Abstract
The cardiovascular system plays a key role in the transport of nutrients, ensuring a continuous supply of all cells of the body with the metabolites necessary for life. The blood supply to the brain is carried out by the large arteries located on its surface, which branch into smaller arterioles that penetrate the cerebral cortex and feed the capillary bed, thereby forming an extensive branching network. The formation of blood vessels is carried out via vasculogenesis and angiogenesis, which play an important role in both embryo and adult life. The review presents approaches to modeling various aspects of both the formation of vascular networks and the construction of the formed arterial tree. In addition, a brief description of models that allows one to study the blood flow in various parts of the circulatory system and the spatiotemporal metabolite distribution in brain tissues is given. Experimental study of these issues is not always possible due to both the complexity of the cardiovascular system and the mechanisms through which the perfusion of all body cells is carried out. In this regard, mathematical models are a good tool for studying hemodynamics and can be used in clinical practice to diagnose vascular diseases and assess the need for treatment.
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References
Abbott NJ, Ronnback L, Hansson E (2006) Astrocyte-endothelial interactions at the blood-brain barrier. Nat Rev Neurosci 7(1):41. https://doi.org/10.1038/nrn1824
Abbott NJ, Patabendige AA, Dolman DE et al (2010) Structure and function of the blood-brain barrier. Neurobiol Dis 37(1):13. https://doi.org/10.1016/j.nbd.2009.07.030
Abdi P, Vahidi B (2022) A vasculogenesis model based on flow-induced stresses on endothelial cells. AIP Adv 12(5):055306. https://doi.org/10.1063/5.0087884
Akbarpour Ghazani M, Nouri Z, Saghafian M et al (2020) Mathematical modeling reveals how the density of initial tumor and its distance to parent vessels alter the growth trend of vascular tumors. Microcirculation 27(1):e12584. https://doi.org/10.1111/micc.12584
Alastruey J, Parker KH, Peiro J et al (2007) Modelling the circle of Willis to assess the effects of anatomical variations and occlusions on cerebral flows. J Biomech 40(8):1794. https://doi.org/10.1016/j.jbiomech.2006.07.008
Alastruey J, Moore SM, Parker KH et al (2008) Reduced modelling of blood flow in the cerebral circulation: coupling 1-D, 0-D and cerebral auto-regulation models. Int J Numer Meth Fluids 56(8):1061. https://doi.org/10.1002/fld.1606
Alberding JP, Secomb TW (2021) Simulation of angiogenesis in three dimensions: application to cerebral cortex. PLoS Comput Biol 17(6):e1009164. https://doi.org/10.1371/journal.pcbi.1009164
Ambrosi D, Bussolino F, Preziosi L (2005) A review of vasculogenesis models. Journal of Theoretical Medicine 6:145376. https://doi.org/10.1080/1027366042000327098
Arciero J, Causin P, Malgaroli F (2017) Mathematical methods for modeling the microcirculation. AIMS Biophysics 4:362. https://doi.org/10.3934/biophy.2017.3.362
Audebert C, Bekheit M, Bucur P et al (2017) Partial hepatectomy hemodynamics changes: experimental data explained by closed-loop lumped modeling. J Biomech 50:202. https://doi.org/10.1016/j.jbiomech.2016.11.037
Augustin CM, Gsell MAF, Karabelas E et al (2021) A computationally efficient physiologically comprehensive 3D–0D closed-loop model of the heart and circulation. Comput Methods Appl Mech Eng 386:114092. https://doi.org/10.1016/j.cma.2021.114092
Bahrami S, Norouzi M (2018) A numerical study on hemodynamics in the left coronary bifurcation with normal and hypertension conditions. Biomech Model Mechanobiol 17(6):1785. https://doi.org/10.1007/s10237-018-1056-1
Balogh P, Bagchi P (2019) The cell-free layer in simulated microvascular networks. J Fluid Mech 864:768. https://doi.org/10.1017/jfm.2019.45
Berg P, Stucht D, Janiga G et al (2014) Cerebral blood flow in a healthy Circle of Willis and two intracranial aneurysms: computational fluid dynamics versus four-dimensional phase-contrast magnetic resonance imaging. J Biomech Eng 136(4). https://doi.org/10.1115/1.4026108
Bhat SM, Badiger VA, Vasishta S et al (2021) 3D tumor angiogenesis models: recent advances and challenges. J Cancer Res Clin Oncol 147(12):3477. https://doi.org/10.1007/s00432-021-03814-0
Blanco PJ, de Queiroz RA, Feijoo RA (2013) A computational approach to generate concurrent arterial networks in vascular territories. Int J Numer Method Biomed Eng 29(5):601. https://doi.org/10.1002/cnm.2547
