Temporomandibular joint loading generated during bilateral static bites at molars and premolars

Abstract

The aim of this study was to investigate the features of the loading vectors of the temporomandibular joint (TMJ) generated during bilateral static bites at the molars and at the premolars, and to determine the major factors affecting the difference between the two loading vectors. We computed the subjects’ estimated and theoretical minimum TMJ loadings under the two different bite conditions by applying the subjects’ bite-force and electromyographic (EMG) data to a two-dimensional (2D) standard model of the jaw based on a rigid-body spring model of the TMJ. For a molar bite, (1) the estimated loading vector was not equal to its theoretical minimum; (2) the TMJ-loading/bite-force ratio, describing the proportion of TMJ loading, was relatively small, 0.477 on average; and (3) the estimated loading vector pointed in the direction of the central part of the articular disk’s intermediate zone. For a premolar bite, on the other hand, (1) the estimated loading vector was nearly equal to its theoretical minimum; (2) the TMJ-loading/bite-force ratio was relatively large, 0.904 on average; and (3) the estimated loading vector pointed at the superior portion of the articular disk’s intermediate zone. The differences between the TMJ-loading vectors for molar and premolar bites originated primarily from changes in the bite-point location.

Appendices

Appendix 1: Review of our previous studies

1.1 Sensitivity analysis

Previously, we have demonstrated how forces produced by the masticatory muscles⁴ and the digastric muscle⁵ contribute to the generation of bite-force and TMJ loading in a 2D standard model (Figs. 1, 2). The values of muscle forces (expressed in N) which were derived from the data of Korioth and Hannam (modified from task 15 of Fig. 2 in [18]), were as follows: masseter, F _m = 142.98; anterior temporalis, F _at = 58.40; lateral pterygoid, F _l = 23.90; posterior temporalis, F _pt = 25.70; and digastric, F _d = 18.00. We refer to the values of the above muscle-forces as “reference values” in this appendix. The bite-force was assumed to be applied to the first molar [Itoh et al. ⁴ (1996); Abe et al. ⁵ (2002)] [also 11, 12]

First, we computed F _o and F _j when each muscle force {F _m, F _at, F _l} except F _pt was modified from 50 to 150% of its reference value in the model (Fig. 2) [Itoh et al. ⁴ (1996)]. Figure 6 shows the relationship between F _o and F _j when each muscle force {F _m, F _at, F _l} was modified within the above range. [Modified from Fig. 5 of Itoh et al. ⁴ (1996).] Here, the point of intersection in Fig. 6 indicates the computed values of F _o and F _j when each muscle force {F _m, F _at, F _l} was set at the above reference-value. Hence, our result demonstrated that the masseter and anterior-temporalis muscles generate TMJ loading as well as the bite-force, whereas the lateral pterygoid muscle does not contribute to the generation of bite-force, but affects TMJ loading [Itoh et al. ⁴ (1996); 12].

Fig. 6

Sensitivity analysis. Relationship between bite-force (F _o) and TMJ-loading (F _j), when each muscle force {F _m, F _at, F _l} was modified from 50 to 150% of its reference value {F _m = 142.98 N, F _at = 58.40 N, F _l = 23.90 N} [Itoh et al. ⁴ (1996)] [also, 11, 12] in a 2D standard model (Fig. 2) except the posterior temporalis and digastric forces (F _pt and F _d). Dashed line: masseter (F _m); dotted line: anterior temporalis (F _at); solid line: lateral pterygoid (F _l) [Modified from Fig. 5 of Itoh et al. ⁴ (1996).]

Second, we calculated the bite-force and TMJ loading (F _o and F _j) by substituting also the posterior-temporalis force (F _pt) at F _pt = 0 N and at F _pt = 25.70 N (reference value) into an identical model (Fig. 2), so that we obtained the following: F _o = 129.50 N, F _j = 97.30 N, and the TMJ-loading/bite-force ratio, F _j/F _o = 0.751, when F _pt is zero [Itoh et al. ⁴ (1996)] [also 11, 12] (point of intersection in Fig. 6); and F _o = 137.24 N, F _j = 88.20 N, and F _j/F _o = 0.642, when F _pt is the reference value (F _pt = 25.70 N) (M. Abe, unpublished data, 2000). Hence, the results demonstrated that the posterior-temporalis muscle slightly increases bite-force and slightly decreases TMJ loading.

