s of posts on dentin Ching-Chang Kwok-Hung
stress distribution
in pulpless
teeth
Ko, BDS, IvIS,~ Chung-Sheng Chu, MS,b Chung, BDS, PhDCand Maw-Chang Lee, PhDd
National Yang-Ming Medical College, Taipei, Taiwan, Republic of China; and the University of Michigan School of Dentistry, Ann Arbor, Mieh. A finite element analysis was carried out to study the roles of posts in reducing dentin stress in pulpless teeth. Two-dimensional plane strain models of the midlabiolingual section of a human maxillary central incisor were first analyzed. The results showed that the gold alloy post reduced maximal dentin stress by as much as 30%. However, the integrity of the dentin was compromised and the effects of the post were likely to be exaggerated in such models. In an effort to correct for these problems, plane stress models with side plates and axisymmetric models were analyzed. Posts were found to reduce maximal dentin stress by only 3% to 8% when the teeth were subjjected to masticatory and traumatic loadings in these latter models. Although posts reduced maximal dentin stress by as much as 20% when the teeth were loaded vertically, teeth such as incisors and canines normally are not subjected to vertical loadings, Thus the reinforcement effects of posts seem to be doubtful in these teeth. (J PROSTRET DENT 1992;68:421-7.)
T ..
radltlonally, most endodontically treated teeth are reinforced with posts, but there is little supporting evidence for the pr0cedure.l Although in vitro mechanical testings have shown that posts increase the fracture loads of pulpless teeth,2, 3 more recent studies have failed to confirm the findings.l* 4s5Teeth with posts were found to show no greater rigidity than those with conservative root therapy in an in vitro study,6 and a clinical survey7 and photoelastic study” did not support this conventional practice. Controversy exists regarding the advisability of restoring pulpless teeth with posts.g-ll Finite element analysis is a very powerful and popular numerical method in stress analysis. It has been applied in dental mechanics for nearly two decades.12,I3 Davy et al.14 analyzed two-dimensional plane strain finite element models of the midlabiolingual section of the central incisor and concluded that dentin stress was gredter in the intact tooth than in the post-restored tooth, even though no data were reported. Since the post was overrepresented in such a model, its role in reducing dentin stress was likely to be exaggerated. In an attempt to correct this problem, modified
Supported in part by research grant no. NSC79-0412-BOlO-23 from the National Science Council, Taipei, Taiwan, Republic of China. *Former Graduate Student, Department of Biomedical Engineering, National Yang-Ming Medical College: Currently, Graduate Student, School of Dentistry, University of Michigan. “Former Graduate Student, Department of Biomedical Engineering, National Yang-Ming Medical College. CAssociate Professor, School of Dentistry, National Yang-Ming Medical College. dAssociate Professor, Department of Biomedical Engineering, National Yang-Ming Medical College. 10/l/36067
THE JOURNAL OF PROSTHETIC DENTISTRY
plane stress and axisymmetric models were analyzed. The results from various models were compared to better understand the roles of posts in reinforcing pulpless teeth. METHODS Plane strain
models
The geometry of the midlabiolingual section of the human maxillary central incisor was adopted from Wheeler.ls Two plane strain models were built. One, with post placement, included cortical and sponge bones, dentin, periodontal ligament, enamel, composite resin, gold alloy post (6-degree tapering), gingiva, and gutta-percha (Fig. 1). The other post was replaced by resin (section D), cement (section C), and gutta-percha. All materials were treated as homogeneous, isotropic, and linear elastic. Their properties were largely adopted from those detailed in the literature (Table I). The models were divided into 687 quadrilateral and triangular elements with 656 nodes (Fig. 2). The cement layer between the post and dentin was treated as part of the dentin because of its thinness and the likeness in Young’s moduli of the elasticity of dentin and cement. All materials were assumed to be rigidly bound together. Three loading forces were applied to the models (Fig. 2). Fl represented the masticatory force and was 45 degrees lingual to the incisal edge. F2 simulated the traumatic force and acted horizontally at the labial crown. F3 was a vertical force acting at the incisal edge. All three forces were as-
sumed to be of 1 N acting uniformly across a thickness of 1 mm. The bone below the apex was assumed to be fixed in all degrees of freedom (Fig. 2). Plane
stress
models
The plane stress model consisted of a central slice and a side plate. The central slice was the same as the plane strain
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Table I. Material properties used in the finite element
models
Material
Gold alloy post Steel post Dentin Enamel Cortical bone Sponge bone Gingiva Periodontal ligament Gutta-percha Zinc phosphate cement Composite resin
Young’s modulus ((=a)
Poisson’s ratio
77 200 18.6 41 13.7 1.37
0.33 0.33 0.31 0.30 0.30 0.30
19 26 19 13 25 *
0.30 0.45 0.45 0.35 0.28
19 19 19 19 *
0.0196 0.0689 0.00069 22.0 8.3
Reference No.
