- Email: [email protected]
Mass properties of the human mandible
Journal of Biomechanics 35 (2002) 975–978
Technical note
Mass properties of the human mandible Futang Zhang, Christopher C. Peck, Alan G. Hannam* Department of Oral Health Sciences, Faculty of Dentistry, The University of British Columbia, 2199 Wesbrook Mall, Vancouver, BC, Canada V6T 1Z3 Accepted 20 March 2002
Abstract Computer simulation of human masticatory dynamics requires specification of the jaw’s mass properties. These are difficult to estimate, especially in living subjects. Here, we used calibrated computed tomography (CT) to determine the properties of eight osseous jaw specimens with adult dentitions. When the CT numbers were converted to mineral densities, the mean estimated jaw mass was 13% greater than the mean wet weight. Putative bone marrow accounted for an extra 7% of mass. The mean bone densities for the sample were very consistent (1.7270.02 g/cm3). The mass and geometric centers were close (mean linear difference 0.4370.18 mm), and were always located anteroposteriorly between the second and third molars. The largest moment of inertia (MI) occurred around the jaw’s superoinferior axis, and the smallest around its transverse axis. Bone marrow added an extra 7% to the MIs. There were linear relationships between the mandibular length (expressed three dimensionally), the actual and estimated masses, and the moments of inertia. Our study suggests non-invasive imaging (such as magnetic resonance) and even direct linear measurement, may be adequate to estimate jaw mass properties in living humans. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Mass property; Mandible; Dynamic modeling
1. Introduction While dynamic modeling is a useful way to study cause and effect in the human masticatory system (Otten, 1987; Koolstra and van Eijden, 1995, 1997a, b; Hannam et al., 1997; Langenbach and Hannam, 1999), it requires specification of the jaw’s mass properties. Previously, these have been estimated from excised human tissue (Koolstra and van Eijden, 1995), an approach which does not lend itself to studying living subjects with different jaw shapes and sizes. Recently, we developed a method to estimate pig jaw mass properties by means of computed tomography (CT) (Zhang et al., 2001). This non-destructive approach permits the application of dynamic modeling to living animals. Essentially, the study showed that jaw mass and geometric centers were very close. It suggested mass properties could be predicted not only from the reconstructed bone volumes, and estimates of mean bone density, but also from simple measurements of jaw dimensions.
*Corresponding author. Tel: +1-604-822-3750; fax: +1-604-8223594. E-mail address: [email protected] (A.G. Hannam).
In the present study, we used the same approach to see whether these findings also apply to human mandibles. If so, CT scanning, magnetic resonance (MR) imaging, and direct linear measurements of jaw dimensions could all be used to estimate jaw mass properties in living humans. This would permit dynamic simulation in specific cases for which physiological information such as jaw motion, muscle activity, and bite force is obtainable.
2. Materials and methods Mass properties were estimated in eight adult dentate human mandibles of unknown origin and gender. Use of this archived material complied with the requirements of The University of British Columbia’s Ethical Review Committee. Details of CT scanning, image processing and the calculation of mass properties have been reported previously (Zhang et al., 2001). In brief, the specimens were weighed dry, and after 48 h hydration. They were submerged in water during imaging. Coronal scans (1 mm intervals and 0.43 mm 0.43 mm pixel sizes) were obtained with a Toshiba Xpress SX scanner (Toshiba
0021-9290/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 0 2 ) 0 0 0 5 7 - X
976
F. Zhang et al. / Journal of Biomechanics 35 (2002) 975–978
Corporation, Tokyo, Japan) operating at 100 KV and 150 mA. The images included KH2PO4 calibration solutions at different concentrations to permit expression of bone mineral density (BMD) as a function of pixel value (Lampmann et al., 1984). The images were filtered so that structures equal in density to, or less dense than, water could be excluded, i.e. they disclosed wet bone without bone marrow (see Zhang et al., 2001 for explanation). Bone segmentation, jaw surface reconstruction, landmark identification and measurement were carried out with commercial software (3DVIEWNIX 1.2, University of Pennsylvania Medical Center, Philadelphia, PA). Another program (Calimage–Calculate Image, Craniofacial Laboratory, The University of British Columbia) performed image matrix operations and mass property calculations (see Zhang et al., 2001). Moments of inertia were referenced to a coordinate system with its x-axis directed transversely from left to right (viewed frontally), its y-axis directed superoinferiorly, its z-axis anteroposteriorly, its x–z plane parallel to the dental occlusal plane, and its y– z plane in the midline (Fig. 1). We also segmented the non-mineralized component in each CT section, assigned pixel densities of 1 g/cm3 (Zhang et al., 2001), and recalculated the mass properties to estimate the contribution of simulated bone marrow. Cephalometric mandibular length was defined by the three-dimensional distance between condylion
Fig. 1. Coordinate system used to express moments of inertia. The transverse axis is represented by x, the superoinferior by y, and the anteroposterior by z. The axes z, y lie in the mid-sagittal plane; x, z are parallel to the dental occlusal plane.
