distribution around dental implants: Influence of tructure, length of implants, and height of mandible H. J. A. F. Bosman,
Meijer,
DDSa
J. H. Kuiper,
MSC,~
F. J. M.
Starmans,
DSc,”
and
PhDd
University of Utrecht, Faculty Orthopaedics, Nijmegen, The
of Medicine, Netherlands
Utrecht;
and University
of Nijmegen,
Institute
for
The stress distribution around dental implants was investigated by use of a two-dimensional model of the mandible with two implants. A vertical load of 100 N was imposed on abutments or the bar connection. The stress was calculated for a number of superstructures under different loading conditions with the help of the finite element method. The length of the implants and the height of the mandible were also varied. A model with solitary abutments showed a more uniform distribution of the stress when compared with a model with connected abutments. The largest compressive stress was also less in the model without the bar. Using shorter implants did not have a large influence on the stress around the implants. When the height of the mandible was reduced, a substantially larger stress was found in the bone around the implants because of a larger overall deformation of thelowerjaw. (J PROSTHET DENT 1992;68:96-102.)
ack of retention and stability of a denture is one of the major complaints of edentulous patients. The use of endosseous implants, by which the prosthesis can be supported, is a successful and popular therapy. Nevertheless, the use of implants is not without problems. Biomechanical influences play an important role in the longevity of bone around imp1ants.l Forces on the denture such as during chewing are transferred to the implants and lead to stress in the bone surrounding the implants. Too much stress can lead to bone resorption, which eventually causes loosening of the implant. Soltesz and Siegele2 demonstrated that regions of stress concentration seen in a laboratory model coincided with resorption zones observed in the bone of the jaws of dogs provided with the same type of implants investigated in the laboratory model. Several investigations have already been performed to select implant configurations that will cause a minimum of stress when being loaded.3-7 Other studies have been published that deal with the influence of the interface (bone-implant) condition on the amount of stress.s g The reported stress calculations were restricted to one implant in a small section of bone. No study has been published to determine the influence of the length of the
Supported in part by IBM Nederland aAssistant Professor, Department of Prosthodontics, and Special Dental bResearch Engineer, Biomechanics jmegen. CResearch Engineer, Department of Prosthodontics, and Special Dental dProfessor and Chairman, Department gery, Prosthodontics, and Special Utrecht.
10/l/27224
96
N.V. contract No. 473104. Oral-Maxillofacial Surgery, Care, University of Utrecht. Section, University of NiOral-Maxillofacial Surgery, Care, University of Utrecht. of Oral-Maxillofacial SurDental Care, University of
implant, the height of the mandible, and the type of superstructure on the stresspattern. In the present study the stressdistribution was calculated in a two-dimensionalmodel representing the frontal section of an edentulousmandible provided with two implants. Bending of the structures was permitted in this model by omitting support at the bottom and instead fastening at the left and right side. These aspectsmake the model a more realistic representation of the clinical situation.
The influence of (1) the presenceof a bar between the implants, (2) the length of the implants, and (3) the height of the mandible on the distribution and the amount of stressin the bone surrounding the implants wasanalyzed. MATERIAL
AND
METHODS
A two-dimensionalmodel was constructed as a first approach to investigate the relative effect of severalfactors. A section of an edentulous mandible was composed.The mandiblewastranslated to a pieceof bone70mm in length, which is approximately the distance between the left and right mandibular angles(Fig. 1). The behavior of the mandible wasrepresentedby a two-dimensionalmodel. The finite element method (FEM) was used to calculate the stressin the bone surrounding implants by which the set of stress-determiningequations could be calculated. The basisof the FEM is a model divided into small elements. Each elementis consideredinterconnected to each of its neighboring elementsat a number of discrete points called nodes.The displacementof each of the nodesmust be calculated to determine the stress throughout the structure. The large number of equations to be solved necessitated the useof a computer.lo Investigation by usingFEM is fa-
JULY1992
VOLUME69
NUMBER
1
STRESS
DISTRIBUTION
AROUND
DENTAL
IMPLANTS
7 lmm
15 m m
i 1
I
I
4 mm
70 mm pd
Fig. 1. Translation implants connected
= tort.
bone
of edentulous with bar.
m
mandible
vorable to other methods because the dimensions and the properties of all composite materials can easily be simulated and thus varied. Stress and displacement can be calculated. The finite element program MARCll was used. FEM enables including material thickness and properties of structures in the third dimension (plane-strain). The influence of the cortical bone at the labial and lingual side of the section of bone was factored by means of a side plate.i2 The 3 mm thick side plate connected the upper and lower cortical layer. The model was provided with two cylindrical implants. The implants had the mechanical properties of titanium. The materials were regarded as isotropic and linearly elastic. The dimensions and properties of the model were similar to a human mandible (Table I). A fixed bond between bone and implant along the entire interface was assumed. The close apposition of titanium and bone and the ingrowth of bone in the surface structure of titanium means that under any subsequent loading, the interface moves as a unit without relative motion between the bone and titanium. The element distribution of a model with connected abutments is shown in Fig. 2. To simulate the clinical situation the model was not supported at the bottom, but the left and the right side of the model were fastened to allow bending of the mandible.
THE
JOURNAL
OF PROSTHETIC
DENTISTRY
into
q
spong.
two-dimensional
bone
model
with
two
Fig. 2. Element distribution of model with connected abutments (mandible of 15 mm height and implants 11 mm in length).
