobiol 15: 525–542. doi: 10.1007/s10237-015-0708-7
    [30] Pan Q, Wang R, Reglin B, et al. (2014) A one-dimensional mathematical model for studying the pulsatile flow in microvascular networks. J Biomech Eng 136: 011009.
    [31] Linninger AA, Gould IG, Marinnan T, et al. (2013) Cerebral microcirculation and oxygen tension in the human secondary cortex. Ann Biomed Eng 41: 2264–2284. doi: 10.1007/s10439-013-0828-0
    [32] Gould I, Tsai P, Kleinfeld D, et al. (2017) The capillary bed offers the largest hemodynamic resistance to the cortical blood supply. J Cereb Blood Flow Metab 37: 52–68. doi: 10.1177/0271678X16671146
    [33] Fry BC, Roy TK, Secomb TW (2013) Capillary recruitment in a theoretical model for blood flow regulation in heterogeneous microvessel networks. Physiol Rep 1: e00050.
    [34] Su SW, Catherall M, Payne S (2012) The influence of network structure on the transport of blood in the human cerebral microvasculature. Microcirculation 19: 175–187. doi: 10.1111/j.1549-8719.2011.00148.x
    [35] Safaeian N, David T (2013) A computational model of oxygen transport in the cerebrocapillary levels for normal and pathologic brain function. J Cereb Blood Flow Metab 33: 1633–1641. doi: 10.1038/jcbfm.2013.119
    [36] Soltani M, Chen P (2013) Numerical modeling of interstitial fluid flow coupled with blood flow through a remodeled solid tumor microvascular network. PLoS One 8: e67025. doi: 10.1371/journal.pone.0067025
    [37] Sefidgar M, Soltania M, Raahemifarc K, et al. (2015) Numerical modeling of drug delivery in a dynamic solid tumor microvasculature. Microvasc Res 99: 43–56. doi: 10.1016/j.mvr.2015.02.007
    [38] Levine H, Pamuk S, Sleeman B, et al. (2002) Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma. Bull Math Biol 63: 801–863.
    [39] Rieger H, Welter M (2015) Integrative models of vascular remodeling during tumor growth. Wiley Interdiscip Rev Syst Biol Med 7: 113–129. doi: 10.1002/wsbm.1295
    [40] Erbertseder K, Reichold J, Flemisch B, et al. (2012) A coupled discrete/continuum model for describing cancer-therapeutic transport in the lung. PLoS One 7: e31966. doi: 10.1371/journal.pone.0031966
    [41] Dash RK, Bassingthwaighte JB (2006) Simultaneous blood-tissue exchange of oxygen, carbon dioxide, bicarbonate, and hydrogen ion. Ann Biomed Eng 34: 1129–1148. doi: 10.1007/s10439-005-9066-4
    [42] Moschandreou TE, Ellis CG, Goldman D (2011) Influence of tissue metabolism and capillary oxygen supply on arteriolar oxygen transport: a computational model. Math Biosci 232: 1–10. doi: 10.1016/j.mbs.2011.03.010
    [43] Welter M, Rieger H (2016) Computer simulations of the tumor vasculature: applications to interstitial fluid flow, drug delivery, and Oxygen supply. Adv Exp Med Biol 936: 31–72. doi: 10.1007/978-3-319-42023-3_3
    [44] Preziosi L, Tosin A (2009) Multiphase and multiscale trends in cancer modelling. Math Model Nat Phenom 4: 1–11.
    [45] Cornelis F, Sautd O, Cumsillea P, et al. (2013) In vivo mathematical modeling of tumor growth from imaging data: Soon to come in the future? Diagn Interv Imaging 94: 593–600. doi: 10.1016/j.diii.2013.03.001
    [46] Wu J, Xu S, Long Q, et al. (2008) Coupled modeling of blood perfusion in intravascular, interstitial spaces in tumor microvasculature. J Biomech 41: 996–1004. doi: 10.1016/j.jbiomech.2007.12.008
    [47] Ye GF, Moore TW, Jaron D (1993) Contributions of oxygen dissociation and convection to the behavior of a compartmental oxygen transport model. Microvasc Res 46: 1–18. doi: 10.1006/mvre.1993.1031
    [48] Piechnik SK, Chiarelli PA, Jezzard P (2008) Modelling vascular reactivity to investigate the basis of the relationship between cerebral blood volume and flow under CO2 manipulation. Neuroimage 39: 107–118. doi: 10.1016/j.neuroimage.2007.08.022
    [49] Fantini S (2014) Dynamic model for the tissue concentration and oxygen saturation of hemoglobin in relation to blood volume, flow velocity, and oxygen consumption: Implications for functional neuroimaging and coherent hemodynamics spectroscopy (CHS). Neuroimage 85: 202–221. doi: 10.1016/j.neuroimage.2013.03.065
    [50] Ursino M, Lodi CA (1998) Interaction among autoregulation, CO2 reactivity, and intracranial pressure: a mathematical model. Am J Physiol 274: H1715–H1728.
