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Global biomechanical model for dental implants
Journal of Biomechanics 44 (2011) 1059–1065
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Global biomechanical model for dental implants a,b ¨ Stig Hansson a,n, Johanna Loberg , Ingela Mattisson a, Elisabet Ahlberg b a b
Astra Tech AB, SE-431 21 M¨ olndal, Sweden Department of Chemistry, University of Gothenburg, SE-41296 Gothenburg, Sweden
a r t i c l e i n f o
abstract
Article history: Accepted 7 February 2011
The osseointegration of titanium dental implants is a complex process and there is a need for systematization of the factors influencing anchoring of implant. A common way of analyzing the strength of the fixation in bone is by measuring the torque required to remove the implants after healing. In this paper, a global biomechanical model is introduced and derived for removal torque situations. In this model, a gap is allowed to form between the bone and the implant and the size of the gap at fracture is a function of the surface roughness and can be shown to be directly related to the mean slope of the surface. The interfacial shear strength increases almost linearly with the mean slope and was also found to increase with an increase in the 2D surface roughness parameter, Ra. Besides the surface roughness, the design of the implant, the bone anatomy and the bone quality were shown to influence the interfacial shear strength. The Global biomechanical model can be used as a tool for optimizing the implant design and the surface topography to obtain high anchoring strength. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Dental implants Mean slope Interfacial shear strength Removal torque Surface roughness
1. Introduction Animal studies have shown that the interfacial shear strength of bone anchored implants can be increased by providing the implant with a rough surface (Gotfredsen et al., 1992, 1995; Hahn and Palich, 1970; Wennerberg et al., 1996a). Assuming that surface roughness can be regarded as consisting of pits, Hansson and Norton (1999) found that the theoretical interfacial shear strength depends on the pit size, the pit shape and the packing density of the pits. Based on a finite element study of two different surface topographies, Simmons et al. (2001) suggested that differences in osseointegration rate can be explained by differences in local tissue strains. A main theme in the pursuit of maximizing the interfacial shear strength has been to correlate the interfacial shear strength obtained in animal studies with the values of different surface roughness parameters (Ellingsen et al., ¨ 2004; Johansson et al., 2004; Loberg et al., 2010; Sul et al., 2002; Steinemann and Straumann, 1984; Wennerberg et al., 1995, 1996a, 1996b, 1996c). Doubts have however been expressed about the existence of an unequivocal relationship between the values of a set of surface roughness parameters and expected interfacial shear strength (Hansson, 2000). Recently, Hansson et al. (2010) found that, under some assumptions regarding the mechanics of the implant–bone interface, the theoretical interfacial shear strength can be expressed by one single 2D surface
n
Corresponding author. Tel.: +46 31 7763120; fax: + 46 31 3568380. E-mail address: [email protected] (S. Hansson).
0021-9290/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2011.02.002
roughness parameter – the absolute mean slope. The results of the model, defined for a profile, showed that the interfacial shear strength could also be estimated using the 3D surface roughness parameter Sdq (root-mean-square of the surface slope) (Hansson et al., 2010). The mathematical model applied was called the Local biomechanical model (referred to as the Local model) and can be used to evaluate the effects of different surface topographies on interfacial shear strength. One key assumption made in the Local model was that 100% bone-to-implant contact is maintained when the interface is exposed to shear forces. However, considering the design of the implant and the bone anatomy, a gap between the bone and surface on theoretical grounds can be assumed to arise. The aims of the present study were (1) derive a mathematical model that estimates the bone–implant interfacial shear strength when a gap between the implant and bone is permitted to arise during shear situations. This model could then be used to simulate removal torque situations. (2) Investigate the effects in gap size caused by variations in mean height of profile roughness (Ra). (3) Investigate changes in interfacial shear strength induced by variations in surface roughness parameters (mean slope and Ra), structural parameters (implant diameter and implant wall thickness) and anatomic parameters (bone support and bone quality). The results of the Global biomechanical model (referred to as the Global model) are discussed in correlation to results obtained for the Local model presented in Hansson et al. (2010). In a forthcoming paper, the model will be transposed to real surfaces and the correlation between interfacial shear strength and 3D surface roughness parameters under removal torque situations will be established.
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Sdq S
Nomenclature a b c Eb Ei Fc Fsh G H Ish L P Ra Rsl
implant radius (m) radius of bone cylinder (m) radius of implant bore (m) modulus of elasticity of bone (Pa) modulus of elasticity of implant material (Pa) compressive force induced by the load (N) shear force induced by the load (N) gap size (m) height of the roughness element (m) interfacial shear strength (Pa) profile length (m) pressure (Pa) mean height of profile roughness (m) mean slope of profile (dimensionless)
a eyb eyi sbcf srb sri syb syi tbf ub ui
root-mean-square of surface slope (dimensionless) horizontal length of implant part of the pit or protrusion in contact with bone (m) inclination angle of surface element ( 1 ) circumferential strain in bone at the interface (dimensionless) circumferential strain in implant at the interface (dimensionless) compressive strength of bone (Pa) radial stress in bone at the interface (Pa) radial stress in implant at the interface (Pa) circumferential stress in bone at the interface (Pa) circumferential stress in implant at the interface (Pa) shear strength of bone (Pa) Poisson’s ratio for bone (dimensionless) Poisson’s ratio for implant (dimensionless)
2. Development of a Global biomechanical model For the establishment of a mathematical model the following assumptions were made, the first four of them being almost identical to those made in the establishment of the Local model (Hansson et al., 2010). Concerning the nature of the interfacial bone the reader is referred to Hansson et al. (2010). (1) The bone is a continuum material and the bone-to-implant contact in the unloaded state is 100%. In the derivation of the Local model it was assumed that the bone-to-implant contact was 100% also in the loaded state. (2) The bone is isotropic and has a uniform compressive strength, sbcf, and uniform shear strength, tbf, within a zone containing the surface roughness. Since the bone strength probably increases with increase in distance from the implant surface (Hansson and Norton, 1999), the assumed values of these entities should increase with increasing dimensions of the roughness. (3) The surface roughness consists of pits and of protrusions (Hansson and Norton, 1999) (Fig. 1a). (4) The interfacial shear strength, Ish, is brought about by interlocking. Bone grows into the pits in the implant surface and creates retention (Hansson and Norton, 1999). In the same way roughness asperities protrude into the bone and retention is brought about by compressive forces between roughness elements and bone. The adhesion strength and friction between surface elements and bone are zero. (5) For occasional protrusions of the roughness into the bone, the failure mode is compressive fracture of the bone, i.e. the surface protrusions are assumed not to yield. Assuming a width of 1 length unit for the protrusion in Fig. 1a, fracture due to compression occurs when the compressive force, Fc1, reaches Fc1 ¼
sbcf ðH1 GÞ sin a1
σ σ α
ð1Þ
where sbcf is the compressive strength of the bone, H1 the height of the protrusion, H1 G the height of the part of the protrusion, which is in contact with bone, and a1 the slope angle of the protrusion. The component of the force Fc1 directed perpendicularly to the implant surface is sbcfS1, where S1 is the length of the part of the protrusion in contact with bone. This force will push the bone away from the implant surface and elastically compress the implant. As a consequence, a gap, G, will arise between implant and bone (Fig. 1a). The horizontal component of the force Fc1, sbcf(H1 G),
Fig. 1. (a) Surface roughness consisting of pits and protrusions with a gap, G, between the implant surface and the bone. (b) A surface roughness consisting of identical repeating elements with a gap, G, between the implant surface and the bone. (c) A cross section of a cylindrical implant with a cylindrical bore surrounded by a cortical bone cylinder.
