×

zbMATH — the first resource for mathematics

How to model a muscle’s active force-length relation: a comparative study. (English) Zbl 1439.74212
Summary: The active isometric force-length relation (FLR) of mammalian skeletal muscle is one of the most investigated characteristics throughout biomechanics. Numerous experiments have been conducted that reveal insights on the mechanisms of muscle contraction. However, the entity of molecular processes is yet not fully understood. Modelers thus rely on a rather descriptive characterization of experimental findings. Starting with the well-known, piece-wise linear formulation by A. M. Gordon et al. [“The variation in isometric tension with sarcomere length in vertebrate muscle fibres”, J. Physiol. 184, No. 1, 170–192 (1966; doi:10.1113/jphysiol.1966.sp007909)], a variety of structurally distinguishable FLR models have been developed. Five decades later, the original idea was taken up to derive the first purely physiological FLR formulation, based on sliding filament and cross-bridge theory. This derivation offers us the opportunity to contrast a broad variety of 19 distinct FLR models. By comparing their ability to fit experimental data, we deduce qualitative as well as quantitative acceptance criteria such as symmetry, normalization, complexity, and physiological interpretability. Resultant, different models comprise different advantages. The new piece-wise linear model is the overall most favorable, a further piece-wise exponential model is mathematically more robust, a polynomial model of fourth order has the best optimization properties, and a certain purely exponential model is the computationally cheapest. This work gives a detailed overview, as well as a mathematical/physiological assessment of existing FLR models, and serves as a guideline for modelers to choose a proper formulation based on individual requirements.

MSC:
74L15 Biomechanical solid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Blix, M., Die Länge und die Spannung des Muskels IV (german text), Skand. Arch. Für Physiol., 5, 1, 173-206 (1894)
[2] Bagust, J.; Knott, S.; Lewis, D. M.; Luck, J. C.; Westerman, R. A., Isometric contractions of motor units in a fast twitch muscle of the cat, J. Physiol., 231, 1, 87-104 (1973)
[3] Banus, M. G.; Zetlin, A. M., The relation of isometric tension to length in skeletal muscle, J. Cell. Comp. Physiol., 12, 3, 403-420 (1938)
[4] Brown, I. E.; Loeb, G. E., Measured and modeled properties of mammalian skeletal muscle. I. the effects of post-activation potentiation on the time course and velocity dependencies of force production, J. Muscle Res. Cell Motil., 20, 5-6, 443-456 (1999)
[5] Colle, J., Influences de l’excitant et du milieu extérieur sur les propiétés élastiques du muscle de grenouille (french text), Arch. Internat. Physiol. Biochim. Biophys., 31, 194-213 (1929)
[6] Debler, C., Zur Dynamick des quergestreiften Muskels, Z. Biol., 97, 93-98 (1936)
[7] Délèze, J. B., The mechanical properties of the semitendinosus muscle at lengths greater than its length in the body, J. Physiol., 158, 1, 154-164 (1961)
[8] Edman, K. A.P., The relation between sarcomere length and active tension in isolated semitendinosus fibres of the frog, J. Physiol., 183, 2, 407-417 (1966)
[9] Evans, C. L.; Hill, A. V., The relation of length to tension development and heat production on contraction in muscle, J. Physiol., 49, 1-2, 10-16 (1914)
[10] Gollapudi, S. K.; Lin, D. C., Experimental determination of sarcomere force-length relationship in type-I human skeletal muscle fibers, J. Biomech., 42, 13 (2009), epub
[11] Guschlbauer, C., Characterisation of the biomechanical, passive, and active properties of femur-tibia joint leg muscles in the stick insectel carausius morosus (2009), Universität zu Köln, (Ph.D. thesis)