Blinder P, Shih AY, Rafie C et al (2010) Topological basis for the robust distribution of blood to rodent neocortex. Proc Natl Acad Sci U S A 107(28):12670. https://doi.org/10.1073/pnas.1007239107
Boas SEM, Palm MM, Koolwijk P et al (2013) Computational modeling of angiogenesis: towards a multi-scale understanding of cell-cell and cell-matrix interactions. Springer, Berlin Heidelberg, p 161
Bokkers RP, van Laar PJ, van der Zwan A et al (2010) Mixed perfusion: a combined blood supply to the brain tissue by multiple arteries. J Neuroradiol 37(4):201. https://doi.org/10.1016/j.neurad.2010.01.035
Borzov AG, Mukhin SI, Sosnin NV (2012) Conservative algorithm of substance transport over a closed graph of cardiovascular system 27(5):413. https://doi.org/10.1515/rnam-2012-0023
Bryngelson SH, Gueniat F, Freund JB (2019) Irregular dynamics of cellular blood flow in a model microvessel. Phys Rev E 100(1–1):012203. https://doi.org/10.1103/PhysRevE.100.012203
Bui AV, Manasseh R, Liffman K et al (2010) Development of optimized vascular fractal tree models using level set distance function. Med Eng Phys 32(7):790. https://doi.org/10.1016/j.medengphy.2010.04.014
Caforio F, Augustin CM, Alastruey J et al (2022) A coupling strategy for a first 3D–1D model of the cardiovascular system to study the effects of pulse wave propagation on cardiac function. Comput Mech 70(4):703. https://doi.org/10.1007/s00466-022-02206-6
Cai Y, Zhang J, Wu J et al (2015) Oxygen transport in a three-dimensional microvascular network incorporated with early tumour growth and preexisting vessel cooption: numerical simulation study. Biomed Res Int 2015:476964. https://doi.org/10.1155/2015/476964
Capoccia M (2015) Development and characterization of the arterial windkessel and its role during left ventricular assist device assistance. Artif Organs 39(8):E138. https://doi.org/10.1111/aor.12532
Carmeliet P (2003) Angiogenesis in health and disease. Nat Med 9(6):653. https://doi.org/10.1038/nm0603-653
Carmeliet P, Jain RK (2000) Angiogenesis in cancer and other diseases. Nature 407(6801):249. https://doi.org/10.1038/35025220
Cassot F, Lauwers F, Fouard C et al (2006) A novel three-dimensional computer-assisted method for a quantitative study of microvascular networks of the human cerebral cortex. Microcirculation 13(1):1. https://doi.org/10.1080/10739680500383407
Casulli V, Dumbser M, Toro EF (2012) Semi-implicit numerical modeling of axially symmetric flows in compliant arterial systems. Int J Numer Method Biomed Eng 28(2):257. https://doi.org/10.1002/cnm.1464
Cebral JR, Castro MA, Soto O et al (2003) Blood-flow models of the circle of Willis from magnetic resonance data. J Eng Math 47(3):369. https://doi.org/10.1023/b:engi.0000007977.02652.02
Chaplain MA (2000) Mathematical modelling of angiogenesis. J Neurooncol 50(1–2):37. https://doi.org/10.1023/a:1006446020377
Chen X, Zhuang J, Huang H et al (2021) Fluid-structure interactions (FSI) based study of low-density lipoproteins (LDL) uptake in the left coronary artery. Sci Rep 11(1):4803. https://doi.org/10.1038/s41598-021-84155-3
Cherevko AA, Yankova GS, Maltseva SV et al (2016) Construction and investigation of 3D vessels net of the brain according to MRI data using the method of variation of scanning plane. J Phys: Conf Ser 722(1):012029. https://doi.org/10.1088/1742-6596/722/1/012029
Cheung CY, Ong S, Ikram MK et al (2014) Retinal vascular fractal dimension is associated with cognitive dysfunction. J Stroke Cerebrovasc Dis 23(1):43. https://doi.org/10.1016/j.jstrokecerebrovasdis.2012.09.002
Chico TJA, Kugler EC (2021) Cerebrovascular development: mechanisms and experimental approaches. Cell Mol Life Sci 78(9):4377. https://doi.org/10.1007/s00018-021-03790-1
Correa-Alfonso CM, Withrow JD, Domal SJ et al (2022) A mesh-based model of liver vasculature: implications for improved radiation dosimetry to liver parenchyma for radiopharmaceuticals. EJNMMI Phys 9(1):28. https://doi.org/10.1186/s40658-022-00456-0
Correa-Alfonso CM, Withrow JD, Domal SJ et al (2023) Intra-brain vascular models within the ICRP mesh-type adult reference phantoms for applications to internal dosimetry. Phys Med Biol 68(10). https://doi.org/10.1088/1361-6560/acc926
Crookston NR, Fung GSK, Frey EC (2019) Development of a customizable hepatic arterial tree and particle transport model for use in treatment planning. IEEE Trans Radiat Plasma Med Sci 3(1):31. https://doi.org/10.1109/trpms.2018.2842463
Curtis LT, Wu M, Lowengrub J et al (2015) Computational modeling of tumor response to drug release from vasculature-bound nanoparticles. PLoS ONE 10(12):e0144888. https://doi.org/10.1371/journal.pone.0144888
