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Parameter estimation and experimental design for Hill-type muscles: impulses from optimization-based modeling. (English) Zbl 07264570
Summary: The benefits of optimization-based modeling for parameter estimation of Hill-type muscle models are demonstrated. Therefore, we examined the model and data of M. Günther et al. [Biol. Cybern. 97, No. 1, 63–79 (2007; Zbl 1125.92007)], who analyzed isometric, concentric, and quick-release contractions of a piglet calf muscle. We found that the isometric experiments are suitable for derivative-based parameter estimation while the others did not provide any additional value. During the estimation process, certain parameters had to be fixed. We give possible reasons and provide impulses for modelers. Subsequently, unnecessarily complex or deprecated model parts were exchanged and the new model was fitted to the data. In order to be able to provide a reliable estimation of the whole parameter set, we propose two isometric and two quick-release experiments, which are real-life feasible and together allow an identification of all parameters based on a local sensitivity analysis. These experiments can be used as qualitative guidelines for practitioners to reduce the experimental effort when estimating parameters for macroscopic muscle models.
92C10 Biomechanics
92-05 Experimental work for problems pertaining to biology
Full Text: DOI
[1] Guschlbauer, C.; Scharstein, H.; Büschges, A., The extensor tibiae muscle of the stick insect: biomechanical properties of an insect walking leg muscle, J. Exp. Biol., 210, Pt. 6, 1092-1108 (2007), URL https://doi.org/10.1242/jeb.02729
[2] Rome, L. C.; Sosnicki, A. A.; Goble, D. O., Maximum velocity of shortening of three fibre types from horse soleus muscle: Implications for scaling with body size, J. Physiol., 431, 1, 173-185 (1990), URL https://doi.org/10.1113/jphysiol.1990.sp018325
[3] Sellers, W. I.; Manning, P. L., Estimating dinosaur maximum running speeds using evolutionary robotics, Proc. R. Soc. B, 274, 1626, 2711-2716 (2007), URL https://doi.org/10.1098/rspb.2007.0846
[4] Blümel, M.; Guschlbauer, C.; Daun-Gruhn, S.; Hopper, S. L.; Büschges, A., Hill-type muscle model parameters determined from experiments on single muscles show large animal-to-animal variation, Biol. Cybernet., 106, 10, 559-571 (2012), URL https://doi.org/10.1007/s00422-012-0530-6
[5] Günther, M.; Schmitt, S.; Wank, V., High-frequency oscillations as a consequence of neglected serial damping in Hill-type muscle models, Biol. Cybernet., 97, 1, 63-79 (2007), URL https://doi.org/10.1007/s00422-007-0160-6 · Zbl 1125.92007
[6] Hof, A. L.; van den Berg, J. W., EMG to force processing II: Estimation of parameters of the Hill muscle model for the human triceps surae by means of a calfergometer, J. Biomech., 14, 11, 759-770 (1981), URL https://doi.org/10.1016/0021-9290(81)90032-4
[7] Siebert, T.; Rode, C.; Herzog, W.; Till, O.; Blickhan, R., Nonlinearities make a difference: comparison of two common hill-type models with real muscle, Biol. Cybernet., 98, 2, 133-143 (2008), URL https://doi.org/10.1007/s00422-007-0197-6 · Zbl 1149.92302
[8] Yu, T. F.; Wilson, A. J., A passive movement method for parameter estimation of a musculo-skeletal arm model incorporating a modified hill muscle model, Comput. Methods Programs Biomed., 114, 3 (2014), URL https://doi.org/10.1016/j.cmpb.2013.11.003
