Technology and Health Care, 2 (1994) 193-207 0928-7329/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved
193
Three-dimensional animation of the temporomandibular joint M. Krebs
a,
L.M. Gallo
a,*,
R.L. Airoldi
a, D. Meier b, P. Boesiger b, S. Palla a
Clinic for Masticatory Disorders and Complete Dentures, Center for oral medicine, University of Zurich, Plattenstrasse 11, CH-8028 Zurich, Switzerland Institute of Biomedical Engineering and Medical1nformatics, University of Zurich and Swiss Federal Institute of Technology, Gloriastrasse 35, CH-8092 Zurich, Switzerland
a
b
Received 8 July 1994; accepted 12 July 1994
Abstract The aim of this investigation was to develop a method to reconstruct three-dimensionally the temporomandibular joint (TMJ) by means of magnetic resonance (MR) tomograms and to combine this reconstruction with jaw motion data, recorded with six degrees of freedom by means of the opto-electronic system Jaws-3D, in order to analyse the movement of the whole condyle within the fossa during opening and closing, protrusive and laterotrusive movements. The three-dimensional reconstruction of the TMJ was calculated and animated on a graphics workstation. The condyle - fossa distance during jaw movements was computed for every condylar point and visualized by shading the surface of the condyle with pseudo colors. Furthermore, the location of the minimum condyle-fossa distance was calculated and displayed in a plane graph representing the condylar surface. For five subjects without any past or present history of myoarthropathies of the masticatory system the resulting patterns were analyzed and compared. Key words: Temporomandibular joint; Biomechanics; Condylar movements; Magnetic resonance imaging; 3D image analysis
1. Introduction
Condylar movements have been studied extensively in the past by means of jaw tracking devices [1-8], high-speed cinematography [9,10], fluoroscopy [9-11] and more recently magnetic resonance (MR) [12,13]. All these methods have several limitations. Jaw tracking devices allow us to
* Corresponding
author. Tel. +41-1-257 3226/3231, Fax
+41-1-2620268.
SSDI0928-7329(94)00021-2
reconstruct the movement of preselected condylar points but not the shape of the fossa. Thus, in case of abnormal condylar paths it can be very difficult to diagnose whether these are caused by irregularities in the fossa or, for instance, by movements irregularities caused by the disc. Moreover, the use of a single, intracondylar point does not allow us to deduce the movement of the whole condyle as this has an asymmetrical shape. The movement of the condyle relative to the fossa can be represented by means of fluoroscopy but only in one plane, unless condylar movements are
194
recorded simultaneously with a lateral and an antero-posterior projection. However, fluoroscopy cannot be used to analyse condylar movements in healthy individuals because of the hazard of X-ray exposure. The use of high-speed cinematography is limited to cadaver studies. Modern MR imaging methods such as Echo Planar Imaging produce a dynamic image of a section but not of the whole spatial structure [14]. Pseudo-dynamical imaging of the temporomandibular joint is obtained by taking a series of· tomographic images with the mandible held at different degrees of mouth opening [15-17]. The images are stored on a video tape and then played back sequentially like a cartoon. Despite the beautiful animations obtained, yielding supplementary information about the movement of the disc, this method remains two-dimensional and does not allow us to analyse the movement of the whole condyle in the fossa. Furthermore, it is not suitable for studying functional movements: transient processes, such as those occurring during clicking, may be lost. Three-dimensional modeling has been used to model the joint structures and thus to describe their form and spatial orientation [13,18,19]. First attempts to combine the motion data obtained by means of a mandibular tracking system with the structural information recorded with a tomographic system have been presented [20,21]. This approach allowed us for the first time to study three-dimensionally and dynamically the relationship of the whole joint surfaces to each other. Since the movement of the mandible is a complex process consisting of rotatory and translatory movements and the unloaded mandible can be considered as a quasi-rigid body, motion recordings to animate the three-dimensionally reconstructed joints require a tracking system with 6 degrees of freedom (3 rotations and 3 translations). The aim of this study was to develop a method to combine the motion data obtained with the opto-electronic recording system Jaws-3D [4,5,8, 22-24] with a set of magnetic resonance sections of the temporomandibular joint (TMJ) in order to obtain a three-dimensional reconstruction and animation of the TMJ bone structures. Since with this approach movement recording is independent
M. Krebs et al. / Technology and Health Care 2 (1994) 193 - 207
of image recording, different motion data can be combined with the same set of MR images to study a variety of functional jaw movements. 2. Experimental procedures
2.1. Subjects
To test the validity of the method, condylar movements were recorded in five fully dentate subjects (3 males and 2 females, age 20 to 24 years) without any past or present history of myoarthropathies of the masticatory system (MAP) - i.e. of craniomandibular disorders. To exclude a myoarthropathy the subjects, selected from the staff and students of the dental school, had to fill out a questionnaire. They were asked whether they have or have had pain or sounds in the TMJ, pain and/or fatigue in the masticatory muscles, impaired jaw mobility, facial pain, headache and toothache. The subjects free of anamnestic symptoms were examined clinically by the third author who also made the motion recordings. The examination included measurement of the mandibular mobility (active and passive), palpation and auscultation of the temporomandibular joints and palpation of the masticatory, neck and shoulder muscles. The occlusion was examined for wear facets. When these were present the provocation test [25] was performed. The criteria for a subject to be included in the study were: 1. pain-free active mouth opening larger than 40 mm (including overbite), protrusion and laterotrusion larger than 7 mm [26], 2. difference between active and passive opening of less than 2 mm [27-29], 3. TMJ, masticatory and neck muscles not tender to palpation, 4. absence of TMJ sounds and 5. a negative provocation test. The clinical examination has been described in detail [8,30]. An informed verbal consent to participate in the study was obtained from all subjects. 2.2. MR imaging
MR imaging was performed on a Philips 1.5 T system Gyroscan ACS-II (Philips Medical Systems, NL-5680 DA Best, Netherlands). First, scout
195