S. Hansson et al. / Journal of Biomechanics 44 (2011) 1059–1065
will contribute to the bone–implant interfacial shear strength and since the gap lowers the value of H1 G, the interfacial shear strength will be reduced. If the protrusions are densely packed, the spaces between them should be regarded as pits – see below. (6) For bone plugs protruding into the pits, the failure mechanism will either be shear fracture or compressive fracture of the bone, depending on (i) the relationship between bone shear strength and bone compressive strength, (ii) the geometry of the pit and (iii) the size of the gap between implant and bone. Also in this case the material constituting the surface roughness is assumed not to yield. When the implant–bone interface is subjected to shear a compressive reaction force, Fc2, will arise at one of the slopes of the pit in Fig. 1a. The horizontal component of this force is Fc2 sin a2, where a2 is the slope angle of the pit. Assuming a width of 1 length unit for the pit in Fig. 1a, shear fracture occurs when Fc2 sin a2 ¼ 2tbf S2
ð2Þ
where S2 is the length of the part of the pit, which is in contact with bone. Since S2 ¼(H2 G)/tan a2 shear fracture occurs when Fc2 ¼
2tbf ðH2 GÞ sin a2 tan a2
ð3Þ
Compressive fracture occurs when Fc2 reaches Fc2 ¼
sbcf ðH2 GÞ sin a2
ð4Þ
This means that the shift between shear fracture and compressive fracture of the bone plug protruding into the pit occurs when 2tbf ðH2 GÞ s ðH2 GÞ ¼ bfc sin a2 tan a2 sin a2
ð5Þ
tbf tan a2 ¼ sbcf 2
srb ¼ p
Thus if
a2 4arctan
shear strength will be calculated for an idealized removal torque test with a cylindrical, unthreaded, titanium implant with a regular surface roughness (Fig. 1b). The implant has a cylindrical bore and is assumed to be osseointegrated in a cortical bone cylinder (Fig. 1c). This model is axisymmetric. Just before fracture the bone cylinder will be subjected to outwardly directed and the implant to inwardly directed forces. These forces have components directed perpendicularly to the implant surface and components in parallel with the implant surface. Since the bone, with exception for the portion immediately adjacent to the implant surface, and the implant material is assumed to be in the linearly elastic state the principle of superposition applies (Timoshenko and Goodier, 1982). Thus the effects of the force resultants are the sum of the effects of the force components directed perpendicularly to the implant surface and the effects of the force components in parallel with the implant surface. The force components in parallel with the implant surface will give rise to pure shear of the implant and the bone. In case of pure shear there is no change in volume (Timoshenko and Goodier, 1982). Since complete symmetry prevails a consequence is that the force components in parallel with the implant surface will not result in a change in shape of the bone cylinder or implant. Thus the analysis of changes in shape can be restricted to considering effects of the force components directed perpendicularly to the implant surface. Just before fracture these force components will subject the bone cylinder to an outwardly directed pressure, p, which will expand it and contribute to the formation of a gap between the implant surface and bone. The size of this contribution to the gap is eyba, where eyb is the circumferential strain in the bone at the interface and a the implant radius (Fig. 1c). The pressure will also compress the implant giving rise to another contribution to the gap – eyia, where eyi is the circumferential strain in the implant at the interface (Fig. 1c). The magnitude of p is P s s ðHGÞ S ð7Þ ¼ bcf p ¼ bcf 2H L where H is the height of the roughness elements and L the profile length considered (Fig. 1b). The stress and strain states in the bone (Fig. 1c) at the interface are according to Timoshenko and Goodier (1982):
which means that
2tbf
sbcf
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ð6Þ
failure occurs through shear, else failure occurs through compression. In the following it is assumed that the dominating failure mode is a compressive fracture. For more information see Hansson et al. (2010). (7) It is assumed that the bone exhibits an idealized stress–strain curve consisting of a linearly elastic phase followed by a plastic phase where the stress in the plastic phase is constant. It is further assumed that the displacement of the implant surface in relation to the interface bone is so big at fracture that the deformation of the bone, immediately adjacent to the implant surface, lies in the plastic region of the stress–strain curve irrespective of slope angle a. The deformation of the rest of the bone is assumed to lie in the linearly elastic region.
syb ¼ p eyb ¼
ð8Þ
a2 b2 a2
2 b2 a þ b2 1þ 2 ¼ p 2 2 a b a
ð9Þ
p a2 þb2 þub 2 2 Eb b a
ð10Þ
syb ub srb Eb
¼
The stress and strain states in the implant (Fig. 1c) at the interface become
sri ¼ p syi ¼ eyi ¼
c2 a2 p a2 p pðc2 þ a2 Þ ¼ ða2 c2 Þa2 a2 c2 a2 c2
syi ui sri Ei
¼
p ðc2 þ a2 Þ þ u i Ei a2 c2
ð11Þ
ð12Þ
The gap, G, between implant and bone caused by the pressure becomes 2.1. Roughness consisting of identical repeating elements The interfacial shear strength is sometimes calculated in removal torque tests (Ellingsen et al., 2004; Franke Stenport and Johansson, 2003; Johansson et al., 2004; Steinemann and Straumann, 1984). In the following, the theoretical interfacial
G ¼ aðeyb eyi Þ ¼ pC
ð13Þ
where C¼a
1 a2 þ b2 1 c2 þ a2 þ u u þ i b Eb b2 a2 Ei a2 c2
ð14Þ
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Assume a gap value
Calculate the pressure, p
p=
σbcf ΣSn
σ
σ σ
L
α
Calculate the gap
Calculate the interface shear strength,Ish
Ish=
σbcf Σ (Hn-G) L
Fig. 2. Irregular surface roughness with a gap between the implant surface and the bone. A suggested iterative procedure to calculate the interfacial shear strength is also shown.
Eq. (7) combined with Eq. (13) gives the following expression for the gap: G¼
sbcf CH 2H þ sbcf C
ð15Þ
Considering a section of the implant surface in Fig. 1b with a width of 1 length unit the contribution of each roughness peak to resist shear is sbcf(H G). Each roughness peak occupies a length of 2H/tan a. This means that the interfacial shear strength becomes Ish ¼
sbcf ðHGÞ sbcf tan aðHGÞ ¼ ð2H=tan aÞ 2H
ð16Þ
will be calculated for the surface roughness in Fig. 1b using different values of the roughness parameters Rsl (mean slope) and Ra. Furthermore the gap will be calculated as a function of the Ra value. It should be noted that Ra ¼H/4 and Rsl ¼tan a. Using the same equations the effect of implant diameter, implant wall thickness, thickness of supporting bone and bone quality will be calculated. The bone quality is expressed as the value of the modulus of elasticity, Eb. The compressive strength of the bone, sbcf, is assumed to depend on Eb according to the expression obtained by Keller et al. (1990):
sbcf ¼ k E1:24 b
ð19Þ 5
2.2. Irregular surface roughness A theoretical estimation of the interfacial shear strength produced by an irregular surface roughness is less straightforward. The following iterative procedure is recommended (Fig. 2): (1) assume a size of the gap between implant and bone at shear fracture based on a study of a profile of the roughness. (2) Calculate the pressure that pushes the bone away from the implant surface and compresses the implant using P s Sn p ¼ bcf ð17Þ L where Sn is the length of the part of the nth roughness element in contact with bone. (3) Calculate a new value of the gap using the calculated pressure. (4) Calculate a new value of the pressure using Eq. (17). (5) Continue until the size of the gap and the pressure have converged. (6) Calculate the interfacial shear strength by means of P ðHn GÞ s Ish ¼ bcf ð18Þ L where Hn G is the height of the part of the nth roughness element in contact with bone.
