Peripheral nerves are extremely complex biological structures. The knowledge of their response to stretch is crucial to better understand physiological and pathological states (e.g., due to overstretch). Since their mechanical response is deterministically related to the nature of the external stimuli, theoretical and computational tools were used to investigate their behaviour. In this work, a Yeoh-like polynomial strain energy function was used to reproduce the response of ^{2} > 0.9). Finally, bi-dimensional in silico models were provided to reduce computational time of more than 90% with respect to the performances of fully three-dimensional models.

Peripheral nerves are extremely complex biological structures which bridge the central nervous system with the periphery of the body (

Since the mechanical response of peripheral nerves is deterministically related to the external stimuli (e.g., traction force), anatomical studies (

The logic flow of the text is the following: first, the suitability of the proposed SEF was tested both for a porcine peripheral specimen and for different animal species (lobster, rabbit,

A peroneal nerve was dissected from a posterior limb of a Large White pig (∼10 months old), which was slaughtered in conformity with the Italian National Regulation and frozen until experiments. Before experiments, the nerve specimen (96 mm long with a cross-sectional area of ∼6.39 mm^{2}) was gradually defrosted and re-hydrated for about one hour at room temperature in a bath of aqueous saline solution isotonic to the blood (0.9% sodium chloride) to minimize the time dependence of the tissue hydration. The length of the specimen between clamps was 69 mm, and its physiological characteristics were kept by regularly spraying saline moisture on its external surface in order to maintain the initial level of hydration. Stretching experiments were carried out at room temperature (∼25 ± 1 °C), by using an Instron R4464 testing machine (Instron Corporation, Canton, MA, USA) with a standard load cell (Instron load cell, cell type 2525–808, max force 10 N, accuracy 0.25% Full Scale Output; (Instron Corporation, Canton, MA, USA) as shown in

A scheme of the experimental framework used to stretch the nervous specimen (magnification in A). The nerve was fixed between two clamps and stretched through the movement of a transverse sliding beam of a testing material machine (isolated from the environment). A load cell recorded both displacements and forces, which were further elaborated to provide the digital stress/stretch curve (B).

More specifically, the nerve was stretched (velocity

The peripheral nervous tissue (PNT) was modelled as a homogeneous and incompressible material and, according to previous literature (_{1}, _{2}, _{3} ∈ ℜ were scalar coefficients. The Cauchy stress tensor was expressed in function of both the first strain invariant (_{1}) and the deformation gradient ^{T} were, respectively, the unit tensor and the transposed of the deformation gradient. Since _{1} = ^{T}), and ^{[9]}(_{1}, _{2}, _{3}) = 3_{3}^{9} + (2_{2} − 18_{3})^{7} + 9_{3}^{6} + (27_{3} − 6_{2} + _{1})^{5} + (2_{2} − 18_{3})^{4} + (−27_{3} + 6_{2} − _{1})^{2} + (36_{3} − 4_{2})_{3}.

Experimental data were collected for five consecutive extensions and their mean values were reported as a function of stretch and used to represent the behaviour of the specimen. ^{2} function to be maximized for each extension. More specifically, guess values for [_{1}, _{2}] were chosen (i.e., [10.00, 0]; [12.99, 0]; [0, 12.40]; [0, 11.99]; [10.00, 0]), while [_{3}], was allowed to vary in the range 6,000–9,000 KPa. To explore the sensitiveness of final vales of _{1} and _{2} to changes of _{3}, the difference _{1}(_{3}) − _{2}(_{3}) was plotted (for a constant ^{2} ≃ 0.99), as shown in _{3} over ^{2} was studied for constant values of _{1}, _{2}, as shown in _{1} and _{2} values was analyzed to test their eventual independency as well as the correlation between these values and the _{3} constant.

Finally, the sensitiveness of the stress function with respect to _{1}, _{2}, _{3} was expressed as: _{i}) = (_{maxci} − _{minci})∕_{maxci} (_{maxci} and _{minci} were respectively the maximum and the minimum values of stress for the maximum and minimum values of _{i}, when the other constant had optimal values.

In silico models were implemented to reproduce the ^{2} =

An in silico model of the right connective of

The solid model of

Experimental data were collected for five consecutive extensions and their mean values were reported as a function of stretch in

^{2} ≃ 0.998) with _{1} = 5.89 KPa, _{2} = 5.89 KPa and _{3} = 7.75 MPa, while both the range of variation of experimental values and the errors between the mean values and the fitting curve were calculated (_{1} and _{2} were almost equal (i.e., error less than 2.422⋅10^{−5} KPa), and had the following values 4.99 KPa , 6.49 KPa, 6.20 KPa, 5.99 KPa, 4.99 KPa, while the _{3} parameter resulted in 8.13 MPa, 7.95 MPa, 7.46 MPa, 7.61 MPa, 8.21 MPa. These values were inserted in ^{2} of 0.995, 0.995, 0.996, 0.996, 0.995). To test possible further cross correlations among numerical constants in _{1} values were plotted versus the corresponding value of _{2} for each extension, while the values of _{3} were also plotted versus _{1}, _{2}. This procedure showed a positive and very strong correlation in the first case (i.e., ^{2} > 0.99, as expected), while a weak correlation in the second one ^{2} = 0.49, as shown in _{1}, _{2}, _{3} constants resulted respectively in 10.79, 0.32, 5.38 (

