Tooth-Implant Life Cycle Design

Chapter 21 Tooth-Implant Life Cycle Design Tomasz Łodygowski, Marcin Wierszycki, Krzysztof Szajek, Wiesław Hȩdzelek, and Rafał Zagalak Abstract. Dental restorations with the application of implants are very effective and commonly used in dental treatment. However, for some percent of patients, diverse complications can be observed. These problems can be caused by mechanical reasons such as loosening of the retaining screws or fracture and cracking of the dental implant components. These problems suggest the need for permanent modernization and development of dental implants. This paper describes selected aspects of the life cycle design process of the tooth-implant system Osteoplant. The authors would like to present what they mean by implant life cycle design as one part of the whole Digital Product Development (DPD) process. The sequential stages of this process are described and the tools and methods are discussed. The attention is focused on numerical simulations the mechanical behavior of dental implants and genetically based optimization algorithms. The tools and methodology of FE simulations of implant behavior are described. The whole process of optimization of a dental implant system is explained, and a self-developed optimization tool based on a genetic algorithm is presented. These processes are crucial for modern design procedure beyond the life sciences industry. Tomasz Łodygowski, Marcin Wierszycki, Krzysztof Szajek Department of Structural Mechanics, Poznan, University of Technology, ul. Piotrowo 5, 65 246 Poznan, Poland e-mail: [email protected], [email protected], [email protected] Wiesław Hȩdzelek University of Medical Sciences, ul. Fredry 10, 61-701 Poznan, Poland e-mail: [email protected] Rafal Zagalak Foundation of University of Medical Sciences, ul. Teczowa 3, 60-275 Poznan e-mail: [email protected] M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 391–422. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com  392 T. Łodygowski et al. 21.1 Introduction Dental implants are a commonly applied treatment method of dental restorations. They are used in prosthetic dentistry to support restorations that resemble a tooth or group of teeth. The current significant position of this treatment method is a result of 50 years of research and development. In the 19th and the first half of the 20th century, there were many examples of metal alloys and porcelain formulations for dental implants. However the long-term success rates have been very poor. The real milestone in implantology was the year 1952. For the first time, Dr. Per Ingvar Branemark’s team from the University of Lund in Sweden used titanium metal as implant material. They observed that an implant screwed into bone had fused to the bone after several months. He called this phenomenon osseointegration. This discovery was the foundation of great success with dental implants. Currant development of dental implants requires close collaboration between doctors and engineers and uses a wide range of engineering methods and tools, e.g. computer-aided techniques. Furthermore, this cooperation requires deep understanding of certain medical issues by the designers. From an engineering viewpoint, the crucial issues are: the dental implant placement procedure and the mechanical behavior of the implant. Dental implant placement is a two-stage procedure. The first stage is a surgical placement of the implant, which can be done with a local anesthetic. The second stage is mounting of the tooth prosthesis. In the first step the implant is screwed or taped into a surgically prepared site in the gum. Next, the gum tissue is closed over the implant. The second step is simply the wait for osseointegration. The implant remains under the gum for 3 to 6 months. While the surface of the implant is fused into bone as a result of the osseointegration process, the patient can continue to wear his denture. In the third step, the implant is exposed by removing a small amount of gum tissue. After this the custom abutment is attached to the implant by a screw. At this point the screw, connecting abutment and implants, are torqued to a pre-determined torque moment, in order to achieve the correct pre-load value of the screw. This step is very important and crucial from a mechanical viewpoint, because it prevents future loosening of the screw but also causes significant stresses in the screw and the other implant components. At the end of the dental implant placement procedure, a ceramic crown is cemented over the abutment. During all these stages, computer-based tool are used. A 3D digital image such as computer tomography provides data which can be used to create a customized crown, manufacture precision drilling guides and virtually plan the surgical procedures. There are several types of dental implants. This paper is focused on the modern implant system Osteoplant, which was created and has been developed over 10 years by Foundation of University of Medical Sciences in Poznan since. The dental implantology is efficient and common but at the same time demands complicated treatment methods. The key direction of Osteoplant system development has been to simplify surgery and prosthetic procedures by reducing the number of necessary tools and instruments for complex implant-prosthetic surgery. Due to the smooth cooperation of experienced dentists, dental and engineering research laboratories, and medical and technical universities, Osteoplant is an affordable, safe and innovative implant system. 21 Tooth-Implant Life Cycle Design 393 Osteoplant Hex is two-component implant system consisting of the abutment and root with a connecting titanium screw (Fig. 1). The root of the implant is installed in the prepared hole drilled into the jawbone as described above. Implants differ in diameter (3,5 mm, 4,0 mm, 4,5 mm) and in length (9-16 mm) but generally size is similar to natural tooth. The permanent crown is placed on the abutment but apart from its basic function, it may also be used as the basis fore a more complicated prosthetic structure. The implant system goes beyond the implant itself. The final success of a dentistry treatment procedure requires not only a well-considered and designed implant, but also tools and instruments like drills, insertion tools, countersinks, bone taps, implant drivers and sockets. The design, engineering and prototyping of dental implant system is supported with computer techniques, in particular computeraided design (CAD), computer-aided manufacturing (CAM), and computer-aided engineering (CAE). These computer and information techniques support engineers in tasks such as analysis, simulation, design, manufacture and planning, crucial aspects of the Digital Product Development (DPD) process. These tasks compromise an implant life cycle. Fig. 21.1 The Osteoplant dental implant 21.2 Motivation Numerous clinical observations of patients, who received the dental restorations implants, drew our attention to the risk of both early and late complications (Goodacre et al., 2003). These complications have both biological (failure of osseointegration or bone breaking, perforation and infections) and mechanical aspects. In many cases these mechanical problems are caused by biological phenomena such as bone remodelling or complex dysfunction of occlusion and mastication (Genna, 2004; 394 T. Łodygowski et al. Khraisat et al., 2002; Koczorowski and Surdacka, 2006; Teoh, 2000). In the late phase, the most frequent complications are loosening of the retaining screw, fracture and cracking of the dental implant components. The loosening of the retaining screw causes patient discomfort in the implant usage but can also be a source of much more serious problems. The reason for this complication is a defective static work of the mechanical implant system. Deep understanding of the mechanical behavior of implants and the possibility of numerical simulation are a prerequisite for minimizing the threat of screw loosening. The cracking of dental implant components always leads to much more serious complications and makes further treatment extremely difficult (Fig. 2). In extreme