    [51] Spronck B, Martens EG, Gommer ED, et al. (2012) A lumped parameter model of cerebral blood flow control combining cerebral autoregulation and neurovascular coupling. Am J Physiol Heart Circ Physiol 303: H1143–H1153. doi: 10.1152/ajpheart.00303.2012
    [52] Payne S (2006) A model of the interaction between autoregulation and neural activation in the brain. Math Biosci 204: 260–281. doi: 10.1016/j.mbs.2006.08.006
    [53] Diamond SG, Perdue KL, Boas DA (2009) A cerebrovascular response model for functional neuroimaging including dynamic cerebral autoregulation. Math Biosci 220: 102–117. doi: 10.1016/j.mbs.2009.05.002
    [54] Muller LO, Toro EF (2014) A global multiscale mathematical model for the human circulation with emphasis on the venous system. Int J Numer Method Biomed Eng 30: 681–725. doi: 10.1002/cnm.2622
    [55] Gutierrez G (2004) A mathematical model of tissue-blood carbon dioxide exchange during hypoxia. Am J Respir Crit Care Med 169: 525–533. doi: 10.1164/rccm.200305-702OC
    [56] Vazquez AL, Masamoto K, Kim SG (2008) Dynamics of oxygen delivery and consumption during evoked neural stimulation using a compartment model and CBF and tissue P(O2) measurements. Neuroimage 42: 49–59. doi: 10.1016/j.neuroimage.2008.04.024
    [57] Barrett MJ, Suresh V (2013) Extra permeability is required to model dynamic oxygen measurements: evidence for functional recruitment? J Cereb Blood Flow Metab 33: 1402–1411. doi: 10.1038/jcbfm.2013.74
    [58] Fang Q, Sakadžić S, Ruvinskaya L, et al. (2008) Oxygen advection and diffusion in a three dimensional vascular anatomical network. Opt Express 16: 17530–17541. doi: 10.1364/OE.16.017530
    [59] Lorthois S, Cassot F, Lauwers F (2011) Simulation study of brain blood flow regulation by intra-cortical arterioles in an anatomically accurate large human vascular network: Part I: methodology and baseline flow. Neuroimage 54: 1031–1042. doi: 10.1016/j.neuroimage.2010.09.032
    [60] Boas DA, Jones SR, Devor A, et al. (2008) A vascular anatomical network model of the spatio-temporal response to brain activation. Neuroimage 40: 1116–1129. doi: 10.1016/j.neuroimage.2007.12.061
    [61] Takahashi T (2014) Microcirculation in fractal branching networks. Springer Japan.
    [62] Ganesan P, He S, Xu H (2010) Analysis of retinal circulation using an image-based network model of retinal vasculature. Microvasc Res 80: 99–109. doi: 10.1016/j.mvr.2010.02.005
    [63] El-Bouri WK, Payne SJ (2015) Multi-scale homogenization of blood flow in 3-dimensional human cerebral microvascular networks. J Theor Biol 380: 40–47. doi: 10.1016/j.jtbi.2015.05.011
    [64] Gagnon L, Sakadžić S, Lesage F, et al. (2015) Multimodal reconstruction of microvascular-flow distributions using combined two-photon microscopy and Doppler optical coherence tomography. Neurophotonics 2: 015008–015008. doi: 10.1117/1.NPh.2.1.015008
    [65] Tsoukias NM, Goldman D, Vadapalli A, et al. (2007) A computational model of oxygen delivery by hemoglobin-based oxygen carriers in three-dimensional microvascular networks. J Theor Biol 248: 657–674. doi: 10.1016/j.jtbi.2007.06.012
    [66] Park CS, Payne SJ (2016) Modelling the effects of cerebral microvasculature morphology on oxygen transport. Med Eng Phys 38: 41–47. doi: 10.1016/j.medengphy.2015.09.004
    [67] Gorodnova NO, Kolobov AV, Mynbaev OA, et al. (2016) Mathematical modeling of blood flow alteration in microcirculatory network due to angiogenesis. Lobachevskii J Math 37: 541–549. doi: 10.1134/S199508021605005X
    [68] Cristini V, Kassab GS (2005) Computer modeling of red blood cell rheology in the microcirculation: a brief overview. Ann Biomed Eng 33: 1724–1727. doi: 10.1007/s10439-005-8776-y
    [69] Sriram K, Intaglietta M, Tartakovsky DM (2014) Non-Newtonian flow of blood in arterioles: consequences for wall shear stress measurements. Microcirculation 21: 628–639. doi: 10.1111/micc.12141
    [70] Kim S, Ong PK, Yalcin O, et al. (2009) The cell-free layer in microvascular blood flow. Biorheology 46: 181–189.
