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A biomechanical study of the human periodontal ligament
A BIOMECHANICAL STUDY OF THE PERIODONTAL LIGAMENT U.
HUMAN
MANDEL
Department of Oral Pathology, Royal Dental College, Department of Oral Medicine and Surgery. University Hospital (Rigshospitalet) and Department of Pathology, University Hospital (Hvidovre Hospital), Copenhagen. Denmark P. DALGAARD Statistical Unit, University
Hospital
(Rigshospitalet),
Copenhagen,
Denmark
and A. Department
of Connective
VIID~K
Tissue Biology, Institute of Anatomy,
University
of Aarhus, Denmark.
Abstract-The mechanical properties of the normal human periodontal ligament (PDL) were investigated at eight different root levels. One miliimetre transverse sections of teeth, PDL and alveolar bone of mandibular premolars were examined in a materials testing machine. During testing bone was supported by metal rings and teeth by metal cylinders of individually adjusted sizes. Having corrected for differences of size and width of the PDL the influence of root level was estimated using a multivariate analysis of variance. The shear strength was almost constant at the upper part of the root, diminishing in apical direction. The shear extensibility and the relative failure energy in shear were higher at the middle of the root, diminishing coronally and apically. Only the elastic stiffness did not vary significantly along the root. These results demonstrate that in order to compare the mechanical properties of PDL care should be taken to compare areas at the same root level.
INTRODUCTION
The periodontal ligament (PDL) is the fibrous connective tissue that occupies the periodontal space between the root of a tooth and its bony socket (Berkowitz and Moxham, 1982). Several approaches have been used to study the biomechanical properties of the tooth, PDL and bone complex: The effects of small physiological loads (up to 20N) on teeth with normal and experimentally altered periodontal tissues have been studied in animals and humans (for review see Berkowitzand Moxham, 1982). The load and corresponding deformation required to extract teeth in rats have been measured on normal and experimentally altered periodontium (Chiba and Ohkawa, 1980). Yamada (1970) using specimens from human autopsies also measured the force required to pull a tooth directly from its socket, and in addition the corresponding ultimate tensile strength (UTS) (extractive force in kgmm-’ surface area of embedded part of the root) of the PDL. However, when a tooth is pulled directly from its socket, the fibers would rupture unevenly due to the root curvature and to the different vertical alignment of the fibers. It is moreover likely that friction will be generated between bone and tooth. Atkinson and Ralph (1975) and Ralph (1976) studied the UTS (N mm-‘) of small samples of PDL in
Received 26 June 1985: in revised form
24 February
1986.
humans using teeth on which remained a fragment of PDL and bone after extraction. Thus, the fibers ruptured in a step-like manner caused by the curved nature of the ligament attachment (Ralph, 1980). This method was improved by testing 1 mm thick transverse sections of teeth with PDL and bone (Ralph, 1982). By using thin sections it was considered that the direction of load (extrusion) would bring all the fiber bundles into play together thus avoiding the problem of the curved ligament attachment. Considerably more information about the biomechanical properties of a tissue can be obtained when the whole stress-strain curve is recorded with sufficient resolution instead of only the parameters for the point of maximum stress, where the specimen is already in the process of breaking (Viidik, 1980). This point is also far beyond the range of physiological functioning. Furthermore, it has been shown that collagen is responsible for the strength in complex soft tisslues (Oxlund and Andreassen, 1980). When collagen is present in meshworks, its mechanical characteristics are primarily dependent on the arrangement of the fiber bundles and fibers (Viidik, 1978). Since the arrangement of collagen fibers of PDL vary along the root (Sloan, 1979). it is thus relevant to study the mechanical properties of the PDL in detail along the root. We have devised a method to study the mechanical properties of PDL. In the present study the mechanical properties of 1 mm transverse sections of intact human
638
U. MANDEL. P. DALGAARD and A.
PDL at different root levels wereanalyzed. Mechanical properties were calculated from both loaddeformation and stress-strain tunes. In addition a histologic examination was made of tissue blocks close to the specimens used for mechanical testing. This method can be used for future studies of the PDL during various normal, healing and pathological conditions.
VIIDIK
PDL
Fig. I. Schematic drawing of specimen preparation from a 1 mm horizontal section of the dissected mandibular block,
The section was split along the vertical lines. Each specimen MATERIALS ASD iMETHODS
consisted of tooth, PDL and alveolar bone.
Mandibular teeth and their supporting bone were obtained at autopsy from twenty human males, 23-55yr old (mean 36.2) from the Medico-legal Institute, University of Copenhagen. Fifteen had died in accidents, four from heart attacks and one from heart disease in combination with alcohol and barbiturate poisoning. Only teeth in normal occlusion and without periodontal disease, i.e. no pocket depth exceeding 5 mm, were included. Right or left sides of the mandibles were chosen at random. An incision was laid along basis mandibulae from the right to the left angulus. The face skin was dissected from the mandible. A rectangular block containing the canine, first and second premolars and first molar was cut from between the canine and the front teeth, and between the first and second molar and along nervus mandibularis with a circular saw. Viidik er al. (1965) found no effect of postmortal storage on the mechanical properties of the intact rabbit anterior cruciatc ligament within 96 h ofstorage. Accordingly, all blocks were removed within 62 h of death and stored for no more than 10 h in Ringer’s solution, pH 7.4 at room temperature until testing, except two blocks, which were frozen in the same solution at - 18°C. The frozen blocks were thawed at 37’C and tested within 7 h. Radiographic examination of each block revealed an odontoma and marginal bone loss, which excluded three of the nonfrozen blocks. Each block was mounted in wax (Kerr) along the tooth crowns, along the region of the mandibular nerve and along either the canine or first molar, allowing predetermined cutting planes. Four to seven 1.05 + 0.1 mm (mean It S.D.) horizontal sections (Ralph, 1982) were cut from each block with a MicroTrenn sectioning machine (Hofer, Switzerland) using careful water refrigeration so that heating of the PDL and thereby denaturation of its collagen fibers was avoided. The sections were split interdentally (Fig. 1).