[12] Henk, L. M.; Granzier, M.; Pollack, G. H., The descending limb of the force-sarcomere length relation of the frog revisited, J. Physiol., 421, 1, 595-615 (1990)
[13] Herzog, W.; Kamal, S.; Clarke, D., Myofilament lengths of cat skeletal muscle: theoretical considerations and functional implications, J. Biomech., 25, 8, 945-948 (1992)
[14] Herzog, W.; Leonard, T. R.; Renaud, J. M.; Wallace, J.; Chaki, G.; Bornemisza, S., Force-length properties and functional demands of cat gastrocnemius, soleus and plantaris muscle, J. Biomech., 25, 11, 1329-1335 (1992)
[15] Herzog, W.; ter Keurs, E. D.J., Force-length relation of in-vivo human rectus femoris muscles, Eur. J. Physiol., 411, 6, 642-647 (1988)
[16] Julian, F. J.; Sollins, M. R., Sarcomere length-tension relations in living rat papillary muscle, Circ. Res., 37, 3, 299-308 (1975)
[17] Muhl, Z. F., Active length tension relation and the effect of muscle pinnation on fiber lengthening, J. Morphol., 173, 3, 285-292 (1982)
[18] Pavlov, I.; Novinger, R.; Rassier, D. E., The mechanical behavior of individual sarcomeres of myofibrils isolated from rabbit psoas muscle, Am. J. Cell Physiol., 297, 5, C1211-C1219 (2009)
[19] Ralston, H. J.; Inman, V. T.; Strait, L. A.; Shaffrath, M. D., Mechanics of human isolated voluntary muscle, Am. J. Physiol., 151, 2, 612-620 (1947)
[20] Ramsey, R. W.; Street, S. F., The isometric length-tension diagram of isolated skeletal muscle fibers of the frog, J. Cell. Comp. Physiol., 15, 1, 11-34 (1940)
[21] Rassier, D. E.; Macintosh, B. R.; Herzog, W., Length dependence of active force production in skeletal muscle, J. Appl. Physiol., 86, 5, 1445-1457 (1999)
[22] Senay, L. C., Effect of ischemia on the length-tension diagram of mammalian skeletal muscle, Am. J. Physiol., 188, 1, 113-117 (1956)
[23] Stevens, J. C.; Dickinson, V.; Jones, N. B., Mechanical properties of human skeletal muscle from in vitro studies of biopsies, Med. Biol. Eng. Comput., 18, 1, 1-9 (1980)
[24] ter Keurs, H. E. D. J.; Iwazumi, T.; Pollack, G. H., The sarcomere length-tension relation in skeletal muscle, J. Gen. Physiol., 72, 4, 565-592 (1978)
[25] ter Keurs, H. E.D. J.; Rijnsburger, W. H.; van Heuningen, R.; Nagelsmit, M. J., Tension development and sarcomere length in rat cardiac trabeculae, Circ. Res., 46, 5, 703-714 (1980)
[26] Wilkie, D. R., The mechanical properties of muscle, Br. Med. Bull., 12, 3, 177-182 (1956)
[27] Williams, C. D.; Salcedo, M. K.; Irving, T. C.; Daniel, T. L., The length-tension curve in muscle depends on lattice spacing, Proc. R. Soc. B: Biol. Sci., 280, 1766, 7 (2013)
[28] Woittiez, R. D.; Huijing, P. A.; Boom, H. B.K.; Rozendal, R. H., A three-dimensional muscle model: A quantified relation between form and function of skeletal muscles, J. Morphol., 182, 1, 95-113 (1984)
[29] Zuurbier, C. J.; Heslinga, J. W.; Lee-de Groot, M. B.E.; Van der Laarse, W. J., Mean sarcomere length-force relationship of rat muscle fibre bundles, J. Biomech., 28, 1, 83-87 (1995)
[30] Huxley, A. F.; Niedergerke, R., Structural changes in muscle during contraction; interference microscopy of living muscle fibres, Nature, 173, 4412, 971-973 (1954)
[31] Huxley, H.; Hanson, J., Changes in the cross-striations of muscle during contraction and stretch and their structural interpretation, Nature, 173, 4412, 973-976 (1954)
[32] Huxley, A. F., Muscle structure and theories of contraction, Prog. Biophy. Biophys. Chem., 7, 1, 255-318 (1957)
[33] Pollack, G. H., The cross-bridge theory, Physiol. Rev., 63, 3, 1049-1113 (1983)
[34] Gordon, A. M.; Huxley, A. F.; Julian, F. J., The length-tension diagram of single vertebrate striated muscle fibres, Proc. Physiol. Soc., 21, 1, 28P-30P (1964)
[35] 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)