[9] de Zee, M.; Heinen, F.; Sørensen, S. N.; King, M.; Lewis, M.; Rasmussen, J., Parameter estimations of the Hill model in subject-specific musculoskeletal models, (Proceedings of the 16th International Symposium on Computer Simulation in Biomechanics, Gold Coast, Australia (2016)), 31-32, URL https://vbn.aau.dk/files/260011454/ISCSB2017_de_zee_et_al_final.pdf, http://isbweb.org/tgcs/iscsb-2017/tgcs2017_book_of_abstracts.pdf
[10] Cavallaro, E.; Rosen, J.; Perry, J. C.; Burns, S.; Hannaford, B., Hill-based model as a myoprocessor for a neural controlled powered exoskeleton arm - parameters optimization, (Proceedings of the 2005 IEEE International Conference on Robotics and Automation Barcelona, Spain, April (2005)), 4514-4519, URL https://doi.org/10.1109/ROBOT.2005.1570815
[11] Bock, H. G., Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen (1987), Rheinische Friedrich-Wilhelms-Universität Bonn, (Dissertation) · Zbl 0622.65064
[12] Myers, C. A.; Laz, P. J.; Shelburne, K. B.; Davidson, B. S., A probabilistic approach to quantify the impact of uncertainty propagation in musculoskeletal simulations, Ann. Biomed. Eng., 43, 5, 1098-1111 (2015), URL https://doi.org/10.1007/s10439-014-1181-7
[13] Walter, S., Structured higher-order algorithmic differentiation in the forward and reverse mode with application in optimum experimental design (2012), Humboldt University of Berlin, URL https://doi.org/10.18452/16514
[14] Haeufle, D. F.B.; Günther, M.; Wunner, G.; Schmitt, S., Quantifying control effort of biological and technical movements: An information-entropy-based approach, Phys. Rev. E, 89, 1, Article 012716 pp. (2014), (7 pages), URL https://doi.org/10.1103/PhysRevE.89.012716
[15] Rockenfeller, R.; Günther, M., Extracting low-velocity concentric and eccentric dynamic muscle properties from isometric contraction experiments, Math. Biosci., 278, 1, 77-93 (2016), URL https://doi.org/10.1016/j.mbs.2016.06.005 · Zbl 1346.92016
[16] Hatze, H., Myocybernetic Control Models of Skeletal Muscle (1981), University of South Africa · Zbl 0635.92003
[17] A.V. Hill, The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves, in: Proceedings of the Physiological Society, vol. 1 (1), 1910, pp. iv-vii.
[18] Rockenfeller, R.; Günther, M., Hill equation and hatze’s muscle activation dynamics complement each other: enhanced pharmacological and physiological interpretability of modelled activity-pCa curves, J. Theoret. Biol., 431, 1, 11-24c (2017), URL https://doi.org/10.1016/j.jtbi.2017.07.023 · Zbl 1388.92015
[19] Zajac, F. E., Muscle and tendon: Properties, models, scaling, and application to biomechanics and motor control, Crit. Rev. Biomed. Eng., 17, 4, 359-411 (1989), URL https://www.ncbi.nlm.nih.gov/pubmed/2676342
[20] Rockenfeller, R.; Günther, M.; Schmitt, S.; Götz, T., Comparative sensitivity analysis of muscle activation dynamics, (Computational and Mathematical Methods in Medicine (2015)), URL https://doi.org/10.1155/2015/585409 · Zbl 1335.92013
[21] Rockenfeller, R.; Günther, M., How to model a muscle’s active force-length relation: A comparative study, Comput. Methods Appl. Mech. Eng., 313, 1, 321-336b (2017), URL https://doi.org/10.1016/j.cma.2016.10.003 · Zbl 1439.74212