    [71] Pries AR, Secomb TW (2003) Rheology of the microcirculation. Clin Hemorheol Microcirc 29: 143–148.
    [72] Ganesan P, He S, Xu H (2011) Modelling of pulsatile blood flow in arterial trees of retinal vasculature. Med Eng Phys 33: 810–823. doi: 10.1016/j.medengphy.2010.10.004
    [73] Secomb TW, Pries AR (2013) Blood viscosity in microvessels: experiment and theory. C R Phys 14: 470–478. doi: 10.1016/j.crhy.2013.04.002
    [74] Pries AR, Secomb TW (2005) Microvascular blood viscosity in vivo and the endothelial surface layer. Am J Physiol Heart Circ Physiol 289: H2657–H2664. doi: 10.1152/ajpheart.00297.2005
    [75] Lipowsky HH, Usami S, Chien S (1980) In vivo measurements of "apparent viscosity" and microvessel hematocrit in the mesentery of the cat. Microvasc Res 19: 297–319. doi: 10.1016/0026-2862(80)90050-3
    [76] Haynes RH (1960) Physical basis of the dependence of blood viscosity on tube radius. Am J Physiol 198: 1193–1200.
    [77] Neofytou P (2004) Comparison of blood rheological models for physiological flow simulation. Biorheology 41: 693–714.
    [78] Hellums JD, Nair PK, Huang NS, et al. (1996) Simulation of intraluminal gas transport processes in the microcirculation. Ann Biomed Eng 24: 1–24. doi: 10.1007/BF02649700
    [79] Hellums JD (1977) The resistance to oxygen transport in the capillaries relative to that in the surrounding tissue. Microvasc Res 13: 131–136. doi: 10.1016/0026-2862(77)90122-4
    [80] Formaggia L, Quarteroni A (2009) Cardiovascular mathematics: modeling and simulation of the circulatory system. Milan: Springer-Verlag.
    [81] Pries AR, Ley K, Claassen M, et al. (1989) Red cell distribution at microvascular bifurcations. Microvasc Res 38: 81–101. doi: 10.1016/0026-2862(89)90018-6
    [82] Canic S, Kim EH (2003) Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels. Math Method Appl Sci 26: 1161–1186. doi: 10.1002/mma.407
    [83] Gagnon L, Smith AF, Boas DA, et al. (2016) Modeling of cerebral oxygen transport based on in vivo microscopic imaging of microvascular network structure, blood flow, and oxygenation. Front Comput Neurosci 1.
    [84] Nair PK, Huang NS, Hellums JD, et al. (1990) A simple model for prediction of oxygen transport rates by flowing blood in large capillaries. Microvasc Res 39: 203–211. doi: 10.1016/0026-2862(90)90070-8
    [85] Pandey HM, Negi DS, Bisht MS (2011) The study of mathematical modelling of human blood circulatory system. Int J Math Scis Appl 1: 425–434.
    [86] Sharan M, Popel AS (2001) A two-phase model for flow of blood in narrow tubes with increased effective viscosity near the wall. Biorheology 38: 415–428.
    [87] Namgung B, Ju M, Cabrales P, et al. (2013) Two-phase model for prediction of cell-free layer width in blood flow. Microvasc Res 85: 68–76. doi: 10.1016/j.mvr.2012.10.006
    [88] Chebbi R (2015) Dynamics of blood flow: modeling of the Fåhræus-Lindqvist effect. J Biol Phys 41: 313–326. doi: 10.1007/s10867-015-9376-1
    [89] Verma SR, Srivastava A (2014) Analytical study of a two-phase model for steady flow of blood in a circular tube. IJERA 12: 1–10.
    [90] Das B, Johnson PC, Popel AS (1998) Effect of nonaxisymmetric hematocrit distribution on non-Newtonian blood flow in small tubes. Biorheology 35: 69–87. doi: 10.1016/S0006-355X(98)00018-3
    [91] Gupta BB, Nigam KM, Jaffrin MY (1982) A three-layer semi-empirical model for flow of blood and other particular suspensions through narrow tubes. J Biomech Eng 104: 129. doi: 10.1115/1.3138326
    [92] Walburn FJ, Schneck DJ (1976) A constitutive equation for whole human blood. Biorheology 13: 201–210.
