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2012, 9(1): 175-198. doi: 10.3934/mbe.2012.9.175

A quasi-lumped model for the peripheral distortion of the arterial pulse

1. 

Department of Materials Science, University of Ioannina, GR 451 10, Ioannina, Greece, Greece, Greece

2. 

Department of Cardiology, Medical School, University of Ioannina, GR 451 10, Ioannnina, Greece

Received  November 2010 Revised  April 2011 Published  December 2011

As blood circulates through the arterial tree, the flow and pressure pulse distort. Principal factors to this distortion are reflections form arterial bifurcations and the viscous character of the flow of the blood. Both of them are expounded in the literature and included in our analysis. The nonlinearities of inertial effects are usually taken into account in numerical simulations, based on Navier-Stokes like equations. Nevertheless, there isn't any qualitative, analytical formula, which examines the role of blood's inertia on the distortion of the pulse. We derive such an analytical nonlinear formula. It emanates from a generalized Bernoulli's equation for an an-harmonic, linear, viscoelastic, Maxwell fluid flow in a linear, viscoelastic, Kelvin-Voigt, thin, cylindrical vessel. We report that close to the heart, convection effects related to the change in the magnitude of the velocity of blood dominate the alteration of the shape of the pressure pulse, while at remote sites of the vascular tree, convection of vorticity, related to the change in the direction of the velocity of blood with respect to a mean axial flow, prevails. A quantitative comparison between the an-harmonic theory and related pressure measurements is also performed.
Citation: Panagiotes A. Voltairas, Antonios Charalambopoulos, Dimitrios I. Fotiadis, Lambros K. Michalis. A quasi-lumped model for the peripheral distortion of the arterial pulse. Mathematical Biosciences & Engineering, 2012, 9 (1) : 175-198. doi: 10.3934/mbe.2012.9.175
References:
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J. Alastruey, K. H. Parker, J. Peiró and S. J. Sherwin, Analysing the pattern of pulse waves in arterial networks: A time-domain study,, J. Engrg. Math., 64 (2009), 331.  doi: 10.1007/s10665-009-9275-1.  Google Scholar

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J. Alastruey, On the mechanics underlying the reservoir-excess separation in systemic arteries and their implications for pulse wave analysis,, Cardiovasc. Eng., 10 (2010), 176.  doi: 10.1007/s10558-010-9109-9.  Google Scholar

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S. Čanić , J. Tambača, G. Guidoboni, A. Mikelić , C. J. Hartley and D. Rosenstrauch, Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow,, SIAM J. Appl. Math., 67 (2006), 164.   Google Scholar

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L. Cardamone, A. Valentin, J. F. Eberth and J. D. Humphrey, Origin of axial prestretch and residual stress in arteries,, Biomechan. Model. Mechanobiol., 8 (2009), 431.  doi: 10.1007/s10237-008-0146-x.  Google Scholar

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C. H. Chen, E. Nevo, B. Fetics, P. H. Pak, F. C. P. Yin, W. L. Maughan and D. A. Kass, Estimation of central aortic pressure waveform by mathematical transformation of radial tonometry pressure: Validation of generalized transfer function,, Circulation, 95 (1997), 1827.   Google Scholar

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S. Melchionna, M. Bernaschi, S. Succi, E. Kaxiras, F. J. Rybicki, D. Mitsouras, A. U. Coskun and C. L. Feldman, Hydrokinetic approach to large-scale cardiovascular blood flow,, Comput. Phys. Comm., 181 (2010).  doi: 10.1016/j.cpc.2009.10.017.  Google Scholar

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show all references

References:
[1]

J. Alastruey, K. H. Parker, J. Peiró and S. J. Sherwin, Analysing the pattern of pulse waves in arterial networks: A time-domain study,, J. Engrg. Math., 64 (2009), 331.  doi: 10.1007/s10665-009-9275-1.  Google Scholar

[2]