Normal biting as in chewing was simulated by vertically applied loads of 100 N.13 Koolstra et al.14 reported that the largest possible biting forces in the incisal and canine region were found approximately 35 degrees relative to the horizontal plane in the dorsal direction. In case of a more horizontal direction, bite forces soon decreased to a much lower level. Different loading conditions of the implants were analyzed under various circumstances. 1. Seven conditions were analyzed for two geometrically different models (Fig. 3), one with a height of the mandible of 15 mm and another with a height of 7 mm. In the latter situation the implants had a length of 3 mm and a spongious bone layer of 2 mm.
97
MEIJER
1
4 !
a vertical load uniformly a bar as connector between
dlstrlbuted on 1 tkm implants.
a vertical
dlstrlbuted
load
a vertical load a bar present.
uniformly
on one
ET AL
on
abutment, I
h
!
i
I
load on one atutment, / 5 a vertical no bar present.
I
I
i
6 a vertical bar
I
load uniformly distributed on and extensions (10 mn on both sides).
a vertical load a bar present.
Fig.
on one extension,
3. Analyzed
loading
conditions
Table
and designs.
I.
Dimensions
and properties
of FE-model
Mandible Height
-
=
path of evaluation
A---B-=-C---D Fig. 4. Path in bone stress was evaluated.
around
left implant
along
which
2. The influence of the length of the implant was determined by analyzing implants with lengths of 13, 9, 7, 5, 3, and 1 mm. The calculations were limited to conditions 1 and 2 (Fig. 3). 3. The influence of the height of the mandible was determined by modifying the height of the model to 13,11,9, and 7 mm. The length of the implant was set at 3 mm in this example. The calculations were limited to conditions 1 and 2 (Fig. 3). The stress pattern in all situations was evaluated along a path in the bone around the left implant, which started at the upper left corner (A) and ended, via the apex of the implant (B and C), at the upper right corner (0) (Fig. 4). With respect to bone remodeling, maximum and minimum principal stresses may be important. Moderate stress values induce bone formation, but when they become too large they cause resorption. Therefore the stresses plotted in the figures are (1) the maximum and minimum principal stresses as evaluated at each material point or are the extreme stresses, which are the highest maximum and (2)
98
15 mm 12 mm
Width Thickness of cortical layer Cranial Labial Caudal Lingual Properties of cortical bone Modulus of elasticity Poisson’s ratio Properties of spongious bone Modulus of elasticity Poisson’s ratio Dimensions of implant Length Diameter Dimensions of abutment Length Diameter Dimensions of metal bar Length Height Width Distance between metal bar and bone Properties of implant Modulus of elasticity Poisson’s ratio Properties of bar: Modulus of elasticity Poisson’s ratio
lmm lmm 4mm 2mm 13,700
MPa
0.30 1,370 MPa 0.30 11.0 mm 3.5 mm
4.5 mm 3.5 mm 21.5 mm 2.5 mm 1.5 mm 4.5 mm 103,400
MPa
0.35 103,400 MPa 0.35
the lowest minimum principal stress values as found in the entire model. Because examples 1, 2, 3, and 6 have symmetric conditions, stresses were calculated for only half of the model. The symmetric condition was used to determine the
JULY
1992
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DISTRIBUTION
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model with bar print.
stres-s
snost
(MPa)
-26’
path
15
mm mandible,
prlnc.
along
left Implant
max principal stress
+
mln
I 012345678
’
I
’
’
(mm) principal
+
stress
Infl. length
11 mm implants
stress
(MPa)
I
length model
of Implant,
with
1
J
I 9
of Implant bar
+
I 10
I 11
I 12
model
without
15 mm mandible
on stress distribu-
: most
negative
stress
(MPa)
-,.1,,, I E
-*Oh 6
along
20
15
left
max prlnclpal stress
Implant +
26
mln prlnclpal stress 5
stresses for the other half. Because two-dimensional models were used, an assumption was made for the third outof-plane direction. Plane-strain models were chosen, which were sufficient for the comparison between different models.
conditions
Fig. 5, A shows the minimum and maximum principal stress around the left implant in loading condition 1. The stress was negative at any location, that is, only compressive stress. The most negative stress was located at the neck of the implant in the upper cortical layer (-18.2 MPa). This stress was only found at the distal side of the implant and not at the mesial side. Fig. 4, B shows the results of loading condition 2. The most negative stress was -11.3 MPa, almost equal at the distal and mesial side. The most negative stress of the seven loading conditions for two different heights (15 and 7 mm) of the mandible was always found at the neck of the implant in the upper cortical layer.
JOURNAL
OF PROSTHETIC
6
7
a
9
height
Fig. 5. A, Stress distribution around left implant in condition 1; 11 mm implants in 15 mm high mandible; vertical load of 100 N distributed on bar. B, Stress distribution around left implant in condition 2; 11 mm implants in 15 mm high mandible; vertical load of 100 N distributed on abutments (no bar).
THE
,
(mm)
11 mm Implants
RESULTS Different loading
..,.,
ci
10
path
15 mm mandible,
15
bar
0
-
I 14
(MPa) :
0
’ 13
(mm)
Fig. 6. Influence of length of implant tion (height of mandible 1.5 mm).
model without bar 6j
stress
/
5
-
negative
DENTISTRY
+ lnfi. height
mandible,
modal
with
10
11
of mandlble bar
-++
model
13
14
15
wlthout
bar
3 mm implants
Fig. 7. Influence of height of mandible tion (length of implants 3 mm).
Influence
12
(mm)
of the length
on stress distribu-
of the implant
The influence of the variation in length of the implant can be seen in Fig. 6, depicting the most negative stress as a function of the length of the implant. To facilitate analyzing the plot, points belonging to the same condition were connected with a line. In the model with connected abutments (condition 1) it can be seen that the shorter the implant the more negative was the stress value: from -17.9 MPa with an implant of 13 mm to -23.2 MPa with an implant of 1 mm. The same phenomenon occurred in the model without the bar (condition 2), although the stress was never as high as in the model with the bar: from -11.0 MPa with an implant of 13 mm to -15.3 MPa with an implant of 1 mm.
