An extensible, flexible, multiscale, and multiphysics model for nonisometric skeletal muscle behavior is presented. The skeletal muscle chemoelectromechanical model is based on a bottom-up approach modeling the entire excitation-contraction pathway by strongly coupling a detailed biophysical model of a half-sarcomere to the propagation of action potentials along skeletal muscle fibers and linking cellular parameters to a transversely isotropic continuum-mechanical constitutive equation describing the overall mechanical behavior of skeletal muscle tissue. Since the multiscale model exhibits separable time scales, a special emphasis is placed on employing computationally efficient staggered solution schemes. Further, the implementation builds on the open-source software library OpenCMISS and uses state-of-the-art parallelization techniques taking advantage of the unique anatomical fiber architecture of skeletal muscles. OpenCMISS utilizes standardized data structures for geometrical aspects (FieldML) and cellular models (CellML). Both standards are designed to allow for a maximum flexibility, reproducibility, and extensibility. The results demonstrate the model’s capability of simulating different aspects of nonisometric muscle contraction and efficiently simulating the chemoelectromechanical behavior in complex skeletal muscles such as the tibialis anterior muscle. 1. Introduction Skeletal muscles’ ability to actively generate force in a controlled fashion allows us to consciously move our body. The force generation is achieved through complex processes on multiple scales and multiple parts of the musculoskeletal system, for example, neural stimuli generation, depolarization at neuromuscular junctions, force generation within skeletal muscle sarcomeres, force transmission to the tendons, and sensory feedback to the nervous system. These processes are extremely complex, strongly coupled with each other, and by far not fully understood. Like in many other research areas, detailed simulation frameworks appealing to realistic models can provide an effective tool to investigate functional and structural interrelations of skeletal muscle force generation. An improved understanding of the physiological mechanisms may also lead to a better understanding of mechanisms behind musculoskeletal disorders. State-of-the-art simulations taking into account the force generating capabilities of skeletal muscles are subject to either phenomenological descriptions using discrete [1–4] or continuum-mechanical models [5–7]. The most commonly used skeletal muscle modeling
References
[1]
G. J. van Ingen Schenau, M. F. Bobbert, G. J. Ettema, J. B. de Graaf, and P. A. Huijing, “A simulation of rat edl force output based on intrinsic muscle properties,” Journal of Biomechanics, vol. 21, no. 10, pp. 815–824, 1988.
[2]
M. G. Pandy, “Computer modeling and simulation of human movement,” Annual Review of Biomedical Engineering, vol. 3, pp. 245–273, 2001.
[3]
M. Günther, S. Schmitt, and V. Wank, “High-frequency oscillations as a consequence of neglected serial damping in Hill-type muscle models,” Biological Cybernetics, vol. 97, no. 1, pp. 63–79, 2007.
[4]
M. Günther and S. Schmitt, “A macroscopic ansatz to deduce the Hill relation,” Journal of Theoretical Biology, vol. 263, no. 4, pp. 407–418, 2010.
[5]
P. Meier and R. Blickhan, “FEM-simulation of skeletal muscle: the influence of inertia during activation and deactivation,” in Skeletal Muscle Mechanics: From Mechanisms to Function, W. Herzog, Ed., chapter 12, pp. 207–233, John Wiley & Sons, 2000.
[6]
S. S. Blemker, P. M. Pinsky, and S. L. Delp, “A 3D model of muscle reveals the causes of nonuniform strains in the biceps brachii,” Journal of Biomechanics, vol. 38, no. 4, pp. 657–665, 2005.
[7]
O. R?hrle and A. J. Pullan, “Three-dimensional finite element modelling of muscle forces during mastication,” Journal of Biomechanics, vol. 40, no. 15, pp. 3363–3372, 2007.
[8]
F. E. Zajac, “Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control,” Critical Reviews in Biomedical Engineering, vol. 17, no. 4, pp. 359–411, 1989.
[9]
S. S. Blemker and S. L. Delp, “Three-dimensional representation of complex muscle architectures and geometries,” Annals of Biomedical Engineering, vol. 33, no. 5, pp. 661–673, 2005.
[10]
L. R. Smith, G. Meyer, and R. L. Lieber, “Systems analysis of biologicalnetworks in skeletal muscle function,” Wiley Interdisciplinary Reviews, vol. 5, no. 1, pp. 55–71, 2013.
