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Biomechanical analysis of stress distribution in the temporomandibular joint
ARTICLE IN PRESS Ann Anat 189 (2007) 329—335
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Biomechanical analysis of stress distribution in the temporomandibular joint Rainer Breul Anatomische Anstalt der LMU Mu ¨ nchen, Pettenkoferstraße 11, D-80336 Mu ¨ nchen, Germany Received 21 December 2006; accepted 15 January 2007
KEYWORDS Temporomandibular joint; Biomechanical analysis; Stress
Summary The positions of the head of the mandible, the articular disc and the outline of the temporal surface are digitised from sagittal MRT-scans of the mandibular joint of a 32-year-old subject in five different positions of occlusion. The stress distribution in the joint is calculated on the basis of these data. For each position of the condyle, the momentary point of rotation in the head of the mandible and the tangent attached to the temporal surface are determined. The line connecting these two points indicates the direction of the resulting bearing force. Furthermore, the extension of the area available to the force transmission is estimated. By means of these parameters, the stress distribution is calculated independently of the position. The analyses show that the mandibular joint is slightly eccentrically loaded in all positions. The increase in stress is in all cases oriented caudo-ventrally. The results are verified in an anatomical specimen of the articular tuberculum. The trabecular structures as well as the subchondral bone-lamella of the articular tuberculum are functionally adapted to the analysed stress situations. & 2007 Elsevier GmbH. All rights reserved.
Introduction In the past years a series of fundamental studies on the biomechanics of the mandibular joint have appeared (Molitor, 1969; Barbenel, 1974; Brehnan et al., 1981; Kubein-Meesenburg, 1985; Kang et al., 1993; Tanaka et al., 1994). Nevertheless, there is still a lively discussion on the loading situation Tel.: +40 (0) 89 5160 4816.
E-mail address: [email protected].
in the mandibular joint. In vivo investigations (Hylander, 1979) are extremely difficult complex because of the mechanical complexity of the joint and its neuro-muscular regulation (Kubein-Meesenburg, 1985; Kubein-Meesenburg et al., 1991). Most studies are therefore restricted to experimental or analytical methods. Molitor (1969), in his very detailed biomechanical analysis, compared the mandible to a balance beam resting with its momentary point of rotation on the articular tuberculum. The suspension of the beam is ensured
0940-9602/$ - see front matter & 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.aanat.2007.02.005
ARTICLE IN PRESS 330 by the masticatory muscles, which, according to the lever principle, balance out the mastication forces on the alimentary bolus and the bearing force on the articular tuberculum. Molitor showed by vectorial analyses that the direction of the bearing force is more or less rostral with respect to the shape of the ascending ramus of the mandible and the momentary position of the condyle on the articular tuberculum. He was further able to prove theoretically that the mandibular joint with its multiple muscle functions can in case of a certain specific combination of the single muscle forces be free of forces during mastication. The teams of Kubein-Meesenburg et al. (1990) and Na ¨gerl et al. (1991) have intensively studied the physico-cinematical and physico-statical requirements of the mandibular movements during mastication. They have also characterized the anatomical structures of the mandibular joint under differential-geometric aspects. They mention two fundamentally different functional states of the joint. On the one hand, the joint serves to transmit forces from the mandible to the maxilla; on the other hand, it represents a space for movements of the condyle. Since the force is transmitted by the bolus in the sense of a hypomochlion, the joint is force free. The above studies in contrast to observations of Hylander (1979) and Brehnan (1981) in macacas. They were able to prove by direct measurements in the mandibular joint that considerable forces are exerted on the joint during the phase of occlusion as well as during the process of mastication. Dos Santos (1995) further showed in a biomechanical model, that during strong occlusion on splints 30–45% of the total muscle force is transmitted to the mandibular joint. Because of the partially contrary statements on the loading condition of the mandibular joint, the present study aims at analysing the kinds of stresses acting on the mandibular joint by magnetic resonance tomography under defined occlusional conditions. The results are morphologically verified in an anatomical specimen.