rings (Figs 2a. b). A metal cylinder was placed on the tooth part for load application. Twenty seven sets of steel rings and 22 metal cylinders ofdifferent sizes were prepared so that the two parts of any specimen could be clamped and loaded respectively, ensuring that only the PDL was deformed. This ‘sandwich’ was mounted with a ‘rig’ (Fig. 2c) into a materials testing machine (Instron type TT-BM-L), and the specimen immersed in a buffered Ringer’s solution in order to avoid drying during testing. The deformation speed was 0.2 mm min-‘. The PDL was loaded until breaking and the load-deformation curve (Fig. 3) recorded continuously. Only extrusive loading tests were performed, i.e. in the direction a tooth is pulled out of its socket. The specimen was examined in a stereo microscope after the mechanical testing. Any specimen with a fracture in tooth or bone was discarded. Data were thus obtained for 134 specimens from seventeen first and sixteen second premolars of seventeen blocks. The width of the PDL (LW) was calculated as an average from two times eight measurements of the width of the periodontal space from photographs (x 20) of each specimen. The measurement points were evenly distributed along the periodontal space. The specimens were divided into eight groups (quantiles) according to their locations with respect to the root, i.e. root lecel in per cent of the whole root length (Fig. 4). The root length was defined
Lamina compacta
Load
F
mm
LE
BP \
was split with a high speed water
cooled bur, and lamina spongiosa was split with a surgical knife. A few specimens were rejected because of damage to them caused by this procedure. Each specimen consisted thus of a first or second premolar with surrounding PDL and alveolar bone. Mechanical testing
A method was developed to evade deformation of tooth and bone during testing: the bone part of the specimen was fastened between two flat stainless steel
LB
tanp
L i; Oeformation
Fig. 3. Load-deformation curve ofa specimen from a second mandibular premolar (31?,; root level) of a 35 yr old man (for explanation see text).
b i
Fig. 2. Schematic drawing and photographs of a specimen fastened in the mounting equipment of a materials testing machine (Instron). (a, b) Specimen fastened between two steel rings. A metal cylinder placed on the tooth part provides load transferring. Note that the ‘sandwich’ allows deformation of the PDL pretenting deformation of tooth and bone. (c) The ‘rig’ for mounting of the ‘sandwich’ in the Instron.
639
641
The human periodontal ligament
tation as well as signs of inflammation. anchylosis or other pathological conditions
Ligomnlwdlh mm I
0.25 0.20 1
necrosis,
Sratisrical methods $
.k)_)‘-3
-_)a’
Fig. 4. Schematic drawing of the position of the samples along a mandibular premolar root. The samples were divided into eight groups according to their location on the root in per cent of the total root length, i.e. root IeveLThecut-ve illustrates the width of the periodontal ligament in mm along the root.
The inhuence of root level on the measured mechanical parameters and the ligament width (LW) and stress area (SA) was estimated using a three-way (person, tooth and root level) multivariate analysis of variance (MANOVA). To analyse whether load varied proportionally to SA and deformation proportionally to LW an analysis was performed of whether the structure of the covariance matrix could be described by these proportionalities plus errors in the variables. The significance level was 0.05. Detailed description of the statistical analysis is given in the Appendix.
RESULTS
as the shortest distance between the facial bone margin (0 “/A)and the root apex (1009;) measured on radiographs. The area of the PDL that was loaded in each test specimen, i.e. stress area (SA), was calculated as the height of each specimen multiplied by circumference of the PDL. PDL circumference was read from photographs (x 20) by a digitizer into a calculator (Hewlett-Packard 9830 system). The load (F )-deformation (AI) curves were analysed by the samecomputing system. Shear stress values were calculated from load values by reducing these with SA. Shear strain values were approximated as deformation per unit LW (instead of tan (Al/W’)). The following parameters were calculated from the load-deformation curves (Fig. 3): maximum shear load value; dl~_, shear deforF DIIII, mation at maximum Load; tanfi, elastic stiffnesstangent of the angle between the linear region of the load-deformation curve and the x-axis; Failure energy in shear (W’), energy measured as the area between the load-deformation curve and the x-axis to the point of breaking (shaded area in Fig. 3). In addition from the stress-strain curves (Fig. 5): rmnx, maximum shear stress value; yrn.,, strain at maximum shear stress; tan u, elastic stiffness-tangent of the angle between the linear region of the stress-strain curve and the xaxis; relative failure energy in shear (WA), energy measured as the area between the stress-strain curve and the x-axis to the point of breaking.
Histologic etxluation
Tissue blocks next to and in between specimens used for mechanical testing were fixed in 10% neutral buffered formaldehyde demineralized in 10% solution of EDTA at pH 6.9, dehydrated and embedded in paraffin. Sections werecut at 5 pm perpendicular to the tooth axis. The following staining techniques were used: hematoxylin-eosin, van Gieson, reticulin, picroSirius plus polarization microscopy, aldehyde fuchsin and orcein. Each section was examined in the light microscope. Special attention was paid to fiber orien-
The load-deformation curves obtained (Fig. 3) were similar to those described for other connective tissues (Viidik, 1980). A toe-part, which was convex towards the deformation axis and became gradually steeper, was followed by a fairly linear segment (L,L,,). Before the point of maximum load (F,,) was reached there was some levelling off towards the deformation axis. Finally, with increased deformation the load decreased and the specimen broke at BP. The analysis showed no significant difference with respect to the effect of root level on the PDL’s width (LW), stress area (SA) and mechanical properties between the first and second premolars (p > 0.6). LW was 0.19 mm (range 0.1 t-O.44 mm) differing sign,% cantly along the root (p < O.OCOS), being smallest at the middle of the root increasing in apical and coronal directions (Fig. 4). This pattern is in agreement with the results of Coolidge (1937) and Ralph and Jefferies (1984)The mean stress area (SA) was 15.9 mm2 (range 8.4-23.5 mm*), largest at the basis of the root decreasing in apical direction with root tapering (p < 0.0005). The maximum shear load value (F,,,,,) was assumed proportional to the stress area (SA) and the deformation (AI,-,) was assumed proportional to the width of the PDL (LW) since the analysis of the covariance matrix (Table 2, Appendix) revealed no significant deviation from this hypothesis (p > 0.7). The load-deformation curves were therefore transformed into stress-strain curves by reducing load values by SA deformation values by LW. The infiuence of root level on the mechanical parameters calculated from load-deformarion curues (Table 1) was significant for the maximum shear load value (F,,,) (p < O.OOOS), the elastic stiffness (tan 8) (p < O.OOOS)and the failure energy in shear (W,) (p < O.OOOS).Each of these values was higher at the upper part of the root, decreasing considerably and almost linearly in apical direction with the steepest slope at the apical third of the root. The deformation (AIF,,) did not vary significantly along the root (p > 0.7).
