0099-2399/92/1811-0540/$03.00/0 JOURNAL OF ENDODONTICS Copyright © 1992 by The American Association of Endodontists

Printed in U.S.A, VOL. 18, NO. 11, NOVEMBER1992

A Comparison of Intracanal Stresses in a PostRestored Tooth Utilizing the Finite Element Method Johnny G. Cailleteau, MS, DDS, MS, Monty R. Rieger, MS, PhD, and J. Ed Akin, PhD

The finite element method was used to compare stresses along the inner canal wall in four twodimensional models of an average maxillary central incisor. The four models evaluated were an intact incisor, an endodontically treated incisor, an endodontically treated crown-restored incisor, and a cylindrical post and crown-restored incisor. A horizontal static force, 1 Newton in magnitude, was applied to the lingual surface of each model and the maximum tensile, compressive, and shear stresses were calculated using the general purpose finite element program PAFEC 75. Results indicate that the stress patterns within the root are altered as a result of post insertion. Specifically, the maximum bending stresses are associated with the apical termination of the post, and post placement does not result in a uniform distribution of stress along the canal wall.

The finite element method utilizes a mathematical model which approximates the geometry of the object to be analyzed. The concept of the finite element method can be illustrated by the mechanics of a simple spring (9). The deflection of the spring (lengthening or shortening) is proportional to the magnitude of the applied force and stiffness of the spring as described by the following equation: Force (F) = Stiffness (K) x Deflection (A) Structures also behave in a manner analogous to the spring described previously. However, the stiffness or resistance to deformation under load, governed primarily by the geometry and modulus of elasticity of the object, is difficult to determine for complicated shapes. The finite element method attempts to simplify these calculations by allowing the operator to subdivide the structure into a number of smaller pieces, or elements. These elements may consist of triangular or quadrilateral shapes whose sides are connected by nodal points. The location of these nodes is defined using a two- or threedimensional coordinate system, thus providing a method for specifying the shape and size of the object. Nodal points also function by supplying points for applying forces or moments to the structure. Additionally, adjoining elements are connected through shared nodes. Subdivision of the structure in this manner facilitates determination of the structural stiffness and, ultimately, deflections and stresses. Calculation of these stresses allows the investigator to determine areas of high stresses or large deformations which could lead to fracture or failure of the structure. In addition, material properties can be assigned to individual elements, thus providing a method of analyzing structures containing more than one material. Although previous efforts (6, 7) have been aimed at determination of the maximum stresses occurring within the restored tooth, none of the investigations have explored the alteration in stress patterns along the canal wall when a post is placed. Examination of stresses and patterns along the canal wall may aid in determining the uniformity of stress distribution within a tooth, and may identify stress concentrations which could lead to crack formation and ultimate fracture failure. Therefore, the purpose of this study was to utilize the finite element method to determine the effects of a cylindrical stainless steel post on the stress levels and patterns occurring along the inner canal wall in a two-dimensional model of a maxillary central incisor.

Techniques for the restoration of endodontically treated teeth have been the subject of discussion and investigation by dental practitioners and researchers for many years. Of the proposed techniques, perhaps the most popular is the use of intraradicular retention via a cemented or threaded steel post. However, although posts are frequently placed within the roots of teeth to aid in retention of the crown, controversy exists concerning their effects on the structural integrity and stress distribution within the tooth. In attempting to describe the structural effects of posts, investigators have used several types of laboratory techniques. Initial investigations of the stresses within post-restored teeth were in vitro tests which utilized loads generated by testing machines to determine the force required for fracture (1, 2). Photoelastic analysis of birefringent materials has also been used to acquire information about the stress levels within teeth (3, 4). Although photoelastic analysis provides visual evidence of stress patterns within the model, difficulty in modeling objects consisting of more than one material is a significant disadvantage of this technique. A procedure originally proposed in 1956 by Turner et al. (5), and used extensively in engineering design for many years, has also recently been used to analyze stresses in teeth and dental prostheses (6-8). This method is known as the finite element method. 540

Finite Element Stress Analysis

Vol. 18, No. 11, November 1992

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X-Y ResUalntl

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541

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(t FiG 1. Model composition and direction of applied force. FIG 2. Pulp model.