J. Alastruey, On the mechanics underlying the reservoir-excess separation in systemic arteries and their implications for pulse wave analysis,, Cardiovasc. Eng., 10 (2010), 176.  doi: 10.1007/s10558-010-9109-9.  Google Scholar

[3]

M. Anand and K. R. Rajagopal, A shear thinning viscoelastic fluid model for describing the flow of blood,, Inter. J. Cardiovasc. Med. Science, 4 (2004), 59.   Google Scholar

[4]

M. C. Aoi, C. T. Kelley, V. Novak and M. S. Olufsen, Optimization of a mathematical model of cerebral autoregulation using patient data,, in, (2009), 181.   Google Scholar

[5]

D. Bessems, C. C. Giannopapa, M. C. M. Rutten and F. N. van de Vosse, Experimental validation of a time-domain based wave propagation model of blood flow in viscoelastic vessels,, J. Biomech., 41 (2008), 284.  doi: 10.1016/j.jbiomech.2007.09.014.  Google Scholar

[6]

R. Caflisch, G. Majda, C. Peskin and G. Strumolo, Distortion of the arterial pulse,, Math. Biosci., 51 (1980), 229.  doi: 10.1016/0025-5564(80)90102-9.  Google Scholar

[7]

S. Čanić , J. Tambača, G. Guidoboni, A. Mikelić , C. J. Hartley and D. Rosenstrauch, Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow,, SIAM J. Appl. Math., 67 (2006), 164.   Google Scholar

[8]

L. Cardamone, A. Valentin, J. F. Eberth and J. D. Humphrey, Origin of axial prestretch and residual stress in arteries,, Biomechan. Model. Mechanobiol., 8 (2009), 431.  doi: 10.1007/s10237-008-0146-x.  Google Scholar

[9]

C. H. Chen, E. Nevo, B. Fetics, P. H. Pak, F. C. P. Yin, W. L. Maughan and D. A. Kass, Estimation of central aortic pressure waveform by mathematical transformation of radial tonometry pressure: Validation of generalized transfer function,, Circulation, 95 (1997), 1827.   Google Scholar

[10]

S. C. Cowin, Polar fluids,, Physics of Fluids, 11 (1968), 1919.  doi: 10.1063/1.1692219.  Google Scholar

[11]

E. Crépeau and M. Sorine, A reduced model of pulsatile flow in an arterial compartment,, Chaos Soliton Fractals, 34 (2007), 594.   Google Scholar

[12]

G. Dai, M. R. Kaazempur-Mofrad, S. Natarajan, Y. Zhang, S. Vaughn, B. R. Blackman, R. D. Kamm, G. Garcia-Cardena and M. A. Gimbrone, Jr., Distinct endothelial phenotypes evoked by arterial waveforms derived from atherosclerosis-susceptible and -resistant regions of human vasculature,, Proc. Nat. Acad. Sci., 101 (2004), 14871.  doi: 10.1073/pnas.0406073101.  Google Scholar

[13]

K. DeVault, P. A. Gremaud, V. Novak, M. S. Olufsen, G. Vernières and P. Zhao, Blood flow in the circle of Willis: Modeling and calibration,, Multiscale Model. Simul., 7 (2008), 888.  doi: 10.1137/07070231X.  Google Scholar

[14]

J. Filo and A. Zauŝková, 2D Navier-Stokes equations in a time dependent domain with Neumann type boundary conditions,, J. Math. Fluid Mech., 12 (2010), 1.  doi: 10.1007/s00021-008-0274-1.  Google Scholar

[15]

Y. C. Fung, "Biomechanics, Mechanical Properties of Living Tissues,", Springer-Verlag, (1993).   Google Scholar

[16]

L. Grinberg, T. Anor, E. Cheever, J. R. Madsen and G. E. Karniadakis, Simulation of the human intracranial arterial tree,, Phil. Trans. R. Soc. A, 367 (2009), 2371.  doi: 10.1098/rsta.2008.0307.  Google Scholar

[17]