642
U. MANDEL,P. DALGAARDand A. VIIDIK Root level had a significant influence on the followparameters (Fig. Sax): the maximum shear stress value (T,,,.=)(p c O.Ol), the shear strain at maximum stress (‘/r,J (p C 0.001) and the relative failure energy in shear (I+‘,) (p < 0.0005). T,,,_ was almost constant at the upper part of the root diminishing in the apical third of the root. ;._ values were higher at the middle of the root diminishing almost equally in apical and coronal direction. The WAvalues were also higher at the middle of the root diminishing in coronal and apical direction, however, decreasing mostly in apical direction. Although tana values showed no significant difference between root levels (p z 0.2), there tended to be some increase in the coronal third of the root (Fig. 5d). No significant difference was observed from the hypothesis that the effect of root level on T,,, were proportional to the effect of root level on tan a and WA, and that the effect of root level on tan z were inversely proportional to the effect of root level on yr,., (p > 0.4). The histologic examination of the tissue blocks close to the specimens used for mechanical testing showed that the PDL fibers were normal. No pathological changes of the PDL were found in any block. ing stress-strain
DISCUSSION
B c ,^
0,
aE
E 02 s .Y
Ltl
In the present study, we have developed a method to study the mechanical properties of the PDL of teeth and utilized this method to evaluate the mechanical properties at different root levels. Admittedly, due to the complex geometry of the PDL, no method of mechanical testing is ideal. Measurement of the force required to extract the whole tooth can be used for estimation of the strength of the whole ligament apparatus, but not for evaluation of the properties of the ligament as a tissue. For such measurements thecommonly used method to test specimens cut to a defined size must be used. The error in this approach is that some fibers in threedimensional meshworks are cut (cf. Viidik, 1978). In this case, however, a PDL fiber must have an angle of 66-83” (mean 79”) to the horizontal plane to be cut completely. This methodological error in the evaluation of the ligament as tissue is thus small compared to extracting whole teeth. In the latter case a significant number of fibers rupture before others are loaded at all; moreover when extracting intact teeth, quantification to stress-strain relationship is not possible. Therefore, the technique to use specimens cut to a welldefined size was employed to evaluate the properties of the PDL as a tissue. The fibers of the PDL have been shown to be grouped into highly organized bundles which are arranged in a complex three-dimensional network (Sloan, 1979). According to Viidik (1980), the behaviour of a three-dimensional meshwork is that of its components, to which the deformation of the geomet-
643
The human periodontal ligament
stress. a)
(bl
ElOSllC sllffncsr Nlmm2
(dl 4.0
IO-
0%
100%
Rootlevel
Fig. 5. The maximum shear stress value (T,,, ) (a), the strain at maximum shear stress (Vr,) (b), the relati\e failure energy in shear (W‘) (c) and the elastic stiffness (tana) (d) calculated from stress-strain curves for mandibular premolars at eight different root levels (mean + 2 S.E.M.).
rical pattern is added. The shape of the loaddeformation curve (Fig. 3) is thus dependent on the degree the fiber meshwork can be deformed (the toepart), before the fibers themselves (at La) start to deform (Viidik, 1980). Both load-deformation and stress-strain parameters were calculated. In general, load-deformation parameters are dependent upon size and width of the specimens. Stress-strain curves are obtained from load-deformation curves by dividing load by crosssectional area and deformation by original ligament length (Viidik, 1979). Stress-strain values allow comparison of the mechanical properties from specimens of different sizes and tissues. In this study stress area (SA) replaced ‘cross-sectional area’, and the PDL’s
width (LW) replaced ‘original ligament length’. SA describes the area of the PDL that is loaded in each specimen, but SA gives no information of the number of PDL collagen fibers that are loaded in each specimen. The direction of load was parallel to the tooth axis. Generally, the direction of deformation should be of the same direction as the ‘original ligament length’, which it was here only for the beginning of a test. These approximations were allowed because the analysis of the covariance matrix supported the hypothesis that there was proportionality between values for load and SA and between values for deformation and LW It was found that the size dependent functional strength as deduced from load-deformation par-
6-U
U.
MANDEL,
P. DALGAARD and A. VIIDIK
ameters was larger at the basis of the root (16.3 O,, root level) diminishing in the apical direction. Only deformation was constant along the root. The periodontium seemed to be able to withstand larger stresses at the basis of the root, this ability declining in the apical direction. Studying the PDL as a material (tissue) the following stress-strain parameters were examined. The maximum shear stress (T,,,) was found to be about 3.0 N mm-‘. This value was in the same order of magnitudeas the tensilestrengthof2.4 Nmmv2 found by Ralph (1982), but higher than the tensile strength of 1.4 Nmm-* quoted by Yamada (1970). Yamada measured maximum load values by extraction of whole teeth. By this method the fibers would rupture unevenly due to the root curvature and to the different vertical alignment of the fibers. The fibers of the most vertical direction (the apical fibers) would rupture first followed by the more horizontal fibers thus resulting in lower maximum stress values since the load values were reduced by the whole root area. Tmox was found to be almost constant at the upper part of the root, diminishing at the apical third of the root. The lower T,,, values in the apical third of the root could be explained by either weaker or fewer load bearing fibers in this area. The maximum sheur srruin due (yr,J was found to be higher at the middle of the root diminishing in apical and coronal direction. This could be the result of either more extensible fibers or more likely of different geometrical configuration of fibers in the middle part of the root. Fibers arranged into meshworks have a different load-deformation curve than fibers arranged more parallel to the direction of testing even though the fibers have the same tensile properties (Viidik, 1978). A greater strain could be the result from an increased rearrangement of the meshes into the direction of tension application before the fibers themselves (and the factors holding them together) begin to absorb the main part of tension. Be relativefailure energy in shear ( W,,) was found to be higher at the middle of the root decreasing in the coronal and apical directions, and most in the latter direction. This parameter gives a precise and useful estimation of how much energy a tissue can absorb without breaking (Viidik, 1978). Estimated primarily from WA the PDL as a material shows the best mechanical properties at the mid part of the root. In conclusion, the present study has shown that the mechanical properties of the normal PDL, when corrected for differences of size and width, vary at different root levels, suggesting dependence upon either the fibers themselves, their number, or geometrical configuration. In order to compare the mechanical properties of PDL it is thus necessary besides comparing stress-strain data (an influence ofarea and width of the PDL) to compare areas at the same root level, or to correct for differences between root levels. The present method for evaluating the mechanical properties of the
PDL. is considered to be of interest for future studies of the PDL during various normal, healing and pathological conditions.
Acknowledgement-We thank Tckn. dr. Mart MBpi, Associate Professor at Chalmers Institute of Technology (Giiteborg, Sweden) for valuable discussionsin conjunction with thedesign ofthe project. Wealso thank the Medico-legal Institute. University of Copenhagen for providing autopsies, and the Department of Dental Materials, Royal Dental College. Copenhagen for helpful advice and assistancein the construction of the mechanical components. Supported by idrzttens Forskning&d Grant Number 84101.