Materials and methods Proton-weighted MRT-sequences of the mandibular joint of a 32-year-old subject were scanned under defined conditions using modified coils for the mandibular joint (Fig. 1). By this method, it was possible to present all anatomical structures rich in contrast. The resolution was 0.4 mm. The head of the subject was fixed in a bearing of foam rubber to avoid moving artefacts during the acquisition
R. Breul
Figure 1. MRT-image of a proton-weighted sequence (Siemens Vision, 1.5 T, TE/TR 20/2000) of the mandibular joint of a 32-year-old subject in complete occlusion. In the centre of the image: articular tuberculum, articular disc and head of the mandible.
period of 3 min and 20 s. The orientation of the acquisition plane was right-angled to the middle of the articular tuberculum. The scan thickness was 2 mm. A total of five scans of the mandibular joint were taken. The distance of the incisor teeth of 0, 6, 9, 12 and 15 mm was kept constant by an individually shaped form of Permagum. The form at the same time served as a bolus. During the acquisition period the subject was asked to exert a constant pressure on the bolus. The MRT-scans were processed by an image processing system (Adobe Photoshop 3.0). The following parameters (Fig. 2) are determined from the position of the articular surfaces: (1) A tangent T is attached to that particular location on the articular tuberculum, which is nearest to the condyle. (2) The momentary point of rotation D in the articular condyle is located with the help of the three secants S1–S3. (3) The line between T and D indicates the orientation of the bearing load A on the articular tuberculum. (4) The lengths of the sections a and b on both sides of the intersection point of A are measured. The sections together with the articular disc contribute to the force transmission in the mandibular joint. (5) The following force components are calculated from the parameters a, b and A for
ARTICLE IN PRESS Biomechanical analysis of stress distribution in the temporomandibular joint
331
slightly eccentric position of PRA the stress diagram shows a stress value on Pa, which is almost twice that on Pb. The joint is subjected to higher stresses. If PRA assumes an extremely eccentric position, the stress values are very high on Pa while they are almost equal to zero on Pb. Especially high and eccentric stresses lead to destruction of the articular cartilage.
Results
Figure 2. Scheme of the measurement of the MRTimages for determining the parameters of the stress distribution analyses: T ¼ tangent attached to the temporal surface, S1–S3 secants for determining the momentary point of rotation D of the head of the mandible, PRA ¼ resultant bearing force on the articular tuberculum, the connection between D and T determines the direction of PRA, a and b ¼ sections for the force transmission to the articular tuberculum.
formulating the stress profile: P a ¼ A2ð2b aÞ=ða þ bÞ2 as lateral border of section a,
P b ¼ A2ð2a bÞ=ða þ bÞ2 as lateral border of section b,
PRA ¼ A4ða3 b3 Þ=ða þ bÞ4 , as force component at the intersection point of the bearing force A. These three approaches take into account the hypothesis of Navier (Schreyer, 1957), according to which the increase in all force components is linearly limited in case of an eccentric position of A. Since the absolute value of the bearing force A is unknown, it is replaced by the unit vector. The absolute value is thereby changed to a relative value, which still allows for determination the kind of stress distribution in the joint. The force components PI are further divided into their normal components and projected on the circular outline to present the stress distribution. The results of the analyses of the stress distribution are differentiated with respect to the position of PRA. In case of a centred position of PRA the normal components PIN (Fig. 3a) are almost equal, leading to an even stress distribution. In case of a
In the first analysis, the stress distribution in the mandibular joint is calculated and graphically presented (Fig. 3a and b) in complete occlusion (distance between upper and lower incisors: 0 mm). The condyle lies on the posterior and lower third of the articular tuberculum; the bearing force runs slightly inclined in a rostro-occipital direction. The sections are a ¼ 65 units and b ¼ 51 units. The stress distribution is slightly eccentric and increases in a caudo-ventral direction. In the second analysis, the distance between the incisors is 6 mm; the condyle is positioned anterior to and below the articular tuberculum (Fig. 3c). The sections are a ¼ 72 units and b ¼ 38 units. The stress distribution is slightly eccentric and increases in a caudo-ventral direction. In the third analysis (Fig. 4), the distance between the incisors is 9 mm; the condyle has almost the same position as in the second analysis. The sections are a ¼ 51 and b ¼ 42 units. The stress distribution is slightly eccentric on a relatively higher level of stress values and increases in a caudo-ventral direction as well. In the fourth analysis (Fig. 5), the distance between the incisors is 12 mm; the condyle has passed the lower vertex on the articular tuberculum ventrally. The sections are a ¼ 58 units and b ¼ 55 units. The stress distribution is almost central; a slight increase is observed in a ventral direction. In the fifth analysis (Figs. 6–8), the distance between the incisors is 15 mm; the condyle has the same position as in the fourth analysis. The sections are a ¼ 62 units and b ¼ 58 units. The stress distribution is almost centre and increases slightly ventrally.
