Muscle and joint forces under variable equilibrium states of the mandible

Abstract

It is well established that subjects without molars have reduced ability to comminute foods. However, epidemiological studies have indicated that the masticatory system is able to functionally adapt to the absence of posterior teeth. This supports the shortened dental arch concept which, as a prosthetic option, recommends no replacement of missing molars. Biomechanical modeling, however, indicates that using more anterior teeth will result in a larger temporomandibular joint load per unit of bite force. In contrast, changing bite from incisor to molar position increases the maximum possible bite force and reduces joint loads. There have been few attempts, however, to determine realistic joint loads and corresponding muscular effort during generation of occlusal forces similar to those used during chewing with intact or shortened dental arches. Therefore, joint and cumulative muscle loads generated by vertical bite forces of submaximum magnitude moving from canine to molar region, were calculated. Calculations were based on intraoral measurement of the feedback-controlled resultant bite force, simultaneous electromyograms, individual geometrical data of the skull, lines of action, and physiological cross-sectional areas of all jaw muscles. Compared to premolar and canine biting, bilateral and unilateral molar bites reduced cumulative muscle and joint loads in a range from 14% to 33% and 25% to 53%, respectively. During unilateral molar bites, the ipsilateral joints and contralateral muscles were about 20% less loaded than the opposing ones. In conclusion, unilateral or bilateral molar biting at chewing-like force ranges caused the least muscle and joint loading.

Appendix

Mathematical method

Force law

Muscle forces were specified according to the law ${\overrightarrow F_i} = P \cdot {A_i} \cdot \cos {\alpha_i}\{ c\;{U_{rel,i}} + (1 - c)\;U_{rel,i}^2\} {\overrightarrow e_i}$ $({\overrightarrow F_{\max, i}} = P \cdot {A_i} \cdot \cos {\alpha_i}{\overrightarrow e_i})$. This equation extends the previously used law ${\overrightarrow F_i} = P \cdot {A_i} \cdot U_{\hbox{rel,i}}\;{\overrightarrow e_i}$ [21] by a correction factor for the pennation angle α_i (for α ≤ 15°:0.97 ≤ cosα ≤ 1) and parameter c characterizing the degree of nonlinearity between the muscle force and the relative electric muscle activity U _rel. This approach fulfills the following boundary conditions:

• 1. F_i(U_rel = 0)/F_max,i = 0 (no electric activation corresponds to zero muscle force)

• 2. ${F_i}\left( {{U_{\rm{rel}}} = {1}} \right)/{F_{{ \max },i}} = c + \left( {{1} - c} \right) = {1}$ (maximum muscle force at maximum activation)

Determination of parameters P and c

Due to model assumptions and measurement errors it cannot be expected that the balance of momentum of the muscle forces $\overrightarrow F_i$ and bite forces $\overrightarrow B_i$ with respect to the intercondylar axis (y-axis), to which the joint forces do not contribute, will be satisfied exactly. The torques of these forces with respect to the y-axis will rather sum up to a resulting error torque ΔM_y ≠ 0. In order to satisfy the balance of momentum exactly and, in addition, to gain an individual constant parameter set (P,c) for each subject (physiological properties of the muscles do not change under natural conditions for the individual subject), the following two-stage procedure was accomplished: In a first step, on the basis of 36 clenching tasks with bite force magnitudes ranging from 50 to 400 N, the best parameter set (P,c)_opt, for which the objective function $f\left( {P,c} \right) = {\sum\limits_{n = 1}^{36} {\left( {\Delta My,n/F{\hbox{res}},n} \right)}^2}$ with ${F_{{\rm{res}},n}} = \left| {\sum\limits_{j = 1}^3 {{{\overrightarrow B }_{j,n}}} } \right|$ reached a minimum value (minimization of the squared relative error torques), was computed via optimization. With the force law F _i = f(P,c,U _rel,i ), all muscle forces would then be given by the measured EMG activities of the masticatory muscles. However, even for this optimal constant parameter set (P,c)_opt small error torques remain.

Principally, for the second step, two possibilities exist to continue:

• 1. Accepting small error torques ΔM_y due to inevitable (presumably minor) errors in measurement and/or assumptions, which are present even for the best parameter set. The advantage of this choice is that P _opt and c _opt are kept constant; the disadvantage is that no (exact) equilibrium can be achieved, i.e., the computed forces cannot represent the exact distribution of joint and muscle forces which balance the clenching forces.

• 2. Exactly fulfilling the equilibrium conditions by varying the parameter P (P _var), i.e., not using the constant P _opt computed by the optimization. The advantage of this approach is that the computed force distribution represents muscle and joint forces exactly balancing the bite forces. The disadvantage of this method is that P is variable. In the present work, option (2) was chosen. For c, the value of c_opt was maintained (Table 3).

  Table 3 Equations used for the calculation of muscle and joint forces and footnote with used abbreviations