M. Krebs et al. / Technology and Health Care 2 (]994) 193 - 207
images with 5 transversal slices with 6 mm thickness and 1 mm gap were used to localize the TMJ. In a second step oblique sagittal slices were planned perpendicular to the longitudinal condilar axis. The caudal-cranial angle of these sections varied between 10 and 15°. This orientation provided a good contour definition for the reconstruction of the joint and an optimal geometrical resolution in the plane of the major mandibular movement. The interesting volume, i.e. the TMJ and the extraorally placed spheres, used as a geometrical reference system to combine the motion data with the MR images, was divided into two stacks. The TMJ stack consisted of 14 contiguous slices while that of the reference spheres of 12 contiguous slices. All slices were 2 mm thick. With a field of view of 10 cm by 10 cm and an initial resolution of 128 by 256, pixel sizes of 0.8 mm by 0.4 mm were measured using a pair of circular surface coils with a diameter of 12 cm. The pixels were interpolated to a final resolution of 256 by 256. Optimal contrast between bone and soft tissues was achieved using an FFE sequence (gradient recalled echo) with a repetition time of 300 ms, an echo time of 9 ms, an excitation angle of 20° and 6 averages. The total examination time for one joint and the corresponding reference system was about 8 minutes. 2.3. Movement tracking system To record mandibular motion, the opto-electronic tracking system Jaws-3D [5,8,22-24] was used. This allows data acquisition with six degrees of freedom, considering the mandibula as a rigid body. The system consists of three one-dimensional CCD cameras with 2048 array elements connected by means of an adapter board to an IBM compatible personal computer. The cameras determined the spatial position of two triplets of LEDs, the so called triangular target frames, disposed at the vertices of two triangles rigidly connected to the maxillary and mandibular dental arches by means of custom made metal splints (Fig. 1). These were fixed to the buccal surface of the canine and first premolar and did not interfere with the dental occlusion [5]. The maxillary LEDs defined a head-related co-ordinate system
Fig. 1. Subject with triangular target frames.
or frame {H} and the mandibular LEDs a mandible-related frame {M}. This setup permitted the determination of absolute movements of the head ITH(t) as well as of the mandible ITM(t) in a global frame {J}. The homogeneous coordinate transforms ATB are matrices describing rotation as well as translation [31] of the type: A BrU
A BR12
A B r 13
A
A BR21
A B r 22
A B r 23
A
A B r 31
A B r 32
A B r 33
A
0
0
0
PBORGl PBORG2 PBORG3
1
where the columns of the orthonormal 3 X 3 submatrix ARB with the elements BArij represent the unit vectors of the frame {B} expressed with respect to {A} and the three-dimensional vector
M. Krebs et at. / Technology and Health Care 2 (1994) 193 - 207
196
with the elements ApBORGi represents the origin of {B} in {A}. Thus, a point Bp in frame {B} is expressed in {A} as A P by means of the matrix multiplication
A PBORG
A
o
P BORG
1
) . (B[ )
By computing the relative motion of the mandible with respect to the head, the head movements are eliminated and the mandibular movement obtained: HTM(t)
=HTAt) .ITM(t) = (ITH(t)r 1 .ITM(t)
Fig. 2. Monobloc with the reference bars.
(1)
For every sample time t; - corresponding to one position measurement of the tracking system - a transform matrix HTM(t) was computed, describing the relative position of the mandible with respect to the head. The sequence of transform matrices represented the whole movement with a sampling rate of 70 Hz. By using this equation the trajectory of any mandibular point could be calculated relatively to the head co-ordinate system
(2) The co-ordinates of the mandibular points relative to the mandibular frame {B} are represented by the vector P.
2.4. Combination of form and movement For the combination of the MR images set with the Jaws-3D movement recordings, all MR and Jaws-3D data must be expressed in the same frame. Thus, a homogeneous co-ordinate transform HTMR , which transformed the contours of condyle and fossa obtained from the MR images, was needed to express all data in the Jaws-3D frame {H}. This \\-as obtained by means of a third reference frame common to the MR as well as to the Jaws-3D frame. Three non collinear landmarks, which could be identified in the MR images as well as with the tracking system Jaws-3D,
were used to define the common reference system. Since no adequate anatomic reference points exist, an artificial reference system was used. The main part of this system consisted of three PVC (polyvinylchloride) spheres of 10 mm diameter which were filled with MR contrast medium Magnevist (Schering AG, D-13342 Berlin) with a dilution of 0.5 millimol per liter. The spheres were arranged into a triangle with vertices of 30 mm side length between the centers and mounted on a round carbon fiber bar with 8 mm diameter (Art. 100.0006, Suter Kunststoffe AG, CH-3303 Jegenstorf, Switzerland) (Fig. 2). The carbon fiber carried also a triangular target frame with LEDs which was attached in a repeatable way to the bar by means of a precision attachment (McCollum 22.03, Cendres and Metaux SA, CH-2501 Biel, Switzerland). This construction was called the reference bar. Two reference bars (one for each side) were connected together by means of a face bow made of acrylic and carbon fibers. This was fixed to an acrylic monobloc-like splint (Fig. 2) used to secure the reference bar to the dental arches. The individually adjustable face bow allowed us to position the reference prism laterally to the TMJ (Fig. 3). In order to determine the spatial relationship between the reference spheres and the LED triangular target frame, the reference bar was calibrated by measuring the spatial position of the centers of the spheres and of the LEDs by means of a three-dimensional coordinate measuring machine (UMM 850, Carl
M. Krebs et af. / Technology and Health Care 2 (J994) 193 - 207
197
Zeiss, D-73447 Oberkochen, Germany) with a precision of 0.5 JLm. The center of the spheres defined a new frame {S} while the LED-triangle defined a co-ordinate system {L}. By using the MR images and the calibration data, the constant transforms STMR(t MR ) between the spheres frame {S} and the MR image frame at MR recording time and LTS between {S} and {L} were calculated. Since the MR images were taken with the subject biting on the monoblock the mandible was in a slightly open position. Thus, the position of the mandible relative to the maxilla had to be determined prior to the jaw movements recording. This was done as follows. Two metal splints carrying the maxillary and mandibular triangular target frame were glued by means of cyanoacrylate (Sekundenkleber No. 1733, Renfert GmbH and
Fig. 4. Subject with triangular target frames and monobloc with reference bars.
