2013, 10(2): 295-318. doi: 10.3934/mbe.2013.10.295

Longitudinal displacement in viscoelastic arteries: A novel fluid-structure interaction computational model, and experimental validation

1. 

Department of Mathematics, University of Houston, 4800 Calhoun Rd, Houston, TX 77204, United States

Received  July 2012 Revised  September 2012 Published  January 2013

Recent in vivo studies, utilizing ultrasound contour and speckle tracking methods, have identified significant longitudinal displacements of the intima-media complex, and viscoelastic arterial wall properties over a cardiac cycle. Existing computational models that use thin structure approximations of arterial walls have so far been limited to models that capture only radial wall displacements. The purpose of this work is to present a simple fluid-struture interaction (FSI) model and a stable, partitioned numerical scheme, which capture both longitudinal and radial displacements, as well as viscoelastic arterial wall properties. To test the computational model, longitudinal displacement of the common carotid artery and of the stenosed coronary arteries were compared with experimental data found in literature, showing excellent agreement. We found that, unlike radial displacement, longitudinal displacement in stenotic lesions is highly dependent on the stenotic geometry. We also showed that longitudinal displacement in atherosclerotic arteries is smaller than in healthy arteries, which is in line with the recent in vivo measurements that associate plaque burden with reduced total longitudinal wall displacement.
    This work presents a first step in understanding the role of longitudinal displacement in physiology and pathophysiology of arterial wall mechanics using computer simulations.
Citation: Martina Bukač, Sunčica Čanić. Longitudinal displacement in viscoelastic arteries: A novel fluid-structure interaction computational model, and experimental validation. Mathematical Biosciences & Engineering, 2013, 10 (2) : 295-318. doi: 10.3934/mbe.2013.10.295
References:
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Å. R. Ahlgren, M. Cinthio, S. Steen, H. W. Persson, T. Sjöberg and K. Lindström, Effects of adrenaline on longitudinal arterial wall movements and resulting intramural shear strain: A first report, Clin. Physiol. Funct. Imaging, 29 (2009), 353-359.

[2]

R. L. Armentano, J. G. Barra, J. Levenson, A. Simon and R. H. Pichel, Arterial wall mechanics in conscious dogs: Assessment of viscous, inertial, and elastic moduli to characterize aortic wall behavior, Circulation Research, 76 (1995), 468-478.

[3]

R. Armentano, J. L. Megnien, A. Simon, F. Bellenfant, J. Barra and J. Levenson, Effects of hypertension on viscoelasticity of carotid and femoral arteries in humans, Hypertension, 26 (1995), 48-54.

[4]

S. Badia, F. Nobile and C. Vergara, Fluid-structure partitioned procedures based on Robin transmission conditions, J. Comput. Phys., 227 (2008), 7027-7051. doi: 10.1016/j.jcp.2008.04.006.

[5]

S. Badia, A. Quaini and A. Quarteroni, Splitting methods based on algebraic factorization for fluid-structure interaction, SIAM J. Sci. Comput., 30 (2008), 1778-1805. doi: 10.1137/070680497.

[6]

J. Bouthier, A. Benetos, A. Simon, J. Levenson and M. Safar, Pulsed Doppler evaluation of diameter, blood velocity and blood flow of common carotid artery in sustained essential hypertension, J. Cardiovasc. Pharmacol., 7 (1985), S99-S104.

[7]

M. Bukač, S. Čanić, R. Glowinski, J. Tambača and A. Quaini, Fluid-structure interaction in blood flow capturing non-zero longitudinal structure displacement, Journal of Computational Physics. DOI: http://dx.doi.org/10.1016/j.bbr.2011.03.031 (2012).

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B. E. Bulwer and J. M. Rivero, "Echocardiography Pocket Guide: The Transthoracic Examination," Jones & Bartlett Learning, 2010.

[9]

C. Bussy, P. Boutouyrie, P. Lacolley, P. Challande and S. Laurent, Intrinsic stiffness of the carotid arterial wall material in essential hypertensives, Hypertension, 35 (2000), 1049-1054.

[10]

S. Čanić, J. Tambaca, G. Guidoboni, A. Mikelic, 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. doi: 10.1137/060651562.