Discussion The present biomechanical analysis of the loading conditions in the mandibular joint is based on the assumption that the head of the mandible is
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Figure 3. (a) Example for a centric strain of the joint. The intersection of the resultant bearing force PRA divides the bearing line ab into two sections a and b of equal length. This leads to an even stress distribution of the force components PI (stress profile on the baseline) and PIN (stress profile on the circular outline): Pa and Pb ¼ lateral borders of the stress profile. (b) Example of a slightly eccentric stress distribution. Section a is shorter than section b. PRA does not meet ab in the centre. (c) Example of a highly eccentric stress distribution. PRA is clearly eccentric; section a is considerably shorter. The values of the force components PI increase towards PA.
supported by the articular tuberculum during occlusion. The bearing force is the resultant of the forces of all muscles inserting in the mandibular joint. It causes a pressure in the moment of occlusion that is transmitted to the articular tuberculum through the articular disc. The assumptions made are supported by in vivo measurements in the mandibular joint and the condyle in macacas (Hylander, 1979; Boyd et al., 1990), which showed considerable pressure forces. For the presented analyses of the stress distribution it is essential to know the parameters for the direction of the resulting bearing force RA as well as the lengths of the sections a and b. They can be determined from the MRT-scans of the mandibular joint for the different distances of the incisors of 0, 6, 9, 12 and 15 mm with the help of the rules of
Pauwels (1965, 1973) and Kummer (1968, 1985). The five stress analyses show that the mandibular joint is subjected to a more or less eccentric stress distribution in all positions. Under eccentric stress, the resultant force in the joint has a caudo-ventral direction. Since our investigation did not include an exact measurement of the pressure force in the mandibular joint, the pressure force is uniformly assumed as 100%. This method is justified because it allows for evaluation of the kind of loading (centre or eccentric) for the stress analysis when the sections a and b are known for the force transmission. Higher or lower forces only shift the level of the stress profile but do not change the type of stress distribution. As the positions 1–5 show, the force transmission between the corresponding components takes place in a
ARTICLE IN PRESS Biomechanical analysis of stress distribution in the temporomandibular joint
Figure 4. Position of the head of the mandible in complete occlusion (incisor distance: 0 mm). T ¼ tangent to the temporal surface, D momentary point of rotation of the head of the mandible, sections a and b for force transmission to the articular tuberculum. PRA ¼ resultant bearing force on the articular tuberculum. See below: Stress distribution on the articular tuberculum incomplete occlusion (incisor distance: 0 mm). The intersection of the resultant bearing force PRA divides the line ab into two sections a and b causing a slightly eccentric stress distribution of the force components PI (in the stress profile on the base line) and PIN (in the stress profile on the circular outline). PA and PB ¼ lateral borders of the stress profile.
333
Figure 6. Position of the head of the mandible for an incisor distance of 9 mm. For further explanations see Fig. 4a. See below: Stress distribution on the articular tuberculum for an incisor distance of 9 mm. The stress values increase slightly caudo-ventrally.
Figure 7. Position of the head of the mandible for an incisor distance of 12 mm. For further explanations see Fig. 4a. See below: Stress distribution on the articular tuberculum for an incisor distance of 12 mm. The stresses increase slightly caudo-ventrally. Figure 5. Position of the head of the mandible for an incisor distance of 6 mm. For further explanations see Fig. 4a. See below: Stress distribution on the articular tuberculum for an incisor distance of 6 mm. The stress values increase slightly caudo-ventrally.
comparatively small area in the lower region of the articular tuberculum, which is referred to as major force-transmitting zone by Kubein-Meesenburg et al. (1988).
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Figure 9. 2 mm section of a tuberculum articulare with pronounced subchondral plate and trabecular structures, V ¼ ventral, d ¼ dorsal, F ¼ Fossa mandibulae.