    [93] Zaman A, Ali N, Sajid M, et al. (2016) Numerical and analytical study of two-layered unsteady blood flow through catheterized artery. PLoS One 11: e0161377. doi: 10.1371/journal.pone.0161377
    [94] Secomb TW, Pries AR (2011) The microcirculation: physiology at the mesoscale. J Physiol 589: 1047–1052. doi: 10.1113/jphysiol.2010.201541
    [95] AlMomani TD, Vigmostad SC, Chivukula VK, et al. (2012) Red blood cell flow in the cardiovascular system: a fluid dynamics perspective. Crit Rev Biomed Eng 40: 427–440. doi: 10.1615/CritRevBiomedEng.v40.i5.30
    [96] Chung B, Johnson PC, Popel AS (2007) Application of Chimera grid to modelling cell motion and aggregation in a narrow tube. Int J Numer Meth Fluids 53: 105–128. doi: 10.1002/fld.1251
    [97] Ye T, Phan-Thien N, Lim CT (2016) Particle-based simulations of red blood cells-A review. J Biomech 49: 2255–2266. doi: 10.1016/j.jbiomech.2015.11.050
    [98] Tsubota K, Wada S (2010) Effect of the natural state of an elastic cellular membrane on tank-treading and tumbling motions of a single red blood cell. Phys Rev E Stat Nonlin Soft Matter Phys 81: 011910. doi: 10.1103/PhysRevE.81.011910
    [99] Nakamura M, Bessho S, Wada S (2013) Spring-network-based model of a red blood cell for simulating mesoscopic blood flow. Int J Numer Method Biomed Eng 29: 114–128. doi: 10.1002/cnm.2501
    [100] Hariprasad DS, Secomb TW (2014) Two-dimensional simulation of red blood cell motion near a wall under a lateral force. Phys Rev E Stat Nonlin Soft Matter Phys 90: 053014. doi: 10.1103/PhysRevE.90.053014
    [101] Pozrikidis C (2003) Numerical simulation of the flow-induced deformation of red blood cells. Ann Biomed Eng 31: 1194–1205. doi: 10.1114/1.1617985
    [102] Gounley J, Peng Y (2015) Computational modeling of membrane viscosity of red blood cells. Commun Comput Phys 17: 1073–1087. doi: 10.4208/cicp.2014.m355
    [103] Eggleton CD, Popel AS (1998) Large deformation of red blood cell ghosts in a simple shear flow. Phys Fluids 10: 1834–1845. doi: 10.1063/1.869703
    [104] Bagchi P (2007) Mesoscale simulation of blood flow in small vessels. Biophys J 92: 1858–1877. doi: 10.1529/biophysj.106.095042
    [105] Sun C, Munn LL (2005) Particulate nature of blood determines macroscopic rheology: A 2-D lattice Boltzmann analysis. Biophys J 88: 1635–1645. doi: 10.1529/biophysj.104.051151
    [106] Li X, Sarkar K (2008) Front tracking simulation of deformation and buckling instability of a liquid capsule enclosed by an elastic membrane. 227: 4998–5018.
    [107] Pan TW, Wang T (2009) Dynamical simulation of red blood cell rheology in microvessels. Int J Numer Anal Mod 6: 455–473.
    [108] Bessonov N, Crauste F, Fischer S, et al. (2011) Application of hybrid models to blood cell production in the bone marrow. Math Model of Nat Phenom 6: 2–12.
    [109] Volpert V, Bessonov N, Sequeira A, et al. (2015) Methods of blood flow modelling. Math Model Nat Phenom 11: 1–25.
    [110] Munn LL, Dupin MM (2008) Blood cell interactions and segregation in flow. Ann Biomed Eng 36: 534–544. doi: 10.1007/s10439-007-9429-0
    [111] Volpert V, Pujo-Menjouet L (2016) Blood cell dynamics: half of a century of modelling. Math Model Nat Phenom 11: 92–115. doi: 10.1051/mmnp/201611106
    [112] Imai Y, Omori T, Shimogonya Y, et al. (2016) Numerical methods for simulating blood flow at macro, micro, and multi scales. J Biomech 49: 2221–2228. doi: 10.1016/j.jbiomech.2015.11.047
    [113] Liu Y, Liu WK (2006) Rheology of red blood cell aggregation by computer simulation. J Comput Phys 220: 139–154. doi: 10.1016/j.jcp.2006.05.010
    [114] Liu Y, Zhang L, Wang X, et al. (2004) Coupling of Navier-Stokes equations with protein molecular dynamics and its application to hemodynamics. Int J Numer Meth Fluids 46: 1237–1252. doi: 10.1002/fld.798
    [115] Bagchi P, Johnson PC, Popel AS (2005) Computational fluid dynamic simulation of aggregation of deformable cells in a shear flow. J Biomech Eng 127: 1070–1080. doi: 10.1115/1.2112907
    [116] Le DV, White J, Peraire J, et al. (2009) An implicit immersed boundary method for three-dimensional fluid-membrane interactions. J Comput Phys 228: 8427–8445. doi: 10.1016/j.jcp.2009.08.018
    [117] Wu T, Feng JJ (2013) Simulation of malaria-infected red blood cells in microfluidic channels: Passage and blockage. Biomicrofluidics 7: 44115. doi: 10.1063/1.4817959
    [118] Ju M, Ye SS, Namgung B, et al. (2015) A review of numerical methods for red blood cell flow simulation. Comput Methods Biomech Biomed Engin 18: 130–140. doi: 10.1080/10255842.2013.783574
    [119] Shaw S, Ganguly S, Sibanda P, et al. (2014) Dispersion characteristics of blood during nanoparticle assisted drug delivery process through a permeable microvessel. Microvasc Res 92: 25–33. doi: 10.1016/j.mvr.2013.12.007
    [120] Shipley RJ, Chapman SJ (2010) Multiscale modelling of fluid and drug transport in vascular tumours. Bull Math Biol 72: 1464–1491. doi: 10.1007/s11538-010-9504-9
    [121] Pittman RN (2011) Regulation of Tissue Oxygenation. Morgan & Claypool Life Sciences.