Fig. 3. Subject with the monobloc. The reference spheres are positioned laterally to the TMJ.
Co., D-78224 Singen) to the upper and lower teeth respectively. The monoblock was inserted and the subject was asked to bite into it (Fig. 4). The relative position HTM(t MR ) of the condyle with respect to the fossa was determined by recording the position of the maxillary and mandibular LEDs by means of Jaws-3D. Thereafter the upper LED target frame was removed and fixed to the reference bar and the position of the reference LEDs with respect to the mandibular LEDs was recorded. This registration allowed to compute MTL. Finally, the monoblock was removed and the subject was asked to perform the jaw movements. For each set of LED co-ordinates, delivered every 14 ms, a transform matrix HTM(t) was computed, describing for every sample time the relative position of the condyle to the fossa. Therefore, the transformation from the MR to
M. Krebs et al. / Technology and Health Care 2 (1994) 193 - 207
198
the Jaws-3D co-ordinate system is the product of four partial transforms HT -HT .MT .LT .sT MR M L S MR'
(3)
If all data must be described in the head-related Jaws-3D-system {H}, the use of (3) gives
(4) In order to reconstruct the static - i.e. the not animated - points of the fossa, the transform HTM(t MR ), obtained by the Jaws-3D static recording with the subject biting on the monoblock, was used. Animated condylar points were transformed by means of the time dependent matrix HTM(t). 2.5. Image processing
The determination of the centers of the reference spheres was straightforward, since their contours were very well defined as the spheres were filled with MR contrast medium. The segmentation was done automatically. After manual choice of one of the three reference spheres, the region of interest was convoluted with a three-dimensional Gauss mask G:
(5)
F=I®G.
The three-dimensional G(x,y,zer) was defined as
Gauss
distribution
and was implemented as a series of one-dimensional convolutions. The width of the gaussian curve was chosen with er corresponding to the radius of the reference spheres, expressed in number of pixels. For an image size of 256 pixels, a field of view of 100 mm and a sphere diameter of 10 mm, this yielded a er value of 13. In order to obtain the same resolution in z direction as in the image planes parallel to xy, the step width to displace the convolution mask was equalized to that in x and y direction. A following maximum search in the convoluted image F yielded the co-ordinates of the required sphere center. To accelerate the process and diminish memory allo-
Fig. 5. MR image taken with the FFE technique. The bony outline of fossa and condyle contrast well with the intra- and periarticular soft tissues.
cation, the maximum search was performed immediately after the convolution of each plane. For the extraction and vectorial description of the surfaces of condyle and fossa as well as for the determination of the centers of the reference spheres a segmentation of the MR images was necessary. MR images are typically inadequate to represent compact bone structures because these generate almost no echo signal with respect to soft tissues: while the fossa appears dark, the condyle shows a thin dark contour corresponding to the compact bone with the cancellous bone appearing lighter inside (Fig. 5). For these reasons a contour editor was implemented, in which the curves were defined by specifying manually driving points through which spline functions were drawn. The driving points of the spline functions yielded the vector description of condyle and fossa, from which the surfaces were reconstructed by triangulation [32]. 2.6. Experimental protocol (A) Error in spheres localization In order to investigate the validity of the algorithms to determine the centers of the reference
199
M Krebs et al./Technology and Health Care 2 (1994) 193-207
spheres on the MR images, the reference system alone was imaged four times with angles of 0, 20, 40 and 60° between the MR slices and the plane defined by the three spheres. The distances between the centers of the spheres were computed and averaged over the four recordings. Comparison of the mean distances with the exact distances of 30.0 mm provided the geometrical error of the MR system. (B) Error in monobloc repositioning To test the accuracy in repositioning the monoblock one LEDs triangular target frame (XYZ co-ordinate system) was fixed to the reference bar and another triangular target frame (UVW co-ordinate system) to the lower metal splint. With the subject biting in maximum intercuspation the spatial orientation between the LEDs was measured 30 times with Jaws-3D. Each recording lasted 5 s. The LEDs positions were averaged over this period. Between the recordings the monoblock was removed and repositioned. The standard deviation of the co-ordinates of the origins of the UVW co-ordinate system as well as the standard deviation of the orientation angles of the co-ordinate system expressed the error in the monoblock repositioning. The orientation angles a, f3 and 'Y were determined considering first a rotation of the XYZ system around the Z, then around the Y and at last around the X-axis to reach the orientation of the UVW system [31]. Finally, the position in the XYZ system of a preselected condylar point with UVW co-ordinates 20, 94 and -18 mm respectively was calculated for each trial (for the preselection procedure see [5] and [8]). These co-ordinates measured the influence of the monoblock repositioning error in the condylar area. (C) Condylar movements At first MR images of the TMJ and of the reference spheres were taken with the face bow in place and the subject biting into the monoblock splint using the previously described protocol. Thereafter the monobloc was removed and the subject was taken to the dental school where the previously adjusted metal splints were glued to the upper and lower teeth by means of cyanoacry-