3. Results and discussion The bone–implant interfacial shear strength is a function of surface roughness, modulus of elasticity of implant material, bone support, bone quality, implant diameter and wall thickness of the implant. In the following these dependences will be addressed separately. By applying Eqs. (7)–(16) the interfacial shear strength
where k¼5.11 10 . The implant material was assumed to be titanium with a modulus of elasticity of 107 GPa. Poisson’s ratios for both bone and titanium were assumed to be 0.3 (Hansson and Ekestubbe, 2004; Hart et al., 1992). 3.1. Surface roughness parameters The theoretical interfacial shear strength, Ish, increases with increasing mean slope, Rsl, (Fig. 3a) in a similar way as for the Local model (Hansson et al., 2010), shown as dotted line in Fig. 3a. In the Local model a direct proportionality between Ish and Rsl is obtained, Eq. (20), since no gap is assumed to evolve during the loading situation. The values of Ish are much lower in the Global model due to the decreased fraction of direct implant–bone contact. The fact that both the Local and Global biomechanical models show increased Ish with increase in Rsl illustrates the importance of the mean slope: Ish ¼
sbcf Rsl 2
ð20Þ
Ish was also found to rapidly increase with increasing values of the Ra parameter for the Global model (Fig. 3b). This is in sharp contrast to the results of the Local model, where Ish was found to be independent of the Ra value (Hansson et al., 2010) (shown as dotted line in Fig. 3b). The difference between the Local and Global models is due to the lower fraction of direct implant–bone contact in the Global model. In fact, Ish can be calculated using the Local model and the fraction of implant–bone contact resulting from the gap formation. The increase in Ish with increasing Ra value agrees well with an analysis made on removal torque values in rabbit, where a linear relationship between removal torque and ¨ the Sa parameter was found (Loberg et al., 2010). In this analysis
S. Hansson et al. / Journal of Biomechanics 44 (2011) 1059–1065
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14 Local
Ish (MPa)
8
12
60
10
50
8
40 6 30 4
20
2
10
0
0 0
0.2
0.4
0.6
Rsl Local 50
20
40
15
30
10
20
5
10
Ish (MPa)
Ish (MPa)
Global 25
0
0 0
1
2
3
4
5
6
Ra (μm) Fig. 3. (a) Interfacial shear strength (Ish) as a function of the mean slope (Rsl) for a surface roughness consisting of identical repeating elements as shown in Fig. 1b. Ra ¼ 1.5 mm, implant diameter ¼4 mm, implant wall thickness ¼0.5 mm, bone support ¼ 3 mm and Eb ¼ 15 MPa. The results for the Local model are shown as dotted line. (b) Interfacial shear strength as a function of the Ra value for a surface roughness consisting of identical repeating elements as shown in Fig. 1b. a ¼201, implant diameter¼4 mm, implant wall thickness ¼0.5 mm, bone support ¼3 mm and Eb ¼ 15 MPa. The results for the Local model are shown as dotted line.
G (μm)
10
70
Ish (MPa)
Global 12
6 4 2 0 0
1
2
3
4
5
6
Ra (μm) Fig. 4. Gap size (G) at fracture as a function of the Ra value for a surface roughness consisting of identical repeating elements as shown in Fig. 1b. a ¼201, implant diameter¼ 4 mm, implant wall thickness¼ 0.5 mm, bone support ¼ 3 mm and Eb ¼ 15 MPa.
is hardly any contact between the bone and the implant. As Ra increases, the fraction of direct contact between bone and implant increases and therefore also the Ish increases even though the gap size at fracture increases. For example, a gap of 4.6 mm will develop on a surface with Ra ¼ 1.5 mm, which means that the gap/height ratio is 77% while for Ra ¼0.5 mm the corresponding values will be 1.8 mm and 91%, respectively. Thus, the Global model predicts that if a nano-roughness is superimposed on a micro-roughness, the contribution of the nano-roughness to the interfacial shear strength will be minor since this roughness will be mainly located in the gap. This is in contrast to the results of the Local model where all sizes of roughness contribute to the Ish (Hansson et al., 2010) since 100% implant to bone contact was assumed. These results clearly show that from a biomechanical point of view the micro-roughness is most important. Thus when the Global model is applied for evaluation of rough implant surfaces the Rsl parameter should be calculated for the bigger features constituting the topography. This can be achieved by, in addition to the 50 mm Gaussian high pass filter that is often used (Wennerberg and Albrektsson, 2000), also applying a low pass Gaussian filter of suitable size, for example 2–4 mm.
3.2. Structural implant parameters the Sa values varied between 0.6 and 1.6 mm and it is interesting to note that a nearly linear relationship between the Ra value and Ish is obtained for the Global model within the same range of Ra values. The Sa parameter is the 3D equivalent to the 2D parameter Ra. Removal torque studies on rabbit have shown that the highest fixation strength appears at Sa equal to 1.5 mm (Wennerberg, 1996). Similar results were obtained for miniature pigs (Buser et al., 1999), where the removal torque value was found to level off at Ra values larger than 2 mm. These studies indicate that other factors come into role at higher surface roughness. The change in the Ish with Ra in the Global model is due the formation of the gap. In Fig. 4, the gap size is shown as a function of surface roughness. At small Ra values the gap is almost as big as the peak-to-valley value of the surface, which means that there
The theoretical interfacial shear strength, Ish, also depends on the design of the implant. This is illustrated by keeping the implant wall thickness constant while changing the implant diameter (Fig. 5a). The Ish is found to decrease with increase in implant diameter due to decreasing stiffness of the implant, i.e. the size of the gap increases with a decrease in implant stiffness. This is also illustrated by increasing wall thickness at constant implant diameter (Fig. 5b). As the implant becomes more massive the Ish reaches a constant value, which depends on the modulus of elasticity of the implant material, Ei. The fact that the design of the implant has large effects on the interfacial shear strength means that caution should be taken when comparing results of removal torque studies with different implant designs and with implants of different materials since the resistance to gap formation can be very different.
