zbMATH — the first resource for mathematics

Comparative sensitivity analysis of muscle activation dynamics. (English) Zbl 1335.92013
Comput. Math. Methods Med. 2015, Article ID 585409, 16 p. (2015); corrigendum ibid. 2017, Article ID 6752731, 2 p. (2017).
Summary: We mathematically compared two models of mammalian striated muscle activation dynamics proposed by Hatze and Zajac. Both models are representative for a broad variety of biomechanical models formulated as ordinary differential equations (ODEs). These models incorporate parameters that directly represent known physiological properties. Other parameters have been introduced to reproduce empirical observations. We used sensitivity analysis to investigate the influence of model parameters on the ODE solutions. In addition, we expanded an existing approach to treating initial conditions as parameters and to calculating second-order sensitivities. Furthermore, we used a global sensitivity analysis approach to include finite ranges of parameter values. Hence, a theoretician striving for model reduction could use the method for identifying particularly low sensitivities to detect superfluous parameters. An experimenter could use it for identifying particularly high sensitivities to improve parameter estimation. Hatze’s nonlinear model incorporates some parameters to which activation dynamics is clearly more sensitive than to any parameter in Zajac’s linear model. Other than Zajac’s model, Hatze’s model can, however, reproduce measured shifts in optimal muscle length with varied muscle activity. Accordingly we extracted a specific parameter set for Hatze’s model that combines best with a particular muscle force-length relation.

92C10 Biomechanics
Algorithm 862
Full Text: DOI
[1] Scovil, C. Y.; Ronsky, J. L., Sensitivity of a Hill-based muscle model to perturbations in model parameters, Journal of Biomechanics, 39, 11, 2055-2063, (2006)
[2] Lehman, S. L.; Stark, L. W., Three algorithms for interpreting models consisting of ordinary differential equations: sensitivity coefficients, sensitivity functions, global optimization, Mathematical Biosciences, 62, 1, 107-122, (1982) · Zbl 0504.93022
[3] Chan, K.; Saltelli, A.; Tarantola, S.; Andradóttir, S.; Healy, K.; Withers, D.; Nelson, B., Sensitivity analysis of model output: variance-based methods make the difference, Proceedings of the 29th Conference on Winter Simulation (WSC ’97), IEEE Computer Society
[4] Saltelli, A.; Chan, K. S. E., Sensitivity Analysis, (2000), New York, NY, USA: John Wiley & Sons, New York, NY, USA
[5] Zajac, F. E., Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control, Critical Reviews in Biomedical Engineering, 17, 4, 359-411, (1989)
[6] Hatze, H., A myocybernetic control model of skeletal muscle, Biological Cybernetics, 25, 2, 103-119, (1977) · Zbl 0346.92011
[7] Kistemaker, D. A.; van Soest, A. J.; Bobbert, M. F., Length-dependent [Ca] sensitivity adds stiffness to muscle, Journal of Biomechanics, 38, 9, 1816-1821, (2005)
[8] van Soest, A., Jumping from structure to control: a simulation study of explosive movements [Ph.D. thesis], (1992), Amsterdam, The Netherlands: Vrije Universiteit, Amsterdam, The Netherlands
[9] van Soest, A. J.; Bobbert, M. F., The contribution of muscle properties in the control of explosive movements, Biological Cybernetics, 69, 3, 195-204, (1993)
[10] Cole, G.; van den Bogert, A.; Herzog, A. W.; Gerritsen, K., Modelling of force production in skeletal muscle undergoing stretch, Journal of Biomechanics, 29, 1091-1104, (1996)
[11] Günther, M.; Schmitt, S.; Wank, V., High-frequency oscillations as a consequence of neglected serial damping in Hill-type muscle models, Biological Cybernetics, 97, 1, 63-79, (2007) · Zbl 1125.92007
[12] Haeufle, D. F. B.; Günther, M.; Bayer, A.; Schmitt, S., Hill-type muscle model with serial damping and eccentric force-velocity relation, Journal of Biomechanics, 47, 6, 1531-1536, (2014)
[13] Hatze, H., Myocybernetic Control Models of Skeletal Muscle, (1981), University of South Africa · Zbl 0635.92003
[14] Kistemaker, D. A.; van Soest, A. J.; Bobbert, M. F., Is equilibrium point control feasible for fast goal-directed single-joint movements?, Journal of Neurophysiology, 95, 5, 2898-2912, (2006)
[15] Kistemaker, D. A.; van Soest, A. J.; Bobbert, M. F., A model of open-loop control of equilibrium position and stiffness of the human elbow joint, Biological Cybernetics, 96, 3, 341-350, (2007) · Zbl 1161.92307
[16] Vukobratovic, R. T. M., General Sensitivity Theory, (1962), New York, NY, USA: Elsevier, New York, NY, USA
[17] Dickinson, R. P.; Gelinas, R. J., Sensitivity analysis of ordinary differential equation systems—a direct method, Journal of Computational Physics, 21, 2, 123-143, (1976) · Zbl 0333.65038
[18] ZivariPiran, H., Efficient simulation, accurate sensitivity analysis and reliable parameter estimation for delay differential equations [Ph.D. thesis], (2009), Toronto, Canada: University of Toronto, Toronto, Canada
[19] Bader, B. W.; Kolda, T. G., Algorithm 862: matlab tensor classes for fast algorithm prototyping, ACM Transactions on Mathematical Software, 32, 4, 635-653, (2006) · Zbl 1230.65054
[20] Scherzer, O., Mathematische Modellierung—Vorlesungsskript, (2009), Universität Wien
[21] Saltelli, A.; Annoni, P., How to avoid a perfunctory sensitivity analysis, Environmental Modelling and Software, 25, 12, 1508-1517, (2010)
[22] Frey, H. C.; Mokhtari, A.; Danish, T., Evaluation of selected sensitivity analysis methods based upon application to two food safety process risk models, (2003), Computational Laboratory for Energy, North Carolina State University
[23] Benker, H., Differentialgleichungen mit MATHCAD und MATLAB, 1, (2005), Berlin , Germany: Springer, Berlin , Germany
[24] Hatze, H., A general mycocybernetic control model of skeletal muscle, Biological Cybernetics, 28, 3, 143-157, (1978) · Zbl 0367.92002
[25] Mörl, F.; Siebert, T.; Schmitt, S.; Blickhan, R.; Günther, M., Electro-mechanical delay in hill-type muscle models, Journal of Mechanics in Medicine and Biology, 12, 5, (2012)
[26] Siebert, T.; Till, O.; Blickhan, R., Work partitioning of transversally loaded muscle: experimentation and simulation, Computer Methods in Biomechanics and Biomedical Engineering, 17, 3, 217-229, (2014)
[27] Roszek, B.; Baan, G. C.; Huijing, P. A., Decreasing stimulation frequency-dependent length-force characteristics of rat muscle, Journal of Applied Physiology, 77, 5, 2115-2124, (1994)
[28] Langland, R. H.; Shapiro, M. A.; Gelaro, R., Initial condition sensitivity and error growth in forecasts of the 25 January 2000 east coast snowstorm, Monthly Weather Review, 130, 4, 957-974, (2002)
[29] Sunar, M.; Belegundu, A. D., Trust region methods for structural optimization using exact second order sensitivity, International Journal for Numerical Methods in Engineering, 32, 2, 275-293, (1991) · Zbl 0825.73474
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.