[36] 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) · Zbl 1149.92302
[37] Rode, C.; Siebert, T.; Tomalka, A.; Blickhan, R., Myosin filament sliding through the Z-disc relates striated muscle fibre structure to function, Proc. R. Soc. B, 283, 1826, 9 Pages (2016)
[38] Buchanan, T. S.; Lloyd, D. G.; Manal, K.; Besier, T. F., Estimation of muscle forces and joint moments using a forward-inverse dynamics model, Med. Sci. Sports Exerc., 37, 11, 1911-1916 (2005)
[39] Morgan, D. L., From sarcomeres to whole muscles, J. Exp. Biol., 115, 69-78 (1985)
[40] Winters, J. M., How detailed should muscle models be to understand multi-joint movement coordination?, Hum. Mov. Sci., 14, 4-5, 401-442 (1995)
[41] Ettema, G. J.C.; Huijing, P. A., Effects of distribution of muscle fiber length on active length-force characteristics of rat gastrocnemius medialis, Anat. Rec., 239, 4, 414-420 (1994)
[42] Gans, C.; Bock, W. J., The functional significance of muscle architecture-a theoretical analysis, Ergeb. Anat. Entwickl., 38, 1, 115-142 (1965)
[43] Lieber, R. L.; Fridén, J., Functional and clinical significance of skeletal muscle architecture, Muscle and Nerve, 23, 11, 1647-1666 (2000)
[44] Wickiewicz, T. L.; Roy, R. R.; Powel, P.; Edgerton, V., Muscle architecture of the human lower limb, Clinic. Orthop. Relat. Res., 179, 179, 275-283 (1983)
[45] Scott, S. H.; Winter, D. A., A comparison of three muscle pennation assumptions and their effect on isometric and isotonic force, J. Biomech., 24, 2, 163-167 (1991)
[47] Julian, F. J.; Sollins, M. R.; Moss, R. L., Sarcomere length non-uniformity in relation to tetanic responses of stretched skeletal muscle fibres, Proc. R. Soc. B, 200, 1138, 109-116 (1978)
[48] Morgan, D. L., New insights into the behavior of muscle during active lengthening, Biophys. J., 57, 2, 209-221 (1990)
[49] Leonard, T. R.; DuVall, M.; Herzog, W., Force enhancement following stretch in a single sarcomere, Am. J. Cell Physiol., 299, 6, C1389-C1401 (2010)
[50] Burkholder, T. J.; Lieber, R. L., Sarcomere length operating range of vertebrate muscles during movement, J. Exp. Biol., 204, Pt 9 (2001), 1539-1536
[51] Herzog, W.; Abrahamse, S. K.; ter Keurs, H. E. D. J., Theoretical determination of force-length relations of intact human skeletal muscles using the cross-bridge model, Eur. J. Physiol., 416, 1-2, 113-119 (1990)
[52] Meijer, K.; Grootenboer, H. J.; Koopman, B. F.J. M.; Huijing, P. A., Fully isometric length-force curves of rat muscle differ from those during and after concentric contraction, J. Appl. Biomech., 13, 2, 164-181 (1997)
[53] MacIntosh, B. R.; MacNaughton, M. B., The length dependence of muscle active force: considerations for parallel elastic properties, J. Appl. Physiol., 98, 5, 1666-1673 (2005)
[54] Rode, C.; Siebert, T.; Blickhan, R., Titin-induced force enhancement and force depression: A ‘sticky-spring’ mechanism in muscle contractions?, J. Theoret. Biol., 259, 2, 350-360 (2009) · Zbl 1402.92042
[55] Winters, T. M.; Takahashi, M.; Lieber, R. L.; Ward, S. R., Whole muscle length-tension relationships are accurately modeled as scaled sarcomeres in rabbit hindlimb muscles, J. Biomech., 44, 1, 109-115 (2011)
[56] Huijing, P. A., Muscle, the motor of movement: properties in function, experiment and modelling, J. Electromyogr. Kinesiol., 8, 2, 61-77 (1998)
[57] Mohammed, G. A.; Hou, M., Optimization of active muscle force-length models using least squares curve fitting, IEEE Trans. Biomed. Eng., 63, 3, 630-635 (2016)
[58] Edman, K. A.P.; Reggiani, C., The sarcomere length-tension relation of frog skeletal muscle has no plateau, J. Physiol., 353, 1, 62 (1984)
[59] Schoenberg, M.; Podolsky, R. J., Length-force relation of calcium activated muscle fibers, Science, 4030, 7, 52-54 (1976)
