zbMATH — the first resource for mathematics

A continuum model for excitation-contraction of smooth muscle under finite deformations. (English) Zbl 1325.92012
Summary: The main focus in most of the continuum based muscle models is the mechanics of muscle contraction while other physiological processes governing muscle contraction, e.g., cell membrane excitation and activation, are ignored. These latter processes are essential to initiate contraction and to determine the amount of generated force, and by excluding them, the developed model cannot replicate the true behavior of the muscle in question. The aim of this study is to establish a thermodynamically and physiologically consistent framework which allows us to model smooth muscle contraction by including cell membrane excitability and kinetics of myosin phosphorylation, along with dynamics of smooth muscle contraction. The model accounts for these processes through a set of coupled dissipative constitutive equations derived by applying first principles. To show the performance of the derived model, it is evaluated for two different cases: a chemo-mechanical study of pig taenia coli cells where the excitation process is excluded, and an electro-chemo-mechanical study of rat myometrium. The results show that the model is able to replicate important aspects of the smooth muscle excitation-contraction process.

92C10 Biomechanics
92C40 Biochemistry, molecular biology
Full Text: DOI
[1] Arner, A., Mechanical characteristics of chemically skinned guinea-pig taenia coli, Pflügers Arch., 395, 4, 277-284, (1982)
[2] Barnett, M. W.; Larkman, P. M., The action potential, Pract. Neurol., 7, 192-197, (2007)
[3] Blemker, S. S.; Pinsky, P. M.; Delp, S. L., A 3d model of muscle reveals the causes of nonuniform strains in the biceps brachii, J. Biomech., 38, 657-665, (2005)
[4] Böl, M.; Abilez, O. J.; Assar, A. N.; Zarins, C. K.; Kuhl, E., In vitro/in silico characterization of active and passive stresses in cardiac muscle, Int. J. Multiscale Comput. Eng., 10, 171-188, (2012)
[5] Böl, M.; Schmitz, A.; Nowak, G.; Siebert, T., A three-dimensional chemo-mechanical continuum model for smooth muscle contraction, J. Mech. Behav. Biomed. Mater., 13, 215-229, (2012)
[6] Burdyga, T.; Wray, S.; Noble, K., In situ calcium signaling, Ann. N. Y. Acad. Sci., 1101, 85-96, (2007)
[7] Bursztyn, L.; Eytan, O.; Jaffa, A. J.; Elad, D., Mathematical model of excitation-contraction in a uterine smooth muscle cell, Am. J. Physiol.: Cell Physiol., 292, C1816-C1829, (2007)
[8] Cheng, D. K., Field and Wave Electromagnetics, vol. 2, (1989), Addison-Wesley New York
[9] Clayton, J. D., Nonlinear Mechanics of Crystals, vol. 177, (2011), Springer · Zbl 1209.74001
[10] DiCarlo, A., 2008. Elementary mechanics of muscular exercise. In: The Mathematics of Growth & Remodelling of Soft Biological Tissues, p. 14.
[11] Ehret, A. E.; Böl, M.; Itskov, M., A continuum constitutive model for the active behaviour of skeletal muscle, J. Mech. Phys. Solids, 59, 625-636, (2011) · Zbl 1270.74138
[12] Gabella, G., Arrangement of smooth muscle cells and intramuscular septa in the taenia coli, Cell Tissue Res., 184, 195-212, (1977)
[13] Gestrelius, S.; Borgström, P., A dynamic model of smooth muscle contraction, Biophys. J., 50, 157-169, (1986)
[14] Gordon, A.; Huxley, A. F.; Julian, F., The variation in isometric tension with sarcomere length in vertebrate muscle fibres, J. Physiol., 184, 170-192, (1966)
[15] Guggenheim, E., The conceptions of electrical potential difference between two phases and the individual activities of ions, J. Phys. Chem., 33, 842-849, (1929)
[16] Gurtin, M. E.; Fried, E.; Anand, L., The mechanics and thermodynamics of continua, (2010), Cambridge University Press, Cambridge.