G. A. Holzapfel, T. C. Gasser and M. Stadler, A structural model for the viscoelastic behavior of arterial walls: Continuum formulation and finite element analysis,, Eur. J. Mech. Solids, 21 (2002), 441.  doi: 10.1016/S0997-7538(01)01206-2.  Google Scholar

[18]

G. A. Holzapfel, T. C. Gasser and R. W. Ogden, A new constitutive framework for arterial wall mechanics and a comparative study of material models. Soft tissue mechanics,, J. Elast., 61 (2000), 1.  doi: 10.1023/A:1010835316564.  Google Scholar

[19]

G. A. Holzapfel and R. W. Ogden, Constitutive modelling of passive myocardium: A structurally based framework for material characterization,, Phil. Trans. R. Soc. A, 367 (2010), 3445.  doi: 10.1098/rsta.2009.0091.  Google Scholar

[20]

J. D. Humphrey and C. A. Taylor, Intracranial and abdominal aortic aneurysms: Similarities, differences and need for a new class of computational models,, Ann. Rev. Biomed. Eng., 10 (2008), 221.  doi: 10.1146/annurev.bioeng.10.061807.160439.  Google Scholar

[21]

D. D. Joseph, M. Renardy and J.-C. Saut, Hyperbolicity and change of type in the flow of viscoelastic fluids,, Arch. Rat. Mech. Anal., 87 (1985), 213.   Google Scholar

[22]

D. K. Ku, D. P. Giddens, C. K. Zarins and S. Glagov, Pulsatile flow and atherosclerosis in the human carotid bifurcation. Positive correlation between plaque location and low oscillating shear stress,, Arteriosclerosis, 5 (1985), 293.  doi: 10.1161/01.ATV.5.3.293.  Google Scholar

[23]

Z. Lei, C. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids,, Arch. Rat. Mech. Anal., 188 (2008), 371.  doi: 10.1007/s00205-007-0089-x.  Google Scholar

[24]

J. Lighthill, "Mathematical Biofluiddynamics,", Based on the lecture course delivered to the Mathematical Biofluiddynamics Research Conference of the National Science Foundation held from July 16–20, (1973).   Google Scholar

[25]

F. Lin and P. Zhang, On the initial-boundary value problem of the incompressible viscoelastic fluid system,, Commun. Pure Appl. Math., 61 (2008), 539.  doi: 10.1002/cpa.20219.  Google Scholar

[26]

M. Lopez de Haro, J. A. P. del Rio and S. Whitaker, Flow of Maxwell fluids in porous media,, Transport in Porous Media, 25 (1996), 167.   Google Scholar

[27]

I. Masson, P. Boutouyrie, S. Laurent, J. D. Humphrey and M. Zidi, Characterization of arterial wall mechanical behavior and stresses from human clinical data,, J. Biomech., 41 (2008), 2618.  doi: 10.1016/j.jbiomech.2008.06.022.  Google Scholar

[28]

K. S. Matthys, J. Alastruey, J. Peiro, A. W. Khir, P. Segers, P. R. Verdonck, K. H. Parker and S. J. Sherwin, Pulse wave propagation in a model human arterial network: Assessment of 1-D numerical simulations against in vitro measurements,, J. Biomech., 40 (2007), 3476.  doi: 10.1016/j.jbiomech.2007.05.027.  Google Scholar

[29]

S. Melchionna, M. Bernaschi, S. Succi, E. Kaxiras, F. J. Rybicki, D. Mitsouras, A. U. Coskun and C. L. Feldman, Hydrokinetic approach to large-scale cardiovascular blood flow,, Comput. Phys. Comm., 181 (2010).  doi: 10.1016/j.cpc.2009.10.017.  Google Scholar

[30]

W. W. Nichols and M. F. O'Rourke, "McDonald's Blood Flow in Arteries,", Arnold Publishers, (1998).   Google Scholar

[31]