cases, overloading of the implant structure is the likely cause of these complications, but in many cases the character of loading and fracture indicates material fatigue as the basic reason (Wierszycki et al. 2006a; Wierszycki 2007; Zagalak, 2005). Our previous studies based on numerical simulations of the dental implant system (Kźkol et al., 2002) confirmed this observation. Research on implants in their real conditions is practically impossible and laboratory tests are very expensive. FEA tools however, can identify the reason for mechanical damage of the implant (Zienkiewicz and Taylor, 2005). CAE tools are widely used in the medical industry. Their use during the Osteoplant implant life cycle design reduces product development costs and time and more importantly, helps to improve the safety, comfort and durability of the dental implant system. In particular, the predictive capability of FEA tools has led us to the point where much of the design verification can be done using computer simulations rather than physical prototype testing. Fig. 21.2 Dental implant: after implantation (a) and damaged (b) Numerical simulation of dental implants has been carried out using a series of different FE models. For analyses of strength and fatigue, simplified axisymmetric models were used. For full simulation of implant structure behavior, a geometrically complex three dimensional model was necessary. All FEM calculations were carried out by commercial code Abaqus/Standard and Abaqus/Explicit. For the fatigue calculations, the fe - safe module was used. The numerical models were created directly on the basis of CAD models. During development of a dental implant system, some small and clear modifications and validations can be proposed using CAD and CAE tools. However, the major modifications need qualitatively improved methods. 21 Tooth-Implant Life Cycle Design 395 An optimization method which can be used in any work on dental implant system improvement must fulfil several specific requirements. First, this method must be based on results of FEA; therefore it should operate directly on values of the objective function instead of derivatives or other knowledge. Second, it can be used in the case of strong nonlinear problems and it should be resistant to possible discontinuities. The algorithm should also permit defining any type of constraints and finally, it should be easy or possible to join it with the environment of the FEA tool. One such technique is an optimization algorithm based on a genetic algorithm (GA). The optimization procedure is done by using self-developed software and Abaqus/CAE environment. In this tool, Abaqus solvers are used to evaluate individuals. This approach can also be applied for any, nonlinear problems, such as realistic simulation of the mechanical behavior of implant system. The CAD, CAE, CAM, CMM software and GA were used simultaneously during the implant system design procedure. The CAD models of implants, tools and instruments were created using CAD software. These full parametric models were starting points for whole tooth-implant life cycle design tasks, such as FE analyses, optimizations with GA, manufacture and planning. The created geometry of the implant was then transferred from the CAD system to a CAE environment. After FE calculation and optimization procedures, the modified geometry was transferred back to CAD software. Finally, the instructions for Computer Numerical Control (CNC) were prepared using CAM software. Digital data were used for validation procedures using the Cordinate-Measuring Machine (CMM), as well as documentation and marketing materials. In this paper the attention is focused on numerical simulation of the mechanical behavior of dental implants using FEA software and optimization based on genetic algorithms. 21.3 Numerical Simulation of Dental Implantsn 21.3.1 Model The following chapters describe the key stages of the FE model preparation procedure: geometry creation, material definition, mechanical assembly simulation, load and a description of the boundary conditions which are the necessary steps to continue the fatigue analysis. 21.3.1.1 Geometry and Mesh The geometry of a dental implant, in particular the screw connection, is very complex. For full simulation of implant structure behavior, a geometrically complex three dimensional model is necessary (Fig. 3b). This model, which includes a spiral thread, takes into consideration several important aspects, such as full simulation of kinematics of an implant, a description of the multiaxial state of stress and, most 396 T. Łodygowski et al. Fig. 21.3 Axisymmetric (a) and 3D (b) model of dental implant (CB - cancellous bone, CC cortical bone, R - root of an implant, S - screw, A - abutment) important, simulation of screw loosening. The most interesting result will be the relationship between torque moment, the friction coefficient and the loosening or fatigue life of the screw under cyclic loads. Unfortunately, a three-dimensional model is very large (ca. 600K DOF). The design process, especially if optimization procedures are taken into consideration, requires a huge number of analyses. The crucial aspect is the characteristic of the numerical model of the implant which is used in the optimization process. In practice, the time of calculation cannot exceed several dozen minutes. This limitation means that a fully three-dimensional model of implant and typical modeling approaches cannot be used (Wierszycki et al., 2006b). For this study a special approach was proposed. Due to the fact that most parts of an implant are axisymmetric, an axisymmetric concept of modelling is a good option for simplification. The Abaqus/Standard offers special types of solid elements (CCL), which are capable of modelling the nonlinear asymmetric deformation of an axisymmetric structure in a very efficient way. These elements use standard isoparametric interpolation in the radial - symmetry axis plane, combined with the Fourier interpolation with respect to the angle of revolution (Zienkiewicz and Taylor, 2005). We called this approach 2.5D (Fig. 3a). The asymmetric deformation is assumed to be symmetric with respect to the radial - symmetry axis plane at an angle equal to 0 or π (Abaqus Manuals, 2007). The creation of an implant model with these elements is a two stage procedure. In the first step an axisymmetric model of an implant was created (Fig. 4a.). This, 2D model definition has a fully parametric geometry description. The final model of the implant has been created automatically by revolving an axisymmetric model about its axis of symmetry (Fig. 4). The threads of the screw and implant body were simplified to axisymmetric, parallel rings. This 21 Tooth-Implant Life Cycle Design 397 approach reduced the geometry description of an implant model from three dimensional to two dimensional, and at the same time enabled automatic modification of the shape of implant components during the design loop. Fig. 21.4 The automatic creation of a 2.5D (b) model based on an axisymmetric model (a) The mesh of the axisymmetric model was generated automatically. In the model of an implant, a special type of axisymmetric solid elements was used. The number of these elements along the circumferential direction is the compromise between time of calculation and accuracy of results. The several series of test analyses were carried out to evaluate which number of these element is optimal. The models with only classic three-dimensional solid elements (C3D), mixed classic and axisymmetric, and only axisymmetric elements were evaluated and compared. The details of these models are shown in Tab. 1. The maximum values of the equivalent Mises stresses at characteristic notches and global bending stiffness of the whole implant structure were used to compare the results. The results of comparable studies are shown in Fig. 5. In the results of this evaluation, the model with two axisymmetric and eight classic three-dimensional solid elements was used in optimization process analyses (Fig. 4b.). Thanks to this concept, the size of the problem (ca. 100K DOF) and the cost of the calculation were significantly reduced. In the case of fatigue calculation, the implant model was fixed into the small part of the jaw bone (Fig. 3a). The geometry of bone was simplified to the regular quasi-cylindrical domain. This simplification is justified for the function of bone in this model. The bone constitutes boundary conditions of an implant. Both types of bone were taken into consideration: cortical and cancellous. The cancellous bone is modelled using axisymmetric solid elements. The cortical bone is modelled as a layer of axisymmetric shell elements which are defined on the nodes of solid elements. 