(midpoint between the superior and posterior borders of the right bony condyle) and gnathion (midpoint between the anterior and inferior borders of the chin, see Jacobson, 1995). This jaw size factor thus included anteroposterior, vertical and transverse jaw dimensions. Linear regression curves were used to describe relationships between mandibular length, real and estimated mass properties.
3. Results There was a linear relationship between pixel values and BMD (r ¼ 0:995) i.e. BMD ¼ 0:012 Pixel Value þ 1:005: The estimated jaw mass was 13% more than the wet weight as shown in Table 1 (mean EM/ WW=1.1370.02; coefficient of variation, CV 1.77%), and 28% more than the dry weight (mean EM/ DW=1.2870.02; CV 1.56%). Inclusion of putative bone marrow added 7% extra mass (mean EMM/ EM=1.0770.02). The estimated mass with marrow was 36% greater than the measured dry weight of the jaw
Table 1 Descriptive statistics for measured jaw weights (dry weights, DW; wet weight, WW, g), estimated masses (estimated mass, EM; estimated mass with marrow, EMM, g), calculated mean bone density (MBD, g/ cm3), MBD with marrow (MBDM, g/cm3), mass and geometric center difference (CD, mm), CD with marrow (CDM, mm), moments of inertia without (Ixx, Iyy, Izz, g cm2) and with marrow (IxxM, IyyM, IzzM, g cm2), and ratios between these variables Measurement
Mean
SD
DW WW DW/WW EM EMM EMM/EM MBD MBDM MBDM/MBD EM/DW EMM/DW EM/WW EMM/WW CD CDM Ixx IxxM Ixx/IxxM Iyy IyyM Iyy/IyyM Izz IzzM Iz/IzzM
80.18 90.85 1.13 102.32 108.90 1.07 1.72 1.65 0.96 1.28 1.36 1.13 1.20 0.43 0.56 776.65 825.69 1.07 1482.03 1582.61 1.07 1250.08 1337.97 1.07
14.46 16.08 0.01 18.33 19.02 0.02 0.02 0.03 0.01 0.02 0.04 0.02 0.04 0.18 0.15 235.61 238.71 0.03 421.59 439.68 0.03 356.12 364.77 0.03
F. Zhang et al. / Journal of Biomechanics 35 (2002) 975–978
(mean EMM/DW=1.3670.04). The estimated mean bone density of the jaw was very consistent (1.7270.02 g/cm3, CV 1.16%). The mean difference between the mass and geometric center for the sample was 0.4370.18 mm, and the inclusion of putative bone marrow had little effect (paired t-test P > 0:05). Moments of inertia were smallest around the jaw’s transverse axis, and largest around its superoinferior axis. Simulated bone marrow accounted for 7% increase in moments of inertia around all three axes (Table 1). The mass centers were always located on the midsagittal plane between the second and third molar teeth, lying in the upper third of the distance from their occlusal surfaces to the lower border of the mandible. Relationships between mandibular length and all CT derived masses and moments of inertia, as well as direct
Fig. 2. Weights (dry weights, DW; wet weight, WW, g), masses (estimated mass, EM; estimated mass with marrow, EMM, g), and moments of inertia (Ixx, Iyy, Izz, g cm2) plotted against mandibular length. Linear regression lines have been fitted. The data include equations and coefficients of determination (R2 ). (A) Relationships between mandibular length, all weights, and CT-derived masses. (B) Relationships between mandibular length and moments of inertia around the three axes. The data suggest mass properties can be predicted by simple anatomical measurements, and the constants describing their relationships.
