# Novel In Silico Strategies to Model the In Vivo Nerve Scarring Around Implanted Parylene C Devices
## 2 Methods  
Parylene C double devices (20 mm long, 200 \(\upmu\)m wide, and 10 \(\upmu\)m thick) were longitudinally implanted within the tibial branch of the sciatic nerve of female Sprague Dawley rats (250-300 g, \(n\) = 6-8 for each group) [56] and resulted in the formation of a scar tissue capsule. The thickness of this capsule formed around these implanted Parylene C devices was examined at 0, 24, 48, 96, 336, 672, 1344, 2688, and 5376 h post-implant by light microscopy in 4\(x\) images and measured as the distance between each side of the device and the closest myelinated axon though ImageJ software (version v1.51, National Institute of Health, USA). The evolution of the scar tissue thickness over time was assumed to be modelled through an unknown continuum function \(s(t):A\to B\), where \(A,B\subseteq R^{+}\). However, here, the subset \(S=\{s(t_{1}),\ldots,s(t_{9})\}\) of mean experimental values (with standard deviation) coming from the whole experimental set [56] was used. Let \(P_{i}(\bar{\alpha},t)\) be a polynomial function defined as follows:  
\[P_{i}(\bar{\alpha},t)=\{\sum_{i=0}^{n}\alpha_{i}t^{i}|\alpha_{0} \ldots,\alpha_{n}\in R\} \tag{1}\]  
while two supplementary functions are defined as follows:  
\[g(t)=abs(t) \tag{2}\] \[h(t)=ln(t) \tag{3}\]  
where \(t\) is the time, while \(abs(t)\) and \(ln(t)\) are the absolute value and the natural logarithm of \(t\), respectively. Therefore, the operator \(M\) is defined as follows:  
\[M[P_{i}(\bar{\alpha},t),F_{j}(\bar{\beta},t),\gamma]=P_{i}(\bar{ \alpha},t)+\gamma F_{j}(\bar{\beta},t) \tag{4}\]  
where \(\gamma\in\mathbb{R}\) and \(i\) and \(j\) are the maximum degree of the polynomial functions. The vectors \(\bar{\alpha}=[\alpha_{0},\ldots,\alpha_{i}]^{T}\) and \(\bar{\beta}=[\beta_{0},\ldots,\beta_{j}]^{T}\) include the coefficients of the polynomial functions. In addition,  
\[F_{j}(\bar{\beta},t)=(h\circ g\circ P_{j})(\bar{\beta},t) \tag{5}\]  
where \"\(\circ\)\" is the composition operator among functions. As a consequence, the evolution over time of the scar tissue thickness was modelled through the approximated function:  
\[q(t,i,j,\bar{\alpha},\bar{\beta},\gamma)=M[P_{i}(\bar{\alpha},t),F_{j}(\bar{ \beta},t),\gamma] \tag{6}\]  
Because of the polynomial nature of \(P_{i}(\bar{\alpha},t)\) and the cardinality of the experimental data set (\(n=9\)), the minimum degree of polynomials resulting in Runge instability was \(i=5\)[57]. Thus, in this work, the attention was narrowed to candidate functions with \(i=1,\ldots,5\) and \(j=1,\ldots,5\), while, in general, \(i\
eq j\) and \(\bar{\alpha}\
eq\bar{\beta}\). However, to reduce the number of elements of vectors \(\bar{\alpha},\bar{\beta}\) the candidate functions were chosen in order to satisfy the further constraints \(\alpha_{i}=\beta_{i}\) for \(i\leq j\). To test the ability of Equation (6) to model the experimental set of data, the standard statistics \(RMSE(i,j),R^{2}(i,j)\) were computed for each set of best \(\bar{\alpha},\bar{\beta}\), resulting from a standard optimisation procedure (nonlinear least-squares method, least absolute residuals, Levemberg-Marquardt algorithm) in Matlab (Mathworks Inc. Academic licence). In particular, each element of a \(5~{}\times~{}5\) matrix \(W\) was defined as follows:  
\[w(i,j)=log_{10}[RMSE(i,j)] \tag{7}\]  
where each \(RMSE(i,j)\) was the value of the \(RMSE\) (Round Mean Square Error) related to the fitting performances of the candidate function \(q(t,i,j,\bar{\alpha},\bar{\beta})\), defined for the best set of parameters \(\bar{\alpha}\),\(\bar{\beta}\). Therefore, each element of a twin matrix \( LRR\) was defined as follows:  
\[lrr(i,j)=log_{10}[R^{2}(i,j)] \tag{8}\]  
while the primary matrix \(RR\) was defined through  
\[rr(i,j)=R^{2}(i,j) \tag{9}\]  
where each \(R^{2}(i,j)\) was the value of the standard \(R^{2}\) (coefficient of determination) related to the fitting performances of the candidate function \(q(t,i,j,\bar{\alpha},\bar{\beta})\), defined for the best set of parameters \(\bar{\alpha}\),\(\bar{\beta}\).