398 T. Łodygowski et al. Table 21.1 Selected parameters of comparable simulation of 3D analyses with the use of cylindrical (CCL-x) and 3D solid elements (C3D-x) CCL-1 CCL-2 CCL-4 CCL-16 CCL-32 Floating Minimum Required Number Number Number point oper- memory diskspace of equa- of incre- of SDI ations per required tions ments iteration Number Wallclock of EI time [-] 6.51E+009 2.81E+010 1.61E+011 9.48E+011 5.15E+012 [-] 7 13 15 20 27 [MB] 47.31 87.25 176.93 447.38 1157.12 [MB] 150.73 366.96 1034.24 2969.60 8427.52 [-] 56226 93588 168312 317760 616656 [-] 7 7 9 13 13 [-] 33 32 47 84 82 [sec] 160 401 1565 8730 47685 Fig. 21.5 Reduction of time calculation as the result of a semianalitycal approach (red color - chosen configuration) 21.3.1.2 Material Properties The implant is made of medical titanium alloy, the mechanical properties of which are nonlinear. The description was based on certificates of conformity (ASTM F13698, ISO 5832 PT 2-93) and literature data (Niinomi, 1998) (Table 2). Table 21.2 Material properties of implant model components Implant’s body (Tita- Screw (6AL-4V- Abutment (6ALnium Grade 4) ELI) 4V-ELI) Young’s modulus [MPa] Poisson ratio Yield [MPa] Tensile Yield [MPa] 105 200 0.19 615.2 742.4 105 200 0.19 832.3 1004.0 105 200 0.19 802.8 970.4 21 Tooth-Implant Life Cycle Design 399 The jaw is composed of two kinds of bones, cancellous and cortical (Fig. 3a). The problem of describing the constitutive law of a bone is very complex. The mechanical characteristics and internal microstructure of cortical and cancellous bones are nonhomogeneous, anisotropic and variable in time. Changes in bone characteristics are caused by the phenomenon of tissue remodelling. It is very difficult to take these aspects into consideration in implant models. In finite element analysis, many concepts of mechanical properties of bone description could be applied, starting with the very simple, linearly elastic isotropic, going through the more complicated, transversely isotropic or orthotropic, and ending with the very complex, nonlinear anisotropic. The assumed material characteristics of the jaw bones are linearly elastic, homogeneous and isotropic and are discussed later (Tab. 4). This simplification is justifiable due to the role which the bone plays in fatigue analysis of an implant. The most important here is the influence of bone loss around an implant as well as bone flexibility on boundary conditions and implant fatigue life (Snauwaert et al., 2000; Zagalak et al., 2005). The best sources of fatigue material characteristics are the experimental tests of smooth specimens for constant strain amplitudes between fixed strain limits. This sources need many technically advanced resources, are expensive and complex. Some fatigue material data can be found in the literature. Also manufacturer’s certificates which are published by titanium manufacturers and fatigue researchers provide basic information about fatigue strength for specific endurance (Akahori et alo., 2005; Baptista et al., 2004, Guilhermeet al., 2005). These data are useful for stress-life analysis but are insufficient for strain-life approach. For strain-life fatigue analysis, six additional material properties are required: cyclic strain hardening coefficient K ′ , cyclic strain hardening exponent n′ , fatigue strength exponent b, fatigue ductility exponent c, fatigue strength coefficient σ ′f , and fatigue ductility coefficient ε ′f . There are several approximated relationships to obtain these data. All of them are based on some physical interpretations of fatigue properties or relationships between fatigue properties and other well known physical parameters of materials. In this study the so-called Seeger (Draper, 1999) method was used. The fatigue strength coefficient σ ′f and cyclic strain hardening coefficient K ′ are approximated with the help of the rescaling conventional monotonic ultimate tensile stress σu (Eq. 1 and 2). For titanium alloys: σ ′f = 1.67σu ′ K = 1.61σu (21.1) (21.2) Seeger’s method is a modification of the universal slopes method and assumes that the slopes of elastic and plastic asymptotes of strain-life curve are the same for specific kinds of alloys. For titanium alloys b = −0.095 and c = −0.69 (Draper, 1999). Similarly, the cyclic strain hardening exponent and fatigue ductility coefficient were assumed as constants, n′ = 0.11 and ε ′f = 0.35 = 0.11 (Draper, 1999). The fatigue data of implant titanium alloys are shown in Tab. 3. 400 T. Łodygowski et al. Table 21.3 Fatigue material data K’[MPa] n’ b c σ ′f [MPa] ε ′f Implant body Abutment Screw 1195.26 0.11 -0.095 -0.69 1239.81 0.35 1561.7 0.11 -0.095 -0.69 1619.9 0.35 1616.44 0.11 -0.095 -0.69 1676.68 0.35 21.3.1.3 Mechanical Assembly The implant system is seemingly simple, but in fact it is quite a complex mechanical system (Kakol et all., 2002; Merz et all., 2000; Sakaguchi et all., 1993; Bozkaya and Müftü, 2005). The most important aspect of the numerical simulation of implant mechanical behavior is modelling of tightening. For this purpose, it was necessary to define the multiple contact surfaces between a root, an abutment and a screw. The friction characteristic on these surfaces is one of the key parameters influencing preload axial force, reduction of implant components mobility, resistance to screw loosening, but also fatigue life of a whole implant (Kakol et al., 2002). For the friction coefficient, a value ranging between 0.1 (as in a special finished surface) and 0.5 (as in dry titanium to titanium friction) may be found in literature. In the present analyses the variant values of friction coefficients were considered. The influence of a friction coefficient on fatigue life and screw loosening is substantial and has been evaluated. In the case of the three-dimensional model, the simulation of tightening can be defined in the way that describes a real physical process. The analyses were performed using the displacement control technique. The screw was subjected to a rotational displacement equal to 1 radian, which in turn was applied to the reference point of a rigid body defined on the top surface nodes of the screw head. This approach creates a very complex, nonlinear and strong discontinuous contact problem. If this analysis is performed using the implicit code, as Abaqus/Standard, a lot of small increments and equilibrium iterations are generally required to reach a converged solution. In the case of the axisymmetric model, the tightening is simulated in a simplified way. The middle part of the screw was subjected to temperature loading (Figure 6). The orthotropic thermal expansion property of the screw material was defined in such a way, that shrinking occurred only in the direction of the screw axis. The value of the axial force in a tightened screw was calculated from the empirical formula (Eq. 3). This equation is based on the assumption that the tightening moment Md is balanced by the sum of moments involved by friction on the threads and surfaces of contact between the screw and abutment: Md = N P + π µgd2 sec α π d2 − µg P sec α d2 D2 + µf 2 2 (21.3) 21 Tooth-Implant Life Cycle Design 401 where d2 is the diameter of the thread, P is the pitch, D2 is the effective diameter of head bolts, α stands for the angle of the thread, and the µg and µ f are the coefficients of friction on surfaces of the threads and between screw and abutment, respectively (Korewa, 1969). As is clearly seen, the axial force in the screw depends on the friction coefficient and torque moment. Finally, the effect of the assumed force (200N) was replaced by an equivalent temperature field (Wierszycki, 2007). This formula was validated based on results of the full simulation of tightening carried out using a three-dimensional implant model. 21.3.1.4 Loads The external loads of an implant model were applied in the second step of the simulation (Fig. 6). The values and directions of forces were assumed according to a physiologically proven scheme. The real loading of an implant is never axial. The vertical component of it is estimated at 600 N and the horizontal at 100 N (MericskeStern et al., 1995; Mericske-Stern et al., 1996; Mericske-Stern, 1998; Morneburg and Pröschel, 2002; Morneburg and Pröschel, 2003). To estimate the most detrimental distribution of stresses, only the maximal realistic components of mastication force were taken into account. For the fatigue calculations, it is necessary to define the character of load changeability in the shape of a load-time curve, the socalled load signal. In an applied low-cycled scheme of 24-hour loads, the average values were 60 N (Bielicki et al., 1975; Hêdzelek et al., 2004). Fig. 21.6 Scheme of implant loads (H - horizontal bending component of load, N - axial force in a tightened screw) 402 T. Łodygowski et al. 21.3.1.5 Boundary Conditions The boundary conditions of the implant model are defined by bone conditions around the implant body. The first, most simple definition of boundary conditions assumed that all degrees of freedom at the bottom part of the implant body were fixed. This assumption seemed to have its explanation in dental practice, where no movements of implants under physiological load are acceptable. It is acceptable if simulation is focused only on the behavior of the screw. However, the difference between rigid fixing and even high stiffness can be significant, especially in cyclic loading and the context of fatigue damage. In this case the stiffness of bone should be taken into consideration. In this approach, the boundary conditions of implants are replaced by a model of a small part of the bone (Fig. 3a). The geometry of a small part of the jaw surrounding the implant is very simplified, but it enables us to take into consideration the finite stiffness of boundary conditions and the changes in implant fixing conditions as well. The changing stiffness of the bone and bone loss phenomena are very important from a mechanical viewpoint. The bone loss phenomenon especially has a significant influence on implant behavior, stress distribution and therefore fatigue damage. The degree of encasement and osseointegration of the implant may not be 100%. It depends on bone quality, microstrains developed during healing and function, and the location of the implant in the jaw. This percentage may decrease to as low as 50%, due to phenomena of bone remodelling. In these analyses three levels of osseointegration were considered (Fig. 7). Fig. 21.7 Levels of dental implant osseointegration in the jaw bone: a) no. 1, b) no. 2, c) no. 3 In the case of the first level, the implant body is fully fixed in the jaw bone (Fig. 7a). In the next two, the degree of implant body embedding decreases to 75 and 21 Tooth-Implant Life Cycle Design 403 50%, respectively (Fig. 7b and Fig. 7c). The top level of cortical bone was assumed as: (a) level no. 1 - entirely embedded to a bone (100%), (b) level no. 2 - 3mm below the conical implant surface (75%), (c) level no. 3 - 6mm below the conical implant surface (50%). The choice of levels of osseointegration expressed in this assumed geometry is based on the following assumptions: (a) the first level is a target position after a restoration treatment and was taken into considerations in all simulations, (b) the second level corresponds to bone removal which uncovers the first outer thread and where the hexagonal abutment changes into the inner thread, (c) the third level corresponds to those boundary conditions of an implant body for which bending acts on the weakest cross-section. Table 21.4 Fatigue material data Bone Cortical all schemes Cancellous 1st scheme (D1) 2nd scheme (D2) 3rd scheme (D3) 4th scheme (D4) Young’s modulus [MPa] 13 000 9 500 5 500 1 600 690 Bone remodelling is a very complex phenomenon, a result of changing the strain state in bone microstructure. The characteristic of bone changes permanently. The microstructure of cancellous bone is adapted to variant loads. The cancellous bone of jaw and maxilla is rebuilt quite fast; bone tissue is replaced by new tissue in a few months (Milewski, 2002). In order to take into consideration the effects of bone remodeling, a simplified approach was used. The four different values of the Young modulus of cancellous bone according to the Lek-holma i Zarba (Zagalak, 2003) classification were used in FEA models of an implant system (Tab. 4). The highest value of the Young modulus corresponds to the case in which cancellous is completely replaced by cortical bone. All mechanical characteristics of both cancellous and cortical bones were taken from the literature (Milewski, 2002; Misch, 1999). 21.3.2 FE Analyses - Strength Analyses The strength analyses of the implant system were carried out for service and exceptional loading. The most adverse load conditions are assumed. The applied 404 T. Łodygowski et al. horizontal force H triggers bending of the screw. Significant plastic strains appear on the notches of the screw, specifically on its thread and its head, only in cases of the most detrimental values of friction coefficient and torque moments. Nevertheless, from the point of view of a static work of the system, the stresses do not exceed the safe level. These results shown that distribution of stresses in implant components should not cause cracking. Local accumulations of stress, concentrated in a small area, and achieving the plastic yield of titanium, cannot be the source of observed mechanical failures. The results obtained also show a negligible effect of the stiffness of cancellous bone on stress distribution in the whole implant. No significant differences in stress values or concentration zones can be observed. However, the significant differences can be observed for the various levels of implant encasement (Fig. 8). For level no. 1, the stress concentration zones are observed mainly in the upper part of the screw in the plane of action of the bending (Fig. 8a). For level no. 2 the stress concentration zone covers the whole screw. The highest values of stresses are observed in the lower part of the screw. The areas with high values stresses are extended to include the first coil of outer thread of the implant body too. The stresses in the screw significantly increased as well (Fig. 8b). For Level no. 3, the zone of high stress value covers virtually the entire length of the screw as well as all coils of the outer and inner thread of the implant body and the thread of the screw in the same plane (Fig. 8c). It is worth mentioning that the plastic strains which appeared in the notches of the screw, specifically on its thread and its head, are in all cases on a safe level from the point of view of the static work of the whole system (Kakol et all, 2002). 21.3.3 Fatigue Calculations - Strain-Life Approach In the fatigue life calculation, nine cases of loading, three cases of bone configurations and three cases of jaw bone stiffness were taken into consideration. The estimation of the fatigue is always based on results of the finite element analysis; therefore all fatigue calculations were preceded by the FE analyses of implant models (Wierszycki et al., 2006a)(Fig. 9). Because the significant plastic strain occurred in case of a specific combination of friction coefficient and torque moment the fatigue calculations were carried out using the strain-life fatigue approach. This approach uses algorithms which incorporate formulae based on both elastic and plastic strain to estimate the fatigue life (Wierszycki et al., 2006a). This approach is based on the strain-life equation (Eq. 4) which is a function of the maximum shear strain γm ax and strain normal to the maximum shear strain εn : σ ′f   c b Δ γmax Δ εmax 2N f + 1.75ε ′f 2N f + = 1.65 2 2 E (21.4) where σ ′f is the fatigue strength coefficient, ε ′f is the fatigue ductility coefficient and N f stands for the number of cycles. Used algorithms are based on the stress results 21 Tooth-Implant Life Cycle Design 405 Fig. 21.8 Huber-Mises equivalent stress distribution after bending for three levels of osseointegration: a) no. 1, b) no. 2, c) no. 3 obtained from the finite element analysis (Fig. 7), variations in loading, hysteresis loop cycle closure, and material properties. Elastic stresses from the finite element analysis model are transformed into elastic-plastic stresses by means of the so called biaxial Neuber’s rule (Eq. 5). This relationship is defined as a hyperbola and is expressed by the following equation: Δ σ Δ ε = Kt2 Sk2 E (21.5) where Δ σ and Δ ε are the amplitudes of true stresses and strains. The nominal stress-strain product is defined by elastic stress concentration factor Kt , nominal elastic stress sk and Young modulus E. During fatigue calculation, this formulae (Eq. 3) is used to calculate the principal stresses and strains for each node separately. The rainflow cycle counting algorithm is used to extract fatigue cycles. For biaxial 406 T. Łodygowski et al. Fig. 21.9 Algorithm of fatigue calculations based on FEA results. s - nominal stress, e nominal strain, Pk = P(t) - load signal, σ Δ - true stress amplitude, εΔ - true strain amplitude, E - Young’s modulus, K’ - cyclic strain hardening coefficient, n’ - cyclic strain hardening exponent, b - fatigue strength exponent, c - fatigue duc-tility exponent, σ ’ f - fatigue strength coefficient, ε ’ f - fatigue ductility coefficient, N f - number of cycles fatigue methods, a critical plane procedure is used to calculate the orientation of the most damaged planes at nodes. Based on these data, the fatigue life is calculated using strain-life formulation (Eq. 4) (fe-safe Manuals, 2005; Draper, 1999). In order to estimate service life of an implant, a designed life is defined. Fe - safe calculates the factor FOS (Factor Of Strength) by which the stresses at each node can be increased or reduced to give the required life (fe safe Manuals, 2005). This kind of result is the most interesting and useful in design and optimization processes. The required level of stresses can be treated as an objective function. Fatigue life of an implant screw was calculated for nine separate cases of loading, three cases of boundary condition schemes and three cases of jaw cancellous bone density. For all of these cases, the same cyclic scheme of loading was assumed. The following assumptions were made: 21 Tooth-Implant Life Cycle Design 407 Fig. 21.10 Cyclic scheme of loading - daily load signal (a) (b) (c) (d) (e) maximum value of load at the physiological process of grinding food - 200 N, average value of load during chewing - 20 N, number of occlusal contacts during meals - 20 1/min, average duration of three meals during a day: 15min, 20 min i 15 min, number of occlusal contacts, associated with the action of saliva swallowing in the periods between meals - 2 ů 105 1/year. A twenty-four-hour changeability scheme was assumed as a signal (Fig. 10), while the number of days corresponding to four years was assumed as the number of cycles. In addition to detailed guidance for the overall design conclusions may be drawn based on fatigue life calculations. The FOS distribution (Fig. 11) analysis for particular cases indicates the axial forces in the screw and the changes in the scheme of boundary conditions which have the greatest influence on fatigue changes. For different bone density and at the same time divergent stiffness of boundary conditions, significant differences of stress distributions present in the screw are noticeable (Fig. 12, 13 and 14). This does not lead to serious fatigue changes. For axial forces above 600 N, there is a noticeable increase in the areas endangered by fatigue failure. The degree of required stress reduction reaches ca. 30%. In the most unfavorable load case, the maximal axial force value is the result of a high torque moment and a very small friction coefficient on a screw thread. It is important to pay attention to the danger of increasing the tightening forces in an implant screw. In two-part implants, this high tightening force is motivated by biological and medical aspects. However, the increase in torsion moments and decrease in friction coefficients pose a danger for fatigue life of implant components. 408 T. Łodygowski et al. Fig. 21.11 The scale of FOS distribution Fig. 21.12 FOS distribution in the screw - level no. 1 and bone stiffness D1 (a), D2 (b), D3 (c), D4 (d) Fig. 21.13 FOS distribution in the screw - level no. 2 and bone stiffness D1 (a), D2 (b), D3 (c), D4 (d) 21 Tooth-Implant Life Cycle Design 409 Fig. 21.14 FOS distribution in the screw - level no. 3 and bone stiffness D1 (a), D2 (b), D3 (c), D4 (d) 21.3.4 Screw Loosing Simulation The simulation of screw loosening can only be performed using a 3D implant model (Fig. 3b). This simulation needs a FE model which can reflect a real physical characteristic of screw connection. In the first step of this simulation, the abutment and implant body were tightened by the screw described previously. The screw was rotated until the expected torque moment and axial force in the screw were obtained. Four values of tightening (0, 15, 25 and 35 Ncm) and two friction coefficients (0.1 and 0.2) were considered. In the second step of simulation, the torsional cyclic load (asymmetric force) was applied to the top of the abutment to initiate a small movement of the implant system. This kind of analysis is strongly nonlinear and extremely discontinuous due to contact conditions and the quasi-static character of the implant response. In this case it is very difficult to obtain any converged solution using implicit code. For simulations of screw loosening, dynamic analysis using an explicit code, like Abaqus/Explicit was carried out. Unfortunately, in this case the time of simulation of the whole screw loosening process is unacceptable. Due to this fact the simplified approach was proposed. The four separated simulations for different values of tightening were treated as four consecutive stages of the same screw loosening process. The obtained results, e.g. permanent rotation of the screw, clearly showed that this approach can reflect screw loosening phenomena. To measure a resistance to screw loosening, the rate of the frictional dissipation energy accumulated over each stage of the same screw loosening process was chosen. The energy dissipated by contact friction forces between the contact surfaces is as follows: 410 T. Łodygowski et al.  t 0   S  tT · ġslT dS dt (21.6) where gslT is the velocity field, tT is the frictional traction and S stands for the boundary (contact surfaces). This energy is a measure of the work to be performed during the screw loosening process. In Fig. 15 the change of the frictional energy dissipation is plotted for three different stages of the screw loosening process. It is clearly seen that dissipation of this energy increases when tightening moment decreases at the same time as a result of screw loosening. During design and optimization processes, each proposed modification can be evaluated in this way for screw loosening resistance. Fig. 21.15 Gradient of energy dissipated through frictional effects during screw loosening (torque moment M = 0/15/25/35Ncm and friction coeff. n = 0.2) 21.4 Optimization Using Genetic Based Algorithms The numerical analyses of dental implant provide a satisfying estimation of mechanical behaviour for many different services and unexpected cases. They enable verification of existing systems and validation of new ones. Some small and clear modifications of existing dental implants are often easy to develop and validate using FEA. However, major modification needs qualitatively new methods of optimization. The optimization of the implant system was divided into several stages. The various objectives are considered: strength and fatigue life of the implant, fixation in a bone and minimizing of the screw loosening phenomenon. 21 Tooth-Implant Life Cycle Design 411 21.4.1 Optimization Algorithm - Genetic Algorithm The optimization of a dental implant is very complex and relatively difficult. The model which is the basis (starting point) for the optimization is very developed, and estimating the implant behaviour requires consideration of material nonlinearity, complex contact definition and prestress of the assembly screw. The level of complexity and large number of design parameters result in many local minima in design space. Thus, the optimization algorithm has to search optimal solutions globally, based on a limited set solution in the design space. Moreover, problems with FE analysis can occur and should be take into account; most frequent are rebuilding and convergence problems. The complexity of the geometry and large number of design parameters combination make it impossible to predict all problems with the geometry. As a consequence, for some configurations of design parameters which are in the feasible range, proposed shapes could be incorrect. Additionally, problem with mesh generation can occur, which can also be treated as a rebuild problem. Reasons for no convergence in analyses are usually caused by too large plastic strain or by contact problems. Both can occur for points in the feasible region of design space and have to be expected during the optimization process. One of the optimization algorithms which can fulfil all these requirements is a genetic algorithm. The genetic algorithm has been inspired by evolutionary biology and uses techniques such as inheritance, mutation, selection and crossover to find a better solution. Despite being a powerful optimization tool, the genetic algorithm uses simple rules, which makes it easy to implement. In the module presented in this work, the galileo (GPL) library was used. Fig. 21.16 Genetic algorithm processing scheme GA operates on individuals, which are the abstract representations of a real solution. In this optimization approach, a chromosome was constructed over the alphabet 0,1 as a binary string. The main idea of GA processing consists in generation of a set of initial solutions and use of evolutionary operators to improve them in successive iterations, called generations. An initial set of individuals, called an initial 412 T. Łodygowski et al. population, is usually randomly created. Every new population is subjected to evaluation. The evaluation process consists of assigning a fitness value to each individual. This value describes solution quality and is calculated according to an objective function and results of FE analyses. Fitness values are the basis of the selection process. The selection mechanism is responsible for improvment of next population quality. During selection, statistically only the most fitted individuals are chosen in order to allow them to take part in the creation of the next population. In the next stage, the selected individuals are subjected to crossover and mutation. The operators use selected chromosomes in order to create new design parameter configurations. The primary function of crossover is mixing the parent individual chromosomes and creating new offspring. In a classic form, crossover consists of sampling a chromosome partition point, dividing parent chromosomes and swapping obtained parts between them. The mutation is responsible for random distortion of chromosomes, which keeps the diversity of population on a sufficient level and helps to prevent from a local minimum. In the binary encoded chromosomes, the mutation samples gene changes its value to the opposite. In the last stage the new population replaces the previous one. Usually, GA ends when either maximum fitness value for the best fitted individual is obtained or a maximum number of generations is achieved. The general chart-flow of the genetic algorithm is presented in Fig. 16. 21.4.2 Goal This study was expected to identify a new dental implant shape which would lead to lower principal stresses in comparison with the current values. In this FE model, geometry and material properties are the same as for real dental implants. The loads that were applied come from the literature (Zygalak, 2003). For the presented configuration, the maximum principal stresses are greater than 625 MPa and are localized in the screw corner (Fig. 17). 21.4.3 Design Parameters The geometry of this two-component implantology system is quite complex and the choice of correct design parameters is crucial. The thread was excluded from consideration at this stage of the work. Six geometrical parameters were defined as real variables (Fig. 18). All of them refer to the upper part of the implant. The design parameters were encoded into the binary string. There were determined for each design parameter range and number of bits for encoding. The ranges come from both the geometry limitations and manufacturer requirements and are listed in Tab. 5. The larger the number of bits for encoding, the better the accuracy; on the other hand, the total increase of optimization time is substantial. In this, for angle 21 Tooth-Implant Life Cycle Design 413 Fig. 21.17 Maximal principal stresses in the initial dental implant design Table 21.5 Ranges and number of bits for design parameter encoding Name Symbol Min value Max value Range Bits Resolution Screw head diameter Screw head conic surface opening angle Screw head height Screw diameter Hexagonal slot height Hexagonal slot conic surface opening angle A 0.95 1.6 0.65 3 0.093 B 0.0 80.0 80.0 4 5.33 C D E 1.5 1.0 0.05 6.4 1.5 2.65 4.9 0.5 2.6 3 3 3 0.7 0.071 0.371 F 11.0 90.0 79.0 4 5.27 variables, four bits strings were constructed, whereas for all the rest only three bits were used. The choice of particular design parameters based on the range influence on final results. All the binary strings reference to the design variables are joined into a chromosome. In this work the twenty bit string is used as follows: D D E E E F F F F chi =[gA1 , gA2 , gA3 |, gB4 , gB5 , gB6 , gB7 , |gC8 , gC9 , gC10 |gD 11 , g12 , g13 |g14 , g15 , g16 |g17 , g18 , g19 , g20 ] (21.7) The superscripts denote the design parameters symbols in accordance with Tab. 5. 21.4.4 Constraints The material used in the FE model simulates the elastic - plastic behavior. To some extent it is allowed to exceed the plastic stress limit in the dental implant, but 414 T. Łodygowski et al. Fig. 21.18 Design parameters the plastic strain cannot be too large. In FE analysis it is assumed that maximum equivalent plastic strain cannot be higher than 10  t( 2 p pl pl ε = ε |0 + ε̇ l : ε̇ p ldt (21.8) 3 0 The ε pl |0 donates initial equivalent plastic strain, which was equal to zero in this case whereas ε̇ p l donates an equivalent plastic strain rate. Principal stress reduction, which was set as a goal, directs changes in individual in opposite direction to this constraint. Thus, no stress or plastic strain constraints were necessary. In case of a rebuild or convergence problem, the death penalty method is applied. The individual is eliminated from further processing. In the evaluation of a single individual, the FE model of dental implant, which is time-consuming significantly limits the population size. On the other hand, the population has to be large enough to provide sufficient diversity and represent design space properly. In the presented work, a population of forty individuals was used. The number of epochs was not limited a priori. The optimization has been stopped, based on monitored results. The signal to break the procedure was sent after a few epochs without any new proposal for a better solution. 