S. Hansson et al. / Journal of Biomechanics 44 (2011) 1059–1065
14
7
12
6
10
5
8
Ish (MPa)
Ish (MPa)
1064
6 4
4 3 2
2
1
0 0
1
2
3
4 5 6 7 8 Implant diameter (mm)
9
10
0
11
0
5
10
15
Eb (GPa)
8 8 7 7 6
5 Ish (MPa)
Ish (MPa)
6
4 3 2
5 4 3 2
1
1
0 0
0.5 1 1.5 Implant wall thickness (mm)
2
0 0
2
4 6 8 Bone support (mm)
10
Fig. 5. (a) Interfacial shear strength (Ish) as a function of the implant diameter for a surface roughness consisting of identical repeating elements as shown in Fig. 1b. Ra ¼ 1.5 mm, a ¼ 201, implant wall thickness ¼ 0.5 mm, bone support ¼ 3 mm and Eb ¼15 MPa. (b) Interfacial shear strength (Ish) as a function of implant wall thickness for a surface roughness consisting of identical repeating elements as shown in Fig. 1b. Ra ¼1.5 mm, a ¼ 201, implant diameter ¼4 mm, bone support ¼3 mm and Eb ¼15 MPa.
Fig. 6. (a) Interfacial shear strength (Ish) as a function of the modulus of elasticity of the bone (Eb) for a surface roughness consisting of identical repeating elements as shown in Fig. 1b. Ra ¼1.5 mm, a ¼ 201, implant diameter¼ 4 mm, implant wall thickness¼ 0.5 mm and bone support ¼ 3 mm. (b) Interfacial shear strength (Ish) as a function of the bone support for a surface roughness consisting of identical repeating elements as shown in Fig. 1b. Ra ¼ 1.5 mm, a ¼ 201, implant diameter¼ 4 mm, implant wall thickness ¼0.5 mm and Eb ¼15 MPa.
3.3. Anatomic parameters
expected. On the very bottom of the thread there is an additional effect, represented by the thread depth, increasing the bone support and thereby increasing the interfacial shear strength. On the thread flanks the resistance to gap formation exerted by the bone is greater, which will increase the interfacial shear strength. The opposite thread flank will further increase this resistance. The magnitude of this increase in resistance can be estimated by means of, for example, finite element analysis. Most dental implants in the market place are threaded while the implants in this study were designed as unthreaded cylinders. However, the static behavior regarding interfacial shear strength is very similar for these two types of implant designs, which means that the findings in this study also apply to threaded implants. The adhesion strength and friction between surface elements and bone were assumed to be zero; the reason for this was that the contribution to bone–implant interfacial shear strength caused by interlocking between the asperities of surface roughness and bone was in focus. By assuming a certain
The bone quality, expressed as modulus of elasticity and compressive strength, had a significant effect on the interfacial shear strength. An increased value of the bone’s modulus of elasticity resulted in an increase in Ish (Fig. 6a). A stronger bone support, here expressed as an increase in the thickness of the bone cylinder, also yielded increased Ish (Fig. 6b). What is presented in this paper is a mathematical model with idealized assumptions, which has to be considered when evaluating the real situation. The bone-to-implant contact is never 100%. Furthermore only a fraction of the implant surface is in contact with cortical bone. Concerning bone support there is a difference in static behavior between a cylindrical and threaded implant, which affects the interfacial shear strength. This will be discussed with reference to Fig. 7. On the very top of the thread, the same effect on interfacial shear strength as for a cylindrical implant can be
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Acknowledgment Financial support from the Swedish Research Council (200521028-35344-27) is gratefully acknowledged.
References
Fig. 7. Qualitative description of how the resistance to gap formation, exerted by the surrounding bone, varies along the profile of a threaded implant.
coefficient of friction the effect of friction can be superimposed on the interfacial shear strength caused by interlocking. The same applies to adhesion. To optimize the interfacial shear strength for a given situation, all parameters addressed in this study need to be considered. The dentist needs to know the quality of the bone and relate that to the design of the implant. An optimization of bone support, implant size and wall thickness must be made irrespective of the surface roughness.
4. Conclusions The Global model introduced in this paper shows that the gap formed between the bone and the implant considerably lowers the interfacial shear strength. This is expected since the fraction of direct bone-to-implant contact is lowered and it is in fact shown that the Local model is applicable if only the fraction of the surface in direct contact with bone is taken into account. According to the Global model the interfacial shear strength increases with increase in values of the mean slope and the 2D surface roughness parameter, Ra. The Global model also shows the importance of implant design on the interfacial shear strength. Besides the surface roughness, parameters such as bone support, bone quality, modulus of elasticity of implant material, implant diameter and wall thickness of the implant were shown to be important. For the application to dental implants this model gives a tool for optimizing the implant for specific conditions. For a given bone anatomy and bone quality, the optimization can be made by changes in surface roughness, implant size and implant wall thickness.
Conflict of interest statement ¨ Stig Hansson, Johanna Loberg and Ingela Mattisson are employed at Astra Tech AB, a company working with development, production and marketing of dental implants.
Buser, D., Nydegger, T., Oxland, T., Cochran, D.L., Schenk, R.K., Hirt, H.P., Sne´tivy, D., Nolte, L.-P., 1999. Interface shear strength of titanium implants with a sandblasted and acid-etched surface: a biomechanical study in the maxilla of miniature pigs. J. Biomed. Mater. Res. A 2, 75–83. Ellingsen, J.E., Johansson, C.B., Wennerberg, A., Holmen, A., 2004. Improved retention and bone-to-implant contact with fluoride-modified titanium implants. Int. J. Oral Maxillofac. Impl. 5, 659–666. Franke Stenport, V., Johansson, C.B., 2003. Enamel matrix derivative and titanium implants: an experimental pilot study in the rabbit. J. Clin. Periodontol. 4, 359–363. Gotfredsen, K., Nimb, L., Hjorting-Hansen, E., Jensen, J.S., Holmen, A., 1992. Histomorphometric and removal torque analysis for TiO2-blasted titanium implants: an experimental study on dogs. Clin. Oral. Impl. Res. 2, 77–84. Gotfredsen, K., Wennerberg, A., Johansson, C., Skovgaard, L.T., Hjorting-Hansen, E., 1995. Anchorage of TiO2-blasted, HA-coated, and machined implants: an experimental study with rabbits. J. Biomed. Mater. Res. 10, 1223–1231. Hahn, H., Palich, W., 1970. Preliminary evaluation of porous metal surfaced titanium for orthopedic implants. J. Biomed. Mater. Res. 4, 571–577. Hansson, S., 2000. Surface roughness parameters as predictors of anchorage strength in bone: a critical analysis. J. Biomech. 33, 1297–1303. Hansson, S., Ekestubbe, A., 2004. Area moments of inertia as a measure of the mandible stiffness of the implant patient. Clin. Oral Impl. Res., 450–458. ¨ Hansson, S., Loberg, J., Mattisson, I., Ahlberg, E., 2010. Characterisation of titanium dental implants: II. Local biomechanical model. Open Biomater. J. 2, 36–52. Hansson, S., Norton, M., 1999. The relation between surface roughness and interfacial shear strength for bone-anchored implants. A mathematical model. J. Biomech. 8, 829–836. Hart, R.T., Hennebel, V.V., Thongpreda, N., van Buskirk, W.C., Anderson, R.C., 1992. Modeling the biomechanics of the mandible: A three-dimensional finite element study. J. Biomech. 