Cury LFM, Maso Talou GD, Younes-Ibrahim M et al (2021) Parallel generation of extensive vascular networks with application to an archetypal human kidney model. R Soc Open Sci 8(12):210973. https://doi.org/10.1098/rsos.210973
Daub JT, Merks RM (2013) A cell-based model of extracellular-matrix-guided endothelial cell migration during angiogenesis. Bull Math Biol 75(8):1377. https://doi.org/10.1007/s11538-013-9826-5
David T, Alzaidi S, Farr H (2009) Coupled autoregulation models in the cerebro-vasculature. J Eng Math 64(4):403. https://doi.org/10.1007/s10665-009-9274-2
Devault K, Gremaud PA, Novak V et al (2008) Blood flow in the circle of Willis: modeling and calibration. Multiscale Model Simul 7(2):888. https://doi.org/10.1137/07070231X
Dorr A, Sahota B, Chinta LV et al (2012) Amyloid-beta-dependent compromise of microvascular structure and function in a model of Alzheimer’s disease. Brain 135(Pt 10):3039. https://doi.org/10.1093/brain/aws243
Dwidmuthe PD, Dastane GG, Mathpati CS et al (2021) Study of blood flow in stenosed artery model using computational fluid dynamics and response surface methodology. The Canadian Journal of Chemical Engineering 99(S1):S820. https://doi.org/10.1002/cjce.23991
Esmaily Moghadam M, Vignon-Clementel IE, Figliola R et al (2013) A modular numerical method for implicit 0D/3D coupling in cardiovascular finite element simulations. J Comput Phys 244:63. https://doi.org/10.1016/j.jcp.2012.07.035
Espinoza I, Peschke P, Karger CP (2013) A model to simulate the oxygen distribution in hypoxic tumors for different vascular architectures. Med Phys 40(8):081703. https://doi.org/10.1118/1.4812431
Farina A, Rosso F, Fasano A (2021) A continuum mechanics model for the fåhræus-lindqvist effect. J Biol Phys 47(3):253. https://doi.org/10.1007/s10867-021-09575-8
Farina A, Fasano A, Rosso F (2023) A theoretical model for the fåhræus effect in medium-large microvessels. J Theor Biol 558:111355. https://doi.org/10.1016/j.jtbi.2022.111355
Figueroa XF, Lillo MA, Gaete PS et al (2013) Diffusion of nitric oxide across cell membranes of the vascular wall requires specific connexin-based channels. Neuropharmacology 75:471. https://doi.org/10.1016/j.neuropharm.2013.02.022
Formaggia L, Lamponi D, Tuveri M et al (2006) Numerical modeling of 1D arterial networks coupled with a lumped parameters description of the heart. Comput Methods Biomech Biomed Engin 9(5):273. https://doi.org/10.1080/10255840600857767
Fredrich T, Welter M, Rieger H (2018) Tumorcode?: a framework to simulate vascularized tumors. Eur Phys J E Soft Matter 41(4):55. https://doi.org/10.1140/epje/i2018-11659-x
Frey D, Livne M, Leppin H et al (2021) A precision medicine framework for personalized simulation of hemodynamics in cerebrovascular disease. Biomed Eng Online 20(1):44. https://doi.org/10.1186/s12938-021-00880-w
Gabrys E, Rybaczuk M, Kedzia A (2005) Fractal models of circulatory system. Symmetrical and asymmetrical approach comparison. Chaos, Solitons & Fractals 24(3):707. https://doi.org/10.1016/j.chaos.2004.09.087
Gacche RN, Meshram RJ (2014) Angiogenic factors as potential drug target: efficacy and limitations of anti-angiogenic therapy. Biochim Biophys Acta 1:161. https://doi.org/10.1016/j.bbcan.2014.05.002
Gagnon L, Sakadzic S, Lesage F et al (2015) Quantifying the microvascular origin of bold-FMRI from first principles with two-photon microscopy and an oxygen-sensitive nanoprobe. J Neurosci 35(8):3663. https://doi.org/10.1523/JNEUROSCI.3555-14.2015
Gaidzik F, Pathiraja S, Saalfeld S et al (2021) Hemodynamic data assimilation in a subject-specific circle of willis geometry. Clin Neuroradiol 31(3):643. https://doi.org/10.1007/s00062-020-00959-2
Gao J, Wang Y, Ding Q (2022) Comparison of the clinical value of transcranial doppler ultrasound and computed tomography angiography for diagnosing ischemic cerebrovascular disease. J Int Med Res 50(6):3000605211047718. https://doi.org/10.1177/03000605211047718
Georg M, Hahn HK, Preusser T et al (2004) Global constructive optimization of vascular systems. The IEEE Transactions on Medical Imaging
Ghigo AR, Fullana JM, Lagrée PY (2017) A 2D nonlinear multiring model for blood flow in large elastic arteries. J Comput Phys 350:136. https://doi.org/10.1016/j.jcp.2017.08.039
Gkontra P, El-Bouri WK, Norton KA et al (2019) Dynamic changes in microvascular flow conductivity and perfusion after myocardial infarction shown by image-based modeling. J Am Heart Assoc 8(7):e011058. https://doi.org/10.1161/JAHA.118.011058
Gompper G, Fedosov DA (2016) Modeling microcirculatory blood flow: current state and future perspectives. Wiley Interdiscip Rev Syst Biol Med 8(2):157. https://doi.org/10.1002/wsbm.1326