21.4.5 Genetic Algorithm Settings A ranking based method of selection is used. In contrast to the classic method of selection, roulette - wheel, this method sets the probability of being selected according to the position in the ranking instead of fitness. As a consequence, the probability increases of choosing an individual with fitness much lower than the maximum and results in higher diversity in the late stage of the optimization process. The 21 Tooth-Implant Life Cycle Design 415 one - point crossover method is applied. The selected individuals are joined into couples; their chromosomes are divided into two substrings at random positions and swapped between parent individuals. The new couple of offspring represent combinations of both parent features. In order to keep sufficient diversity during the optimization process, not the all couples were subjected to crossover. The intensity of crossover is determined with the crossover rate, which is constant and equal to 0.75. In the last stage of recombination, random distortions are introduced to chromosomes. The probability of each gene distortion is controlled by the mutation rate. In this work, the constant value of 0.02 was kept during whole optimization process. When the gene to mutate is selected, its value is changed to the opposite, from 0 to 1 and vice versa. The classic form of population replacement was used. The whole previous population was completely replaced with the new one. This method allows for increasing the diversity and searches the design space more extensively in comparison with the alternative methods, e.g. steady-state replacement. Unfortunately, it also moderates the convergence and increases the number of necessary FE analyses. The small population size which was used limits the variation of design parameters, therefore diversity is an important factor. 21.4.6 Incorporation of the Genetic Algorithm into Abaqus/CAE Genetic algorithm processes chromosomes which represent various combinations of design parameters. Each new chromosome has to be evaluated in order to assess its quality. The great advantage of the genetic algorithm is that evaluation processes can be fully elaborated in an exterior procedure. The procedure returns fitness values based on chromosomes. The tool employed in the evaluation procedure has to provide the possibility to rebuild the FE model for any design parameter configuration. Moreover, some procedures like meshing have to be created automatically. Thus, the reliability and capacity of the rebuilding tool are crucial. In this work, the Abaqus/CAE environment in cooperation with Abaqus/Standard code are used. Abaqus/CAE provides an efficient environment for user scripting. An objectoriented scripting language, called python, is available, allowing the user to add new procedures into the kernel and graphical user interface (GUI). In order to carry out the dental implant optimization, a new module has been created with full functional graphical user interface (Fig. 19). The optimization driver was imported as an external library into the Abaqus/CAE environment. The open source library called Galileo was used. The implemented module is divided into two main sections. The first organizes genetic optimization and employs mostly Galileo library, whereas the second provides the tool for chromosome evaluation. Every chromosome created in the reproduction process is a starting point for the evaluation procedure. In the first stage each chromosome is divided into substrings, which reference design parameters and are encoded. Finally, a set of real design parameter values is obtained and provides the basis for model rebuilding. All design parameters refer to the geometry of the dental 416 T. Łodygowski et al. Fig. 21.19 Optimization module for Abaqus/CAE implants. Abaqus/CAE kernel script is created and run in order to change them in the model. The obtained geometry is controlled and if the shape is invalid, rebuild error is raised. The weakest point of the procedure is the creation of a mesh. The mesh is generated automatically and due to large number of analyses, it is not possible to control them manually. It is assumed that any errors in mesh result in higher principal stresses and eliminate the solution during selection. Based on modified geometry, the FE analysis definition is created and sent to the Abaqus/Standard solver. In the case of presented work, additional analysis has to be carried out to provide results for the axisymmertic model. The spatial dental FE model was built based on results for axisymmetric models with additional data such as lateral force magnitude. The module waits for analysis termination until maximum time is not exceeded. If the analysis takes too long, the solver stops and no convergence error is raised. This limitation can also eliminate well fitted solutions but is necessary to precede optimization. The analysis results and all information about its proceeding are stored in an output database, which can be read with the use of python script. All necessary information is extracted from the output database and objective function is calculated. Finally, fitness value returns to the optimization driver. The whole flow-chart of the individual evaluation procedure is presented in Fig. 20. Fig. 21.20 Individual evaluation procedure 21 Tooth-Implant Life Cycle Design 417 In case of the rebuild error, a zero fitness value is returned for the individual. Zero fitness value guarantees that invalid solutions will be eliminated during the selection process. 21.4.7 Results The optimization was performed using 8 cpus in computations (Fujitsu-Siemens PRIMERGY TX300 S3). During the whole optimization process over, 1600 FEA analyses were carried out. The total time of optimization was ca. 250 hours. Fig. 21.21 Maximal principal stress value for successive generated individuals The evolution of the optimization process is presented in Fig. 21. It shows the changes of the maximum principal stresses in the upper part of the implant screw for successively generated solutions. The average value of the principal stress is drawn by a solid (red) curve. It can be observed that subsequent designs are being improved - the maximum principal stress is decreasing. Starting with a value of 625 MPa, the principal stress is reduced to 150 MPa. The graph does not consist of the first stage when the initial population was established. More than 320 FE analyses were necessary to find the forty correct individuals. If starting with the first population of random individuals the better solution is founded. Moreover, the reduction of principal stress is meaningful and equals 475 MPa, more than 75% of initial value. In the next generations, the design parameters from the best solution (peaks) were promoted more strong and thus they strongly influenced the population improvement. This results in a 418 T. Łodygowski et al. constant decrease of both average and minimum value of the principal stresses. After the 20th generation, the best solution is established and no further essential reduction of stress is observed. Because of the chosen reproduction strategy, diversity within the population is sufficient and even in the last monitored population, the individuals represent wide range of fitness values. Fig. 21.22 Initial (a) and new (b, c) shapes of dental implant Two proposals for new shapes are shown in Fig. 22. The lowest principal stresses were obtained for shape b) and equal 150MPa. The next proposal (Fig. 22c) provides reduction to 180MPa but still has hexagonal slot instead of the most optimal solution. The haxagonal slot plays an important role in preventing screw loosening, thus the solution b) will not be considered in further analyses. There are a few clear tendencies easy to explore. In the first place, the screw head is wider than the initial one and the opening angle of the screw head conic surface is moderated. Together with modification of the opening angle of the hexagonal slot, the conic surface makes clamping of the abutment and the body more efficient. Moreover, the moderation of the opening angle of the screw head conic surface reduces the stress concentration in the interior corner of the screw. All these modifications make the joint stiffer and reduce relative rotation between the abutment and the body. The implant starts to behave as a cantilever (Fig. 23). The final implant incorporates the body to carry lateral load. As a result the displacement of the points of the upper edge are reduced almost twice from 0.085 mm to 0.045 mm. The change of dental implant behavior also results in more uniform stress distribution (Fig. 24). The contact pressure is realized on the whole contact surfaces, 21 Tooth-Implant Life Cycle Design 419 Fig. 21.23 Deformation of initial (a) and final (b) dental implant (the displacements are scaled 40 times) Fig. 21.24 Mises stress contour of initial (a) and final (b) dental implant in contrast to the initial implant. As a result, the screw is bent less. All proposals of the new shapes were verified with the use of full three dimensional FE models. The results confirmed that the FE model built with cylindrical and solid elements provides compatible values and can be used for further optimization. 