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Mathematical Models of the Cerebral Microcirculation in Health and Pathophysiology

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Quantitative Approaches to Microcirculation

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 36))

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Abstract

The cerebral microcirculation is a highly complex interconnected network of vessels that cover a wide range of length scales and respond over a wide range of time scales. It plays a critical role in the delivery of nutrients to tissue and the removal of metabolic waste products. Impairments to the cerebral microcirculation are implicated in many cerebrovascular and neurodegenerative diseases. Despite this, until around 15 years ago, little quantitative information was known about its anatomy and geometry in humans; the advent of new data since then has led to a substantial body of work to model blood flow and oxygen transport within the microcirculation and surrounding tissue. Until around 10 years ago, it was also thought to be a largely passive bed with control of cerebral blood flow (CBF) occurring upstream (in the arteriolar bed) and with limited implications in the impairment of cerebral blood flow, either in healthy or in diseased states. However, it is now known that there is significant control of CBF at this length scale through the action of pericytes. Given the difficulties involved in imaging this vascular bed in humans, models of the cerebral microcirculation have an important role to play in gaining a better understanding of CBF and its behaviour in different conditions. In this chapter, the development and application of models of the cerebral microcirculation are thus described, together with potential avenues for future exploration.

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References

  1. Abbott, N.J., Patabendige, A.A., Dolman, D.E., Yusof, S.R., Begley, D.J.: Structure and function of the blood–brain barrier. Neurobiol. Dis. 37(1), 13–25 (2010). https://doi.org/10.1016/j.nbd.2009.07.030

  2. Alberding, J.P., Secomb, T.W.: Simulation of angiogenesis in three dimensions: application to cerebral cortex. PLoS Comput. Biol. 17(6), e1009164 (2021)

    Google Scholar 

  3. Álvarez-Sabín, J., Maisterra, O., Santamarina, E., Kase, C.S.: Factors influencing haemorrhagic transformation in ischaemic stroke. Lancet Neurol. 12(7), 689–705 (2013)

    Google Scholar 

  4. Angleys, H., Østergaard, L., Jespersen, S.N.: The effects of capillary transit time heterogeneity (CTH) on brain oxygenation. J. Cereb. Blood Flow Metab. 35(5), 806–817 (2015)

    Google Scholar 

  5. Attwell, D., Mishra, A., Hall, C.N., O’Farrell, F.M., Dalkara, T.: What is a pericyte? J. Cereb. Blood Flow Metab. 36(2), 451–455 (2016)

    Google Scholar 

  6. Banaji, M., Tachtsidis, I., Delpy, D., Baigent, S.: A physiological model of cerebral blood flow control. Math. Biosci. 194(2), 125–173 (2005)

    Google Scholar 

  7. Beard, D.A.: Computational framework for generating transport models from databases of microvascular anatomy. Ann. Biomed. Eng. 29(10), 837–843 (2001). https://doi.org/10.1114/1.1408920

  8. Blinder, P., Tsai, P.S., Kaufhold, J.P., Knutsen, P.M., Suhl, H., Kleinfeld, D.: The cortical angiome: an interconnected vascular network with noncolumnar patterns of blood flow. Nat Neurosci. 16(7), 889–897 (2013). https://doi.org/10.1038/nn.3426

  9. Boas, D.A., Jones, S.R., Devor, A., Huppert, T.J., Dale, A.M.: A vascular anatomical network model of the spatio-temporal response to brain activation. Neuroimage 40(3), 1116–1129 (2008)

    Google Scholar 

  10. Bordoni, L., Li, B., Kura, S., Boas, D.A., Sakadžić, S., Østergaard, L., Frische, S., Gutiérrez-Jiménez, E.: Quantification of capillary perfusion in an animal model of acute intracranial hypertension. J. Neurotrauma 38(4), 446–454 (2021)

    Google Scholar 

  11. Brown, W.R., Thore, C.R.: Cerebral microvascular pathology in ageing and neurodegeneration. Neuropathol. Appl. Neurobiol. 37(1), 56–74 (2011)

    Google Scholar 

  12. Cassot, F., Lauwers, F., Fouard, C., Prohaska, S., Lauwers-Cances, V.: A novel three-dimensional computer-assisted method for a quantitative study of microvascular networks of the human cerebral cortex. Microcirculation 13(1), 1–18 (2006)

