A viscoelastic Eshelby inclusion model and analysis of the Cell-in-Gel system

We develop a viscoelastic generalization of the elastic Eshelby inclusion solution, where the inclusion and surrounding matrix are two different viscoelastic solids and the inclusion's eigenstrain is a time-periodic oscillatory input. The solution exploits the Correspondence Principle of Linear Viscoelasticity and a Discrete Fourier Transform to efficiently capture the steady-state oscillatory behavior of the 3-D mechanical fields. The approach is illustrated here in the context of the recently-developed in vitro Cell-in-Gel system, where an isolated live cardiomyocyte (the inclusion) is paced to contract periodically within a soft hydrogel (the matrix), for the purpose of studying the effect of mechanical load on biochemical signals that regulate contractility. The addition of viscoelasticity improves the fidelity of our previous elastic Eshelby inclusion analysis of the Cell-in-Gel system by accounting for the time-varying fields and the resulting hysteresis and dissipated mechanical energy. This mathematical model is used to study the parametric sensitivities of the relative stiffness of the inclusion, the inclusion's aspect ratio (slenderness), and the cross-link density of the hydrogel matrix.