420 T. Łodygowski et al. 21.5 Conclusions The dental implant system life cycle design is supported with many computer techniques that support engineers in many tasks, such as drafting, 3D design, analyses, simulations, manufacture, planning, documentation creating and preparing marketing materials preparing as well. The FEA simulation of the behavior of a dental implant system was presented. The results of simulation showed that computational modelling and 3D simulation enables realistic prediction of implant mechanical behavior under restoration treatment as well as service loads. On the basis of the results of fatigue analysis, it can be claimed that material fatigue is the basic reason for the observed complications and must be taken into consideration during the dental implant design cycle. Furthermore the application as a measure of loosening resistance, the energy dissipated through frictional effects, can lead to practical recommendations in dentistry. For screw loosening simulation, the modelling of tightening is a crucial task that must be carried out in such a way that it describes a real physical process as much as possible. Only 3D modelling allows for a full simulation of the kinematics of the implant, and, in consequence, the possibility of screw loosening. The presented optimization approach enables us to successfully incorporate FEA and genetic optimization procedures into the design cycle. The OSTEOPLANT dental implant has been optimized. The new shape has been found and it provided the solution to both substantial principal stresses and displacement reductions. One of the crucial design goals was the minimization of fatigue damage of the implant. The obtained level of principal stress in key parts of the implant connection enables us to reduce or even to eliminate the risk of fatigue damage. The study and the final results give evidence that this method is efficient and can be used during the design process. These stages of dental implant design clearly showed that CAD, CAM and CAE tools are the foundation for efficient and robust engineering and prototyping activities for medical devices. Acknowledgements. Financial support for this research was provided by grant N R13 0020 06. This help is kindly acknowledged. References [1] Abaqus Manuals, SIMULIA, Pawtucket (2007) [2] Akahori, T., Niinomi, M., Fukui, H., Ogawa, M., Toda, H.: Improvement in fatigue characteris-tics of newly developed beta type titanium alloy for biomedical applications by thermo-mechanical treatments. Materials Science and Engineering C25, 248–254 (2005) [3] Baptista, C.A.R.P., Schneider, S.G., Taddei, E.B., da Silva, H.M.: Fatigue behavior of arc melted Ti-13Nb-13Zr alloy. International Journal of Fatigue 26, 967–973 (2004) [4] Bielicki, J., Prośba-Mackiewicz, M., Bereznowski, Z.: Chewing forces and their measurement (in polish). Protetyka Stomatologiczna 25, 241–251 (1975) 21 Tooth-Implant Life Cycle Design 421 [5] Bozkaya, D., Müftü, S.: Mechanics of the taper integrated screwed-in (TIS) abutments used in dental implants. Journal of Biomechanics 38, 87–97 (2005) [6] Draper, J.: Modern metal fatigue analysis. HKS, Inc. Pawtucket (1999) [7] fe-safe Manuals, Safe Technology Limited, U.K. (2005) [8] Genna, F.: Shakedown, self-stresses, and unilateral contact in a dental implant problem. European Journal of Mechanics A/Solids 23, 485–498 (2004) [9] Goodacre, C.J., Bernal, G., Rungcharassaeng, K., Kan, J.Y.: Clinical complication with implants and implant prostheses. International Journal of Prosthodontics 90, 121–129 (2003) [10] Guilherme, A.S., Henriques, G.E.P., Zavanelli, R.A., Mesquita, M.F.: Surface roughness and fatigue performance of commercially pure titanium and Ti-6Al-4V alloy after different polishing protocols. Journal of Prosthetic Dentistry 93(4), 378–385 (2005) [11] Hȩdzelek, W., Zagalak, R., Łodygowski, T., Wierszycki, M.: Biomechanical studies of the parts of prosthetic implants using finite element method (in polish). Protetyka Stomatologiczna 51(1), 23–29 (2004) [12] Ka̧kol, W., Łodygowski, T., Wierszycki, M.: Numerical analysis of dental implant fatigue. Acta of Bioengineering and Biomechanics 4(1), 795–796 (2002) [13] Khraisat, A., Stegaroiu, R., Nomura, S., Miyakawa, O.: Fatigue resistance of two implant/abutment joint designs. Journal of Prosthetic Dentistry 88(6), 604–610 (2002) [14] Koczorowski, R., Surdacka, A.: Evaluation of bone loss at single-stage and two-stage implant abutments of fixed partial dentures. Advances in Medical Sciences 51, 43–46 (2006) [15] Korewa, W., Zygmunt, K.: Basics of Machine Construction (in polish). Wydawnictwo Nauko-wo-Techniczne, Warszawa (1969) [16] Mericske-Stern, R., Assal, P., Mericske, E., Burgin, W.: Occlusal force and tactile sensibility measured an partially edentulous patients with ITI implants. The International Journal of Oral and Maxillofacial Implants 3, 345–353 (1995) [17] Mericske-Stern, R., Assal, P., Buergin, W.: Simultaneous force measurements in 3 dimensions on oral endosseous implants in vitro and in vivo. Clinical Oral Implants Research 7, 378–386 (1996) [18] Mericske-Stern, R.: Three-Dimensional Force Measurements With Mandibular Overdentures Connected to Implants by Ball-Shaped Retentive Anchors. A Clinical Study. The International Journal of Oral and Maxillofacial Implants 13, 36–43 (1998) [19] Merz, B.R., Hunenbart, S., Belser, U.C.: Mechanics of the implant-abutment connection: an 8-degree taper compared to a butt joint connection. The International Journal of Oral and Maxillofacial Implants 15(4), 519–526 (2000) [20] Milewski, G.: Strength aspects of biomechanical interaction bone tissue-implant in dental bio-mechanics (in polish). Zeszyty Naukowe Politechniki Krakowskiej seria Mechanika 89 (2002) [21] Misch, C.E.: Contemporary Implant Dentistry. Mosby St. Louis (1999) [22] Morneburg, T.R., Pröschel, P.A.: In vivo forces on implants influenced by occlusal scheme and food consistency. International Journal of Prosthodontics 16, 481–486 (2003) [23] Morneburg, T.R., Pröschel, P.A.: Measurment of masticatory forces and implant load: a methodologic clinical study. International Journal of Prosthodontics 15, 20–27 (2002) [24] Niinomi, M.: Mechanical properties of biomedical titanium alloys. Materials Science and Engineering 243, 231–236 (1998) 422 T. Łodygowski et al. [25] Sakaguchi, R.L., Borgersen, S.E.: Nonlinear finite element contact analysis of dental implant components. The International Journal of Oral and Maxillofacial Implants 7, 655–661 (1993) [26] Snauwaert, K., Duyck, J., van Steeberghe, D., Quirynen, M., Naert, I.: Time dependent failure rate and marginal bone loss of implant supported prosthese: a 15-year follow-up study. Clinical Oral Investigations 4, 13–20 (2000) [27] Wierszycki, M.: Numerical analysis of strength of the dental implants and the human spine motion segment (in polish). PhD Thesis, Poznan University of Technology, Poznan (2007) [28] Wierszycki, M., Ka̧kol, W., Łodygowski, T.: Fatigue algorithm for dental implant. Foundations of Civil and Environmental Engineering 7, 363–380 (2006) [29] Wierszycki, M., Ka̧kol, W., Łodygowski, T.: Numerical complexity of selected biomechanical problems. Journal of Theoretical and Applied Mechanics 44(4), 797–818 (2006) [30] Zagalak, R.: Evaluation of mechanical properties of two dental implants Osteoplant (in polish). PhD Thesis, University of Medical Sciences in Poznan, Poznan (2003) [31] Zagalak, R., Hȩdzelek, W., Łodygowski, T., Wierszycki, M.: Influence of the loss of bone and the loss their density on risk of fracture of the implant - a study using finite element (in polish). Implantoprotetyka 6(1), 3–7 (2007) [32] Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method. Elsevier, Amsterdam (2005)