#10 Stress Analysis of an Abutment Tooth with Extracoronal Magnetic Attachment -Introduction of Nonlinear Property into Three-Dimensional Finite Element Method-

R. Kanbara, Y. Nakamura, Y. Ohno, A. Ando, H. Kumano, T. Masuda, M. Sakane,Ｙ. Tanaka
	Department of Removable Prosthodontics, School of Dentistry, Aichi-Gakuin University

Introduction

　The importance of understanding and defining the mechanical properties of supporting tissues is of interest in prosthodontic investigation. The finite element method (FEM) has been commonly used for dental application analysis in recent years. However, it has proven difficult to analyze soft tissues such as the oral mucosa and periodontal ligament of abutment teeth using currently available methods. This includes a simple theoretical analysis, a model experiment, and the FEM for elastic displacement due to their viscoelastic properties. It has been reported that the dynamics in denture supporting mucosa during occlusion is complex. The oral mucosa and the periodontal ligament demonstrate a viscoelastic property specific to the soft tissues. During load application, these tissues, unlike other tissues, show a non-linear, phase proportioned complex elastic displacement. The initial displacement is rapid in the low-load condition, but then, slows as the load increases (secondary displacement), thus demonstrating a two-phase characteristic.

Objective

　The purpose of the present study was to construct a Finite Element Model demonstrating incorporated nonlinear behavior to simulating physiologic tissue viscoelasticity for evaluation of complex oral tissues including the oral mucosa and abutment tooth periodontal ligament.

Methods

1. Introduction of Nonlinear Property

	A mathematically described nonlinear property was introducted into a Finite Element Analysis model to simulate oral mucosa and periodontal ligament using he following methods, and a stress analysis was performed: Analysis model Fig. 1 shows the analysis model created by Ando from our department. The three-dimensional FEM model was constructed from patient CT data according to the method developed by Ando for accurate morphologic simulation. A fixed partial denture with attachments was designed on the analysis model, and connective crowns incorporating extracoronal magnetic attachments were placed on the abutment left lower canine, first and second premolars for the loss of left lower first and second molars. The comprising elements type designations were pentahedral element for the periodontal ligament, and tetrahedral for the other tissues. The point total of the analysis model was 27,083, and a total element count was 118,085.
		Fig 1： Analysis Model

	1) Material Properties (Mucosa and Periodontal ligament)
		Material constants for oral mucosa and periodontal ligament under load application were determined prior in order to simulate analysis model comlex tissue behavior. Material constants were set to for automatic control by the computer program to account for load weight. This permitted simulation of the human behavior by the model. Prior reports by Kishi (Fig 2) and Ogita (Fig 3) were used as reference for the respective measurement of oral mucosa and periodontal ligament. Fig. 4 shows the sequential process. 　　The denture was removed from the analysis model shown in Fig. 1, and loads were directly applied on to the oral mucosa and abutments to create the program. Patran 2007 r1b (MSC) was used as the modeling software, and Marc 2008 r1 was used for analysis solution solving.

	(1) Analysis 1
			The measurement conditions shown in references (Fig 2 and 3) were reproduced as closely as possible to simulate the initial displacement. ① Mucosa 　　Fig. 5 shows the load conditions of the oral mucosa.The load applied was 0.1 kg / cm2, and the loading range was 60 mm2. The surface load was applied, and maximum vertical displacement of the oral mucosa was measured. Material properties (Young modulus and Poison ratios) of the oral mucosa were determined so that those values were close to the actual measurement value found in human subjects under the same loading condition (82.5 m). The oral mucosal elements stress value of maximum vertical displacement was measured as well(von Mises equivalent stress). The results are shown in Table 1.

② Periodontal ligament
　　Fig. 6 shows the load condition of the periodontal ligament. A 100 gf contact load was applied on the crown of abutment left lower canine in the tooth axial direction. The maximum vertical displacement was measured, and the material constant was determined so that the value was close to the actual measurement value in humans (10.0 μm). The maximum stress value in the periodontal ligament element　 was also measured (von Mises equivalent stress). The results are shown in Table 2.