977
measured weights, were essentially linear, with coefficients of determination (R2 ) >0.85 (Fig. 2).
4. Discussion Specification of the mandible’s mass and mass center is not trivial, for the instantaneous mass constitutes the masses of all hard and soft tissues being moved, and could conceivably include the tongue. Previous assumptions have included the mass of the bony jaw only (Langenbach and Hannam, 1999), or estimates made with tissue blocks including adherent soft tissue (Koolstra and van Eijden, 1995). While imaging approaches such as CT or MR imaging have potential usefulness for estimating mass properties in living subjects, CT has the drawback of radiation cost. If this is accepted, however, close approximations of mass properties can be made by assigning densities to the voxels describing the structure (Smith et al., 1995). Overestimations of bone mass by CT scanning with uniform calibrators such as potassium dihydrogen phosphate solutions have been reported, to range from 12% to 15% (Cheng et al., 1995; Zhang et al., 2001). Here, we found a consistent 13% overestimation of the jaw’s wet weight (cf. Zhang et al., 2001). Most likely, this is attributable to the uniform density of the solution used for calibration (compared with the inhomogeneous structure of bone) and the assumed linearity between pixel values and solution concentrations (the solution at maximum concentration was still less dense than bone). Also, volume averaging of bone and water can occur when pixel sizes exceed the bone components within them. It is difficult to estimate which of these factors had the most profound effect in this instance. With a 13% correction factor, however, CT seems to be a practical way to estimate bone mass in the human jaw. Notably, there was a highly linear relationship between the estimated jaw mass and the mandibular length (the size factor itself accounted for over 85% of all weights and masses). Thus, a good estimation of mass can also be made by measuring the jaw’s length alone. It is possible to define the mass centers of irregular objects by direct experiment, e.g. by suspending the jaw in various orientations. Koolstra and van Eijden (1995) however sectioned a cadaver jaw into elements, assumed the mass distribution was homogeneous, and used the element locations to obtain the mass center. In effect, the present imaging adopts this principle, as the small elements permitted a refined estimation of regional bone density. As the difference between the mass and geometric centers was very small, the human jaw’s mass seem homogeneously distributed with respect to its geometric center. This finding confirms our previous findings in pigs, and suggests that imaging modalities revealing jaw shape alone such as MR could also be
978
F. Zhang et al. / Journal of Biomechanics 35 (2002) 975–978
used to estimate mass properties. In addition, since the mass center in the adult dentate human mandible lies in a relatively fixed location anatomically, in most cases, it can be approximated (to within a few mm) by simple linear measurement of conventional radiographic images, and the conversion constants provided here. Koolstra and van Eijden (1995) used equally sized jaw segments to calculate moments of inertia. In related studies, Hannam et al. (1997), Langenbach and Hannam (1999), and Peck et al. (2000) used inertial moments derived from a finite-element model of the human jaw developed by Korioth et al. (1992). The latter was based on CT scans, and included elements with properties specific to various jaw regions. Both approaches are somewhat similar to the present method. It remains difficult to validate calculated moments of inertia, but in the present context, the moment of inertia (MI) for each element was the product of its mass and the squared distance from the center of the element to the mass center; thus, any MI estimated for each element was theoretically valid provided the element’s mass was valid. As the total MI equals the sum of the moments of inertia of all constituent elements, its validity ultimately depends on that of the total mass calculated. Thus, a 13% overestimation of the mass for the adult dentate human mandible might be expected to affect its moments of inertia similarly in all three axes. The relative magnitudes of the human jaw’s moments of inertia are intuitively predictable. The largest occurred around its superoinferior axis, due to the long anteroposterior, and second-longest, transverse dimensions. The smallest moment occurred around the anteroposterior axis, due to the small vertical and transverse dimensions. The mean width of the jaw was actually slightly greater than its mean length (data not reported), so the smallest MI would be expected to occur around the transverse axis. Like others (Cheng et al., 1995; Smith et al., 1995), we assumed variations in jaw density would have a major effect on the mass properties. We found the mean bone density to be quite consistent, however, so it might be expected that the mass and moments of inertia are proportional to jaw size. The general similarity of our mass and moments of inertia proportions to estimations made by Koolstra and van Eijden (1995) lends credence to the idea that the mandible’s diagonal length is a primary determinant of its mass properties. Koolstra and van Eijden’s (1995) jaw mass of 440 g (which included attached soft tissue) was about four times heavier than the median jaw mass in our sample (105 g), yet the proportions of their mass properties are about the same as ours. In conclusion, it seems for modeling
purposes, that any non-invasive imaging technique such as MR (which reveals bone surface morphology), or direct measurement alone, are adequate for approximating jaw mass properties in living humans. This opens up the possibility for modeling individual jaws, including those which are asymmetric or partly resected.
