Smooth muscle contraction: Mechanochemical formulation for homogeneous finite strains

Abstract

Chemical kinetics of smooth muscle contraction affect mechanical properties of organs that function under finite strains. In an effort to gain further insight into organ physiology, we formulate a mechanochemical finite strain model by considering the interaction between mechanical and biochemical components of cell function during activation. We propose a new constitutive framework and use a mechanochemical device that consists of two parallel elements: (i) spring for the cell stiffness; (ii) contractile element for the sarcomere. We use a multiplicative decomposition of cell elongation into filament contraction and cross-bridge deformation, and suggest that the free energy be a function of stretches, four variables (free unphosphorylated myosin, phosphorylated cross-bridges, phosphorylated and dephosphorylated cross-bridges attached to actin), chemical state variable driven by Ca2+-concentration, and temperature. The derived constitutive laws are thermodynamically consistent. Assuming isothermal conditions, we specialize the mechanical phase such that we recover the linear model of Yang et al. [2003a. The myogenic response in isolated rat cerebrovascular arteries: smooth muscle cell. Med. Eng. Phys. 25, 691-709]. The chemical phase is also specialized so that the linearized chemical evolution law leads to the four-state model of Hai and Murphy [1988. Cross-bridge phosphorylation and regulation of latch state in smooth muscle. Am. J. Physiol. 254, C99-C106]. One numerical example shows typical mechanochemical effects and the efficiency of the proposed approach. We discuss related parameter identification, and illustrate the dependence of muscle contraction (Ca2+-concentration) on active stress and related stretch. Mechanochemical models of this kind serve the mathematical basis for analyzing coupled processes such as the dependency of tissue properties on the chemical kinetics of smooth muscle.

A continuum model for skeletal muscle contraction at homogeneous finite deformations

“…(6) for their true counterparts. Following Stålhand et al (2008), the free energy of the model is assumed to be additively decomposed in the following way:…”

“…To guarantee a physically reasonable behavior, the model must fulfill requirements like dissipation and objectivity. This is not implicitly guaranteed when the force is postulated as above and may lead to restrictions in the model, see Ambrosi and Pezzuto (2012) and Stålhand et al (2008).…”

“…This is done by applying fundamental principles in mechanics using a previously presented framework (see Stålhand et al 2008, andStålhand et al 2011). This framework implicitly guarantees the model to be dissipative and thermodynamically consistent.…”

Stress-sensitive tissue regeneration in viscoelastic biomaterials subjected to modulated tensile strain

“…In this study, strain-activated tissue regeneration is stimulated by subjecting viscoelastic biomaterials that contain uniformly dispersed anchorage-dependent cells to harmonic tensile stress. The formalism for scalar cross-phenomena originates from a consideration of the mechanochemistry of materials [10] and the corresponding rate of entropy generation in solids [11], but slight modification is necessary to include a contribution from timeindependent strain to stress-sensitive reactions. Hence, scalar stress-kinetic couplings are reformulated in terms of the magnitude of the 2 nd -rank strain tensor, not the velocity gradient tensor or the corresponding symmetric rate-of-strain tensor that is typical for fluids.…”

“…Hence, scalar stress-kinetic couplings are reformulated in terms of the magnitude of the 2 nd -rank strain tensor, not the velocity gradient tensor or the corresponding symmetric rate-of-strain tensor that is typical for fluids. The strain-energy function represents another option to characterize the effect of deformation on biochemical kinetics [11,12]. A strain-energy-dependent source term for bone cell proliferation that monitors tissue rigidity was proposed by Harrigan and Hamilton [13] which becomes activated when strain energy density increases above a predetermined threshold [12].…”