Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

A fractional approach to 3D artery simulation under a regular pulse load

1 Tecnológico Nacional de México/ITS de Cajeme, subdirección de posgrado e investigación, Carretera internacional a Nogales Km. 2 S/N, Ciudad Obregón, Sonora, México
2 Pontificia Universidad Católica de Valparaíso, Escuela de Ingeniería Mecánica, Chile. Avenida Los Carrera 01567, Quilpué, Valparaíso, Chile

Special Issues: Recent Advances in Biomedical and Mechanical Engineering and Related Sciences

For the diagnosis and treatment of many pathologies related to arteries, it is necessary to known their mechanical behavior. Previous investigation implement multi-layer structural models for arterial walls based on a Fung model, which can be problematic with the material stability in the convergence sense for finite element methods, issue avoided with a large number of terms in the prony series and the inclusion of relaxation function. On the other hand, this solution increase significantly the computer cost for the solution finding. In this research was implement a 3D simulation of the aorta artery, composed of three different layers that allow identifying how are distributed the stress-strain state caused by the flow pressure. A vectorized geometry was created based on medical tomography images and a fractional linear-standard viscoelastic constitutive model for solids was developed and validated. For the model adjustment was used creep-relaxation experiment data and a set of parameters, in the frequency domain, from a previous calculated complex modulus. The mechanical simulated behavior of the artery section proof that the fractional model showns an accurate representation of the simulated phenomenon, and a lower convergence time.
  Article Metrics

Keywords Zener; fractional; hyperelastic; biomechanics; soft tissues

Citation: Juan Palomares-Ruiz, Efrén Ruelas, Flavio Muñoz, José Castro, Angel Rodríguez. A fractional approach to 3D artery simulation under a regular pulse load. Mathematical Biosciences and Engineering, 2020, 17(3): 2516-2529. doi: 10.3934/mbe.2020138