Finally, we calculated the bite-force and TMJ loading (F _o and F _j) by substituting also the digastric force (F _d) at F _d = 0 N and at F _d = 18.00 N (reference value) into the 2D model except F _pt (Figs. 1, 2), so that we obtained the following: F _o = 128.34 N, F _j = 96.10 N, and F _j/F _o = 0.741, when F _d is zero; and F _o = 104.72 N, F _j = 100.45 N, and F _j/F _o = 0.958, when F _d is the reference value. Hence, the results demonstrated that the digastric muscle slightly decreases the bite-force and slightly increases the TMJ loading [Abe et al. ⁵ (2002)].

1.2 Control analysis of TMJ loading

On the basis of the above results, the masseter and anterior-temporalis forces (F _m and F _at) were treated as parameters, whereas the lateral pterygoid force (F _l) was fixed at the reference value. The bite-force (F _o) was fixed at a computed value at which the rigid-body mandible reaches an equilibrium state. In other words, the TMJ-loading vector (F _j) and F _at were computed from the simultaneous equations by increasing F _m while keeping F _l and bite-force (F _o) constant [11, 12, 14, 15]. This yielded a linear trajectory of F _j as shown in Fig. 7 [11, 12]. Our results demonstrated that (1) the magnitude of F _j is controlled by adjusting F _m and F _at (Fig. 7); (2) the minimized TMJ-loading vector (F _j,min) is found along the trajectory, pointing in the general direction of the intermediate zone of the disk (Fig. 7); and (3) the direction (φ _j,min) of F _j,min is completely independent of the bite and lateral-pterygoid forces.

Fig. 7

Control analysis. Linear trajectory of TMJ-loading vector (F _j), during a bilateral bite at first molar, which was computed by changing the combination of masseter and anterior-temporalis forces (F _m and F _at), under the condition that both the lateral-pterygoid and bite-forces (F _l and F _o) were constant

Next, we investigated the variation of the direction (φ _j,min) due to anatomical and morphological differences, first by translating the points of application of the masseter and temporalis force vectors, and then by changing the direction of these vectors in a morphologically permissible range [12]. The results were summarized as follows: (1) φ _j,min was relatively sensitive to the point of application and direction of the masseter-force vector, as well as the horizontal position of the point of application of the temporalis-force vector; and (2) the variation of φ _j,min lay within an anatomically acceptable limit.

Appendix 2

Table 6 Spring model of articular disk: geometric data of the mandibular fossa and condyle, and spring constants for individual springs simulating the articular disk (see Fig. 1)

Appendix 3: Equations of static equilibrium

F _m :

magnitude of masseter force vector

φ _m :

direction of masseter force vector

(x _m, z _m):

point of application of masseter force vector (initial position)

(x _ma, z _ma):

point of application of masseter force vector (displaced position)

F _t :

magnitude of resultant temporalis force vector

φ _t :

direction of resultant temporalis force vector

(x _t, z _t):

point of application of temporalis force vector (initial position)

(x _ta, z _ta):

point of application of temporalis force vector (displaced position)

F _l :

magnitude of lateral-pterygoid force vector

φ _l :

direction of lateral-pterygoid force vector

(x _l, z _l):

point of application of lateral pterygoid force vector (initial position)

(x _la, z _la):

point of application of lateral pterygoid force vector (displaced position)

F _o :

magnitude of bite-force vector

φ _o :

direction of bite-force vector

(x _o, z _o):

point of application of bite-force vector (initial position)

(x _oa, z _oa):

point of application of bite-force vector (displaced position)

F _j :

TMJ-loading vector

F _jx :

x-component of TMJ-loading vector

F _jz :

z-component of TMJ-loading vector

φ _j :

direction of TMJ-loading vector

(x _j, z _j):

point of application of TMJ-loading vector (initial position)

(x _ja, z _ja):

point of application of TMJ-loading vector (displaced position)

F _p,i (i = 1–24):

magnitude of compressive force of the ith spring

φ _p,i (i = 1–24):

direction of compressive force of the ith spring

k _i (i = 1–24):

spring constant of the ith spring

d _i (i = 1–24):

displacement of the ith spring

(x _p,i , z _p,i ) (i = 1–24):

point of application of compressive force of the ith spring on the condyle (initial position)

(x _pa,i , z _pa,i ) (i = 1–24):

point of application of compressive force of the ith spring on the condyle (displaced position)

(x _q,i , z _q,i ) (i = 1–24):

point of application of compressive force of the ith spring on the fossa (initial position)

α _plane :

inclination of the occlusal plane, which was set as α _plane = 10.0° in this paper

(x _g, z _g):

bite-point (initial position)

u :

magnitude of translation of the mandible relative to the maxilla

θ :

rotation angle of the mandible relative to the maxilla

n :

number of springs, which was set as n = 24 in this paper

Three equations of equilibrium:

$$ F_{{\text{m}}} \,\cos \,\phi _{{\text{m}}} + {\text{ }}F_{{\text{t}}} \,\cos \,\phi _{{\text{t}}} + {\text{ }}F_{{\text{l}}} \,\cos \,\phi _{{\text{l}}} {\text{ }} + {\text{ }}F_{{\text{o}}} \,\cos \,\phi _{{\text{o}}} {\text{ }} + {\sum\limits_{i = 1}^n {(F_{{{\text{p}},i}} \,\cos \,\phi _{{{\text{p}},i}} ){\text{ }} = {\text{ }}0} }, $$

(4)

$$ F_{{\text{m}}} \,\sin \,\phi _{{\text{m}}} {\text{ }} + {\text{ }}F_{{\text{t}}} \,\sin \,\phi _{{\text{t}}} + {\text{ }}F_{{\text{l}}} \,\sin \,\phi _{{\text{l}}} {\text{ }} + {\text{ }}F_{{\text{o}}} \,\sin \,\phi _{{\text{o}}} + {\sum\limits_{i = 1}^n {(F_{{{\text{p}},i}} \,\sin \,\phi _{{{\text{p}},i}} ){\text{ }} = {\text{ }}0,} } $$

(5)

$$ \begin{aligned}{} & F_{{\text{m}}} (z_{{{\text{ma}}}} \,\cos \,\phi _{{\text{m}}} - x_{{{\text{ma}}}} \,\sin \,\phi _{{\text{m}}} ){\text{ }} + {\text{ }}F_{{\text{t}}} (z_{{{\text{ta}}}} \,\cos \,\phi _{{\text{t}}} {\text{ }} - {\text{ }}x_{{{\text{ta}}}} \,\sin \,\phi _{{\text{t}}} ){\text{ }} + {\text{ }}F_{{\text{l}}} (z_{{{\text{la}}}} \,\cos \,\phi _{{\text{l}}} {\text{ }} - {\text{ }}x_{{{\text{la}}}} \,\sin \,\phi _{{\text{l}}} ) \\ & \quad + {\text{ }}F_{{\text{o}}} (z_{{{\text{oa}}}} \,\cos \,\phi _{{\text{o}}} - x_{{{\text{oa}}}} \,\sin \,\phi _{{\text{o}}} ){\text{ }} + {\sum\limits_{i = 1}^n {\{ F_{{{\text{p}},i}} (z_{{{\text{pa}},i}} \,\cos \,\phi _{{{\text{p}},i}} - x_{{{\text{pa}},i}} \,\sin \,\phi _{{{\text{p}},i}} )\} {\text{ }} = {\text{ }}0.} }{\text{ }} \\ \end{aligned} $$

(6)

During a bite, the mandible rotates and translates infinitesimally. The points of application of muscle force vectors and spring compressive forces, then, also become dislocated to a slight degree. Letting θ and u be the rotation angle and the magnitude of translation of the mandible relative to the maxilla, respectively, the displaced position of the point of application of the spring compressive force can be expressed approximately as:

