Mandible Movement Pattern

The movement of the mandible is rather complicated even if only focused on the sagittal plane, in which it is featured of combined sliding along the temporal fossa-eminence surface with rotating during the normal open-close; the jaw movement therefrom used to be simplified to a pure rotation hinged on the TMJ, which has been widely applied in the dental prosthesis as a fixed revolving center [68]. In light of the achievable opening range which was considerably restricted in the vertical distance compared to that obtained under the fixed center opted to be positioned on an average ICR, the pure hinge on the posterior TMJ was considered oversimplified to properly model the jaw movement.

The general model recruiting three variables that capably assesses travelling distance of any point on the jaw facilitates to reveal the inherent pattern which is likely to exist in the way of the correlations in the inside TMJ concerning the jaw movement between each pair. The variables are apparently varying along with the jaw depression in terms of the vertical displacement; and as formulated in Eq. (3-1) of the position for the IP and CP, the ratios of the horizontal displacement versus vertical one, plus the angle versus translation, are herein introduced to define the corresponding relationship during the jaw movement, denoted as ݎ௧ ൌௗ௫_ௗ௬Ǣݎ௔ൌௗ௫_థ respectively.

Theoretically, with a set of definite values of the correlation, the mandible can be exactly positioned. Usually, the ratios that have been investigated in the literature are given in mean values which are so averaged across the whole trajectory on the IP with respect to that on the CP that only outlining a general mandibular movement trace; but each ratio can be finely valued to spread into the divided phases along the overall travelling distance. Live recordings capable to provide the correlated trajectories under the synchronous time frame are herein preferable to acquire the stepwise ratios. Unfortunately, current resources (AG500 machine) could not support non-invasive tracking of the condylar trace directly in vivo; as an alternative way, two more points on the molars that could define a spatial position of a rigid-body together with the incisor were recruited in trajectory tracking for further computation of the joint variables. In fact, the information of simultaneous trajectories on two points whose positions can be universally expressed in Eq. (3-16) is adequate to solve each instant variable, then through subtraction each corresponding item that is shown in Eq. (3-17) to obtain the ratios in Eq. (3-18).

ቈ ܲ௑ଵǡଶ ை ܲ_௒ଵǡଶ ை ቉ ൌ ቈ ݀ݔ ൅ ܮ௉ଵǡଶ…‘•ሺߙ ൅ ߶ሻ ݀ݕ ൅ ܮ_௉ଵǡଶ•‹ሺߙ ൅ ߶ሻ቉ (3-16) ቈ ܲ௑ଶ ை _{െ ܲ} ௑ଵ ை ܲ_௒ଶ ை _{െ ܲ} ௒ଵ ை ቉ ൌ ൤ ሺܮ௉ଶെ ܮ௉ଵሻ …‘•ሺߙ ൅ ߶ሻ ሺܮ௉ଶെ ܮ௉ଵሻ •‹ሺߙ ൅ ߶ሻ൨ (3-17) ߶ ൌ –ƒିଵ ைܲ௒ଶെ ܲை ௒ଵ ܲ_௑ଶ ை _{െ ܲ} ௑ଵ ை െ ߙ (3-18)

Data in the literature that disclosed the rhythm based on each model are also referenced in the form of either giving the corresponding ratios or providing pairs of trajectories on the IP and the CP that were found to appear coincidently with the measurement of the jaw morphology by referencing Eq. (3-12). Since the relationship describing the decomposed magnitudes is barely mentioned in the literature to be present in a unified concept of TMJ rhythm, selected data basically offering ratios of rotation versus slides are found to not entirely conform to the same way those defined in the context. Only couples of articles explicitly indicated the contribution correspondingly from components of the rotation and the translation to the overall movement regarding the incisor opening displacement; so ratios of

ݎ௧ and ݎ௔ are managed to be transformed to adapt those partial apparatuses as provided, for

example, the proportions, either of which is attributed to rotation and translation respectively as stated in the literature is formulated in Eq. (3-19).