  • 1. Y. Fung, Biomechanics: Mechanical properties of living tissues, 2nd edition, Springer-Verlag, New York, 1981.
  • 2. Y. Fung, Biomechanics: Motion, Flow, Stress, and Growth, 2nd edition, Springer-Verlag, New York, 1990.
  • 3. Y. Fung, K. Fronek, P. Patitucci, Pseudoelasticity of arteries and the choice of its mathematical expression, Am. J. Physiol., 237 (1979), H620-631.
  • 4. G. Holzapfel, T. Gasser, R. Ogden, A new constitutive framework for arterial wall mechanics and a comparative study of material models, J. Elast., 61 (2000), 1-48.
  • 5. R. Ogden, Non-linear Elastic Deformations, 1st edition, Dover Publications, 1997.
  • 6. G. Holzapfel, R. Ogden, Constitutive modelling of arteries, Proceed. Royal Soc. A, 466 (2010), 1551-1597.
  • 7. J. Humphrey, G. Holzapfel, Mechanics, mechanobiology, and modeling of human abdominal aorta and aneurysms, J. Biomechan., 45 (2012), 805-814.
  • 8. T. Gasser, M. Auer, F. Labruto, J. Swedenborg, J. Roy, Biomechanical rupture risk assessment of abdominal aortic aneurysms: Model complexity versus predictability of finite element simulations, European J. Vascul. Endovascul. Surg., 40 (2010), 176-185.
  • 9. G. Holzapfel, R. Ogden, Modelling the layer-specific 3D residual stresses in arteries, with an application to the human aorta, J. Royal Soc. Interf., 7 (2010), 787-799.
  • 10. D. Balzani, S. Brinkhues, G. Holzapfel, Constitutive framework for the modeling of damage in collagenous soft tissues with application to arterial walls, Comput. Methods Appl. Mechan. Eng., 11 (2012), 139-151.
  • 11. A. Gholipour, M. Ghayesh, A. Zander, R. Mahajan, Three-dimensional biomechanics of coronary arteries, Int. J. Eng. Sci., 130 (2018), 93-114.
  • 12. A. Gholipour, M. Ghayesh, A. Zander, Nonlinear biomechanics of bifurcated atherosclerotic coronary arteries, Int. J. Eng. Sci., 133 (2018), 60-83.
  • 13. F. Meral, T. Royston, R. Magin, Fractional calculus in viscoelasticity: An experimental study, Commun. Nonlinear Sci. Numer. Simulat., 4 (2010), 939-945.
  • 14. J. Mauro, Y. Mauro, On the Prony series representation of stretched exponential relaxation, Physica A Statist. Mechan. Appl., 506 (2018), 75-87.
  • 15. A. Tayeb, A. Makrem, Z. Abdelmalek, H. Adel, B. Jalel, M. Ichchou, On the nonlinear viscoelastic behavior of rubber-like materials: Constitutive description and identification, Int. J. Mechan. Sci., 130 (2017), 437-447.
  • 16. K. Adolfsson, M. Enelund, P. Olsson, On the fractional order model of viscoelasticity, Mechan. Time-Depend. Mater., 9 (2005), 15-34.
  • 17. R. Bagley, P. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983), 201-210.
  • 18. Y. Peng, J. Zhao, Y. Li, A wellbore creep model based on the fractional viscoelastic constitutive equation, Petrol. Explor. Development, 44 (2017), 1038-1044.
  • 19. D. Nagehan, T. Ergin, Non-integer viscoelastic constitutive law to model soft biological tissues to in-vivo indentation, Acta Bioeng. Biomech., 16 (2014), 13-21.
  • 20. R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Mathemat. Appl., 12 (2010), 1586-1593.
  • 21. C. Wex, C. Bruns, A. Stoll, Fractional Kelvin-Voight Model for Liver Tissue in the Frequency and Time Domain, Scottish J. Arts Soc. Sci. Scient. Studies, 11 (2014), 69-78.
  • 22. C. Wex, M. Fröhlich, K. Brandstädter, C. Bruns, A. Stoll, Experimental analysis of the mechanical behavior of the viscoelastic porcine pancreas and preliminary case study on the human pancreas, J. Mechan. Behav. Biomed. Mater., 41 (2015), 199-207.
  • 23. P. Smyth, I. Green, Fractional calculus model of articular cartilage based on experimental stressrelaxation, Mechan. Time-Depend. Mater., 19 (2015), 209-228.
  • 24. A. Freed, K. Diethelm, Fractional calculus in biomechanics: A 3D viscoelastic model using regularized fractional derivative kernels with application to the human calcaneal fat pad, Biomechan. Model. Mechanobiol., 5 (2006), 203-215.
  • 25. G. Davis, M. Kohandel, S. Sivaloganathan, G. Tenti, The constitutive properties of the brain paraenchyma: Part 2. Fractional derivative approach, Biomechan. Model. Mechanobiol., 28 (2006), 455-459.
  • 26. D. Craiem, F. Rojo, J. Atienza, G. Guinea, R. Armentano, Fractional calculus applied to model arterial viscoelasticity, Latin Am. Appl. Res., 38 (2008), 141-145.
  • 27. D. Craiem, F. Rojo, J. Atienza, R. Armentano, G. Guinea, Fractional-order viscoelasticity applied to describe uniaxial stress relaxation of human arteries, Phys. Med. Biol., 53 (2008), 4543.
  • 28. D. Craiem, R. Armentano, A fractional derivative model to describe arterial viscoelasticity, Biorheology, 44 (2007), 251-263.
  • 29. S. Müller, M. Kästner, J. Brummund, V. Ulbricht, On the numerical handling of fractional viscoelastic material models in a FE analysis, Computat. Mechan., 51 (2013), 999-1012.
  • 30. O. Agrawal, A general finite element formulation for fractional variational problems, J. Math. Analy. Appl., 12 (2008), 1-12.
  • 31. I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, 1st edition, Academic Press, 1998.
  • 32. C. Martin, W. Sun, T. Pham, J. Elefteriades, Predictive biomechanical analysis of ascending aortic aneurysm rupture, Acta Biomater., 9 (2013), 9392-9400.
  • 33. A. Tsamis, J. Krawiec, D. Vorp, Elastin and collagen fibre microstructure of the human aorta in ageing and disease: A review, J. Royal Soc. Interf., 10 (2013), 20121004.
  • 34. J. Palomares-Ruiz, M. Rodriguez, J. Castro, A. Rodriguez, Fractional viscoelastic models applied to biomechanical constitutive equations, Rev. Mex. Fis., 61 (2015), 261-267.
  • 35. M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel Progr. Fract. Differ. Appl., 1 (2015), 1-13.
  • 36. B. Xu, H. Li, Y. Zhang, A new definition of fractional derivative without singular kernel J. Biomechan. Eng., 135 (2013), 054501.
  • 37. J. Palomares, M. Rodriguez, J. Castro, Determinación del orden fraccional en el modelo Zener para caracterizar los efectos biomecánicos ocasionados por el flujo sanguíneo Rev. Int. Metod. Numer. Dis., 33 (2017), 10-17.
  • 38. K. Volokh, Challenge of biomechanics, Molecul. Cellul. Biomechan., 10 (2013), 107-135.
  • 39. J. Navarro, D. Sánchez, L. Quijano, J. Briceño, Un sistema presión-volumen para la medición de propiedades mecánicas de vasos cardíacos menores, Revista de ingeniería, 37 (2012), 31-37.
  • 40. P. Richardson, Biomechanics of plaque rupture: Progress, problems, and new frontiers, Ann. Biomed. Eng., 30 (2002), 524-536.
  • 41. G. Sommer, G. Holzapfel, 3D constitutive modeling of the biaxial mechanical response of intact and layer-dissected human carotid arteries, J. Mechan. Behav. Biomed. Mater., 5 (2012), 116-128.


Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved