zbMATH — the first resource for mathematics

A 3D-geometric model for the deformation of a transversally loaded muscle. (English) Zbl 1397.92071
Summary: Very recent measurements provided data on the ratio between the change in length of the active fibres and the lifting height of a mass compressing the muscle belly transversally to the direction of fibre contraction. In this study, we present additional data of the change in the third (unloaded) muscle dimension, extracted from the same contraction experiments. Using this data set for validation, we verify whether body models of two different geometries, cylindrical or ellipsoidal, can explain the three-dimensional deformation of a contracting muscle, when volume constancy is required as a constraint. Presetting the contractile length change and using this constraint, an additional equation is needed for model predictions. To that, we minimise the sum of the squared and weighted circumference length changes. With a specific set of the three penalty weights, it turns out that the ellipsoid model can explain the three-dimensional deformation. The set of penalty weights can also be interpreted as an anisotropic stiffness distribution of the connective tissue of the muscle belly. In various loading situations, our ellipsoidal model may help to predict the corresponding deformation scenarios or to calculate the stiffness distribution from measured load and deformation data.

92C10 Biomechanics
35Q92 PDEs in connection with biology, chemistry and other natural sciences
Full Text: DOI
[1] Almqvist, G.; Berndt, B., Gauss, landen, Ramanujan, the arithmetic-geometric Mean, ellipses, \(\pi\), and the ladies diary, Am. math. mon., 95, 585-608, (1988) · Zbl 0665.26007
[2] Azizi, E.; Roberts, T.J., Biaxial strain and variable stiffness in aponeuroses, J. physiol., 587, Pt 17, 4309-4318, (2009)
[3] Baskin, R.; Paolini, P., Volume change and pressure development in muscle during contraction, Am. J. physiol., 213, 4, 1025-1030, (1967)
[4] Böl, M.; Reese, S., Micromechanical modelling of skeletal muscles based on the finite element method, Comput. methods biomech. biomed. eng., 11, 5, 489-504, (2008)
[5] Boriek, A.; Zhu, D.; Zeller, M.; Rodarte, J., Inferences on force transmission from muscle fiber architecture of the canine diaphragm, Am. J. physiol.—regul. integrative comp. physiol., 280, 1, R156-R165, (2001)
[6] Callister, R.; Callister, R.; Peterson, E., Design and control of the head retractor muscle in a turtle, pseudemys (trachemys) scripta: I. architecture and histochemistry of single muscle fibers, J. comp. neurol., 325, 3, 405-421, (1992)
[7] Cull-Candy, S.; Miledi, R.; Nakajima, Y.; Uchitel, O., Visualization of satellite cells in living muscle fibres of the frog, Proc. R. soc. London B, 209, 1177, 563-568, (1980)
[8] Eldred, E.; Garfinkel, A.; Hsu, E.; Ounjian, M.; Roy, R.; Edgerton, V., The physiological cross-sectional area of motor units in the cat tibialis anterior, Anat. rec., 235, 3, 381-389, (1993)
[9] Eldred, E.; Ounjian, M.; Roy, R.; Edgerton, V., Tapering of the intrafascicular endings of muscle fibers and its implications to relay of force, Anat. rec., 236, 2, 390-398, (1993)
[10] Epstein, M.; Herzog, W., Theoretical models of skeletal muscle—biological and mathematical considerations, (1998), Wiley & Sons Chichester
[11] Huijing, P.; Baan, G., Extramuscular myofascial force transmission within the rat anterior tibial compartment: proximo-distal differences in muscle force, Acta physiol. scand., 173, 3, 297-311, (2001)
[12] Iwamoto, M.; Miki, K.; Yang, K.H., Development of a finite element model of the human shoulder to investigate the mechanical responses and injuries in side impact, JSME int. J. ser. C, 44, 4, 1072-1081, (2001)
[13] Johansson, T.; Meier, P.; Blickhan, R., A finite-element model for the mechanical analysis of skeletal muscles, J. theor. biol., 206, 1, 131-149, (2000)
[14] Kardel, T., Niels Stensen’s geometrical theory of muscle contraction (1667): a reappraisal, J. biomech., 23, 10, 953-965, (1990)
[15] Lee, D.; Glueck, M.; Khan, A.; Fiume, E.; Jackson, K., A survey of modeling and simulation of skeletal muscle, ACM trans. graphics, 28, 4, 1-13, (2010)
[16] Linder-Ganz, E.; Yarnitzky, G.; Yizhar, Z.; Siev-Ner, I.; Gefen, A., Real-time finite element monitoring of sub-dermal tissue stresses in individuals with spinal cord injury: toward prevention of pressure ulcers, Ann. biomed. eng., 37, 2, 387-400, (2009)
[17] Mackerle, J., Finite element crash simulations and impact-induced injuries: an addendum. A bibliography (1998-2002), Shock vib. dig., 6, 321-334, (2003)
[18] McMahon, T.A., Muscles, reflexes, and locomotion, (1984), Princeton University Press Princeton, NJ