Influence
of the height
of the mandible
The results of varying the height of the mandible are plotted in Fig. 7. Again, only the most negative stress is
99
MEIJER
situation
rcdel mst (MaI
1
pj
4
1
i
3i-j
4
Fig. 8. Displacement of bony parts of model with connected abutments (A) and of model with solitary abutments (B) as consequence of vertical loading. Undeformed state is pointed out by dotted line (magnification of displacement X200).
given. The height of the lower jaw is set along the horizontal axis; the most negative stress along the vertical axis. The points belonging to the same situation are connected with a line. The stronger the reduction of the mandible, the higher was the stress. Tensile stress was not found in any of the situations or was negligibly small. DISCUSSION This study was performed to gain more insight in the influence of several variables on the stress distribution in the mandibular bone. Because of the assumptions made for the configuration of the two-dimensional model and the properties of the bone, the values of stress provided by the model are not necessarily identical to actual ones. Moreover it is not known at which amount of stress biologic changes such as resorption or deposition of the bony structures take place. However, the present model study is suitable for comparison of relative stress distributions between various implant situations. From the stress distribution in Fig. 5, it can be concluded that the largest compressive stress is situated in the upper cortical layer. This observation was also made by other authors.2, 3,5-g:i5, I6 This observation is not surprising, because the largest stress is often found near structure surfaces. Moreover, the cortical layer with a relative high modulus of elasticity strengthens this effect. In Fig. 5, B the stress is equally distributed at the distal and mesial side of the implant. If a bar is placed between the abutments, the stress is situated only at the distal side of the implant (Fig. 5, A). This location may be explained by the bending of the mandible as a result of the vertical load. In case of a structure with solitary abutments they are able to approach each other when the lower jaw bends. The stress is equally high in that situation at the mesial and distal side. In case of a structure with connected abutments, however, the bar does
100
j
7 ‘rrl
i-l
1
of 15 m m heiqht negative stress
ET AL
PC&l of 1 rim height most negative stress IFWit)
-18.2
-53.8
-11.3
-26.6
-19.4
-56.3
I
-27.5
-41.9
-107.5
Fig. 9. Most negative stress found in model of 15 mm height and of 7 mm height.
prevent movement of the abutments. When the structure is loaded, the mandible deforms, but the abutments are kept in place by the bar. This causes an extreme amount of stress at the distal sides of the connected implants. This explanation was confirmed by studying displacements of the same model (Fig. 8). Fig. 9 shows the results of different loading conditions and different designs of the superstructure. Apart from the most negative stress values seen in Fig. 5, the results of five other conditions are presented. In Fig. 9 it can be noticed that differences in stress are small between the conditions where the vertical load is distributed on the bar, where the vertical load is distributed on the abutments (bar present), and where the vertical load is distributed on the bar and extensions (conditions 1, 3, and 6). From a clinical viewpoint, conditions 4, 5, and 7 have an unfavorable prospect because of the unequal distribution of the chewing force on the superstructure. However, condition 5 does not show a disadvantageous effect from the point of view of stress distribution. The most negative stress is found if one extension is loaded with bar and extensions (condition 7). Fig. 9 shows also the results of the seven loading conditions of a mandible of 7 mm height with implants 3 mm long. The amount of compressive stress is much larger compared with a mandible of 15 mm height. This is partly because of a reduction of the length of the implant, but also because a larger deformation of the lower jaw as the mandible will bend more easily. When the stresses of the different loading conditions of the 7 mm mandible are compared with each other, it is found that the proportions are roughly the same as in the mandible of 15 mm height. The length of the implant does not have much influence on the amount of stress (Fig. 6) because the stress value mainly depends on bending of the mandible. Fig. 7 shows the influence of the height of the mandible. There is a strong increase in stress concentration when the
JULY
1992
VOLUME
68
NUMBER
1
STRESS
DISTRIBUTION
AROUND
DENTAL
IMPLANTS
normal
stress
ear
stresses
Fig. 10. Normal and shear stresses at point P. (See Appendix.)
Fig.
height of the mandible is reduced, which is caused by a larger deformation of the lower jaw. This can also be noticed in Fig. 9 comparing the several results. Comparing Figs. 6 and 7 reveals that the amount of stress depends more on the deformation of the jaw than on the length of the implant. Although the dimensions and properties of this two-dimensional model and the support of it at both ends simulates the human mandible only to a limited extent, it does give insight in the geometric behavior of bone as a result of chewing force. This study reveals that stress analysis of the bone around dental implants is incomplete if the model constructed does not account for deformation of the mandible. Given the interesting results of the stress analysis of the two-dimensional study, a three-dimensional model of the entire lower jaw is recommended for further investigation. A better translation of the boundary conditions would be possible. A horizontal load could be simulated by a threedimensional model in a direction corresponding with the clinical situation. Next to an improved biomechanical analysis, more clinical research and biologic research are necessary to predict the behavior of bone in the presence of a certain amount of stress.
stressbecauseof the overall deformation of the bone as a reaction to loading.