REFERENCES Anderson. T. W. (1958) Testing the general linear hypothesis; analysis of variance. An fntrodwetion to Multivariote Statistical Analysis. pp. 178-227. John Wiley, New York. Atkinson, H. F. and Ralph, W. J. (1975) In ritro strength of the human periodontal ligament. J. dent. Res. 56,48-52. Berkowitz, B. K. B. and Moxham, B. J. (1982) Introductory remarks. The effects of external forces on the periodontal ligament. The response to axial loads. The effects of external forces on the periodontal ligament. The response to horizontal loads. The Periodontal Ligament in Health and Disease (Edited by Berkowitz, 8. K. B., Moxham, B. J. and Newmann, H. N.), pp. 1, 253-275. Pergamon Press, Oxford. Chiba, M. and Ohkawa, S. (1980) Measurement of the tensile strength of the periodontium in the rat mandibular first molar. Archs oral Biol. 25, 569-572. Coolidge, E. D. (1937) The thickness of the human periodontal membrane. J. Am. dent. Ass. dent. Cosmos 24, 1260-l 270. Oxlund, H. and Andreassen. T. (1980) The role of hyaluronic acid, collagen and elastin in the mechanical properties of connective tissues. J. Anur. 131, 61 l-620. Ralph, W. J. (1976) A study of the tooth support system. Thesis, University of Melbourne. Ralph, W. J. (1980)The in oitro rupture of human periodontal ligament. J. Biomechanics 13, 369-373. Ralph, W. J. (1982) Tensile behaviour of the periodontal ligament. J. periodont. Res. 17, 423-426. Ralph, W. J. and Jefferies, J. R. (1984) The minimal width of the periodontal space. J. oral Rehabil. II, 415-418. Sloan, P. (1979) Collagen fiber architecture in the periodontal ligament. J. R. Sot. Med. 72. 188-191. Viidik, A. (1978) On the correlation between structure and mechanical function of soft connective tissues. Verb. anat. ties., Jena, 72, 75-89.
Viidik, A. (1979) Biomechanical behaviour of soft connective tissues. Progress in Biomechanics, Nato Advanced Study Institute-Ankara (Edited by Akkas, N.), pp. 75-l 13. Sijthoff and Noordhoff, Alpen aan den Rinjn. Viidik, A. (1980) Mechanical properties of parallel fibered collagenous tissues. Dependence between structure and function in collagenous tissues.Biologyofcollagen(Edited by Viidik, A. and Vuust, J.), pp. 238-241, 271-274, Academic Press, New York. Viidik. A., Sandquist, L. and MBgi, M. (1965) Influence of postmortal slorage on rensile strength characteristics and histology of rabbit ligaments. Acta orthop. stand. suppl. 79. Yamada, H. (1970) Strength of Biological Materials (Edited by Evans, F. G.), pp. 14-15. 153-154. Williams &Wilkins, Baltimore.
The human
periodontal
645
individuals, for which data were only present for one tooth were omitted here. (The means are given + 1.96 times their standard error. and transformed back to original scale.) In transforming load-deformation values into stress-strain values it is supposed that there is proportionality between values for load and SA and between values for deformation and LW, i.e.. that the stress-strain values vary independently of SA and Lu’. but in fact significant correlations were present. This can be explained, however, if SA and LW are measured with error, since then the same error would enter into the transformed values. Such correlations should have a specific structure and therefore it can be tested whether they might be entirely due to errors in LW and SA. To say this more precisely denote the mechanical parameters and the f.R’ and SA parameters respectively
~PPESDIX.SThTISTICALA>ALYSIS The mechanical parameters as well as PDL width (Lu’) and stress area (SA) are analysed as a function of individuals (I), tooth type i T) and root level IL), the focus being on the latter factor. Proportionality relations between mechanical parameters are examined, and finally, the validity of transforming load-deformation parameters into stress-strain parameters is assessed. The framework is a MANOVA model with six variables described by three factors (Anderson. 1958). The data are written in a matrix layout, where the rows
, ylb’(i))
y(i) = (y”‘(i).
ligament
are the logarithmated
observations of rmoi, y,__, tan 2, W,, t&l’ and SA respectively for each specimen (i). The v(i) are assumed realisations of independent multivariate normals .t Ii) with covariance matrix Z. The mean values are described
s”‘(i) = (f”‘(i).
, #“(i))r
x”‘(i) = (jJS’(i), ,.‘h+))? Now if
by EJ’/’ = &+
a’1’
x”‘(i) = x?‘(i) + e(i)
Here e.g. x’::r denotes a vector with identical elements for specimens from the same tooth and individual. In the above model, the etTect of root level is assumed to be the same for both types of teeth. This was tested by extending the model uith terms 1”’ LIT, yielding no significant improvement. By the structure of the mechanical parameters it can be expected that
then s”‘(i) = x:‘(i)-
where C is the matrix describing how In LW and In SA enter into the transformation of the mechanical parameters and the asterisked variables denote the values if LW and SA were measured exactly
the quantities
do not vary with proportionally to and yr_,.. Testing combinattons of significant result. D”’ = d”‘, D”’ = since
D”’
= ).<3’_(Y(1t_l.“‘)
D”’
= P
C&(i)
-().(LI+yUl)
root level: tana and WA should change the ratio, respectively the product, of rnrrl the hypothesis that the corresponding . the a”“s were constants gave a nonThis a‘ilows a conditional analvsis uoon dl*’ with respect to the effect of ;oot I&$
Assuming x:‘(i), x’,“(i) and c(i) to be independent with covariance matrices A,. AZ and I- respectively the structure of the covariance matrix is the following
I
111 =jlrrr+~‘l’+y,xd”‘+y,xd”
As this hypothesis contains sixteen (10 + 3 + 3) parameters compared to 21 in the model with Z unconstrained, the - 2 In (likelihood ratio) is approximately chi-square distributed with 5 degrees of freedom. The test came out non-significant and the estimated values of A,, Ar and r are presented in Table 2 along with the empirical covariance matrix 2. Note, however, that since almost all variation in In LW is estimated to be random, there is not much information on the relation between LW and the mechanical parameters.
with a’/’ the same as before. In the conditional model the significance of the effect of root level was tested for each variable separately (see Results for details). For presentation adjusted means are calculated, i.e. the expected values at different root levels for a tooth with a blxr value equal to the average of those occurring in the present study and corresponding averages of d”’ and d”‘. Two
Table 2. The
empirical
covariance
matrix
(2)
and
the estimates
t = 0.0406 0.0013 0.0276 0.0215 0.0098 0.0022
0.0338 0.0346 0.0191 0.0041 0.0063
0.0661 0.0011 0.0146 0.0074
0.05 I 1 0.0086 0.0047
A, = 0.0319 0.005 I 0.0145 0.0140
0.0269 0.0245 0.0160
0.0430 0.0043
0.038 1
Ar = 0.0011 0.0015
0.0033
r = o.ofii 0.0021
0.0064
values
0.0123 O.OOOS
of A,, A, and f
0.0096
HUMAN
MANDEL
Department of Oral Pathology, Royal Dental College, Department of Oral Medicine and Surgery. University Hospital (Rigshospitalet) and Department of Pathology, University Hospital (Hvidovre Hospital), Copenhagen. Denmark P. DALGAARD Statistical Unit, University
Hospital
(Rigshospitalet),
Copenhagen,
Denmark
and A. Department
of Connective
VIID~K
Tissue Biology, Institute of Anatomy,
University
of Aarhus, Denmark.