[19] Meier, P.; Blickhan, R., FEM-simulation of skeletal muscle: the influence of inertia during activation and deactivation, (), 207-223, (Chapter 12)
[20] Otten, E.; Hulliger, M., A finite-elements approach to the study of functional architecture in skeletal muscle, Zoology, 98, 233-241, (1995)
[21] Ounjian, M.; Roy, R.; Eldred, E.; Garfinkel, A.; Payne, J.; Armstrong, A.; Toga, A.; Edgerton, V., Physiological and developmental implications of motor unit anatomy, J. neurobiol., 22, 5, 547-559, (1991)
[22] Pogogeff, I.; Murray, M., Form and behavior of adult Mammalian skeletal muscle in vitro, Anat. rec., 95, 3, 321-335, (1946)
[23] Ramanujan, S., Modular equations and approximations to \(\pi\), Q. J. math. (Oxford), 45, 355-372, (1914) · JFM 45.1249.01
[24] Ramanujan, S., Notebooks 2 volumes, (1957), Tata Institute of Fundamental Research Bombay
[25] Ramanujan, S., Collected papers, (1962), Chelsea New York
[26] Richmond, F.; MacGillis, D.; Scott, D., Muscle-fiber compartmentalization in cat splenius muscles, J. neurophysiol., 53, 4, 868-885, (1985)
[27] Röhrle, O.; Pullan, A., Three-dimensional finite element modelling of muscle forces during mastication, J. biomech., 40, 15, 3363-3372, (2007)
[28] Scheepers, F.; Parent, R.; Carlson, W.; May, S., Anatomy-based modeling of the human musculature, (), 163-172
[29] Segal, S.; Cunningham, S.; Jacobs, T., Motor nerve topology reflects myocyte morphology in hamster retractor and epitrochlearis muscles, J. morphol., 246, 2, 103-117, (2000)
[30] 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. cybern., 98, 2, 133-143, (2008) · Zbl 1149.92302
[31] Siebert, T.; Sust, M.; Thaller, S.; Tilp, M.; Wagner, H., An improved method to determine neuromuscular properties using force laws – from single muscle to applications in human movements, Hum. movement sci., 26, 2, 320-341, (2007)
[32] Siebert, T., Till, O., Blickhan, R., 2010. Muscle force is influenced by lateral muscle loading: experiment and simulation. In: 6th World Congress of Biomechanics (WCB). Singapore, p. 444.
[33] Siebert, T., Till, O., Blickhan, R. Work partitioning of transversally loaded muscle: experimentation and simulation. Comput. Methods Biomech. Biomed. Eng., submitted for publication. · Zbl 1407.92022
[34] Stensen (Stenonis), N., 1667. Elementorum Myologiæ Specimen, seu Muscili Descriptio Geometrica, vol. 2. Stellae, Florence, pp. 61-111 (see Kardel, T., 1990. Niels Stensen’s geometrical theory of muscle contraction (1667): a reappraisal. J. Biomech. 23 (10), 953-965).
[35] Swammerdam, J., 1663. Earliest known experimental evidence for volume constancy of skeletal muscle during contraction. After McMahon, T.A., 1984. Muscles, Reflexes, and Locomotion. Princeton University Press, Princeton, NJ; first published by quotation not before 1669, see also Kardel, T., 1990. Niels Stensen’s geometrical theory of muscle contraction (1667): a reappraisal. J. Biomech. 23 (10), 953-965.
[36] van Leeuwen, J.; Spoor, C., Modelling mechanically stable muscle architectures, Philos. trans. R. soc. London B, 336, 1277, 275-292, (1992)
[37] Van Loocke, M.; Lyons, C.G.; Simms, C.K., Viscoelastic properties of passive skeletal muscle in compression: stress-relaxation behaviour and constitutive modelling, J. biomech., 41, 7, 1555-1566, (2008)
[38] van Rooij, L.; Bours, R.; van Hoof, J.; Mihm, J.; Ridella, S.; Bass, C.; Crandall, J., The development, validation and application of a finite element upper extremity model subjected to air bag loading, Stapp car crash J., 47, 55-78, (2003)
[39] Verver, M.; van Hoof, J.; Oomens, C.; Wismans, J.; Baaijens, F., A finite element model of the human buttocks for prediction of seat pressure distributions, Comput. methods biomech. biomed. eng., 7, 4, 193-203, (2004)
[40] Wagner, H.; Siebert, T.; Ellerby, D.; Marsh, R.; Blickhan, R., ISOFIT: a model-based method to measure muscle-tendon properties simultaneously, Biomech. modeling mechanobiol., 4, 1, 10-19, (2005)
[41] Woittiez, R.; Huijing, P.; Boom, H.; Rozendal, R., A three-dimensional muscle model: a quantified relation between form and function of skeletal muscles, J. morphol., 182, 1, 95-113, (1984)
[42] Yaniv, Y.; Juhaszova, M.; Wang, S.; Fishbein, K.; Zorov, D.; Sollott, S., Analysis of mitochondrial 3D-deformation in cardiomyocytes during active contraction reveals passive structural anisotropy of orthogonal short axes, Plos one, 6, 7, (2011)
[43] Young, M.; Paul, A.; Rodda, J.; Duxson, M.; Sheard, P., Examination of intrafascicular muscle fiber terminations: implications for tension delivery in series-fibered muscles, J. morphol., 245, 2, 130-145, (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.