M A T E R I A L S AND M E T H O D S Four two-dimensional models of an upper central incisor were created using the data of Wheeler (10). These models were buccolingual cross-sectional representations of a central incisor, 24 m m in length. All models included the periodontal ligament (width --- 0.175 m m (11)), cancellous bone, and cortical bone as illustrated in Fig. 1. The outer dimensions were held constant while the inner dimensions were altered to reflect treatment rendered. Model 1 represents an intact tooth, model 2 an endodontically treated tooth, model 3 a crown-restored endodontically treated tooth, and model 4 a post and crown-restored endodontically treated tooth. All models were restrained at the apical extent of the surrounding bone to prevent movement in the X (horizontal) or Y (vertical) directions (Fig. 1). All models were created using triangular and quadrilateral isoparametric elements containing six or eight nodes, respectively. Model 1 (pulp model), shown in Fig. 2, was an intact maxillary central incisor. The pulp chamber was constructed so that the diameter of the apical portion, 0.9 m m from the root end of the tooth, was the size of a #30 file (0.30 mm). The coronal orifice was modeled as a diameter of 1.3 mm. A smooth taper from the apical end to the coronal orifice was fabricated to form the remainder of the canal which was filled with pulp tissue. Model 2 (endo model), shown in Fig. 3, was similar to model 1 except that it approximated a tooth that had been endodontically treated, including an access preparation. The configuration of the canal space was constructed to simulate the final shape after endodontic instrumentation. The preparation was terminated 0.9 m m from the apex of the tooth with a master apical file #45. The coronal orifice opening was modeled as if enlarged to a #6 Gates Glidden drill (-#140 file), and a gradual taper from the apical end to the orifice opening was constructed to represent the final shape of the canal. The canal was filled with gutta-percha to within 0.9 mm of the root apex. Amalgam was placed into the access and extended to a point 1 m m coronal to the cervical bone. Model 3 (the crown model) had the same characteristics as model 2 except that it was restored with a full coverage, nonprecious metal crown. A cylindrical 1.5 m m in diameter

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FiG 3. Endo/crown model.

stainless steel post was placed into the canal of model 4 (post model) shown in Fig. 4, with 5.0 m m o f gutta-percha remaining in the apical segment. The coronal portion of the post was surrounded by amalgam, and a full coverage metal crown was the final restoration. The program used for the stress calculations was PAFEC 75 (Program for Automatic Finite Element Calculations; Pafec Ltd., Strelley, Nottingham) and was installed on a SUN (SUN Microsystems, Mountainview, CA) workstation. The final element mesh and corresponding stress graphs were printed using a Hewlett-Packard Laser Jet III (Hewlett-Packard Inc., Sunnyvale, CA). The following assumptions were made in conducting this analysis. The material properties (modulus of elasticity and Poisson's ratio) used for each of the components (enamel, dentin, etc.) were assumed to have identical properties in all

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Cailleteau et al.

Journal of Endodontics

MAXIMUM STRESS Hodzocael./r~ua/Force

f

Shear

LEGEND ~/j o.ow..o ENDOMODEL 1

~a ComprNa~e

PtJLPMODEL

-0.471 !i~i~i!ii~ii~;i;i~i~i;ili~ii~i~!i~!!~i~il .0 -0. J

0.263 0.533

Ter,.sile 3

-0'.4 ~.2

-6'.6

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FIG 5. Maximum intracanal stresses. TABLE 2. Percentage of change in maximum stresses along the canal wall (pulp as baseline)

FiG 4. Post model.