[60] 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) · Zbl 1125.92007
[61] Pedotti, A.; Krishnan, V. V.; Stark, L., Optimization of muscle-force sequencing in human locomotion, Math. Biosci., 38, 1-2, 56-76 (1978)
[62] van Soest, A. J.; Bobbert, M. F., The contribution of muscle properties in the control of explosive movements, Biol. Cybernet., 69, 3, 195-204 (1993)
[63] Bahler, A. S., Modeling of mammalian skeletal muscle, IEEE Trans. Biomed. Eng., BME-15, 4, 249-257 (1968)
[64] White, S. C.; Winter, D. A., Predicting muscle forces in gait from EMG signals and musculotendon kinematics, J. Electromyogr. Kinesiol., 2, 4, 217-231 (1993)
[65] Edman, K. A.P.; Reggiani, C., The sarcomere length-tension relation determined in short segments of intact muscle fibres of the frog, J. Physiol., 385, 1, 709-732 (1987)
[66] Hatze, H., Myocybernetic Control Models of Skeletal Muscle (1981), University of South Africa · Zbl 0635.92003
[67] Rosen, J.; Fuchs, M. B.; Arcan, M., Performances of Hill-type and neural network muscle models-toward a myosignal-based exoskeleton, Comput. Biomed. Res., 32, 5, 415-439 (1999)
[68] Otten, E., A myocybernetic model of the jaw system of the rat, J. Neurosci. Methods, 21, 2-4, 287-302 (1987)
[69] van der Linden, B. J.J. J., Mechanical modeling of skeletal muscle functioning (1998), Universiteit Twente, (Ph.D. thesis)
[70] Winters, J. M.; Stark, L., Analysis of fundamental human movement patterns through the use of in-depth antagonistic muscle models, IEEE Trans. Biomed. Eng., BME-32, 10, 826-839 (1985)
[71] Blümel, M.; Hooper, S. L.; Guschlbauer, C.; White, W. W.; Büschges, A., Determining all parameters necessary to build Hill-type muscle models from experiments on single muscles, Biol. Cybernet., 106, 10, 543-558 (2012)
[73] Freivalds, A., Incorporation of Active Elements into The Articulated total Body Model (1985), Defense Technical Information Center
[74] Bahill, A. T., Bioengineering: Biomedical, Medical and Clinical Engineering (1981), Prentice Hall
[75] DeWoody, Y.; Martin, C. F.; Schovanec, L., A forward dynamic model of gait with application to stress analysis of bone, Math. Comput. Modelling, 33, 1-3, 121-143 (2001) · Zbl 0960.92005
[76] Lloyd, D. G.; Besier, T. F., An EMG-driven musculoskeletal model to estimate muscle forces and knee joint moments in vivo, J. Biomech., 36, 6, 765-776 (2003)
[77] Zajac, F. E., Muscle and tendon: Properties, models, scaling, and application to biomechanics and motor control, CRC Crit. Rev. Biomed. Eng., 17, 4, 359-411 (1989)
[78] Millard, M.; Uchida, T.; Seth, A.; Delp, S. L., Flexing computational muscle: Modeling and simulation of musculotendon dynamics, J. Biomech. Eng., 135, 2, 12 pages (2013)
[79] Levenberg, K., A method for the solution of certain problems in least squares, Quart. Appl. Math., 2, 164-168 (1944) · Zbl 0063.03501
[80] Marquart, D., An algorithm for least-squares estimation of nonlinear parameters, SIAM J. Appl. Math., 11, 2, 431-441 (1963) · Zbl 0112.10505
[81] Shirangi, M. G.; Emerick, A. A., An improved TSVD-based Levenberg-Marquardt algorithm for history matching and comparison with Gauss-Newton, J. Pet. Sci. Eng., 143, 1, 258-271 (2016)
[82] Strutz, T., Data Fitting and Uncertainty: A Practical Introduction to Weighted Least Squares and Beyond, Vol. 1 (2016), Springer
[83] Hawkins, D. M., The problem of overfitting, J. Chem. Inf. Comput. Sci., 44, 1, 1-12 (2004)
[84] Transtrum, M. K.; MAchta, B. B.; Sethna, J. P., Why are nonlinear fits to data so challenging?, Phys. Rev. Lett., 104, 6, 6-12 (2010)
[85] Noble, M. I., Enhancement of mechanical performance of striated muscle by stretch during contraction, Exp. Physiol., 77, 4, 539-552 (1992)
