2014, 11(5): 1199-1214. doi: 10.3934/mbe.2014.11.1199

Effect of residual stress on peak cap stress in arteries

1. 

St. Olaf College, 1520 St. Olaf Ave, Northfield, MN 55057, United States

Received  January 2014 Revised  April 2014 Published  June 2014

Vulnerable plaques are a subset of atherosclerotic plaques that are prone to rupture when high stresses occur in the cap. The roles of residual stress, plaque morphology, and cap stiffness on the cap stress are not completely understood. Here, arteries are modeled within the framework of nonlinear elasticity as incompressible cylindrical structures that are residually stressed through differential growth. These structures are assumed to have a nonlinear, anisotropic, hyperelastic response to stresses in the media and adventitia layers and an isotropic response in the intima and necrotic layers. The effect of differential growth on the peak stress is explored in a simple, concentric geometry and it is shown that axial differential growth decreases the peak stress in the inner layer. Furthermore, morphological risk factors are explored. The peak stress in residually stressed cylinders is not greatly affected by changing the thickness of the intima. The thickness of the necrotic layer is shown to be the most important morphological feature that affects the peak stress in a residually stressed vessel.
Citation: Rebecca Vandiver. Effect of residual stress on peak cap stress in arteries. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1199-1214. doi: 10.3934/mbe.2014.11.1199
References:
[1]

A. Akyildiz, L. Speelman, H. Nieuwstadt, S. J. W. Van Der and F. Gijsen, Influence of plaque geometry on peak cap stress,, Proceedings of the ASME 2011 Summer Bioegnineering ConferenceArtery Research, 5 (2011), 159.  doi: 10.1016/j.artres.2011.10.047.  Google Scholar

[2]

A. Akyildiz, L. Speelman, H. van Brummelen, M. Gutiérrez, R. Virmani, A. van der Lugt, A. Van Der Steen, J. Wentzel and F. Gijsen, Effects of intima stiffness and plaque morphology on peak cap stress,, Biomedical Engineering Online, 10 (2011), 1.  doi: 10.1186/1475-925X-10-25.  Google Scholar

[3]

R. Baldewsing, C. De Korte, J. Schaar, F. Mastik and Van Der Steen, Finite element modeling and intravascular ultrasound elastography of vulnerable plaques: Parameter variation,, Ultrasonics, 42 (2004), 723.  doi: 10.1016/j.ultras.2003.11.017.  Google Scholar

[4]

S. Barrett, M. Sutcliffe, S. Howarth, Z. Li and J. Gillard, Experimental measurement of the mechanical properties of carotid atherothrombotic plaque fibrous cap,, Journal of Biomechanics, 42 (2009), 1650.  doi: 10.1016/j.jbiomech.2009.04.025.  Google Scholar

[5]

E. Falk, K. S. Prediman and F. Valenin, Coronary Plaque Disruption,, Circulation, 92 (1995), 657.  doi: 10.1161/01.CIR.92.3.657.  Google Scholar

[6]

G. Finet, J. Ohayon and G. Rioufol, Biomechanical interaction between cap thickness, lipid core composition and blood pressure in vulnerable coronary plaque: Impact on stability or instability,, Coronary artery disease, 15 (2004), 13.  doi: 10.1097/00019501-200402000-00003.  Google Scholar

[7]

A. Goriely and R. Vandiver, On the mechanical stability of growing arteries,, IMA Journal of Applied Mathematics, 75 (2010), 549.  doi: 10.1093/imamat/hxq021.  Google Scholar

[8]

G. Holzapfel, Nonlinear Solid Mechanics: A Continuum Approach for Engineering,, John Wiley & Sons Ltd. 2000., (2000).   Google Scholar

[9]

G. Holzapfel, T. Gasser and R. Ogden, Comparison of a multi-layer structural model for arterial walls with a Fung-type model, and issues of material stability,, Journal of Biomechanical Engineering, 126 (2004), 264.  doi: 10.1115/1.1695572.  Google Scholar