Acknowledgements We are grateful to Dr. Vlad Stanescu for permitting access to specimens in the Department of Anatomy. We thank Ms. Joy Scott for her assistance with every aspect of this experiment. This investigation was supported by the Canadian Institutes of Health Research.
References Cheng, S., Toivanen, J.A., Suominen, H., Toivanen, J.T., Timonen, J., 1995. Estimation of structural and geometrical properties of cortical bone by computerized tomography in 78-year-old women. Journal of Bone and Mineral Research 10, 139–148. Hannam, A.G., Langenbach, G.E., Peck, C.C., 1997. Computer simulation of jaw biomechanics. In: McNeill, C. (Ed.), Science and Practice of Occlusion. Quintessence Publishing Co., Inc., Chicago, pp. 187–194. Jacobson, A., 1995. McNamara analysis. In: Jacobson, A. (Ed.), Radiographic Cephalometry. Quintessence Publishing Co., Inc., Chicago, pp. 113–126. Koolstra, J.H., van Eijden, T.M., 1995. Biomechanical analysis of jawclosing movements. Journal of Dental Research 74, 1564–1570. Koolstra, J.H., van Eijden, T.M., 1997a. Dynamics of the human masticatory muscles during a jaw open-close movement. Journal of Biomechanics 30, 883–889. Koolstra, J.H., van Eijden, T.M., 1997b. The jaw open-close movements predicted by biomechanical modelling. Journal of Biomechanics 30, 943–950. Korioth, T.W., Romilly, D.P., Hannam, A.G., 1992. Three-dimensional finite element stress analysis of the dentate human mandible. American Journal of Physical Anthropology 88, 69–96. Lampmann, L.E.H., Duursma, S.A., Ruys, J.H.J., 1984. CT Densitometry in Osteoporosis. Maritinus Nijhoff Publishers, Hingham. Langenbach, G.E., Hannam, A.G., 1999. The role of passive muscle tensions in a three-dimensional dynamic model of the human jaw. Archives of Oral Biology 44, 557–573. Otten, E., 1987. A myocybernetic model of the jaw system of the rat. Journal of Neuroscience Methods 21, 287–302. Peck, C.C., Langenbach, G.E., Hannam, A.G., 2000. Dynamic simulation of muscle and articular properties during human wide jaw opening. Archives of Oral Biology 45, 963–982. Smith, K.E., Bhatia, G., Vannier, M.W., 1995. Assessment of mass properties of human head using various three-dimensional imaging modalities. Medical and Biological Engineering and Computing 33, 278–284. Zhang, F., Langenbach, G.E., Hannam, A.G., 2001. Mass properties of the pig mandible. Journal of Dental Research 80, 325–337.