[11]
P. R. Shorten, P. O'Callaghan, J. B. Davidson, and T. K. Soboleva, “A mathematical model of fatigue in skeletal muscle force contraction,” Journal of Muscle Research and Cell Motility, vol. 28, no. 6, pp. 293–313, 2007.
[12]
B. Hernández-Gascón, J. Grasa, B. Calvo, and J. Rodríguez, “A 3D electro-mechanical continuum model for simulating skeletal muscle contraction,” Journal of Theoretical Biology, pp. 108–118, 2013.
[13]
J. W. Fernandez, M. L. Buist, D. P. Nickerson, and P. J. Hunter, “Modelling the passive and nerve activated response of the rectus femoris muscle to a flexion loading: a finite element framework,” Medical Engineering and Physics, vol. 27, no. 10, pp. 862–870, 2005.
[14]
M. B?l, R. Weikert, and C. Weichert, “A coupled electromechanical model for the excitation-dependent contraction of skeletal muscle,” Journal of the Mechanical Behavior of Biomedical Materials, vol. 4, no. 7, pp. 1299–1310, 2011.
[15]
O. R?hrle, J. B. Davidson, and A. J. Pullan, “Bridging scales: a three-dimensional electromechanical finite element model of skeletal muscle,” SIAM Journal on Scientific Computing, vol. 30, no. 6, pp. 2882–2904, 2008.
[16]
O. R?hrle, “Simulating the electro-mechanical behavior of skeletalmuscles,” IEEE Computing in Science and Engineering, vol. 12, no. 6, pp. 48–58, 2010.
[17]
O. R?hrle, M. Sprenger, E. Ramasamy, and T. Heidlauf, “Multiscale skeletal muscle modeling: from cellular level to a multi-segmentSkeletal muscle model of the upper limb,” in Computer Models in Biomechanics, G. A. Holzapfel and E. Kuhl, Eds., pp. 103–116, Springer, Amsterdam, The Netherlands, 2013.
[18]
O. R?hrle, J. B. Davidson, and A. J. Pullan, “A physiologically based, multi-scale model of skeletal muscle structure and function,” Frontiers in Physiology, vol. 3, 2012.
[19]
C. Bradley, A. Bowery, R. Britten et al., “OpenCMISS: a multi-physics & multi-scale computational infrastructure for the VPH/Physiome project,” Progress in Biophysics and Molecular Biology, vol. 107, no. 1, pp. 32–47, 2011.
[20]
G. R. Christie, P. M. F. Nielsen, S. Blackett, C. P. Bradley, and P. J. Hunter, “Fieldml: concepts and implementation,” Philosophical Transactions of the Royal Society A, vol. 367, no. 1895, pp. 1869–1884, 2009.
[21]
A. Garny, D. P. Nickerson, J. Cooper et al., “CellML and associated tools and techniques,” Philosophical Transactions of the Royal Society A, vol. 366, no. 1878, pp. 3017–3043, 2008.
[22]
C. M. Lloyd, M. D. B. Halstead, and P. F. Nielsen, “CellML: its future, present and past,” Progress in Biophysics and Molecular Biology, vol. 85, no. 2-3, pp. 433–450, 2004.
[23]
A. J. Fuglevand, D. A. Winter, and A. E. Patla, “Models of recruitment and rate coding organization in motor-unit pools,” Journal of Neurophysiology, vol. 70, no. 6, pp. 2470–2488, 1993.
[24]
F. Negro and D. Farina, “Decorrelation of cortical inputs and motoneuron output,” Journal of Neurophysiology, vol. 106, no. 5, pp. 2688–2697, 2011.
[25]
J. Bonet and R. D. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge University Press, 2nd edition, 2008.
[26]
A. J. M. Spencer, “Theory of invariants,” in Continuum Physics, A. Eringen, Ed., vol. 1, pp. 239–353, Academic Press, New York, NY, USA, 1971.
[27]
A. J. M. Spencer, Deformations of Fibre-Reinforced Materials, Oxford University Press, 1972.
[28]
G. A. Holzapfel, Nonlinear Solid Mechanics, John Wiley & Sons, West Sussex, England, 2000.
[29]
Y. Zheng, A. F. T. Mak, and B. Lue, “Objective assessment of limb tissue elasticity: development of a manual indentation procedure,” Journal of Rehabilitation Research and Development, vol. 36, no. 2, pp. 71–85, 1999.