$$ \begin{aligned}{} x_{{{\text{ma}}}} {\text{ }} & = {\text{ }}x_{{\text{m}}} {\text{ }} + {\text{ }}u\,\cos \,\alpha _{{{\text{plane}}}} {\text{ }} - \theta (z_{{\text{m}}} {\text{ }} - {\text{ }}z_{{\text{g}}} ), \\ z_{{{\text{ma}}}} {\text{ }} & = {\text{ }}z_{{\text{m}}} {\text{ }} + {\text{ }}u\,\sin \,\alpha _{{{\text{plane}}}} {\text{ }} - \theta (x_{{\text{m}}} {\text{ }} - {\text{ }}x_{{\text{g}}} ), \\ x_{{{\text{ta}}}} {\text{ }} & = {\text{ }}x_{{\text{t}}} {\text{ }} + {\text{ }}u\,\cos \,\alpha _{{{\text{plane}}}} {\text{ }} - \theta (z_{{\text{t}}} {\text{ }} - {\text{ }}z_{{\text{g}}} ), \\ z_{{{\text{ta}}}} {\text{ }} & = {\text{ }}z_{{\text{t}}} {\text{ }} + {\text{ }}u\,\sin \,\alpha _{{{\text{plane}}}} {\text{ }} - \theta (x_{{\text{t}}} {\text{ }} - {\text{ }}x_{{\text{g}}} ), \\ x_{{{\text{la}}}} {\text{ }} & = {\text{ }}x_{{\text{l}}} {\text{ }} + {\text{ }}u\,\cos \,\alpha _{{{\text{plane}}}} {\text{ }} - \theta (z_{{\text{l}}} {\text{ }} - {\text{ }}z_{{\text{g}}} ), \\ z_{{{\text{la}}}} {\text{ }} & = {\text{ }}z_{{\text{l}}} {\text{ }} + {\text{ }}u\,\sin \,\alpha _{{{\text{plane}}}} {\text{ }} - \theta (x_{{\text{l}}} {\text{ }} - {\text{ }}x_{{\text{g}}} ), \\ x_{{{\text{oa}}}} {\text{ }} & = {\text{ }}x_{{\text{o}}} {\text{ }} + {\text{ }}u\,\cos \,\alpha _{{{\text{plane}}}} {\text{ }} - \theta (z_{{\text{o}}} {\text{ }} - {\text{ }}z_{{\text{g}}} ), \\ z_{{{\text{oa}}}} {\text{ }} & = {\text{ }}z_{{\text{o}}} {\text{ }} + {\text{ }}u\,\sin \,\alpha _{{{\text{plane}}}} {\text{ }} - \theta (x_{{\text{o}}} {\text{ }} - {\text{ }}x_{{\text{g}}} ), \\ x_{{{\text{pa}},i}} {\text{ }} & = {\text{ }}x_{{{\text{p}},i}} {\text{ }} + {\text{ }}u\,\cos \,\alpha _{{{\text{plane}}}} {\text{ }} - \theta (z_{{{\text{p}},i}} {\text{ }} - {\text{ }}z_{{\text{g}}} ), \\ z_{{{\text{pa}},i}} {\text{ }} & = {\text{ }}z_{{{\text{p}},i}} {\text{ }} + {\text{ }}u\,\sin \,\alpha _{{{\text{plane}}}} {\text{ }} - \theta (x_{{{\text{p}},i}} {\text{ }} - {\text{ }}x_{{\text{g}}} ). \\ \end{aligned} $$

(7)

The compressive force (F _p,i ) for each spring is described by Hooke’s law as:

$$ F_{{{\text{p}},i}} = \left\{ \begin{aligned}{} & k_{i} \cdot d_{i} (d_{i} > 0) \\ & 0\begin{array}{*{20}c} {{}} & {{}} \\ \end{array} (d_{i} \le 0) \\ \end{aligned} \right., $$

(8)

where d _i can be expressed approximately, due to the very slight amount of displacement, as:

$$ d_{i} = {\sqrt {(x_{{\text{p}}, i} - x_{{\text{q}}, i} )^{2} + (z_{{\text{p}},i} - z_{{\text{q}}, i} )^{2} } } - {\sqrt {(x_{{\text{pa}},i} - x_{{\text{qa}},i} )^{2} + (z_{{\text{pa}},i} - z_{{\text{qa}}, i} )^{2} } }. $$

(9)

The direction (φ _p,i ) of the spring-force vector is expressed as:

$$ \phi _{{{\text{p}},i}} {\text{ }} = {\text{ }}\tan ^{{-1}} { {\frac{{(z_{{{\text{pa}},i}} - z_{{{\text{q}},i}} )}} {{(x_{{{\text{pa}},i}} - x_{{{\text{q}},i}} )}}} }. $$

(10)

The TMJ-loading vector, F _j ≡ (F _jx , F _jz )^T, and its direction (φ _j) are given as

$$ F_{{{\text{j}x}}} {\text{ }} = {\text{ }}{\sum\limits_{i = 1}^n {(F_{{{\text{p}},i}} \cos \phi _{{{\text{p}},i}} ),} }{\text{ }}F_{{{\text{j}z}}} {\text{ }} = {\sum\limits_{i = 1}^n {(F_{{{\text{p}},i}} \,\sin \,\phi _{{{\text{p}},i}} ),\quad \phi _{{\text{j}}} {\text{ }} = {\text{ }}\tan ^{{-1}} {\left( {\frac{{F_{{\text{j}z}} }} {{F_{{\text{j}x}} }}} \right)}.} }{\text{ }} $$

(11)

The above non-linear equations 4–9 can be solved numerically by using the Newton–Raphson method. This computational procedure was described in detail by Itoh and Hayashi [15].

Appendix 4

Table 7 EMG activities and forces of masseter and temporalis muscles for a molar bite (all subjects)

Table 8 EMG activities and forces of masseter and temporalis muscles for a premolar bite (all subjects)