CONCLUSIONS The assumptions made for the composition of the model and its boundary conditions suggest reservations in translating the results of the FEM model to the clinical situation. The results indicate the following. The main stress peaks arise around the neck of the implant in the upper cortical layer. The presence of a bar between the abutments causes more negative stress when the structure is vertically loaded, compared with a situation in which no bar is present. Application of extensions does not improve the stress distribution. The possibility of an extreme amount of stress exists when the superstructure is not uniformly loaded. The length of the implant has little influence on the amount of stress in vertical loading. The height of the mandible has a large influence on the amount of
THE
JOURNAL
OF
PROSTHETIC
DENTISTRY
11. Front of infinitesimal material cubewith stresses. (See Appendix.)
REFERENCES 1. Skalak
R. Biomechanical
considerations
in osseointegrated
prostheses.
J PROSTHETDENT~~~~;~%~~~-~. 2. Soltesz U, Siegele D. Principal characteristics of the stress distribution in the jaw caused by dental implants. In: Huiskes R, Van Campen D, De Wijn J, eds. Biomechanics: principles and applications. The Hague, The Netherlands: Martinus Nijhoff, 1982:439-44. 3. Siegele D, Soltesz U, Scheicher H. Dental implants with flexible inserts-a possibility to improve the stress distribution in the jaw. In: Perren SM, Scheicher E, eds. Biomechanics: current interdisciplinary research. Dordrecht, 1985. 4. Soltesz U, Siegele D, Riedmueller J. Die Spannungsverteilungen urn ein stufenformiges Implantat im Modellversuch und im Vergleich zu einfachen Grundformen. Dtsch Zahnarztl Z 1981;36:571-8. 5. Siegele D, Soltesz U. Numerische Untersuchungen rum Einfluss der Implantatform auf die Beanspruchung des Kieferknochens. Z Zahnarztl Implantol 1987;3:161-9. 6. Rieger MR, Fareed K, Adams WK, Tanquist RA. Bone stress distribution for three endosseous implants. J PROSTHET DENT 1989;61: 223-8. 7. Kitoh M, Suetsugu T, Murakami Y, Tabata T. A biomathematical study on implant design and stress distribution. Bull Tokyo Med Dent Univ 1978;25:269-76. 8. Borchers L, Reichart P. Three-dimensional stress distribution at different stages of interface development. J Dent Res 1983;62: 155-9. 9. Soltesz U, Siegele D, Riedmueller J, Schulz P. Stress concentration and bone resorption in the jaw for dental implants with shoulders. In: Lee AJC, Albrektsson T, Branemark P-I, eds. Clinical applications of biomaterials. New York: John Wiley & Sons Inc. 1982. JA. The finite element method of stress analysis and 10. Von Fraunhofer its application to dental problems. In: Scientific aspects of dental materials, London and Boston: Butterworths, 1975:41-4. 11. MARC general purpose finite element program, revision K3. Palo Alto: MARC Analysis Research Corporation, 1988: user’s manuals A-F. 12. Huiskes R, Chao EYS. A survey of finite element analysis in orthopedic biomechanics: the first decade. J Biomech 1983;16:385-409. 13. Haraldson T, Car&on GE. Bite force and oral function in patients with osseointegrated oral implants. Stand J Dent Res 1977;85:200-8. 14. Koolstra JH, Van Eijden TMGJ, Weijs WA, Naeije. A three-dimensional model of the hurean masticatory system predicting maximum possible bite forces. J Biomech 1988;21:563-76. 15. Soltesz U, Siegele D. Einfluss der Steifigkeit des Implantat materials auf die im Knochen erzeugten Spannungen. Dtsch Zahnarztl Z 1984;39: 183-6. 16. Kitoh M, Matsushita Y, Yamaue S, Ikeda H, Suetsugu T. The stress distribution of the hydroxyapatite implant under the vertical load by the two-dimensional finite element method. J Oral Implantol1988;14:6571.
101
XEIJER
Reprint requests to: DR. H. J. A. MEIJER DEPT. ORAL-MAXILLOFACIAL SURGERY, SPECIAL DENTAL CARE UNIVERSITY HOSPITAL GRONINGEN P.O. Box 30.001 9700 RD GRONINGEN THE NETHERLANDS
Contributing
PROSTHODONTICS
AND
author
DDS, PhD, ResearchAssociate,Department of Oral-Maxillofacial Surgery, Prosthodontics and Special Dental Care, University of Utrecht, Faculty of Medicine, Utrecht, The Netherlands. W. H. A. Steen,
APPENDIX For a better understandingof the figurespresentedin the article, a short supplement with respect to the concept of stressis presented. A stressacting at a point on a surface is defined asthe force acting on a small areaof the surface at that point divided by the area. Stressis a force per unit of the area. The so-callednormal stressand shearstresses can be distinguished (Fig. 10). Normal stressequalsthe force per unit of area in the direction perpendicular to the surface, whereas the shear stressesequal forces per unit of area that lay in the plane of the surface.To study the stressstate in a material point inside a body that is loaded by forces at its exterior, it is
102
ET AL
helpful to consider an infinitesimally small material cube at that point, with its facesperpendicular to somearbitrary chosencoordinate system.As the material of the cube may be deformed, the surrounding material loads the cube by stressesat its faces. Cutting the cube out in thought, one candistinguish at onesideof the cubethree normal stresses and six shear stressesin the coordinate directions as sketchedin Fig. 11.The stresseson the oppositefacesat the other sideof the cube are equal in size but acting in the oppositedirection to keep the cube in force equilibrium. This is true for the infinitesimal cube consideredhere but generally not for a cube of finite dimensions.Considerationof equilibrium of sucha finite cube is not relevant for the goal of this appendix. Only six componentsare neededto completely describethe stressstate at the material point under considerationas Sxy = Syx, Syz = Szy and SZX = SXZ due to moment equilibrium. Note that positive normal stressesload the cube in tension and negative normal stressesin compression. To judge the stressstate at some point, the so-called maximum and minimum principal stressesare used.These are the maximum and minimum normal stressesof all possible normal stressesthat can be distinguished at a material point. They can be determined by choosingthe local coordinate system such that all shear stresseson the correspondingcube becomezero.