3, 261–286. ¨ Johansson, C.B., Wennerberg, A., Bostrom-Junemo, K., Holmen, A., Hansson, S., 2004. In vivo comparison of TiO2 blasted and fluoride modified implants in rabbit bone. In: Proceedings of the 7th World Biomaterials Congress, Sydney. Keller, T.S., Mao, Z., Spengler, D.M., 1990. Young’s modulus, bending strength, and tissue physical properties of human compact bone. J. Orthop. Res. 4, 592–603. ¨ Loberg, J., Petersson, I., Hansson, S., Ahlberg, E., 2010. Surface characterisation of titanium dental implants. I. Critical assessment of surface roughness parameters. Open Biomater. J. 2, 18–35. Simmons, C.A., Meguid, S.A., Pilliar, R.M., 2001. Differences in osseointegration rate due to implant surface geometry can be explained by local tissue strains. J. Orthop. Res. 19, 187–194. Steinemann, S.G., Straumann, F., 1984. Ankylotische verankerung von implantaten. Monatsschr. Zahnmed., 682–687. Sul, Y., Johansson Carina, B., Jeong, Y., Wennerberg, A., Albrektsson, T., 2002. Resonance frequency and removal torque analysis of implants with turned and anodized surface oxides. Clin. Oral. Impl. Res. 3, 252–259. Timoshenko, S.P., Goodier, J.N., 1982. Theory of Elasticity. McGraw-Hill Book Co, Singapore, pp. 8–11, 68–71. Wennerberg, A., 1996. On surface rougness and implant incorporation. PhD Thesis ¨ ¨ Goteborg University, Goteborg. Wennerberg, A., Albrektsson, T., 2000. Suggested guidelines for the topographic evaluation of implant surfaces. Int. J. Oral Maxillofac. Impl. 15, 2000. Wennerberg, A., Albrektsson, T., Andersson, B., 1996a. Bone tissue response to commercially pure titanium implants blasted with fine and coarse particles of aluminum oxide. Int. J. Oral Maxillofac. Impl. 1, 38–45. Wennerberg, A., Albrektsson, T., Andersson, B., Krol, J.J., 1995. A histomorphometric and removal torque study of screw-shaped titanium implants with three different surface topographies. Clin. Oral. Impl. Res. 1, 24–30. Wennerberg, A., Albrektsson, T., Johansson, C., Andersson, B., 1996b. Experimental study of turned and grit-blasted screw-shaped implants with special emphasis on effects of blasting material and surface topography. Biomaterials 1, 15–22. Wennerberg, A., Albrektsson, T., Lausmaa, J., 1996c. Torque and histomorphometric evaluation of c.p. titanium screws blasted with 25- and 75-mm-sized particles of Al2O3. J. Biomed. Mater. Res. 2, 251–260.
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Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Global biomechanical model for dental implants a,b ¨ Stig Hansson a,n, Johanna Loberg , Ingela Mattisson a, Elisabet Ahlberg b a b
Astra Tech AB, SE-431 21 M¨ olndal, Sweden Department of Chemistry, University of Gothenburg, SE-41296 Gothenburg, Sweden
a r t i c l e i n f o
abstract
Article history: Accepted 7 February 2011
The osseointegration of titanium dental implants is a complex process and there is a need for systematization of the factors influencing anchoring of implant. A common way of analyzing the strength of the fixation in bone is by measuring the torque required to remove the implants after healing. In this paper, a global biomechanical model is introduced and derived for removal torque situations. In this model, a gap is allowed to form between the bone and the implant and the size of the gap at fracture is a function of the surface roughness and can be shown to be directly related to the mean slope of the surface. The interfacial shear strength increases almost linearly with the mean slope and was also found to increase with an increase in the 2D surface roughness parameter, Ra. Besides the surface roughness, the design of the implant, the bone anatomy and the bone quality were shown to influence the interfacial shear strength. The Global biomechanical model can be used as a tool for optimizing the implant design and the surface topography to obtain high anchoring strength. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Dental implants Mean slope Interfacial shear strength Removal torque Surface roughness
1. Introduction Animal studies have shown that the interfacial shear strength of bone anchored implants can be increased by providing the implant with a rough surface (Gotfredsen et al., 1992, 1995; Hahn and Palich, 1970; Wennerberg et al., 1996a). Assuming that surface roughness can be regarded as consisting of pits, Hansson and Norton (1999) found that the theoretical interfacial shear strength depends on the pit size, the pit shape and the packing density of the pits. Based on a finite element study of two different surface topographies, Simmons et al. (2001) suggested that differences in osseointegration rate can be explained by differences in local tissue strains. A main theme in the pursuit of maximizing the interfacial shear strength has been to correlate the interfacial shear strength obtained in animal studies with the values of different surface roughness parameters (Ellingsen et al., ¨ 2004; Johansson et al., 2004; Loberg et al., 2010; Sul et al., 2002; Steinemann and Straumann, 1984; Wennerberg et al., 1995, 1996a, 1996b, 1996c). Doubts have however been expressed about the existence of an unequivocal relationship between the values of a set of surface roughness parameters and expected interfacial shear strength (Hansson, 2000). Recently, Hansson et al. (2010) found that, under some assumptions regarding the mechanics of the implant–bone interface, the theoretical interfacial shear strength can be expressed by one single 2D surface
n
Corresponding author. Tel.: +46 31 7763120; fax: + 46 31 3568380. E-mail address: [email protected] (S. Hansson).
0021-9290/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2011.02.002
roughness parameter – the absolute mean slope. The results of the model, defined for a profile, showed that the interfacial shear strength could also be estimated using the 3D surface roughness parameter Sdq (root-mean-square of the surface slope) (Hansson et al., 2010). The mathematical model applied was called the Local biomechanical model (referred to as the Local model) and can be used to evaluate the effects of different surface topographies on interfacial shear strength. One key assumption made in the Local model was that 100% bone-to-implant contact is maintained when the interface is exposed to shear forces. However, considering the design of the implant and the bone anatomy, a gap between the bone and surface on theoretical grounds can be assumed to arise. The aims of the present study were (1) derive a mathematical model that estimates the bone–implant interfacial shear strength when a gap between the implant and bone is permitted to arise during shear situations. This model could then be used to simulate removal torque situations. (2) Investigate the effects in gap size caused by variations in mean height of profile roughness (Ra). (3) Investigate changes in interfacial shear strength induced by variations in surface roughness parameters (mean slope and Ra), structural parameters (implant diameter and implant wall thickness) and anatomic parameters (bone support and bone quality). The results of the Global biomechanical model (referred to as the Global model) are discussed in correlation to results obtained for the Local model presented in Hansson et al. (2010). In a forthcoming paper, the model will be transposed to real surfaces and the correlation between interfacial shear strength and 3D surface roughness parameters under removal torque situations will be established.
