Abstract
In this work we employ integer- and fractional-order viscoelastic models in a one-dimensional blood flow solver, and study their behavior by presenting an in-silico study on a large patient-specific cranial network. The use of fractional-order models is motivated by recent experimental studies indicating that such models provide a new flexible alternative to fitting biological tissue data. This is attributed to their inherent ability to control the interplay between elastic energy storage and viscous dissipation by tuning a single parameter, the fractional order α, as well as to account for a continuous viscoelastic relaxation spectrum. We perform simulations using four viscoelastic parameter data-sets aiming to compare different viscoelastic models and highlight the important role played by the fractional order. Moreover, we carry out a detailed global stochastic sensitivity analysis study to quantify uncertainties of the input parameters that define each wall model. Our results confirm that the effect of fractional models on hemodynamics is primarily controlled by the fractional order, which affects pressure wave propagation by introducing viscoelastic dissipation in the system.
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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.
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.
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.
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.
Craiem, D. O., and R. L. Armentano. A fractional derivative model to describe arterial viscoelasticity. Biorheology 44:251–263, 2007.
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.
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.
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.
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.
Eringen, A. C. Mechanics of Continua. Huntington, NY: Robert E. Krieger, 1980.
Formaggia, L., A. Quarteroni, and A. Veneziani. Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System, Vol. 1. New York: Springer, 2009.
Fung, Y. Biomechanics: Mechanical Properties of Living Tissues. New York: Springer, 1993.
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.
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.
Lubich, C., and A. Schädle. Fast convolution for nonreflecting boundary conditions. SIAM J. Sci. Comput. 24:161–182, 2002.
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.
Magin, R. L. Fractional Calculus in Bioengineering. Redding: Begell House, 2006.
Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. London: Imperial College Press, 2010.
Näsholm, S. P., and S. Holm. On a fractional Zener elastic wave equation. Fract. Calc. Appl. Anal. 16:26–50, 2013.
Podlubny, I. Fractional Differential Equations, Vol. 198. San Diego: Academic Press, 1998.
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.
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.
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.
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.
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.
Shu, C. W. Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9(6):1073–1084, 1988.
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.
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.
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.
Witthoft, A., and G. E. Karniadakis. A bidirectional model for communication in the neurovascular unit. J. Theor. Biol. 311:80–93, 2012.
Xiu, D., and D. E. Karniadakis. The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24:619–644, 2002.
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.
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This work was supported by the DOE Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4), the NSF Directorate of Mathematical Sciences (DMS), and the DOE/INCITE program.
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Associate Editor Aleksander S. Popel oversaw the review of this article.
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Perdikaris, P., Karniadakis, G.E. Fractional-Order Viscoelasticity in One-Dimensional Blood Flow Models. Ann Biomed Eng 42, 1012–1023 (2014). https://doi.org/10.1007/s10439-014-0970-3
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DOI: https://doi.org/10.1007/s10439-014-0970-3