    Google Scholar 

  13. Chen, X., Józsa, T.I., Payne, S.J.: Computational modelling of cerebral oedema and osmotherapy following ischaemic stroke. Comput. Biol. Med. 151, 106226 (2022)

    Google Scholar 

  14. Daher, A., Payne, S.: A network-based model of dynamic cerebral autoregulation. Microvasc. Res. 147, 104503 (2023)

    Google Scholar 

  15. Deshpande, A., Jamilpour, N., Jiang, B., Michel, P., Eskandari, A., Kidwell, C., Wintermark, M., Laksari, K.: Automatic segmentation, feature extraction and comparison of healthy and stroke cerebral vasculature. NeuroImage: Clin. 30, 102573 (2021)

    Google Scholar 

  16. Duvernoy, H.M., Delon, S., Vannson, J.: Cortical blood vessels of the human brain. Brain Res. Bull. 7(5), 519–579 (1981)

    Google Scholar 

  17. El-Bouri, W.K., Payne, S.J.: Multi-scale homogenization of blood flow in 3-dimensional human cerebral microvascular networks. J. Theor. Biol. 380, 40–47 (2015)

    Google Scholar 

  18. El-Bouri, W.K., Payne, S.J.: A statistical model of the penetrating arterioles and venules in the human cerebral cortex. Microcirculation 23(7), 580–590 (2016)

    Google Scholar 

  19. El-Bouri, W.K., Payne, S.J.: Investigating the effects of a penetrating vessel occlusion with a multi-scale microvasculature model of the human cerebral cortex. NeuroImage 172, 94–106 (2018)

    Google Scholar 

  20. El-Bouri, W.K., MacGowan, A., Józsa, T.I., Gounis, M.J., Payne, S.J.: Modelling the impact of clot fragmentation on the microcirculation after thrombectomy. PLoS Comput. Biol. 17(3), e1008515 (2021)

    Google Scholar 

  21. Ertürk, A., Bradke, F.: High-resolution imaging of entire organs by 3-dimensional imaging of solvent cleared organs (3DISCO). Exp. Neurol. 242, 57–64 (2013)

    Google Scholar 

  22. Fang, Q., Sakadzic, S., Ruvinskaya, L., Devor, A., Dale, A.M., Boas, D.A.: Oxygen advection and diffusion in a three-dimensional vascular anatomical network. Opt. Express 16, 17530–17541 (2008)

    Google Scholar 

  23. Gagnon, L., Sakadžić, S., Lesage, F., Musacchia, J.J., Lefebvre, J., Fang, Q., Yücel, M.A., Evans, K.C., Mandeville, E.T., Cohen-Adad, J., et al.: Quantifying the microvascular origin of BOLD-fMRI from first principles with two-photon microscopy and an oxygen-sensitive nanoprobe. J. Neurosci. 35(8), 3663–3675 (2015)

    Google Scholar 

  24. Gagnon, L., Smith, A.F., Boas, D.A., Devor, A., Secomb, T.W., Sakadžić, S.: Modeling of cerebral oxygen transport based on in vivo microscopic imaging of microvascular network structure, blood flow, and oxygenation. Front. Comput. Neurosci. 10, 82 (2016)

    Google Scholar 

  25. Gkontra, P., El-Bouri, W.K., Norton, K.A., Santos, A., Popel, A.S., Payne, S.J., Arroyo, A.G.: Dynamic changes in microvascular flow conductivity and perfusion after myocardial infarction shown by image-based modeling. J. Am. Heart Assoc. 8(7), e011058 (2019)

    Google Scholar 

  26. Goirand, F., Le Borgne, T., Lorthois, S.: Network-driven anomalous transport is a fundamental component of brain microvascular dysfunction. Nat. Commun. 12(1), 7295 (2021)

    Google Scholar 

  27. Gould, I.G., Linninger, A.A.: Hematocrit distribution and tissue oxygenation in large microcirculatory networks. Microcirculation 22(1), 1–18 (2015)

    Google Scholar 

  28. Gould, I.G., Tsai, P., Kleinfeld, D., Linninger, A.: The capillary bed offers the largest hemodynamic resistance to the cortical blood supply. J. Cereb. Blood Flow Metab. 37(1), 52–68 (2017)