[10]

G. Holzapfel, G. Sommer, M. Auer, P. Regitnig and R. Ogden, Layer-specific 3D residual deformations of human aortas with non-atherosclerotic intimal thickening,, Annals of Biomedical Engineering, 35 (2007), 530.  doi: 10.1007/s10439-006-9252-z.  Google Scholar

[11]

G. Holzapfel, G. Sommer, C. Gasser and P. Regitnig, Determination of layer-specific mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling,, American Journal of Physiology-Heart and Circulatory Physiology, 289 (2005).  doi: 10.1152/ajpheart.00934.2004.  Google Scholar

[12]

P. Kalita and R. Schaefer, Mechanical models of artery walls,, Archives of Computational Methods in Engineering, 15 (2008), 1.  doi: 10.1007/s11831-007-9015-5.  Google Scholar

[13]

E. Lee, Elastic-plastic deformation at finite strains,, J. Appl. Mech., 36 (1968), 1.  doi: 10.1115/1.3564580.  Google Scholar

[14]

R. Lee, A. Grodzinsky, E. Frank, R. Kamm and F. Schoen, Structure-dependent dynamic mechanical behavior of fibrous caps from human atherosclerotic plaques,, Circulation, 83 (1991), 1764.  doi: 10.1161/01.CIR.83.5.1764.  Google Scholar

[15]

R. Lee, S. Richardson, H. Loree, A. Grodzinsky, S. Gharib, F. Schoen and N. Pandian, Prediction of mechanical properties of human atherosclerotic tissue by high-frequency intravascular ultrasound imaging. An in vitro study,, Arteriosclerosis, 12 (1992), 1.  doi: 10.1161/01.ATV.12.1.1.  Google Scholar

[16]

M. Li, J. Beech-Brandt, L. John, P. Hoskins and W. Easson, Numerical analysis of pulsatile blood flow and vessel wall mechanics in different degrees of stenoses,, Journal of Biomechanics, 40 (2007), 3715.  doi: 10.1016/j.jbiomech.2007.06.023.  Google Scholar

[17]

A. Li, S. Howarth and R. Trivedi, Stress analysis of carotid plaque rupture based on in vivo high resolution MRI,, Journal of Biomechanics, 39 (2006), 2611.  doi: 10.1016/j.jbiomech.2005.08.022.  Google Scholar

[18]

S. Liu and Y. Fung, Zero-stress states of arteries,, Journal of Biomechanical Engineering, 110 (1988), 82.  doi: 10.1115/1.3108410.  Google Scholar

[19]

H. Loree, R. Kamm, R. Stringfellow and R. Lee, Effects of fibrous cap thickness on peak circumferential stress in model atherosclerotic vessels,, Circulation research, 71 (1992), 850.  doi: 10.1161/01.RES.71.4.850.  Google Scholar

[20]

H. Loree, B. Tobias, L. Gibson, R. Kamm, D. Small and R. Lee, Mechanical properties of model atherosclerotic lesion lipid pools,, Arteriosclerosis, 14 (1994), 230.  doi: 10.1161/01.ATV.14.2.230.  Google Scholar

[21]

M. Naghavi, P. Libby, E. Falk, S. Casscells, S. Litovsky, J. Rumberger, J. Badimon, C. Stefanadis, P. Moreno and P. Pasterkamp, From vulnerable plaque to vulnerable patient a call for new definitions and risk assessment strategies: Part I,, Circulation, 108 (2003), 1664.  doi: 10.1161/01.CIR.0000087480.94275.97.  Google Scholar

[22]

J. Ohayon, N. Mesnler, A. Brolsat, J. Toczek, L. Rlou and Pl Tracqui, Elucidating atherosclerotic vulnerable plaque rupture by modeling cross substitution of ApoE mouse and human plaque components stiffnesses,, Biomech. Model. Mechanobiol., 11 (2011), 801.  doi: 10.1007/s10237-011-0353-8.  Google Scholar