late (Sekundenkleber No. 1733, Renfert GmbH and Co., D-78224 Singen). Mter attaching the LED triangular target frames to the upper and lower metal splint, the monobloc with the face bow and the reference bar was reinserted into the mouth and the subject asked to firmly bite on it. The co-ordinates of the maxillary and mandibular LEDs triangular target frames in relation to the reference system were recorded by means of Jaws-3D as previously described. The monobloc was removed and the subject, who sat in a dental chair without head rest, was asked to perform 10 series of 5 opening/closing movements, 10 lateral movements (laterotrusions) to the right side, 10 to the left side and 10 protrusive movements. The opening/ closing movements had to start and end in maximum intercuspation (maximum tooth contact position) and were performed with a 1 Hz frequency i.e. one open-close cycle in one second. The velocity of the open-close cycle was paced by a vertical LED array placed in front of the subject. In each series of 5 open-close movements only the third or fourth one was recorded i.e. the first cycle performed at the paced velocity. Thus, 10 open-close cycles were recorded for analysis. The laterotrusive and protrusive movements were performed at deliberate speed, i.e. without pacing, with the teeth slightly separated. 2. 7. Data analysis
The distances between the surfaces of condyle and fossa were used to analyse variations in the joint space as a function of time. The functions Xi and Zi were calculated for every time step ti as a weighted average of the co-ordinates X ji and Z ji along the condylar anteroposterior axis and the main axis respectively - of the points Pi nearer than the minimum distance min plus a c constant, chosen as 2 mm. The weighting was inversely proportional to the square of the distance d ji :
E Xi
=
Xji
. d2. J JI . I ' "lor every J• such th at d ji < mm
Edh j
+ c.
200
The same formula was used for the z-coordinate. To visualize how the position of the minimum distance between condyle and fossa changed during jaw movements, the computed value was projected for every movement step on the condylar surface and plotted graphically in a planar coordinate system in which the medio-Iateral and antero-posterior co-ordinates of the condyle were displayed versus time. The medio-lateral axis was parallel to the main condylar axis and the antero-posterior axis perpendicular to it. The origin of the co-ordinate system was given by the position of the minimum distance between condyle and fossa at the beginning of the movement, e.g. at maximum intercuspation for opening/closing movements. The resulting line provided the path of minimum distance. In order to study the variations in the distance condyle-fossa from mesial to lateral during movements, the minimum distance to the fossa was computed for every triangulation point of the
M. Krebs et al. / Technology and Health Care 2 (1994) 193 - 207
condylar surface and every movement step. This calculation took about 15 min for a movement of one or two seconds and produced a detailed distance map of the condyle-fossa distance in function of the time. Reloading this distance map during the animation allowed a pseudo-coloring of the condyle. Colors from red via yellow to green and blue were used to indicate the distance condyle-fossa in millimeters (0-12 mm). This representation was done statically and dynamically for each recording. The three-dimensional reconstruction of the TMJ was represented on a graphics workstation (IRIS 4D /310GTX, Silicon Graphics, Mountain View, CA 94043) utilizing rendering algorithms and could be interactively rotated in real time. A transparent representation of the fossa allowed an optimal view of the condyle. The computer hardware used was able to draw up to 100000 polygons including z-buffering and Gouraud-shading. By combining the 3-D reconstruction with a Jaws3D movement recording an animation of the TMJ
Fig. 6. Two views of a 3-D model of the temporomandibular joint representing the condyle position at maximum intercuspation i.e. with teeth in maximum contact. Left: top-front aspect. Right: lateral aspect.
M Krebs et al./Technology and Health Care 2 (1994) 193-207
was obtained, which could be played either with full time resolution or after res amp ling in real time with 25 images per second.
201
were 29.35 mm, 30.0 mm and 29.63 mm for the three distances dAB' d AC and d BC (expected distance 30.00 mm). 3.2. Error in monobloc repositioning
3. Results 3.1. Error in spheres localization The distances between the three reference spheres are shown in Table 1. They were calculated from four MR recordings with angles 0, 20, 40 and 60 degrees between the MR slices and the plane defined by the spheres. The mean values Table 1 Measured distances between the centers of the three reference spheres in mm Angle
dAB
d AC
d BC
0° 20° 40° 60° Mean
29.43 29.43 29.49 29.05 29.35
30.11 30.11 30.10 29.67 30.00
29.68 29.68 29.58 29.59 29.63
The standard deviations of the co-ordinates of the origins of the UVW-system in the XYZ-systern were 0.09 mm, 0.11 mm and 0.08 mm for the x o' Yo and Zo co-ordinates respectively. The standard deviations of each orientation angle were 0.24°, 0.08° and 0.24° for ao, f3 0 and 'Yo respectively. These values represented the mean orientation error. With these values the error in the computation of the position of the preselected condylar point with co-ordinates (20/94/ - 18 mm) in the UVW-systems did not exceed 0.13 mm (x cp 0.12 mm, Ycp 0.12 mm and zcp 0.13 mm), giving a mean spatial orientation error of 0.19 mm. I
3.3. Condylar movements Fig. 6 shows the model of the TMJ at maximum intercuspation from a top-front and a lateral as-
Fig. 7. Sequence of selected images of the position of the condyle during an opening and closing sequence. Position of the condyle in maximum intercuspation (beginning of opening) (image 1), at maximum opening (image 4) and again in maximum intercuspation (end of closing) (image 8).