Abstract-The mechanical properties of the normal human periodontal ligament (PDL) were investigated at eight different root levels. One miliimetre transverse sections of teeth, PDL and alveolar bone of mandibular premolars were examined in a materials testing machine. During testing bone was supported by metal rings and teeth by metal cylinders of individually adjusted sizes. Having corrected for differences of size and width of the PDL the influence of root level was estimated using a multivariate analysis of variance. The shear strength was almost constant at the upper part of the root, diminishing in apical direction. The shear extensibility and the relative failure energy in shear were higher at the middle of the root, diminishing coronally and apically. Only the elastic stiffness did not vary significantly along the root. These results demonstrate that in order to compare the mechanical properties of PDL care should be taken to compare areas at the same root level.
INTRODUCTION
The periodontal ligament (PDL) is the fibrous connective tissue that occupies the periodontal space between the root of a tooth and its bony socket (Berkowitz and Moxham, 1982). Several approaches have been used to study the biomechanical properties of the tooth, PDL and bone complex: The effects of small physiological loads (up to 20N) on teeth with normal and experimentally altered periodontal tissues have been studied in animals and humans (for review see Berkowitzand Moxham, 1982). The load and corresponding deformation required to extract teeth in rats have been measured on normal and experimentally altered periodontium (Chiba and Ohkawa, 1980). Yamada (1970) using specimens from human autopsies also measured the force required to pull a tooth directly from its socket, and in addition the corresponding ultimate tensile strength (UTS) (extractive force in kgmm-’ surface area of embedded part of the root) of the PDL. However, when a tooth is pulled directly from its socket, the fibers would rupture unevenly due to the root curvature and to the different vertical alignment of the fibers. It is moreover likely that friction will be generated between bone and tooth. Atkinson and Ralph (1975) and Ralph (1976) studied the UTS (N mm-‘) of small samples of PDL in
Received 26 June 1985: in revised form
24 February
1986.
humans using teeth on which remained a fragment of PDL and bone after extraction. Thus, the fibers ruptured in a step-like manner caused by the curved nature of the ligament attachment (Ralph, 1980). This method was improved by testing 1 mm thick transverse sections of teeth with PDL and bone (Ralph, 1982). By using thin sections it was considered that the direction of load (extrusion) would bring all the fiber bundles into play together thus avoiding the problem of the curved ligament attachment. Considerably more information about the biomechanical properties of a tissue can be obtained when the whole stress-strain curve is recorded with sufficient resolution instead of only the parameters for the point of maximum stress, where the specimen is already in the process of breaking (Viidik, 1980). This point is also far beyond the range of physiological functioning. Furthermore, it has been shown that collagen is responsible for the strength in complex soft tisslues (Oxlund and Andreassen, 1980). When collagen is present in meshworks, its mechanical characteristics are primarily dependent on the arrangement of the fiber bundles and fibers (Viidik, 1978). Since the arrangement of collagen fibers of PDL vary along the root (Sloan, 1979). it is thus relevant to study the mechanical properties of the PDL in detail along the root. We have devised a method to study the mechanical properties of PDL. In the present study the mechanical properties of 1 mm transverse sections of intact human
638
U. MANDEL. P. DALGAARD and A.
PDL at different root levels wereanalyzed. Mechanical properties were calculated from both loaddeformation and stress-strain tunes. In addition a histologic examination was made of tissue blocks close to the specimens used for mechanical testing. This method can be used for future studies of the PDL during various normal, healing and pathological conditions.
VIIDIK
PDL
Fig. I. Schematic drawing of specimen preparation from a 1 mm horizontal section of the dissected mandibular block,
The section was split along the vertical lines. Each specimen MATERIALS ASD iMETHODS
consisted of tooth, PDL and alveolar bone.
Mandibular teeth and their supporting bone were obtained at autopsy from twenty human males, 23-55yr old (mean 36.2) from the Medico-legal Institute, University of Copenhagen. Fifteen had died in accidents, four from heart attacks and one from heart disease in combination with alcohol and barbiturate poisoning. Only teeth in normal occlusion and without periodontal disease, i.e. no pocket depth exceeding 5 mm, were included. Right or left sides of the mandibles were chosen at random. An incision was laid along basis mandibulae from the right to the left angulus. The face skin was dissected from the mandible. A rectangular block containing the canine, first and second premolars and first molar was cut from between the canine and the front teeth, and between the first and second molar and along nervus mandibularis with a circular saw. Viidik er al. (1965) found no effect of postmortal storage on the mechanical properties of the intact rabbit anterior cruciatc ligament within 96 h ofstorage. Accordingly, all blocks were removed within 62 h of death and stored for no more than 10 h in Ringer’s solution, pH 7.4 at room temperature until testing, except two blocks, which were frozen in the same solution at - 18°C. The frozen blocks were thawed at 37’C and tested within 7 h. Radiographic examination of each block revealed an odontoma and marginal bone loss, which excluded three of the nonfrozen blocks. Each block was mounted in wax (Kerr) along the tooth crowns, along the region of the mandibular nerve and along either the canine or first molar, allowing predetermined cutting planes. Four to seven 1.05 + 0.1 mm (mean It S.D.) horizontal sections (Ralph, 1982) were cut from each block with a MicroTrenn sectioning machine (Hofer, Switzerland) using careful water refrigeration so that heating of the PDL and thereby denaturation of its collagen fibers was avoided. The sections were split interdentally (Fig. 1).