Moreover, ^{2} = 0.981) for _{1}, _{2} = 0.2 MPa and _{3} = 43.7 MPa, with errors (between data and theoretical curve) ranging from −0.38 to +0.58 MPa (^{2} = 0.971) for _{1}, _{2} = 0.14 MPa and _{3} = 0, with errors in the range −0.03, + 0.05 MPa (^{2} = 0.972 for _{1}, _{2} = 0.0081 KPa and _{3} = 0.0054 KPa), with errors ranging between −34.41 and +25.31 KPa (

(A) Stress/stretch curve for a nerve of rabbit and theoretical approximation. (B) Error (MPa) between data and approximation for rabbit. (C) Stress/stretch curve for a lobster nerve. (D) Error (MPa) between data and approximation for lobster. (E) Stress/stretch curve for a connective nerve of _{1}, _{2}, _{3} for different animal species.

In

(A) Displacements in

Furthermore, the stress/stretch curves resulting from three-dimensional models (elliptic and re-scaled circular cylinders) (^{−6} (

A rectangular slice coming from the axisymmetric re-scaled circular cylinder was used to approximate the nerve specimen (

(A) Displacements in

The performances of the three-dimensional in silico model of the

(A) Comparison between theoretical and in silico stress/stretch curves. (B) Percentage error between theoretical and in silico curves: the error is zero up to

The bidimensional approximation of the

(A) Comparison between theoretical and in silico stress/stretch curves. (B) Percentage error between theoretical and in silico curves: the error is zero up to

In this work, a unified approach was proposed to reproduce the stress/stretch behaviour of neural structures across different animal species (vertebrate/invertebrate) through a classic formulation involving a strain energy function. The chosen function was in Yeoh-like form (_{1} and _{2} were almost equal for each extension, as well as for different values of the _{3} coefficient. Indeed, a maximum error of 2.422⋅10^{−5} KPa (i.e., 4⋅10^{−4}%) was found for all extensions (^{−3} MPa (i.e., 1.70%) for rabbit, 1.9⋅10^{−2} KPa (i.e., 1.42%) for lobster, and 1⋅10^{−7} KPa (i.e., 1⋅10^{−3}%) for _{1} and _{2} were assumed to be equal in these animal models. However, the sensitivity of the stress function to _{2} (_{1} and _{3} (respectively _{1} and _{3} seemed to mainly affect the behaviour of the stress function, which was able to reproduce the elastic response of several tissues, ranging from

Further cross connections between _{1}, _{2} and _{3} were investigated (^{2} = 0.49 and scattered values). Therefore, no important mutual stiffening effects (e.g., due to dehydration) were found during experiments. This was in agreement with the hypothesis of mutual independence between these two constants. As a consequence, ^{2} > 0.99) experimental data for the large white pig model (both the mean curve and each extension) and for other animal species. However, the percentage errors was around 8–10%, since numerical oscillations arose between theoretical curve and interpolated data.

The in silico elliptic cylinder, reproducing the mean surface of the nervous specimen, was able to closely replicate the theoretical behaviour for longitudinal and transversal stretches. Indeed, the percentage error with respect to the theoretical predictions ranged between 0.2% and −0.4% for the axial stretch (^{−6} for the transversal one), but also the nodal stress had the same distribution and the same values, clustered around the theoretical value (

Literature studies on neural-like (

The shape of the stress/stretch curve varied across different species, (e.g., lobster and

In this work, a Yeoh-like SEF was proposed to reproduce the mechanical response of neural structures for a wide range of stretches (from

Raw data.

On the left side, the evolution of both constants when _{3} varied. On the right side, the evolution of their difference for variations of _{3}.

For variations of _{3} the problem has a single solution. In particular, for lobster, the solution is the minimum one maximizing the ^{2} value within the allowed range (i.e., _{3} > 0).

(A) Global frontal view and extraction of the mean border lines (corresponding to the normalized values = 1) deriving from natural profiles. (B) Global lateral view of specimen and extraction of the mean border lines (corresponding to the normalized values = 1) deriving from natural profiles. (C) Cross section and solid approximation of the nervous specimen built up from the previous mean frontal and lateral border lines. Computational constraints reproduced the experimental boundary conditions.

This cylinder is axisymmetric, thus a single bidimensional slice is representative for the global behaviour of the solid.

(A) Comparison between stress for three-dimensional elliptic and three-dimensional circular approximations when the stretch increased up to

The authors thank the company “Desideri Luciano s.r.l” for biological specimens and Dr. Cesare Temporin for his valuable technical assistance in handling and dissection of peripheral nerves.

The authors declare there are no competing interests.

The following information was supplied relating to ethical approvals (i.e., approving body and any reference numbers):

No ethical problem or restrictions apply in this case, since porcine posterior limbs were provided by a private company (“DESIDERI LUCIANO SRL Via Abruzzi—56025 Pontedera (PI), Italy”), for research uses. This company continuously slaughter large white pigs for alimentary use and in conformity of the standard Italian Regulation. The nervous specimens were extracted from the whole posterior limbs within the Department of Veterinary Science of the University of Pisa and frozen until it was time for the experiments.

The following information was supplied regarding data availability:

The raw data has been supplied as a