JULY
1992
VOLUME
68
NUMBER
1
Meijer,
DDSa
J. H. Kuiper,
MSC,~
F. J. M.
Starmans,
DSc,”
and
PhDd
University of Utrecht, Faculty Orthopaedics, Nijmegen, The
of Medicine, Netherlands
Utrecht;
and University
of Nijmegen,
Institute
for
The stress distribution around dental implants was investigated by use of a two-dimensional model of the mandible with two implants. A vertical load of 100 N was imposed on abutments or the bar connection. The stress was calculated for a number of superstructures under different loading conditions with the help of the finite element method. The length of the implants and the height of the mandible were also varied. A model with solitary abutments showed a more uniform distribution of the stress when compared with a model with connected abutments. The largest compressive stress was also less in the model without the bar. Using shorter implants did not have a large influence on the stress around the implants. When the height of the mandible was reduced, a substantially larger stress was found in the bone around the implants because of a larger overall deformation of thelowerjaw. (J PROSTHET DENT 1992;68:96-102.)
ack of retention and stability of a denture is one of the major complaints of edentulous patients. The use of endosseous implants, by which the prosthesis can be supported, is a successful and popular therapy. Nevertheless, the use of implants is not without problems. Biomechanical influences play an important role in the longevity of bone around imp1ants.l Forces on the denture such as during chewing are transferred to the implants and lead to stress in the bone surrounding the implants. Too much stress can lead to bone resorption, which eventually causes loosening of the implant. Soltesz and Siegele2 demonstrated that regions of stress concentration seen in a laboratory model coincided with resorption zones observed in the bone of the jaws of dogs provided with the same type of implants investigated in the laboratory model. Several investigations have already been performed to select implant configurations that will cause a minimum of stress when being loaded.3-7 Other studies have been published that deal with the influence of the interface (bone-implant) condition on the amount of stress.s g The reported stress calculations were restricted to one implant in a small section of bone. No study has been published to determine the influence of the length of the
Supported in part by IBM Nederland aAssistant Professor, Department of Prosthodontics, and Special Dental bResearch Engineer, Biomechanics jmegen. CResearch Engineer, Department of Prosthodontics, and Special Dental dProfessor and Chairman, Department gery, Prosthodontics, and Special Utrecht.
10/l/27224
96
N.V. contract No. 473104. Oral-Maxillofacial Surgery, Care, University of Utrecht. Section, University of NiOral-Maxillofacial Surgery, Care, University of Utrecht. of Oral-Maxillofacial SurDental Care, University of
implant, the height of the mandible, and the type of superstructure on the stresspattern. In the present study the stressdistribution was calculated in a two-dimensionalmodel representing the frontal section of an edentulousmandible provided with two implants. Bending of the structures was permitted in this model by omitting support at the bottom and instead fastening at the left and right side. These aspectsmake the model a more realistic representation of the clinical situation.
The influence of (1) the presenceof a bar between the implants, (2) the length of the implants, and (3) the height of the mandible on the distribution and the amount of stressin the bone surrounding the implants wasanalyzed. MATERIAL
AND
METHODS
A two-dimensionalmodel was constructed as a first approach to investigate the relative effect of severalfactors. A section of an edentulous mandible was composed.The mandiblewastranslated to a pieceof bone70mm in length, which is approximately the distance between the left and right mandibular angles(Fig. 1). The behavior of the mandible wasrepresentedby a two-dimensionalmodel. The finite element method (FEM) was used to calculate the stressin the bone surrounding implants by which the set of stress-determiningequations could be calculated. The basisof the FEM is a model divided into small elements. Each elementis consideredinterconnected to each of its neighboring elementsat a number of discrete points called nodes.The displacementof each of the nodesmust be calculated to determine the stress throughout the structure. The large number of equations to be solved necessitated the useof a computer.lo Investigation by usingFEM is fa-
JULY1992
VOLUME69
NUMBER
1
STRESS
DISTRIBUTION
AROUND
DENTAL
IMPLANTS
7 lmm
15 m m
i 1
I
I
4 mm
70 mm pd
Fig. 1. Translation implants connected
= tort.
bone
of edentulous with bar.
m
mandible
vorable to other methods because the dimensions and the properties of all composite materials can easily be simulated and thus varied. Stress and displacement can be calculated. The finite element program MARCll was used. FEM enables including material thickness and properties of structures in the third dimension (plane-strain). The influence of the cortical bone at the labial and lingual side of the section of bone was factored by means of a side plate.i2 The 3 mm thick side plate connected the upper and lower cortical layer. The model was provided with two cylindrical implants. The implants had the mechanical properties of titanium. The materials were regarded as isotropic and linearly elastic. The dimensions and properties of the model were similar to a human mandible (Table I). A fixed bond between bone and implant along the entire interface was assumed. The close apposition of titanium and bone and the ingrowth of bone in the surface structure of titanium means that under any subsequent loading, the interface moves as a unit without relative motion between the bone and titanium. The element distribution of a model with connected abutments is shown in Fig. 2. To simulate the clinical situation the model was not supported at the bottom, but the left and the right side of the model were fastened to allow bending of the mandible.
THE
JOURNAL
OF PROSTHETIC
DENTISTRY
into
q
spong.
two-dimensional
bone
model
with
two
Fig. 2. Element distribution of model with connected abutments (mandible of 15 mm height and implants 11 mm in length).
