Abstract
Pulmonary arteries constitute a low-pressure network of vessels, often characterized as a bifurcating tree with heterogeneous vessel mechanics. Understanding the vascular complexity and establishing homeostasis is important to study diseases such as pulmonary arterial hypertension (PAH). The onset and early progression of PAH can be traced to changes in the morphometry and structure of the distal vasculature. Coupling hemodynamics with vessel wall growth and remodeling (G&R) is crucial for understanding pathology at distal vasculature. Accordingly, the goal of this study is to provide a multiscale modeling framework that embeds the essential features of arterial wall constituents coupled with the hemodynamics within an arterial network characterized by an extension of Murray’s law. This framework will be used to establish the homeostatic baseline characteristics of a pulmonary arterial tree, including important parameters such as vessel radius, wall thickness and shear stress. To define the vascular homeostasis and hemodynamics in the tree, we consider two timescales: a cardiac cycle and a longer period of vascular adaptations. An iterative homeostatic optimization, which integrates a metabolic cost function minimization, the stress equilibrium, and hemodynamics, is performed at the slow timescale. In the fast timescale, the pulsatile blood flow dynamics is described by a Womersley's deformable wall analytical solution. Illustrative examples for symmetric and asymmetric trees are presented that provide baseline characteristics for the normal pulmonary arterial vasculature. The results are compared with diverse literature data on morphometry, structure, and mechanics of pulmonary arteries. The developed framework demonstrates a potential for advanced parametric studies and future G&R and hemodynamics modeling of PAH.
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References
Ambrosi D, Ateshian GA, Arruda EM et al (2011) Perspectives on biological growth and remodeling. J Mech Phys Solids 59:863–883. https://doi.org/10.1016/j.jmps.2010.12.011
Avolio AP (1980) Multi-branched model of the human arterial system. Med Biol Eng Comput 18:709–718. https://doi.org/10.1007/BF02441895.
Baek S, Gleason RL, Rajagopal KR, Humphrey JD (2007a) Theory of small on large: potential utility in computations of fluid-solid interactions in arteries. Comput Methods Appl Mech Eng 196:3070–3078. https://doi.org/10.1016/j.cma.2006.06.018
Baek S, Valentín A, Humphrey JD (2007b) Biochemomechanics of cerebral vasospasm and its resolution: II. Constitutive relations and model simulations. Ann Biomed Eng 35:1498. https://doi.org/10.1007/s10439-007-9322-x
Banks J, Booth FV, MacKay EH et al (1978) The physcial properties of human pulmonary arteries and veins. Clin Sci Mol Med 55:477–484
Bernal M, Urban MW, Rosario D et al (2011) Measurement of biaxial mechanical properties of soft tubes and arteries using piezoelectric elements and sonometry. Phys Med Biol 56:3371–3386. https://doi.org/10.1088/0031-9155/56/11/012
Briones AM, Salaices M, Vila E (2007) Mechanisms underlying hypertrophic remodeling and increased stiffness of mesenteric resistance arteries from aged rats. J Gerontol Series A, Biol Sci Med Sci 62(7): 696–706. https://doi.org/10.1093/gerona/62.7.696
Burrowes KS, Hunter PJ, Tawhai MH, Kelly S (2005) Anatomically based finite element models of the human pulmonary arterial and venous trees including supernumerary vessels. J Appl Physiol 99:731–738. https://doi.org/10.1152/japplphysiol.01033.2004
Caro CG, Saffman PG (1965) Extensibility of blood vessels in isolated rabbit lungs. J Physiol 178:193–210. https://doi.org/10.1113/jphysiol.1965.sp007623