Models Type of Stress

TABLE 1. Material properties Material

Modulus of Elasticity (E, MN/m 2)

Cortical bone Cancellous bone Periodontal ligament Dentin Enamel Post material Amalgam Gutta-percha Crown material Pulp tissue

13,800.0 345.0 50.0 18,600.0 41,400.0 19,030.0 13,720.0 0.69 77,200.0 0.003

Pulp Poisson's Ratio (~) 0.26 0.38 0.30 0.32 0.30 0.305 0.33 0.45 0.33 0.45

(6)* (6) (12) (6) (8) (13) (14) (7) (15) (7)

* Numbersin parenthesesdenotereferences.

Shear Compression Tensile

RESULTS The maximum tensile, compressive, and shear stresses located along the inner canal wall of each of the models are shown in Fig. 5. The horizontal lingual load produced the

Endo (%)

Post (%)

-5.5 -26.3 +47.9

-4.6 -25.8 +48.3

+12.2 -1.1 + 102.7

Intracanal Stress Distribution Tensile Stress (FacialCanal Wall) 0.60.5-

..~"~

"~" 0.4-

E 0.3-

z

directions (i.e. isotropic), and the relationship of stress and strain (i.e. modulus of elasticity) was assumed to be linear for all materials. Finally, cement interfaces were considered to be of insignificant thickness. Table 1 lists the material properties used in this study. Each mode/was statically loaded with 1 Newton of force perpendicular to the lingual surface as illustrated in Fig. 1. This load was not considered to be a force consistent with clinical observations but was chosen for the purpose of inducing stresses within the respective structures so that relative comparisons could be made. Using PAFEC 75 and the loading described above, stresses were calculated along the inner canal wall. The maximum tensile, maximum compressive, and the maximum shear stresses were compared among the four models using bar and line graphs.

Crown (%)

0.2 0.1,

r

0 -0.1 10

12

14

16 18 ;~0 Distance from Incisal Edge

[ --~- Crown -+-- Endo

--'~"" Post ~

22

Pulp

24

I

FiG 6. Tensile stress distribution along facial canal wall.

largest tensile and shear stress in the post model while the largest compressive stresses were within the post and pulp models. Table 2 illustrates the percentage of change in maximum stresses considering the pulp model as baseline. The endo and crown models demonstrated a decrease in shear and compressive stresses of 5 and 25%, respectively, while there was an increase of approximately 48% in the maximum tensile stress. The post model exhibited an increase of 12% in shear stress and a decrease of 1% in compressive stress. In addition, there was a 102% increase in the maximum tensile stress in the post model as compared with the pulp model.

Vol. 18, No. 11, November 1992

Figure 6 illustrates the variation of tensile stress levels on the facial canal wall as a function of distance from the incisal edge. All models exhibited fluctuation in stress levels along the length of the canal. Within the post model, tensile stress increased sharply at the termination of the post and represented the maximum bending stress within this model. In contrast, the pulp, crown, and endo models demonstrated a gradual increase to the maximum stress which occurred approximately 2 m m below the orifice level. The addition of a crown to the endo model did not appreciably alter the pattern or magnitude of stresses within the canal for the load tested. DISCUSSION Stresses are produced as a result of loads imposed on a structure. The distribution or pattern of these stresses is the result of the angle of the load and the geometry of the object. In addition, notches or imperfections (inclusions or cracks) present within the material may cause localized increases in the magnitude of the stresses, known as stress concentrations. These stress concentrations can contribute to the failure of the material through crack formation and an increased likelihood of fatigue failure. The results of the present investigation indicate that placement of a post within the tooth alters the pattern of stress along the root canal wall as compared with an intact tooth. As seen in Fig. 6 the pulp and crown/endo models demonstrated a gradual increase in stress levels to a maximum which occurred approximately 2 m m apical to the canal orifice. In contrast, the post-restored model demonstrated a decreased level of stress along the coronal facial portion of the root surface which peaked abruptly near the apical end of the post. This area of peak stresses within the post model appears to be the result of the maximum bending occurring at the apex of the post. This result is in agreement with the previous work of Williams and Edmunson (6) and Davy et al. (7). The resulting stress patterns within the post model contradict the belief that posts strengthen the tooth by evenly distributing the external force acting on the tooth. Instead, the pattern of stresses within the post-restored tooth may actually result from stiffening the coronal segment of the tooth where the post is placed. As previously noted, the maximum bending stress present in the nonposted models was located within 2 m m of the orifice. However, when the post was placed, the maximum stresses were shifted to the apical end of the post. The addition of a post altered the flexure of the tooth by stiffening the coronal posted section and shifting the flexure point to a more apical level as compared with the pulp and crown/endo models. The effect of this stiffening would force the nonposted apical portion of the tooth to deform at the post apex, resulting in the increase in stress seen in Fig. 6. The cyclic loading and unloading of an incisor during mastication requires that the fatigue life of the root dentin be considered. A fatigue failure is characterized by cycles of stress and strain which result in a deterioration of the material and progressive cracking eventually producing a fracture (16). Since the maximum bending stresses occur in connection with the apex of the post, any inclusions or defects within the wall of the dentin near the apical end of the post would create stress concentrations which ~ould increase the risk of a fatigue