	(2) Program for Material Constant alteration
			A program was specifically created for conversion of the material constant for oral mucosa and periodontal ligament during the analysis. This program was used as a user subroutine of the analysis solver Marc 2008 r1 in order to reflect the program in the analysis. The program was reflected in the analysis through Intel (R) Visual Fortran complier 10.1. 　　Element numbers of the oral mucosa and the periodontal ligament were assigned to the program so that those material constants can be changed automatically. Stress value at the subject elements was extracted as von Mises stress. The program was set so that the stress values shown in Tables 1 and 2 were used as material constants until the stress reached a conversion point, and then material constants changed when further stress was applied. This program automatically changes the material constant when the further stress was applied. The results suggested that the stress values arising in the oral mucosa and the periodontal ligament automatically change when they exceeded the value in the initial displacement.

	(3) Analysis 2
			A second analysis was performed using the program described above to reproduce the when higher loading exceeding the first analysis loading was applied. The analysis conditions were identical to the first analysis conditions except for the new higher loading force evaluated. ① Mucosa 　　The load applied on the oral mucosa was 3.0 kg / cm2. The maximum vertical displacement of the oral mucosa was measured in the same manner as the first analysis, and material constants of the oral mucosa were determined so that those values were close to the actual measurement value in humans under the same loading condition (210.0 m). The results are shown in Table 3.


			② Periodontal ligament 　　The load applied on the abutment was 500 gf. The maximum vertical displacement of the abutment was measured, and material constants of the periodontal ligament were determined so that those values were close to the actual measurement value in humans under the same loading condition (20.0 m). The results are shown in Table 4. The program allows the simulation of the initial displacement when a load of < 0.1 kg / cm2 and < 100 gf were applied on the oral mucosa and the periodontal ligament, respectively, and the secondary displacement when a heavier load was applied. Stress-strain curve obtained by the computation history of the analysis was used to check if the material property conversion in the second analysis was performed using the stress value obtained in the first analysis.

2. Stress analysis

A load was applied on the analysis shown in Fig. 1 using the material nonlinear program developed in the present study. The detailed conditions for the analysis were as follows: The analysis type used was elastic stress analysis. General purpose finite element program Marc 2008 r1 was used as an analysis solver.

	1) Boundary conditions
		Coulomb’s friction coefficient μ=0.01 was applied as a contact condition at a contact point between a denture and retainer. Fig. 7 shows the constraint condition and he loading condition.. A compete constraint was applied to the inferior border of the mandible in the X, Y, and Z directions.Vertical surface pressure loads of 10, 20, and 30 N were applied on the occlusal plane of a denture. The loading range was entire occlusal surface of the denture.

	2) Material properties
		Table 5 shows components and mechanical property values of the analysis model. The same material constant was used for a crown, attachment, and metal frame. The material constants of the oral mucosa and periodontal ligament were changed by the program to reflect the analysis, and the initial and secondary displacements were reproduced.

Results

1. Introduction of nonlinear property
	1) Comparison of analysis result and actual measurement 　　Nonlinear properties were introduced into the finite element model for the analysis. The comparison was made between actual measurements in humans described in the references (Fig 2 and 3) and analysis results.
		① Mucosa 　　Fig 8 shows the vertical displacement of the oral mucosa. ② Periodontal ligament 　　Fig 9 shows the vertical displacement of the abutment. 　　 The initial and secondary displacements of the oral mucosa and the periodontal ligament were simulated to be close.

2. Stress analysis results
Load was applied to the analysis model shown in Fig. 1 using the material nonlinear program, followed by the analysis. The analysis results of the displacement amount of a denture and abutment were as follows:
	1) Displacement of an abutment 　　 Fig 10 shows the displacement of an abutment under three loading conditions. The displacement amount increased with increasing applied load, but no commensurate increase against the load was observed.
	2) Displacement of a denture 　　 Fig 11 shows the vertical displacement of the inferior border of a denture under three loading conditions. The displacement amount increased with increasing applied load, but no commensurate increase against the load was observed.

Discussion

1. Finite element method
　　There have been many reports on the stress analysis in dentistry. Although the analysis subjects vary from hard tissues to implants, few reports are available on the stress analysis of the soft tissues. The dearth of soft tissue analysis studies suggests and inherent difficulty in the evaluation and simulation of viscoelasticity and soft tissues behavior. An elastic element with true elastic properties and alternative elements can be used to simulate the viscoelasticity of the soft tissues. Solid biologic foundational information of oral mucosa and periodontal ligament tissues is required to accurately use these methods. However, few reports are available on physiologic behavior of the oral mucosa and the periodontal ligament. In the present study, the difference in the displacement between the oral mucosa and the periodontal ligament under loading conditions were evaluated. The material constant was changed using adaptational programming with a user subroutine utilizing loading conditions based upon actual body measurements for close simulation of oral mucosa and periodontal ligament behavior. This study will be of great value to future stress analyses investigations of intraoral soft tissues.