    Google Scholar 

  29. Graff, B.J., Payne, S.J., El-Bouri, W.K.: The ageing brain: investigating the role of age in changes to the human cerebral microvasculature with an in silico model. Front. Aging Neurosci. 13, 632521 (2021)

    Google Scholar 

  30. Guibert, R., Fonta, C., Plouraboué, F.: Cerebral blood flow modeling in primate cortex. J. Cereb. Blood Flow Metab. 30(11), 1860–1873 (2010)

    Google Scholar 

  31. Hall, C.N., Reynell, C., Gesslein, B., Hamilton, N.B., Mishra, A., Sutherland, B.A., O’Farrell, F.M., Buchan, A.M., Lauritzen, M., Attwell, D.: Capillary pericytes regulate cerebral blood flow in health and disease. Nature 508(7494), 55–60 (2014)

    Google Scholar 

  32. Hartmann, D.A., Coelho-Santos, V., Shih, A.Y.: Pericyte control of blood flow across microvascular zones in the central nervous system. Annu. Rev. Physiol. 84, 331–354 (2022)

    Google Scholar 

  33. Hartung, G., Vesel, C., Morley, R., Alaraj, A., Sled, J., Kleinfeld, D., Linninger, A.: Simulations of blood as a suspension predicts a depth dependent hematocrit in the circulation throughout the cerebral cortex. PLoS Comput. Biol. 14(11), e1006549 (2018)

    Google Scholar 

  34. Hartung, G., Badr, S., Mihelic, S., Dunn, A., Cheng, X., Kura, S., Boas, D.A., Kleinfeld, D., Alaraj, A., Linninger, A.A.: Mathematical synthesis of the cortical circulation for the whole mouse brain–part ii: microcirculatory closure. Microcirculation 28(5), e12687 (2021)

    Google Scholar 

  35. Hartung, G., Badr, S., Moeini, M., Lesage, F., Kleinfeld, D., Alaraj, A., Linninger, A.: Voxelized simulation of cerebral oxygen perfusion elucidates hypoxia in aged mouse cortex. PLoS Comput. Biol. 17(1), e1008584 (2021)

    Google Scholar 

  36. Hashemi, Z., Rahnama, M.: Numerical simulation of transient dynamic behavior of healthy and hardened red blood cells in microcapillary flow. Int. J. Numer. Method Biomed. Eng. 32(11), e02763 (2016)

    Google Scholar 

  37. Holmes, M.H.: Introduction to Perturbation Methods, vol. 20. Springer Science & Business Media, Berlin (2012)

    Google Scholar 

  38. Horton, N.G., Wang, K., Kobat, D., Clark, C.G., Wise, F.W., Schaffer, C.B., Xu, C.: In vivo three-photon microscopy of subcortical structures within an intact mouse brain. Nat. Photonics 7(3), 205–209 (2013)

    Google Scholar 

  39. Jespersen, S.N., Østergaard, L.: The roles of cerebral blood flow, capillary transit time heterogeneity, and oxygen tension in brain oxygenation and metabolism. J. Cereb. Blood Flow Metab. 32(2), 264–277 (2012)

    Google Scholar 

  40. Józsa, T.I., Padmos, R.M., El-Bouri, W.K., Hoekstra, A.G., Payne, S.J.: On the sensitivity analysis of porous finite element models for cerebral perfusion estimation. Ann. Biomed. Eng. 49(12), 3647–3665 (2021)

    Google Scholar 

  41. Józsa, T.I., Padmos, R.M., Samuels, N., El-Bouri, W., Hoekstra, A.G., Payne, S.J.: A porous circulation model of the human brain for in silico clinical trials in ischaemic stroke. Interface Focus 11(1), 20190127 (2021)

    Google Scholar 

  42. Krüger, T., Varnik, F., Raabe, D.: Efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice Boltzmann finite element method. Comput. Math. Appl. 61(12), 3485–3505 (2011)

    Google Scholar 

  43. Linninger, A., Gould, I., Marinnan, T., Hsu, C.Y., Chojecki, M., Alaraj, A.: Cerebral microcirculation and oxygen tension in the human secondary cortex. Ann. Biomed. Eng. 41, 2264–2284 (2013)

