*2.4. Learning the Constitutive Model of Porcine Carotid Tissue*

One of the most intricate experimental procedures in the framework of solid mechanics is perhaps that of constitutive modeling of soft living tissues. Here, we will employ data previously obtained and presented in [14] for the constitutive modeling of porcine carotid tissue.

What is remarkable in soft living tissue modeling is, on one hand, the heterogeneity and anisotropy of the tissue; and on the other, the large differences between experimental values found in different specimens. For instance, the behavior of porcine carotid tissue—whose interest is to serve as a proxy of that of humans—differs strongly if the sample is extracted from proximal (i.e., close to the heart) positions of the vessel or if it is extracted from distal positions. Additionally, there is a strong anisotropy regarding circumferential versus longitudinal behavior.

In [14], a traditional fitting procedure was accomplished so as to determine the best fitting model for these data. Taking the mean of the experimental results arising from [14] as the only plausible reference solution—this is standard experimental procedure in the literature—what we did first was to determine by TDA the set of experimental results neighboring this (mean) reference solution. A GENERIC model was then determined for each one of these neighbor results. Three different approaches were then compared: i) a GENERIC model whose terms are computed as the mean values of each of the GENERIC models for each neighboring curve (the closest ones in the data manifold); and either ii) ordinary or iii) local Kriging interpolation techniques among these neighbors of the terms of a new GENERIC model. The result of our approach is a new GENERIC-TDA model whose integration in (pseudo-)time produces a prediction of the tissue behavior.

#### 2.4.1. Experimental tests

To introduce to the reader the most significant details in the experimental models used for our numerical analysis, here we show a brief description of the sample's harvesting and tensile test protocols performed in [14]. The interested reader is referred to this article for a precise description of the experimental campaign.

We consider nine female pigs of 3.5 ± 0.45 months (mean ± SD). The experiments on these swine were approved by the Ethical Committee for Animal Research of the University of Zaragoza. All procedures were carried out in accordance with the Principles of Laboratory Animal Care (86/609/EEC Norm, incorporated into Spanish legislation through the RD 1021/2005).

For each one of the left and right carotids, proximal and distal regions were considered for mechanical testing, as mentioned before. At each location, circumferential and longitudinal strips, approximately 3-mm-wide and 11-mm-long, and 5-mm wide and 15-mm long, respectively, were cut. A total of 14 carotid specimens with 47 and 49 valid tests were performed for the proximal and distal zones, respectively. A minimum number of two test strips along each direction was accomplished for each specimen.

Simple tension tests of the carotid strips were performed in a high-precision-drive Instron Microtester 5548 system, see Figure 3. The procedure is properly described in [14], and consisted of the Instron Microtester 5548 System with two clamps holding the sample. The samples are subjected to the tensile test under a humidity-controlled environment to prevent sample drying.

The applied force was measured with a 5-N load cell with a minimal resolution of 0.001 N, and the axial strain was measured using a noncontact Instron 2663-281 video-extensometer equipped with a high-performance digital camera with a megapixel sensor (0.5 m ± 0.5% )

**Figure 3.** Experimental setup. Instron Microtester 5548 System with two clamps holding the sample. The samples are subjected to the tensile test under a humidity-controlled environment to prevent sample drying.

Different loading and unloading cycles were applied that correspond to 60, 120, and 240 kPa (50%, 100%, and 200% of the estimated physiological stress state in the artery) at 30%/min of strain rate, which can be considered as quasi-static. Therefore, these experiments serve for the hyperelastic modeling of soft tissues, but not for their viscous characterization. The resulting GENERIC model will therefore be purely Hamiltonian under these conditions.

For each one of the fourteen carotids, proximal and distal measurements are obtained, thus giving a total of 28 datasets. Each dataset includes circumferential and longitudinal stresses and stretches, *z* = {*σc*, *λc*, *σ*<sup>ℓ</sup> , *λ*ℓ}. Due to the quasi-static nature of the experiments, time is actually a pseudo-time. In addition to these 14 stress–stretch curves for each location, a fifteenth curve is obtained by computing the mean value of the first 14. This curve will be taken as a sort of reference for comparison purposes, see Figure 4.

**Figure 4.** Experimental (**a**) distal-circumferential, (**b**) distal-longitudinal, (**c**) proximal-circumferential, and (**d**) proximal-longitudinal stress–stretch curves. The continuous blue line represents the mean values of the 14 experimental results, while the dashed lines represent the neighboring experiments, as found by TDA techniques.