Grinberg L, Anor T, Cheever E et al (2009) Simulation of the human intracranial arterial tree. Philos Trans A Math Phys Eng Sci 367(1896):2371. https://doi.org/10.1098/rsta.2008.0307
Guan D, Liang F, Gremaud PA (2016) Comparison of the Windkessel model and structured-tree model applied to prescribe outflow boundary conditions for a one-dimensional arterial tree model. J Biomech 49(9):1583. https://doi.org/10.1016/j.jbiomech.2016.03.037
Guibert R, Fonta C, Plouraboue F (2010) Cerebral blood flow modeling in primate cortex. J Cereb Blood Flow Metab 30(11):1860. https://doi.org/10.1038/jcbfm.2010.105
Guy AA, Justin AW, Aguilar-Garza DM et al (2020) 3D printable vascular networks generated by accelerated constrained constructive optimization for tissue engineering. IEEE Trans Biomed Eng 67(6):1650. https://doi.org/10.1109/TBME.2019.2942313
Haselden WD, Kedarasetti RT, Drew PJ (2020) Spatial and temporal patterns of nitric oxide diffusion and degradation drive emergent cerebrovascular dynamics. PLoS Comput Biol 16(7):e1008069. https://doi.org/10.1371/journal.pcbi.1008069
Heck TAM, Vaeyens MM, Van Oosterwyck H (2015) Computational models of sprouting angiogenesis and cell migration: towards multiscale mechanochemical models of angiogenesis. Math Model Nat Phenom 10(1):108. https://doi.org/10.1051/mmnp/201510106
Hladky SB, Barrand MA (2018) Elimination of substances from the brain parenchyma: efflux via perivascular pathways and via the blood-brain barrier. Fluids Barriers CNS 15(1):30. https://doi.org/10.1186/s12987-018-0113-6
Hoksbergen AWJ, Fülesdi B, Legemate DA et al (2000) Collateral configuration of the circle of Willis. Stroke 31(6):1346. https://doi.org/10.1161/01.STR.31.6.1346
Ii S, Kitade H, Ishida S et al (2020) Multiscale modeling of human cerebrovasculature: a hybrid approach using image-based geometry and a mathematical algorithm. PLoS Comput Biol 16(6):e1007943. https://doi.org/10.1371/journal.pcbi.1007943
Kafarov ES, Miltykh I, Dmitriev AV et al (2023) Anatomical variability of kidney arterial vasculature based on zonal and segmental topography. Heliyon 9(4):e15315. https://doi.org/10.1016/j.heliyon.2023.e15315
Karch R, Neumann F, Neumann M et al (2000) Staged growth of optimized arterial model trees. Ann Biomed Eng 28(5):495. https://doi.org/10.1114/1.290
Karch R, Neumann F, Neumann M et al (2003) Voronoi polyhedra analysis of optimized arterial tree models. Ann Biomed Eng 31(5):548. https://doi.org/10.1114/1.1566444
Kashkooli FM, Abazari MA, Soltani M et al (2022) A spatiotemporal multi-scale computational model for FDG PET imaging at different stages of tumor growth and angiogenesis. Sci Rep 12(1):10062. https://doi.org/10.1038/s41598-022-13345-4
Keelan J, Chung EML, Hague JP (2019) Development of a globally optimised model of the cerebral arteries. Phys Med Biol 64(12):125021. https://doi.org/10.1088/1361-6560/ab2479
Kim HJ, Rundfeldt HC, Lee I et al (2023) Tissue-growth-based synthetic tree generation and perfusion simulation. Biomech Model Mechanobiol 22(3):1095. https://doi.org/10.1007/s10237-023-01703-8
Kirst C, Skriabine S, Vieites-Prado A et al (2020) Mapping the fine-scale organization and plasticity of the brain vasculature. Cell 180(4):780. https://doi.org/10.1016/j.cell.2020.01.028
Kleinstreuer N, Dix D, Rountree M et al (2013) A computational model predicting disruption of blood vessel development. PLoS Comput Biol 9(4):e1002996. https://doi.org/10.1371/journal.pcbi.1002996
Kopylova VS, Boronovskiy SE, Nartsissov YR (2017) Fundamental principles of vascular network topology. Biochem Soc Trans 45(3):839. https://doi.org/10.1042/BST20160409
Kopylova VS, Boronovskiy SE, Nartsissov YR (2018) Tree topology analysis of the arterial system model. J Phys: Conf Ser 1141:012027. https://doi.org/10.1088/1742-6596/1141/1/012027
Kopylova VS, Boronovskiy SE, Nartsissov YR (2019) Fundamental constraints of vessels network architecture properties revealed by reconstruction of a rat brain vasculature. Math Biosci 315:108237. https://doi.org/10.1016/j.mbs.2019.108237
Kopylova VS, Boronovskiy SE, Nartsissov YR (2019) Multiparametric topological analysis of reconstructed rat brain arterial system. Physical Biology 16(5):056002. https://doi.org/10.1088/1478-3975/ab2704
Kopylova VS, Boronovskiy SE, Nartsissov YR (2020) Application of fractal analysis to evaluate the rat brain arterial system. Biophysics 65(3):495. https://doi.org/10.1134/S0006350920030100
Kopylova VS, Boronovskiy SE, Nartsissov YR (2022) Estimation of oxygen and glucose concentration distribution in the rat brain arterial system. Mathematical Biology and Bioinformatics 17(2):386. https://doi.org/10.17537/2022.17.386
