Non-integer viscoelastic constitutive law to model soft biological tissues to in-vivo indentation

• Autorzy: Demirci N. , Tönük E.

Identyfikatory

DOI

10.5277/ABB-00005-2014-03

• Języki publikacji: EN

Abstrakty

EN

Purpose: During the last decades, derivatives and integrals of non-integer orders are being more commonly used for the description of constitutive behavior of various viscoelastic materials including soft biological tissues. Compared to integer order constitutive relations, non-integer order viscoelastic material models of soft biological tissues are capable of capturing a wider range of viscoelastic behavior obtained from experiments. Although integer order models may yield comparably accurate results, non-integer order material models have less number of parameters to be identified in addition to description of an intermediate material that can monotonically and continuously be adjusted in between an ideal elastic solid and an ideal viscous fluid.Methods: In this work, starting with some preliminaries on non-integer (fractional) calculus, the “spring-pot”, (intermediate mechanical element between a solid and a fluid), non-integer order three element (Zener) solid model, finally a user-defined large strain non-integer order viscoelastic constitutive model was constructed to be used in finite element simulations. Using the constitutive equation developed, by utilizing inverse finite element method and in vivo indentation experiments, soft tissue material identification was performed. Results: The results indicate that material coefficients obtained from relaxation experiments, when optimized with creep experimental data could simulate relaxation, creep and cyclic loading and unloading experiments accurately.Conclusions: Non-integer calculus viscoelastic constitutive models, having physical interpretation and modeling experimental data accurately is a good alternative to classical phenomenological viscoelastic constitutive equations.

Słowa kluczowe

EN

fractional calculus; indentation tests; inverse finite element analysis; soft tissue constitutive relation; viscoelasticity

PL

rachunek niecałkowitego rzędu; tkanka miękka; lepkosprężystość

• Wydawca: Oficyna Wydawnicza Politechniki Wrocławskiej
• Czasopismo: Acta of Bioengineering and Biomechanics
• Rocznik: 2014
• Tom: Vol. 16, no. 4
• Strony: 13--21
• Opis fizyczny: Bibliogr. 36 poz., rys., tab., wykr.

Twórcy

autor

Demirci N.

• Department of Mechanical Engineering, Middle East Technical University, Ankara, Turkey

autor

Tönük E.

• Department of Mechanical Engineering, Middle East Technical University, Ankara, Turkey