[23]

J. Ohayon, O. Dubreuil, P. Tracqui, S. Le Floc'h, G. Rioufol, L. Chalabreysse, F. Thivolet, R. Pettigrew and G. Finet, Influence of residual stress/strain on the biomechanical stability of vulnerable coronary plaques: Potential impact for evaluating the risk of plaque rupture,, American Journal of Physiology-Heart and Circulatory Physiology, 293 (2007).  doi: 10.1152/ajpheart.00018.2007.  Google Scholar

[24]

J. Ohayon, G. Finet, A. Gharib, D. Herzka, P. Tracqui, J. Heroux, G. Rioufol, M. Kotys, A. Elagha and R. Pettigrew, Necrotic core thickness and positive arterial remodeling index: Emergent biomechanical factors for evaluating the risk of plaque rupture,, American Journal of Physiology-Heart and Circulatory Physiology, 295 (2008).  doi: 10.1152/ajpheart.00005.2008.  Google Scholar

[25]

A. Rachev, Theoretical study of the effect of stress-dependent remodeling on arterial geometry under hypertensive conditions,, Journal of Biomechanics, 30 (1997), 819.  doi: 10.1016/S0021-9290(97)00032-8.  Google Scholar

[26]

E. Rodriguez, A. Hoger and A. McCulloch, Stress-dependent finite growth in soft elastic tissues,, Journal of Biomechanics, 27 (1994), 455.  doi: 10.1016/0021-9290(94)90021-3.  Google Scholar

[27]

U. Sadat, Z. Teng and J. Gillard, Biomechanical structural stresses of atherosclerotic plaques,, Expert Review of Cardiovascular Therapy, 8 (2010), 1469.  doi: 10.1586/erc.10.130.  Google Scholar

[28]

L. Taber, Biomechanical growth laws for muscle tissue,, Journal of Theoretical Biology, 193 (1998), 201.  doi: 10.1006/jtbi.1997.0618.  Google Scholar

[29]

L. Taber, A model for aortic growth based on fluid shear and fiber stresses,, Journal of Biomechanical Engineering, 120 (1998), 348.  doi: 10.1115/1.2798001.  Google Scholar

[30]

L. Taber and D. Eggers, Theoretical study of stress-modulated growth in the aorta,, Journal of Theoretical Biology, 180 (1996), 343.  doi: 10.1006/jtbi.1996.0107.  Google Scholar

[31]

L. Taber and J. Humphrey, Stress-modulated growth, residual stress, and vascular heterogeneity,, Journal of Biomechanical Engineering, 123 (2001), 528.  doi: 10.1115/1.1412451.  Google Scholar

[32]

D. Tang, Z. Teng, G. Canton, C. Yang, M. Ferguson, X. Huang, J. Zheng, P. Woodard and C. Yuan, Sites of rupture in human atherosclerotic carotid plaques are associated with high structural stresses,, Stroke, 40 (2009), 3258.   Google Scholar

[33]

D. Tang, C. Yang, J. Zheng, P. Woodard, G. Sicard, J. Saffitz and C. Yuan, 3D MRI-based multicomponent FSI models for atherosclerotic plaques,, Annals of Biomedical Engineering, 32 (2004), 947.   Google Scholar

[34]

Z. Teng, G. Canton, C. Yuan, M. Ferguson, C. Yang, X. Huang, J. Zheng, P. Woodard and D. Tang, 3D critical plaque wall stress is a better predictor of carotid plaque rupture sites than flow shear stress: An in vivo MRI-based 3D FSI study,, Journal of Biomechanical Engineering, 132 (2010).   Google Scholar

[35]

R. Vaishnav and J. Vossoughi, Residual stress and strain in aortic segments,, Journal of Biomechanics, 20 (1987), 235.  doi: 10.1016/0021-9290(87)90290-9.  Google Scholar

[36]