Kreft M, Luksic M, Zorec TM et al (2013) Diffusion of D-glucose measured in the cytosol of a single astrocyte. Cell Mol Life Sci 70(8):1483. https://doi.org/10.1007/s00018-012-1219-7
Kretowski M, Rolland Y, Bezy-Wendling J et al (2003) Physiologically based modeling of 3-D vascular networks and CT scan angiography. IEEE Trans Med Imaging 22(2):248. https://doi.org/10.1109/TMI.2002.808357
Krock BL, Skuli N, Simon MC (2011) Hypoxia-induced angiogenesis: good and evil. Genes Cancer 2(12):1117. https://doi.org/10.1177/1947601911423654
Lee TR, Yoo SS, Yang J (2017) Generalized plasma skimming model for cells and drug carriers in the microvasculature. Biomech Model Mechanobiol 16(2):497. https://doi.org/10.1007/s10237-016-0832-z
Lee TR, Hong JA, Yoo SS et al (2018) A computational modeling of blood flow in asymmetrically bifurcating microvessels and its experimental validation. Int J Numer Method Biomed Eng 34(6):e2981. https://doi.org/10.1002/cnm.2981
Leontidis G, Al-Diri B, Wigdahl J et al (2015) Evaluation of geometric features as biomarkers of diabetic retinopathy for characterizing the retinal vascular changes during the progression of diabetes. Conf Proc IEEE Eng Med Biol Soc 2015:5255. https://doi.org/10.1109/EMBC.2015.7319577
Li P, Pan Q, Jiang S et al (2021) Development of novel fractal method for characterizing the distribution of blood flow in multi-scale vascular tree. Front Physiol 12:711247. https://doi.org/10.3389/fphys.2021.711247
Liang F, Fukasaku K, Liu H et al (2011) A computational model study of the influence of the anatomy of the circle of Willis on cerebral hyperperfusion following carotid artery surgery. Biomed Eng Online 10:84. https://doi.org/10.1186/1475-925X-10-84
Linninger AA, Gould IG, Marrinan T et al (2013) Cerebral microcirculation and oxygen tension in the human secondary cortex. Ann Biomed Eng 41(11):2264. https://doi.org/10.1007/s10439-013-0828-0
Lloyd BA, Szczerba D, Rudin M et al (2008) A computational framework for modelling solid tumour growth. Philos Trans A Math Phys Eng Sci 366(1879):3301. https://doi.org/10.1098/rsta.2008.0092
Lopes D, Puga H, Teixeira JC et al (2019) Influence of arterial mechanical properties on carotid blood flow: comparison of CFD and FSI studies. Int J Mech Sci 160:209. https://doi.org/10.1016/j.ijmecsci.2019.06.029
Lucker A, Secomb TW, Weber B et al (2017) The relative influence of hematocrit and red blood cell velocity on oxygen transport from capillaries to tissue. Microcirculation 24(3). https://doi.org/10.1111/micc.12337
Lucker A, Secomb TW, Barrett MJP et al (2018a) The relation between capillary transit times and hemoglobin saturation heterogeneity. Part 2: capillary networks. Front Physiol 9:1296. https://doi.org/10.3389/fphys.2018.01296
Lucker A, Secomb TW, Weber B et al (2018b) The relation between capillary transit times and hemoglobin saturation heterogeneity. Part 1: Theoretical models. Front Physiol 9:420. https://doi.org/10.3389/fphys.2018.00420
Lykov K, Li X, Lei H et al (2015) Inflow/outflow boundary conditions for particle-based blood flow simulations: application to arterial bifurcations and trees. PLoS Comput Biol 11(8):e1004410. https://doi.org/10.1371/journal.pcbi.1004410
Mackenzie J (2022) A graph theoretic exploration of coronary vascular trees. https://doi.org/10.48550/arXiv.2207.14624
Malossi ACI, Blanco PJ, Crosetto P et al (2013) Implicit coupling of one-dimensional and three-dimensional blood flow models with compliant vessels. Multiscale Modeling & Simulation 11(2):474. https://doi.org/10.1137/120867408
Manganotti J, Caforio F, Kimmig F et al (2021) Coupling reduced-order blood flow and cardiac models through energy-consistent strategies: modeling and discretization. Advanced Modeling and Simulation in Engineering Sciences 8(1):21. https://doi.org/10.1186/s40323-021-00206-4
Meneses LDM, Brito PF, Rocha BM et al (2017) Construction of arterial networks considering a power law with exponent dependent on bifurcation level. In: Torres I, Bustamante J, Sierra DA (eds) VII Latin American Congress on Biomedical Engineering CLAIB 2016, Bucaramanga, Santander, Colombia, October 26th -28th, 2016. Springer Singapore, p 545. https://doi.org/10.1007/978-981-10-4086-3_137
Mirramezani M, Diamond SL, Litt HI et al (2019) Reduced order models for transstenotic pressure drop in the coronary arteries. J Biomech Eng 141(3):0310051. https://doi.org/10.1115/1.4042184
Moledina S, de Bruyn A, Schievano S et al (2011) Fractal branching quantifies vascular changes and predicts survival in pulmonary hypertension: a proof of principle study. Heart 97(15):1245. https://doi.org/10.1136/hrt.2010.214130