rings (Figs 2a. b). A metal cylinder was placed on the tooth part for load application. Twenty seven sets of steel rings and 22 metal cylinders ofdifferent sizes were prepared so that the two parts of any specimen could be clamped and loaded respectively, ensuring that only the PDL was deformed. This ‘sandwich’ was mounted with a ‘rig’ (Fig. 2c) into a materials testing machine (Instron type TT-BM-L), and the specimen immersed in a buffered Ringer’s solution in order to avoid drying during testing. The deformation speed was 0.2 mm min-‘. The PDL was loaded until breaking and the load-deformation curve (Fig. 3) recorded continuously. Only extrusive loading tests were performed, i.e. in the direction a tooth is pulled out of its socket. The specimen was examined in a stereo microscope after the mechanical testing. Any specimen with a fracture in tooth or bone was discarded. Data were thus obtained for 134 specimens from seventeen first and sixteen second premolars of seventeen blocks. The width of the PDL (LW) was calculated as an average from two times eight measurements of the width of the periodontal space from photographs (x 20) of each specimen. The measurement points were evenly distributed along the periodontal space. The specimens were divided into eight groups (quantiles) according to their locations with respect to the root, i.e. root lecel in per cent of the whole root length (Fig. 4). The root length was defined
Lamina compacta
Load
F
mm
LE
BP \
was split with a high speed water
cooled bur, and lamina spongiosa was split with a surgical knife. A few specimens were rejected because of damage to them caused by this procedure. Each specimen consisted thus of a first or second premolar with surrounding PDL and alveolar bone. Mechanical testing
A method was developed to evade deformation of tooth and bone during testing: the bone part of the specimen was fastened between two flat stainless steel
LB
tanp
L i; Oeformation
Fig. 3. Load-deformation curve ofa specimen from a second mandibular premolar (31?,; root level) of a 35 yr old man (for explanation see text).
b i
Fig. 2. Schematic drawing and photographs of a specimen fastened in the mounting equipment of a materials testing machine (Instron). (a, b) Specimen fastened between two steel rings. A metal cylinder placed on the tooth part provides load transferring. Note that the ‘sandwich’ allows deformation of the PDL pretenting deformation of tooth and bone. (c) The ‘rig’ for mounting of the ‘sandwich’ in the Instron.
639
641
The human periodontal ligament
tation as well as signs of inflammation. anchylosis or other pathological conditions
Ligomnlwdlh mm I
0.25 0.20 1
necrosis,
Sratisrical methods $
.k)_)‘-3
-_)a’
Fig. 4. Schematic drawing of the position of the samples along a mandibular premolar root. The samples were divided into eight groups according to their location on the root in per cent of the total root length, i.e. root IeveLThecut-ve illustrates the width of the periodontal ligament in mm along the root.
The inhuence of root level on the measured mechanical parameters and the ligament width (LW) and stress area (SA) was estimated using a three-way (person, tooth and root level) multivariate analysis of variance (MANOVA). To analyse whether load varied proportionally to SA and deformation proportionally to LW an analysis was performed of whether the structure of the covariance matrix could be described by these proportionalities plus errors in the variables. The significance level was 0.05. Detailed description of the statistical analysis is given in the Appendix.
RESULTS
as the shortest distance between the facial bone margin (0 “/A)and the root apex (1009;) measured on radiographs. The area of the PDL that was loaded in each test specimen, i.e. stress area (SA), was calculated as the height of each specimen multiplied by circumference of the PDL. PDL circumference was read from photographs (x 20) by a digitizer into a calculator (Hewlett-Packard 9830 system). The load (F )-deformation (AI) curves were analysed by the samecomputing system. Shear stress values were calculated from load values by reducing these with SA. Shear strain values were approximated as deformation per unit LW (instead of tan (Al/W’)). The following parameters were calculated from the load-deformation curves (Fig. 3): maximum shear load value; dl~_, shear deforF DIIII, mation at maximum Load; tanfi, elastic stiffnesstangent of the angle between the linear region of the load-deformation curve and the x-axis; Failure energy in shear (W’), energy measured as the area between the load-deformation curve and the x-axis to the point of breaking (shaded area in Fig. 3). In addition from the stress-strain curves (Fig. 5): rmnx, maximum shear stress value; yrn.,, strain at maximum shear stress; tan u, elastic stiffness-tangent of the angle between the linear region of the stress-strain curve and the xaxis; relative failure energy in shear (WA), energy measured as the area between the stress-strain curve and the x-axis to the point of breaking.
Histologic etxluation
Tissue blocks next to and in between specimens used for mechanical testing were fixed in 10% neutral buffered formaldehyde demineralized in 10% solution of EDTA at pH 6.9, dehydrated and embedded in paraffin. Sections werecut at 5 pm perpendicular to the tooth axis. The following staining techniques were used: hematoxylin-eosin, van Gieson, reticulin, picroSirius plus polarization microscopy, aldehyde fuchsin and orcein. Each section was examined in the light microscope. Special attention was paid to fiber orien-
The load-deformation curves obtained (Fig. 3) were similar to those described for other connective tissues (Viidik, 1980). A toe-part, which was convex towards the deformation axis and became gradually steeper, was followed by a fairly linear segment (L,L,,). Before the point of maximum load (F,,) was reached there was some levelling off towards the deformation axis. Finally, with increased deformation the load decreased and the specimen broke at BP. The analysis showed no significant difference with respect to the effect of root level on the PDL’s width (LW), stress area (SA) and mechanical properties between the first and second premolars (p > 0.6). LW was 0.19 mm (range 0.1 t-O.44 mm) differing sign,% cantly along the root (p < O.OCOS), being smallest at the middle of the root increasing in apical and coronal directions (Fig. 4). This pattern is in agreement with the results of Coolidge (1937) and Ralph and Jefferies (1984)The mean stress area (SA) was 15.9 mm2 (range 8.4-23.5 mm*), largest at the basis of the root decreasing in apical direction with root tapering (p < 0.0005). The maximum shear load value (F,,,,,) was assumed proportional to the stress area (SA) and the deformation (AI,-,) was assumed proportional to the width of the PDL (LW) since the analysis of the covariance matrix (Table 2, Appendix) revealed no significant deviation from this hypothesis (p > 0.7). The load-deformation curves were therefore transformed into stress-strain curves by reducing load values by SA deformation values by LW. The infiuence of root level on the mechanical parameters calculated from load-deformarion curues (Table 1) was significant for the maximum shear load value (F,,,) (p < O.OOOS), the elastic stiffness (tan 8) (p < O.OOOS)and the failure energy in shear (W,) (p < O.OOOS).Each of these values was higher at the upper part of the root, decreasing considerably and almost linearly in apical direction with the steepest slope at the apical third of the root. The deformation (AIF,,) did not vary significantly along the root (p > 0.7).