[30]
B. Markert, W. Ehlers, and N. Karajan, “A general polyconvex strainenergy function for fiber-reinforced materials,” Proceedings in Applied Mathematics and Mechanics, vol. 5, no. 1, pp. 245–246, 2005.
[31]
D. Hawkins and M. Bey, “A comprehensive approach for studying muscle-tendon mechanics,” Journal of Biomechanical Engineering, vol. 116, no. 1, pp. 51–55, 1994.
[32]
A. E. Ehret, M. B?l, and M. Itskov, “A continuum constitutive model for the active behaviour of skeletal muscle,” Journal of the Mechanics and Physics of Solids, vol. 59, no. 3, pp. 625–636, 2011.
[33]
B. R. MacIntosh, P. F. Gardiner, and A. J. McComas, Skeletal Muscle: Form and Function, Human Kinetics, 2nd edition, 2006.
[34]
A. V. Hill, “The heat of shortening and the dynamic constants of muscle,” Proceedings of the Royal Society B, vol. 126, no. 843, pp. 136–195, 1938.
[35]
R. H. Adrian and L. D. Peachey, “Reconstruction of the action potential of frog sartorius muscle,” The Journal of Physiology, vol. 235, no. 1, pp. 103–131, 1973.
[36]
W. Wallinga, S. L. Meijer, M. J. Alberink, M. Vliek, E. D. Wienk, and D. L. Ypey, “Modelling action potentials and membrane currents of mammalian skeletal muscle fibres in coherence with potassium concentration changes in the T-tubular system,” European Biophysics Journal, vol. 28, no. 4, pp. 317–329, 1999.
[37]
E. Ríos, M. Karhanek, J. Ma, and A. González, “An allosteric model of the molecular interactions of excitation- contraction coupling in skeletal muscle,” The Journal of General Physiology, vol. 102, no. 3, pp. 449–481, 1993.
[38]
S. M. Baylor and S. Hollingworth, “Model of sarcomeric Ca2+ movements, including ATP Ca2+ binding and diffusion, during activation of frog skeletal muscle,” The Journal of General Physiology, vol. 112, no. 3, pp. 297–316, 1998.
[39]
M. V. Razumova, A. E. Bukatina, and K. B. Campbell, “Stiffness-distortion sarcomere model for muscle simulation,” Journal of Applied Physiology, vol. 87, no. 5, pp. 1861–1876, 1999.
[40]
M. V. Razumova, A. E. Bukatina, and K. B. Campbell, “Different myofilament nearest-neighbor interactions have distinctive effects on contractile behavior,” Biophysical Journal, vol. 78, no. 6, pp. 3120–3137, 2000.
[41]
K. B. Campbell, M. V. Razumova, R. D. Kirkpatrick, and B. K. Slinker, “Myofilament kinetics in isometric twitch dynamics,” Annals of Biomedical Engineering, vol. 29, no. 5, pp. 384–405, 2001.
[42]
K. B. Campbell, M. V. Razumova, R. D. Kirkpatrick, and B. K. Slinker, “Nonlinear myofilament regulatory processes affect frequency-dependent muscle fiber stiffness,” Biophysical Journal, vol. 81, no. 4, pp. 2278–2296, 2001.
[43]
A. F. Huxley and R. Niedergerke, “Structural changes in muscle during contraction: interference microscopy of living muscle fibres,” Nature, vol. 173, no. 4412, pp. 971–973, 1954.
[44]
C. Y. Scovil and J. L. Ronsky, “Sensitivity of a Hill-based muscle model to perturbations in model parameters,” Journal of Biomechanics, vol. 39, no. 11, pp. 2055–2063, 2006.
[45]
O. Till, T. Siebert, C. Rode, and R. Blickhan, “Characterization of isovelocity extension of activated muscle: a Hill-type model for eccentric contractions and a method for parameter determination,” Journal of Theoretical Biology, vol. 255, no. 2, pp. 176–187, 2008.
[46]
A. V. Hill, First and Last Experiments in Muscle Mechanics, UniversityPress Cambridge, 1970.
[47]
B. R. Epstein and K. R. Foster, “Anisotropy in the dielectric properties of skeletal muscle,” Medical and Biological Engineering and Computing, vol. 21, no. 1, pp. 51–55, 1983.