[11]

S. Čanić, M. Bukač and B. Muha, Stability of the kinematically coupled $\beta$-scheme for simulation of fluid-structure interaction problems in hemodynamics, Submitted, (2012).

[12]

P. Ciarlet, "Mathematical Elasticity: Theory of Shells," North-Holland, Amsterdam, 2000.

[13]

M. Cinthio, Å. R. Ahlgren, J. Bergkvist, T. Jansson, H. W. Persson and K. Lindstrom, Longitudinal movements and resulting shear strain of the arterial wall, Am. J. Physiol. Heart Circ. Physiol., 291 (2006), H394-H402.

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R. S. C. Cobbold, "Foundations of Biomedical Ultrasound," Oxford University Press, USA, 2007.

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V. Deplano and M. Siouffi, Experimental and numerical study of pulsatile flows through stenosis: Wall shear stress analysis, Journal of Biomechanics, 32 (1999), 1081-1090.

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M. M. Dizaji, M. Maerefat and S. Rahgozar, Estimation of carotid artery pulse wave velocity by doppler ultrasonography, J. Tehran Heart Cent, 4 (2009), 91-96.

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J. T. Dodge, B. G. Brown, E. L. Bolson and H. T. Dodge, Lumen diameter of normal human coronary arteries. Influence of age, sex, anatomic variation, and left ventricular hypertrophy or dilation, Circulation, 86 (1992), 232-246.

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A. M. Fallon, L. P. Dassi, U. M. Marzec, S. R. Hanson and A. P. Yoganathan, Procoagulant properties of flow fields in stenotic and expansive orifices, Ann. Biomed. Eng., 36 (2008), 1-13.

[21]

L. Formaggia, A. Quarteroni and A. Veneziani, "Cardiovascular Mathematics," Springer-Verlag Italia, Milano, 2009. doi: 10.1007/978-88-470-1152-6.

[22]

Y. C. Fung, "Biomechanics: Circulation," Second Edition. Springer-Verlag, New York, 1984.

[23]

R. Glowinski, "Finite Element Methods for Incompressible Viscous Flow," in "Handbook of Numerical Analysis" (Eds. P.G.Ciarlet and J.-L.Lions), 9 North-Holland, Amsterdam, (2003).

[24]

G. Guidoboni, R. Glowinski, N. Cavallini and S. Čanić, Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow, J. Comput. Phys., 228 (2009), 6916-6937. doi: 10.1016/j.jcp.2009.06.007.

[25]

T. Hozumi, K. Yoshida, T. Akasaka, Y. Asami, Y. Kanzaki, Y. Ueda, A. Yamamuro, T. Takagi and J. Yoshikawa, Value of acceleration flow and the prestenotic to stenotic coronary flow velocity ratio by transthoracic color Doppler echocardiography in noninvasive diagnosis of restenosis after percutaneous transluminal coronary angioplasty, J. Am. Coll. Cardiol., 35 (2000), 164-168.

[26]

T. J. R. Hughes, W. K. Liu and T. K. Zimmermann, Lagrangian-eulerian finite element formulation for incompressible viscous flows, Comput. Methods Appl. Mech. Eng., 29 (1981), 329-349. doi: 10.1016/0045-7825(81)90049-9.

[27]

J. D. Humphrey and S. L. Delange, "An Introduction to Biomechanics: Solids and Fluids, Analysis and Design," Springer-Verlag, New York, 2004.

[28]

J. D. Humphrey, "Cardiovascular Solid Mechanics: Cells, Tissues, and Organs," Springer-Verlag, New York, 2002.

[29]

E. L. Johnson, P. G. Yock, V. K. Hargrave, J. P. Srebro, S. M. Manubens, W. Seitz and T. A. Ports, Assessment of severity of coronary stenoses using a Doppler catheter. Validation of a method based on the continuity equation, Circulation, 80 (1989), 625-635.

[30]

M. Juonala, M. J. Jarvisalo, N. Maki-Torkko, M. Kahonen, J. S. A. Viikari and O. T. Raitakari, Risk factors identified in childhood and decreased carotid artery elasticity in adulthood: The cardiovascular risk in young finns study, Circulation, 112 (2005), 1486-1493.

