Hydrostatic Pressure

A structural-based model for the effective (tissue-level) mechanical behavior of the RVFW was developed as follows. We idealize the RV myocardium as primarily composed of two major material phases: a dense fibrous phase and a non-fibrous substance. The fibrous phase is subdivided into two mechanically relevant fiber constituents: myo- and collagen fibers. Here, we make a further distinction between two sub-groups of collagen fibers: the “large” perimysial collagen fibers that were observed to run nearly parallel to myofibers (Fig. 3a), and the “fine” endomysial collagen fiber network that surrounds individual myofibers as well as interconnects extensively with the large collagen fibers (Figs. 3a,b) (Borg and Caulfield 1981 (link); Borg et al. 1983 (link)). Also, the non-fibrous phase, referred to as the matrix phase, includes remaining mechanically insignificant substances such as vasculature, neural tissues, myofibroblasts, etc. This phase together with the fluid phase are assumed to be responsible for the incompressibility of the tissue and are combined into one single phase.
Based on these structural idealizations, we first consider a representative tissue element (RTE) of RVFW treated as a 3D continuum. The RTE is assumed to be large enough to represent the properties associated with the myocardial microstructure in an average sense, yet small compared to the characteristic length scale of the RVFW (typical RTE size was 3mm × 3mm × 0.5 mm). Upon application of external loading, a material point X in the reference configuration of the RTE moves to a new point x = X(x,t) at time t in the deformed configuration of the RTE. The deformation gradient tensor F = Gradx serves to characterize the deformation of the material. The tissue-level right Cauchy-Green tensor is defined as C = F^TF, and the tissue-level Green-Lagrange strain tensor is given by E = (C − I) / 2, where I is the identity tensor.
Ignoring any time-dependency effects, each constituent phase is assumed to exhibit a pseudo-hyperelastic behavior (Fung 1993 ), so that the effective mechanical behavior of the RVFW in loading can be described by a strain energy function Ψ(C). In this work, we assume the affine kinematics for tissue constituents, i.e. the fibers and matrix phases undergo the same deformation applied to the tissue (Fan and Sacks 2014 (link); Lanir 1983b (link); Lee et al. 2015 (link)). The total energy function can be written as the sum of the mechanical contribution of the ground matrix and embedded fibers, as
$$\mathrm{\Psi }(\mathbf{C})={\mathrm{\varphi }}^{\mathrm{g}}{\mathrm{\Psi }}^{\mathrm{g}}(\mathbf{C})+{\mathrm{\varphi }}^{\mathrm{m}}{\mathrm{\Psi }}^{\mathrm{m}}(\mathbf{C})+{\mathrm{\varphi }}^{\mathrm{c}}{\overline{\mathrm{\Psi }}}^{\mathrm{c}}(\mathbf{C})={\mathrm{\varphi }}^{\mathrm{g}}{\mathrm{\Psi }}^{\mathrm{g}}(\mathbf{C})+{\mathrm{\varphi }}^{\mathrm{m}}{\mathrm{\Psi }}^{\mathrm{m}}(\mathbf{C})+{\mathrm{\varphi }}^{\mathrm{c}}[{\mathrm{\Psi }}^{\mathrm{c}}(\mathbf{C})+{\mathrm{\Psi }}^{\mathrm{m}-\mathrm{c}}(\mathbf{C})],$$
where Ψ^g, Ψ^m and Ψ̄^c are the strain energy functions associated with the ground matrix phase, myofibers, and collagen fibers. The energy term Ψ̄^c has contributions from large (perimysial) collagen fibers and fine (endomysial) collagen fiber network (Fig. 3b). The latter is hypothesized to drive the interaction (coupling) between myofiber and (large) collagen fibers. Therefore, two terms Ψ^c and Ψ^m-c in (6) are designated to capture the mechanical contribution of large collagen fibers and their interaction with myofibers. Note that ϕ^c in (6) represents the total volume fraction of both groups of collagen fibers (as we were able to measure in our experimental analysis). The accurate calculation of amount of energy generated in each group of collagen fibers will be still due to the knowledge of sub-fraction of each collagen group which is absorbed in the respective term in the form (6). Such information is not currently available, however the form (6) is still useful to estimate the contribution of each fiber group to the total energy.
At the tissue level, the second Piola-Kirchhoff stress tensor S can be described in terms of the energy function Ψ(C), through
$$\mathbf{S}=2\frac{\partial \mathrm{\Psi }}{\partial \mathbf{C}}-\mathrm{p}{\mathbf{C}}^{-1}={\mathbf{S}}^{\mathrm{g}}+{\mathbf{S}}^{\mathrm{m}}+{\mathbf{S}}^{\mathrm{c}}+{\mathbf{S}}^{\mathrm{m}-\mathrm{c}},$$
where S^g = 2ϕ^g∂Ψ^g (C) / ∂C −pC⁻¹, S^m = 2ϕ^m ∂Ψ^m (C) / ∂C, Sⁱ = 2ϕ^c ∂Ψⁱ(C) / ∂C (i=c, m-c), and p is the unknown hydrostatic pressure to enforce det(C)= 1. In the following, we describe the behavior of each constituent phase.