M. S. Olufsen, C. S. Peskin, W. Y. Kim, E. M. Pedersen, A. Nadim and J. Larsen, Numerical simulation and experimental validation of blood flow in arteries with structured tree outflow conditions,, Ann. Biomed. Eng., 28 (2000), 1281.  doi: 10.1114/1.1326031.  Google Scholar

[32]

R. W. Ogden, "Nonlinear Elastic Deformations," 2nd edition,, Dover Publications, (1997).   Google Scholar

[33]

R. W. Ogden and G. Saccomandi, Introducing mesoscopic information into constitutive equations for arterial walls,, Biomechan. Model. Mechanobiol., 6 (2007), 333.  doi: 10.1007/s10237-006-0064-8.  Google Scholar

[34]

M. F. O'Rourke and J. Hashimoto, Mechanical factors in arterial aging: A clinical perspective,, J. Amer. Coll. Cardiol., 50 (2007), 1.  doi: 10.1016/j.jacc.2007.11.003.  Google Scholar

[35]

K. H. Parker, An introduction to wave intensity analysis,, Med. Biol. Eng. Comput., 47 (2009), 175.  doi: 10.1007/s11517-009-0439-y.  Google Scholar

[36]

T. J. Pedley, Mathematical modelling of arterial fluid dynamics. Mathematical modelling of the cardiovascular system,, J. Engrg. Math., 47 (2003), 419.  doi: 10.1023/B:ENGI.0000007978.33352.59.  Google Scholar

[37]

A. D. Polyanin, "Handbook of Linear Partial Differential Equations for Engineers and Scientists,", Chapman & Hall/CRC, (2002).   Google Scholar

[38]

S. R. Pope, L. M. Ellwein, C. L. Zapata, V. Novak, C. T. Kelley and M. S. Olufsen, Estimation and identification of parameters in a lumped cerebrovascular model,, Math. Biosci. Eng., 6 (2009), 93.  doi: 10.3934/mbe.2009.6.93.  Google Scholar

[39]

A. Quarteroni, Cardiovascular mathematics,, in, (2007), 479.   Google Scholar

[40]

R. Quintanilla and K. R. Rajagopal, On Burgers fluids,, Math. Meth. Appl. Sci., 29 (2006), 2133.  doi: 10.1002/mma.760.  Google Scholar

[41]

E. A. Rosei, G. Mancia, M. F. O'Rourke, M. J. Roman, M. E. Safar, H. Smulyan, J. G. Wang, I. B. Wilkinson, B. Williams and C. Vlachopoulos, Central blood pressure measurements and antihypertensive therapy,, Hypertension, 50 (2007), 154.  doi: 10.1161/HYPERTENSIONAHA.107.090068.  Google Scholar

[42]

M. Roper and M. P. Brenner, A nonperturbative approximation for the moderate Reynolds number Navier-Stokes equations,, Proc. Nat. Acad. Sci. USA, 106 (2009), 2977.  doi: 10.1073/pnas.0810578106.  Google Scholar

[43]

F. J. Rybicki, S. Melchionna, D. Mitsouras, A. U. Coskun, A. G. Whitmore, M. Steigner, L. Nallamshetty, F. G. Welt, M. Bernaschi, M. Borkin, J. Sircar, E. Kaxiras, S. Succi, P. H. Stone and C. L. Feldman, Prediction of coronary artery plaque progression and potential rupture from 320-detector row prospectively ECG-gated single heart beat CT angiography: Lattice Boltzmann evaluation of endothelial shear stress,, Int. J. Cardiovasc. Imaging, 25 (2009), 289.  doi: 10.1007/s10554-008-9418-x.  Google Scholar

[44]

A. B. Shvartsburg, "Impulse Time-Domain Electromagnetics of Continuous Media,", Birkhäuser Boston, (1999).   Google Scholar

[45]

J. Stalhand, Determination of human arterial wall parameters from clinical data,, Biomechan. Model. Mechanobiol., 8 (2009), 141.  doi: 10.1007/s10237-008-0124-3.  Google Scholar

[46]

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