202
M Krebs et al. / Technology and Health Care 2 (J994) 193 - 207 anterior - posterior
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193
Three-dimensional animation of the temporomandibular joint M. Krebs
a,
L.M. Gallo
a,*,
R.L. Airoldi
a, D. Meier b, P. Boesiger b, S. Palla a
Clinic for Masticatory Disorders and Complete Dentures, Center for oral medicine, University of Zurich, Plattenstrasse 11, CH-8028 Zurich, Switzerland Institute of Biomedical Engineering and Medical1nformatics, University of Zurich and Swiss Federal Institute of Technology, Gloriastrasse 35, CH-8092 Zurich, Switzerland
a
b
Received 8 July 1994; accepted 12 July 1994
Abstract The aim of this investigation was to develop a method to reconstruct three-dimensionally the temporomandibular joint (TMJ) by means of magnetic resonance (MR) tomograms and to combine this reconstruction with jaw motion data, recorded with six degrees of freedom by means of the opto-electronic system Jaws-3D, in order to analyse the movement of the whole condyle within the fossa during opening and closing, protrusive and laterotrusive movements. The three-dimensional reconstruction of the TMJ was calculated and animated on a graphics workstation. The condyle - fossa distance during jaw movements was computed for every condylar point and visualized by shading the surface of the condyle with pseudo colors. Furthermore, the location of the minimum condyle-fossa distance was calculated and displayed in a plane graph representing the condylar surface. For five subjects without any past or present history of myoarthropathies of the masticatory system the resulting patterns were analyzed and compared. Key words: Temporomandibular joint; Biomechanics; Condylar movements; Magnetic resonance imaging; 3D image analysis
1. Introduction
Condylar movements have been studied extensively in the past by means of jaw tracking devices [1-8], high-speed cinematography [9,10], fluoroscopy [9-11] and more recently magnetic resonance (MR) [12,13]. All these methods have several limitations. Jaw tracking devices allow us to
* Corresponding
author. Tel. +41-1-257 3226/3231, Fax
+41-1-2620268.
SSDI0928-7329(94)00021-2
reconstruct the movement of preselected condylar points but not the shape of the fossa. Thus, in case of abnormal condylar paths it can be very difficult to diagnose whether these are caused by irregularities in the fossa or, for instance, by movements irregularities caused by the disc. Moreover, the use of a single, intracondylar point does not allow us to deduce the movement of the whole condyle as this has an asymmetrical shape. The movement of the condyle relative to the fossa can be represented by means of fluoroscopy but only in one plane, unless condylar movements are
194
recorded simultaneously with a lateral and an antero-posterior projection. However, fluoroscopy cannot be used to analyse condylar movements in healthy individuals because of the hazard of X-ray exposure. The use of high-speed cinematography is limited to cadaver studies. Modern MR imaging methods such as Echo Planar Imaging produce a dynamic image of a section but not of the whole spatial structure [14]. Pseudo-dynamical imaging of the temporomandibular joint is obtained by taking a series of· tomographic images with the mandible held at different degrees of mouth opening [15-17]. The images are stored on a video tape and then played back sequentially like a cartoon. Despite the beautiful animations obtained, yielding supplementary information about the movement of the disc, this method remains two-dimensional and does not allow us to analyse the movement of the whole condyle in the fossa. Furthermore, it is not suitable for studying functional movements: transient processes, such as those occurring during clicking, may be lost. Three-dimensional modeling has been used to model the joint structures and thus to describe their form and spatial orientation [13,18,19]. First attempts to combine the motion data obtained by means of a mandibular tracking system with the structural information recorded with a tomographic system have been presented [20,21]. This approach allowed us for the first time to study three-dimensionally and dynamically the relationship of the whole joint surfaces to each other. Since the movement of the mandible is a complex process consisting of rotatory and translatory movements and the unloaded mandible can be considered as a quasi-rigid body, motion recordings to animate the three-dimensionally reconstructed joints require a tracking system with 6 degrees of freedom (3 rotations and 3 translations). The aim of this study was to develop a method to combine the motion data obtained with the opto-electronic recording system Jaws-3D [4,5,8, 22-24] with a set of magnetic resonance sections of the temporomandibular joint (TMJ) in order to obtain a three-dimensional reconstruction and animation of the TMJ bone structures. Since with this approach movement recording is independent
M. Krebs et al. / Technology and Health Care 2 (1994) 193 - 207
of image recording, different motion data can be combined with the same set of MR images to study a variety of functional jaw movements. 2. Experimental procedures
2.1. Subjects
To test the validity of the method, condylar movements were recorded in five fully dentate subjects (3 males and 2 females, age 20 to 24 years) without any past or present history of myoarthropathies of the masticatory system (MAP) - i.e. of craniomandibular disorders. To exclude a myoarthropathy the subjects, selected from the staff and students of the dental school, had to fill out a questionnaire. They were asked whether they have or have had pain or sounds in the TMJ, pain and/or fatigue in the masticatory muscles, impaired jaw mobility, facial pain, headache and toothache. The subjects free of anamnestic symptoms were examined clinically by the third author who also made the motion recordings. The examination included measurement of the mandibular mobility (active and passive), palpation and auscultation of the temporomandibular joints and palpation of the masticatory, neck and shoulder muscles. The occlusion was examined for wear facets. When these were present the provocation test [25] was performed. The criteria for a subject to be included in the study were: 1. pain-free active mouth opening larger than 40 mm (including overbite), protrusion and laterotrusion larger than 7 mm [26], 2. difference between active and passive opening of less than 2 mm [27-29], 3. TMJ, masticatory and neck muscles not tender to palpation, 4. absence of TMJ sounds and 5. a negative provocation test. The clinical examination has been described in detail [8,30]. An informed verbal consent to participate in the study was obtained from all subjects. 2.2. MR imaging
MR imaging was performed on a Philips 1.5 T system Gyroscan ACS-II (Philips Medical Systems, NL-5680 DA Best, Netherlands). First, scout
195