Normal biting as in chewing was simulated by vertically applied loads of 100 N.13 Koolstra et al.14 reported that the largest possible biting forces in the incisal and canine region were found approximately 35 degrees relative to the horizontal plane in the dorsal direction. In case of a more horizontal direction, bite forces soon decreased to a much lower level. Different loading conditions of the implants were analyzed under various circumstances. 1. Seven conditions were analyzed for two geometrically different models (Fig. 3), one with a height of the mandible of 15 mm and another with a height of 7 mm. In the latter situation the implants had a length of 3 mm and a spongious bone layer of 2 mm.
97
MEIJER
1
4 !
a vertical load uniformly a bar as connector between
dlstrlbuted on 1 tkm implants.
a vertical
dlstrlbuted
load
a vertical load a bar present.
uniformly
on one
ET AL
on
abutment, I
h
!
i
I
load on one atutment, / 5 a vertical no bar present.
I
I
i
6 a vertical bar
I
load uniformly distributed on and extensions (10 mn on both sides).
a vertical load a bar present.
Fig.
on one extension,
3. Analyzed
loading
conditions
Table
and designs.
I.
Dimensions
and properties
of FE-model
Mandible Height
-
=
path of evaluation
A---B-=-C---D Fig. 4. Path in bone stress was evaluated.
around
left implant
along
which
2. The influence of the length of the implant was determined by analyzing implants with lengths of 13, 9, 7, 5, 3, and 1 mm. The calculations were limited to conditions 1 and 2 (Fig. 3). 3. The influence of the height of the mandible was determined by modifying the height of the model to 13,11,9, and 7 mm. The length of the implant was set at 3 mm in this example. The calculations were limited to conditions 1 and 2 (Fig. 3). The stress pattern in all situations was evaluated along a path in the bone around the left implant, which started at the upper left corner (A) and ended, via the apex of the implant (B and C), at the upper right corner (0) (Fig. 4). With respect to bone remodeling, maximum and minimum principal stresses may be important. Moderate stress values induce bone formation, but when they become too large they cause resorption. Therefore the stresses plotted in the figures are (1) the maximum and minimum principal stresses as evaluated at each material point or are the extreme stresses, which are the highest maximum and (2)
98
15 mm 12 mm
Width Thickness of cortical layer Cranial Labial Caudal Lingual Properties of cortical bone Modulus of elasticity Poisson’s ratio Properties of spongious bone Modulus of elasticity Poisson’s ratio Dimensions of implant Length Diameter Dimensions of abutment Length Diameter Dimensions of metal bar Length Height Width Distance between metal bar and bone Properties of implant Modulus of elasticity Poisson’s ratio Properties of bar: Modulus of elasticity Poisson’s ratio
lmm lmm 4mm 2mm 13,700
MPa
0.30 1,370 MPa 0.30 11.0 mm 3.5 mm
4.5 mm 3.5 mm 21.5 mm 2.5 mm 1.5 mm 4.5 mm 103,400
MPa
0.35 103,400 MPa 0.35
the lowest minimum principal stress values as found in the entire model. Because examples 1, 2, 3, and 6 have symmetric conditions, stresses were calculated for only half of the model. The symmetric condition was used to determine the
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model with bar print.
stres-s
snost
(MPa)
-26’
path
15
mm mandible,
prlnc.
along
left Implant
max principal stress
+
mln
I 012345678
’
I
’
’
(mm) principal
+
stress
Infl. length
11 mm implants
stress
(MPa)
I
length model
of Implant,
with
1
J
I 9
of Implant bar
+
I 10
I 11
I 12
model
without
15 mm mandible
on stress distribu-
: most
negative
stress
(MPa)
-,.1,,, I E
-*Oh 6
along
20
15
left
max prlnclpal stress
Implant +
26
mln prlnclpal stress 5
stresses for the other half. Because two-dimensional models were used, an assumption was made for the third outof-plane direction. Plane-strain models were chosen, which were sufficient for the comparison between different models.
conditions
Fig. 5, A shows the minimum and maximum principal stress around the left implant in loading condition 1. The stress was negative at any location, that is, only compressive stress. The most negative stress was located at the neck of the implant in the upper cortical layer (-18.2 MPa). This stress was only found at the distal side of the implant and not at the mesial side. Fig. 4, B shows the results of loading condition 2. The most negative stress was -11.3 MPa, almost equal at the distal and mesial side. The most negative stress of the seven loading conditions for two different heights (15 and 7 mm) of the mandible was always found at the neck of the implant in the upper cortical layer.
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6
7
a
9
height
Fig. 5. A, Stress distribution around left implant in condition 1; 11 mm implants in 15 mm high mandible; vertical load of 100 N distributed on bar. B, Stress distribution around left implant in condition 2; 11 mm implants in 15 mm high mandible; vertical load of 100 N distributed on abutments (no bar).
THE
,
(mm)
11 mm Implants
RESULTS Different loading
..,.,
ci
10
path
15 mm mandible,
15
bar
0
-
I 14
(MPa) :
0
’ 13
(mm)
Fig. 6. Influence of length of implant tion (height of mandible 1.5 mm).
model without bar 6j
stress
/
5
-
negative
DENTISTRY
+ lnfi. height
mandible,
modal
with
10
11
of mandlble bar
-++
model
13
14
15
wlthout
bar
3 mm implants
Fig. 7. Influence of height of mandible tion (length of implants 3 mm).
Influence
12
(mm)
of the length
on stress distribu-
of the implant
The influence of the variation in length of the implant can be seen in Fig. 6, depicting the most negative stress as a function of the length of the implant. To facilitate analyzing the plot, points belonging to the same condition were connected with a line. In the model with connected abutments (condition 1) it can be seen that the shorter the implant the more negative was the stress value: from -17.9 MPa with an implant of 13 mm to -23.2 MPa with an implant of 1 mm. The same phenomenon occurred in the model without the bar (condition 2), although the stress was never as high as in the model with the bar: from -11.0 MPa with an implant of 13 mm to -15.3 MPa with an implant of 1 mm.