[31]

J. Krejza, M. Arkuszewski, S. E. Kasner, J. Weigele, A. Ustymowicz, R. W. Hurst, B. L. Cucchiara and S. R. Messe, Carotid artery diameter in men and women and the relation to body and neck size, Stroke, 37 (2006), 1103-1105.

[32]

V. S. Lee, B. S. Hertzberg, M. A. Kliewer and B. A. Carroll, Assessment of stenosis: Implications of variability of Doppler measurements in normal-appearing carotid arteries, Radiology, 212 (1999), 493-498.

[33]

A. Leuprecht, K. Perktold, M. Prosi, T. Berk, W. Trubel and H. Schima, Numerical study of hemodynamics and wall mechanics in distal end-to-side anastomoses of bypass grafts, J. Biomech., 35 (2002), 225-236.

[34]

R. A. Levine, A. Jimoh, E. G. Cape, S. McMillan, A. P. Yoganathan and A. E. Weyman, Pressure recovery distal to a stenosis: Potential cause of gradient "Overestimation'' by doppler echocardiography, J. Am. Coll. Cardiol., 13 (1989), 706-715.

[35]

A. Manzoni1, A. Quarteroni and G. Rozza, Model reduction techniques for fast blood flow simulation in parametrized geometries, International Journal for Numerical Methods in Biomedical Engineering, 28 (2012), 604-625. doi: 10.1002/cnm.1465.

[36]

K. M. J. Marques, H. J. Spruijt, C. Boer, N. Westerhof, C. A. Visser and F. C. Visser, The diastolic flow-pressure gradient relation in coronary stenoses in humans, J. Am. Coll. Cardiol., 39 (2002), 1630-1635.

[37]

J. M. Meinders, L. Kornet and A. P. G. Hoeks, Assessment of spatial inhomogeneities in intima media thickness along an arterial segment using its dynamic behavior, Am. J. Physiol. Heart Circ. Physiol., 285 (2003), H384-H391.

[38]

M. Mokhtari-Dizaji, M. Montazeri and H. Saberi, Differentiation of mild and severe stenosis with motion estimation in ultrasound images, Ultrasound Med. Biol., 32 (2006), 1493-1498.

[39]

W. W. Nichols and F. O. R. Michael, "McDonald's Blood Flow in Arteries: Theoretical, Experimental and Clinical Principles," Hodder Arnold London, UK, 2005.

[40]

M. Persson, Å. Rydén Ahlgren, T. Jansson, A. Eriksson, H. W. Persson and K. Lindstrom, A new non-invasive ultrasonic method for simultaneous measurements of longitudinal and radial arterial wall movements: First in vivo trial, Clin. Physiol. Funct. Imaging, 23 (2003), 247-251.

[41]

R. A. Ribeiro, J. A. S. Ribeiro, O. A. Rodrigues Filho, A. G. Caetano and V. P. S. Fazan, Common carotid artery bifurcation levels related to clinical relevant anatomical landmarks, Int. J. Morphol., 24 (2006), 413-416.

[42]

E. M. Rohren, M. A. Kliewer, B. A. Carroll and B. S. Hertzberg, A spectrum of doppler waveforms in the carotid and vertebral arteries, AJR. Am. J. Roentgenol., 181 (2003), 1695-1704.

[43]

S. K. Samijo, J. M. Willigers, R. Barkhuysen, P. Kitslaar, R. S. Reneman, P. J. Brands and A. P. G. Hoeks, Wall shear stress in the human common carotid artery as function of age and gender, Cardiovascular Research, 39 (1998), 515-522.

[44]

S. Svedlund and L. M. Gan, Longitudinal common carotid artery wall motion is associated with plaque burden in man and mouse, Atherosclerosis, 217 (2011), 120-124.

[45]

S. Svedlund and L. M. Gan, Longitudinal wall motion of the common carotid artery can be assessed by velocity vector imaging, Clin. Physiol. Funct. Imaging, 31 (2011), 32-38.

[46]

J. Tambača, S. Čanić and A. Mikelić, Effective model of the fluid flow through elastic tube with variable radius, Grazer Math. Ber., 348 (2005), 91-112.

[47]

E. M. Urbina, S. R. Srinivasan, R. L. Kieltyka, R. Tang, M. G. Bond, W. Chen and G. S. Berenson, Correlates of carotid artery stiffness in young adults: The Bogalusa heart study, Atherosclerosis, 176 (2004), 157-164.