*Estimated.
loadings (Fl and F2) could be applied to the model.r7 Both Fl and F2 loadings were uniform in magnitude and acted in the faciolingual plane along the circumference of the model. Their net forces were 1 N in magnitude. F3 acted vertically at the incisal edge and was also 1 N in magnitude. Software
Fig. 1. The midfaciolingual section of the human maxillary central incisor with post restoration. In the case of no post restoration, section D of the post was replaced by resin, section C by cement, and the rest by gutta-percha (g.p.); p.l., periodontal ligament.
model except for its thickness, which was approximated by the post diameter and ranged from 0.54 to 2.0 mm.i6 The side plate,r6 added on top of the central slice, simulated the out-of-plane dentin connections. Its thickness was approximated by aDf4 - d, where D and d denoted the widths of the tooth and the post, respectively. Fl and F2 loadings on the central slice were similar to those in the plane strain models except that they were 1 N acting uniformly across a thickness of 2 mm. Axisymmetric
models
The facial half of the plane strain model was slightly modified to become the axisymmetric model. The gingiva was eliminated, and the shapes of resin and the incisal end of the post were modified (Fig. 3). A steel post was also simulated by increasing Young’s modulus of the elasticity of the post to 200 GPa. Harmonic elements were used so that nonaxisymmetric
422
and hardware
ANSYS (Swanson Analysis Systems, Houston, Pa), a general-purpose finite element analysis program, was executed on a minicomputer (Iris 4D/20, Silicon Graphics Inc., Mountain View, Calif.). Maximal and minimal principal stresses as well as von Mises’ equivalent stress were calculated. Maximal and minimal principal stresses stand for the maximal tensile and compressive stresses, respectively. Von Mises’ equivalent stress is equal to 3/@times the shear stress on the ? x ?Zy + z = 0 planes (octahedral shear stress) and is often used as the yield criterion of metals.‘s RESULTS Plane strain
models
When the pulpless incisor without a post was subjected to the masticatory loading (Fl), dentin equivalent stress was concentrated on the coronal and middle thirds of the root, with the facial side being slightly greater (Fig. 4, a). The maximal tensile stress was concentrated on the lingual side (Fig. 4, b), while maximal compressive stress was on the facial side (Fig. 4, c). With post placement, dentin stresses were reduced substantially while the post was stressed significantly (Fig. 5). The reductions in peak dentin stress reached 30% (Table II). Under the traumatic loading (F2), the distribution of equivalent stress in the dentin of a tooth with no post was similar to Fl loading except that the lingual side had the greater value. As expected, maximal tensile and compres-
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Fig. 2. Finite element mesh of the plane strain model of human maxillary central incisor. Three different loadings are shown. F2 represented the mast,icatory force and was 45 degrees lingual to the incisal edge. F.2 simulated the traumatic force and acted horizontally at the facial crown. F3 was a vertical force acting at the incisal edge. All three forces were assumed to be of 1 N acting uniformly across a thickness of 1 mm. All nodes at Y = 0 were assumed to be fixed.
Fig. 3. Finite element mesh of axisymmetric model of human maxillary central incisor. Post and resin both were two elements wide. Fl and F2 were uniformly distributed along the circumference of the model and always acted in the faciolingual plane. The net forces of all three loadings were 1 N.
matic loading, and the reductions in dentin stress as a result of post placement were less than 8%. sive stresses were concentrated on the facial and lingual sides, respectively. All peak dentin stresses were also reduced substantially in the presence of the post (Table II). When the tooth with no post was subjected to the vertical loading (F3), equivalent (Fig. 6, a) and compressive stresseswere concentrated at the cervical dentin and at the crown where the concentrated force was loaded. With post placement, both dentin stresses were concentrated around the apex of the post and were reduced drastically in other parts of the dentin (Fig. 6, b).
Plane stress models Under masticatory loading, dentin equivalent stress was distributed more uniformly in the central slice (Fig. 7, a) than in the plane strain model. This was also true for the maximal tensile and compressive stresses. Post placement reduced peak stress in the dentin by less than 6% (Fig. 7, b and Table III). Similar results were observed under trau-
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Axisymmetric
models
Under masticatory (Fig. 8, a) and traumatic loadings, dentin stress distributions were similar to those of the plane stress models. With post placement, peak dentin stresses were reduced by no more than 6.2 % (Fig. 8, b and Table IV). When Young’s modulus of the elasticity of the post was nearly tripled to simulate a steel post, the reductions were up to 11% (Table IV). Under vertical loadings, however, both peak dentin equivalent and compressive stresses were reduced by as much as 20 % for a gold alloy post and by 35 % for a steel post (Table IV).
DISCUSSION To simplify model construction and thus save computational time and costs, two-dimensional finite element models have been used extensively to model three-dimen-
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Y
b
b
Fig. 4. Stress contours in dentin of plane strain model without post restoration under masticatory loading (Fl). Van Mises’ equivalent stress (a), maximal tensile stress (b), and maximal compressive stress (c) are shown. Shape of root canal is somewhat simplified in this and subsequent figures. *Denotes the location of peak stress; A, 0.6 MPa; B, 0.8 MPa; C, 1.0 MPa; D, 1.2 MPa.
Table II. Comparisons of peak dentin stresses (MPa) predicted by plane strain models with and without post placement Loading
Fl
F2
Post
-
Gold
Gold
Equivalent stress
Tensile stress
1.36 0.98
1.00
(28%)"
(30%)
1.43
1.40
1.29
0.98
1.10 (14%)
(30%)
Compressive stress
-1.56 -1.14 (27%) -1.71 -1.15 (32%)
Locations of peak stress are shown in Figs. 4 and 5. Fl and F2 loadings are shown in Fig. 2; both were 1 N/mm. Von Mises’ equivalent, maximal tensile, and maximal compressive are listed. *Percent reduction with post placement.
stresses
sional objects in biomechanics. Good two-dimensional models, of course, can often provide valuable insights into three-dimensional problems.22 Our plane strain models predicted that peak dentin stresses were reduced by as much as 30% under masticatory and traumatic loadings in .the presence of a post (Table II). This is consistent with the finding of Davy et a1.,14 even though they did not present any data. Hunter et al6 recently carried out photoelastic studies on two-dimen-
424
Fig. 5. Stress contours in dentin of plane strain model with post restoration under masticatory loading. All dentin stresses were substantially lowered compared with Fig. 4. Same notations as in Fig. 4.