[22] Rockenfeller, R.; Günther, M., Inter-filament spacing mediates calcium binding to troponin: A simple geometric-mechanistic model explains the shift of force-length maxima with muscle activation, J. Theoret. Biol., 454, 1, 240-252 (2018), URL https://doi.org/10.1016/j.jtbi.2018.06.009 · Zbl 1406.92116
[23] Stephenson, D. G.; Wendt, I. R., Length dependence of changes in sarcoplasmic calcium concentration and myofibrillar calcium sensitivity in striated muscle fibres, J. Muscle Res. Cell Motil., 5, 3, 243-272 (1984), URL https://doi.org/10.1007/BF00713107
[24] Hill, A. V., The heat of shortening and the dynamic constants of muscle, Proc. R. Soc. Lond B, 126, 843, 136-195 (1938), URL https://doi.org/10.1098/rspb.1938.0050
[25] Chow, J. W.; Darling, W. G., The maximum shortening velocity of muscle should be scaled with activation, J. Appl. Physiol., 86, 3, 1025-1031 (1985), URL https://doi.org/10.1152/jappl.1999.86.3.1025
[26] Petrofsky, J. S.; Phillips, C. A., The influence of temperature, initial length and electrical activity on the force-velocity relationship of the medial gastrocnemius muscle of the cat, J. Biomech., 14, 5, 297-306 (1981), URL https://doi.org/10.1016/0021-9290(81)90039-7
[27] Till, O.; Siebert, T.; Rode, C.; Blickhan, R., Characterization of isovelocity extension of activated muscle: A Hill-type model for eccentric contractions and a method for parameter determination, J. Theoret. Biol., 225, 2, 176-187 (2008), URL https://doi.org/10.1016/j.jtbi.2008.08.009 · Zbl 1400.92053
[28] Brown, I. E.; Scott, S. H.; Loeb, G. E., Mechanics of feline soleus: II. Design and validation of a mathematical model, J. Muscle Res. Cell Motil., 17, 2, 221-233 (1996), URL https://doi.org/10.1007/BF00124244
[29] Bauer, I., Numerische Verfahren zur Lösung von Anfangswertaufgaben und zur Generierung von ersten und zweiten Ableitungen mit Anwendungen bei Optimierungsaufgaben in Chemie und Verfahrenstechnik (1999), Heidelberg University, URL https://doi.org/10.11588/heidok.00001513
[30] Bock, H. G., Numerical Treatment of Inverse Problems in Chemical Reaction Kinetics, 102-125 (1981), Springer Berlin Heidelberg, URL https://doi.org/10.1007/978-3-642-68220-9_8
[31] Bischof, C.; Carle, A.; Hovland, P.; Khademi, P.; Mauer, A., ADIFOR 2.0 users’ guide (revision d), ((1998)), URL https://doi.org/10.2172/93483
[32] Körkel, S., Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen (2002), Heidelberg University, URL https://doi.org/10.11588/heidok.00002980 · Zbl 1011.62076
[33] Khan, K. A.; Barton, P. I., Generalized derivatives for solutions of parametric ordinary differential equations with non-differentiable right-hand sides, J. Optim. Theory Appl., 163, 2, 355-386 (2014), URL https://doi.org/10.1007/s10957-014-0539-1 · Zbl 1304.49035
[34] Xu, X.; Antsaklis, P. J., Optimal control of switched systems based on parameterization of the switching instants, IEEE Trans. Automat. Control, 49, 1, 2-16 (2004), URL https://doi.org/10.1109/TAC.2003.821417 · Zbl 1365.93308
[35] Hairer, E.; Nørsett, S. P.; Wanner, G., (Solving Ordinary Differential Equations I. Solving Ordinary Differential Equations I, Springer Series in Computational Mathematics, vol. 8 (1993), Springer: Springer Berlin), URL https://doi.org/10.1007/978-3-540-78862-1 · Zbl 0789.65048
[36] Tolsma, J.; Barton, P., Hidden discontinuities and parametric sensitivity calculations, SIAM J. Sci. Comput., 23, 6, 1861-1874 (2002), URL https://doi.org/10.1137/S106482750037281X · Zbl 1009.65055