Figure 8. Position of the head of the mandible for an incisor distance of 15 mm. For further explanations see Fig. 4a. See below: Stress distribution on the articular tuberculum for an incisor distance of 15 mm. The stresses increase slightly caudo-ventrally.
Dos Santos (1995) came to the result that of an assumed total force of 100%, 31% have to be attributed to the mandibular joint and 69% to the occluding teeth. Observations of occlusion splints led to the same conclusions. No analysis, however, succeeded in proving a complete unloading of the mandibular joint under equilibrium conditions. Klesper et al. (1987, 1989) found clear densitometric indications that in more than 98% of the mandibular joints investigated the subchondral bone showed the highest density in this region. One personally made finding (Fig. 9) shows as well that there is a pronounced subchondral plate, to which some trabecular structures lead, which transmit the pressure forces from the articular tuberculum to the cranial base. Takamura et al. (1980) and Oberg et al. (1971) observed distinct arthrotic changes in this region with a thinning out and perforations of the disc as a sign of high stress. Koop (1976) was able to find glycosaminoglycans in the anterior part of the articular disc, which only occur in pressure-exposed parts of joints. Our analyses as well as the results of the investigations cited above make clear that the mandibular joint is subjected to pressure forces during occlusion as well as during mastication. The specific geometrical conditions of the articular surfaces in the different positions during the
process of occlusion lead to a more or less eccentric stress distribution in the joint (Fig. 9). Despite the results presented, it has to be taken into account that the mandibular joint is, in the mechanical connotation, part of an underdetermined system because of the masticatory muscles inserting in the mandible. In such a system the bearing force in the mandibular joint can, according to Molitor (1969), be equal to zero in the process of occlusion since certain combinations of forces can free the joint from forces. Observations in patients, having the head of the mandible resected, support this finding. These patients are able to develop a physiologically appropriate masticatory pressure. The masticatory apparatus seems to be able to fulfil this special task to a variable extent because of its exceptional muscular construction. The two functional states mentioned in the introduction (Kubein-Meesenburg et al., 1990; Na ¨gerl et al., 1991) refer to this special ability.
References Barbenel, J.C., 1974. The mechanics of the temporomandibular joint – a theoretical and electromyographical study. J. Oral Rehab. 1, 19–27. Boyd, R.L., Gibbs, C.H., Mahan, P.E., Tichmond, A.F., Laskin, J., 1990. Temporomandibular forces measured at the condyle of macaca arctoides. Am. J. Orthod. Dentofac. Orthop. 97, 472–479. Brehnan, K., Boyd, R.L., Laskin, J., Gibbs, C.H., Mahan, P., 1981. Direct measurement of loads at the temporomandilular joint in macaca arctoides. J. Dent. Res. 60, 1820–1824. Dos Santos Jr., J.D., 1995. Vectorial analysis of the equilibrium of forces transmitted to TMJ and occlusal biteplane splints. J. Oral Rehabil. 4, 301–310. Hylander, L.W., 1979. An experimental analysis of temporomandibular joint reaction force in macaques. Am. J. Phys. Anthro. 51, 433–456. Kang, Q.S., Updike, D.P., Salathe, E.P., 1993. Kinematic analysis of the human temporomandibular joint. Ann. Biomed. Eng. 21, 699–707.
ARTICLE IN PRESS Biomechanical analysis of stress distribution in the temporomandibular joint Klesper, B., Kummer, B., Pape, H.D., 1987. Mo ¨glichkeiten der Computerdensitometrie am Beispiel des menschlichen Kiefergelenks. Fortschr. Kiefer-Gesichtschir. 32, 52–53. Klesper, B., Pape, H.D., Kummer, B., 1989. Densitometrische Untersuchungen der kno ¨chernen Strukturen des menschlichen Kiefergelenkes. Dtsch. Zahna ¨rztl. Z. 44, 766–768. Koop, S., 1976. Topographical distribution of sulphated glycosaminoglycans in human temporomandibular joint disks. A histochemical study of an autopsy material. J. Oral. Pathol. 5, 265–276. Kubein-Meesenburg, D., 1985. Die kraniale Grenzfunktion des stomatognaten Systems des Menschen. Hanser, Mu ¨nchen, Wien. Kubein-Meesenburg, D., Na ¨gerl, H., Klamt, B., 1988. The biomechanical relation between incisal and condylar guidance in man. J. Biomech. 21 (12), 997–1009. Kubein-Meesenburg, D., Na ¨gerl, H., Fangha ¨nel, J., 1990. Elements of a general theory of joints. Anat. Anz. 170, 301–308. Kubein-Meesenburg, D., Nagerl, H., Fanghanel, J., Thieme, K.M., Klamt, B., Schwestka-Polly, R., 1991. The general even mandibular movements as couple movements in neuromuscular guided mechanism. Dtsch. Stomatol. 41 (9), 332–336. Kummer, B., 1968. Die Beanspruchung des menschlichen Hu ¨ftgelenks. I. Allgemeine Problematik. Z. Anat. Entw-Gesch. 127, 277–285.