2. Introduction of nonlinear property
　　Two phase behavior of the oral mucosa and the periodontal ligament can be simulated by using the program in which the stress value of the element under load application was set to be a turning point of the material constant. This method allows simulation approaching human physiologic tissue boundaries. This program also allows the simulation of both oral mucosa and the periodontal ligament tissues by different analysis models and using other materials to show complex behaviors. The methods in the present study was based on the previous efforts by Ishida (The First Department of Prosthodontics, School of Dentistry, Aichi-Gakuin University).
1) Mucosa
　　Initial and secondary displacements of the oral mucosa against load were simulated by finite element method using two material constants. However, due to the complex human behavior as reported by Kishi (Fig 2), no significant change was observed in the displacement with an increase in the load above 1.5 kg/cm2. The program used in the present study was designed for simulation of two-phase human tissue behavior by change of the material constant value. The results shown in Table 9 suggest a minor difference in the displacement of tissues.
Further studies are required to simulate complex human tissue behavior more accurately by the subdivision of displacements against different loads based upon the program and setting the material constant for each displacement.
2) Periodontal ligament
　　Initial and secondary displacements of the periodontal ligament against load were simulated by finite element method using two material constants.
The analysis of the periodontal ligament was performed in the program using two material constants to compare the results with Ogita’s report shown in Fig 9 . A minor difference was noted in the displacement between 100 gf and 300 gf. Unlike the oral mucosa where the load is directly applied, the load is transmitted to the periodontal ligament through an abutment. In the analyses 1 and 2, the material constants were determined to be close to the actual measurements in humans. In contrast, there are minor variations in stress generated in the periodontal ligament element depending on the morphology of abutment roots under 300 gf load application.
The results suggest the necessity of subdivision of displacements against different loads and the setting of the material constant for each displacement in the periodontal ligament.

3. Stress analysis results
　　Although the greater displacement was observed at 30 N load application than 10 N, an increase in the displacement was not in proportion to the load as was the case in material linear analysis. This is considered to be due to the fact that stress arose in the oral mucosa and periodontal ligament elements by load appication reached a turning point, resulting in an increase in rigidity of elements and an inhibition of displacement. The rigidity of the oral mucosa and periodontal membrane increased by using this system, allowing the application of heavy loads that could not be used in the conventional material linear analysis. This system provides a more practical stress analysis.

Conclusion

　A material nonlinear property was introduced to the finite element method in order to accurately simulate the viscoelastic behavior of oral mucosa and the periodontal ligament. The present study demonstrated the analysis results and the availability of the material constant changing program can be effectively used in material nonlinear analysis. The following conclusions were drawn: 1. Two-phase behavior similar to that found in humans was simulated by introducing a material nonlinear property into the oral mucosa and the periodontal ligament estimations of the finite element model, allowing practical simulation. 2. The analysis of the mechanical properties demonstrated a decrease in the displacement of oral mucosa and periodontal ligament with an increase of Young’s modulus and Poisson’s ratio.

References

1. Akihiro Ando, Yoshinori Nakamura, Ryo Kambara., et al. Stress Analysis of the Surrounding Tissue of a Extracoronal Attachment using Three Dimensional Finite Element Method, The Journal of the Japanese Society of Magnetic Applications in Dentistry. Vol.18, No.1: 2009
2. Takashi Ishida, Improvement of Finite Element Method through the introducing the Nonlinear Property into the Visco-Elastic Tissues and Sliding Mode into the Contact Factor, The Aichi-Gakuin journal of dental science 39(1): 2001
3. Masataka Kishi. The relationship between the pressure surface of the alveolar mucosa and the artesian displacement, Dental Journal 72. 6: 1972
4. Kunihisa Ogita. Measurement of Three-Dimensional Movement of Anterior teeth. J. Japan Prosthodont. Soc. 27 : 1983

Discussion Board