J. Valenta, J. Svoboda, D. Valerianova and K. Vitek, Residual strain in human atherosclerotic coronary arteries and age related geometrical changes,, Biomedical Materials and Engineering, 9 (1999), 311.   Google Scholar

[37]

Y. Vengrenyuk, S. Carlier, S. Xanthos, L. Cardos, P. Ganatos, R. Virmani, S. Einav, L. Gilchrist and S. Weinbaum, A hypothesis for vulnerable plaque rupture due to stress-induced debonding around cellular microcalcifications in thin fibrous caps,, PNAS, 103 (2006), 14678.  doi: 10.1073/pnas.0606310103.  Google Scholar

show all references

References:
[1]

A. Akyildiz, L. Speelman, H. Nieuwstadt, S. J. W. Van Der and F. Gijsen, Influence of plaque geometry on peak cap stress,, Proceedings of the ASME 2011 Summer Bioegnineering ConferenceArtery Research, 5 (2011), 159.  doi: 10.1016/j.artres.2011.10.047.  Google Scholar

[2]

A. Akyildiz, L. Speelman, H. van Brummelen, M. Gutiérrez, R. Virmani, A. van der Lugt, A. Van Der Steen, J. Wentzel and F. Gijsen, Effects of intima stiffness and plaque morphology on peak cap stress,, Biomedical Engineering Online, 10 (2011), 1.  doi: 10.1186/1475-925X-10-25.  Google Scholar

[3]

R. Baldewsing, C. De Korte, J. Schaar, F. Mastik and Van Der Steen, Finite element modeling and intravascular ultrasound elastography of vulnerable plaques: Parameter variation,, Ultrasonics, 42 (2004), 723.  doi: 10.1016/j.ultras.2003.11.017.  Google Scholar

[4]

S. Barrett, M. Sutcliffe, S. Howarth, Z. Li and J. Gillard, Experimental measurement of the mechanical properties of carotid atherothrombotic plaque fibrous cap,, Journal of Biomechanics, 42 (2009), 1650.  doi: 10.1016/j.jbiomech.2009.04.025.  Google Scholar

[5]

E. Falk, K. S. Prediman and F. Valenin, Coronary Plaque Disruption,, Circulation, 92 (1995), 657.  doi: 10.1161/01.CIR.92.3.657.  Google Scholar

[6]

G. Finet, J. Ohayon and G. Rioufol, Biomechanical interaction between cap thickness, lipid core composition and blood pressure in vulnerable coronary plaque: Impact on stability or instability,, Coronary artery disease, 15 (2004), 13.  doi: 10.1097/00019501-200402000-00003.  Google Scholar

[7]

A. Goriely and R. Vandiver, On the mechanical stability of growing arteries,, IMA Journal of Applied Mathematics, 75 (2010), 549.  doi: 10.1093/imamat/hxq021.  Google Scholar

[8]

G. Holzapfel, Nonlinear Solid Mechanics: A Continuum Approach for Engineering,, John Wiley & Sons Ltd. 2000., (2000).   Google Scholar

[9]

G. Holzapfel, T. Gasser and R. Ogden, Comparison of a multi-layer structural model for arterial walls with a Fung-type model, and issues of material stability,, Journal of Biomechanical Engineering, 126 (2004), 264.  doi: 10.1115/1.1695572.  Google Scholar

[10]

G. Holzapfel, G. Sommer, M. Auer, P. Regitnig and R. Ogden, Layer-specific 3D residual deformations of human aortas with non-atherosclerotic intimal thickening,, Annals of Biomedical Engineering, 35 (2007), 530.  doi: 10.1007/s10439-006-9252-z.  Google Scholar

[11]

G. Holzapfel, G. Sommer, C. Gasser and P. Regitnig, Determination of layer-specific mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling,, American Journal of Physiology-Heart and Circulatory Physiology, 289 (2005).  doi: 10.1152/ajpheart.00934.2004.  Google Scholar