    Google Scholar 

  44. Linninger, A., Hartung, G., Badr, S., Morley, R.: Mathematical synthesis of the cortical circulation for the whole mouse brain-part i. Theory and image integration. Comput. Biol. Med. 110, 265–275 (2019)

    Google Scholar 

  45. Lorthois, S., Cassot, F., Lauwers, F.: 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 (2011). https://doi.org/10.1016/j.neuroimage.2010.09.032

  46. Lorthois, S., Cassot, F., Lauwers, F.: Simulation study of brain blood flow regulation by intra-cortical arterioles in an anatomically accurate large human vascular network. part ii: flow variations induced by global or localized modifications of arteriolar diameters. NeuroImage 54(4), 2840–2853 (2011)

    Google Scholar 

  47. Marchand, P.J., Lu, X., Zhang, C., Lesage, F.: Validation of red blood cell flux and velocity estimations based on optical coherence tomography intensity fluctuations. Sci. Rep. 10(1), 1–10 (2020)

    Google Scholar 

  48. Milanovic, S., Shaw, K., Hall, C., Payne, S.: Investigating the role of pericytes in cerebral autoregulation: a modeling study. Physiol. Meas. 42(5), 054003 (2021)

    Google Scholar 

  49. Moeini, M., Lu, X., Avti, P.K., Damseh, R., Bélanger, S., Picard, F., Boas, D., Kakkar, A., Lesage, F.: Compromised microvascular oxygen delivery increases brain tissue vulnerability with age. Sci. Rep. 8(1), 8219 (2018)

    Google Scholar 

  50. Mokhtarudin, M.J.M., Payne, S.J.: The study of the function of aqp4 in cerebral ischaemia–reperfusion injury using poroelastic theory. Int. J. Numer. Method Biomed. Eng. 33(1), e02784 (2017)

    Google Scholar 

  51. Mokhtarudin, M.J.M., Naim, W.N.W.A., Shabudin, A., Payne, S.J.: Multiscale modelling of brain tissue oxygen and glucose dynamics in tortuous capillary during ischaemia-reperfusion. Appl. Math. Model. 109, 358–373 (2022)

    Google Scholar 

  52. Padmos, R.M., Józsa, T.I., El-Bouri, W.K., Konduri, P.R., Payne, S.J., Hoekstra, A.G.: Coupling one-dimensional arterial blood flow to three-dimensional tissue perfusion models for in silico trials of acute ischaemic stroke. Interface Focus 11(1), 20190125 (2021)

    Google Scholar 

  53. Park, C.S., Payne, S.J.: A generalized mathematical framework for estimating the residue function for arbitrary vascular networks. Interface Focus 3(2), 20120078 (2013)

    Google Scholar 

  54. Payne, S.J.: Cerebral Blood Flow and Metabolism: A Quantitative Approach. World Scientific, Singapore (2018)

    Google Scholar 

  55. Payne, S., Józsa, T.I., El-Bouri, W.K.: Review of in silico models of cerebral blood flow in health and pathology. Prog. Biomed. Eng. 5(2), 022003 (2023)

    Google Scholar 

  56. Penta, R., Gerisch, A.: Investigation of the potential of asymptotic homogenization for elastic composites via a three-dimensional computational study. Comput. Vis. Sci. 17, 185–201 (2015)

    Google Scholar 

  57. Peyrounette, M., Davit, Y., Quintard, M., Lorthois, S.: Multiscale modelling of blood flow in cerebral microcirculation: details at capillary scale control accuracy at the level of the cortex. PLoS One 13(1), e0189474 (2018)

    Google Scholar 

  58. Pias, S.C.: How does oxygen diffuse from capillaries to tissue mitochondria? Barriers and pathways. J. Physiol. 599(6), 1769–1782 (2021)

    Google Scholar 

  59. Price, G.F., Chernyavsky, I.L., Jensen, O.E.: Advection-dominated transport past isolated disordered sinks: step** beyond homogenization. Proc. R. Soc. A 478(2262), 20220032 (2022)

    Google Scholar 

  60. Pries, A.R., Secomb, T.W.: Microvascular blood viscosity in vivo and the endothelial surface layer. Am. J. Physiol. Heart Circ. Physiol. 289(6), H2657–H2664 (2005)

    Google Scholar 

  61. Pries, A., Secomb, T., Gaehtgens, P., Gross, J.: Blood flow in microvascular networks. Experiments and simulation. Circ. Res. 67(4), 826–834 (1990)

