# Multilayered Artificial Dura-Mater Models for a Minimally Invasive Brain Surgery Simulator
## 4 Evaluation of Dura-Mater Models
### Viscoelastic Measurements  
Dura-mater is known as a composite material that has viscoelastic properties [26, 27]. Its viscoelasticity was measured to check if the developed dura-mater models show similar properties compared with those of the actual dura-mater. In this study, we used a three-element viscoelastic model to express the mechanical properties of dura-mater models. The three-element viscoelastic model has one spring and a Voigt model (Figure 5b). The three-element viscoelastic model exhibits a solid-like behavior in the condition of constant stress and a stress relaxation phenomenon over time in the condition of constant strain.  
A rheometer employed with the three-element viscoelastic model was employed to quantify the viscoelasticity of dura-mater models. Storage and loss moduli were obtained from the measurements  
\begin{table}
\begin{tabular}{c} Hard non-woven fabric \\ Thickness: \\ \(120\pm 10\)\(\mu\) \\ mFiber diameter: \\ \(40\pm 10\)\(\mu\)m \\ \end{tabular}
\end{table}
Table 1: Fabric materials for dura-mater models.  
Figure 5: Viscoelastic measurements: (**a**) Fabricated dura-mater model; (**b**) Three-element viscoelastic model; (**c**) Measurement setup for viscoelasticity of dura-mater models.  
by a rheometer (PZ-Rheo NDS-1000, Syscom, Tokyo, Japan) with a probe that had 10 mm in diameter (Figure 5c). The storage modulus \(G^{\prime}\) and loss modulus \(G^{\prime\prime}\) were measured by the rheometer based on the following Equations (1) to (4).  
\[G^{\prime}=\left(\frac{\tau_{max}}{\gamma_{max}}\right)\cos\delta \tag{1}\]  
\[G^{\prime\prime}=\left(\frac{\tau_{max}}{\gamma_{max}}\right)\sin\delta \tag{2}\]  
\[\gamma(t)=\frac{u(t)}{d} \tag{3}\]  
\[\tau(t)=\frac{F(t)}{A} \tag{4}\]  
where \(\delta\) represents the phase difference, \(\gamma\) represents the shear strain, \(u\) shows the displacement, \(d\) indicates the sample thickness, \(\tau\) indicates the applied stress, \(F\) indicates the reactive force, and \(A\) represents the sample area. Strain with a frequency of 3 Hz and an amplitude of 5% was applied to the samples, and the storage modulus \(G^{\prime}\) and loss modulus \(G^{\prime\prime}\) were calculated from Equations (1)-(3). Measurement of six different samples were carried out: porcine dura-mater and five different dura-mater models (Figure 4a-e). The measurement was repeated five times in each sample, and the highest and lowest values were eliminated from the results.  
Figure 6 indicates the storage modulus \(G^{\prime}\) and loss modulus \(G^{\prime\prime}\) measured in each sample. The dura-mater model in Figure 4a yields relatively similar \(E^{\prime}\) and \(E^{\prime\prime}\) values, while the dura-mater model in Figure 4b,c had similar properties with those of the porcine dura-mater from the viewpoint of \(G^{\prime}\). The results indicate that the dura-mater models with two-layered fabrics yielded better results compared with the three-layered fabrics from the viewpoint of viscoelasticity. The differences in the results may be attributed to the differences in thickness in the acrylic rubber.