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Sdq S
Nomenclature a b c Eb Ei Fc Fsh G H Ish L P Ra Rsl
implant radius (m) radius of bone cylinder (m) radius of implant bore (m) modulus of elasticity of bone (Pa) modulus of elasticity of implant material (Pa) compressive force induced by the load (N) shear force induced by the load (N) gap size (m) height of the roughness element (m) interfacial shear strength (Pa) profile length (m) pressure (Pa) mean height of profile roughness (m) mean slope of profile (dimensionless)
a eyb eyi sbcf srb sri syb syi tbf ub ui
root-mean-square of surface slope (dimensionless) horizontal length of implant part of the pit or protrusion in contact with bone (m) inclination angle of surface element ( 1 ) circumferential strain in bone at the interface (dimensionless) circumferential strain in implant at the interface (dimensionless) compressive strength of bone (Pa) radial stress in bone at the interface (Pa) radial stress in implant at the interface (Pa) circumferential stress in bone at the interface (Pa) circumferential stress in implant at the interface (Pa) shear strength of bone (Pa) Poisson’s ratio for bone (dimensionless) Poisson’s ratio for implant (dimensionless)
2. Development of a Global biomechanical model For the establishment of a mathematical model the following assumptions were made, the first four of them being almost identical to those made in the establishment of the Local model (Hansson et al., 2010). Concerning the nature of the interfacial bone the reader is referred to Hansson et al. (2010). (1) The bone is a continuum material and the bone-to-implant contact in the unloaded state is 100%. In the derivation of the Local model it was assumed that the bone-to-implant contact was 100% also in the loaded state. (2) The bone is isotropic and has a uniform compressive strength, sbcf, and uniform shear strength, tbf, within a zone containing the surface roughness. Since the bone strength probably increases with increase in distance from the implant surface (Hansson and Norton, 1999), the assumed values of these entities should increase with increasing dimensions of the roughness. (3) The surface roughness consists of pits and of protrusions (Hansson and Norton, 1999) (Fig. 1a). (4) The interfacial shear strength, Ish, is brought about by interlocking. Bone grows into the pits in the implant surface and creates retention (Hansson and Norton, 1999). In the same way roughness asperities protrude into the bone and retention is brought about by compressive forces between roughness elements and bone. The adhesion strength and friction between surface elements and bone are zero. (5) For occasional protrusions of the roughness into the bone, the failure mode is compressive fracture of the bone, i.e. the surface protrusions are assumed not to yield. Assuming a width of 1 length unit for the protrusion in Fig. 1a, fracture due to compression occurs when the compressive force, Fc1, reaches Fc1 ¼
sbcf ðH1 GÞ sin a1
σ σ α
ð1Þ
where sbcf is the compressive strength of the bone, H1 the height of the protrusion, H1 G the height of the part of the protrusion, which is in contact with bone, and a1 the slope angle of the protrusion. The component of the force Fc1 directed perpendicularly to the implant surface is sbcfS1, where S1 is the length of the part of the protrusion in contact with bone. This force will push the bone away from the implant surface and elastically compress the implant. As a consequence, a gap, G, will arise between implant and bone (Fig. 1a). The horizontal component of the force Fc1, sbcf(H1 G),
Fig. 1. (a) Surface roughness consisting of pits and protrusions with a gap, G, between the implant surface and the bone. (b) A surface roughness consisting of identical repeating elements with a gap, G, between the implant surface and the bone. (c) A cross section of a cylindrical implant with a cylindrical bore surrounded by a cortical bone cylinder.
S. Hansson et al. / Journal of Biomechanics 44 (2011) 1059–1065
will contribute to the bone–implant interfacial shear strength and since the gap lowers the value of H1 G, the interfacial shear strength will be reduced. If the protrusions are densely packed, the spaces between them should be regarded as pits – see below. (6) For bone plugs protruding into the pits, the failure mechanism will either be shear fracture or compressive fracture of the bone, depending on (i) the relationship between bone shear strength and bone compressive strength, (ii) the geometry of the pit and (iii) the size of the gap between implant and bone. Also in this case the material constituting the surface roughness is assumed not to yield. When the implant–bone interface is subjected to shear a compressive reaction force, Fc2, will arise at one of the slopes of the pit in Fig. 1a. The horizontal component of this force is Fc2 sin a2, where a2 is the slope angle of the pit. Assuming a width of 1 length unit for the pit in Fig. 1a, shear fracture occurs when Fc2 sin a2 ¼ 2tbf S2
ð2Þ
where S2 is the length of the part of the pit, which is in contact with bone. Since S2 ¼(H2 G)/tan a2 shear fracture occurs when Fc2 ¼
2tbf ðH2 GÞ sin a2 tan a2
ð3Þ
Compressive fracture occurs when Fc2 reaches Fc2 ¼
sbcf ðH2 GÞ sin a2
ð4Þ
This means that the shift between shear fracture and compressive fracture of the bone plug protruding into the pit occurs when 2tbf ðH2 GÞ s ðH2 GÞ ¼ bfc sin a2 tan a2 sin a2
ð5Þ
tbf tan a2 ¼ sbcf 2
srb ¼ p
Thus if
a2 4arctan
shear strength will be calculated for an idealized removal torque test with a cylindrical, unthreaded, titanium implant with a regular surface roughness (Fig. 1b). The implant has a cylindrical bore and is assumed to be osseointegrated in a cortical bone cylinder (Fig. 1c). This model is axisymmetric. Just before fracture the bone cylinder will be subjected to outwardly directed and the implant to inwardly directed forces. These forces have components directed perpendicularly to the implant surface and components in parallel with the implant surface. Since the bone, with exception for the portion immediately adjacent to the implant surface, and the implant material is assumed to be in the linearly elastic state the principle of superposition applies (Timoshenko and Goodier, 1982). Thus the effects of the force resultants are the sum of the effects of the force components directed perpendicularly to the implant surface and the effects of the force components in parallel with the implant surface. The force components in parallel with the implant surface will give rise to pure shear of the implant and the bone. In case of pure shear there is no change in volume (Timoshenko and Goodier, 1982). Since complete symmetry prevails a consequence is that the force components in parallel with the implant surface will not result in a change in shape of the bone cylinder or implant. Thus the analysis of changes in shape can be restricted to considering effects of the force components directed perpendicularly to the implant surface. Just before fracture these force components will subject the bone cylinder to an outwardly directed pressure, p, which will expand it and contribute to the formation of a gap between the implant surface and bone. The size of this contribution to the gap is eyba, where eyb is the circumferential strain in the bone at the interface and a the implant radius (Fig. 1c). The pressure will also compress the implant giving rise to another contribution to the gap – eyia, where eyi is the circumferential strain in the implant at the interface (Fig. 1c). The magnitude of p is P s s ðHGÞ S ð7Þ ¼ bcf p ¼ bcf 2H L where H is the height of the roughness elements and L the profile length considered (Fig. 1b). The stress and strain states in the bone (Fig. 1c) at the interface are according to Timoshenko and Goodier (1982):
which means that
2tbf
sbcf
1061
ð6Þ
failure occurs through shear, else failure occurs through compression. In the following it is assumed that the dominating failure mode is a compressive fracture. For more information see Hansson et al. (2010). (7) It is assumed that the bone exhibits an idealized stress–strain curve consisting of a linearly elastic phase followed by a plastic phase where the stress in the plastic phase is constant. It is further assumed that the displacement of the implant surface in relation to the interface bone is so big at fracture that the deformation of the bone, immediately adjacent to the implant surface, lies in the plastic region of the stress–strain curve irrespective of slope angle a. The deformation of the rest of the bone is assumed to lie in the linearly elastic region.