sional central incisor models. While they concluded that conservative enlargement of the root canal might render post placement unnecessary for a largely intact tooth, they also found that posts with moderate diameters, comparable to those in our models, substantially reinforced pulpless teeth, a finding in line with the present results. However, a plane strain model assumes that deformations (and strains) occur (for instance) in the x-y plane only and this model is ideal for simulating an infinitely long (in the z direction) object under transverse loadings that are uniform along the z axis. The post in the midfaciolingual section of a pulpless incisor is in its widest dimension and the dentin is in its narrowest (Fig. 1). A plane strain model of such a cross section, although used widely in the literature,13, 14,lg, 2othus tends to exaggerate the role of the post in reducing dentin stress. To compensate for the overrepresentation of posts in the plane strain models, modified plane strain models were built. In addition to the central slice, which modeled the midfaciolingual section of the pulpless incisor, the plane stress model consisted of a side plate that represented the out-of-plane dentin connections. Thus the post-restored tooth appears to be better represented by this model. Similar models have been used in orthopedic biomechanics with satisfactory results.21z 22 Because of the added dentin (that is, the side plate), the reductions in dentin stress in the presence of the post were, as expected, lowered substantially to only 3 % to 8 % under the masticatory and traumatic loadings (Table III). To
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a
Fig. 6. Contours of equivalent stress in dentin of plane strain models subjected to vertical loading (F3). Dentin stress was substantially lower with post restoration (b) than without (a). A, 0.10 MPa; B, 0.15 MPa; C, 0.20 MPa; L?, 0.25 MPa; E, 0.30 MPa; F, 0.35 MPa.
demonstrate the importance of the side plate, Ko16 has shown that when Young’s modulus of the elasticity of the side plate was reduced, the percent reduction in dentin stress caused by the post increased. It could be anticipated that without side plates, the results from plane stress models would approach those from plane strain models. The plane stress model is nevertheless still two-dimensional. Although the crown of the central incisor is certainly not symmetric around its long axis, its root is nearly so. We thus further modeled the pulpless incisor as axisymmetric. Even though only two-dimensional elements were used, these models were essentially three-dimensional in nature. Although not anatomically exact, they resembled the root of the incisor closely. The results (Table IV) confirmed those of the plane stress models. Even in the presence of a steel post, the reductions in dentin stress were only 6% to 11% under masticatory and traumatic loadings. These consistent results further support our belief that plane
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DENTISTRY
b \
Fig. 7. Contours of dentin equivalent stress in central slice of plane stress models subjected to masticatory loading. Dentin stress was slightly lower with post restoration (b) than without (a). A, 0.1 MPa; B, 0.2 MPa; C, 0.3 MPa.
Table III. Comparisons of peak dentin stresses (MPa) predicted by plane stress models with and without post placement Loading Fl
F2
Equivalent stress
Tensile stress
Gold
0.325 0.313 (3.7%)*
0.254 0.240 (5.5%)
-0.323 -0.315 (2.5%)
Gold
0.296 0.273 (7.8%)
0.274 0.265 (3.3%)
-0.313 -0.294
Post -
Compressive stress
(6.1%)
Fl and F2 loadings are shown in Fig. 2; both were 1 N. Stresses in the central slice are shown. *Percent reduction with post placement.
strain models are not good models for pulpless teeth. This conclusion is similar to that drawn by Huiskes and Chaoz2 concerning the femoral component of total hip replacement.
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Table IV. Comparisons of peak dentin stresses (MPa) predicted by axisymmetric models with and without post placement Loading
Fl
Post
Gold Steel
F2
Gold Steel
F3
Gold Steel
Equivalent stress
Tensile stress
Compressive stress
0.76 0.73 (4.0%)’ 0.69 (9.2%)
0.75 0.72 (4.0%) 0.69 (8.0%)
-0.81 -0.76 (6.2%) -0.72 (11%)
0.62 0.60 (3.2%) 0.58 (6.4%)
0.67 0.64 (4.5%) 0.62 (7.5%)
-0.67 -0.64 (4.5%) -0.62 (7.5%)
0.31 0.25 (19%) 0.20 (35%)
0.031f 0.027 (13%) 0.023 (26%)
-0.33 -0.26 (21%) -0.21 (36%)
Loadings are shown in Fig. 3; all net forces were 1 N. *Percent reduction with post placement. tNote that tensile stresses under F3 loadings were one order lower than other stresses.
Fig. 8. Contours of dentin equivalent stress in axisymmetric models subjected to masticatory loading. Dentin stress was slightly lower with post restoration (b) than without (a). A, 0.1 MPa; B, 0.3 MPa; C, 0.5 MPa; D, 0.7 MPa.
