# A Critical Review of Human Jaw Biomechanical Modeling
## 2 Materials and Methods  
The mathematic modeling of mandible biomechanics is certainly an ongoing and fascinating field of bioengineering. Indeed, it has captured the attention and the curiosity of clinicians and engineers since 1970s [57] when the first experiments were performed. Nowadays, the use of any kind of model is only investigative, with no applications to diagnostic protocols [58], via cause-effect scenarios: the aim is to rapidly obtain the outputs desired by varying the boundary conditions. In light of this, the evident complexity of the human jaw should be taken into account in terms of its geometry, mechanical properties, muscle activations and so on. Moreover, each patient has a specific conformation from a morphological point of view, which can be described using only seven morphometric measurements, as stated by [PERSON] et al. [59]. Since the jaw shape is extremely intricate, ascertaining its geometry is achieved using a CT scan, which is successively refined or modified [60]. Secondly, the entire coupling, composed of muscles, ligaments and bone, must be described according to its mechanical and physical properties. For instance, the anisotropic property of bone has a great influence on the stress-strain regime [61] but causes greater computation efforts. Hence, numerical models are based on the trade-off between realistic behavior and the equation resolution time. Muscle, joint and tendon modeling, instead, is essentially based on the multibody theory, in which the joints are characterized using Equation (1):  
\[\{\Phi(q,t)\}=0 \tag{1}\]  
This indicates the set of algebraical constraints on the generalized coordinate vector \(q\) at the interval of time \(t\). These can be divided into time-independent constraints like anatomical ones or constraints as a function of time, such as the muscle length during the performed action. The choice of the coordinates \(q\) depends on the specific investigation, but they usually refer to the condyle process, muscle insertions and biting force origins. Furthermore, the latter are referred to according to the \(x,y\) and \(z\) axes, outlining, respectively, the anterior/posterior, lateral and inferior-superior directions [62]. By amending Equation (1) twice with respect to the time, we obtain:  
\[\big{[}\Phi_{q}\big{]}\Big{(}\dot{\dot{q}}\Big{)}=[\gamma] \tag{2}\]  
where \(\gamma\) refers to the constraint's quadratic velocity tensor, equal to:  
\[\gamma=-\Big{(}\Phi_{q}\dot{\dot{q}}\Big{)}_{q}\dot{\dot{q}}-2\Phi_{q,t}\dot{ \dot{q}}-\Phi_{t,t} \tag{3}\]  
in which \(\Phi_{\text{q}}\) is the constraint's Jacobian matrix; \(\dot{q}\) and \(\ddot{\dot{q}}\) are the first and second derivatives of coordinate \(q\), corresponding, respectively, to the velocity and acceleration vector of the generalized coordinates; \(\Phi_{\text{q},t}\) is the constraint's Jacobian matrix's first partial time derivative and \(\Phi_{\text{t},t}\) the constraint vector's first partial time derivative. These relationships are thus coupled with the motion laws (Equation (4)), providing an algebraic-differential system of equations (DAE):  
\[[M]\Big{(}\dot{\dot{q}}\Big{)}=\Big{(}F^{i}\Big{)}+(F^{e})=F \tag{4}\]  
\[ \begin{bmatrix}M&\Phi_{q}^{T}\\ \Phi_{q}&0\end{bmatrix} \Bigg{\{} \begin{array}{c}\dot{\dot{\mathbf{q}}}\\ \dot{\lambda}\end{array} \Bigg{\}}=\Bigg{\{} \begin{array}{c}F\\ \gamma\end{array} \Bigg{\}} \tag{5}\]  
where \([M]\) is the mass matrix, and \(F^{i}\) and \(F^{e}\) are vectors of all the internal (muscles, ligaments) and external forces (gravity, ground reaction). Finally, \(\lambda\) represents the Lagrange multiplier vector, indicating the joint reaction forces. The mathematical system reported in (5) can be approached using both forward and inverse analysis. The first technique is adopted when the forces are determined using electromyography [63], and the position vector must be evaluated: \(\ddot{\dot{q}}\) and \(\lambda\) are therefore solved in Equation (5) and successively integrated to obtain the position and velocity vector.