[12]

P. Kalita and R. Schaefer, Mechanical models of artery walls,, Archives of Computational Methods in Engineering, 15 (2008), 1.  doi: 10.1007/s11831-007-9015-5.  Google Scholar

[13]

E. Lee, Elastic-plastic deformation at finite strains,, J. Appl. Mech., 36 (1968), 1.  doi: 10.1115/1.3564580.  Google Scholar

[14]

R. Lee, A. Grodzinsky, E. Frank, R. Kamm and F. Schoen, Structure-dependent dynamic mechanical behavior of fibrous caps from human atherosclerotic plaques,, Circulation, 83 (1991), 1764.  doi: 10.1161/01.CIR.83.5.1764.  Google Scholar

[15]

R. Lee, S. Richardson, H. Loree, A. Grodzinsky, S. Gharib, F. Schoen and N. Pandian, Prediction of mechanical properties of human atherosclerotic tissue by high-frequency intravascular ultrasound imaging. An in vitro study,, Arteriosclerosis, 12 (1992), 1.  doi: 10.1161/01.ATV.12.1.1.  Google Scholar

[16]

M. Li, J. Beech-Brandt, L. John, P. Hoskins and W. Easson, Numerical analysis of pulsatile blood flow and vessel wall mechanics in different degrees of stenoses,, Journal of Biomechanics, 40 (2007), 3715.  doi: 10.1016/j.jbiomech.2007.06.023.  Google Scholar

[17]

A. Li, S. Howarth and R. Trivedi, Stress analysis of carotid plaque rupture based on in vivo high resolution MRI,, Journal of Biomechanics, 39 (2006), 2611.  doi: 10.1016/j.jbiomech.2005.08.022.  Google Scholar

[18]

S. Liu and Y. Fung, Zero-stress states of arteries,, Journal of Biomechanical Engineering, 110 (1988), 82.  doi: 10.1115/1.3108410.  Google Scholar

[19]

H. Loree, R. Kamm, R. Stringfellow and R. Lee, Effects of fibrous cap thickness on peak circumferential stress in model atherosclerotic vessels,, Circulation research, 71 (1992), 850.  doi: 10.1161/01.RES.71.4.850.  Google Scholar

[20]

H. Loree, B. Tobias, L. Gibson, R. Kamm, D. Small and R. Lee, Mechanical properties of model atherosclerotic lesion lipid pools,, Arteriosclerosis, 14 (1994), 230.  doi: 10.1161/01.ATV.14.2.230.  Google Scholar

[21]

M. Naghavi, P. Libby, E. Falk, S. Casscells, S. Litovsky, J. Rumberger, J. Badimon, C. Stefanadis, P. Moreno and P. Pasterkamp, From vulnerable plaque to vulnerable patient a call for new definitions and risk assessment strategies: Part I,, Circulation, 108 (2003), 1664.  doi: 10.1161/01.CIR.0000087480.94275.97.  Google Scholar

[22]

J. Ohayon, N. Mesnler, A. Brolsat, J. Toczek, L. Rlou and Pl Tracqui, Elucidating atherosclerotic vulnerable plaque rupture by modeling cross substitution of ApoE mouse and human plaque components stiffnesses,, Biomech. Model. Mechanobiol., 11 (2011), 801.  doi: 10.1007/s10237-011-0353-8.  Google Scholar

[23]

J. Ohayon, O. Dubreuil, P. Tracqui, S. Le Floc'h, G. Rioufol, L. Chalabreysse, F. Thivolet, R. Pettigrew and G. Finet, Influence of residual stress/strain on the biomechanical stability of vulnerable coronary plaques: Potential impact for evaluating the risk of plaque rupture,, American Journal of Physiology-Heart and Circulatory Physiology, 293 (2007).  doi: 10.1152/ajpheart.00018.2007.  Google Scholar

[24]