    Google Scholar 

  62. Reichold, J., Stampanoni, M., Keller, A.L., Buck, A., Jenny, P., Weber, B.: Vascular graph model to simulate the cerebral blood flow in realistic vascular networks. J. Cereb. Blood Flow Metab. 29(8), 1429–1443 (2009)

    Google Scholar 

  63. Riddle, D.R., Sonntag, W.E., Lichtenwalner, R.J.: Microvascular plasticity in aging. Ageing Res. Rev. 2(2), 149–168 (2003)

    Google Scholar 

  64. Safaeian, N., David, T.: A computational model of oxygen transport in the cerebrocapillary levels for normal and pathologic brain function. J. Cereb. Blood Flow Metab. 33(10), 1633–1641 (2013)

    Google Scholar 

  65. Schmid, F., Tsai, P.S., Kleinfeld, D., Jenny, P., Weber, B.: Depth-dependent flow and pressure characteristics in cortical microvascular networks. PLoS Comput. Biol. 13(2), e1005392 (2017)

    Google Scholar 

  66. Schmid, F., Barrett, M.J., Jenny, P., Weber, B.: Vascular density and distribution in neocortex. NeuroImage 197, 792–805 (2019)

    Google Scholar 

  67. Schneider, M., Reichold, J., Weber, B., Székely, G., Hirsch, S.: Tissue metabolism driven arterial tree generation. Med. Image Anal. 16(7), 1397–1414 (2012)

    Google Scholar 

  68. Secomb, T.W., Hsu, R., Beamer, N.B., Coull, B.M.: Theoretical simulation of oxygen transport to brain by networks of microvessels: effects of oxygen supply and demand on tissue hypoxia. Microcirculation 7(4), 237–247 (2000)

    Google Scholar 

  69. Sharma, D., Smith, M.: The intensive care management of acute ischaemic stroke. Curr. Opin. Crit. Care 28(2), 157–165 (2022)

    Google Scholar 

  70. Shaw, K., Bell, L., Boyd, K., Grijseels, D., Clarke, D., Bonnar, O., Crombag, H., Hall, C.: Neurovascular coupling and oxygenation are decreased in hippocampus compared to neocortex because of microvascular differences. Nat. Commun. 12(1), 3190 (2021)

    Google Scholar 

  71. Smith, A.F., Doyeux, V., Berg, M., Peyrounette, M., Haft-Javaherian, M., Larue, A.E., Slater, J.H., Lauwers, F., Blinder, P., Tsai, P., et al.: Brain capillary networks across species: a few simple organizational requirements are sufficient to reproduce both structure and function. Front. Physiol. 10, 233 (2019)

    Google Scholar 

  72. Su, S.W., Catherall, M., Payne, S.: The influence of network structure on the transport of blood in the human cerebral microvasculature. Microcirculation 19(2), 175–187 (2012)

    Google Scholar 

  73. Tahir, A.M., Chowdhury, M.E., Khandakar, A., Rahman, T., Qiblawey, Y., Khurshid, U., Kiranyaz, S., Ibtehaz, N., Rahman, M.S., Al-Maadeed, S., et al.: Covid-19 infection localization and severity grading from chest x-ray images. Comput. Biol. Med. 139, 105002 (2021)

    Google Scholar 

  74. Tong, Z., Catherall, M., Payne, S.J.: A multiscale model of cerebral autoregulation. Med. Eng. Phys. 95, 51–63 (2021)

    Google Scholar 

  75. Tsai, P.S., Friedman, B., Ifarraguerri, A.I., Thompson, B.D., Lev-Ram, V., Schaffer, C.B., **ong, Q., Tsien, R.Y., Squier, J.A., Kleinfeld, D.: All-optical histology using ultrashort laser pulses. Neuron 39(1), 27–41 (2003)

    Google Scholar 

  76. Tsai, P.S., Kaufhold, J.P., Blinder, P., Friedman, B., Drew, P.J., Karten, H.J., Lyden, P.D., Kleinfeld, D.: Correlations of neuronal and microvascular densities in murine cortex revealed by direct counting and colocalization of nuclei and vessels. J. Neurosci. 29(46), 14553–14570 (2009)