[37] Kircheis, R., Structure exploiting parameter estimation and optimum experimental design methods and applications in microbial enhanced oil recovery (2015), Heidelberg University, URL https://doi.org/10.11588/heidok.00022098 · Zbl 1318.65067
[38] Schlöder, J. P., Numerische Methoden zur Behandlung hochdimensionaler Aufgaben der Parameteridentifizierung (1988), Rheinische Friedrich-Wilhelms-Universität Bonn, (Dissertation) · Zbl 0639.65036
[39] Bock, H. G.; Körkel, S.; Schlöder, J. P., Parameter estimation and optimum experimental design for nonlinear differential equation models, (Model Based Parameter Estimation: Theory and Applications. Model Based Parameter Estimation: Theory and Applications, Contributions in Mathematical and Computational Sciences, Vol. 4 (2013)), 1-30, URL https://doi.org/10.1007/978-3-642-30367-8_1 · Zbl 1269.65014
[40] Pukelsheim, F., Optimal Design of Experiments (1993), John Wiley & Sons, URL https://doi.org/10.1137/1.9780898719109 · Zbl 0834.62068
[41] Gill, P. E.; Murray, W.; Saunders, M. A., SNOPT: an SQP algorithm for large-scale constrained optimization, SIAM Rev., 47, 1, 99-131 (2005), URL https://doi.org/10.1137/S0036144504446096 · Zbl 1210.90176
[42] Bard, Y., Nonlinear Parameter Estimation (1974), Academic Press · Zbl 0345.62045
[43] Maganaris, C. N.; Paul, J. P., Load-elongation characteristics of in vivo human tendon and aponeurosis, J. Exp. Biol., 203, Pt 4, 751-756 (2000), URL https://jeb.biologists.org/content/203/4/751
[44] Mörl, F.; Siebert, T.; Schmitt, S.; Blickhan, R.; Günther, M., Electro-mechanical delay in Hill-type muscle models, J. Mech. Med. Biol., 12, 5, 85-102 (2012), URL https://doi.org/10.1142/S0219519412500856
[45] Rockenfeller, R., On the application of mathematical methods in Hill-type muscle modeling: Stability, sensitivity and optimal control (2016), Universität Koblenz-Landau, URL https://kola.opus.hbz-nrw.de/frontdoor/index/index/docId/1273
[46] Hatze, H., A myocybernetic control model of skeletal muscle, Biol. Cybernet., 25, 2, 103-119 (1977), URL https://doi.org/10.1007/BF00337268 · Zbl 0346.92011
[47] Gordon, A. M.; Huxley, A. F.; Julian, F. J., The variation in isometric tension with sarcomere length in vertebrate muscle fibers, J. Physiol., 184, 1, 170-192 (1966), URL https://doi.org/10.1113/jphysiol.1966.sp007909
[48] Piazzesi, G.; Lucii, L.; Lombardi, V., The size and the speed of the working stroke of muscle myosin and its dependence on the force, J. Physiol., 545, Pt. 1, 145-151 (2002), URL https://doi.org/10.1113/jphysiol.2002.028969
[49] Günther, M.; Haeufle, D. F.B.; Schmitt, S., The basic mechanical structure of the skeletal muscle machinery: One model for linking microscopic and macroscopic scales, J. Theoret. Biol., 456, 1, 137-167 (2018), URL https://doi.org/10.1016/j.jtbi.2018.07.023 · Zbl 1406.92027
[50] Haeufle, D. F.B.; Günther, M.; Bayer, A.; Schmitt, S., Hill-type muscle model with serial damping and eccentric force-velocity relation, J. Biomech., 47, 6, 1531-1536a (2014)
[51] Rockenfeller, R.; Götz, T., Optimal control of isometric muscle dynamics, J. Math. Fund. Sci., 47, 1, 12-30 (2015), URL http://dx.doi.org/10.5614/j.math.fund.sci.2015.47.1.2
[52] Rockenfeller, R.; Günther, M., Math. Biosci., 291, 1, 56-58a (2017), URL https://doi.org/10.1016/j.mbs.2017.04.001
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