642
U. MANDEL,P. DALGAARDand A. VIIDIK Root level had a significant influence on the followparameters (Fig. Sax): the maximum shear stress value (T,,,.=)(p c O.Ol), the shear strain at maximum stress (‘/r,J (p C 0.001) and the relative failure energy in shear (I+‘,) (p < 0.0005). T,,,_ was almost constant at the upper part of the root diminishing in the apical third of the root. ;._ values were higher at the middle of the root diminishing almost equally in apical and coronal direction. The WAvalues were also higher at the middle of the root diminishing in coronal and apical direction, however, decreasing mostly in apical direction. Although tana values showed no significant difference between root levels (p z 0.2), there tended to be some increase in the coronal third of the root (Fig. 5d). No significant difference was observed from the hypothesis that the effect of root level on T,,, were proportional to the effect of root level on tan a and WA, and that the effect of root level on tan z were inversely proportional to the effect of root level on yr,., (p > 0.4). The histologic examination of the tissue blocks close to the specimens used for mechanical testing showed that the PDL fibers were normal. No pathological changes of the PDL were found in any block. ing stress-strain
DISCUSSION
B c ,^
0,
aE
E 02 s .Y
Ltl
In the present study, we have developed a method to study the mechanical properties of the PDL of teeth and utilized this method to evaluate the mechanical properties at different root levels. Admittedly, due to the complex geometry of the PDL, no method of mechanical testing is ideal. Measurement of the force required to extract the whole tooth can be used for estimation of the strength of the whole ligament apparatus, but not for evaluation of the properties of the ligament as a tissue. For such measurements thecommonly used method to test specimens cut to a defined size must be used. The error in this approach is that some fibers in threedimensional meshworks are cut (cf. Viidik, 1978). In this case, however, a PDL fiber must have an angle of 66-83” (mean 79”) to the horizontal plane to be cut completely. This methodological error in the evaluation of the ligament as tissue is thus small compared to extracting whole teeth. In the latter case a significant number of fibers rupture before others are loaded at all; moreover when extracting intact teeth, quantification to stress-strain relationship is not possible. Therefore, the technique to use specimens cut to a welldefined size was employed to evaluate the properties of the PDL as a tissue. The fibers of the PDL have been shown to be grouped into highly organized bundles which are arranged in a complex three-dimensional network (Sloan, 1979). According to Viidik (1980), the behaviour of a three-dimensional meshwork is that of its components, to which the deformation of the geomet-
643
The human periodontal ligament
stress. a)
(bl
ElOSllC sllffncsr Nlmm2
(dl 4.0
IO-
0%
100%
Rootlevel
Fig. 5. The maximum shear stress value (T,,, ) (a), the strain at maximum shear stress (Vr,) (b), the relati\e failure energy in shear (W‘) (c) and the elastic stiffness (tana) (d) calculated from stress-strain curves for mandibular premolars at eight different root levels (mean + 2 S.E.M.).
rical pattern is added. The shape of the loaddeformation curve (Fig. 3) is thus dependent on the degree the fiber meshwork can be deformed (the toepart), before the fibers themselves (at La) start to deform (Viidik, 1980). Both load-deformation and stress-strain parameters were calculated. In general, load-deformation parameters are dependent upon size and width of the specimens. Stress-strain curves are obtained from load-deformation curves by dividing load by crosssectional area and deformation by original ligament length (Viidik, 1979). Stress-strain values allow comparison of the mechanical properties from specimens of different sizes and tissues. In this study stress area (SA) replaced ‘cross-sectional area’, and the PDL’s
width (LW) replaced ‘original ligament length’. SA describes the area of the PDL that is loaded in each specimen, but SA gives no information of the number of PDL collagen fibers that are loaded in each specimen. The direction of load was parallel to the tooth axis. Generally, the direction of deformation should be of the same direction as the ‘original ligament length’, which it was here only for the beginning of a test. These approximations were allowed because the analysis of the covariance matrix supported the hypothesis that there was proportionality between values for load and SA and between values for deformation and LW It was found that the size dependent functional strength as deduced from load-deformation par-
6-U
U.
MANDEL,
P. DALGAARD and A. VIIDIK
ameters was larger at the basis of the root (16.3 O,, root level) diminishing in the apical direction. Only deformation was constant along the root. The periodontium seemed to be able to withstand larger stresses at the basis of the root, this ability declining in the apical direction. Studying the PDL as a material (tissue) the following stress-strain parameters were examined. The maximum shear stress (T,,,) was found to be about 3.0 N mm-‘. This value was in the same order of magnitudeas the tensilestrengthof2.4 Nmmv2 found by Ralph (1982), but higher than the tensile strength of 1.4 Nmm-* quoted by Yamada (1970). Yamada measured maximum load values by extraction of whole teeth. By this method the fibers would rupture unevenly due to the root curvature and to the different vertical alignment of the fibers. The fibers of the most vertical direction (the apical fibers) would rupture first followed by the more horizontal fibers thus resulting in lower maximum stress values since the load values were reduced by the whole root area. Tmox was found to be almost constant at the upper part of the root, diminishing at the apical third of the root. The lower T,,, values in the apical third of the root could be explained by either weaker or fewer load bearing fibers in this area. The maximum sheur srruin due (yr,J was found to be higher at the middle of the root diminishing in apical and coronal direction. This could be the result of either more extensible fibers or more likely of different geometrical configuration of fibers in the middle part of the root. Fibers arranged into meshworks have a different load-deformation curve than fibers arranged more parallel to the direction of testing even though the fibers have the same tensile properties (Viidik, 1978). A greater strain could be the result from an increased rearrangement of the meshes into the direction of tension application before the fibers themselves (and the factors holding them together) begin to absorb the main part of tension. Be relativefailure energy in shear ( W,,) was found to be higher at the middle of the root decreasing in the coronal and apical directions, and most in the latter direction. This parameter gives a precise and useful estimation of how much energy a tissue can absorb without breaking (Viidik, 1978). Estimated primarily from WA the PDL as a material shows the best mechanical properties at the mid part of the root. In conclusion, the present study has shown that the mechanical properties of the normal PDL, when corrected for differences of size and width, vary at different root levels, suggesting dependence upon either the fibers themselves, their number, or geometrical configuration. In order to compare the mechanical properties of PDL it is thus necessary besides comparing stress-strain data (an influence ofarea and width of the PDL) to compare areas at the same root level, or to correct for differences between root levels. The present method for evaluating the mechanical properties of the
PDL. is considered to be of interest for future studies of the PDL during various normal, healing and pathological conditions.
Acknowledgement-We thank Tckn. dr. Mart MBpi, Associate Professor at Chalmers Institute of Technology (Giiteborg, Sweden) for valuable discussionsin conjunction with thedesign ofthe project. Wealso thank the Medico-legal Institute. University of Copenhagen for providing autopsies, and the Department of Dental Materials, Royal Dental College. Copenhagen for helpful advice and assistancein the construction of the mechanical components. Supported by idrzttens Forskning&d Grant Number 84101.
