zbMATH — the first resource for mathematics

Analysis of the force-sharing problem using an optimization model. (English) Zbl 1053.92005
Summary: In biomechanics, one frequently used approach for finding a unique set of muscle forces in the ‘force-sharing problem’ is to formulate and solve a nonlinear optimization problem of the form: \(\min \varphi(f)= \sum(f_i/ \omega_i)^\alpha\) subject to \(Af=b\) and \(f\geq 0\). Solutions to this problem have typically been obtained numerically for complex models, or analytically for specific musculoskeletal geometries. Here, we present simple geometrical methods for analyzing the solution to this family of optimization problems for a general \(n\)-degrees-of-freedom musculoskeletal system.
For example, it is shown that the moment-arm vectors of active \((f_i>0)\) and passive \((f_i=0)\) muscles are separated by a hyperplane through the origin of the moment-arm vector space. For the special case of a system with two degrees-of-freedom, solutions can be readily represented in graphical form. This allows for powerful interpretations of force-sharing calculated using optimization.

92C10 Biomechanics
49N90 Applications of optimal control and differential games
Full Text: DOI
[1] Bernstein, N., The coordination and regulation of movements, (1967), Pergamon Oxford
[2] Crowninshield, R.D.; Brand, R.A., The prediction of forces in joint structures: distribution of intersegmental resultants, Exercise sport sci. R., 9, 159, (1981)
[3] Walmsley, B.; Hodgson, J.A.; Burke, R.E., Forces produced by medial gastrocnemius and soleus muscles during locomotion in freely moving cats, J. neurophysiol., 41, 5, 1203, (1978)
[4] Herzog, W.; Leonard, T.R., Validation of optimization models that estimate the forces exerted by synergistic muscles, J. biomech., 24, Suppl. 1, 31, (1991)
[5] Tsirakos, D.; Baltzopoulos, V.; Bartlett, R., Inverse optimization: functional and physiological considerations related to the force-sharing problem, Crit. rev. biomed. eng., 25, 4-5, 371, (1997)
[6] Crowninshield, R.D.; Brand, R.A., A physiologically based criterion of muscle force prediction in locomotion, J. biomech., 14, 11, 793, (1981)
[7] Herzog, W., Force-sharing among synergistic muscles: theoretical considerations and experimental approaches, Exercise sport sci. R., 24, 173, (1996)
[8] Hardt, D.E., Determining muscle forces in the leg during normal human walking: an application and evaluation of optimization methods, J. biomech. eng., 100, 72, (1978)
[9] Binding, P.; Jinha, A.; Herzog, W., Analytical analysis of the force sharing among synergistic muscles in one and two degree-of-freedom models, J. biomech., 33, 11, 1423, (2000)
[10] van Bolhuis, B.M.; Gielen, C.C.A.M., A comparison of models explaining muscle activation patterns for isometric contractions, Biol. cybernet., 81, 249, (1999) · Zbl 0957.92013
[11] Herzog, W., Individual muscle force estimations using a non-linear optimal design, J. neurosci. meth., 21, 2-4, 167, (1987)
[12] Buchanan, T.S.; Shreeve, D.A., An evaluation of optimization techniques for the prediction of muscle activation patterns during isometric tasks, J. biomech. eng., 118, 4, 565, (1996)
[13] Brand, R.A.; Pedersen, D.R.; Friederich, J.A., The sensitivity of muscle force predictions to changes in physiologic cross-sectional area, J. biomech., 19, 8, 589, (1986)
[14] Davy, D.T.; Audu, M.L., A dynamic optimization technique for predicting muscle forces in the swing phase of gait, J. biomech., 20, 2, 187, (1987)
[15] Herzog, W.; Binding, P., Predictions of antagonistic muscular activity using non-linear optimization, Math. biosci., 111, 2, 217, (1992) · Zbl 0781.92007
[16] Challis, J.H.; Kerwin, D.G., An analytical examination of muscle force estimations using optimization techniques, Proceedings of the institution of mechanical engineers: part H, J. eng. med., 207, 3, 139, (1993)
[17] Collins, J.J., The redundant nature of locomotor optimization laws, J. biomech., 28, 3, 251, (1995)
[18] Raikova, R.T.; Prilutsky, B.I., Sensitivity of predicted muscle forces to parameters of the optimization-based human leg model revealed by analytical and numerical analyses, J. biomech., 34, 1243, (2001)
[19] Herzog, W.; Binding, P., Cocontraction of pairs of antagonistic muscles: analytical solution for planar static non-linear optimization approaches, Math. biosci., 118, 83, 83, (1993) · Zbl 0784.92003
[20] Wismer, D.A.; Chattergy, R., Introduction to non-linear optimization: a problem solving approach, (1979), Elsevier North Holland Inc. New York
[21] R.E. Hughes, D.B. Chaffin, Conditions under which optimization models will not predict coactivation of antagonist muscles, in: Proceedings of the American Society of Biomechanics 12 (1988) 69
[22] Seireg, A.; Arvikar, R.J., A mathematical model for evaluation of forces in lower extremities of the musculoskeletal system, J. biomech., 6, 313, (1973)
[23] Pedotti, A.; Krishnan, V.V.; Stark, L., Optimization of muscle-force sequencing in human locomotion, Math. biosci., 38, 57, (1978)
[24] Dul, J.; Townsend, M.A.; Shiavi, R.; Johnson, G.E., Muscular synergism - II. A minimum-fatigue criterion for load sharing between synergistic muscles, J. biomech., 17, 9, 675, (1984)
[25] Hughes, R.E., Effect of optimization criterion on spinal force estimates during asymmetric lifting, J. biomech., 33, 225, (2000)
[26] Herzog, W., Sensitivity of muscle force estimations to changes in muscle input parameters using non-linear optimization approaches, J. biomech. eng., 114, 2, 267, (1992)
[27] Zajac, F.E.; Gordon, M.E., Determining a muscle’s force and action in multi-articular movement, Exercise sport sci. R., 17, 187, (1989)
[28] Dul, J.; Townsend, M.A.; Shiavi, R.; Johnson, G.E., Muscular synergism - I. on criteria for load sharing between synergistic muscles, J. biomech., 17, 9, 663, (1984)
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.