M. Krebs et al. / Technology and Health Care 2 (]994) 193 - 207
images with 5 transversal slices with 6 mm thickness and 1 mm gap were used to localize the TMJ. In a second step oblique sagittal slices were planned perpendicular to the longitudinal condilar axis. The caudal-cranial angle of these sections varied between 10 and 15°. This orientation provided a good contour definition for the reconstruction of the joint and an optimal geometrical resolution in the plane of the major mandibular movement. The interesting volume, i.e. the TMJ and the extraorally placed spheres, used as a geometrical reference system to combine the motion data with the MR images, was divided into two stacks. The TMJ stack consisted of 14 contiguous slices while that of the reference spheres of 12 contiguous slices. All slices were 2 mm thick. With a field of view of 10 cm by 10 cm and an initial resolution of 128 by 256, pixel sizes of 0.8 mm by 0.4 mm were measured using a pair of circular surface coils with a diameter of 12 cm. The pixels were interpolated to a final resolution of 256 by 256. Optimal contrast between bone and soft tissues was achieved using an FFE sequence (gradient recalled echo) with a repetition time of 300 ms, an echo time of 9 ms, an excitation angle of 20° and 6 averages. The total examination time for one joint and the corresponding reference system was about 8 minutes. 2.3. Movement tracking system To record mandibular motion, the opto-electronic tracking system Jaws-3D [5,8,22-24] was used. This allows data acquisition with six degrees of freedom, considering the mandibula as a rigid body. The system consists of three one-dimensional CCD cameras with 2048 array elements connected by means of an adapter board to an IBM compatible personal computer. The cameras determined the spatial position of two triplets of LEDs, the so called triangular target frames, disposed at the vertices of two triangles rigidly connected to the maxillary and mandibular dental arches by means of custom made metal splints (Fig. 1). These were fixed to the buccal surface of the canine and first premolar and did not interfere with the dental occlusion [5]. The maxillary LEDs defined a head-related co-ordinate system
Fig. 1. Subject with triangular target frames.
or frame {H} and the mandibular LEDs a mandible-related frame {M}. This setup permitted the determination of absolute movements of the head ITH(t) as well as of the mandible ITM(t) in a global frame {J}. The homogeneous coordinate transforms ATB are matrices describing rotation as well as translation [31] of the type: A BrU
A BR12
A B r 13
A
A BR21
A B r 22
A B r 23
A
A B r 31
A B r 32
A B r 33
A
0
0
0
PBORGl PBORG2 PBORG3
1
where the columns of the orthonormal 3 X 3 submatrix ARB with the elements BArij represent the unit vectors of the frame {B} expressed with respect to {A} and the three-dimensional vector
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196
with the elements ApBORGi represents the origin of {B} in {A}. Thus, a point Bp in frame {B} is expressed in {A} as A P by means of the matrix multiplication
A PBORG
A
o
P BORG
1
) . (B[ )
By computing the relative motion of the mandible with respect to the head, the head movements are eliminated and the mandibular movement obtained: HTM(t)
=HTAt) .ITM(t) = (ITH(t)r 1 .ITM(t)
Fig. 2. Monobloc with the reference bars.
(1)
For every sample time t; - corresponding to one position measurement of the tracking system - a transform matrix HTM(t) was computed, describing the relative position of the mandible with respect to the head. The sequence of transform matrices represented the whole movement with a sampling rate of 70 Hz. By using this equation the trajectory of any mandibular point could be calculated relatively to the head co-ordinate system
(2) The co-ordinates of the mandibular points relative to the mandibular frame {B} are represented by the vector P.
2.4. Combination of form and movement For the combination of the MR images set with the Jaws-3D movement recordings, all MR and Jaws-3D data must be expressed in the same frame. Thus, a homogeneous co-ordinate transform HTMR , which transformed the contours of condyle and fossa obtained from the MR images, was needed to express all data in the Jaws-3D frame {H}. This \\-as obtained by means of a third reference frame common to the MR as well as to the Jaws-3D frame. Three non collinear landmarks, which could be identified in the MR images as well as with the tracking system Jaws-3D,
were used to define the common reference system. Since no adequate anatomic reference points exist, an artificial reference system was used. The main part of this system consisted of three PVC (polyvinylchloride) spheres of 10 mm diameter which were filled with MR contrast medium Magnevist (Schering AG, D-13342 Berlin) with a dilution of 0.5 millimol per liter. The spheres were arranged into a triangle with vertices of 30 mm side length between the centers and mounted on a round carbon fiber bar with 8 mm diameter (Art. 100.0006, Suter Kunststoffe AG, CH-3303 Jegenstorf, Switzerland) (Fig. 2). The carbon fiber carried also a triangular target frame with LEDs which was attached in a repeatable way to the bar by means of a precision attachment (McCollum 22.03, Cendres and Metaux SA, CH-2501 Biel, Switzerland). This construction was called the reference bar. Two reference bars (one for each side) were connected together by means of a face bow made of acrylic and carbon fibers. This was fixed to an acrylic monobloc-like splint (Fig. 2) used to secure the reference bar to the dental arches. The individually adjustable face bow allowed us to position the reference prism laterally to the TMJ (Fig. 3). In order to determine the spatial relationship between the reference spheres and the LED triangular target frame, the reference bar was calibrated by measuring the spatial position of the centers of the spheres and of the LEDs by means of a three-dimensional coordinate measuring machine (UMM 850, Carl
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197
Zeiss, D-73447 Oberkochen, Germany) with a precision of 0.5 JLm. The center of the spheres defined a new frame {S} while the LED-triangle defined a co-ordinate system {L}. By using the MR images and the calibration data, the constant transforms STMR(t MR ) between the spheres frame {S} and the MR image frame at MR recording time and LTS between {S} and {L} were calculated. Since the MR images were taken with the subject biting on the monoblock the mandible was in a slightly open position. Thus, the position of the mandible relative to the maxilla had to be determined prior to the jaw movements recording. This was done as follows. Two metal splints carrying the maxillary and mandibular triangular target frame were glued by means of cyanoacrylate (Sekundenkleber No. 1733, Renfert GmbH and
Fig. 4. Subject with triangular target frames and monobloc with reference bars.