Chambers MJ, Colebank MJ, Qureshi MU et al (2020) Structural and hemodynamic properties of murine pulmonary arterial networks under hypoxia-induced pulmonary hypertension. Proc Inst Mech Eng H 234:1312–1329. https://doi.org/10.1177/0954411920944110
Chazova I, Loyd JE, Zhdanov VS et al (1995) Pulmonary artery adventitial changes and venous involvement in primary pulmonary hypertension. Am J Pathol 146:389–397
Clark AR, Tawhai MH (2018) temporal and spatial heterogeneity in pulmonary perfusion: a mathematical model to predict interactions between macro- and micro-vessels in health and disease. ANZIAM J 59:562–580. https://doi.org/10.1017/S1446181118000111
Colebank MJ, Qureshi MU, Rajagopal S et al (2021) A multiscale model of vascular function in chronic thromboembolic pulmonary hypertension. Am J Physiol Heart Circ Physiol 321:H318–H338. https://doi.org/10.1152/ajpheart.00086.2021
Collins R, Maccario JA (1979) Blood flow in the lung. J Biomech 12:373–395. https://doi.org/10.1016/0021-9290(79)90051-4
Elliott FM, Reid L (1965) Some new facts about the pulmonary artery and its branching pattern. Clin Radiol 16:193–198. https://doi.org/10.1016/S0009-9260(65)80042-3
Figueroa CA, Baek S, Taylor CA, Humphrey JD (2009) A computational framework for fluid-solid-growth modeling in cardiovascular simulations. Comput Methods Appl Mech Eng 198:3583–3602. https://doi.org/10.1016/j.cma.2008.09.013
Filonova V, Arthurs CJ, Vignon-Clementel IE, Figueroa CA (2020) Verification of the coupled-momentum method with Womersley’s deformable wall analytical solution. Int J Numer Method Biomed Eng 36:e3266. https://doi.org/10.1002/cnm.3266
Guo X, Kassab GS (2004) Distribution of stress and strain along the porcine aorta and coronary arterial tree. Am J Physiol Heart Circ Physiol 286:H2361–H2368. https://doi.org/10.1152/ajpheart.01079.2003
Hislop A, Reid L (1978) Normal structure and dimensions of the pulmonary arteries in the rat. J Anat 125:71–83
Hollander EH, Wang JJ, Dobson GM et al (2001) Negative wave reflections in pulmonary arteries. Am J Physiol Heart Circ Physiol 281:H895-902
Huang W, Yen RT, McLaurine M, Bledsoe G (1996) Morphometry of the human pulmonary vasculature. J Appl Physiol 81:2123–2133. https://doi.org/10.1152/jappl.1996.81.5.2123
Humbert M, Guignabert C, Bonnet S, et al (2019) Pathology and pathobiology of pulmonary hypertension: state of the art and research perspectives. Eur Respir J 53
Humphrey JD, Rajagopal KR (2002) A constrained mixture model for growth and remodeling of soft tissues. Math Models Methods Appl Sci 12:407–430
Humphrey JD, Dufresne ER, Schwartz MA (2014) Mechanotransduction and extracellular matrix homeostasis. Nat Rev Mol Cell Biol 15:802–812. https://doi.org/10.1038/nrm3896
Humphrey JD, Harrison DG, Figueroa CA et al (2017) Central artery stiffness in hypertension and aging: a problem with cause and consequence. Circ Res 118:379–381. https://doi.org/10.1161/CIRCRESAHA.115.307722.Central
Hunter KS, Lee PF, Lanning CJ et al (2008) Pulmonary vascular input impedance is a combined measure of pulmonary vascular resistance and stiffness and predicts clinical outcomes better than pulmonary vascular resistance alone in pediatric patients with pulmonary hypertension. Am Heart J 155:166–174. https://doi.org/10.1016/J.AHJ.2007.08.014
Huo Y, Kassab GS (2007) A hybrid one-dimensional/Womersley model of pulsatile blood flow in the entire coronary arterial tree. Am J Physiol-Heart Circ Physiol 2007 292:6, H2623–H2633. https://doi.org/10.1152/ajpheart.00987.2006
Ionescu C, Oustaloup A, Levron F, et al (2009) A model of the lungs based on fractal geometrical and structural properties. In: IFAC Proceedings Volumes (IFAC-PapersOnline) 15:994–999. https://doi.org/10.3182/20090706-3-FR-2004.0265