    Google Scholar 

  77. Ursino, M., Lodi, C.A.: A simple mathematical model of the interaction between intracranial pressure and cerebral hemodynamics. J. Appl. Physiol. 82(4), 1256–1269 (1997)

    Google Scholar 

  78. van der Wijk, A.E., Georgakopoulou, T., Majolée, J., van Bezu, J.S., van der Stoel, M.M., van Het Hof, B.J., de Vries, H.E., Huveneers, S., Hordijk, P.L., Bakker, E.N., et al.: Microembolus clearance through angiophagy is an auxiliary mechanism preserving tissue perfusion in the rat brain. Acta Neuropathol. Commun. 8(1), 195 (2020)

    Google Scholar 

  79. Van Rooij, B., Závodszky, G., Azizi Tarksalooyeh, V., Hoekstra, A.: Identifying the start of a platelet aggregate by the shear rate and the cell-depleted layer. J. R. Soc. Interface 16(159), 20190148 (2019)

    Google Scholar 

  80. Ventimiglia, T., Linninger, A.A.: Mesh-free high-resolution simulation of cerebrocortical oxygen supply with fast Fourier preconditioning. Int. J. Numer. Method Biomed. Eng. 39, e3735 (2023)

    Google Scholar 

  81. Walsh, C., Tafforeau, P., Wagner, W., Jafree, D., Bellier, A., Werlein, C., Kühnel, M., Boller, E., Walker-Samuel, S., Robertus, J., et al.: Imaging intact human organs with local resolution of cellular structures using hierarchical phase-contrast tomography. Nat. Methods 18(12), 1532–1541 (2021)

    Google Scholar 

  82. Wang, J., Payne, S.J.: Mathematical modelling of haemorrhagic transformation after ischaemic stroke. J. Theor. Biol. 531, 110920 (2021)

    Google Scholar 

  83. Wang, J., Van Kranendonk, K.R., El-Bouri, W.K., Majoie, C.B., Payne, S.J.: Mathematical modelling of haemorrhagic transformation within a multiscale microvasculature network. Physiol. Meas. 43(5), 055006 (2022)

    Google Scholar 

  84. Weber, B., Keller, A.L., Reichold, J., Logothetis, N.K.: The microvascular system of the striate and extrastriate visual cortex of the macaque. Cereb. Cortex 18(10), 2318–2330 (2008)

    Google Scholar 

  85. Xue, S., Gong, H., Jiang, T., Luo, W., Meng, Y., Liu, Q., Chen, S., Li, A.: Indian-ink perfusion based method for reconstructing continuous vascular networks in whole mouse brain. PLoS One 9(1), e88067 (2014)

    Google Scholar 

  86. Xue, Y., El-Bouri, W.K., Józsa, T.I., Payne, S.J.: Modelling the effects of cerebral microthrombi on tissue oxygenation and cell death. J. Biomech. 127, 110705 (2021)

    Google Scholar 

  87. Zagzoule, M., Marc-Vergnes, J.P.: A global mathematical model of the cerebral circulation in man. J. Biomech. 19(12), 1015–1022 (1986)

    Google Scholar 

  88. Závodszky, G., Van Rooij, B., Azizi, V., Hoekstra, A.: Cellular level in-silico modeling of blood rheology with an improved material model for red blood cells. Front. Physiol. 8, 563 (2017)

    Google Scholar 

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Acknowledgements

SJP is supported by a Yushan Scholarship from the Ministry of Education, Taiwan (#110VV004).

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  1. National Taiwan University, Taipei, Taiwan

    Stephen J. Payne

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  1. Departments of Biomedical Engineering and Neurosurgery, University of Illinois Chicago, Chicago, IL, USA

    Andreas Linninger

  2. Department of Mathematics, University of Oslo, Oslo, Norway, Simula Research Laboratory, Oslo, Norway

    Kent-Andre Mardal

  3. MOX, Department of Mathematics, Politecnico di Milano, Milan, Italy

    Paolo Zunino

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Payne, S.J. (2024). Mathematical Models of the Cerebral Microcirculation in Health and Pathophysiology. In: Linninger, A., Mardal, KA., Zunino, P. (eds) Quantitative Approaches to Microcirculation. SEMA SIMAI Springer Series, vol 36. Springer, Cham. https://doi.org/10.1007/978-3-031-58519-7_1

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