    [122] Popel AS (1989) Theory of oxygen transport to tissue. Crit Rev Biomed Eng 17: 257–321.
    [123] Goldman D (2008) Theoretical models of microvascular oxygen transport to tissue. Microcirculation 15: 795–811. doi: 10.1080/10739680801938289
    [124] Krogh A (1919) The supply of oxygen to the tissues and the regulation of the capillary circulation. J Physiol 52: 457–474. doi: 10.1113/jphysiol.1919.sp001844
    [125] Buxton RB, Frank LR (1997) A model for the coupling between cerebral blood flow and oxygen metabolism during neural stimulation. J Cereb Blood Flow Metab 17: 64–72.
    [126] Buxton RB, Uludag K, Dubowitz DJ, et al. (2004) Modeling the hemodynamic response to brain activation. Neuroimage 23 (Suppl 1): S220–S233.
    [127] Secomb TW (2015) Krogh-cylinder and infinite-domain models for washout of an inert diffusible solute from tissue. Microcirculation 22: 91–98. doi: 10.1111/micc.12180
    [128] Schumacker PT, Samsel RW (1989) Analysis of oxygen delivery and uptake relationships in the Krogh tissue model. J Appl Physiol 67: 1234–1244.
    [129] Lagerlund TD, Low PA (1991) Axial diffusion and Michaelis-Menten kinetics in oxygen delivery in rat peripheral nerve. Am J Physiol 260: R430–R440.
    [130] Piiper J, Scheid P (1991) Diffusion limitation of O2 supply to tissue in homogeneous and heterogeneous models. Respir Physiol 85: 127–136. doi: 10.1016/0034-5687(91)90011-7
    [131] Whiteley JP, Gavaghan DJ, Hahn CE (2003) Mathematical modelling of pulmonary gas transport. J Math Biol 47: 79–99. doi: 10.1007/s00285-003-0196-8
    [132] McGuire BJ, Secomb TW (2001) A theoretical model for oxygen transport in skeletal muscle under conditions of high oxygen demand. J Appl Physiol 91: 2255–2265.
    [133] McGuire BJ, Secomb TW (2003) Estimation of capillary density in human skeletal muscle based on maximal oxygen consumption rates. Am J Physiol Heart Circ Physiol 285: H2382–H2391. doi: 10.1152/ajpheart.00559.2003
    [134] Federspiel WJ, Popel AS (1986) A theoretical analysis of the effect of the particulate nature of blood on oxygen release in capillaries. Microvasc Res 32: 164–189. doi: 10.1016/0026-2862(86)90052-X
    [135] Page TC, Light WR, Hellums JD (1998) Prediction of microcirculatory oxygen transport by erythrocyte/hemoglobin solution mixtures. Microvasc Res 56: 113–126. doi: 10.1006/mvre.1998.2088
    [136] Stathopoulos NA, Nair PK, Hellums JD (1987) Oxygen transport studies of normal and sickle red cell suspensions in artificial capillaries. Microvasc Res 34: 200–210. doi: 10.1016/0026-2862(87)90053-7
    [137] Severns ML, Adams JM (1982) The relation between Krogh and compartmental transport models. J Theor Biol 97: 239–249. doi: 10.1016/0022-5193(82)90101-1
    [138] Chen X, Buerk DG, Barbee KA, et al. (2007) A model of NO/O2 transport in capillary-perfused tissue containing an arteriole and venule pair. Ann Biomed Eng 35: 517–529. doi: 10.1007/s10439-006-9236-z
    [139] Ye GF, Moore TW, Buerk DG, et al. (1994) A compartmental model for oxygen-carbon dioxide coupled transport in the microcirculation. Ann Biomed Eng 22: 464–479. doi: 10.1007/BF02367083
    [140] Hayashi T, Watabe H, Kudomi N, et al. (2003) A theoretical model of oxygen delivery and metabolism for physiologic interpretation of quantitative cerebral blood flow and metabolic rate of oxygen. J Cereb Blood Flow Metab 23: 1314–1323.
