This parameterization of the constitutive model has been shown to be correctly identified. With any initial values, the method provides a single set of parameters leading to a good agreement with the axial and circumferential experimental data, which proves that the model is well parameterized with respect to the available data and that the method is robust. These results confirm that identifying layer-specific fiber angles, based on the constitutive model of Holzapfel, is possible for such arteries using data from only one experimental test of inflation with free axial movement. In our results, the mean fiber angle in the media is close to 45° while fibers are, in average, more circumferentially oriented in the adventitia. This means that the adventitia is found to be circumferentially stiffer than the media. Though it cannot be generalized based on a single example, this result brings some knowledge about the mechanical behavior of these arteries at the layer-scale. The relationship with the actual microstructure remains, however, a pending question [14]. From a global point of view the overall anisotropy of the artery, with the circumferential direction being stiffer, is typical for arteries in general, and is in qualitative agreement with previous studies [11, 12, 14]. 4.2. Seven-parameter model In contrast with the five parameter model, the full two-layer Holzapfel-type model has failed in providing a unique solution to the problem of identification. Our results (see Table 1) show the typical trend of an over-parameterized problem. They confirm that identifying such a model on these experimental data is very easy because the algorithm is always able to capture the experimental data with a close fit. Unfortunately, robustness is very poor because seven families of different possible solutions are obtained. Yet, this issue is not unexpected because the problem of inflating a tube made up of two fiberreinforced layers potentially presents two solutions: one solution with a large fiber angle for the inner layer and small angle for the outer layer, and vice versa. Both solutions are possible as long as their constitutive properties can be different to compensate 3D through-thickness effects. Interestingly, our results clearly show these two alternatives, with a noticeable preference for large media angle and low adventitia angle (80% solutions). Note, also, that the mechanism of compensation is clearly illustrated by the ratio of k1 parameters between media and adventitia, which is inversed in these two situations. To correctly identify the parameters for this model, more abundant and/or relevant experimental data are required. In order to confirm this hypothesis, additional synthetic data are generated by simulating a simple axial tension test on the arterial segment, without any pressurization, recording axial force versus axial stretch. Accordingly, the cost function (see Eq. 2) is enriched with these data: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 sim exp sim exp sim exp 11 11 22 22 1 2 χ λ λ = − + − + − ª º « » ¬ ¼ ¦ ¦ G i i i i i i i i J E p E p E p E p F F (3) where Ȝi is the axial stretch applied during the tension test, with index i ranging over the available experimental (synthetic) data points, Fsim and Fexp are simulated and experimental (synthetic) axial force. Values are scaled so that both sum terms in J are of the same order of magnitude. Using this updated cost function, the same multiple identification runs are performed again. Instead of obtaining seven different families of solutions, the algorithm provides only two (see Table 1). Among these two solutions, the first one, obtained only 14.3% times, reaches two boundaries of the allowed range for the parameters. This solution must obviously be rejected, as the procedure would have diverged or found inconsistent values. The other solution is considered as the true solution. These results show that additional relevant data may easily make the identification procedure robust, discriminating thus the true solution. In this case, a simple tension test added to the inflation/extension test is sufficient. Circumventing the problem of non-uniqueness emphasized above would also be made possible by using different types of additional data, or other strategies. For instance, separate mechanical tests of medial and adventitial layers can be used as was done in [10], or inversion tests in which the arterial segment is turned inside out to reverse the spatial locations of the media and adventitia and then redistribute through-thickness stresses in a different manner (theoretically studied in [2]). Another possible way to access useful experimental information would be to acquire through-thickness data, thanks to the use of optical coherence tomography or confocal microscopy, for instance. Such layer-specific data would help discriminate the true solution. Otherwise, in order to reduce the number of dependent parameters to be identified and help make the solution unique, additional information regarding the fiber angle distributions within each layer (from histology for instance) would be helpful. A part of the work in progress at this time is related to these last two aspects. 16
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