In all our models, however, the root canal was greatly enlarged (diameters at apex and cervix being 0.5 and 1.7 mm, respectively). In other words, the dentin was thinner in our models than in teeth with conservative root canal treatments. Therefore the differences in dentin stress between teeth with posts and those with conservative root canal treatments are expected to be even less than those predicted in the present study. Although only the central incisors were modeled in the present study, our results are likely to apply to other single-rooted teeth. Even though the geometry and material properties were simplified in our models, our results from the plane stress and axisymmetric models are consistent with the in vitro mechanical test findings of Leary et al.6 They subjected human pulpless central incisors and canines to lateral loads and found that teeth with posts had no greater rigidity than those without posts with conservative root canal therapy. The small reinforcement effects of posts as predicted in the present study may not always be able to be demonstrated by experiments, because of the inherent experi426
mental errors as well as the limited sample sizes and thus the limited power of statistical tests.24This may explain the contradictory results from various fracture experiments in the literature,lW5 even though the present models do not predict fractures. Under masticatory and traumatic loading alike, the tooth is subjected to bending. The bending rigidity (area moment of inertia around its long axis) of a circular cylinder is in proportion to the fourth power of its diameter.23 Since the post is located near the centerline of the tooth, its contribution to the bending rigidity of the incisor is expected to be small. This point has been correctly made by Guzy and Nicholls,l and the present results of plane stress and axisymmetric models thus appear to be reasonable from a mechanics point of view. Under vertical loadings (F3), however, posts reduced dentin stress substantially (Table IV). This reduction is because the posts are under compression in vertical loading. As incisors and canines are rarely subjected to vertical loading, the reinforcement effects of posts are doubtful in these teeth. REFERENCES 1. Guzy GE, Nicholls JI. In vitro comparison of intact endodontically treated teeth with and without endo-post reinforcement. J PROSTHET DENT 1979;42:39-44. 2. Kantor ME, Pines MS. A comparative study of restoration techniques for pulpless teeth. J PROSTHET DENT 1977;38:405-12. 3. Trabert KC, Caputo AA, Abou-Rass M. Tooth fracture-a comparison of endodontic and restorative treatments. J Endodont 1978;4:341-5. 4. Lu YC. Comparisons of the fractures of pulpless teeth. Chin Dent J 1987;6:2631.
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5. Trope M, Malts DO, Tronstad L. Resistance to fracture of restored endodontically treated teeth. Endodont Dent Traumatol 1985;1:108-11. 6. Leary JM, Aquilino SA, Svare CW. An evaluation of post length within the elastic limits of dentin. J PROSTHET DENT 1987;57:277-81. 7. Sorensen JA, Martinoff JT. Intracoronal reinforcement and coronal coverage: a study of endodontically treated teeth. J PROSTHETDENT 1984;51:780-4. 8. Hunter AJ, Feiglin B, Williams JF. Effects of post placement on endodontically treated teeth. J PROSTHETDENT 1989;62:166-72. 9. Eissmann HF, Radke RA Jr. Postendodontic restoration. In: Cohen S, Burns RC, eds. Pathways of the pulp. 4th ed. St. Louis: The CV Mosby Co, 1987:640-83. 10. Potashnick SR, Weine FS, Strauss S. Restoration of the endodontically treated tooth. In: Weine FS, ed. Endodontic therapy. 4th ed. St. Louis: The CV Mosby Co, 1989653.98. 11. Zakariasen KL. Preparation for restoration and temporization. In: Walton RE, Torabinejad M, eds. Principles and practice of endodontics. Philadelphia: WB Saunders, 1989:249-65. 12. Farah JW, Craig RG, Sikarskie DC. Photoelastic and finite element stress analysis of a restored axisymmetric first molar. J Biomech 1973;6:511-20. 13. Thresher RW, Saito GE. The stress analysis of human teeth. J Biomech 1973;8:443-9. 14. Davy DT, Dilley GL, Krejci RF. Determination of stress patterns in root-filled teeth incorporating various dowel designs, J Dent Res 1981;60:1301-10. 15. Wheeler RC. An atlas of tooth form. Philadelphia: WB Saunders, 1962. 16. Ko C-C. Stress analysis of pulpless tooth: effects of casting post on dentin stress distribution. Master’s thesis. Taipei, Taiwan: Yang-Ming Medical College, 1989.
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17. DeSalvo GJ, German RW. ANSYS Engineering Analysis System User’s Manual. Houston, Pa: Swanson Analysis Systems, 19&X9:2.25.1-7. 18. Caddell RM. Deformation and fracture of solids. Englewood Cliffs, NJ: Prentice-Hall, 1980:69-73. 19. Reinhardt RA, Krejci RF, Pao YC, Stannard JG. Dentin stresses in post-reconstructed teeth with diminishing bone support. J Dent Res 1983;62:1002-8. 20. Pao YC, Reinhardt RA, Krejci RF. Root stresses with tapered-end post design in periodontally compromised teeth. J PROSTHETDENT 1987;57: 281-6. 21. Svensson NL, Valliappan S, Wood RD. Stress analysis of human femur with implanted Charnley prosthesis, J Biomech 19’77;10:581-8. 22. Huiskes R, Chao EYS. A survey of finite element analysis in orthopaedic bionmchanics: the first decade. J Biomech 1983;16:385-409. 23. Popov EP. Introduction to mechanics of solids. Englewood Clitl’s, NJ: Prentice-Hall, 1968. 24. Zar JH. Biostatistical analysis, 2nd ed. Englewood Cliffs, NJ: PrenticeHall, lL984. 25. Weinstein AM, Klawitter JJ, Cook SD. Implant-bone interface characteristic of bioglass dental implants. J Biomed Mater Res 1980;14:23-9. 26. Williams KR, Edmundson JT, Rees JS. Finite element analysis of restored teeth. Dent Mater 1987;3:200-6.
Reprintrquests to; DR. MAW-CHANG LEE DEPARTMIZNT OF BIOMEDICAL ENGINEERING NATIONAL YANG-MING MEDICAL COLLEGE TAIPEI, TAIWAN 11221, R. 0. C.
427
stress distribution
in pulpless
teeth
Ko, BDS, IvIS,~ Chung-Sheng Chu, MS,b Chung, BDS, PhDCand Maw-Chang Lee, PhDd
National Yang-Ming Medical College, Taipei, Taiwan, Republic of China; and the University of Michigan School of Dentistry, Ann Arbor, Mieh. A finite element analysis was carried out to study the roles of posts in reducing dentin stress in pulpless teeth. Two-dimensional plane strain models of the midlabiolingual section of a human maxillary central incisor were first analyzed. The results showed that the gold alloy post reduced maximal dentin stress by as much as 30%. However, the integrity of the dentin was compromised and the effects of the post were likely to be exaggerated in such models. In an effort to correct for these problems, plane stress models with side plates and axisymmetric models were analyzed. Posts were found to reduce maximal dentin stress by only 3% to 8% when the teeth were subjjected to masticatory and traumatic loadings in these latter models. Although posts reduced maximal dentin stress by as much as 20% when the teeth were loaded vertically, teeth such as incisors and canines normally are not subjected to vertical loadings, Thus the reinforcement effects of posts seem to be doubtful in these teeth. (J PROSTRET DENT 1992;68:421-7.)
T ..
radltlonally, most endodontically treated teeth are reinforced with posts, but there is little supporting evidence for the pr0cedure.l Although in vitro mechanical testings have shown that posts increase the fracture loads of pulpless teeth,2, 3 more recent studies have failed to confirm the findings.l* 4s5Teeth with posts were found to show no greater rigidity than those with conservative root therapy in an in vitro study,6 and a clinical survey7 and photoelastic study” did not support this conventional practice. Controversy exists regarding the advisability of restoring pulpless teeth with posts.g-ll Finite element analysis is a very powerful and popular numerical method in stress analysis. It has been applied in dental mechanics for nearly two decades.12,I3 Davy et al.14 analyzed two-dimensional plane strain finite element models of the midlabiolingual section of the central incisor and concluded that dentin stress was gredter in the intact tooth than in the post-restored tooth, even though no data were reported. Since the post was overrepresented in such a model, its role in reducing dentin stress was likely to be exaggerated. In an attempt to correct this problem, modified
Supported in part by research grant no. NSC79-0412-BOlO-23 from the National Science Council, Taipei, Taiwan, Republic of China. *Former Graduate Student, Department of Biomedical Engineering, National Yang-Ming Medical College: Currently, Graduate Student, School of Dentistry, University of Michigan. “Former Graduate Student, Department of Biomedical Engineering, National Yang-Ming Medical College. CAssociate Professor, School of Dentistry, National Yang-Ming Medical College. dAssociate Professor, Department of Biomedical Engineering, National Yang-Ming Medical College. 10/l/36067
THE JOURNAL OF PROSTHETIC DENTISTRY
plane stress and axisymmetric models were analyzed. The results from various models were compared to better understand the roles of posts in reinforcing pulpless teeth. METHODS Plane strain
models
The geometry of the midlabiolingual section of the human maxillary central incisor was adopted from Wheeler.ls Two plane strain models were built. One, with post placement, included cortical and sponge bones, dentin, periodontal ligament, enamel, composite resin, gold alloy post (6-degree tapering), gingiva, and gutta-percha (Fig. 1). The other post was replaced by resin (section D), cement (section C), and gutta-percha. All materials were treated as homogeneous, isotropic, and linear elastic. Their properties were largely adopted from those detailed in the literature (Table I). The models were divided into 687 quadrilateral and triangular elements with 656 nodes (Fig. 2). The cement layer between the post and dentin was treated as part of the dentin because of its thinness and the likeness in Young’s moduli of the elasticity of dentin and cement. All materials were assumed to be rigidly bound together. Three loading forces were applied to the models (Fig. 2). Fl represented the masticatory force and was 45 degrees lingual to the incisal edge. F2 simulated the traumatic force and acted horizontally at the labial crown. F3 was a vertical force acting at the incisal edge. All three forces were as-
sumed to be of 1 N acting uniformly across a thickness of 1 mm. The bone below the apex was assumed to be fixed in all degrees of freedom (Fig. 2). Plane
stress
models
The plane stress model consisted of a central slice and a side plate. The central slice was the same as the plane strain
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Table I. Material properties used in the finite element
models
Material
Gold alloy post Steel post Dentin Enamel Cortical bone Sponge bone Gingiva Periodontal ligament Gutta-percha Zinc phosphate cement Composite resin
Young’s modulus ((=a)
Poisson’s ratio
77 200 18.6 41 13.7 1.37
0.33 0.33 0.31 0.30 0.30 0.30
19 26 19 13 25 *
0.30 0.45 0.45 0.35 0.28
19 19 19 19 *
0.0196 0.0689 0.00069 22.0 8.3
Reference No.
*Estimated.
loadings (Fl and F2) could be applied to the model.r7 Both Fl and F2 loadings were uniform in magnitude and acted in the faciolingual plane along the circumference of the model. Their net forces were 1 N in magnitude. F3 acted vertically at the incisal edge and was also 1 N in magnitude. Software
Fig. 1. The midfaciolingual section of the human maxillary central incisor with post restoration. In the case of no post restoration, section D of the post was replaced by resin, section C by cement, and the rest by gutta-percha (g.p.); p.l., periodontal ligament.
model except for its thickness, which was approximated by the post diameter and ranged from 0.54 to 2.0 mm.i6 The side plate,r6 added on top of the central slice, simulated the out-of-plane dentin connections. Its thickness was approximated by aDf4 - d, where D and d denoted the widths of the tooth and the post, respectively. Fl and F2 loadings on the central slice were similar to those in the plane strain models except that they were 1 N acting uniformly across a thickness of 2 mm. Axisymmetric
models
The facial half of the plane strain model was slightly modified to become the axisymmetric model. The gingiva was eliminated, and the shapes of resin and the incisal end of the post were modified (Fig. 3). A steel post was also simulated by increasing Young’s modulus of the elasticity of the post to 200 GPa. Harmonic elements were used so that nonaxisymmetric
422
and hardware
ANSYS (Swanson Analysis Systems, Houston, Pa), a general-purpose finite element analysis program, was executed on a minicomputer (Iris 4D/20, Silicon Graphics Inc., Mountain View, Calif.). Maximal and minimal principal stresses as well as von Mises’ equivalent stress were calculated. Maximal and minimal principal stresses stand for the maximal tensile and compressive stresses, respectively. Von Mises’ equivalent stress is equal to 3/@times the shear stress on the ? x ?Zy + z = 0 planes (octahedral shear stress) and is often used as the yield criterion of metals.‘s RESULTS Plane strain
models
When the pulpless incisor without a post was subjected to the masticatory loading (Fl), dentin equivalent stress was concentrated on the coronal and middle thirds of the root, with the facial side being slightly greater (Fig. 4, a). The maximal tensile stress was concentrated on the lingual side (Fig. 4, b), while maximal compressive stress was on the facial side (Fig. 4, c). With post placement, dentin stresses were reduced substantially while the post was stressed significantly (Fig. 5). The reductions in peak dentin stress reached 30% (Table II). Under the traumatic loading (F2), the distribution of equivalent stress in the dentin of a tooth with no post was similar to Fl loading except that the lingual side had the greater value. As expected, maximal tensile and compres-
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25.3
F3
Fig. 2. Finite element mesh of the plane strain model of human maxillary central incisor. Three different loadings are shown. F2 represented the mast,icatory force and was 45 degrees lingual to the incisal edge. F.2 simulated the traumatic force and acted horizontally at the facial crown. F3 was a vertical force acting at the incisal edge. All three forces were assumed to be of 1 N acting uniformly across a thickness of 1 mm. All nodes at Y = 0 were assumed to be fixed.
Fig. 3. Finite element mesh of axisymmetric model of human maxillary central incisor. Post and resin both were two elements wide. Fl and F2 were uniformly distributed along the circumference of the model and always acted in the faciolingual plane. The net forces of all three loadings were 1 N.
matic loading, and the reductions in dentin stress as a result of post placement were less than 8%. sive stresses were concentrated on the facial and lingual sides, respectively. All peak dentin stresses were also reduced substantially in the presence of the post (Table II). When the tooth with no post was subjected to the vertical loading (F3), equivalent (Fig. 6, a) and compressive stresseswere concentrated at the cervical dentin and at the crown where the concentrated force was loaded. With post placement, both dentin stresses were concentrated around the apex of the post and were reduced drastically in other parts of the dentin (Fig. 6, b).
Plane stress models Under masticatory loading, dentin equivalent stress was distributed more uniformly in the central slice (Fig. 7, a) than in the plane strain model. This was also true for the maximal tensile and compressive stresses. Post placement reduced peak stress in the dentin by less than 6% (Fig. 7, b and Table III). Similar results were observed under trau-
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Under masticatory (Fig. 8, a) and traumatic loadings, dentin stress distributions were similar to those of the plane stress models. With post placement, peak dentin stresses were reduced by no more than 6.2 % (Fig. 8, b and Table IV). When Young’s modulus of the elasticity of the post was nearly tripled to simulate a steel post, the reductions were up to 11% (Table IV). Under vertical loadings, however, both peak dentin equivalent and compressive stresses were reduced by as much as 20 % for a gold alloy post and by 35 % for a steel post (Table IV).
DISCUSSION To simplify model construction and thus save computational time and costs, two-dimensional finite element models have been used extensively to model three-dimen-
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Y
b
b
Fig. 4. Stress contours in dentin of plane strain model without post restoration under masticatory loading (Fl). Van Mises’ equivalent stress (a), maximal tensile stress (b), and maximal compressive stress (c) are shown. Shape of root canal is somewhat simplified in this and subsequent figures. *Denotes the location of peak stress; A, 0.6 MPa; B, 0.8 MPa; C, 1.0 MPa; D, 1.2 MPa.
Table II. Comparisons of peak dentin stresses (MPa) predicted by plane strain models with and without post placement Loading
Fl
F2
Post
-
Gold
Gold
Equivalent stress
Tensile stress
1.36 0.98
1.00
(28%)"
(30%)
1.43
1.40
1.29
0.98
1.10 (14%)
(30%)
Compressive stress
-1.56 -1.14 (27%) -1.71 -1.15 (32%)
Locations of peak stress are shown in Figs. 4 and 5. Fl and F2 loadings are shown in Fig. 2; both were 1 N/mm. Von Mises’ equivalent, maximal tensile, and maximal compressive are listed. *Percent reduction with post placement.
stresses
sional objects in biomechanics. Good two-dimensional models, of course, can often provide valuable insights into three-dimensional problems.22 Our plane strain models predicted that peak dentin stresses were reduced by as much as 30% under masticatory and traumatic loadings in .the presence of a post (Table II). This is consistent with the finding of Davy et a1.,14 even though they did not present any data. Hunter et al6 recently carried out photoelastic studies on two-dimen-
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Fig. 5. Stress contours in dentin of plane strain model with post restoration under masticatory loading. All dentin stresses were substantially lowered compared with Fig. 4. Same notations as in Fig. 4.