    [141] Vadapalli A, Pittman RN, Popel AS (2000) Estimating oxygen transport resistance of the microvascular wall. Am J Physiol Heart Circ Physiol 279: H657–H671.
    [142] d'Angelo C (2007) Multiscale modelling of metabolism and transport phenomena in living tissues. Lausanne: PhD Thesis, EPFL.
    [143] Eggleton CD, Vadapalli A, Roy TK, et al. (2000) Calculations of intracapillary oxygen tension distributions in muscle. Math Biosci 167: 123–143. doi: 10.1016/S0025-5564(00)00038-9
    [144] Lucker A, Weber B, Jenny P (2015) A dynamic model of oxygen transport from capillaries to tissue with moving red blood cells. Am J Physiol Heart Circ Physiol 308: H206–H216. doi: 10.1152/ajpheart.00447.2014
    [145] Rees SE, Klaestrup E, Handy J, et al. (2010) Mathematical modelling of the acid-base chemistry and oxygenation of blood: a mass balance, mass action approach including plasma and red blood cells. Eur J Appl Physiol 108: 483–494.
    [146] Dash RK, Bassingthwaighte JB (2004) Blood HbO2 and HbCO2 dissociation curves at varied O2, CO2, pH, 2,3-DPG and temperature levels. Ann Biomed Eng 32: 1676–1693. doi: 10.1007/s10439-004-7821-6
    [147] Dash RK, Bassingthwaighte JB (2010) Erratum to: blood HbO2 and HbCO2 dissociation curves at varied O2, CO2, pH, 2,3-DPG and temperature levels. Ann Biomed Eng 38: 1683–1701. doi: 10.1007/s10439-010-9948-y
    [148] Schacterle RS, Adams JM, Ribando RJ (1991) A theoretical model of gas transport between arterioles and tissue. Microvasc Res 41: 210–228. doi: 10.1016/0026-2862(91)90023-5
    [149] Li Z, Yipintsoi T, Bassingthwaighte JB (1997) Nonlinear model for capillary-tissue oxygen transport and metabolism. Ann Biomed Eng 25: 604–619. doi: 10.1007/BF02684839
    [150] Vadapalli A, Goldman D, Popel AS (2002) Calculations of oxygen transport by red blood cells and hemoglobin solutions in capillaries. Artif Cells Blood Substit Immobil Biotechnol 30: 157–188. doi: 10.1081/BIO-120004338
    [151] Cabrera ME, Saidel GM, Kalhan SC (1998) Role of O2 in regulation of lactate dynamics during hypoxia: mathematical model and analysis. Ann Biomed Eng 26: 1–27. doi: 10.1114/1.28
    [152] Chen HS, Gross JF (1979) Estimation of tissue-to-plasma partition coefficients used in physiological pharmacokinetic models. J Pharmacokinet Biopharm 7: 117–125. doi: 10.1007/BF01059446
    [153] Sharan M, Popel AS (2002) A compartmental model for oxygen transport in brain microcirculation in the presence of blood substitutes. J Theor Biol 216: 479–500. doi: 10.1006/jtbi.2002.3001
    [154] Hsu R, Secomb TW (1989) A Green's function method for analysis of oxygen delivery to tissue by microvascular networks. Math Biosci 96: 61–78. doi: 10.1016/0025-5564(89)90083-7
    [155] Secomb TW, Hsu R, Park EY, et al. (2004) Green's function methods for analysis of oxygen delivery to tissue by microvascular networks. Ann Biomed Eng 32: 1519–1529. doi: 10.1114/B:ABME.0000049036.08817.44
    [156] Secomb TW (2016) A Green's function method for simulation of time-dependent solute transport and reaction in realistic microvascular geometries. Math Med Biol 33: 475–494. doi: 10.1093/imammb/dqv031
    [157] Lamkin-Kennard KA, Buerk DG, Jaron D (2004) Interactions between NO and O2 in the microcirculation: a mathematical analysis. Microvasc Res 68: 38–50. doi: 10.1016/j.mvr.2004.03.001
    [158] Sriram K, Vazquez BY, Yalcin O, et al. (2011) The effect of small changes in hematocrit on nitric oxide transport in arterioles. Antioxid Redox Signal 14: 175–185. doi: 10.1089/ars.2010.3266
    [159] Gundersen SI, Chen G, Palmer AF (2009) Mathematical model of NO and O2 transport in an arteriole facilitated by hemoglobin based O2 carriers. Biophys Chem 143: 1–17. doi: 10.1016/j.bpc.2009.02.005
    [160] Vullo V (2014) Circular Cylinders and Pressure Vessels. Springer.