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Kummer, B., 1985. Einfu ¨hrung in die Biomechanik des Hu ¨ftgelenks. Springer, Berlin, Heidelberg, NY. Molitor, J., 1969. Investigations on the stress of the temporo-maxillary joint. Z. Anat. Entwicklungsgesch. 128 (2), 109–140. Na ¨gerl, H., Kubein-Meesenburg, D., Fangha ¨nel, J., Klamt, K.M., Schwestka-Polly, R.J., 1991. Die posteriore Fu ¨hrung der Mandibula als neuromuskula ¨r gegebene dimere Kette. Dtsch. Stomat. 41, 279–283. Oberg, T., Carlsson, G.E., Fajers, C.M., 1971. The temporomandibular joint. A morphologic study on a human autopsy material. Acta Odontol. Scand. 29, 349–384. Pauwels, F., 1965. Gesammelte Abhandlungen zur funktionellen Anatomie des Bewegungsapparates. Springer, Berlin, Heidelberg, NY. Pauwels, F., 1973. Atlas zur kranken und gesunden Hu ¨fte. Springer, Berlin, Heidelberg, NY. Schreyer, C., 1957. Praktische Baustatik. Teubner, Stuttgart. Tanaka, E., Tanne, K., Sakuda, M., 1994. A threedimensional finite element model of the mandible including the TMJ and its application to stress analysis in the TMJ during clenching. Med. Eng. Phys. 16, 316–322. Takamura, N., Kudo, E., Hamamoto, Y., 1980. Chronic rheumatoid arthritis accompanied by various arteritis – 2 autopsy cases. Nippon Rinsho. 38 (12), 4714–4717.
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Biomechanical analysis of stress distribution in the temporomandibular joint Rainer Breul Anatomische Anstalt der LMU Mu ¨ nchen, Pettenkoferstraße 11, D-80336 Mu ¨ nchen, Germany Received 21 December 2006; accepted 15 January 2007
KEYWORDS Temporomandibular joint; Biomechanical analysis; Stress
Summary The positions of the head of the mandible, the articular disc and the outline of the temporal surface are digitised from sagittal MRT-scans of the mandibular joint of a 32-year-old subject in five different positions of occlusion. The stress distribution in the joint is calculated on the basis of these data. For each position of the condyle, the momentary point of rotation in the head of the mandible and the tangent attached to the temporal surface are determined. The line connecting these two points indicates the direction of the resulting bearing force. Furthermore, the extension of the area available to the force transmission is estimated. By means of these parameters, the stress distribution is calculated independently of the position. The analyses show that the mandibular joint is slightly eccentrically loaded in all positions. The increase in stress is in all cases oriented caudo-ventrally. The results are verified in an anatomical specimen of the articular tuberculum. The trabecular structures as well as the subchondral bone-lamella of the articular tuberculum are functionally adapted to the analysed stress situations. & 2007 Elsevier GmbH. All rights reserved.
Introduction In the past years a series of fundamental studies on the biomechanics of the mandibular joint have appeared (Molitor, 1969; Barbenel, 1974; Brehnan et al., 1981; Kubein-Meesenburg, 1985; Kang et al., 1993; Tanaka et al., 1994). Nevertheless, there is still a lively discussion on the loading situation Tel.: +40 (0) 89 5160 4816.
