## 4 Discussion  
Knowledge of the time-dependent mechanical behaviors of materials designed for mimicking native tissues is of paramount importance in the context of mechanobiology and tissue engineering research. Thus, in this study, we developed a methodological approach for extrapolating the characteristic mechanical parameters that describe the viscoelastic behaviors of soft biological tissues and hydrogels. In particular, the proposed approach was based on the analysis of the stress-relaxation experimental curves that we obtained by adopting the nanoindentation technique and controlling the indentation depths to minimize variability in the results [60]. Samples of pig and human lung tissues and a specific GelMA hydrogel were tested, and their viscoelastic properties were quantified and compared. For each sample, several stress-relaxation curves were obtained and analyzed in order to evaluate the relaxation moduli (G\({}_{0}\) and G\({}_{\infty}\)) and the viscosity parameters in a straightforward and consistent way. Moreover, for the optimization of the experimental data-fitting, the genetic algorithm was demonstrated to be an effective and reliable choice. Indeed, the adopted approach allowed us to properly fit the experimental data (Figure 5) and identify the characteristic behaviors of the tested samples. The explanatory stress-relaxation curves (Figure 6) indicated that for all the samples, a holding period of 5 s was sufficient for reaching a complete relaxation of the stresses stored during the instantaneous loading phase at the cell-length scale. Studies conducted on biological samples at the macroscopic scale have shown that after an impulse load, a steady state could be reached in minutes or hours [61; 62; 63], though at the cell-length scale, the materials' responses change, as do the characteristic time-dependent responses. From the comparison of the mean behaviors of the tested samples, faster decreases characterized the pig lung and GelMA samples compared to the human lung (Figure 7). The relaxation behavior, as reported in Equation (13), was determined by the following two \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample** & **G\({}_{0}\) (Pa)** & **G\({}_{\infty}\) (Pa)** \\ \hline Pig lung & 1166.7 \(\pm\) 995.5 & 792.7 \(\pm\) 764.7 \\ \hline Human lung & 929.1 \(\pm\) 751.8 & 702.3 \(\pm\) 548.2 \\ \hline GelMA & 4339.2 \(\pm\) 2689 & 2565 \(\pm\) 1698 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean values and standard deviations of the instantaneous and equilibrium relaxation moduli. Figure 10: Boxplots of the mechanical properties describing the constitutive model of the analyzed samples: relaxation moduli (**a**) and viscosity (**b**) parameters. classes of parameters: (i) the material parameters, \(p_{i}\), that took into account the amount of force reached at the prescribed deformation, and (ii) the time parameters, \(\tau_{iI}\), that controlled the relaxation rate (Figure 8). To identify and compare the potential differences in the relaxation mechanisms of the tested samples, the values of the time parameters were not assigned a priori with default values, as has been reported in the literature [11, 64, 65, 66]. As regards the lung tissue, its mechanics were determined by the synergic interaction of all its constituents [63]. Accordingly, the adopted viscoelastic model highlighted the evident bimodal distribution of the time parameters of the biological samples (Figure 9), and the two peaks in the distribution could have represented the contribution of the most abundant molecular elements that characterized the microstructures of the biological samples, i.e., the ECM, which is made of highly cross-linked collagen and elastin [58], and the cells [59, 67, 68].