    [161] Gonzalez-Fernandez JM, Ermentrout B (1994) On the origin and dynamics of the vasomotion of small arteries. Math Biosci 119: 127–167. doi: 10.1016/0025-5564(94)90074-4
    [162] Achakri H, Rachev A, Stergiopulos N, et al. (1994) A theoretical investigation of low frequency diameter oscillations of muscular arteries. Ann Biomed Eng 22: 253–263. doi: 10.1007/BF02368232
    [163] 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 Circ Physiol 295: H1572–H1579. doi: 10.1152/ajpheart.00262.2008
    [164] Ford Versypt AN, Makrides E, Arciero JC, et al. (2015) Bifurcation study of blood flow control in the kidney. Math Biosci 263: 169–179. doi: 10.1016/j.mbs.2015.02.015
    [165] Carlson BE, Secomb TW (2005) A theoretical model for the myogenic response based on the length-tension characteristics of vascular smooth muscle. Microcirculation 12: 327–338. doi: 10.1080/10739680590934745
    [166] Arciero J, Harris A, Siesky B, et al. (2013) Theoretical analysis of vascular regulatory mechanisms contributing to retinal blood flow autoregulation. Invest Ophthalmol Vis Sci 54: 5584–5593. doi: 10.1167/iovs.12-11543
    [167] Ursino M, Di Giammarco P (1991) A mathematical model of the relationship between cerebral blood volume and intracranial pressure changes: the generation of plateau waves. Ann Biomed Eng 19: 15–42. doi: 10.1007/BF02368459
    [168] Ursino M (1988) A mathematical study of human intracranial hydrodynamics part 1-The cerebrospinal fluid pulse pressure. Ann Biomed Eng 16: 379–401. doi: 10.1007/BF02364625
    [169] Banaji M, Tachtsidis I, Delpy D, et al. (2005) A physiological model of cerebral blood flow control. Math Biosci 194: 125–173. doi: 10.1016/j.mbs.2004.10.005
    [170] Beard DA, Bassingthwaighte JB (2001) Modeling advection and diffusion of oxygen in complex vascular networks. Ann Biomed Eng 29: 298–310. doi: 10.1114/1.1359450
    [171] Jespersen SN, Ostergaard L (2012) The roles of cerebral blood flow, capillary transit time heterogeneity, and oxygen tension in brain oxygenation and metabolism. J Cereb Blood Flow Metab 32: 264–277. doi: 10.1038/jcbfm.2011.153
    [172] Lemon DD, Nair PK, Boland EJ, et al. (1987) Physiological factors affecting O2 transport by hemoglobin in an in vitro capillary system. J Appl Physiol 62: 798–806.
    [173] Goldman D, Popel AS (2000) A computational study of the effect of capillary network anastomoses and tortuosity on oxygen transport. J Theor Biol 206: 181–194. doi: 10.1006/jtbi.2000.2113
    [174] Tsoukias NM, Popel AS (2003) A model of nitric oxide capillary exchange. Microcirculation 10: 479–495. doi: 10.1038/sj.mn.7800210
    [175] Delp MD, Armstrong RB (1988) Blood flow in normal and denervated muscle during exercise in conscious rats. Am J Physiol 255: H1509–H1515.
    [176] Pohl U, De Wit C, Gloe T (2000) Large arterioles in the control of blood flow: role of endothelium-dependent dilation. Acta Physiol Scand 168: 505–510. doi: 10.1046/j.1365-201x.2000.00702.x
    [177] Segal SS (2005) Regulation of blood flow in the microcirculation. Microcirculation 12: 33–45. doi: 10.1080/10739680590895028
    [178] Harris A, Jonescu-Cuypers CP, Kagemann L, et al. (2003) Atlas of ocular blood flow: vascular anatomy, pathophysiology, and metabolism. Butterworth-Heinemann/Elsevier.
    [179] Aletti M, Gerbeau JF, Lombardi D (2016) A simplified fluid-structure model for arterial flow. Application to retinal hemodynamics. Comput Methods Appl Mech Engrg 306: 77–94. doi: 10.1016/j.cma.2016.03.044
    [180] Causin P, Malgaroli F (2016) Mathematical modeling of local perfusion in large distensible microvascular networks. arXiv:161002292 [q-bioTO].