Finite Element Stress Analysis

543

crack formation, thus reducing the useful life of the tooth. Therefore, defects introduced during endodontic treatment or post space preparation could become areas contributing to stress concentrations as a consequence of post placement. Under the conditions of this investigation the maximum tensile stress within the post-restored tooth was larger than the other models, especially the pulp model. Lehman (I 7) has shown that the tensile strength of dentin is much lower than its compressive strength, and, since dentin behaves as a brittle substance in tension (6), the magnitude of the tensile stresses could be responsible for initiating a crack within the canal wall. A cement layer was not included in this study. Leary et al. (18), using strain gauges, speculated that a cement layer would act as a stress breaker. However, their results indicate that there was a significant increase in the strain within the root structure when comparing strain present before post cementation to postcemented strain in two of the four cementing mediums. Clinically, when a post is cemented, pressure is applied to ensure that the post is fully seated. It is likely that cement is displaced from the end of the post and the apical end of the post is actually adjacent to tooth structure as modeled in this investigation. Also, should any cementing medium remain in the area of the post apex, the tensile stresses occurring in this area would likely have a detrimental effect on the integrity of the cement. Furthermore, as stated above, the post's stiffening effects may be more of a factor in determining where the maximum bending stresses are located than the presence of a cement layer. A full coverage nonprecious all-metal restoration was chosen instead of the traditional porcelain fused-to-metal. In this study, the crown was present to act primarily as a point of load application and to transmit the load to the root structure. Since it was used in this manner it was felt that there were no significant differences, structurally, between the porcelain fused-to-metal and the all-metal crown. In addition to the results of this study, there are other factors that should be mentioned concerning post placement. Procedural errors during post space preparation including perforation, disturbance of the apical seal, and removal of the entire gutta-percha filling may significantly affect the overall prognosis of the case. Also, in cases where failure ofendodontic therapy has occurred, retreatment options may be limited to surgery, due to the inability to remove the post from the canal space. Unquestionably there are instances where the addition of a post is necessary to retain a core and crown. In these cases the operator must place a post even though there may be other detrimental effects. However, based on the results of this investigation, placement of a post would not be recommended as a routine procedure after endodontic therapy in a maxillary central incisor if there is sufficient remaining tooth structure to support a restoration. Within the parameters and conditions of this investigation, the following conclusions were drawn concerning the effects of a post placed within a maxillary central incisor: Stress patterns within the root are altered as a result of post insertion. Maximum bending stresses within the canal are associated with the apical termination of the post. Post placement does not result in a uniform distribution of stress along the canal wall.

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Cailleteau et al.