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(7):681. https://doi.org/10.1002/cnm.2622
Muskat JC, Rayz VL, Goergen CJ et al (1985) (2021) Hemodynamic modeling of the circle of Willis reveals unanticipated functions during cardiovascular stress. J Appl Physiol 131(3):1020. https://doi.org/10.1152/japplphysiol.00198.2021
Mynard JP, Nithiarasu P (2008) A 1D arterial blood flow model incorporating ventricular pressure, aortic valve and regional coronary flow using the locally conservative Galerkin (LCG) method. Commun Numer Methods Eng 24(5):367. https://doi.org/10.1002/cnm.1117
Nadeem ZA, Dakhil SF, Abdullah H (2018) Numerical analysis of blood flow through multiple stenosis right coronary artery. Int J Appl Eng Res 13(22):16064
Nakazawa T, Tasaki S, Nakai K et al (2022) Multicellular model of angiogenesis. AIMS. Bioengineering 9(1):44. https://doi.org/10.3934/bioeng.2022004
Nartsissov YR (2021) The effect of flux dysconnectivity functions on concentration gradients changes in a multicomponent model of convectional reaction-diffusion by the example of a neurovascular unit. Defect and Diffusion Forum 413:19. https://doi.org/10.4028/www.scientific.net/DDF.413.19
Nartsissov YR (2021) Validation of a spatial-time concentration gradients estimation by the superposition of sphere sources diffusion fields using the finite element method. Journal of Physics: Conference Series 2090(1):012007. https://doi.org/10.1088/1742-6596/2090/1/012007
Nartsissov YR (2022) Application of a multicomponent model of convectional reaction-diffusion to description of glucose gradients in a neurovascular unit. Front Physiol 13:843473. https://doi.org/10.3389/fphys.2022.843473
Nartsissov YR, Tiukina ES, Boronovskiy SE et al (2013) Computer modeling of spatial-time distribution of metabolite concentrations in phantoms of biological objects by example of rat brain pial. Biofizika 58(5):887
Nekka F, Kyriacos S, Kerrigan C et al (1996) A model of growing vascular structures. Bull Math Biol 58(3):409. https://doi.org/10.1016/0092-8240(95)00343-6
Nicholson C (2001) Diffusion and related transport mechanisms in brain tissue. Rep Prog Phys 64(7):815. https://doi.org/10.1088/0034-4885/64/7/202
Norton KA, Popel AS (2016) Effects of endothelial cell proliferation and migration rates in a computational model of sprouting angiogenesis. Sci Rep 6:36992. https://doi.org/10.1038/srep36992
Ostergaard L, Engedal TS, Moreton F et al (2016) Cerebral small vessel disease: capillary pathways to stroke and cognitive decline. J Cereb Blood Flow Metab 36(2):302. https://doi.org/10.1177/0271678X15606723
Pascalau R, Padurean VA, Bartos D et al (2019) The geometry of the circle of Willis anatomical variants as a potential cerebrovascular risk factor. Turk Neurosurg 29(2):151. https://doi.org/10.5137/1019-5149.JTN.21835-17.3
Perfahl H, Byrne HM, Chen T et al (2011) Multiscale modelling of vascular tumour growth in 3D: the roles of domain size and boundary conditions. PLoS ONE 6(4):e14790. https://doi.org/10.1371/journal.pone.0014790
Peyrounette M, Davit Y, Quintard M et al (2018) 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. https://doi.org/10.1371/journal.pone.0189474
Phillips CM, Lima E, Woodall RT et al (2020) A hybrid model of tumor growth and angiogenesis: in silico experiments. PLoS ONE 15(4):e0231137. https://doi.org/10.1371/journal.pone.0231137
Pillay S, Byrne HM, Maini PK (2017) Modeling angiogenesis: a discrete to continuum description. Phys Rev E 95(1–1):012410. https://doi.org/10.1103/PhysRevE.95.012410
Pries AR, Secomb TW (2014) Making microvascular networks work: angiogenesis, remodeling, and pruning. Physiology (Bethesda) 29(6):446. https://doi.org/10.1152/physiol.00012.2014
Quarteroni A, Veneziani A, Vergara C (2016) Geometric multiscale modeling of the cardiovascular system, between theory and practice. Comput Methods Appl Mech Eng 302:193. https://doi.org/10.1016/j.cma.2016.01.007
Queiroz RABd, Aquino LCMd (2018) Automatic construction of vascular arteriovenous tree geometric model. In: Proceeding Series of the Brazilian Society of Computational and Applied Mathematics. https://doi.org/10.5540/03.2018.006.02.0257
Rahma AG, Abdelhamid T (2023) Hemodynamic and fluid flow analysis of a cerebral aneurysm: a CFD simulation. SN Applied Sciences 5(2):62. https://doi.org/10.1007/s42452-023-05276-0
Rahma AG, Yousef K, Abdelhamid T (2022) Blood flow CFD simulation on a cerebral artery of a stroke patient. SN Applied Sciences 4(10):261. https://doi.org/10.1007/s42452-022-05149-y