Jiang ZL, Kassab GS, Fung YC (1994) Diameter-defined Strahler system and connectivity matrix of the pulmonary arterial tree. J Appl Physiol 76:882–892. https://doi.org/10.1152/jappl.1994.76.2.882
Kaimovitz B, Lanir Y, Kassab GS (2005) Large-scale 3-D geometric reconstruction of the porcine coronary arterial vasculature based on detailed anatomical data. Ann Biomed Eng 33:1517–1535. https://doi.org/10.1007/s10439-005-7544-3
Kamiya A, Bukhari R, Togawa T (1984) Adaptive regulation of wall shear stress optimizing vascular tree function. Bull Math Biol 46:127–137. https://doi.org/10.1016/S0092-8240(84)80038-5
Kopeć G, Moertl D, Jankowski P et al (2013) Pulmonary artery pulse wave velocity in idiopathic pulmonary arterial hypertension. Can J Cardiol 29:683–690. https://doi.org/10.1016/j.cjca.2012.09.019
Krenz GS, Dawson CA (2003) Flow and pressure distributions in vascular networks consisting of distensible vessels. Am J Physiol Heart Circ Physiol 284:H2192–H2203. https://doi.org/10.1152/ajpheart.00762.2002
Li N, Zhang S, Hou J et al (2012) Assessment of pulmonary artery morphology by optical coherence tomography. Heart Lung Circ 21:778–781. https://doi.org/10.1016/J.HLC.2012.07.014
Lindström SB, Satha G, Klarbring A (2015) Extension of Murray’s law including nonlinear mechanics of a composite artery wall. Biomech Model Mechanobiol 14:83–91. https://doi.org/10.1007/s10237-014-0590-8
Liu Y, Kassab GS (2007) Vascular metabolic dissipation in Murray’s law. Am J Physiol Heart Circ Physiol 292:H1336–H1339. https://doi.org/10.1152/ajpheart.00906.2006
Liu D, Wood NB, Witt N et al (2012) Assessment of energy requirement for the retinal arterial network in normal and hypertensive subjects. J Biomech Eng 134:14501–14507. https://doi.org/10.1115/1.4005529
Mackay EH, Banks J, Sykes B, Lee G (1978) Structural basis for the changing physical properties of human pulmonary vessels with age. Thorax 33:335–344. https://doi.org/10.1136/thx.33.3.335
Marquis AD, Jezek F, Pinsky DJ, Beard DA (2021) Hypoxic pulmonary vasoconstriction as a regulator of alveolar-capillary oxygen flux: a computational model of ventilation-perfusion matching. PLoS Comput Biol 17:e1008861
Milnor WR, Conti CR, Lewis KB, O’Rourke MF (1969) Pulmonary arterial pulse wave velocity and impedance in man. Circ Res 25:637–649
Mittal M, Zhou Y, Ung S et al (2005) A computer reconstruction of the entire coronary arterial tree based on detailed morphometric data. Ann Biomed Eng 33:1015–1026
Murray CD (1926) The physiological principle of minimum work applied to the angle of branching of arteries. Proc Natl Acad Sci 12:835–841
Nichols WW, O’Rourke MF, Vlachopoulos C (2011) McDonald’s blood flow in arteries: theoretical, experimental and clinical principles, Sixth. Hodder Arnold
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. https://doi.org/10.1114/1.1326031
Olufsen MS, Hill NA, Vaughan GDA et al (2012) Rarefaction and blood pressure in systemic and pulmonary arteries. J Fluid Mech 705:280–305. https://doi.org/10.1017/jfm.2012.220
Paul R (1980) Chemical energetics of vascular smooth muscle. Compr Physiol 2007:1–16. https://doi.org/10.1002/cphy.cp020209
Pries AR, Secomb TW (2005) Microvascular blood viscosity in vivo and the endothelial surface layer. Am J Physiol Heart Circ Physiol 289:H2657–H2664. https://doi.org/10.1152/ajpheart.00297.2005
Pries AR, Secomb TW, Gaehtgens P (1995) Design principles of vascular beds. Circ Res 77:1017–1023. https://doi.org/10.1161/01.RES.77.5.1017
Pries AR, Reglin B, Secomb TW (2005) Remodeling of blood vessels. Hypertension 46:725–731. https://doi.org/10.1161/01.HYP.0000184428.16429.be
Prins KW, Weir EK, Archer SL et al (2016) Pulmonary pulse wave transit time is associated with right ventricular–pulmonary artery coupling in pulmonary arterial hypertension. Pulm Circ 6:576–585. https://doi.org/10.1086/688879
