Fractional-Order Viscoelasticity in One-Dimensional Blood Flow Models

Annals of Biomedical Engineering

Volumn 42, Issue 5, 2014, Pages 1012-1023

  Perdikaris, Paris ^a   Karniadakis, George Em ^a  

^a BROWN UNIVERSITY   (United States)

Author keywords

1D blood flow; Fractional order constitutive laws; Global stochastic sensitivity; Viscoelasticity

Indexed keywords

ELASTICITY; HEMODYNAMICS; MODELS; STOCHASTIC MODELS; STOCHASTIC SYSTEMS; WAVE PROPAGATION;

BLOOD FLOW; CONSTITUTIVE LAW; FRACTIONAL-ORDER MODELS; GLOBAL STOCHASTIC SENSITIVITY; PRESSURE WAVE PROPAGATION; STOCHASTIC SENSITIVITY ANALYSIS; VISCO-ELASTIC PARAMETERS; VISCOELASTIC DISSIPATION;

VISCOELASTICITY;

ARTICLE; BLOOD FLOW; BLOOD VESSEL WALL; COMPUTER MODEL; COMPUTER SIMULATION; CONCEPTUAL FRAMEWORK; CONTROLLED STUDY; ENERGY; FRACTIONAL KELVIN ZENER MODEL; FRACTIONAL ORDER VOIGT MODEL; FUNG QUASI LINEAR VISCOELASTICITY THEORY; GLOBAL STOCHASTIC SENSITIVITY; HEMODYNAMICS; HUMAN; HYSTERESIS; HYSTERESIS LOOP; INTERNAL CAROTID ARTERY; NUCLEAR MAGNETIC RESONANCE IMAGING; ONE DIMENSIONAL BLOOD FLOW; PRIORITY JOURNAL; STOCHASTIC MODEL; SYSTOLE; UNCERTAINTY; VERTEBRAL ARTERY; VISCOELASTICITY; WAVEFORM;

ANIMALS; AORTA; AORTIC VALVE; CAROTID ARTERIES; COMPUTER SIMULATION; ELASTICITY; FEMORAL ARTERY; HEMODYNAMICS; HUMANS; MODELS, BIOLOGICAL; REGIONAL BLOOD FLOW; SHEEP; SWINE; VISCOSITY;

EID: 84898545946     PISSN: 00906964     EISSN: 15739686     Source Type: Journal    
DOI: 10.1007/s10439-014-0970-3     Document Type: Article

References (32)