Influence
of the height
of the mandible
The results of varying the height of the mandible are plotted in Fig. 7. Again, only the most negative stress is
99
MEIJER
situation
rcdel mst (MaI
1
pj
4
1
i
3i-j
4
Fig. 8. Displacement of bony parts of model with connected abutments (A) and of model with solitary abutments (B) as consequence of vertical loading. Undeformed state is pointed out by dotted line (magnification of displacement X200).
given. The height of the lower jaw is set along the horizontal axis; the most negative stress along the vertical axis. The points belonging to the same situation are connected with a line. The stronger the reduction of the mandible, the higher was the stress. Tensile stress was not found in any of the situations or was negligibly small. DISCUSSION This study was performed to gain more insight in the influence of several variables on the stress distribution in the mandibular bone. Because of the assumptions made for the configuration of the two-dimensional model and the properties of the bone, the values of stress provided by the model are not necessarily identical to actual ones. Moreover it is not known at which amount of stress biologic changes such as resorption or deposition of the bony structures take place. However, the present model study is suitable for comparison of relative stress distributions between various implant situations. From the stress distribution in Fig. 5, it can be concluded that the largest compressive stress is situated in the upper cortical layer. This observation was also made by other authors.2, 3,5-g:i5, I6 This observation is not surprising, because the largest stress is often found near structure surfaces. Moreover, the cortical layer with a relative high modulus of elasticity strengthens this effect. In Fig. 5, B the stress is equally distributed at the distal and mesial side of the implant. If a bar is placed between the abutments, the stress is situated only at the distal side of the implant (Fig. 5, A). This location may be explained by the bending of the mandible as a result of the vertical load. In case of a structure with solitary abutments they are able to approach each other when the lower jaw bends. The stress is equally high in that situation at the mesial and distal side. In case of a structure with connected abutments, however, the bar does
100
j
7 ‘rrl
i-l
1
of 15 m m heiqht negative stress
ET AL
PC&l of 1 rim height most negative stress IFWit)
-18.2
-53.8
-11.3
-26.6
-19.4
-56.3
I
-27.5
-41.9
-107.5
Fig. 9. Most negative stress found in model of 15 mm height and of 7 mm height.
prevent movement of the abutments. When the structure is loaded, the mandible deforms, but the abutments are kept in place by the bar. This causes an extreme amount of stress at the distal sides of the connected implants. This explanation was confirmed by studying displacements of the same model (Fig. 8). Fig. 9 shows the results of different loading conditions and different designs of the superstructure. Apart from the most negative stress values seen in Fig. 5, the results of five other conditions are presented. In Fig. 9 it can be noticed that differences in stress are small between the conditions where the vertical load is distributed on the bar, where the vertical load is distributed on the abutments (bar present), and where the vertical load is distributed on the bar and extensions (conditions 1, 3, and 6). From a clinical viewpoint, conditions 4, 5, and 7 have an unfavorable prospect because of the unequal distribution of the chewing force on the superstructure. However, condition 5 does not show a disadvantageous effect from the point of view of stress distribution. The most negative stress is found if one extension is loaded with bar and extensions (condition 7). Fig. 9 shows also the results of the seven loading conditions of a mandible of 7 mm height with implants 3 mm long. The amount of compressive stress is much larger compared with a mandible of 15 mm height. This is partly because of a reduction of the length of the implant, but also because a larger deformation of the lower jaw as the mandible will bend more easily. When the stresses of the different loading conditions of the 7 mm mandible are compared with each other, it is found that the proportions are roughly the same as in the mandible of 15 mm height. The length of the implant does not have much influence on the amount of stress (Fig. 6) because the stress value mainly depends on bending of the mandible. Fig. 7 shows the influence of the height of the mandible. There is a strong increase in stress concentration when the
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normal
stress
ear
stresses
Fig. 10. Normal and shear stresses at point P. (See Appendix.)
Fig.
height of the mandible is reduced, which is caused by a larger deformation of the lower jaw. This can also be noticed in Fig. 9 comparing the several results. Comparing Figs. 6 and 7 reveals that the amount of stress depends more on the deformation of the jaw than on the length of the implant. Although the dimensions and properties of this two-dimensional model and the support of it at both ends simulates the human mandible only to a limited extent, it does give insight in the geometric behavior of bone as a result of chewing force. This study reveals that stress analysis of the bone around dental implants is incomplete if the model constructed does not account for deformation of the mandible. Given the interesting results of the stress analysis of the two-dimensional study, a three-dimensional model of the entire lower jaw is recommended for further investigation. A better translation of the boundary conditions would be possible. A horizontal load could be simulated by a threedimensional model in a direction corresponding with the clinical situation. Next to an improved biomechanical analysis, more clinical research and biologic research are necessary to predict the behavior of bone in the presence of a certain amount of stress.
stressbecauseof the overall deformation of the bone as a reaction to loading.
CONCLUSIONS The assumptions made for the composition of the model and its boundary conditions suggest reservations in translating the results of the FEM model to the clinical situation. The results indicate the following. The main stress peaks arise around the neck of the implant in the upper cortical layer. The presence of a bar between the abutments causes more negative stress when the structure is vertically loaded, compared with a situation in which no bar is present. Application of extensions does not improve the stress distribution. The possibility of an extreme amount of stress exists when the superstructure is not uniformly loaded. The length of the implant has little influence on the amount of stress in vertical loading. The height of the mandible has a large influence on the amount of
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11. Front of infinitesimal material cubewith stresses. (See Appendix.)