This study was part of a thesis submitted in partial fulfillment of the requirements for a Master of Science degree at The University of Texas, Houston Health Science Center, Dental Branch, Houston, TX. This research was presented in 1989 at the 46t" Annual Meeting of the American Association of Endodontists in New Orleans, I_A, where it was awarded second place in the Graduate Student Section of the Oral Research Seminars. The authors acknowledge the personnel in the Rice University CAESAR lab for their invaluable guidance and technical expertise. Dr. Cailleteau is assistant professor, Department of Oral Health Practice, Endodontic Section, University of Kentucky, College of Dentistry, Lexington, KY. Dr. Rieger is associate professor, Department of Oral Biomaterials, University of Texas, Houston Health Science Center, Dental Branch, Houston, TX. Dr. Akin is professor and chairman, Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX. Address requests for reprints to Dr. Johnny Cailleteau, Department of Oral Health Practice, University of Kentucky, College of Dentistry, Lexington, KY 40536-0084.

References 1. Kantor ME, Pines MS. A comparative study of restorative techniques for pulpless teeth. J Prosthet Dent 1977;36:405-12. 2. Brandal JL, Nicholls JI, Harrington GW. A comparison of three restorative techniques for endodontically treated anterior teeth. J Prosthet Dent 1987;58:161-5. 3. Assif D, Oren E, Marshak BL. Photoelastic analysis of stress transfer by endodontically treated teeth to the supporting structure using different restorative techniques. J Prosthet Dent 1989;61:535-43.

4. Mattison GD. Photoelastic stress analysis of cast-gold endodontic posts. J Prosthet Dent 1982;48:407-11. 5. Turner MJ, Clough RW, Martin HC, and Topp LJ. Stiffness and deflection analysis of complex structures. J Aero Sci 1956;23:805-23. 6. Williams KR, Edmunson JT. A finite element stress analysis of an endodontically restored tooth. Engineer Med 1984;13:167-73. 7. Davy DT, Dilley GL, Krejci RF. Determinations of stress patterns in root filled teeth incorporating various dowel designs. J Dent Res 1981 ;60:1301-10. 8. Thresher RW, Saito GE. The stress analysis of human teeth. J Biomech 1973;6:443-9. 9. Akin JE. Finite element analysis for undergraduates. London: Academic Press, 1986:27-44. 10. Wheeler RC. An atlas of tooth form. Philadelphia: WB Saunders, 1984. 11. Grant DA, Stern IB, Everett FG. Orban's periodontics: a concept-theory and practice. 4th ed. St. Louis: CV Mosby, 1972. 12. Weinstein AM, Klawitter JJ, Cook SD. Implant-bone interface characteristics of bicglass dental implants. J Biomed Mater Res 1980;14:23-29. 13. McGregor CW. Mechanical properties of materials. In; Baumeister T, Marks LS, eds. Standard handbook for mechanical engineers. 7th ed. New York: McGraw-Hill, 1967:5-6. 14. Stanford JW, Weigel KV, Paffenbarger GC, Sweeney WT. Compressive properties of hard tooth tissues and some restorative materials. J Am Dent Assoc 1960;60:746-60. 15. Phillips RW. Skinners science of dental materials. 7th ed. Philadelphia: WB Saunders, 1973. 16. Gere JM, Timenshenko SP. Mechanics of materials. 3rd ed. Boston: PWS-KENT Publishing Co., 1984:116-9. 17. Lehman ML. Tensile strength of human dentin. J Dent Res 1967;46:197-201. 18. Leary JM, Jensen ME, Sheth JJ. Load transfer of posts and cores to roots through cements. J Prosthet Dent 1989;62:298-302.

A Word for the Wise Speaking of tautologies (who was speaking of tautologies?), "general consensus," "fully completed," and "universal panacea" are famous. Fortunately, the scientific literature is free of these grammatical solecisms. Why to utter one would be worse than suffering a traumatic injury. Andrea Wiley