Reymond P, Merenda F, Perren F et al (2009) Validation of a one-dimensional model of the systemic arterial tree. Am J Physiol Heart Circ Physiol 297(1):H208. https://doi.org/10.1152/ajpheart.00037.2009
Rieger H, Welter M (2015) Integrative models of vascular remodeling during tumor growth. Wiley Interdiscip Rev Syst Biol Med 7(3):113. https://doi.org/10.1002/wsbm.1295
Rosales-Alcantar CA, Hernández-Dueñas G (2023) A new two-dimensional blood flow model with arbitrary cross sections. ESAIM: M2AN 57(3):1657. https://doi.org/10.1051/m2an/2023030
Rostamian A, Fallah K, Rostamiyan Y et al (2023) Computational study of the blood hemodynamic inside the cerebral double dome aneurysm filling with endovascular coiling. Sci Rep 13(1):2909. https://doi.org/10.1038/s41598-023-29988-w
Ryu J, Hu X, Shadden SC (2015) A coupled lumped-parameter and distributed network model for cerebral pulse-wave hemodynamics. J Biomech Eng 137(10):101009. https://doi.org/10.1115/1.4031331
Sadeh A, Kazemi A, Bahramkhoo M et al (2023) Computational study of blood flow inside MCA aneurysm with/without endovascular coiling. Sci Rep 13(1):4560. https://doi.org/10.1038/s41598-023-31522-x
Salavati H, Soltani M (2019) The impact of endothelial cells proliferation in a multiscale realistic reproduction of angiogenesis. Biochem Eng J 142:74. https://doi.org/10.1016/j.bej.2018.11.012
Santamaria R, Gonzalez-Alvarez M, Delgado R et al (2020) Remodeling of the microvasculature: may the blood flow be with you. Front Physiol 11:586852. https://doi.org/10.3389/fphys.2020.586852
Schreiner W (1993) Computer generation of complex arterial tree models. J Biomed Eng 15(2):148. https://doi.org/10.1016/0141-5425(93)90046-2
Schwen LO, Preusser T (2012) Analysis and algorithmic generation of hepatic vascular systems. Int J Hepatol 2012:357687. https://doi.org/10.1155/2012/357687
Scianna M, Bell CG, Preziosi L (2013) A review of mathematical models for the formation of vascular networks. J Theor Biol 333:174. https://doi.org/10.1016/j.jtbi.2013.04.037
Seraya IP, Nartsissov YR (2002) Theoretical approach to description of time-dependent nitric oxide effects in the vasculature. Mol Biol Rep 29(1–2):151. https://doi.org/10.1023/a:1020361325884
Seraya IP, Nartsissov YR, Brown G (2003) Mathematical modeling of non-stationary spatial gradients of nitric oxide in the muscle wall of blood vessels. Biofizika 48(1):91
Sharma BD, Yadav PK (2019) A mathematical model of blood flow in narrow blood vessels in presence of magnetic field. National Academy Science Letters 42(3):239. https://doi.org/10.1007/s40009-018-0718-y
Hong LS, Adib MA, Matalif MU et al (2020) Modeling and simulation of blood flow analysis on simplified aneurysm models. IOP Conference Series: Materials Science and Engineering 917(1):012067. https://doi.org/10.1088/1757-899X/917/1/012067
Sherwin SJ, Franke V, Peiró J et al (2003) One-dimensional modelling of a vascular network in space-time variables. J Eng Math 47(3):217. https://doi.org/10.1023/B:ENGI.0000007979.32871.e2
Shi Y, Lawford P, Hose R (2011) Review of zero-D and 1-D models of blood flow in the cardiovascular system. Biomed Eng Online 10:33. https://doi.org/10.1186/1475-925X-10-33
Shin E, Kim JJ, Lee S et al (2018) Hemodynamics in diabetic human aorta using computational fluid dynamics. PLoS ONE 13(8):e0202671. https://doi.org/10.1371/journal.pone.0202671
Shipley RJ, Smith AF, Sweeney PW et al (2020) A hybrid discrete-continuum approach for modelling microcirculatory blood flow. Math Med Biol 37(1):40. https://doi.org/10.1093/imammb/dqz006
Simakov S, Gamilov T (2019) Computational study of the cerebral circulation accounting for the patient-specific anatomical features. In: Petrov IB, Favorskaya AV, Favorskaya MN et al (eds) Smart Modeling for Engineering Systems. Springer International Publishing, p 309. https://doi.org/10.1007/978-3-030-06228-6_25
Steele BN, Olufsen MS, Taylor CA (2007) Fractal network model for simulating abdominal and lower extremity blood flow during resting and exercise conditions. Comput Methods Biomech Biomed Engin 10(1):39. https://doi.org/10.1080/10255840601068638
Sutalo I, Bui A, Ahmed S et al (2009) Modelling of flow through the circle of Willis and cerebral vasculature, vol 13. https://doi.org/10.2495/BIO090081
Sutalo ID, Bui AV, Ahmed S et al (2014) Modeling of flow through the circle of Willis and cerebral vasculature to assess the effects of changes in the peripheral small cerebral vasculature on the inflows. Engineering Applications of Computational Fluid Mechanics 8(4):609. https://doi.org/10.1080/19942060.2014.11083311
