## 2 Materials and Methods
### Experimental Procedure  
The nanoindentation tests were performed using a Piuma nanoindenter (Optics11 Life, Amsterdam, The Netherlands) (Figure 2a). Briefly, the indenter head was mounted on a probe, comprising a reflective cantilever and a spherical tip. Samples with different mechanical properties could be tested by selecting the most suitable combination of cantilever stiffness (\(k_{c}\)) and tip radius (\(R\)). For this study, a probe with \(k_{c}=0.025\) N/m and \(R=25.5\)\(\upmu\)m was selected.  
A standard nanoindentation test consists of 3 distinct phases: (_i_) the loading phase, which begins when the probe touches the surface of the sample and the cantilever starts to bend (Figure 2b); (_ii_) the holding phase, which starts when a set load (\(P\)) or indentation depth (\(\delta\)) is reached and kept constant for a defined time period; and (_iii_) the unloading phase, during which the probe is retracted from the surface of the sample.  
Figure 3a shows a typical nanoindentation input curve, where the set indentation depth needs to be reached and held constant for a specific holding period (\(t=5\) s). Ideally, the set indentation depth should be reached through an impulsive load (\(\delta(t=0)\equiv\delta_{max}\); \(P(t=0)\equiv P_{max}\)), but due to practical limitations, the maximum load (\(P_{max}\)) required to reach the set indentation depth (\(\delta_{max}\)) is often applied within a finite ramp time. Figure 3b shows an explanatory load vs time curve in which the load relaxation portion (from \(P_{0}\) to \(P_{\infty}\)) is isolated in order to be analyzed and to evaluate the viscoelastic parameters. Figure 3c shows the corresponding indentation depth vs load curve.  
Figure 2: (a) Piuma nanoindenter (Optics11 Life). (b) Schematic representation of the cantilever/probe–sample contact, where \(k_{c}\) is the cantilever stiffness, \(\delta\) is the indentation depth, and \(\Delta z\) and \(\Delta d\) are the total displacements covered by the probe and the cantilever bending, respectively.  
The applied load and the indentation depth were derived as follows:  
\[\delta=(z-z_{0})-(d-d_{0})=\Delta z-\Delta d\text{ and } \tag{1}\]  
\[P=k_{c}\cdot\Delta d, \tag{2}\]  
where \(z\) and \(d\) are the displacements along the vertical z-axis of the probe and of the cantilever bending, respectively; \(z_{0}\) and \(d_{0}\) are the respective values at the contact point; and \(\Delta z\) and \(\Delta d\) are the total displacements covered by the probe and the cantilever bending, respectively. The end of the loading phase of the \(\delta\)-\(P\) curve (Figure 3c) was used to derive the shear modulus corresponding to the onset of the relaxation period [54; 55; 56].  
All tests were performed in hydrated conditions (in PBS for the pig lung and GelMA samples and in a cell-culture medium for human lung sample) at a temperature of 37 \({}^{\circ}\)C, and they all controlled the indentation depth (I-mode). In I-mode, the constraint set by a user is a specific indentation depth value to be reached during the loading phase, while the displacements of the probe and of the cantilever bending are recorded. The I-mode allows users to perform stress-relaxation tests, which were essential for evaluating the time-dependent behaviors of the samples. Stress-relaxation tests consist of measuring the load relaxation over time in response to a constant value of indentation depth, which has to be reached rapidly. The change in load, required to maintain the imposed strain field, is measured during the holding phase. The experimental data we obtained were then used to evaluate the viscoelastic properties of the tested samples.  
In our case, the value of \(\delta\) was set to between 1 \(\upmu\)m and 5 \(\upmu\)m, and the indentation rate values (\(\rho\)) were selected to reach this value as fast as possible, i.e., \(\rho\) = 10-20 \(\upmu\)m/s. The holding time was set to be equal to 5 s for all the experiments. For each sample, 50 indentations were performed, and 11 indentation curves were selected for the analysis.