E-mail address: [email protected].
in the mandibular joint. In vivo investigations (Hylander, 1979) are extremely difficult complex because of the mechanical complexity of the joint and its neuro-muscular regulation (Kubein-Meesenburg, 1985; Kubein-Meesenburg et al., 1991). Most studies are therefore restricted to experimental or analytical methods. Molitor (1969), in his very detailed biomechanical analysis, compared the mandible to a balance beam resting with its momentary point of rotation on the articular tuberculum. The suspension of the beam is ensured
0940-9602/$ - see front matter & 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.aanat.2007.02.005
ARTICLE IN PRESS 330 by the masticatory muscles, which, according to the lever principle, balance out the mastication forces on the alimentary bolus and the bearing force on the articular tuberculum. Molitor showed by vectorial analyses that the direction of the bearing force is more or less rostral with respect to the shape of the ascending ramus of the mandible and the momentary position of the condyle on the articular tuberculum. He was further able to prove theoretically that the mandibular joint with its multiple muscle functions can in case of a certain specific combination of the single muscle forces be free of forces during mastication. The teams of Kubein-Meesenburg et al. (1990) and Na ¨gerl et al. (1991) have intensively studied the physico-cinematical and physico-statical requirements of the mandibular movements during mastication. They have also characterized the anatomical structures of the mandibular joint under differential-geometric aspects. They mention two fundamentally different functional states of the joint. On the one hand, the joint serves to transmit forces from the mandible to the maxilla; on the other hand, it represents a space for movements of the condyle. Since the force is transmitted by the bolus in the sense of a hypomochlion, the joint is force free. The above studies in contrast to observations of Hylander (1979) and Brehnan (1981) in macacas. They were able to prove by direct measurements in the mandibular joint that considerable forces are exerted on the joint during the phase of occlusion as well as during the process of mastication. Dos Santos (1995) further showed in a biomechanical model, that during strong occlusion on splints 30–45% of the total muscle force is transmitted to the mandibular joint. Because of the partially contrary statements on the loading condition of the mandibular joint, the present study aims at analysing the kinds of stresses acting on the mandibular joint by magnetic resonance tomography under defined occlusional conditions. The results are morphologically verified in an anatomical specimen.
Materials and methods Proton-weighted MRT-sequences of the mandibular joint of a 32-year-old subject were scanned under defined conditions using modified coils for the mandibular joint (Fig. 1). By this method, it was possible to present all anatomical structures rich in contrast. The resolution was 0.4 mm. The head of the subject was fixed in a bearing of foam rubber to avoid moving artefacts during the acquisition
R. Breul
Figure 1. MRT-image of a proton-weighted sequence (Siemens Vision, 1.5 T, TE/TR 20/2000) of the mandibular joint of a 32-year-old subject in complete occlusion. In the centre of the image: articular tuberculum, articular disc and head of the mandible.
period of 3 min and 20 s. The orientation of the acquisition plane was right-angled to the middle of the articular tuberculum. The scan thickness was 2 mm. A total of five scans of the mandibular joint were taken. The distance of the incisor teeth of 0, 6, 9, 12 and 15 mm was kept constant by an individually shaped form of Permagum. The form at the same time served as a bolus. During the acquisition period the subject was asked to exert a constant pressure on the bolus. The MRT-scans were processed by an image processing system (Adobe Photoshop 3.0). The following parameters (Fig. 2) are determined from the position of the articular surfaces: (1) A tangent T is attached to that particular location on the articular tuberculum, which is nearest to the condyle. (2) The momentary point of rotation D in the articular condyle is located with the help of the three secants S1–S3. (3) The line between T and D indicates the orientation of the bearing load A on the articular tuberculum. (4) The lengths of the sections a and b on both sides of the intersection point of A are measured. The sections together with the articular disc contribute to the force transmission in the mandibular joint. (5) The following force components are calculated from the parameters a, b and A for
ARTICLE IN PRESS Biomechanical analysis of stress distribution in the temporomandibular joint
331
slightly eccentric position of PRA the stress diagram shows a stress value on Pa, which is almost twice that on Pb. The joint is subjected to higher stresses. If PRA assumes an extremely eccentric position, the stress values are very high on Pa while they are almost equal to zero on Pb. Especially high and eccentric stresses lead to destruction of the articular cartilage.