syb ¼ p eyb ¼
ð8Þ
a2 b2 a2
2 b2 a þ b2 1þ 2 ¼ p 2 2 a b a
ð9Þ
p a2 þb2 þub 2 2 Eb b a
ð10Þ
syb ub srb Eb
¼
The stress and strain states in the implant (Fig. 1c) at the interface become
sri ¼ p syi ¼ eyi ¼
c2 a2 p a2 p pðc2 þ a2 Þ ¼ ða2 c2 Þa2 a2 c2 a2 c2
syi ui sri Ei
¼
p ðc2 þ a2 Þ þ u i Ei a2 c2
ð11Þ
ð12Þ
The gap, G, between implant and bone caused by the pressure becomes 2.1. Roughness consisting of identical repeating elements The interfacial shear strength is sometimes calculated in removal torque tests (Ellingsen et al., 2004; Franke Stenport and Johansson, 2003; Johansson et al., 2004; Steinemann and Straumann, 1984). In the following, the theoretical interfacial
G ¼ aðeyb eyi Þ ¼ pC
ð13Þ
where C¼a
1 a2 þ b2 1 c2 þ a2 þ u u þ i b Eb b2 a2 Ei a2 c2
ð14Þ
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Assume a gap value
Calculate the pressure, p
p=
σbcf ΣSn
σ
σ σ
L
α
Calculate the gap
Calculate the interface shear strength,Ish
Ish=
σbcf Σ (Hn-G) L
Fig. 2. Irregular surface roughness with a gap between the implant surface and the bone. A suggested iterative procedure to calculate the interfacial shear strength is also shown.
Eq. (7) combined with Eq. (13) gives the following expression for the gap: G¼
sbcf CH 2H þ sbcf C
ð15Þ
Considering a section of the implant surface in Fig. 1b with a width of 1 length unit the contribution of each roughness peak to resist shear is sbcf(H G). Each roughness peak occupies a length of 2H/tan a. This means that the interfacial shear strength becomes Ish ¼
sbcf ðHGÞ sbcf tan aðHGÞ ¼ ð2H=tan aÞ 2H
ð16Þ
will be calculated for the surface roughness in Fig. 1b using different values of the roughness parameters Rsl (mean slope) and Ra. Furthermore the gap will be calculated as a function of the Ra value. It should be noted that Ra ¼H/4 and Rsl ¼tan a. Using the same equations the effect of implant diameter, implant wall thickness, thickness of supporting bone and bone quality will be calculated. The bone quality is expressed as the value of the modulus of elasticity, Eb. The compressive strength of the bone, sbcf, is assumed to depend on Eb according to the expression obtained by Keller et al. (1990):
sbcf ¼ k E1:24 b
ð19Þ 5
2.2. Irregular surface roughness A theoretical estimation of the interfacial shear strength produced by an irregular surface roughness is less straightforward. The following iterative procedure is recommended (Fig. 2): (1) assume a size of the gap between implant and bone at shear fracture based on a study of a profile of the roughness. (2) Calculate the pressure that pushes the bone away from the implant surface and compresses the implant using P s Sn p ¼ bcf ð17Þ L where Sn is the length of the part of the nth roughness element in contact with bone. (3) Calculate a new value of the gap using the calculated pressure. (4) Calculate a new value of the pressure using Eq. (17). (5) Continue until the size of the gap and the pressure have converged. (6) Calculate the interfacial shear strength by means of P ðHn GÞ s Ish ¼ bcf ð18Þ L where Hn G is the height of the part of the nth roughness element in contact with bone.
3. Results and discussion The bone–implant interfacial shear strength is a function of surface roughness, modulus of elasticity of implant material, bone support, bone quality, implant diameter and wall thickness of the implant. In the following these dependences will be addressed separately. By applying Eqs. (7)–(16) the interfacial shear strength
where k¼5.11 10 . The implant material was assumed to be titanium with a modulus of elasticity of 107 GPa. Poisson’s ratios for both bone and titanium were assumed to be 0.3 (Hansson and Ekestubbe, 2004; Hart et al., 1992). 3.1. Surface roughness parameters The theoretical interfacial shear strength, Ish, increases with increasing mean slope, Rsl, (Fig. 3a) in a similar way as for the Local model (Hansson et al., 2010), shown as dotted line in Fig. 3a. In the Local model a direct proportionality between Ish and Rsl is obtained, Eq. (20), since no gap is assumed to evolve during the loading situation. The values of Ish are much lower in the Global model due to the decreased fraction of direct implant–bone contact. The fact that both the Local and Global biomechanical models show increased Ish with increase in Rsl illustrates the importance of the mean slope: Ish ¼
sbcf Rsl 2
ð20Þ
Ish was also found to rapidly increase with increasing values of the Ra parameter for the Global model (Fig. 3b). This is in sharp contrast to the results of the Local model, where Ish was found to be independent of the Ra value (Hansson et al., 2010) (shown as dotted line in Fig. 3b). The difference between the Local and Global models is due to the lower fraction of direct implant–bone contact in the Global model. In fact, Ish can be calculated using the Local model and the fraction of implant–bone contact resulting from the gap formation. The increase in Ish with increasing Ra value agrees well with an analysis made on removal torque values in rabbit, where a linear relationship between removal torque and ¨ the Sa parameter was found (Loberg et al., 2010). In this analysis
S. Hansson et al. / Journal of Biomechanics 44 (2011) 1059–1065
1063
14 Local
Ish (MPa)
8
12
60
10
50
8
40 6 30 4
20
2
10
0
0 0
0.2
0.4
0.6
Rsl Local 50
20
40
15
30
10
20
5
10
Ish (MPa)
Ish (MPa)
Global 25
0
0 0
1
2
3
4
5
6
Ra (μm) Fig. 3. (a) Interfacial shear strength (Ish) as a function of the mean slope (Rsl) for a surface roughness consisting of identical repeating elements as shown in Fig. 1b. Ra ¼ 1.5 mm, implant diameter ¼4 mm, implant wall thickness ¼0.5 mm, bone support ¼ 3 mm and Eb ¼ 15 MPa. The results for the Local model are shown as dotted line. (b) Interfacial shear strength as a function of the Ra value for a surface roughness consisting of identical repeating elements as shown in Fig. 1b. a ¼201, implant diameter¼4 mm, implant wall thickness ¼0.5 mm, bone support ¼3 mm and Eb ¼ 15 MPa. The results for the Local model are shown as dotted line.
G (μm)
10
70
Ish (MPa)
Global 12
6 4 2 0 0
1
2
3
4
5
6
Ra (μm) Fig. 4. Gap size (G) at fracture as a function of the Ra value for a surface roughness consisting of identical repeating elements as shown in Fig. 1b. a ¼201, implant diameter¼ 4 mm, implant wall thickness¼ 0.5 mm, bone support ¼ 3 mm and Eb ¼ 15 MPa.
is hardly any contact between the bone and the implant. As Ra increases, the fraction of direct contact between bone and implant increases and therefore also the Ish increases even though the gap size at fracture increases. For example, a gap of 4.6 mm will develop on a surface with Ra ¼ 1.5 mm, which means that the gap/height ratio is 77% while for Ra ¼0.5 mm the corresponding values will be 1.8 mm and 91%, respectively. Thus, the Global model predicts that if a nano-roughness is superimposed on a micro-roughness, the contribution of the nano-roughness to the interfacial shear strength will be minor since this roughness will be mainly located in the gap. This is in contrast to the results of the Local model where all sizes of roughness contribute to the Ish (Hansson et al., 2010) since 100% implant to bone contact was assumed. These results clearly show that from a biomechanical point of view the micro-roughness is most important. Thus when the Global model is applied for evaluation of rough implant surfaces the Rsl parameter should be calculated for the bigger features constituting the topography. This can be achieved by, in addition to the 50 mm Gaussian high pass filter that is often used (Wennerberg and Albrektsson, 2000), also applying a low pass Gaussian filter of suitable size, for example 2–4 mm.