Fig. 3. Subject with the monobloc. The reference spheres are positioned laterally to the TMJ.
Co., D-78224 Singen) to the upper and lower teeth respectively. The monoblock was inserted and the subject was asked to bite into it (Fig. 4). The relative position HTM(t MR ) of the condyle with respect to the fossa was determined by recording the position of the maxillary and mandibular LEDs by means of Jaws-3D. Thereafter the upper LED target frame was removed and fixed to the reference bar and the position of the reference LEDs with respect to the mandibular LEDs was recorded. This registration allowed to compute MTL. Finally, the monoblock was removed and the subject was asked to perform the jaw movements. For each set of LED co-ordinates, delivered every 14 ms, a transform matrix HTM(t) was computed, describing for every sample time the relative position of the condyle to the fossa. Therefore, the transformation from the MR to
M. Krebs et al. / Technology and Health Care 2 (1994) 193 - 207
198
the Jaws-3D co-ordinate system is the product of four partial transforms HT -HT .MT .LT .sT MR M L S MR'
(3)
If all data must be described in the head-related Jaws-3D-system {H}, the use of (3) gives
(4) In order to reconstruct the static - i.e. the not animated - points of the fossa, the transform HTM(t MR ), obtained by the Jaws-3D static recording with the subject biting on the monoblock, was used. Animated condylar points were transformed by means of the time dependent matrix HTM(t). 2.5. Image processing
The determination of the centers of the reference spheres was straightforward, since their contours were very well defined as the spheres were filled with MR contrast medium. The segmentation was done automatically. After manual choice of one of the three reference spheres, the region of interest was convoluted with a three-dimensional Gauss mask G:
(5)
F=I®G.
The three-dimensional G(x,y,zer) was defined as
Gauss
distribution
and was implemented as a series of one-dimensional convolutions. The width of the gaussian curve was chosen with er corresponding to the radius of the reference spheres, expressed in number of pixels. For an image size of 256 pixels, a field of view of 100 mm and a sphere diameter of 10 mm, this yielded a er value of 13. In order to obtain the same resolution in z direction as in the image planes parallel to xy, the step width to displace the convolution mask was equalized to that in x and y direction. A following maximum search in the convoluted image F yielded the co-ordinates of the required sphere center. To accelerate the process and diminish memory allo-
Fig. 5. MR image taken with the FFE technique. The bony outline of fossa and condyle contrast well with the intra- and periarticular soft tissues.
cation, the maximum search was performed immediately after the convolution of each plane. For the extraction and vectorial description of the surfaces of condyle and fossa as well as for the determination of the centers of the reference spheres a segmentation of the MR images was necessary. MR images are typically inadequate to represent compact bone structures because these generate almost no echo signal with respect to soft tissues: while the fossa appears dark, the condyle shows a thin dark contour corresponding to the compact bone with the cancellous bone appearing lighter inside (Fig. 5). For these reasons a contour editor was implemented, in which the curves were defined by specifying manually driving points through which spline functions were drawn. The driving points of the spline functions yielded the vector description of condyle and fossa, from which the surfaces were reconstructed by triangulation [32]. 2.6. Experimental protocol (A) Error in spheres localization In order to investigate the validity of the algorithms to determine the centers of the reference
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M Krebs et al./Technology and Health Care 2 (1994) 193-207
spheres on the MR images, the reference system alone was imaged four times with angles of 0, 20, 40 and 60° between the MR slices and the plane defined by the three spheres. The distances between the centers of the spheres were computed and averaged over the four recordings. Comparison of the mean distances with the exact distances of 30.0 mm provided the geometrical error of the MR system. (B) Error in monobloc repositioning To test the accuracy in repositioning the monoblock one LEDs triangular target frame (XYZ co-ordinate system) was fixed to the reference bar and another triangular target frame (UVW co-ordinate system) to the lower metal splint. With the subject biting in maximum intercuspation the spatial orientation between the LEDs was measured 30 times with Jaws-3D. Each recording lasted 5 s. The LEDs positions were averaged over this period. Between the recordings the monoblock was removed and repositioned. The standard deviation of the co-ordinates of the origins of the UVW co-ordinate system as well as the standard deviation of the orientation angles of the co-ordinate system expressed the error in the monoblock repositioning. The orientation angles a, f3 and 'Y were determined considering first a rotation of the XYZ system around the Z, then around the Y and at last around the X-axis to reach the orientation of the UVW system [31]. Finally, the position in the XYZ system of a preselected condylar point with UVW co-ordinates 20, 94 and -18 mm respectively was calculated for each trial (for the preselection procedure see [5] and [8]). These co-ordinates measured the influence of the monoblock repositioning error in the condylar area. (C) Condylar movements At first MR images of the TMJ and of the reference spheres were taken with the face bow in place and the subject biting into the monoblock splint using the previously described protocol. Thereafter the monobloc was removed and the subject was taken to the dental school where the previously adjusted metal splints were glued to the upper and lower teeth by means of cyanoacry-
