## 2 Materials and Methods
### Geometrical Core Concepts
The translational movement variation (\(+\mathfrak{y}_{\beta}\)) in the same direction with the driving force \(\mathfrak{u}_{0}=\mathfrak{u}_{0}(t)\) and with the following movements (displacements) (\(\mathfrak{y}^{\prime}_{\alpha}\)), (\(\mathfrak{y}_{\alpha}\)), and (\(+\mathfrak{y}_{\beta}\)) ensures the fact that the desired orthodontic movement occurred, with the remark that the recommended variation is enclosed between [\(+\mathfrak{y}_{\beta}=(0.5\to 0.9)\cdot\mathfrak{y}^{\prime}_{\alpha}\)]. It is important to mention that \(\mathfrak{y}^{\prime}_{\alpha}=\mathfrak{y}^{\prime}_{\alpha}(\mathfrak{t}_ {\mathrm{f}})\) can be precisely observed by the orthodonticst, but the movement of the tooth's apex (\(\pm\mathfrak{y}_{\beta}\)) can only be approximated, sometimes with a considerable error. Certainly, the (Orf) axis from Figure 1 can be associated with other parabolic shapes defining more complicated volumetric tooth shapes. However, the proposed approximation is accurate enough for the upper central incisor in the context of the orthodontic process dynamics modeling. Figure 2: Possible orthodontic tooth movement [21]. B. Analogical Model of Visible Deformation (\(y^{\prime}_{\alpha}(t,s)\)) in Elastic-Plastic Regime. Scaling the \(K_{y}\) Coefficient  
The proposed analytical approximating solution which models the tooth position deformation during the orthodontic treatment has the following form:  
\[y^{\prime}_{a}(t,s)=y^{\prime}_{00}(t,s)=K_{y}\cdot F_{OS}(s)\cdot[F_{OT}(t)*u_{ 0}(t)], \tag{5}\]  
where the elastic resort force (\(u_{0}(t)\)) from Figure 2 is applied at a distance (\(\widetilde{s}^{\prime}\)) from the visible incisal edge of the tooth (\(y^{\prime}_{\alpha 0}\)) or at a distance (\(s_{\mathrm{f}}+\widetilde{s}\)) from the tooth apex \(y_{\beta 0}=0\). Moreover, the notation \"*\" signifies the convolution product between (\(F_{OT}(t)\)) and (\(u_{0}(t)\)) functions, respectively; (s) represents the second independent variable which highlights the position on the (Or) axis (the \"length\" independent variable). In the case of the functions from Equation (5), the numerical indices show their differentiation order (for (\(F_{OT}(t)\)) and \(u_{0}(t)\) in relation to (t), for FOS(s) in relation to (s) and for \(y^{\prime}_{00}(t,s)\) in relation to both (t) and (s)--the first index being associated to (t)). Practically, the analytical approximating solution can be mathematically interpreted as the solution of a second-order partial differential equation, with two independent variables, (t) and (s). By scaling the proportionality coefficient (\(K_{y}\)), the equality between the observable left member \(y^{\prime}_{00}(t,s)\) and the right-member (which represents the approximation of the biomedical process model) of Equation (5) can be ensured. The elastic force (\(u_{0}(t)\)) of the spring from Equation (5) is approximated by:  
\[u_{0}(t)=\widetilde{u}_{f}+(K_{u}-\widetilde{u}_{f})\cdot(\frac{T_{1u}}{T_{1u }-T_{2u}}\cdot e^{-\frac{t}{T_{1u}}}+\frac{T_{2u}}{T_{2u}-T_{1u}}\cdot e^{- \frac{t}{T_{2u}}}) \tag{6}\]  
where (\(K_{u}\)) represents the maximum value of the (\(u_{0}(t)\)) force, right after its application at \(t=t_{0}\) (t0 signifying the initial moment of the force application which corresponds with the orthodontic treatment start). Further on, the strength (\(u_{0}(t)\)) decreases exponentially towards the asymptote \(\widetilde{u}_{f}\) representing the residual (remament) plastic strength (usually \(\widetilde{u}_{f}=(0.05\div 0.20)\cdot K_{u}\)). In Figure 3, both the variation of the input signal in the biomedical process (\(u_{0}(t)\) force) in relation to time and the variation of the output signal from the biomedical process (\(y^{\prime}_{00}(t,s)\) movement) in relation to both independent variables (t) and (s), are qualitatively presented. Figure 3 is presented in order to highlight in an intuitive manner the variation form of the signal which occurs in the model of the treated biomedical process.