Characterization of the active response of a guinea pig carotid artery

This work presents a characterization of the active response of the carotid artery of guinea pig fetuses through a methodology that encompasses experiments, modeling and numerical simulation. To this end, the isometric contraction test is carried out in ring samples subjected to different levels of KCl concentrations and pre-stretching. Then, a coupled mechanochemical model, aimed at describing the smooth cell behavior and its influence on the passive and active mechanical response of the vascular tissue, is calibrated from the experimental measurements. Due to the complex stress and strain fields developed in the artery, a finite element numerical simulation of the test is performed to fit the model parameters, where those related to the phosphorylation and dephosphorylation activity along with the load-bearing capacity of the myosin cross-bridges are found to be the most predominant when sensitizing the active response. The main strengths of the model are associated with the prediction of the stationary state of the active mechanical response of the tissue through a realistic description of the mechanochemical process carried out at its cellular level.


A.2 Mechanochemical model
The model used to describe the active mechanical response of the arterial wall is based on the mechanochemical model proposed by Murtada et al., 2012, which considers phosphorylated/dephosphory states of cross-bridges due to increment/decrement of intracellular [Ca +2 ] level (chemical contribution) and sliding of actin and myosin fibers, both mechanisms that finally triggers the SMC contraction (mechanical contribution). Different variables of the active behavior defined below are set in the reference space known as a contractile unit (CU), composed by actin and myosin filaments in addition to myosin cross bridges, where each CU is separated by a dense body; see Figure 1. In order to explain as clearly as possible the different variables that compose the mechanochemical model, sub-sections have been created to define its chemical and mechanical contributions and that related to the mechanism of fibre contraction.  Figure 2, where k M LCK corresponds to a shift from a dephosphorylated to a phosphorylated state under both attached and detached conditions such that its value is related to the MLCK activity. In the same way, k M LCP is associated with the rate of transition from a phosphorylated to a dephosphorylated state, representing the MLCP activity. k 3 and k 4 respectively represent attachment and detachment of the cross-bridge cycle, while k 7 is the rate of latch-bridge detachment. The kinetic model is represented by a set of differential equation written as:  To determine the k M LCP value, it is stated that temporal changes of the phosphorylation and dephosphorylation processes do not exist in the steady-state condition, i.e.,ṅM =ṅM p = 0.
Therefore, a constant k M LCP value is determined, depending on the steady-state condition (t → ∞), as: The remaining parameter values (k 3 , k 4 and k 7 ) can be obtained by fitting the model via numerical simulation as shown in Section 3.2.

Contractile unit elongation
According to Figure 1, each CU consists of two actin filaments located above and below a myosin filament (lengths L a and L m ), where the CU length L CU is given by the distance between two consecutive dense bodies located between the myosin filament. Regarding to the myosoin cross-bridges, these are equally spaced by a distance δ. Changes in the CU length are determined by the contribution of two components: u f s associated to the relative filament sliding which can be generated by the myosin power-stroke or external deformation, and u e that takes into account the elastic elongation of the cross-bridges attached to the actin filaments (i.e., u e only exists when there are crossed bridges attached to actin filaments). Therefore, CU elongation is defined by: where In this situation, the optimal overlapped length (L opt o ) remains constant until the analysed actin filament starts to move away from the myosin filament and the next CU appears, taking place the third situation, and this process is repeated again. This behavior can be modelled by a parabolic function, defining an optimal relative sliding filament value u opt f s as: It is also assumed that the active stress is directly proportional to the overlap length (L o ) for which the reference (active stress P 0 )) and the optimum (active stress P opt ) states are considered. From this consideration, it is possible to have an expression for the initial overlap (in the reference state): It is worth mentioning that the filament sliding rate (u f s ) is defined in this model in terms of other variables that have not yet introduced and are described below in the present section.

Active stress
From the definition of the cross-bridge elastic elongation u e (Murtada et al., 2012), an expression of the active stress P a can be established as: where u e is written in terms of the stretch λ and the relative filament slidingū f s according to Expression 5. Adittionally, two terms are defined:L o = L o /L CU which is a function of the relative filament sliding (ū f s ) (see Expression 6), and µ a which is associated with the crossbridge stiffness. It should be noted that Expression 8 involves the chemical part of the model along with the mechanical part, where (n AM P + n AM ) is obtained from a chemical analysis of the phosphorylation/ dephosphorylation process according to Expression 1, while the filament sliding process is described by the rest of the factors appearing in Expression 8.

Displacement rate
In order to get an approach to describe the contractile mechanism, an evolution law of the relative filament sliding (u f s ) is proposed according to the Hill muscle model for which a hyperbolic force-velocity relationship is written as: where α, β 1 and β 2 correspond to fitting parameters, while P LBC is refered to the maximum stress that can be supported by a CU. The second term in the right hand side of Expression 9 has been added to quantify the effect of muscle relaxation. On the other hand, the P c , that operates as driving stress, depends on muscle state (relaxation or contraction): When P cntr c ≤ P a ≤ P relax c , P c is defined by the active stress P a , according to Expression 8.