[86] Nishikawa, K. C.; Monroy, J. A.; Uyeno, T. E.; Yeo, S. H.; Pai, D. K.; Linstedt, S. L., Is titin a ‘winding filament’? a new twist on muscle contraction, Proc. R. Soc. B, 279, 1730, 981-990 (2012)
[87] Rode, C.; Siebert, T.; Herzog, W.; Blickhan, R., The effects of parallel and series elastic components on estimated active cat soleus muscle force, J. Mech. Med. Biol., 9, 1, 105-122 (2009)
[88] Rockenfeller, R.; Günther, M., Determining concentric and eccentric dynamic muscle properties from isometric contraction experiments, Math. Biosci., 278, 1, 77-93 (2016) · Zbl 1346.92016
[89] Schappacher-Tilp, G.; Leonard, T.; Desch, G.; Herzog, W., A novel three-filament model of force generation in eccentric contraction of skeletal muscles, Public Libr. Sci. One, 10, 3, e0117634 (2015)
[90] Craig, R., Structure of A-segments from frog and rabbit skeletal muscle, J. Mol. Biol., 109, 1, 69-81 (1977)
[91] ter Keurs, H. E. D. J.; Luff, A. R.; Luff, Susan E., Contractile Mechanisms in Muscle, Chapter force-Sarcomere-Length Relation and Filament Length in Rat Extensor Digitorum Muscle, 511-525 (1984), Springer US: Springer US Boston, MA
[92] Rinne, H., Taschenbuch der Statistik, Vol. 4 (2008), Harri Verlag GmbH
[93] Herzog, W.; Nigg, B., Biomechanics of The Musculo-skeletal System (2007), John Wiley
[94] 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, 1531-1536 (2014)
[95] Rockenfeller, R., On the application of mathematical methods in hill-type muscle modeling: Stability, sensitivity and optimal control (2016), Universität Koblenz-Landau, (Ph.D. thesis)
[96] Brown, I. E.; Loeb, G. E., Measured and modeled properties of mammalian skeletal muscle: IV. dynamics of activation and deactivation, J. Muscle Res. Cell Motil., 21, 1, 33-47 (2000)
[97] 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)
[98] Cheng, E. J.; Brown, I. E.; Loeb, G. E., Virtual muscle: a computational approach to understanding the effects of muscle properties on motor control, J. Neurosci. Methods, 101, 2, 117-130 (2000)
[99] Gareis, H.; Solomonow, M.; Baratta, R.; Best, R.; D’Ambrosia, R., The isometric length-force models of nine different skeletal muscles, J. Biomech., 25, 8, 903-916 (1992)
[100] Hodson-Tole, E. F.; Wakeling, J. M., The influence of strain and activation on the locomotor function of rat ankle extensor muscles, J. Exp. Biol., 213, 2, 318-330 (2010)
[101] Kaufman, K. R.; An, K.; Chao, E. Y.S., Incorporation of muscle architecture into the muscle length-tension relationship, J. Biomech., 22, 8/9, 943-948 (1989)
[102] Audu, M. L.; Davy, D. T., The influence of muscle model complexity in musculoskeletal motion modeling, J. Biomech. Eng., 107, 2, 147-156 (1985)
[103] Challis, J. H.; Kerwin, D. G., Determining individual muscle forces during maximal activity: model development, parameter determination, and validation, Hum. Mov. Sci., 13, 1, 29-61 (1994)
[104] Gallucci, J. G.; Challis, J. H., Examining the role of the gastrocnemius during the leg curl exercise, J. Appl. Biomech., 18, 1, 15-27 (2002)
[105] Ramírez, A.; Grasa, J.; Alonso, A.; Soteras, F.; Osta, R.; Muños, M. J.; Calvo, B., Active response of skeletal muscle: In vivo experimental results and model formulation, J. Theoret. Biol., 267, 4, 546-553 (2010) · Zbl 1414.92069
[106] Thelen, D. G., Adjustment of muscle mechanics model parameters to simulate dynamic contractions in older adults, ASME J. Biomech. Eng., 125, 1, 70-77 (2003)
[107] Winter, S. L.; Challis, J. H., The expression of the skeletal muscle force-length relationship in vivo: a simulation study, J. Theoret. Biol., 262, 4, 634-643 (2010) · Zbl 1403.92020
[108] Ettema, G. J.C.; Meijer, K., Muscle contraction history: Modified Hill versus an exponential decay model, Biol. Cybernet., 83, 6, 491-500 (2000)
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.