Qureshi MU, Vaughan GDA, Sainsbury C et al (2014) Numerical simulation of blood flow and pressure drop in the pulmonary arterial and venous circulation. Biomech Model Mechanobiol 13:1137–1154. https://doi.org/10.1007/s10237-014-0563-y
Rol N, Timmer EM, Faes TJC et al (2017) Vascular narrowing in pulmonary arterial hypertension is heterogeneous: rethinking resistance. Physiol Rep 5:e13159. https://doi.org/10.14814/phy2.13159
Seyedsalehi S, Zhang L, Choi J, Baek S (2015) Prior distributions of material parameters for bayesian calibration of growth and remodeling computational model of abdominal aortic wall. J Biomech Eng 10(1115/1):4031116
Stacher E, Graham BB, Hunt JM et al (2012) Modern age pathology of pulmonary arterial hypertension. Am J Respir Crit Care Med 186(3):261-72. https://doi.org/10.1164/RCCM.201201-0164OC
Taber LA (1998) An optimization principle for vascular radius including the effects of smooth muscle tone. Biophys J 74:109–114
Tamaddon H, Behnia M, Behnia M, Kritharides L (2016) A new approach to blood flow simulation in vascular networks. Comput Methods Biomech Biomed Eng 19:673–685. https://doi.org/10.1080/10255842.2015.1058926
Taylor MG (1957) An approach to an analysis of the arterial pulse wave I. Oscillations in an attenuating line. Phys Med Biol 1:258–269
Townsley M (2012) Structure and composition of pulmonary arteries, capillaries, and veins. Compr Physiol 2:675–709. https://doi.org/10.1002/cphy.c100081
Truong U, Fonseca B, Dunning J et al (2013) Wall shear stress measured by phase contrast cardiovascular magnetic resonance in children and adolescents with pulmonary arterial hypertension. J Cardiovasc Magn Reson 15:1–9. https://doi.org/10.1186/1532-429X-15-81
Tuder RM (2016) Pulmonary vascular remodeling in pulmonary hypertension. Cell Tissue Res 3(367):643–649. https://doi.org/10.1007/S00441-016-2539-Y
van de Vosse FN, Stergiopulos N (2011) Pulse wave propagation in the arterial tree. Annu Rev Fluid Mech 43:467–499. https://doi.org/10.1146/annurev-fluid-122109-160730
Wang Z, Chesler NC (2011) Pulmonary vascular wall stiffness: an important contributor to the increased right ventricular afterload with pulmonary hypertension. Pulm Circ 1:212–223. https://doi.org/10.4103/2045-8932.83453
Wang Y, Gharahi H, Grobbel MR et al (2020) Potential damage in pulmonary arterial hypertension: an experimental study of pressure-induced damage of pulmonary artery. J Biomed Mater Res Part A. https://doi.org/10.1002/jbm.a.37042
Womersley JR (1957) Oscillatory flow in arteries : the constrained elastic tube as a model of arterial flow and pulse transmission. Phys Med Biol 2:178–187
Yen RT, Tai D, Rong Z, Zhang B (1990) Elasticity of pulmonary blood vessels in human lungs. In: Epstein MAF, Ligas JR (eds) Respiratory biomechanics: engineering analysis of structure and function. Springer-Verlag, New York
Zambrano BA, McLean NA, Zhao X et al (2018) Image-based computational assessment of vascular wall mechanics and hemodynamics in pulmonary arterial hypertension patients. J Biomech 68:84–92. https://doi.org/10.1016/j.jbiomech.2017.12.022
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The funding was provided by National Institutes of Health (Grants No. U01-HL135842 and R01-HL158723).
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Gharahi, H., Filonova, V., Mullagura, H.N. et al. A multiscale framework for defining homeostasis in distal vascular trees: applications to the pulmonary circulation. Biomech Model Mechanobiol 22, 971–986 (2023). https://doi.org/10.1007/s10237-023-01693-7
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DOI: https://doi.org/10.1007/s10237-023-01693-7