# A Quantitative Investigation on the Peripheral Nerve Response within the Small Strain Range
## 1 Introduction  
Peripheral nerves are composed by several cellular elements, as Schwann cells, perineurial cells, and axons (neuronal processes), enveloped by connective tissue [1]. The axons could be originated from centrally or peripheral neurons, while the Schwann cell coats, which envelop axons, are placed within a circumferential layer of perineurial cells. Similarly, the perineurial coat lies within epineurium, which is a soft tissue rich in fibroblasts, collagen fibers, and small blood vessels [2]. Since peripheral nerves have unique anatomical characteristics, due to the very high ratio between their cross sectional area and their distance with respect to the parent cell body [3], they have extrinsic and intrinsic blood supply [4, 5]. More specifically, nutritive arteries, arterioles, and venules, which run with the epineural vessels, belong to the extrinsic system, while longitudinal microvessels within the endoneural space belong to the intrinsic system [3].  
Depolarization potentials travel through neurons packed within nerves, then peripheral nerves link the central nervous system to the periphery of the body. Afferent nerves, link muscles and external sensory receptors to the CNS, while in efferent nerves the information travel in the other direction. Peripheral nerves could be classified as motor nerves, which target muscles, sensory nerves, which target sensory receptors and mixed nerves, in which both motor and sensory fibers are packed together [1, 3, 6].  
Anatomical investigations of peripheral nerves were extensively performed [6; 7] and sometimes they were related to biomechanical considerations [8; 9]. In particular, tissue mechanics was used in combination with finite element analysis [10; 11; 12] to explore the behaviour of nerves in presence of different stimuli [13; 14; 15; 16]. Indeed, a quantitative assessment of the nerve response could be interesting for several applications, as neuroprosthetics [17; 18], neural interfaces design [19; 20] and interaction with biomedical devices (e.g., microneedles [21; 22; 23]).  
In particular conditions, peripheral nerves are elongated only of a small extent, as in case of movements constrained after medical interventions (e.g., immobilization procedures), or in case of neuropathic tremors [24], or for voluntary/involuntary small and quick movements involving articulations [25]. Therefore, this kind of response could be interesting \"per se\", as well as to evaluate the coupling between nerves and biomaterials, as for acute or chronic implantation of medical devices, interacting with the nervous tissue in these conditions [17]. Nevertheless, the nerve behaviour within the small strain range is a less investigated topic, while the response to higher strains was explored in some greater details [13; 14; 15; 16; 25]. In particular, the displacement field arising within the small strain regime (i.e., \(0<\epsilon<<0.1\)) was theoretically predicted together with the effects of the variation of the tissue compressibility on the nerve response. In addition, numerical results were quantitatively compared to experimental data to test the suitability of the presented framework.