sional central incisor models. While they concluded that conservative enlargement of the root canal might render post placement unnecessary for a largely intact tooth, they also found that posts with moderate diameters, comparable to those in our models, substantially reinforced pulpless teeth, a finding in line with the present results. However, a plane strain model assumes that deformations (and strains) occur (for instance) in the x-y plane only and this model is ideal for simulating an infinitely long (in the z direction) object under transverse loadings that are uniform along the z axis. The post in the midfaciolingual section of a pulpless incisor is in its widest dimension and the dentin is in its narrowest (Fig. 1). A plane strain model of such a cross section, although used widely in the literature,13, 14,lg, 2othus tends to exaggerate the role of the post in reducing dentin stress. To compensate for the overrepresentation of posts in the plane strain models, modified plane strain models were built. In addition to the central slice, which modeled the midfaciolingual section of the pulpless incisor, the plane stress model consisted of a side plate that represented the out-of-plane dentin connections. Thus the post-restored tooth appears to be better represented by this model. Similar models have been used in orthopedic biomechanics with satisfactory results.21z 22 Because of the added dentin (that is, the side plate), the reductions in dentin stress in the presence of the post were, as expected, lowered substantially to only 3 % to 8 % under the masticatory and traumatic loadings (Table III). To
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Fig. 6. Contours of equivalent stress in dentin of plane strain models subjected to vertical loading (F3). Dentin stress was substantially lower with post restoration (b) than without (a). A, 0.10 MPa; B, 0.15 MPa; C, 0.20 MPa; L?, 0.25 MPa; E, 0.30 MPa; F, 0.35 MPa.
demonstrate the importance of the side plate, Ko16 has shown that when Young’s modulus of the elasticity of the side plate was reduced, the percent reduction in dentin stress caused by the post increased. It could be anticipated that without side plates, the results from plane stress models would approach those from plane strain models. The plane stress model is nevertheless still two-dimensional. Although the crown of the central incisor is certainly not symmetric around its long axis, its root is nearly so. We thus further modeled the pulpless incisor as axisymmetric. Even though only two-dimensional elements were used, these models were essentially three-dimensional in nature. Although not anatomically exact, they resembled the root of the incisor closely. The results (Table IV) confirmed those of the plane stress models. Even in the presence of a steel post, the reductions in dentin stress were only 6% to 11% under masticatory and traumatic loadings. These consistent results further support our belief that plane
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b \
Fig. 7. Contours of dentin equivalent stress in central slice of plane stress models subjected to masticatory loading. Dentin stress was slightly lower with post restoration (b) than without (a). A, 0.1 MPa; B, 0.2 MPa; C, 0.3 MPa.
Table III. Comparisons of peak dentin stresses (MPa) predicted by plane stress models with and without post placement Loading Fl
F2
Equivalent stress
Tensile stress
Gold
0.325 0.313 (3.7%)*
0.254 0.240 (5.5%)
-0.323 -0.315 (2.5%)
Gold
0.296 0.273 (7.8%)
0.274 0.265 (3.3%)
-0.313 -0.294
Post -
Compressive stress
(6.1%)
Fl and F2 loadings are shown in Fig. 2; both were 1 N. Stresses in the central slice are shown. *Percent reduction with post placement.
strain models are not good models for pulpless teeth. This conclusion is similar to that drawn by Huiskes and Chaoz2 concerning the femoral component of total hip replacement.
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Table IV. Comparisons of peak dentin stresses (MPa) predicted by axisymmetric models with and without post placement Loading
Fl
Post
Gold Steel
F2
Gold Steel
F3
Gold Steel
Equivalent stress
Tensile stress
Compressive stress
0.76 0.73 (4.0%)’ 0.69 (9.2%)
0.75 0.72 (4.0%) 0.69 (8.0%)
-0.81 -0.76 (6.2%) -0.72 (11%)
0.62 0.60 (3.2%) 0.58 (6.4%)
0.67 0.64 (4.5%) 0.62 (7.5%)
-0.67 -0.64 (4.5%) -0.62 (7.5%)
0.31 0.25 (19%) 0.20 (35%)
0.031f 0.027 (13%) 0.023 (26%)
-0.33 -0.26 (21%) -0.21 (36%)
Loadings are shown in Fig. 3; all net forces were 1 N. *Percent reduction with post placement. tNote that tensile stresses under F3 loadings were one order lower than other stresses.
Fig. 8. Contours of dentin equivalent stress in axisymmetric models subjected to masticatory loading. Dentin stress was slightly lower with post restoration (b) than without (a). A, 0.1 MPa; B, 0.3 MPa; C, 0.5 MPa; D, 0.7 MPa.