J. Ohayon, G. Finet, A. Gharib, D. Herzka, P. Tracqui, J. Heroux, G. Rioufol, M. Kotys, A. Elagha and R. Pettigrew, Necrotic core thickness and positive arterial remodeling index: Emergent biomechanical factors for evaluating the risk of plaque rupture,, American Journal of Physiology-Heart and Circulatory Physiology, 295 (2008).  doi: 10.1152/ajpheart.00005.2008.  Google Scholar

[25]

A. Rachev, Theoretical study of the effect of stress-dependent remodeling on arterial geometry under hypertensive conditions,, Journal of Biomechanics, 30 (1997), 819.  doi: 10.1016/S0021-9290(97)00032-8.  Google Scholar

[26]

E. Rodriguez, A. Hoger and A. McCulloch, Stress-dependent finite growth in soft elastic tissues,, Journal of Biomechanics, 27 (1994), 455.  doi: 10.1016/0021-9290(94)90021-3.  Google Scholar

[27]

U. Sadat, Z. Teng and J. Gillard, Biomechanical structural stresses of atherosclerotic plaques,, Expert Review of Cardiovascular Therapy, 8 (2010), 1469.  doi: 10.1586/erc.10.130.  Google Scholar

[28]

L. Taber, Biomechanical growth laws for muscle tissue,, Journal of Theoretical Biology, 193 (1998), 201.  doi: 10.1006/jtbi.1997.0618.  Google Scholar

[29]

L. Taber, A model for aortic growth based on fluid shear and fiber stresses,, Journal of Biomechanical Engineering, 120 (1998), 348.  doi: 10.1115/1.2798001.  Google Scholar

[30]

L. Taber and D. Eggers, Theoretical study of stress-modulated growth in the aorta,, Journal of Theoretical Biology, 180 (1996), 343.  doi: 10.1006/jtbi.1996.0107.  Google Scholar

[31]

L. Taber and J. Humphrey, Stress-modulated growth, residual stress, and vascular heterogeneity,, Journal of Biomechanical Engineering, 123 (2001), 528.  doi: 10.1115/1.1412451.  Google Scholar

[32]

D. Tang, Z. Teng, G. Canton, C. Yang, M. Ferguson, X. Huang, J. Zheng, P. Woodard and C. Yuan, Sites of rupture in human atherosclerotic carotid plaques are associated with high structural stresses,, Stroke, 40 (2009), 3258.   Google Scholar

[33]

D. Tang, C. Yang, J. Zheng, P. Woodard, G. Sicard, J. Saffitz and C. Yuan, 3D MRI-based multicomponent FSI models for atherosclerotic plaques,, Annals of Biomedical Engineering, 32 (2004), 947.   Google Scholar

[34]

Z. Teng, G. Canton, C. Yuan, M. Ferguson, C. Yang, X. Huang, J. Zheng, P. Woodard and D. Tang, 3D critical plaque wall stress is a better predictor of carotid plaque rupture sites than flow shear stress: An in vivo MRI-based 3D FSI study,, Journal of Biomechanical Engineering, 132 (2010).   Google Scholar

[35]

R. Vaishnav and J. Vossoughi, Residual stress and strain in aortic segments,, Journal of Biomechanics, 20 (1987), 235.  doi: 10.1016/0021-9290(87)90290-9.  Google Scholar

[36]

J. Valenta, J. Svoboda, D. Valerianova and K. Vitek, Residual strain in human atherosclerotic coronary arteries and age related geometrical changes,, Biomedical Materials and Engineering, 9 (1999), 311.   Google Scholar

[37]

Y. Vengrenyuk, S. Carlier, S. Xanthos, L. Cardos, P. Ganatos, R. Virmani, S. Einav, L. Gilchrist and S. Weinbaum, A hypothesis for vulnerable plaque rupture due to stress-induced debonding around cellular microcalcifications in thin fibrous caps,, PNAS, 103 (2006), 14678.  doi: 10.1073/pnas.0606310103.  Google Scholar

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