# A Critical Review of Human Jaw Biomechanical Modeling
## 2 Materials and Methods
This procedure is repeated for all the time steps imposed by the analysis. On the other hand, the inverse dynamics uses jaw movements extrapolated experimentally via optical techniques [64] to calculate the forces and time steps applied. Both the methodologies involve sets of equations whose analytical solution is unique only if the number of equations is exactly the same as that of the unknown variables. When the equality is not respected, an optimization problem (Equation (6)) should be considered, characterized by an objective or cost function that may be minimized or maximized according to the set of equality and inequality constraints. \[\left\{ \begin{array}{c}\min f(x)\\ Ax=b\\ A^{\prime}x^{\prime}\leq b^{\prime}\end{array} \right\} \tag{6}\]  
The function \(f(x)\) can measure the difference between numerical and experimental results, individual muscular forces or stress [65]. The most common model for describing muscular activities at the microscopic level is certainly [PERSON]'s [66] and its variants [67], for example, adopting a pennation angle between the tendon traction line and muscular fibers [68] or the viscosity of the muscle [69]. According to his theory [70], a muscular unity can be built as shown in Figure 5. The contractile element (CE) is the active component [71] determined by the cross-bridges formed between the two proteins of the sarcomere unit, myosin and actin [72]. Two non-linear springs are also introduced: the series element (SE) directly connected to the CE, representing the tendon and the elasticity of the myofilament, and the parallel element (PE), indicating the elasticity of connective tissues such as the epimysium, perimysium and endomysium. For this configuration, the following relationships both for the forces (Equation (7)) and length \(L\) (Equation (8)) must be satisfied:  
\[\left\{ \begin{array}{l}F=F^{(PE)}+F^{(SE)}\\ F^{(CE)}=F^{(SE)}\end{array} \right\} \tag{7}\]  
\[\left\{ \begin{array}{l}L=L^{(CE)}+L^{(SE)}\\ L=L^{(PE)}\end{array} \right\} \tag{8}\]  
The forces are a function of the deformed muscle length according to empirical constants and of the peak isometric force. Moreover, the contractile force (F\({}^{(CE)}\)) depends also on the muscle velocity and activation law. The latter (\(a(t)\)) is usually determined experimentally [73] or by solving the first-order ordinary differential equation indicated in (9):  
\[\frac{da(t)}{dt}=\frac{1}{\tau_{rise}}(1-a(t))u(t)+\frac{1}{\tau_{fall}}(a_{ \text{min}}-a(t))(1-u(t)) \tag{9}\]  
where \(\tau_{rise}\) and \(\tau_{fall}\) are the time constants, respectively, for the activation and deactivation of the muscles, whereas \(u(t)\) is the neural excitation included in the range [0, 1]. The activation function instead goes from \(a_{min}\) to 1 [74]. The set of obtained equations is strictly non-linear, inducing complications in their computational resolution. Nevertheless, in hypothesizing a small strain, the governing relationships can be suitably linearized, as applied to the biomechanics of the human tongue by [PERSON] et al. [75]. In conclusion, FEM modeling for the mechanical properties of soft tissues is somewhat complex due to their nonlinear response. The definition of the muscle features significantly influences the results [76], and it thus is indispensable for a complete and correct description of the biomechanical behavior of the human jaw. To achieve this, the modeling should mimic the human body structure as much as possible [77]. On the other hand, the muscle properties are strictly variable and affected by their functional demands [78]. An interesting procedure for calculating them was proposed by [PERSON] et al. [79] using inverse finite element analysis. The approach provided consistent results with the experimental trials conducted using dynamic magnetic resonance imaging, commonly used in the dental field [80]. In this regard, hyperelastic theory [81], transversely isotropic [82; 83] or anisotropic [84; 85; 86], was found to be in a good agreement with the experiments, as confirmed by [PERSON] et al. [87].