REFERENCES Anderson. T. W. (1958) Testing the general linear hypothesis; analysis of variance. An fntrodwetion to Multivariote Statistical Analysis. pp. 178-227. John Wiley, New York. Atkinson, H. F. and Ralph, W. J. (1975) In ritro strength of the human periodontal ligament. J. dent. Res. 56,48-52. Berkowitz, B. K. B. and Moxham, B. J. (1982) Introductory remarks. The effects of external forces on the periodontal ligament. The response to axial loads. The effects of external forces on the periodontal ligament. The response to horizontal loads. The Periodontal Ligament in Health and Disease (Edited by Berkowitz, 8. K. B., Moxham, B. J. and Newmann, H. N.), pp. 1, 253-275. Pergamon Press, Oxford. Chiba, M. and Ohkawa, S. (1980) Measurement of the tensile strength of the periodontium in the rat mandibular first molar. Archs oral Biol. 25, 569-572. Coolidge, E. D. (1937) The thickness of the human periodontal membrane. J. Am. dent. Ass. dent. Cosmos 24, 1260-l 270. Oxlund, H. and Andreassen. T. (1980) The role of hyaluronic acid, collagen and elastin in the mechanical properties of connective tissues. J. Anur. 131, 61 l-620. Ralph, W. J. (1976) A study of the tooth support system. Thesis, University of Melbourne. Ralph, W. J. (1980)The in oitro rupture of human periodontal ligament. J. Biomechanics 13, 369-373. Ralph, W. J. (1982) Tensile behaviour of the periodontal ligament. J. periodont. Res. 17, 423-426. Ralph, W. J. and Jefferies, J. R. (1984) The minimal width of the periodontal space. J. oral Rehabil. II, 415-418. Sloan, P. (1979) Collagen fiber architecture in the periodontal ligament. J. R. Sot. Med. 72. 188-191. Viidik, A. (1978) On the correlation between structure and mechanical function of soft connective tissues. Verb. anat. ties., Jena, 72, 75-89.
Viidik, A. (1979) Biomechanical behaviour of soft connective tissues. Progress in Biomechanics, Nato Advanced Study Institute-Ankara (Edited by Akkas, N.), pp. 75-l 13. Sijthoff and Noordhoff, Alpen aan den Rinjn. Viidik, A. (1980) Mechanical properties of parallel fibered collagenous tissues. Dependence between structure and function in collagenous tissues.Biologyofcollagen(Edited by Viidik, A. and Vuust, J.), pp. 238-241, 271-274, Academic Press, New York. Viidik. A., Sandquist, L. and MBgi, M. (1965) Influence of postmortal slorage on rensile strength characteristics and histology of rabbit ligaments. Acta orthop. stand. suppl. 79. Yamada, H. (1970) Strength of Biological Materials (Edited by Evans, F. G.), pp. 14-15. 153-154. Williams &Wilkins, Baltimore.
The human
periodontal
645
individuals, for which data were only present for one tooth were omitted here. (The means are given + 1.96 times their standard error. and transformed back to original scale.) In transforming load-deformation values into stress-strain values it is supposed that there is proportionality between values for load and SA and between values for deformation and LW, i.e.. that the stress-strain values vary independently of SA and Lu’. but in fact significant correlations were present. This can be explained, however, if SA and LW are measured with error, since then the same error would enter into the transformed values. Such correlations should have a specific structure and therefore it can be tested whether they might be entirely due to errors in LW and SA. To say this more precisely denote the mechanical parameters and the f.R’ and SA parameters respectively
~PPESDIX.SThTISTICALA>ALYSIS The mechanical parameters as well as PDL width (Lu’) and stress area (SA) are analysed as a function of individuals (I), tooth type i T) and root level IL), the focus being on the latter factor. Proportionality relations between mechanical parameters are examined, and finally, the validity of transforming load-deformation parameters into stress-strain parameters is assessed. The framework is a MANOVA model with six variables described by three factors (Anderson. 1958). The data are written in a matrix layout, where the rows
, ylb’(i))
y(i) = (y”‘(i).
ligament
are the logarithmated
observations of rmoi, y,__, tan 2, W,, t&l’ and SA respectively for each specimen (i). The v(i) are assumed realisations of independent multivariate normals .t Ii) with covariance matrix Z. The mean values are described
s”‘(i) = (f”‘(i).
, #“(i))r
x”‘(i) = (jJS’(i), ,.‘h+))? Now if
by EJ’/’ = &+
a’1’
x”‘(i) = x?‘(i) + e(i)
Here e.g. x’::r denotes a vector with identical elements for specimens from the same tooth and individual. In the above model, the etTect of root level is assumed to be the same for both types of teeth. This was tested by extending the model uith terms 1”’ LIT, yielding no significant improvement. By the structure of the mechanical parameters it can be expected that
then s”‘(i) = x:‘(i)-
where C is the matrix describing how In LW and In SA enter into the transformation of the mechanical parameters and the asterisked variables denote the values if LW and SA were measured exactly
the quantities
do not vary with proportionally to and yr_,.. Testing combinattons of significant result. D”’ = d”‘, D”’ = since
D”’
= ).<3’_(Y(1t_l.“‘)
D”’
= P
C&(i)
-().(LI+yUl)
root level: tana and WA should change the ratio, respectively the product, of rnrrl the hypothesis that the corresponding . the a”“s were constants gave a nonThis a‘ilows a conditional analvsis uoon dl*’ with respect to the effect of ;oot I&$
Assuming x:‘(i), x’,“(i) and c(i) to be independent with covariance matrices A,. AZ and I- respectively the structure of the covariance matrix is the following
I
111 =jlrrr+~‘l’+y,xd”‘+y,xd”
As this hypothesis contains sixteen (10 + 3 + 3) parameters compared to 21 in the model with Z unconstrained, the - 2 In (likelihood ratio) is approximately chi-square distributed with 5 degrees of freedom. The test came out non-significant and the estimated values of A,, Ar and r are presented in Table 2 along with the empirical covariance matrix 2. Note, however, that since almost all variation in In LW is estimated to be random, there is not much information on the relation between LW and the mechanical parameters.
with a’/’ the same as before. In the conditional model the significance of the effect of root level was tested for each variable separately (see Results for details). For presentation adjusted means are calculated, i.e. the expected values at different root levels for a tooth with a blxr value equal to the average of those occurring in the present study and corresponding averages of d”’ and d”‘. Two
Table 2. The
empirical
covariance
matrix
(2)
and
the estimates
t = 0.0406 0.0013 0.0276 0.0215 0.0098 0.0022
0.0338 0.0346 0.0191 0.0041 0.0063
0.0661 0.0011 0.0146 0.0074
0.05 I 1 0.0086 0.0047
A, = 0.0319 0.005 I 0.0145 0.0140
0.0269 0.0245 0.0160
0.0430 0.0043
0.038 1
Ar = 0.0011 0.0015
0.0033
r = o.ofii 0.0021
0.0064
values
0.0123 O.OOOS
of A,, A, and f
0.0096