late (Sekundenkleber No. 1733, Renfert GmbH and Co., D-78224 Singen). Mter attaching the LED triangular target frames to the upper and lower metal splint, the monobloc with the face bow and the reference bar was reinserted into the mouth and the subject asked to firmly bite on it. The co-ordinates of the maxillary and mandibular LEDs triangular target frames in relation to the reference system were recorded by means of Jaws-3D as previously described. The monobloc was removed and the subject, who sat in a dental chair without head rest, was asked to perform 10 series of 5 opening/closing movements, 10 lateral movements (laterotrusions) to the right side, 10 to the left side and 10 protrusive movements. The opening/ closing movements had to start and end in maximum intercuspation (maximum tooth contact position) and were performed with a 1 Hz frequency i.e. one open-close cycle in one second. The velocity of the open-close cycle was paced by a vertical LED array placed in front of the subject. In each series of 5 open-close movements only the third or fourth one was recorded i.e. the first cycle performed at the paced velocity. Thus, 10 open-close cycles were recorded for analysis. The laterotrusive and protrusive movements were performed at deliberate speed, i.e. without pacing, with the teeth slightly separated. 2. 7. Data analysis
The distances between the surfaces of condyle and fossa were used to analyse variations in the joint space as a function of time. The functions Xi and Zi were calculated for every time step ti as a weighted average of the co-ordinates X ji and Z ji along the condylar anteroposterior axis and the main axis respectively - of the points Pi nearer than the minimum distance min plus a c constant, chosen as 2 mm. The weighting was inversely proportional to the square of the distance d ji :
E Xi
=
Xji
. d2. J JI . I ' "lor every J• such th at d ji < mm
Edh j
+ c.
200
The same formula was used for the z-coordinate. To visualize how the position of the minimum distance between condyle and fossa changed during jaw movements, the computed value was projected for every movement step on the condylar surface and plotted graphically in a planar coordinate system in which the medio-Iateral and antero-posterior co-ordinates of the condyle were displayed versus time. The medio-lateral axis was parallel to the main condylar axis and the antero-posterior axis perpendicular to it. The origin of the co-ordinate system was given by the position of the minimum distance between condyle and fossa at the beginning of the movement, e.g. at maximum intercuspation for opening/closing movements. The resulting line provided the path of minimum distance. In order to study the variations in the distance condyle-fossa from mesial to lateral during movements, the minimum distance to the fossa was computed for every triangulation point of the
M. Krebs et al. / Technology and Health Care 2 (1994) 193 - 207
condylar surface and every movement step. This calculation took about 15 min for a movement of one or two seconds and produced a detailed distance map of the condyle-fossa distance in function of the time. Reloading this distance map during the animation allowed a pseudo-coloring of the condyle. Colors from red via yellow to green and blue were used to indicate the distance condyle-fossa in millimeters (0-12 mm). This representation was done statically and dynamically for each recording. The three-dimensional reconstruction of the TMJ was represented on a graphics workstation (IRIS 4D /310GTX, Silicon Graphics, Mountain View, CA 94043) utilizing rendering algorithms and could be interactively rotated in real time. A transparent representation of the fossa allowed an optimal view of the condyle. The computer hardware used was able to draw up to 100000 polygons including z-buffering and Gouraud-shading. By combining the 3-D reconstruction with a Jaws3D movement recording an animation of the TMJ
Fig. 6. Two views of a 3-D model of the temporomandibular joint representing the condyle position at maximum intercuspation i.e. with teeth in maximum contact. Left: top-front aspect. Right: lateral aspect.
M Krebs et al./Technology and Health Care 2 (1994) 193-207
was obtained, which could be played either with full time resolution or after res amp ling in real time with 25 images per second.
201
were 29.35 mm, 30.0 mm and 29.63 mm for the three distances dAB' d AC and d BC (expected distance 30.00 mm). 3.2. Error in monobloc repositioning
3. Results 3.1. Error in spheres localization The distances between the three reference spheres are shown in Table 1. They were calculated from four MR recordings with angles 0, 20, 40 and 60 degrees between the MR slices and the plane defined by the spheres. The mean values Table 1 Measured distances between the centers of the three reference spheres in mm Angle
dAB
d AC
d BC
0° 20° 40° 60° Mean
29.43 29.43 29.49 29.05 29.35
30.11 30.11 30.10 29.67 30.00
29.68 29.68 29.58 29.59 29.63
The standard deviations of the co-ordinates of the origins of the UVW-system in the XYZ-systern were 0.09 mm, 0.11 mm and 0.08 mm for the x o' Yo and Zo co-ordinates respectively. The standard deviations of each orientation angle were 0.24°, 0.08° and 0.24° for ao, f3 0 and 'Yo respectively. These values represented the mean orientation error. With these values the error in the computation of the position of the preselected condylar point with co-ordinates (20/94/ - 18 mm) in the UVW-systems did not exceed 0.13 mm (x cp 0.12 mm, Ycp 0.12 mm and zcp 0.13 mm), giving a mean spatial orientation error of 0.19 mm. I
3.3. Condylar movements Fig. 6 shows the model of the TMJ at maximum intercuspation from a top-front and a lateral as-
Fig. 7. Sequence of selected images of the position of the condyle during an opening and closing sequence. Position of the condyle in maximum intercuspation (beginning of opening) (image 1), at maximum opening (image 4) and again in maximum intercuspation (end of closing) (image 8).
202
M Krebs et al. / Technology and Health Care 2 (J994) 193 - 207 anterior - posterior
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