In all our models, however, the root canal was greatly enlarged (diameters at apex and cervix being 0.5 and 1.7 mm, respectively). In other words, the dentin was thinner in our models than in teeth with conservative root canal treatments. Therefore the differences in dentin stress between teeth with posts and those with conservative root canal treatments are expected to be even less than those predicted in the present study. Although only the central incisors were modeled in the present study, our results are likely to apply to other single-rooted teeth. Even though the geometry and material properties were simplified in our models, our results from the plane stress and axisymmetric models are consistent with the in vitro mechanical test findings of Leary et al.6 They subjected human pulpless central incisors and canines to lateral loads and found that teeth with posts had no greater rigidity than those without posts with conservative root canal therapy. The small reinforcement effects of posts as predicted in the present study may not always be able to be demonstrated by experiments, because of the inherent experi426
mental errors as well as the limited sample sizes and thus the limited power of statistical tests.24This may explain the contradictory results from various fracture experiments in the literature,lW5 even though the present models do not predict fractures. Under masticatory and traumatic loading alike, the tooth is subjected to bending. The bending rigidity (area moment of inertia around its long axis) of a circular cylinder is in proportion to the fourth power of its diameter.23 Since the post is located near the centerline of the tooth, its contribution to the bending rigidity of the incisor is expected to be small. This point has been correctly made by Guzy and Nicholls,l and the present results of plane stress and axisymmetric models thus appear to be reasonable from a mechanics point of view. Under vertical loadings (F3), however, posts reduced dentin stress substantially (Table IV). This reduction is because the posts are under compression in vertical loading. As incisors and canines are rarely subjected to vertical loading, the reinforcement effects of posts are doubtful in these teeth. REFERENCES 1. Guzy GE, Nicholls JI. In vitro comparison of intact endodontically treated teeth with and without endo-post reinforcement. J PROSTHET DENT 1979;42:39-44. 2. Kantor ME, Pines MS. A comparative study of restoration techniques for pulpless teeth. J PROSTHET DENT 1977;38:405-12. 3. Trabert KC, Caputo AA, Abou-Rass M. Tooth fracture-a comparison of endodontic and restorative treatments. J Endodont 1978;4:341-5. 4. Lu YC. Comparisons of the fractures of pulpless teeth. Chin Dent J 1987;6:2631.
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5. Trope M, Malts DO, Tronstad L. Resistance to fracture of restored endodontically treated teeth. Endodont Dent Traumatol 1985;1:108-11. 6. Leary JM, Aquilino SA, Svare CW. An evaluation of post length within the elastic limits of dentin. J PROSTHET DENT 1987;57:277-81. 7. Sorensen JA, Martinoff JT. Intracoronal reinforcement and coronal coverage: a study of endodontically treated teeth. J PROSTHETDENT 1984;51:780-4. 8. Hunter AJ, Feiglin B, Williams JF. Effects of post placement on endodontically treated teeth. J PROSTHETDENT 1989;62:166-72. 9. Eissmann HF, Radke RA Jr. Postendodontic restoration. In: Cohen S, Burns RC, eds. Pathways of the pulp. 4th ed. St. Louis: The CV Mosby Co, 1987:640-83. 10. Potashnick SR, Weine FS, Strauss S. Restoration of the endodontically treated tooth. In: Weine FS, ed. Endodontic therapy. 4th ed. St. Louis: The CV Mosby Co, 1989653.98. 11. Zakariasen KL. Preparation for restoration and temporization. In: Walton RE, Torabinejad M, eds. Principles and practice of endodontics. Philadelphia: WB Saunders, 1989:249-65. 12. Farah JW, Craig RG, Sikarskie DC. Photoelastic and finite element stress analysis of a restored axisymmetric first molar. J Biomech 1973;6:511-20. 13. Thresher RW, Saito GE. The stress analysis of human teeth. J Biomech 1973;8:443-9. 14. Davy DT, Dilley GL, Krejci RF. Determination of stress patterns in root-filled teeth incorporating various dowel designs, J Dent Res 1981;60:1301-10. 15. Wheeler RC. An atlas of tooth form. Philadelphia: WB Saunders, 1962. 16. Ko C-C. Stress analysis of pulpless tooth: effects of casting post on dentin stress distribution. Master’s thesis. Taipei, Taiwan: Yang-Ming Medical College, 1989.
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17. DeSalvo GJ, German RW. ANSYS Engineering Analysis System User’s Manual. Houston, Pa: Swanson Analysis Systems, 19&X9:2.25.1-7. 18. Caddell RM. Deformation and fracture of solids. Englewood Cliffs, NJ: Prentice-Hall, 1980:69-73. 19. Reinhardt RA, Krejci RF, Pao YC, Stannard JG. Dentin stresses in post-reconstructed teeth with diminishing bone support. J Dent Res 1983;62:1002-8. 20. Pao YC, Reinhardt RA, Krejci RF. Root stresses with tapered-end post design in periodontally compromised teeth. J PROSTHETDENT 1987;57: 281-6. 21. Svensson NL, Valliappan S, Wood RD. Stress analysis of human femur with implanted Charnley prosthesis, J Biomech 19’77;10:581-8. 22. Huiskes R, Chao EYS. A survey of finite element analysis in orthopaedic bionmchanics: the first decade. J Biomech 1983;16:385-409. 23. Popov EP. Introduction to mechanics of solids. Englewood Clitl’s, NJ: Prentice-Hall, 1968. 24. Zar JH. Biostatistical analysis, 2nd ed. Englewood Cliffs, NJ: PrenticeHall, lL984. 25. Weinstein AM, Klawitter JJ, Cook SD. Implant-bone interface characteristic of bioglass dental implants. J Biomed Mater Res 1980;14:23-9. 26. Williams KR, Edmundson JT, Rees JS. Finite element analysis of restored teeth. Dent Mater 1987;3:200-6.
Reprintrquests to; DR. MAW-CHANG LEE DEPARTMIZNT OF BIOMEDICAL ENGINEERING NATIONAL YANG-MING MEDICAL COLLEGE TAIPEI, TAIWAN 11221, R. 0. C.
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