Technical note
Mass properties of the human mandible Futang Zhang, Christopher C. Peck, Alan G. Hannam* Department of Oral Health Sciences, Faculty of Dentistry, The University of British Columbia, 2199 Wesbrook Mall, Vancouver, BC, Canada V6T 1Z3 Accepted 20 March 2002
Abstract Computer simulation of human masticatory dynamics requires specification of the jaw’s mass properties. These are difficult to estimate, especially in living subjects. Here, we used calibrated computed tomography (CT) to determine the properties of eight osseous jaw specimens with adult dentitions. When the CT numbers were converted to mineral densities, the mean estimated jaw mass was 13% greater than the mean wet weight. Putative bone marrow accounted for an extra 7% of mass. The mean bone densities for the sample were very consistent (1.7270.02 g/cm3). The mass and geometric centers were close (mean linear difference 0.4370.18 mm), and were always located anteroposteriorly between the second and third molars. The largest moment of inertia (MI) occurred around the jaw’s superoinferior axis, and the smallest around its transverse axis. Bone marrow added an extra 7% to the MIs. There were linear relationships between the mandibular length (expressed three dimensionally), the actual and estimated masses, and the moments of inertia. Our study suggests non-invasive imaging (such as magnetic resonance) and even direct linear measurement, may be adequate to estimate jaw mass properties in living humans. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Mass property; Mandible; Dynamic modeling
1. Introduction While dynamic modeling is a useful way to study cause and effect in the human masticatory system (Otten, 1987; Koolstra and van Eijden, 1995, 1997a, b; Hannam et al., 1997; Langenbach and Hannam, 1999), it requires specification of the jaw’s mass properties. Previously, these have been estimated from excised human tissue (Koolstra and van Eijden, 1995), an approach which does not lend itself to studying living subjects with different jaw shapes and sizes. Recently, we developed a method to estimate pig jaw mass properties by means of computed tomography (CT) (Zhang et al., 2001). This non-destructive approach permits the application of dynamic modeling to living animals. Essentially, the study showed that jaw mass and geometric centers were very close. It suggested mass properties could be predicted not only from the reconstructed bone volumes, and estimates of mean bone density, but also from simple measurements of jaw dimensions.
*Corresponding author. Tel: +1-604-822-3750; fax: +1-604-8223594. E-mail address: [email protected] (A.G. Hannam).
In the present study, we used the same approach to see whether these findings also apply to human mandibles. If so, CT scanning, magnetic resonance (MR) imaging, and direct linear measurements of jaw dimensions could all be used to estimate jaw mass properties in living humans. This would permit dynamic simulation in specific cases for which physiological information such as jaw motion, muscle activity, and bite force is obtainable.
2. Materials and methods Mass properties were estimated in eight adult dentate human mandibles of unknown origin and gender. Use of this archived material complied with the requirements of The University of British Columbia’s Ethical Review Committee. Details of CT scanning, image processing and the calculation of mass properties have been reported previously (Zhang et al., 2001). In brief, the specimens were weighed dry, and after 48 h hydration. They were submerged in water during imaging. Coronal scans (1 mm intervals and 0.43 mm 0.43 mm pixel sizes) were obtained with a Toshiba Xpress SX scanner (Toshiba
0021-9290/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 0 2 ) 0 0 0 5 7 - X
976
F. Zhang et al. / Journal of Biomechanics 35 (2002) 975–978
Corporation, Tokyo, Japan) operating at 100 KV and 150 mA. The images included KH2PO4 calibration solutions at different concentrations to permit expression of bone mineral density (BMD) as a function of pixel value (Lampmann et al., 1984). The images were filtered so that structures equal in density to, or less dense than, water could be excluded, i.e. they disclosed wet bone without bone marrow (see Zhang et al., 2001 for explanation). Bone segmentation, jaw surface reconstruction, landmark identification and measurement were carried out with commercial software (3DVIEWNIX 1.2, University of Pennsylvania Medical Center, Philadelphia, PA). Another program (Calimage–Calculate Image, Craniofacial Laboratory, The University of British Columbia) performed image matrix operations and mass property calculations (see Zhang et al., 2001). Moments of inertia were referenced to a coordinate system with its x-axis directed transversely from left to right (viewed frontally), its y-axis directed superoinferiorly, its z-axis anteroposteriorly, its x–z plane parallel to the dental occlusal plane, and its y– z plane in the midline (Fig. 1). We also segmented the non-mineralized component in each CT section, assigned pixel densities of 1 g/cm3 (Zhang et al., 2001), and recalculated the mass properties to estimate the contribution of simulated bone marrow. Cephalometric mandibular length was defined by the three-dimensional distance between condylion
Fig. 1. Coordinate system used to express moments of inertia. The transverse axis is represented by x, the superoinferior by y, and the anteroposterior by z. The axes z, y lie in the mid-sagittal plane; x, z are parallel to the dental occlusal plane.