1. Alastruey, J., 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: 2250-2258, 2011.
2. Atanacković, T. M., S. Konjik, L. Oparnica, and D. Zorica. Thermodynamical restrictions and wave propagation for a class of fractional order viscoelastic rods. Abstr. Appl. Anal. 2011: 1-32, 2011.
3. Bia, D., I. Aguirre, Y. Zócalo, L. Devera, E. Cabrera Fischer, and R. L. Armentano. Regional differences in viscosity, elasticity, and wall buffering function in systemic arteries: pulse wave analysis of the arterial pressure-diameter relationship. Rev. Esp. Cardiol. (Engl. Ed.) 58: 167-174, 2005.
4. Čanić, S., C. J. Hartley, D. Rosenstrauch, J. Tambača, G. Guidoboni, and A. Mikelić. Blood flow in compliant arteries: an effective viscoelastic reduced model, numerics, and experimental validation. Ann. Biomed. Eng. 34(4): 575-592, 2006.
5. Craiem, D. O., and R. L. Armentano. A fractional derivative model to describe arterial viscoelasticity. Biorheology 44: 251-263, 2007.
6. Craiem, D. O., F. J. Rojo, J. M. Atienza, R. L. Armentano, and G. V. Guinea. Fractional-order viscoelasticity applied to describe uniaxial stress relaxation of human arteries. Phys. Med. Biol. 53: 4543, 2008.
7. Craiem, D. O., F. J. Rojo, J. M. Atienza, G. V. Guinea, and R. L. Armentano. Fractional calculus applied to model arterial viscoelasticity. Latin Am. Appl. Res. 38: 141-145, 2008.
8. DeVault, K., P. A. Gremaud, V. Novak, M. S. Olufsen, G. Vernieres, and P. Zhao. Blood flow in the circle of willis: modeling and calibration. SIAM Multiscale Model. Simul. 7(2): 888-909, 2008.
9. Doehring, T. C., A. D. Freed, E. O. Carew, I. Vesely, et al. Fractional order viscoelasticity of the aortic valve cusp: an alternative to quasilinear viscoelasticity. J. Biomech. Eng.-Trans. ASME 127: 700, 2005.
10. Eringen, A. C. Mechanics of Continua. Huntington, NY: Robert E. Krieger, 1980.
11. Formaggia, L., A. Quarteroni, and A. Veneziani. Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System, Vol. 1. New York: Springer, 2009.
12. Fung, Y. Biomechanics: Mechanical Properties of Living Tissues. New York: Springer, 1993.
13. Grinberg, L., E. Cheever, T. Anor, J. R. Madsen, and G. E. Karniadakis. Modeling blood flow circulation in intracranial arterial networks: a comparative 3D/1D simulation study. Ann. Biomed. Eng. 39: 297-309, 2011.
14. López-Fernández, M., C. Lubich, and A. Schädle. Adaptive, fast, and oblivious convolution in evolution equations with memory. SIAM J. Sci. Comput. 30: 1015-1037, 2008.
15. Lubich, C., and A. Schädle. Fast convolution for nonreflecting boundary conditions. SIAM J. Sci. Comput. 24: 161-182, 2002.
16. Lundkvist, A., E. Lilleodden, W. Siekhaus, J. Kinney, L. Pruitt, and M. Balooch. Viscoelastic properties of healthy human artery measured in saline solution by AFM-based indentation technique. In: MRS Proceedings, Vol. 436, 1996.
17. Magin, R. L. Fractional Calculus in Bioengineering. Redding: Begell House, 2006.
18. Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. London: Imperial College Press, 2010.
19. Näsholm, S. P., and S. Holm. On a fractional Zener elastic wave equation. Fract. Calc. Appl. Anal. 16: 26-50, 2013.
20. Podlubny, I. Fractional Differential Equations, Vol. 198. San Diego: Academic Press, 1998.
21. Podlubny, I. Calculation of the Mittag-Leffler function with desired accuracy. http://www. mathworks. com/matlabcentral/fileexchange/8738-mittag-leffler-function, 2012. Accessed 12 September 2012.
22. Raghu, R., I. E. Vignon-Clementel, C. A. Figueroa, and C. A. Taylor. Comparative study of viscoelastic arterial wall models in nonlinear one-dimensional finite element simulations of blood flow. J. Biomech. Eng.-Trans. ASME 133(8): 081003-081003, 2011.
23. Reymond, P., F. Merenda, F. Perren, D. Rüfenacht, and N. Stergiopulos. Validation of a one-dimensional model of the systemic arterial tree. Am. J. Physiol. Heart Circ. Physiol. 297: 208, 2009.
24. Reymond, P., F. Perren, F. Lazeyras, and N. Stergiopulos. Patient-specific mean pressure drop in the systemic arterial tree, a comparison between 1-D and 3-D models. J. Biomech. 45(15): 2499-2505, 2012.
25. Sherwin, S. J., V. Franke, J. Peiro, and K. H. Parker. One-dimensional modelling of a vascular network in space-time variables. J. Eng. Math. 47: 217-250, 2012.
26. Shu, C. W. Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9(6): 1073-1084, 1988.
27. Smith, N. P., A. J. Pullan, and P. J. Hunter. An anatomically based model of transient coronary blood flow in the heart. SIAM J. Appl. Math. 62: 990-1018, 2001.
28. Steele, B. N., D. Valdez-Jasso, M. A. Haider, and M. S. Olufsen. Predicting arterial flow and pressure dynamics using a 1D fluid dynamics model with a viscoelastic wall. SIAM J. Appl. Math. 71(4): 1123-1143, 2011.
29. Valdez-Jasso, D., D. Bia, Y. Zócalo, R. L. Armentano, M. Haider, and M. Olufsen. Linear and nonlinear viscoelastic modeling of aorta and carotid pressure-area dynamics under in vivo and ex vivo conditions. Ann. Biomed. Eng. 39(5): 1-19, 2011.
30. Witthoft, A., and G. E. Karniadakis. A bidirectional model for communication in the neurovascular unit. J. Theor. Biol. 311: 80-93, 2012.
31. Xiu, D., and D. E. Karniadakis. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24: 619-644, 2002.
32. Yang, X., M. Choi, G. Lin, and G. E. Karniadakis. Adaptive anova decomposition of stochastic incompressible and compressible flows. J. Comput. Phys. 231: 1587-1614, 2012.