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A mechanochemical 3D continuum model for smooth muscle contraction under finite strains. (English) Zbl 1411.92020
Summary: This paper presents a modelling framework in which the mechanochemical properties of smooth muscle cells may be studied. The activation of smooth muscles is considered in a three-dimensional continuum model which is key to realistically capture the function of hollow organs such as blood vessels. On the basis of a general thermodynamical framework the mechanical and chemical phases are specialized in order to quantify the coupled mechanochemical process. A free-energy function is proposed as the sum of a mechanical energy stored in the passive tissue, a coupling between the mechanical and chemical kinetics and an energy related purely to the chemical kinetics and the calcium ion concentration. For the chemical phase it is shown that the cross-bridge model of C. M. Hai and R. A. Murphy, [“Cross-bridge phosphorylation and regulation of latch state in smooth muscle”, Am. J. Physiol. Cell Physiol. 254, No. 1, C99–C106 (1988; doi:10.1152/ajpcell.1988.254.1.C99)] is included in the developed evolution law as a special case. In order to show the specific features and the potential of the proposed continuum model a uniaxial extension test of a tissue strip is analysed in detail and the related kinematics and stress-stretch relations are derived. Parameter studies point to coupling phenomena; in particular the tissue response is analysed in terms of the calcium ion level. The model for smooth muscle contraction may significantly contribute to current modelling efforts of smooth muscle tissue responses.

92C10 Biomechanics
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
Ode15s; Simulink
Full Text: DOI
[1] Arner, A., Mechanical characteristics of chemically skinned guinea-pig taenia coli, Pflügers arch., 395, 277-284, (1982)
[2] Balzani, D.; Neff, P.; Schröder, J.; Holzapfel, G.A., A polyconvex framework for soft biological tissues. adjustment to experimental data, Int. J. solid struct., 43, 6052-6070, (2006) · Zbl 1120.74632
[3] Baek, S.; Gleason, R.L.; Rajagopal, K.R.; Humphrey, J.D., Theory of small on large: potential utility in computation of fluid – solid interaction in arteries, Comput. meth. appl. mech. eng., 196, 3070-3078, (2007) · Zbl 1127.74026
[4] Clark, J.M.; Glagov, S., Transmural organization of the arterial media. the lamellar unit revisited, Arteriosclerosis, 5, 19-34, (1985)
[5] Germain, P., The method of virtual power in continuum mechanics. part 2, SIAM J. appl. math., 50, 556-575, (1973) · Zbl 0273.73061
[6] Gestrelius, S.; Borgström, P., A dynamical model of smooth muscle contraction, Biophys. J., 50, 157-169, (1986)
[7] Gurtin, M.E.; Fried, E.; Anand, L., The mechanics and thermodynamics of continua, (2010), Cambridge University Press New York
[8] Hai, C.M.; Murphy, R.A., Cross-bridge phosphorylation and regulation of latch state in smooth muscle, Am. J. physiol., 254, C99-C106, (1988)
[9] Hill, A.V., The heat of shortening and the dynamics constants of muscle, Proc. R. soc. B, 126, 136-195, (1938)
[10] Holzapfel, G.A., Nonlinear solid mechanics. A continuum approach for engineers, (2000), Wiley & Sons Chichester
[11] Holzapfel, G.A.; Gasser, T.C.; Ogden, R.W., A new constitutive framework for arterial wall mechanics and a comparative study of material models, J. elasticity, 61, 1-48, (2000) · Zbl 1023.74033
[12] Holzapfel, G.A.; Gasser, T.C.; Ogden, R.W., Comparison for a multi-layer structural model for arterial walls with a Fung-type model, and issues of material stability, J. biomech. eng., 126, 264-275, (2004)
[13] Humphrey, J.D., Native properties of cardiovascular tissues: guidelines of functional tissue engineering, (), 35-45
[14] Klarbring, A.; Olsson, T.; Stålhand, J., Theory of residual stresses with application to arterial geometry, Arch. mech., 59, 341-364, (2007) · Zbl 1143.74041
[15] Li, Q.; Muragaki, Y.; Hatamura, I.; Ueno, H.; Ooshima, A., Stretch-induced collagen synthesis in cultured smooth muscle cells from rabbit aortic media and a possible involvement of angiotensin II and transforming growth factor-\(\beta\), J. vasc. res., 35, 93-103, (1998)
[16] Murtada, S.I., Kroon, M., Holzapfel, G.A., in press. A calcium-driven mechanochemical model for prediction of force generation in smooth muscle. Biomech. Model. Mechanobio, doi:10.1007/s10237-010-0211-0.
[17] Rachev, A.; Hayashi, K., Theoretical study of the effects of vascular smooth muscle contraction on strain and stress distributions in arteries, Ann. biomed. eng., 27, 459-468, (1999)
[18] Rembold, C.M.; Murphy, R.A., Latch-bridge model in smooth muscle: [ca^2+]i can predict quantitatively predict stress, Am. J. physiol. cell physiol., 259, C251-C257, (1990)
[19] Singer, H.A.; Kamm, K.E.; Murphy, R.A., Estimates of activation in arterial smooth muscle, Am. J. physiol., 251, C465-C473, (1986)
[20] Shampine, L.F.; Reichelt, M.W.; Kierzenka, J.A., Solving index-1 DAEs in MATLAB and simulink, SIAM rev., 41, 538-552, (1999) · Zbl 0935.65082
[21] Stålhand, J.; Klarbring, A.; Holzapfel, G.A., Smooth muscle contraction: mechanochemical formulation for homogeneous finite strains, Prog. biophys. molec. biol., 96, 465-481, (2008)
[22] Yang, J.; Clark, J.W.; Bryan, R.M.; Robertsson, C., The myogenic response in isolated rat cerebrovascular arteries: smooth muscle cell model, Med. eng. phys., 25, 691-709, (2003)
[23] Yang, J.; Clark, J.W.; Bryan, R.M.; Robertsson, C., The myogenic response in isolated rat cerebrovascular arteries: vessel model, Med. eng. phys., 25, 711-717, (2003)
[24] Wolinsky, H.; Glagov, S., A lamellar unit of aortic medial structure and function in mammals, Circ. res., 20, 99-111, (1967)
[25] Zulliger, M.A.; Rachev, A.; Stergiopulos, N., A constitutive formulation of mechanics including vascular smooth muscles, Am. J. physiol. heart circ. physiol., 287, H1335-H1343, (2004)
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