A mechanochemical 3D continuum model for smooth muscle contraction under finite strains.

*(English)*Zbl 1411.92020Summary: 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.

##### MSC:

92C10 | Biomechanics |

92C45 | Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) |

35Q92 | PDEs in connection with biology, chemistry and other natural sciences |

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\textit{J. Stålhand} et al., J. Theor. Biol. 268, 120--130 (2011; Zbl 1411.92020)

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