[48]

R. K. Warriner, K. W. Johnston and R. S. C. Cobbold, A viscoelastic model of arterial wall motion in pulsatile flow: Implications for Doppler ultrasound clutter assessment, Physiol. Meas., 29 (2008), 157-179.

[49]

I. Weinberg., "Carotid Duplex Protocol," Vascular Medicine, 2012.

[50]

J. Wu, B. Min Yun, A. M. Fallon, S. R. Hanson, C. K. Aidun and A. P. Yoganathan, Numerical investigation of the effects of channel geometry on platelet activation and blood damage, Ann. Biomed. Eng., 39 (2011), 897-910.

show all references

References:
[1]

Å. R. Ahlgren, M. Cinthio, S. Steen, H. W. Persson, T. Sjöberg and K. Lindström, Effects of adrenaline on longitudinal arterial wall movements and resulting intramural shear strain: A first report, Clin. Physiol. Funct. Imaging, 29 (2009), 353-359.

[2]

R. L. Armentano, J. G. Barra, J. Levenson, A. Simon and R. H. Pichel, Arterial wall mechanics in conscious dogs: Assessment of viscous, inertial, and elastic moduli to characterize aortic wall behavior, Circulation Research, 76 (1995), 468-478.

[3]

R. Armentano, J. L. Megnien, A. Simon, F. Bellenfant, J. Barra and J. Levenson, Effects of hypertension on viscoelasticity of carotid and femoral arteries in humans, Hypertension, 26 (1995), 48-54.

[4]

S. Badia, F. Nobile and C. Vergara, Fluid-structure partitioned procedures based on Robin transmission conditions, J. Comput. Phys., 227 (2008), 7027-7051. doi: 10.1016/j.jcp.2008.04.006.

[5]

S. Badia, A. Quaini and A. Quarteroni, Splitting methods based on algebraic factorization for fluid-structure interaction, SIAM J. Sci. Comput., 30 (2008), 1778-1805. doi: 10.1137/070680497.

[6]

J. Bouthier, A. Benetos, A. Simon, J. Levenson and M. Safar, Pulsed Doppler evaluation of diameter, blood velocity and blood flow of common carotid artery in sustained essential hypertension, J. Cardiovasc. Pharmacol., 7 (1985), S99-S104.

[7]

M. Bukač, S. Čanić, R. Glowinski, J. Tambača and A. Quaini, Fluid-structure interaction in blood flow capturing non-zero longitudinal structure displacement, Journal of Computational Physics. DOI: http://dx.doi.org/10.1016/j.bbr.2011.03.031 (2012).

[8]

B. E. Bulwer and J. M. Rivero, "Echocardiography Pocket Guide: The Transthoracic Examination," Jones & Bartlett Learning, 2010.

[9]

C. Bussy, P. Boutouyrie, P. Lacolley, P. Challande and S. Laurent, Intrinsic stiffness of the carotid arterial wall material in essential hypertensives, Hypertension, 35 (2000), 1049-1054.

[10]

S. Čanić, J. Tambaca, G. Guidoboni, A. Mikelic, 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. doi: 10.1137/060651562.

[11]

S. Čanić, M. Bukač and B. Muha, Stability of the kinematically coupled $\beta$-scheme for simulation of fluid-structure interaction problems in hemodynamics, Submitted, (2012).

[12]

P. Ciarlet, "Mathematical Elasticity: Theory of Shells," North-Holland, Amsterdam, 2000.

[13]

M. Cinthio, Å. R. Ahlgren, J. Bergkvist, T. Jansson, H. W. Persson and K. Lindstrom, Longitudinal movements and resulting shear strain of the arterial wall, Am. J. Physiol. Heart Circ. Physiol., 291 (2006), H394-H402.

[14]

R. S. C. Cobbold, "Foundations of Biomedical Ultrasound," Oxford University Press, USA, 2007.

[15]

CREATIS., "Biomedical Imaging Laboratory," University of Lyon-INSA, Lyon, France, 2011.

[16]

V. Deplano and M. Siouffi, Experimental and numerical study of pulsatile flows through stenosis: Wall shear stress analysis, Journal of Biomechanics, 32 (1999), 1081-1090.

