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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), 331-351. doi: 10.1007/s10665-009-9275-1.

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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), 59-68.

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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), 284-291. doi: 10.1016/j.jbiomech.2007.09.014.

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R. Caflisch, G. Majda, C. Peskin and G. Strumolo, Distortion of the arterial pulse, Math. Biosci., 51 (1980), 229-260. doi: 10.1016/0025-5564(80)90102-9.

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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), 164-193.

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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), 431-446. doi: 10.1007/s10237-008-0146-x.

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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), 1827-1836.

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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), 221-246. doi: 10.1146/annurev.bioeng.10.061807.160439.

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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), 213-251.

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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), 293-302. doi: 10.1161/01.ATV.5.3.293.

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Z. Lei, C. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Rat. Mech. Anal., 188 (2008), 371-398. doi: 10.1007/s00205-007-0089-x.

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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), 167-192.

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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), 3476-3486. doi: 10.1016/j.jbiomech.2007.05.027.

[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), 462-–472. doi: 10.1016/j.cpc.2009.10.017.

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K. H. Parker, An introduction to wave intensity analysis, Med. Biol. Eng. Comput., 47 (2009), 175-188. doi: 10.1007/s11517-009-0439-y.

[36]

T. J. Pedley, Mathematical modelling of arterial fluid dynamics. Mathematical modelling of the cardiovascular system, J. Engrg. Math., 47 (2003), 419-444. doi: 10.1023/B:ENGI.0000007978.33352.59.

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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), 93-115. doi: 10.3934/mbe.2009.6.93.

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R. Quintanilla and K. R. Rajagopal, On Burgers fluids, Math. Meth. Appl. Sci., 29 (2006), 2133-2147. doi: 10.1002/mma.760.

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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), 154-160. doi: 10.1161/HYPERTENSIONAHA.107.090068.

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M. Roper and M. P. Brenner, A nonperturbative approximation for the moderate Reynolds number Navier-Stokes equations, Proc. Nat. Acad. Sci. USA, 106 (2009), 2977-2982. doi: 10.1073/pnas.0810578106.

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J. Stalhand, Determination of human arterial wall parameters from clinical data, Biomechan. Model. Mechanobiol., 8 (2009), 141-148. doi: 10.1007/s10237-008-0124-3.

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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), 331-351. doi: 10.1007/s10665-009-9275-1.

[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), 176-89. doi: 10.1007/s10558-010-9109-9.

[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), 59-68.

[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 "Proceedings of the 7th IFAC Symposium on Modelling and Control in Biomedical Systems," Vol. 7, (2009), 181-186.

[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), 284-291. doi: 10.1016/j.jbiomech.2007.09.014.

[6]

R. Caflisch, G. Majda, C. Peskin and G. Strumolo, Distortion of the arterial pulse, Math. Biosci., 51 (1980), 229-260. doi: 10.1016/0025-5564(80)90102-9.

[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), 164-193.

[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), 431-446. doi: 10.1007/s10237-008-0146-x.

[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), 1827-1836.

[10]

S. C. Cowin, Polar fluids, Physics of Fluids, 11 (1968), 1919-1927. doi: 10.1063/1.1692219.

[11]

E. Crépeau and M. Sorine, A reduced model of pulsatile flow in an arterial compartment, Chaos Soliton Fractals, 34 (2007), 594-605.

[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), 14871-14876. doi: 10.1073/pnas.0406073101.

[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), 888-909. doi: 10.1137/07070231X.

[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), 1-46. doi: 10.1007/s00021-008-0274-1.

[15]

Y. C. Fung, "Biomechanics, Mechanical Properties of Living Tissues," Springer-Verlag, New York, 1993.

[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), 2371-2386. doi: 10.1098/rsta.2008.0307.

[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), 441-463. doi: 10.1016/S0997-7538(01)01206-2.

[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), 1-48. doi: 10.1023/A:1010835316564.

[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), 3445-3475. doi: 10.1098/rsta.2009.0091.

[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), 221-246. doi: 10.1146/annurev.bioeng.10.061807.160439.

[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), 213-251.

[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), 293-302. doi: 10.1161/01.ATV.5.3.293.

[23]

Z. Lei, C. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Rat. Mech. Anal., 188 (2008), 371-398. doi: 10.1007/s00205-007-0089-x.

[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, at Rensselaer Polytechnic Institute, Troy, New York, Regional Conference Series in Applied Mathematics, No. 17, SIAM, Philadelphia, Pa., 1975.

[25]

F. Lin and P. Zhang, On the initial-boundary value problem of the incompressible viscoelastic fluid system, Commun. Pure Appl. Math., 61 (2008), 539-558. doi: 10.1002/cpa.20219.

[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), 167-192.

[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), 2618-2627. doi: 10.1016/j.jbiomech.2008.06.022.

[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), 3476-3486. doi: 10.1016/j.jbiomech.2007.05.027.

[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), 462-–472. doi: 10.1016/j.cpc.2009.10.017.

[30]

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

[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), 1281-1299. doi: 10.1114/1.1326031.

[32]

R. W. Ogden, "Nonlinear Elastic Deformations," 2nd edition, Dover Publications, Inc., Mineola, New York, 1997.

[33]

R. W. Ogden and G. Saccomandi, Introducing mesoscopic information into constitutive equations for arterial walls, Biomechan. Model. Mechanobiol., 6 (2007), 333-344. doi: 10.1007/s10237-006-0064-8.

[34]

M. F. O'Rourke and J. Hashimoto, Mechanical factors in arterial aging: A clinical perspective, J. Amer. Coll. Cardiol., 50 (2007), 1-13. doi: 10.1016/j.jacc.2007.11.003.

[35]

K. H. Parker, An introduction to wave intensity analysis, Med. Biol. Eng. Comput., 47 (2009), 175-188. doi: 10.1007/s11517-009-0439-y.

[36]

T. J. Pedley, Mathematical modelling of arterial fluid dynamics. Mathematical modelling of the cardiovascular system, J. Engrg. Math., 47 (2003), 419-444. doi: 10.1023/B:ENGI.0000007978.33352.59.

[37]

A. D. Polyanin, "Handbook of Linear Partial Differential Equations for Engineers and Scientists," Chapman & Hall/CRC, Boca Raton, Florida, 2002.

[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), 93-115. doi: 10.3934/mbe.2009.6.93.

[39]

A. Quarteroni, Cardiovascular mathematics, in "International Conference on Mathematics," Vol. I, Eur. Math. Soc., Zürich, (2007), 479-512.

[40]

R. Quintanilla and K. R. Rajagopal, On Burgers fluids, Math. Meth. Appl. Sci., 29 (2006), 2133-2147. doi: 10.1002/mma.760.

[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), 154-160. doi: 10.1161/HYPERTENSIONAHA.107.090068.

[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), 2977-2982. doi: 10.1073/pnas.0810578106.

[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), 289-299. doi: 10.1007/s10554-008-9418-x.

[44]

A. B. Shvartsburg, "Impulse Time-Domain Electromagnetics of Continuous Media," Birkhäuser Boston, Inc., Boston, MA, 1999.

[45]

J. Stalhand, Determination of human arterial wall parameters from clinical data, Biomechan. Model. Mechanobiol., 8 (2009), 141-148. doi: 10.1007/s10237-008-0124-3.

[46]

B. N. Steele, M. S. Olufsen and C. A. Taylor, Fractal network model for simulating abdominal and lower extremity blood flow during resting and exercise conditions, Comp. Meth. Biomech. Biomed. Eng., 10 (2007), 39-51. doi: 10.1080/10255840601068638.

[47]

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