_ோȀ் ൌோ _் ൌ ܮ_ூ௉•‹ሺߙ ൅ ߶ሻ ݀ݕ ൌ ܮ_ூ௉•‹ሺߙ ൅ ߶ሻ ݎ_௔߶ ݎΤ _௧ (3-19) Where: _ோ ൌ ௅಺ುୱ୧୬ሺఈାథሻൈଵ଴଴Ψ ௥ೌథ ௥Τ ା௅೟ ಺ುୱ୧୬ሺఈାథሻǢ ் ൌ ௥ೌథ ௥Τ ൈଵ଴଴Ψ೟ ௥ೌథ ௥Τ ା௅೟ ಺ುୱ୧୬ሺఈାథሻ

In AG recording trials, the subject was sitting in the chair while maintaining an upright head position, and ordered to relax the jaw and open the mouth naturally as usual till up the limit, as much as in consistency with the motion that was described in the literature. This session was repeated for three times in which GUI for AG data processing coded in the MATLAB was then used to calculate the each ratio based on each selected data group; the mean ratio is then averaged over the values of the corresponding item; note the statistical analysis in terms of standard deviation on the same subject is not carried out. The calculated AG data in the form of ݎ௧ and ݎ௔ are then reshaped to quantify the relationship in the definition of ோ and ், since correlation displayed in this way for the overall trace were already known and could be proof-test the AG recordings data. Then the data in the literature that were not present in numerical format undergo a process of discretization before reconstructed into the specified

TMJ rhythm concept with a speculated assumed morphology of the jaw which was not explicitly given. Then the process is followed by the computation of the relationship in the way adopted in the general jaw movement model.

Table 3-2 Mandible movement rhythms on the sagittal plane

Stage Phase †š †› Ԅ Opening I-i 1 0 ሾͲǡ͵ሿ ሾͲǡ͵ሿ 0° I-ii 0.2-0.5 0.5 mm/° ሾ͵ǡ͹ሿ ሾ͵ǡͷሿ 15° II 0.2-0.3 2 mm/° ሾ͹ǡʹͲሿ ሾͲǡ͸ሿ 32° Close I 0.2-0.5 2 mm/° ሾ͵ǡʹͲሿ ሾ͵ǡ͸ሿ 30° II 1 -- ሾͲǡ͵ሿ ሾͲǡ͵ሿ 2°

Table 3-2 lists the stepwise ratios transformed in expression of both definitions that are split into an opening and a closing phase to reveal the likely mandible movement rhythms. According to the resultant ratios, the normal open-close movement also referred to the habitual movement is dominated by mandibular rotation, which accounts for the majority of the jaw opening displacement, in accordance with the discovery from literatures discussed in section 2.1.4.

Each one-way movement can roughly be divided into two phases in terms of the correlation of different value standing for each ratio. Separately, during the opening section, phase I contributes to the starting stage of the jaw opening till the rotational angle to around 15°, in which both rotation and translation are going up linearly on each own, basically approximating to maintain a constant ratio; but can also be finely partitioned with one more ratio that increases proportionally at the very beginning whose duration is too short to neglect. In phase II, the rotation prevails in the depression with an increasing proportion of the instantaneous opening distance, whereas the translation loses the percentage of its contribution at a linearly dropping rate; though the ratio between two values stays approximately invariant as a constant up to the maximal depression, around 2mm/°. The closing section is by and large filled with a symmetrical distribution in terms of each component with respect to the peak point, and can also be depicted by two phases according to the differentiated ratios. The mouth closing movement starts from the maximum opening position with similar values of ratio till phase II; in the very ending part, the rotation hits its bottom value while the translation peaks up to finalize the elevation to occlusion [83]. Figure 3-12 sketches the stepwise relation of vertical displacement of the jaw evaluated on the incisor with respect to rotational and translational movement during depression.

Figure 3-12 Stepwise Phases of the mandible movement on the sagittal plane

The information about the TMJ rhythm in exact values that is chosen here has been obtained on healthy subjects by means of in-vivo 3D tracking systems, which excluded the influence likely coming from the location of different reference points. Subject’s mandible size under our trials with AG can be roughly estimated by calculating the distance between the sensors at rest attached on the teeth (incisor and the other two molars) and attached around ears as reference, which were not ensured to be the rest position of condyles. The unloaded jaw is requested to carry out normal movement up to the maximum mouth opening distance during the recording. Measurement in terms of original tracking apparatus and data post-processing was individualized in different researches, but basically concluded with the indirect calculation from trajectories within accessible oral space. Inconsistent discoveries on the contribution of each apparatus appeared occasionally in previous publications, which may result from the inter-individual variation of the movement pattern among subjects or the intra-individual allowance due to joint morphology, exclusive of factors from the methods adopted in movement recording and data computing.