Estimation of the quasi-linear viscoelastic parameters using a genetic algorithm. (English) Zbl 1187.74145

Summary: The quasi-linear viscoelastic (QLV) theory of Fung has been widely used for the modeling of viscoelastic properties of soft tissues. The essence of Fung’s approach is that the stress relaxation can be expressed in terms of the instantaneous elastic response and the reduced relaxation function. Using the Boltzmann superposition principle, the constitutive equation can be written as a convolution integral of the strain history and reduced relaxation function. In the appropriate models, QLV theory usually consists of five material parameters (two for the elastic response and three for the reduced relaxation function), which must be determined experimentally. However, to be consistent with the assumptions of QLV theory, the material functions should be obtained based on a step change in strain which is not possible to be performed experimentally. It is known that this may result in regression algorithms that converge poorly and yield non-unique solutions with highly variable constants, especially for long ramp times. In this paper, we use the genetic algorithm approach, which is an adaptive heuristic search algorithm premised on the evolutionary ideas of natural selection and genetics, and simultaneously fit the ramping and relaxation experimental data (on ligaments) to the QLV constitutive equation to obtain the material parameters.


74L15 Biomechanical solid mechanics
92C10 Biomechanics
Full Text: DOI


[1] Fung, Y. C., Stress-strain-history relations of soft tissues in simple elongation, (Fung, Y. C.; Perrone, N.; Anliker, M., Biomechanics: Its Foundations and Objectives (1972), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ), 181-208
[2] Fung, Y. C., Biomechanics: Mechanical Properties of Living Tissues (1981), Springer: Springer New York
[3] Woo, S. L.-Y., Mechanical properties of tendons and ligaments. I. Quasi-static and nonlinear viscoelastic properties, Biorheology, 19, 385-396 (1982)
[4] Dortmans, L. J.; Sauren, A. A.; Rousseau, E. P., Parameter estimation using the quasi-linear viscoelastic model proposed by Fung, J. Biomech. Eng., 106, 198-203 (1984)
[5] Nigul, I.; Nigul, U., On algorithms of evaluation of Fung’s relaxation function parameters, J. Biomech., 20, 343-352 (1987)
[6] Myers, B. S.; McElhaney, J. H.; Doherty, B. J., The viscoelastic responses of the human cervical spine in torsion: Experimental limitations of quasi-linear theory, and a method for reducing these effects, J. Biomech., 24, 811-817 (1991)
[7] Kwan, M. K.; Lin, T. H.; Woo, S. L., On the viscoelastic properties of the anteromedial bundle of the anterior cruciate ligament, J. Biomech., 26, 447-452 (1993)
[8] Carew, E. O.; Talman, E. A.; Boughner, D. R.; Vesely, I., Quasi-linear viscoelastic theory applied to internal shearing of porcine aortic valve leaflets, J. Biomech. Eng., 4, 386-392 (1999)
[9] Abramowitch, S. D.; Woo, S. L., An improved method to analyze the stress relaxation of ligaments following a finite ramp time based on the quasi-linear viscoelastic theory, J. Biomech. Eng., 126, 92-97 (2004)
[10] R.S. Rivlin, Forty years of nonlinear continuum mechanics, in: Proceeding of the IX International Congress on Rheology, Mexico, 1984, pp. 1-29; R.S. Rivlin, Forty years of nonlinear continuum mechanics, in: Proceeding of the IX International Congress on Rheology, Mexico, 1984, pp. 1-29
[11] Ogden, R. W., Nonlinear Elastic Deformations (1984), Ellis Harwood Ltd.: Ellis Harwood Ltd. Chichester, England · Zbl 0541.73044
[12] Miller, K.; Chinzei, K., Constitutive modelling of brain tissue: Experiment and theory, J. Biomech., 30, 1115-1121 (1997)
[13] Miller, K.; Chinzei, K., Mechanical properties of brain tissue in tension, J. Biomech., 35, 483-490 (2002)
[14] Drapaca, C. S.; Tenti, G.; Rohlf, K.; Sivaloganathan, S., A quasi-linear viscoelastic constitutive equation for the brain: Application to hydrocephalus, J. Elasticity, 85, 65-83 (2006) · Zbl 1098.74040
[15] Goldberg, D. E., Genetic Algorithms in Search, Optimization and Machine Learning (1989), Addison-Wesley · Zbl 0721.68056
[16] Rechenberg, I., Evolutionstrategie: Optimierung technischer Systeme nach Prinzipien der biologischen Evolution (1973), Fromman-Holzboog: Fromman-Holzboog Stuttgart
[17] Holland, J. H., Adaptation in Natural and Artificial Systems (1975), University of Michigan Press · Zbl 0317.68006
[18] Koza, J. R., Genetic Programming: On the Programming of Computers by Means of Natural Selection (1992), MIT Press: MIT Press Cambridge, MA · Zbl 0850.68161
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