Results
Figure 2. Scheme of the measurement of the MRTimages for determining the parameters of the stress distribution analyses: T ¼ tangent attached to the temporal surface, S1–S3 secants for determining the momentary point of rotation D of the head of the mandible, PRA ¼ resultant bearing force on the articular tuberculum, the connection between D and T determines the direction of PRA, a and b ¼ sections for the force transmission to the articular tuberculum.
formulating the stress profile: P a ¼ A2ð2b aÞ=ða þ bÞ2 as lateral border of section a,
P b ¼ A2ð2a bÞ=ða þ bÞ2 as lateral border of section b,
PRA ¼ A4ða3 b3 Þ=ða þ bÞ4 , as force component at the intersection point of the bearing force A. These three approaches take into account the hypothesis of Navier (Schreyer, 1957), according to which the increase in all force components is linearly limited in case of an eccentric position of A. Since the absolute value of the bearing force A is unknown, it is replaced by the unit vector. The absolute value is thereby changed to a relative value, which still allows for determination the kind of stress distribution in the joint. The force components PI are further divided into their normal components and projected on the circular outline to present the stress distribution. The results of the analyses of the stress distribution are differentiated with respect to the position of PRA. In case of a centred position of PRA the normal components PIN (Fig. 3a) are almost equal, leading to an even stress distribution. In case of a
In the first analysis, the stress distribution in the mandibular joint is calculated and graphically presented (Fig. 3a and b) in complete occlusion (distance between upper and lower incisors: 0 mm). The condyle lies on the posterior and lower third of the articular tuberculum; the bearing force runs slightly inclined in a rostro-occipital direction. The sections are a ¼ 65 units and b ¼ 51 units. The stress distribution is slightly eccentric and increases in a caudo-ventral direction. In the second analysis, the distance between the incisors is 6 mm; the condyle is positioned anterior to and below the articular tuberculum (Fig. 3c). The sections are a ¼ 72 units and b ¼ 38 units. The stress distribution is slightly eccentric and increases in a caudo-ventral direction. In the third analysis (Fig. 4), the distance between the incisors is 9 mm; the condyle has almost the same position as in the second analysis. The sections are a ¼ 51 and b ¼ 42 units. The stress distribution is slightly eccentric on a relatively higher level of stress values and increases in a caudo-ventral direction as well. In the fourth analysis (Fig. 5), the distance between the incisors is 12 mm; the condyle has passed the lower vertex on the articular tuberculum ventrally. The sections are a ¼ 58 units and b ¼ 55 units. The stress distribution is almost central; a slight increase is observed in a ventral direction. In the fifth analysis (Figs. 6–8), the distance between the incisors is 15 mm; the condyle has the same position as in the fourth analysis. The sections are a ¼ 62 units and b ¼ 58 units. The stress distribution is almost centre and increases slightly ventrally.
Discussion The present biomechanical analysis of the loading conditions in the mandibular joint is based on the assumption that the head of the mandible is
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Figure 3. (a) Example for a centric strain of the joint. The intersection of the resultant bearing force PRA divides the bearing line ab into two sections a and b of equal length. This leads to an even stress distribution of the force components PI (stress profile on the baseline) and PIN (stress profile on the circular outline): Pa and Pb ¼ lateral borders of the stress profile. (b) Example of a slightly eccentric stress distribution. Section a is shorter than section b. PRA does not meet ab in the centre. (c) Example of a highly eccentric stress distribution. PRA is clearly eccentric; section a is considerably shorter. The values of the force components PI increase towards PA.