[17]

M. M. Dizaji, M. Maerefat and S. Rahgozar, Estimation of carotid artery pulse wave velocity by doppler ultrasonography, J. Tehran Heart Cent, 4 (2009), 91-96.

[18]

J. T. Dodge, B. G. Brown, E. L. Bolson and H. T. Dodge, Lumen diameter of normal human coronary arteries. Influence of age, sex, anatomic variation, and left ventricular hypertrophy or dilation, Circulation, 86 (1992), 232-246.

[19]

J. Donea, "Arbitrary Lagrangian-Eulerian Finite Element Methods, In: Computational Methods for Transient Analysis," North-Holland, Amsterdam, 1983.

[20]

A. M. Fallon, L. P. Dassi, U. M. Marzec, S. R. Hanson and A. P. Yoganathan, Procoagulant properties of flow fields in stenotic and expansive orifices, Ann. Biomed. Eng., 36 (2008), 1-13.

[21]

L. Formaggia, A. Quarteroni and A. Veneziani, "Cardiovascular Mathematics," Springer-Verlag Italia, Milano, 2009. doi: 10.1007/978-88-470-1152-6.

[22]

Y. C. Fung, "Biomechanics: Circulation," Second Edition. Springer-Verlag, New York, 1984.

[23]

R. Glowinski, "Finite Element Methods for Incompressible Viscous Flow," in "Handbook of Numerical Analysis" (Eds. P.G.Ciarlet and J.-L.Lions), 9 North-Holland, Amsterdam, (2003).

[24]

G. Guidoboni, R. Glowinski, N. Cavallini and S. Čanić, Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow, J. Comput. Phys., 228 (2009), 6916-6937. doi: 10.1016/j.jcp.2009.06.007.

[25]

T. Hozumi, K. Yoshida, T. Akasaka, Y. Asami, Y. Kanzaki, Y. Ueda, A. Yamamuro, T. Takagi and J. Yoshikawa, Value of acceleration flow and the prestenotic to stenotic coronary flow velocity ratio by transthoracic color Doppler echocardiography in noninvasive diagnosis of restenosis after percutaneous transluminal coronary angioplasty, J. Am. Coll. Cardiol., 35 (2000), 164-168.

[26]

T. J. R. Hughes, W. K. Liu and T. K. Zimmermann, Lagrangian-eulerian finite element formulation for incompressible viscous flows, Comput. Methods Appl. Mech. Eng., 29 (1981), 329-349. doi: 10.1016/0045-7825(81)90049-9.

[27]

J. D. Humphrey and S. L. Delange, "An Introduction to Biomechanics: Solids and Fluids, Analysis and Design," Springer-Verlag, New York, 2004.

[28]

J. D. Humphrey, "Cardiovascular Solid Mechanics: Cells, Tissues, and Organs," Springer-Verlag, New York, 2002.

[29]

E. L. Johnson, P. G. Yock, V. K. Hargrave, J. P. Srebro, S. M. Manubens, W. Seitz and T. A. Ports, Assessment of severity of coronary stenoses using a Doppler catheter. Validation of a method based on the continuity equation, Circulation, 80 (1989), 625-635.

[30]

M. Juonala, M. J. Jarvisalo, N. Maki-Torkko, M. Kahonen, J. S. A. Viikari and O. T. Raitakari, Risk factors identified in childhood and decreased carotid artery elasticity in adulthood: The cardiovascular risk in young finns study, Circulation, 112 (2005), 1486-1493.

[31]

J. Krejza, M. Arkuszewski, S. E. Kasner, J. Weigele, A. Ustymowicz, R. W. Hurst, B. L. Cucchiara and S. R. Messe, Carotid artery diameter in men and women and the relation to body and neck size, Stroke, 37 (2006), 1103-1105.

[32]

V. S. Lee, B. S. Hertzberg, M. A. Kliewer and B. A. Carroll, Assessment of stenosis: Implications of variability of Doppler measurements in normal-appearing carotid arteries, Radiology, 212 (1999), 493-498.

[33]

A. Leuprecht, K. Perktold, M. Prosi, T. Berk, W. Trubel and H. Schima, Numerical study of hemodynamics and wall mechanics in distal end-to-side anastomoses of bypass grafts, J. Biomech., 35 (2002), 225-236.