3.2. Structural implant parameters the Sa values varied between 0.6 and 1.6 mm and it is interesting to note that a nearly linear relationship between the Ra value and Ish is obtained for the Global model within the same range of Ra values. The Sa parameter is the 3D equivalent to the 2D parameter Ra. Removal torque studies on rabbit have shown that the highest fixation strength appears at Sa equal to 1.5 mm (Wennerberg, 1996). Similar results were obtained for miniature pigs (Buser et al., 1999), where the removal torque value was found to level off at Ra values larger than 2 mm. These studies indicate that other factors come into role at higher surface roughness. The change in the Ish with Ra in the Global model is due the formation of the gap. In Fig. 4, the gap size is shown as a function of surface roughness. At small Ra values the gap is almost as big as the peak-to-valley value of the surface, which means that there
The theoretical interfacial shear strength, Ish, also depends on the design of the implant. This is illustrated by keeping the implant wall thickness constant while changing the implant diameter (Fig. 5a). The Ish is found to decrease with increase in implant diameter due to decreasing stiffness of the implant, i.e. the size of the gap increases with a decrease in implant stiffness. This is also illustrated by increasing wall thickness at constant implant diameter (Fig. 5b). As the implant becomes more massive the Ish reaches a constant value, which depends on the modulus of elasticity of the implant material, Ei. The fact that the design of the implant has large effects on the interfacial shear strength means that caution should be taken when comparing results of removal torque studies with different implant designs and with implants of different materials since the resistance to gap formation can be very different.
S. Hansson et al. / Journal of Biomechanics 44 (2011) 1059–1065
14
7
12
6
10
5
8
Ish (MPa)
Ish (MPa)
1064
6 4
4 3 2
2
1
0 0
1
2
3
4 5 6 7 8 Implant diameter (mm)
9
10
0
11
0
5
10
15
Eb (GPa)
8 8 7 7 6
5 Ish (MPa)
Ish (MPa)
6
4 3 2
5 4 3 2
1
1
0 0
0.5 1 1.5 Implant wall thickness (mm)
2
0 0
2
4 6 8 Bone support (mm)
10
Fig. 5. (a) Interfacial shear strength (Ish) as a function of the implant diameter for a surface roughness consisting of identical repeating elements as shown in Fig. 1b. Ra ¼ 1.5 mm, a ¼ 201, implant wall thickness ¼ 0.5 mm, bone support ¼ 3 mm and Eb ¼15 MPa. (b) Interfacial shear strength (Ish) as a function of implant wall thickness for a surface roughness consisting of identical repeating elements as shown in Fig. 1b. Ra ¼1.5 mm, a ¼ 201, implant diameter ¼4 mm, bone support ¼3 mm and Eb ¼15 MPa.
Fig. 6. (a) Interfacial shear strength (Ish) as a function of the modulus of elasticity of the bone (Eb) for a surface roughness consisting of identical repeating elements as shown in Fig. 1b. Ra ¼1.5 mm, a ¼ 201, implant diameter¼ 4 mm, implant wall thickness¼ 0.5 mm and bone support ¼ 3 mm. (b) Interfacial shear strength (Ish) as a function of the bone support for a surface roughness consisting of identical repeating elements as shown in Fig. 1b. Ra ¼ 1.5 mm, a ¼ 201, implant diameter¼ 4 mm, implant wall thickness ¼0.5 mm and Eb ¼15 MPa.
3.3. Anatomic parameters
expected. On the very bottom of the thread there is an additional effect, represented by the thread depth, increasing the bone support and thereby increasing the interfacial shear strength. On the thread flanks the resistance to gap formation exerted by the bone is greater, which will increase the interfacial shear strength. The opposite thread flank will further increase this resistance. The magnitude of this increase in resistance can be estimated by means of, for example, finite element analysis. Most dental implants in the market place are threaded while the implants in this study were designed as unthreaded cylinders. However, the static behavior regarding interfacial shear strength is very similar for these two types of implant designs, which means that the findings in this study also apply to threaded implants. The adhesion strength and friction between surface elements and bone were assumed to be zero; the reason for this was that the contribution to bone–implant interfacial shear strength caused by interlocking between the asperities of surface roughness and bone was in focus. By assuming a certain
The bone quality, expressed as modulus of elasticity and compressive strength, had a significant effect on the interfacial shear strength. An increased value of the bone’s modulus of elasticity resulted in an increase in Ish (Fig. 6a). A stronger bone support, here expressed as an increase in the thickness of the bone cylinder, also yielded increased Ish (Fig. 6b). What is presented in this paper is a mathematical model with idealized assumptions, which has to be considered when evaluating the real situation. The bone-to-implant contact is never 100%. Furthermore only a fraction of the implant surface is in contact with cortical bone. Concerning bone support there is a difference in static behavior between a cylindrical and threaded implant, which affects the interfacial shear strength. This will be discussed with reference to Fig. 7. On the very top of the thread, the same effect on interfacial shear strength as for a cylindrical implant can be
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Acknowledgment Financial support from the Swedish Research Council (200521028-35344-27) is gratefully acknowledged.
References
Fig. 7. Qualitative description of how the resistance to gap formation, exerted by the surrounding bone, varies along the profile of a threaded implant.
coefficient of friction the effect of friction can be superimposed on the interfacial shear strength caused by interlocking. The same applies to adhesion. To optimize the interfacial shear strength for a given situation, all parameters addressed in this study need to be considered. The dentist needs to know the quality of the bone and relate that to the design of the implant. An optimization of bone support, implant size and wall thickness must be made irrespective of the surface roughness.
4. Conclusions The Global model introduced in this paper shows that the gap formed between the bone and the implant considerably lowers the interfacial shear strength. This is expected since the fraction of direct bone-to-implant contact is lowered and it is in fact shown that the Local model is applicable if only the fraction of the surface in direct contact with bone is taken into account. According to the Global model the interfacial shear strength increases with increase in values of the mean slope and the 2D surface roughness parameter, Ra. The Global model also shows the importance of implant design on the interfacial shear strength. Besides the surface roughness, parameters such as bone support, bone quality, modulus of elasticity of implant material, implant diameter and wall thickness of the implant were shown to be important. For the application to dental implants this model gives a tool for optimizing the implant for specific conditions. For a given bone anatomy and bone quality, the optimization can be made by changes in surface roughness, implant size and implant wall thickness.
Conflict of interest statement ¨ Stig Hansson, Johanna Loberg and Ingela Mattisson are employed at Astra Tech AB, a company working with development, production and marketing of dental implants.
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