REFERENCES 1. Skalak
R. Biomechanical
considerations
in osseointegrated
prostheses.
J PROSTHETDENT~~~~;~%~~~-~. 2. Soltesz U, Siegele D. Principal characteristics of the stress distribution in the jaw caused by dental implants. In: Huiskes R, Van Campen D, De Wijn J, eds. Biomechanics: principles and applications. The Hague, The Netherlands: Martinus Nijhoff, 1982:439-44. 3. Siegele D, Soltesz U, Scheicher H. Dental implants with flexible inserts-a possibility to improve the stress distribution in the jaw. In: Perren SM, Scheicher E, eds. Biomechanics: current interdisciplinary research. Dordrecht, 1985. 4. Soltesz U, Siegele D, Riedmueller J. Die Spannungsverteilungen urn ein stufenformiges Implantat im Modellversuch und im Vergleich zu einfachen Grundformen. Dtsch Zahnarztl Z 1981;36:571-8. 5. Siegele D, Soltesz U. Numerische Untersuchungen rum Einfluss der Implantatform auf die Beanspruchung des Kieferknochens. Z Zahnarztl Implantol 1987;3:161-9. 6. Rieger MR, Fareed K, Adams WK, Tanquist RA. Bone stress distribution for three endosseous implants. J PROSTHET DENT 1989;61: 223-8. 7. Kitoh M, Suetsugu T, Murakami Y, Tabata T. A biomathematical study on implant design and stress distribution. Bull Tokyo Med Dent Univ 1978;25:269-76. 8. Borchers L, Reichart P. Three-dimensional stress distribution at different stages of interface development. J Dent Res 1983;62: 155-9. 9. Soltesz U, Siegele D, Riedmueller J, Schulz P. Stress concentration and bone resorption in the jaw for dental implants with shoulders. In: Lee AJC, Albrektsson T, Branemark P-I, eds. Clinical applications of biomaterials. New York: John Wiley & Sons Inc. 1982. JA. The finite element method of stress analysis and 10. Von Fraunhofer its application to dental problems. In: Scientific aspects of dental materials, London and Boston: Butterworths, 1975:41-4. 11. MARC general purpose finite element program, revision K3. Palo Alto: MARC Analysis Research Corporation, 1988: user’s manuals A-F. 12. Huiskes R, Chao EYS. A survey of finite element analysis in orthopedic biomechanics: the first decade. J Biomech 1983;16:385-409. 13. Haraldson T, Car&on GE. Bite force and oral function in patients with osseointegrated oral implants. Stand J Dent Res 1977;85:200-8. 14. Koolstra JH, Van Eijden TMGJ, Weijs WA, Naeije. A three-dimensional model of the hurean masticatory system predicting maximum possible bite forces. J Biomech 1988;21:563-76. 15. Soltesz U, Siegele D. Einfluss der Steifigkeit des Implantat materials auf die im Knochen erzeugten Spannungen. Dtsch Zahnarztl Z 1984;39: 183-6. 16. Kitoh M, Matsushita Y, Yamaue S, Ikeda H, Suetsugu T. The stress distribution of the hydroxyapatite implant under the vertical load by the two-dimensional finite element method. J Oral Implantol1988;14:6571.
101
XEIJER
Reprint requests to: DR. H. J. A. MEIJER DEPT. ORAL-MAXILLOFACIAL SURGERY, SPECIAL DENTAL CARE UNIVERSITY HOSPITAL GRONINGEN P.O. Box 30.001 9700 RD GRONINGEN THE NETHERLANDS
Contributing
PROSTHODONTICS
AND
author
DDS, PhD, ResearchAssociate,Department of Oral-Maxillofacial Surgery, Prosthodontics and Special Dental Care, University of Utrecht, Faculty of Medicine, Utrecht, The Netherlands. W. H. A. Steen,
APPENDIX For a better understandingof the figurespresentedin the article, a short supplement with respect to the concept of stressis presented. A stressacting at a point on a surface is defined asthe force acting on a small areaof the surface at that point divided by the area. Stressis a force per unit of the area. The so-callednormal stressand shearstresses can be distinguished (Fig. 10). Normal stressequalsthe force per unit of area in the direction perpendicular to the surface, whereas the shear stressesequal forces per unit of area that lay in the plane of the surface.To study the stressstate in a material point inside a body that is loaded by forces at its exterior, it is
102
ET AL
helpful to consider an infinitesimally small material cube at that point, with its facesperpendicular to somearbitrary chosencoordinate system.As the material of the cube may be deformed, the surrounding material loads the cube by stressesat its faces. Cutting the cube out in thought, one candistinguish at onesideof the cubethree normal stresses and six shear stressesin the coordinate directions as sketchedin Fig. 11.The stresseson the oppositefacesat the other sideof the cube are equal in size but acting in the oppositedirection to keep the cube in force equilibrium. This is true for the infinitesimal cube consideredhere but generally not for a cube of finite dimensions.Considerationof equilibrium of sucha finite cube is not relevant for the goal of this appendix. Only six componentsare neededto completely describethe stressstate at the material point under considerationas Sxy = Syx, Syz = Szy and SZX = SXZ due to moment equilibrium. Note that positive normal stressesload the cube in tension and negative normal stressesin compression. To judge the stressstate at some point, the so-called maximum and minimum principal stressesare used.These are the maximum and minimum normal stressesof all possible normal stressesthat can be distinguished at a material point. They can be determined by choosingthe local coordinate system such that all shear stresseson the correspondingcube becomezero.
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