## 4 Discussion
The sensitivity of \(\hat{k}_{fib}\) to variations of the parameter \(D\) resulted in a highly non-linear and diverging line. The more the value of \(D\) increased, the more the line diverged. In other words, the value of \(D\) heavily affected the curvature of the stress due to the collagen action. The parameter \(K_{1}\) was, then, physically related to the apparent stiffness of the nerve for small strain (in this case, the Young modulus was \(6K_{1}\)), while the parameter \(A\) controlled the linear part of stress increment due to the collagen action. Furthermore, the parameter \(D\) affected the curve steepness and controlled both the delay (with respect to the reference configuration) and the velocity of the nerve stiffening process. Therefore, it was shown, as a matter of principle, that the proposed deterministicand explicit framework, exploiting closed-form equations with three main parameters, was able to reproduce the classic nerve stiffening behavior [2; 7]. As a consequence, first, this framework was used to explore the stiffening behavior of real specimens. Indeed, with reference to the porcine peroneal nerve, theoretical predictions were able to closely reproduce the experiments (\(R^{2}\geq 0.997\)), while residual plots were analyzed to avoid numerical bias (e.g., over-fitting) keeping the value of the \(R^{2}\) statistic artificially high. All residuals (\(\Delta\sigma\)), between theoretical predictions and experimental values, were within 0.1 kPa; that is, the error was about 2.5% in the worst case, confirming that experiments were reproduced with confidence. To further explore the response of the porcine nerve, dimensionless stress and curvature were used, since the dimensionless stress was able to quantify the evolution of stress with respect to the stiffness of nerve for small strain, while the dimensionless curvature was used to measure the nerve stiffening, since it mapped where and how much the dimensionless stress evolved non-linearly because of the action of collagen fibrils. More specifically, the slope of \(\hat{\sigma}_{tot}\) at \(\lambda=1.08\) was on average 14 times greater than the slope at the reference configuration (i.e., at \(\lambda=1\)), while the slope of \(\hat{\sigma}_{mat}\) was practically constant. In addition, \(\hat{\kappa}_{mat}\) was small with respect to \(\hat{\kappa}_{fib}\) (i.e., \(\max\{\hat{\kappa}_{fib}\}/\max\{\hat{\kappa}_{mat}\}\simeq 599.39\)). These results not only suggested that the stiffening process was dominated by the action of the collagen fibrils, confirming previous literature [2; 7; 24], but also provided a quantification of the amount of the stiffening phenomenon (i.e., through the change of the slope by 14 times), as well as a quantification of how much the fibrils' action was able to overcome the contribution of the ground matrix (i.e., about two orders of magnitude). On the contrary, the maximum dimensionless curvature of \(\hat{\sigma}_{tot}\) was lower than the maximum curvature of \(\hat{\sigma}_{fib}\) (i.e, \(\max\{\hat{\kappa}_{fib}\}/\max\{\hat{\kappa}_{tot}\}\simeq 4.21\)). This result suggested a possible lowering action of the ground matrix on the huge stiffening action of the collagen fibrils. Furthermore, since the proposed framework was not intrinsically connected to any kind of animal or human model, it is, as a matter of principle, suitable to explore any kind of nerve. To show this, a different specimen, coming from a different animal model, was investigated. Indeed, with reference to a canine left vagus nerve, theoretical predictions were able to closely reproduce the experiments (\(R^{2}\geq 0.999\)), while all residuals were within 0.32 kPa; that is, the error was less than 1.5%. Furthermore, the slope of \(\hat{\sigma}_{tot}\) at \(\lambda=1.08\) was on average 12 times greater than the slope at the reference state (i.e., at \(\lambda=1\)), while the slope of \(\hat{\sigma}_{mat}\) was practically constant.