[17] Hai, C. M.; Murphy, R. A., Cross-bridge phosphorylation and regulation of latch state in smooth muscle, Am. J. Physiol.: Cell Physiol., 254, C99-C106, (1988)
[18] Hai, C. M.; Murphy, R. A., Regulation of shortening velocity by cross-bridge phosphorylation in smooth muscle, Am. J. Physiol.: Cell Physiol., 255, C86-C94, (1988)
[19] Haust, M. D.; More, R. H.; Movat, H. Z., The role of smooth muscle cells in the fibrogenesis of arteriosclerosis, Am. J. Pathol., 37, 377, (1960)
[20] Herlihy, J. T.; Murphy, R. A., Length-tension relationship of smooth muscle of the hog carotid artery, Circ. Res., 33, 275-283, (1973)
[21] Hodgkin, A. L.; Huxley, A. F., A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117, 500, (1952)
[22] Johansson, T.; Meier, P.; Blickhan, R., A finite-element model for the mechanical analysis of skeletal muscles, J. Theor. Biol., 206, 131-149, (2000)
[23] Keener, J. P.; Sneyd, J., Mathematical Physiology: I: Cellular Physiology, vol. 1, (2008), Springer Verlag, New York, USA · Zbl 1273.92017
[24] Lang, R.; Rattray-Wood, C., A simple mathematical model of the spontaneous electrical activity in a single smooth muscle myocyte, (1996), Academic Press Limited
[25] Meier, P., Blickhan, R., 2000. FEM-simulation of skeletal muscle: the influence of inertia during activation and deactivation. In: Skeletal Muscle Mechanics: From Mechanisms to Function, pp. 207-223.
[26] Murphy, R., Mechanics of vascular smooth muscle. In: Comprehensive Physiology, 1980.
[27] Murtada, S. C.; Arner, A.; Holzapfel, G. A., Experiments and mechanochemical modeling of smooth muscle contractionsignificance of filament overlap, J. Theor. Biol., 297, 176-186, (2012) · Zbl 1336.92034
[28] Murtada, S. I.; Kroon, M.; Holzapfel, G. A., A calcium-driven mechanochemical model for prediction of force generation in smooth muscle, Biomech. Model. Mechanobiol., 9, 749-762, (2010)
[29] Nash, M. P.; Panfilov, A. V., Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias, Progress Biophys. Mol. Biol., 85, 501-522, (2004)
[30] Rihana, S.; Terrien, J.; Germain, G.; Marque, C., Mathematical modeling of electrical activity of uterine muscle cells, Med. Biol. Eng. Comput., 47, 665-675, (2009)
[31] Schmitz, A.; Böl, M., On a phenomenological model for active smooth muscle contraction, J. Biomech., 44, 2090-2095, (2011)
[32] Sharifimajd, B.; Stålhand, J., A continuum model for skeletal muscle contraction at homogeneous finite deformations, Biomech. Model. Mechanobiol., 12, 965-973, (2013)
[33] Stålhand, J.; Klarbring, A.; Holzapfel, G. A., Smooth muscle contractionmechanochemical formulation for homogeneous finite strains, Progress Biophys. Mol. Biol., 96, 465-481, (2008)
[34] Stålhand, J.; Klarbring, A.; Holzapfel, G. A., A mechanochemical 3d continuum model for smooth muscle contraction under finite strains, J. Theor. Biol., 268, 120-130, (2011) · Zbl 1411.92020
[35] Tong, W. C.; Choi, C. Y.; Karche, S.; Holden, A. V.; Zhang, H.; Taggart, M. J., A computational model of the ionic currents, ca^2+ dynamics and action potentials underlying contraction of isolated uterine smooth muscle, PloS one, 6, e18685, (2011)
[36] Uvelius, B., Isometric and isotonic length-tension relations and variations in cell length in longitudinal smooth muscle from rabbit urinary bladder, Acta Physiol. Scand., 97, 1-12, (1976)
[37] Vander, A.; Sherman, J.; Luciano, K. D., Human physiology: the mechanism of body function, (1998), WCB McGraw-Hill Burr Ridge, IL
[38] Walmsley, J. G.; Murphy, R. A., Force-length dependence of arterial lamellar, smooth muscle, and myofilament orientations, Am. J. Physiol.: Heart Circ. Physiol., 253, H1141-H1147, (1987)
[39] Yang, J.; Clark, J. W.; Bryan, R. M.; Robertson, C., The myogenic response in isolated rat cerebrovascular arteriessmooth muscle cell model, Med. Eng. Phys., 25, 691-709, (2003)
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.