(midpoint between the superior and posterior borders of the right bony condyle) and gnathion (midpoint between the anterior and inferior borders of the chin, see Jacobson, 1995). This jaw size factor thus included anteroposterior, vertical and transverse jaw dimensions. Linear regression curves were used to describe relationships between mandibular length, real and estimated mass properties.
3. Results There was a linear relationship between pixel values and BMD (r ¼ 0:995) i.e. BMD ¼ 0:012 Pixel Value þ 1:005: The estimated jaw mass was 13% more than the wet weight as shown in Table 1 (mean EM/ WW=1.1370.02; coefficient of variation, CV 1.77%), and 28% more than the dry weight (mean EM/ DW=1.2870.02; CV 1.56%). Inclusion of putative bone marrow added 7% extra mass (mean EMM/ EM=1.0770.02). The estimated mass with marrow was 36% greater than the measured dry weight of the jaw
Table 1 Descriptive statistics for measured jaw weights (dry weights, DW; wet weight, WW, g), estimated masses (estimated mass, EM; estimated mass with marrow, EMM, g), calculated mean bone density (MBD, g/ cm3), MBD with marrow (MBDM, g/cm3), mass and geometric center difference (CD, mm), CD with marrow (CDM, mm), moments of inertia without (Ixx, Iyy, Izz, g cm2) and with marrow (IxxM, IyyM, IzzM, g cm2), and ratios between these variables Measurement
Mean
SD
DW WW DW/WW EM EMM EMM/EM MBD MBDM MBDM/MBD EM/DW EMM/DW EM/WW EMM/WW CD CDM Ixx IxxM Ixx/IxxM Iyy IyyM Iyy/IyyM Izz IzzM Iz/IzzM
80.18 90.85 1.13 102.32 108.90 1.07 1.72 1.65 0.96 1.28 1.36 1.13 1.20 0.43 0.56 776.65 825.69 1.07 1482.03 1582.61 1.07 1250.08 1337.97 1.07
14.46 16.08 0.01 18.33 19.02 0.02 0.02 0.03 0.01 0.02 0.04 0.02 0.04 0.18 0.15 235.61 238.71 0.03 421.59 439.68 0.03 356.12 364.77 0.03
F. Zhang et al. / Journal of Biomechanics 35 (2002) 975–978
(mean EMM/DW=1.3670.04). The estimated mean bone density of the jaw was very consistent (1.7270.02 g/cm3, CV 1.16%). The mean difference between the mass and geometric center for the sample was 0.4370.18 mm, and the inclusion of putative bone marrow had little effect (paired t-test P > 0:05). Moments of inertia were smallest around the jaw’s transverse axis, and largest around its superoinferior axis. Simulated bone marrow accounted for 7% increase in moments of inertia around all three axes (Table 1). The mass centers were always located on the midsagittal plane between the second and third molar teeth, lying in the upper third of the distance from their occlusal surfaces to the lower border of the mandible. Relationships between mandibular length and all CT derived masses and moments of inertia, as well as direct
Fig. 2. Weights (dry weights, DW; wet weight, WW, g), masses (estimated mass, EM; estimated mass with marrow, EMM, g), and moments of inertia (Ixx, Iyy, Izz, g cm2) plotted against mandibular length. Linear regression lines have been fitted. The data include equations and coefficients of determination (R2 ). (A) Relationships between mandibular length, all weights, and CT-derived masses. (B) Relationships between mandibular length and moments of inertia around the three axes. The data suggest mass properties can be predicted by simple anatomical measurements, and the constants describing their relationships.
977
measured weights, were essentially linear, with coefficients of determination (R2 ) >0.85 (Fig. 2).