[34]

R. A. Levine, A. Jimoh, E. G. Cape, S. McMillan, A. P. Yoganathan and A. E. Weyman, Pressure recovery distal to a stenosis: Potential cause of gradient "Overestimation'' by doppler echocardiography, J. Am. Coll. Cardiol., 13 (1989), 706-715.

[35]

A. Manzoni1, A. Quarteroni and G. Rozza, Model reduction techniques for fast blood flow simulation in parametrized geometries, International Journal for Numerical Methods in Biomedical Engineering, 28 (2012), 604-625. doi: 10.1002/cnm.1465.

[36]

K. M. J. Marques, H. J. Spruijt, C. Boer, N. Westerhof, C. A. Visser and F. C. Visser, The diastolic flow-pressure gradient relation in coronary stenoses in humans, J. Am. Coll. Cardiol., 39 (2002), 1630-1635.

[37]

J. M. Meinders, L. Kornet and A. P. G. Hoeks, Assessment of spatial inhomogeneities in intima media thickness along an arterial segment using its dynamic behavior, Am. J. Physiol. Heart Circ. Physiol., 285 (2003), H384-H391.

[38]

M. Mokhtari-Dizaji, M. Montazeri and H. Saberi, Differentiation of mild and severe stenosis with motion estimation in ultrasound images, Ultrasound Med. Biol., 32 (2006), 1493-1498.

[39]

W. W. Nichols and F. O. R. Michael, "McDonald's Blood Flow in Arteries: Theoretical, Experimental and Clinical Principles," Hodder Arnold London, UK, 2005.

[40]

M. Persson, Å. Rydén Ahlgren, T. Jansson, A. Eriksson, H. W. Persson and K. Lindstrom, A new non-invasive ultrasonic method for simultaneous measurements of longitudinal and radial arterial wall movements: First in vivo trial, Clin. Physiol. Funct. Imaging, 23 (2003), 247-251.

[41]

R. A. Ribeiro, J. A. S. Ribeiro, O. A. Rodrigues Filho, A. G. Caetano and V. P. S. Fazan, Common carotid artery bifurcation levels related to clinical relevant anatomical landmarks, Int. J. Morphol., 24 (2006), 413-416.

[42]

E. M. Rohren, M. A. Kliewer, B. A. Carroll and B. S. Hertzberg, A spectrum of doppler waveforms in the carotid and vertebral arteries, AJR. Am. J. Roentgenol., 181 (2003), 1695-1704.

[43]

S. K. Samijo, J. M. Willigers, R. Barkhuysen, P. Kitslaar, R. S. Reneman, P. J. Brands and A. P. G. Hoeks, Wall shear stress in the human common carotid artery as function of age and gender, Cardiovascular Research, 39 (1998), 515-522.

[44]

S. Svedlund and L. M. Gan, Longitudinal common carotid artery wall motion is associated with plaque burden in man and mouse, Atherosclerosis, 217 (2011), 120-124.

[45]

S. Svedlund and L. M. Gan, Longitudinal wall motion of the common carotid artery can be assessed by velocity vector imaging, Clin. Physiol. Funct. Imaging, 31 (2011), 32-38.

[46]

J. Tambača, S. Čanić and A. Mikelić, Effective model of the fluid flow through elastic tube with variable radius, Grazer Math. Ber., 348 (2005), 91-112.

[47]

E. M. Urbina, S. R. Srinivasan, R. L. Kieltyka, R. Tang, M. G. Bond, W. Chen and G. S. Berenson, Correlates of carotid artery stiffness in young adults: The Bogalusa heart study, Atherosclerosis, 176 (2004), 157-164.

[48]

R. K. Warriner, K. W. Johnston and R. S. C. Cobbold, A viscoelastic model of arterial wall motion in pulsatile flow: Implications for Doppler ultrasound clutter assessment, Physiol. Meas., 29 (2008), 157-179.

[49]

I. Weinberg., "Carotid Duplex Protocol," Vascular Medicine, 2012.

[50]

J. Wu, B. Min Yun, A. M. Fallon, S. R. Hanson, C. K. Aidun and A. P. Yoganathan, Numerical investigation of the effects of channel geometry on platelet activation and blood damage, Ann. Biomed. Eng., 39 (2011), 897-910.

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