Bibliografia

• [1] BAYIN S.Ş., Chapter 14: Fractional Derivatives and Integrals: Differintegrals, [in:] Mathematical Methods in Science and Engineering, John Wiley & Sons, Hoboken, N.J., 2006.
• [2] CAPUTO M., MAINARDI F., A new dissipation model based on memory mechanism, Pure Appl. Goephys., 1971, 91, 134–147.
• [3] CAPUTO M., MAINARDI F., Linear models of dissipation in anelastic solids, Rivista del Nuovo Cimento, 1971, 1, 161– 198.
• [4] CHANG T.K., ROSSIKHIN YU.A., SHITIKOVA M.V., CHAO C.K., Application of fractional-derivative standard linear solid model to impact response of human frontal bone, Theo. Appl. Fract. Mech., 2011, 56, 148–153.
• [5] CRAIEM D., ROJO F.J., ATIENZA J.M., ARMENTANO R.L., GUINEA G.V., Fractional-order viscoelasticity applied to describe uniaxial stress relaxation of human arteries, Phys. Med. Biol., 2008, 53, 4543–4554.
• [6] CRAIEM O.D., ARMENTANO L.R., Arterial Viscoelasticity: A Fractional Derivative Model, Proceedings of the 28th IEEE EMBS Annual International Conference, New York City, USA, 2006, 1098–1101.
• [7] DALIR M., BASHOUR M., Applications of fractional calculus, Appl. Math. Sci., 2010, 4(21), 1021–1032.
• [8] DAVIS G.B., KOHANDEL M., SIVALOGANATHAN S., TENTI G., The constitutive properties of the brain paraenchyma. Part. 2. Fractional derivative approach, Mech. Eng. Phys., 2006, 28, 455–459.
• [9] DAVIS H.T., The Theory of Linear Operators, Bloomington, The Principia Press, Indiana, 1936.
• [10] DEMIRCI N., Formulation and Implementation of a Fractional Order Viscoelastic Material Model into Finite Element Software and Material Model Parameter Identification Using In-vivo Indenter Experiments for Soft Biological Tissues, Thesis submitted to the Graduate School of Natural and Applied Sciences of Middle East Technical University, Ankara, Turkey, 2012.
• [11] DIETHELM K., FORD N.J., FREED A.D., LUCHKO Y., Algorithms for the fractional calculus: A selection of numerical methods, Comp. Meth. Appl. Mech. Eng., 2005, 194, 743– 773.
• [12] DJORDJEVIC D.V., JARIC J., FABRY B., FREDBERG J.J., STAMENOJIC D., Fractional derivatives embody essential features of cell rheological behavior, Ann. Biomed. Eng., 2003, 31, 692–699
• [13] DOEHRING C.T., FREED D.A., CAREW O.E., VESELY I., Fractional order viscoelasticity of the aortic valve cusp: An alternative to quasilinear viscoelasticity, J. Biomech. Eng., 2005, 127, 700–708.
• [14] FREED A.D., DIETHELM K., Fractional calculus in biomechanics: a 3D viscoelastic model using regularized fractional derivative kernels with application to the human calcaneal fat pad, Biomech. Model Mechanobiol., 2006, 5, 203–215.
• [15] FUNG Y.C., Structure and Stress-Strain Relationship of Soft Tissues, Am. Zoologist, 1984, 24, 13–22.
• [16] GEMANT A., A method of analyzing experimental results obtained from elasto-viscous bodies, Phys, 1936, 7, 311– 317.
• [17] GRAHOVAC N.M., ZIGIC M.M., Modeling of the hamstring muscle group by use of fractional derivatives, Comput. Math. Appl., 2010, 59(5), 1695–1700.
• [18] KOELLER R., Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech., 1984, 51, 299–307.
• [19] KOHANDEL M., SIVALOGANATHAN S., TENTI G., DRAKE J., The Constitutive Properties of the Brain Parenchyma. Part 1. Strain Energy Approach, Med. Eng. Phys., 2006, 28, 449– 454.
• [20] MACHADO J.T., KIRYAKOVA V., MAINARDI F., Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simulation, 2011, 16, 1140–1153.
• [21] NUTTING P., A new general law of deformation, J. Franklin Institute, 1921, 191, 679–685.
• [22] OLDHAM K.B., SPANIER J., The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press Inc. California, 1974.
• [23] PETEKKAYA A.T., In vivo indenter experiments on soft biological tissues for identification of material models and corresponding parameters, Thesis submitted to the Graduate School of Natural and Applied Sciences of Middle East Technical University, Ankara, Turkey, 2008.
• [24] PETEKKAYA A.T., TÖNÜK E., In vivo indenter experiments via ellipsoid indenter tips to determine the personal and local in-plane anisotropic mechanical behavior of soft biological tissues, J. Fac. Eng. Arch. Gazi. Univ., 2011, 26(1), 63–72.
• [25] PODLUBNY I., Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus and Appl. Analysis, 2002, 5(4), 367–386.
• [26] PODLUBNY I., Mittag–Leffler Function – File Exchange – Matlab Central, http://www.mathworks. com/matlabcentral/ fileexchange/8738-mittag-leffler-function, Accessed 25 March 2009.
• [27] RABOTNOV Y.N., Equilibrium of an elastic medium with after-effect, J. Appl. Math. Mech., 1948, 12(1), 53–62.
• [28] RABOTNOV Y.N., Elements of Hereditary Solid Mechanics, Mir Publishers, Moscow, 1980.
• [29] ROSS B., Fractional calculus, Math. Mag., 1977, 50(3), 115– 122.
• [30] SCOTT BLAIR G., COPPEN F.M., The subjective judgement of the elastic and plastic properties of soft bodies, “the differential thresholds” for viscosities and compression moduli, Proc. R. Soc., 1939, B, 128, 109–125.
• [31] SCOTT-BLAIR G.W., An Introduction to Biorheology, Elsevier, Amsterdam 1974.
• [32] SCOTT-BLAIR G., A study of the firmness of soft materials based on Nutting’s equation, J. Sci. Instr., 1944, 21, 149–154.
• [33] SCOTT-BLAIR G., Analytical and integrative aspects of the stress–strain–time problem, J. Sci. Instr., 1944, 21, 80–84.
• [34] SUKI B., BARABASI A.L., LUTCHEN K.L., Lung tissue viscoelasticity: A mathematical framework and its molecular basis, J. Appl. Phys., 1994, 76(6), 2749–2759.
• [35] TÖNÜK E., SILVER-THORN M.B., Nonlinear Viscoelastic Material Property Estimation of Lower Extremity Residual Limb Tissues, J. Biomech. Eng., 2004, 126, 289–300.
• [36] WILKIE K.P., DRAPACA C.S., SIVOLAGANATHAN S., A nonlinear viscoelastic fractional derivative model of infant hydrocephalus, Appl. Math. Comput., 2011, 217, 8693– 8704.