    [181] Carichino L, Guidoboni G, Siesky B, et al. (2013) Effect of intraocular pressure and cerebrospinal fluid pressure on the blood flow in the central retinal vessels, In: Causin P, Guidoboni G, Sacco R et al., Editors, Integrated Multidisciplinary Approaches in the Study and Care of the Human Eye, Kugler Publications, 59–66.
    [182] Cassani S, Arciero J, Guidoboni G, et al. (2016) Theoretical predictions of metabolic flow regulation in the retina. J Model Ophthalmol 1.
    [183] Yu DY, Cringle SJ (2001) Oxygen distribution and consumption within the retina in vascularised and avascular retinas and in animal models of retinal disease. Prog Retin Eye Res 20: 175–208. doi: 10.1016/S1350-9462(00)00027-6
    [184] Yu DY, Cringle SJ, Su EN (2005) Intraretinal oxygen distribution in the monkey retina and the response to systemic hyperoxia. Invest Ophthalmol Vis Sci 46: 4728–4733. doi: 10.1167/iovs.05-0694
    [185] Roos MW (2004) Theoretical estimation of retinal oxygenation during retinal artery occlusion. Physiol Meas 25: 1523–1532. doi: 10.1088/0967-3334/25/6/016
    [186] Haugh LM, Linsenmeier RA, Goldstick TK (1990) Mathematical models of the spatial distribution of retinal oxygen tension and consumption, including changes upon illumination. Ann Biomed Eng 18: 19–36. doi: 10.1007/BF02368415
    [187] Cornelissen AJ, Dankelman J, VanBavel E, et al. (2002) Balance between myogenic, flow-dependent, and metabolic flow control in coronary arterial tree: a model study. Am J Physiol Heart Circ Physiol 282: H2224–H2237. doi: 10.1152/ajpheart.00491.2001
    [188] He Z, Lim JK, Nguyen CT, et al. (2013) Coupling blood flow and neural function in the retina: a model for homeostatic responses to ocular perfusion pressure challenge. Physiol Rep 1: e00055.
    [189] Ursino M (1991) Mechanisms of cerebral blood flow regulation. Crit Rev Biomed Eng 18: 255–288.
    [190] Ursino M (1991) A mathematical model of overall cerebral blood flow regulation in the rat. IEEE Trans Biomed Eng 38: 795–807. doi: 10.1109/10.83592
    [191] Nobrega AC, O'Leary D, Silva BM, et al. (2014) Neural regulation of cardiovascular response to exercise: role of central command and peripheral afferents. Biomed Res Int 2014: 478965.
    [192] Sigal IA, Ethier CR (2009) Biomechanics of the optic nerve head. Exp Eye Res 88: 799–807. doi: 10.1016/j.exer.2009.02.003
    [193] Downs JC, Roberts MD, Burgoyne CF, et al. (2009) Multiscale finite element modeling of the lamina cribrosa microarchitecture in the eye. Conf Proc IEEE Eng Med Biol Soc 2009: 4277–4280.
    [194] Grytz R, Girkin CA, Libertiaux V, et al. (2012) Perspectives on biomechanical growth and remodeling mechanisms in glaucoma. Mech Res Commun 42: 92–106. doi: 10.1016/j.mechrescom.2012.01.007
    [195] Newson T, El-Sheikh A (2006) Mathematical modeling of the biomechanics of the lamina cribrosa under elevated intraocular pressures. J Biomech Eng 128: 496–504. doi: 10.1115/1.2205372
    [196] Harris A, Guidoboni G, Arciero JC, et al. (2013) Ocular hemodynamics and glaucoma: the role of mathematical modeling. Eur J Ophthalmol 23: 139–146.
    [197] Guidoboni G, Harris A, Carichino L, et al. (2014) Effect of intraocular pressure on the hemodynamics of the central retinal artery: a mathematical model. Math Biosci Eng 11: 523–546. doi: 10.3934/mbe.2014.11.523
    [198] Cassani S, Harris A, Siesky B, et al. (2015) Theoretical analysis of the relationship between changes in retinal blood flow and ocular perfusion pressure. J Coupled Syst Multiscale Dyn 3: 38–46. doi: 10.1166/jcsmd.2015.1063
    [199] Causin P, Guidoboni G, Harris A, et al. (2014) A poroelastic model for the perfusion of the lamina cribrosa in the optic nerve head. Math Biosci 257: 33–41. doi: 10.1016/j.mbs.2014.08.002
    [200] Biesecker KR, Srienc AI, Shimoda AM, et al. (2016) Glial cell Calcium signaling mediates capillary regulation of blood flow in the retina. J Neurosci 36: 9435–9445. doi: 10.1523/JNEUROSCI.1782-16.2016
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