# Sensitivity Analysis of a Numerical Model for Percutaneous Auxicular Vagus Nerve Stimulation
## 2 Materials and Methods
### Sensitivity Assessment of the Numerical Model  
A sensitivity analysis of the numerical model parameters is critical to model the robustness and validity. It also serves to guide future research on the most influential of these factors. The method used to assess the sensitivity of the numerical model to the parameters' fluctuation and uncertainty is to repeatedly change one parameter at a time while keeping the others fixed (at an average value). We increase the parameter by a given step and we observe the change in model output, which is the stimulation threshold for single axons and the stimulation threshold for 100% recruitment volume for the whole axon population. This figure of merit represents a good quantification of the parameter model and variability [14]. A sensitivity index (SI) is calculated (as percentage) for each parameter as follows:  
\[SI_{parameter}=\frac{O_{max}-O_{min}}{O_{max}} \tag{6}\]  
where \(O_{min}\) and \(O_{max}\) are minimum and maximum output values, respectively, resulting from changing the respective input over its selected range. The output values are defined as the stimulation threshold for single axons and the stimulation threshold for a 100% activation of an axon population, respectively. The largest SI values correspond to a high degree of sensitivity of the output values to the given input. This figure of merit represents a good quantification of the parameter model and variability [14]. Two more quantifications were used for the variation of the stimulation thresholds to achieve 100% axon activation: relative variation of the stimulation thresholds compared to an average value (\(\Delta_{\text{B}}\)) and relative variation between the thresholds of two adjacent inputs (\(\Delta\)). These two quantities show the variation in the stimulation threshold compared to a reference value and between thresholds of two consecutives inputs. The investigated SENN model parameters include the axon fiber diameter, number of axons, axon population region size, and SENN model temperature.  
For each of the above parameters, a range of values, an average (AV), and an incremental step were determined from literature and summarized in Table 1:  
1. [PERSON] et al. [24] provided the number of targeted myelinated A\(\beta\) axons with diameter \(\geq\) 7 \(\upmu\)m [25] of the auricular vagus nerve branch in the left and the right ear for nine evaluated individuals.
2. The model temperature was changed based on recent investigations of the variation in normal ear temperature measured in 2006 individuals [26].  
Figure 1: A numerical model of the percutaneous auricular vagus nerve stimulation (pVNS) application. **(a)** The scheme of the application. **(b)** The auricle, vessels, nerves, and the electrodes. Vessels and nerves are inside the ear and not on the surface of the ear. **(c)** The locations and shapes of the nerves: Nerve 1, N1 and Nerve 2, N2. Electrode 1, E1 and Electrode 2, E2, are the stimulating electrodes. **(d)** The details of the axon population N2 (refer to Table 1 for the number of axons and Figure 1c for the Nerve 2 location) and Electrode 2 (refer to Figure 1c for the Electrode 2 location).  
The ear conductivity was changed from 0.1 S/m to 0.7 S/m based on Reference [27]. The needle penetration depth of E1 and E2 inside the ear was changed from 1.5 to 0.8 mm in steps of 0.1 mm [12]. The lateral position of the electrodes (E1 and E2) was also changed from the original position by \(\pm\)0.1 mm in the y and z directions, with a maximum of (\(\pm\)0.1, \(\pm\)0.1) mm [12].