supported by the articular tuberculum during occlusion. The bearing force is the resultant of the forces of all muscles inserting in the mandibular joint. It causes a pressure in the moment of occlusion that is transmitted to the articular tuberculum through the articular disc. The assumptions made are supported by in vivo measurements in the mandibular joint and the condyle in macacas (Hylander, 1979; Boyd et al., 1990), which showed considerable pressure forces. For the presented analyses of the stress distribution it is essential to know the parameters for the direction of the resulting bearing force RA as well as the lengths of the sections a and b. They can be determined from the MRT-scans of the mandibular joint for the different distances of the incisors of 0, 6, 9, 12 and 15 mm with the help of the rules of
Pauwels (1965, 1973) and Kummer (1968, 1985). The five stress analyses show that the mandibular joint is subjected to a more or less eccentric stress distribution in all positions. Under eccentric stress, the resultant force in the joint has a caudo-ventral direction. Since our investigation did not include an exact measurement of the pressure force in the mandibular joint, the pressure force is uniformly assumed as 100%. This method is justified because it allows for evaluation of the kind of loading (centre or eccentric) for the stress analysis when the sections a and b are known for the force transmission. Higher or lower forces only shift the level of the stress profile but do not change the type of stress distribution. As the positions 1–5 show, the force transmission between the corresponding components takes place in a
ARTICLE IN PRESS Biomechanical analysis of stress distribution in the temporomandibular joint
Figure 4. Position of the head of the mandible in complete occlusion (incisor distance: 0 mm). T ¼ tangent to the temporal surface, D momentary point of rotation of the head of the mandible, sections a and b for force transmission to the articular tuberculum. PRA ¼ resultant bearing force on the articular tuberculum. See below: Stress distribution on the articular tuberculum incomplete occlusion (incisor distance: 0 mm). The intersection of the resultant bearing force PRA divides the line ab into two sections a and b causing a slightly eccentric stress distribution of the force components PI (in the stress profile on the base line) and PIN (in the stress profile on the circular outline). PA and PB ¼ lateral borders of the stress profile.
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Figure 6. Position of the head of the mandible for an incisor distance of 9 mm. For further explanations see Fig. 4a. See below: Stress distribution on the articular tuberculum for an incisor distance of 9 mm. The stress values increase slightly caudo-ventrally.
Figure 7. Position of the head of the mandible for an incisor distance of 12 mm. For further explanations see Fig. 4a. See below: Stress distribution on the articular tuberculum for an incisor distance of 12 mm. The stresses increase slightly caudo-ventrally. Figure 5. Position of the head of the mandible for an incisor distance of 6 mm. For further explanations see Fig. 4a. See below: Stress distribution on the articular tuberculum for an incisor distance of 6 mm. The stress values increase slightly caudo-ventrally.
comparatively small area in the lower region of the articular tuberculum, which is referred to as major force-transmitting zone by Kubein-Meesenburg et al. (1988).
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R. Breul
Figure 9. 2 mm section of a tuberculum articulare with pronounced subchondral plate and trabecular structures, V ¼ ventral, d ¼ dorsal, F ¼ Fossa mandibulae.
Figure 8. Position of the head of the mandible for an incisor distance of 15 mm. For further explanations see Fig. 4a. See below: Stress distribution on the articular tuberculum for an incisor distance of 15 mm. The stresses increase slightly caudo-ventrally.
Dos Santos (1995) came to the result that of an assumed total force of 100%, 31% have to be attributed to the mandibular joint and 69% to the occluding teeth. Observations of occlusion splints led to the same conclusions. No analysis, however, succeeded in proving a complete unloading of the mandibular joint under equilibrium conditions. Klesper et al. (1987, 1989) found clear densitometric indications that in more than 98% of the mandibular joints investigated the subchondral bone showed the highest density in this region. One personally made finding (Fig. 9) shows as well that there is a pronounced subchondral plate, to which some trabecular structures lead, which transmit the pressure forces from the articular tuberculum to the cranial base. Takamura et al. (1980) and Oberg et al. (1971) observed distinct arthrotic changes in this region with a thinning out and perforations of the disc as a sign of high stress. Koop (1976) was able to find glycosaminoglycans in the anterior part of the articular disc, which only occur in pressure-exposed parts of joints. Our analyses as well as the results of the investigations cited above make clear that the mandibular joint is subjected to pressure forces during occlusion as well as during mastication. The specific geometrical conditions of the articular surfaces in the different positions during the
process of occlusion lead to a more or less eccentric stress distribution in the joint (Fig. 9). Despite the results presented, it has to be taken into account that the mandibular joint is, in the mechanical connotation, part of an underdetermined system because of the masticatory muscles inserting in the mandible. In such a system the bearing force in the mandibular joint can, according to Molitor (1969), be equal to zero in the process of occlusion since certain combinations of forces can free the joint from forces. Observations in patients, having the head of the mandible resected, support this finding. These patients are able to develop a physiologically appropriate masticatory pressure. The masticatory apparatus seems to be able to fulfil this special task to a variable extent because of its exceptional muscular construction. The two functional states mentioned in the introduction (Kubein-Meesenburg et al., 1990; Na ¨gerl et al., 1991) refer to this special ability.
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