4. Discussion Specification of the mandible’s mass and mass center is not trivial, for the instantaneous mass constitutes the masses of all hard and soft tissues being moved, and could conceivably include the tongue. Previous assumptions have included the mass of the bony jaw only (Langenbach and Hannam, 1999), or estimates made with tissue blocks including adherent soft tissue (Koolstra and van Eijden, 1995). While imaging approaches such as CT or MR imaging have potential usefulness for estimating mass properties in living subjects, CT has the drawback of radiation cost. If this is accepted, however, close approximations of mass properties can be made by assigning densities to the voxels describing the structure (Smith et al., 1995). Overestimations of bone mass by CT scanning with uniform calibrators such as potassium dihydrogen phosphate solutions have been reported, to range from 12% to 15% (Cheng et al., 1995; Zhang et al., 2001). Here, we found a consistent 13% overestimation of the jaw’s wet weight (cf. Zhang et al., 2001). Most likely, this is attributable to the uniform density of the solution used for calibration (compared with the inhomogeneous structure of bone) and the assumed linearity between pixel values and solution concentrations (the solution at maximum concentration was still less dense than bone). Also, volume averaging of bone and water can occur when pixel sizes exceed the bone components within them. It is difficult to estimate which of these factors had the most profound effect in this instance. With a 13% correction factor, however, CT seems to be a practical way to estimate bone mass in the human jaw. Notably, there was a highly linear relationship between the estimated jaw mass and the mandibular length (the size factor itself accounted for over 85% of all weights and masses). Thus, a good estimation of mass can also be made by measuring the jaw’s length alone. It is possible to define the mass centers of irregular objects by direct experiment, e.g. by suspending the jaw in various orientations. Koolstra and van Eijden (1995) however sectioned a cadaver jaw into elements, assumed the mass distribution was homogeneous, and used the element locations to obtain the mass center. In effect, the present imaging adopts this principle, as the small elements permitted a refined estimation of regional bone density. As the difference between the mass and geometric centers was very small, the human jaw’s mass seem homogeneously distributed with respect to its geometric center. This finding confirms our previous findings in pigs, and suggests that imaging modalities revealing jaw shape alone such as MR could also be
978
F. Zhang et al. / Journal of Biomechanics 35 (2002) 975–978
used to estimate mass properties. In addition, since the mass center in the adult dentate human mandible lies in a relatively fixed location anatomically, in most cases, it can be approximated (to within a few mm) by simple linear measurement of conventional radiographic images, and the conversion constants provided here. Koolstra and van Eijden (1995) used equally sized jaw segments to calculate moments of inertia. In related studies, Hannam et al. (1997), Langenbach and Hannam (1999), and Peck et al. (2000) used inertial moments derived from a finite-element model of the human jaw developed by Korioth et al. (1992). The latter was based on CT scans, and included elements with properties specific to various jaw regions. Both approaches are somewhat similar to the present method. It remains difficult to validate calculated moments of inertia, but in the present context, the moment of inertia (MI) for each element was the product of its mass and the squared distance from the center of the element to the mass center; thus, any MI estimated for each element was theoretically valid provided the element’s mass was valid. As the total MI equals the sum of the moments of inertia of all constituent elements, its validity ultimately depends on that of the total mass calculated. Thus, a 13% overestimation of the mass for the adult dentate human mandible might be expected to affect its moments of inertia similarly in all three axes. The relative magnitudes of the human jaw’s moments of inertia are intuitively predictable. The largest occurred around its superoinferior axis, due to the long anteroposterior, and second-longest, transverse dimensions. The smallest moment occurred around the anteroposterior axis, due to the small vertical and transverse dimensions. The mean width of the jaw was actually slightly greater than its mean length (data not reported), so the smallest MI would be expected to occur around the transverse axis. Like others (Cheng et al., 1995; Smith et al., 1995), we assumed variations in jaw density would have a major effect on the mass properties. We found the mean bone density to be quite consistent, however, so it might be expected that the mass and moments of inertia are proportional to jaw size. The general similarity of our mass and moments of inertia proportions to estimations made by Koolstra and van Eijden (1995) lends credence to the idea that the mandible’s diagonal length is a primary determinant of its mass properties. Koolstra and van Eijden’s (1995) jaw mass of 440 g (which included attached soft tissue) was about four times heavier than the median jaw mass in our sample (105 g), yet the proportions of their mass properties are about the same as ours. In conclusion, it seems for modeling
purposes, that any non-invasive imaging technique such as MR (which reveals bone surface morphology), or direct measurement alone, are adequate for approximating jaw mass properties in living humans. This opens up the possibility for modeling individual jaws, including those which are asymmetric or partly resected.
Acknowledgements We are grateful to Dr. Vlad Stanescu for permitting access to specimens in the Department of Anatomy. We thank Ms. Joy Scott for her assistance with every aspect of this experiment. This investigation was supported by the Canadian Institutes of Health Research.
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