[48]
F. L. H. Gielen, W. Wallinga-de Jonge, and K. L. Boon, “Electrical conductivity of skeletal muscle tissue: experimental results from different muscles in vivo,” Medical and Biological Engineering and Computing, vol. 22, no. 6, pp. 569–577, 1984.
[49]
A. J. Pullan, M. L. Buist, and L. K. Cheng, Mathematically Modellingthe Electrical Activity of the Heart: From Cell to Body Surfaceand Back Again, World Scientific, 2005.
[50]
J. Sundnes, B. F. Nielsen, K. A. Mardal, X. Cai, G. T. Lines, and A. Tveito, “On the computational complexity of the bidomain and the monodomain models of electrophysiology,” Annals of Biomedical Engineering, vol. 34, no. 7, pp. 1088–1097, 2006.
[51]
B. F. Nielsen, T. S. Ruud, G. T. Lines, and A. Tveito, “Optimal monodomain approximations of the bidomain equations,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 276–290, 2007.
[52]
J. B. Davidson, Biophysical modelling of skeletal muscle [Ph.D. thesis], University of Auckland, Auckland, New Zealand, 2009.
[53]
J. Sundnes, G. T. Lines, and A. Tveito, “An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso,” Mathematical Biosciences, vol. 194, no. 2, pp. 233–248, 2005.
[54]
O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, vol. 1, Butterworth-Heinemann, 2005.
[55]
C. Linder, M. Tkachuk, and C. Miehe, “A micromechanically motivated diffusion-based transient network model and its incorporation into finite rubber viscoelasticity,” Journal of the Mechanics and Physics of Solids, vol. 59, no. 10, pp. 2134–2156, 2011.
[56]
M. Tkachuk and C. Linder, “The maximal advance path constraint for the homogenization of materials with random network microstructure,” Philosophical Magazine, vol. 92, no. 22, pp. 2779–2808, 2012.
[57]
V. M. Spitzer and D. G. Whitlock, “The visible human dataset: the anatomical platform for human simulation,” The Anatomical Record, vol. 253, no. 2, pp. 49–57, 1998.
[58]
W. Herzog and T. R. Leonard, “Force enhancement following stretching of skeletal muscle: a new mechanism,” Journal of Experimental Biology, vol. 205, no. 9, pp. 1275–1283, 2002.
[59]
D. Labeit, K. Watanabe, C. Witt et al., “Calcium-dependent molecular spring elements in the giant protein titin,” Proceedings of the National Academy of Sciences of the United States of America, vol. 100, no. 23, pp. 13716–13721, 2003.
[60]
C. Rode, T. Siebert, and R. Blickhan, “Titin-induced force enhancement and force depression: a “sticky-spring” mechanism in muscle contractions?” Journal of Theoretical Biology, vol. 259, no. 2, pp. 350–360, 2009.
[61]
B. Sharafi and S. S. Blemker, “A micromechanical model of skeletal muscle to explore the effects of fiber and fascicle geometry,” Journal of Biomechanics, vol. 43, no. 16, pp. 3207–3213, 2010.
[62]
B. Sharafi and S. S. Blemker, “A mathematical model of force transmission from intrafascicularly terminating muscle fibers,” Journal of Biomechanics, vol. 44, no. 11, pp. 2031–2039, 2011.
[63]
K. Hoyt, T. Kneezel, B. Castaneda, and K. J. Parker, “Quantitative sonoelastography for the in vivo assessment of skeletal muscle viscoelasticity,” Physics in Medicine and Biology, vol. 53, no. 15, pp. 4063–4080, 2008.
[64]
M. van Loocke, C. G. Lyons, and C. K. Simms, “A validated model of passive muscle in compression,” Journal of Biomechanics, vol. 39, no. 16, pp. 2999–3009, 2006.
[65]
M. Munteanu and L. F. Pavarino, “Decoupled schwarz algorithms for implicit discretizations of nonlinear monodomain and bidomain systems,” Mathematical Models and Methods in Applied Sciences, vol. 19, no. 7, pp. 1065–1097, 2009.
[66]
S. G?ktepe and E. Kuhl, “Electromechanics of the heart: a unified approach to the strongly coupled excitation-contraction problem,” Computational Mechanics, vol. 45, no. 2-3, pp. 227–243, 2010.