Sweeney PW, Walsh C, Walker-Samusel S et al (2022) A three-dimensional, discrete-continuum model of blood flow in microvascular networks. bioRxiv p 2022.11.23.517681. https://doi.org/10.1101/2022.11.23.517681
Sykova E, Nicholson C (2008) Diffusion in brain extracellular space. Physiol Rev 88(4):1277. https://doi.org/10.1152/physrev.00027.2007
Szczerba D, Szekely G (2005) Simulating vascular systems in arbitrary anatomies. Med Image Comput Comput Assist Interv 8(Pt 2):641. https://doi.org/10.1007/11566489_79
Talou GDM, Safaei S, Hunter PJ et al (2021) Adaptive constrained constructive optimisation for complex vascularisation processes. Sci Rep 11(1):6180. https://doi.org/10.1038/s41598-021-85434-9
Tamis A, Drapaca CS (2021) Modeling no biotransport in brain using a space-fractional reaction-diffusion equation. Front Physiol 12:644149. https://doi.org/10.3389/fphys.2021.644149
Tang H, Hu N, Yuan Y et al (2019) Accelerated time-of-flight magnetic resonance angiography with sparse undersampling and iterative reconstruction for the evaluation of intracranial arteries. Korean J Radiol 20(2):265. https://doi.org/10.3348/kjr.2017.0634
Terman D, Chen L, Hannawi Y (2021) Mathematical modeling of cerebral capillary blood flow heterogeneity and its effect on brain tissue oxygen levels. J Theor Biol 527:110817. https://doi.org/10.1016/j.jtbi.2021.110817
Tian FB, Zhu L, Fok PW et al (2013) Simulation of a pulsatile non-Newtonian flow past a stenosed 2D artery with atherosclerosis. Comput Biol Med 43(9):1098. https://doi.org/10.1016/j.compbiomed.2013.05.023
Ursino M, Giannessi M (2010) A model of cerebrovascular reactivity including the circle of Willis and cortical anastomoses. Ann Biomed Eng 38(3):955. https://doi.org/10.1007/s10439-010-9923-7
Vahidkhah K, Balogh P, Bagchi P (2016) Flow of red blood cells in stenosed microvessels. Sci Rep 6:28194. https://doi.org/10.1038/srep28194
Vandamme TF (2014) Use of rodents as models of human diseases. J Pharm Bioallied Sci 6(1):2. https://doi.org/10.4103/0975-7406.124301
Vigneshwaran V, Sands GB, LeGrice IJ et al (2019) Reconstruction of coronary circulation networks: a review of methods. Microcirculation 26(5):e12542. https://doi.org/10.1111/micc.12542
Vilanova G, Colominas I, Gomez H (2017) A mathematical model of tumour angiogenesis: growth, regression and regrowth. J R Soc Interface 14(126). https://doi.org/10.1098/rsif.2016.0918
Vilanova G, Bures M, Colominas I et al (2018) Computational modelling suggests complex interactions between interstitial flow and tumour angiogenesis. J R Soc Interface 15(146). https://doi.org/10.1098/rsif.2018.0415
Vrselja Z, Brkic H, Mrdenovic S et al (2014) Function of circle of Willis. J Cereb Blood Flow Metab 34(4):578. https://doi.org/10.1038/jcbfm.2014.7
Wang Z, Zhao G, Wang T et al (2015) Three-dimensional numerical simulation of the effects of fractal vascular trees on tissue temperature and intracelluar ice formation during combined cancer therapy of cryosurgery and hyperthermia. Appl Therm Eng 90:296. https://doi.org/10.1016/j.applthermaleng.2015.06.103
Westerhof N, Lankhaar JW, Westerhof BE (2009) The arterial windkessel. Med Biol Eng Comput 47(2):131. https://doi.org/10.1007/s11517-008-0359-2
Wymer DT, Patel KP, Burke RWF et al (2020) Phase-contrast MRI: physics, techniques, and clinical applications. Radiographics 40(1):122. https://doi.org/10.1148/rg.2020190039
Xia F, Wu C, Sinefeld D et al (2018) In vivo label-free confocal imaging of the deep mouse brain with long-wavelength illumination. Biomed Opt Express 9(12):6545. https://doi.org/10.1364/BOE.9.006545
Xing CY, Tarumi T, Liu J et al (2017) Distribution of cardiac output to the brain across the adult lifespan. J Cereb Blood Flow Metab 37(8):2848. https://doi.org/10.1177/0271678X16676826
Yankova G, Tur D, Parshin D et al (2021) Cerebral arterial architectonics and CFD simulation in mice with type 1 diabetes mellitus of different duration. Sci Rep 11(1):3969. https://doi.org/10.1038/s41598-021-83484-7
Zhang H, Fu P, Liu Y et al (2022) Large-depth three-photon fluorescence microscopy imaging of cortical microvasculature on nonhuman primates with bright AIE probe in vivo. Biomaterials 289:121809. https://doi.org/10.1016/j.biomaterials.2022.121809
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Kopylova, V., Boronovskiy, S. & Nartsissov, Y. Approaches to vascular network, blood flow, and metabolite distribution modeling in brain tissue. Biophys Rev 15, 1335–1350 (2023). https://doi.org/10.1007/s12551-023-01106-0
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DOI: https://doi.org/10.1007/s12551-023-01106-0