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Hindawi Applied Bionics and Biomechanics Volume 2019, Article ID 3768695, 14 pages https://doi.org/10.1155/2019/3768695 Research Article Kriging Surrogate Model for Resonance Frequency Analysis of Dental Implants by a Latin Hypercube-Based Finite Element Method 1 1 2 Liu Chu , Jiajia Shi , and Eduardo Souza de Cursi Department of Transportation, Nantong University, Nantong, China Département Mécanique, Institut National des Sciences Appliquées de Rouen, Rouen, France Correspondence should be addressed to Jiajia Shi; firstname.lastname@example.org Received 4 October 2018; Revised 3 February 2019; Accepted 13 March 2019; Published 10 April 2019 Academic Editor: Le Ping Li Copyright © 2019 Liu Chu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The dental implantation in clinical operations often encounters diﬃculties and challenges of failure in osseointegration, bone formulation, and remodeling. The resonance frequency (RF) can eﬀectively describe the stability of the implant in physical experiments or numerical simulations. However, the exact relationship between the design variables of dental implants and RF of the system is correlated, complicated, and dependent. In this study, an appropriate mathematical model is proposed to evaluate and predict the implant stability and performance. The model has merits not only in the prediction reliability and accuracy but also in the compatibility and ﬂexibility, in both experimental data and numerical simulation results. The Kriging surrogate model is proposed to present the numerical relationship between RF and material parameters of dental implants. The Latin Hypercube (LH) sampling method as a competent and sophisticated method is applied and combined with the ﬁnite element method (FEM). The methods developed in this paper provide helpful guidance for designers and researchers in the implantation design and surgical plans. stability oﬀers helpful information to keep reliability in the 1. Introduction individual treatment [15, 16]. The histological examination is one of the traditional The dental implant as a predictable and reliable treatment has been widely applied in the rehabilitation of edentulous invasive approaches. Noninvasive methods are required for the observation and measurement of implant stability [17, patients . The implantation fails sometimes caused by 18]. The Periotest and the Osstell Mentor system test are some complicated reasons in oral environment . Fortu- two typical noninvasive methods for the measurement of nately the stability of the implants can be introduced to suc- implant stability in diagnosis. In the Periotest, the interfacial cessfully predict such a failure in most cases by numerical computing or experimental testing . Usually, the stability damping characteristics between the implant and the sur- rounding tissue are evaluated [19, 20]. However, the Periotest of implantation can be concluded in two categories: the pri- is often criticized for its lack of prognostic accuracy and poor mary and the secondary stability. The former is largely asso- sensitivity. On the contrary, the Osstell Mentor system is based ciated with osseointegration, while the secondary stability is on resonance frequency analysis (RFA), which appears more highly corresponding with the bone formulation and remod- eling in the process of healing [4–7]. Bone material quality, competent of assessing the implantation stability [21–25]. The RFA was ﬁrstly introduced for dental applications in geometry characteristics of implants, and cortical bone can 1996 and then developed in both experimental and computa- aﬀect the primary stability [8–12]. The secondary stability tional aspects [26–31]. It provides an eﬀective way to study obtains from the bone apposition surrounding the interface the relationship between the stiﬀness of the implant-bone of implants and bone [13, 14]. The quantiﬁcation of implant 2 Applied Bionics and Biomechanics W2 interface and its surrounding local structures during the heal- Dental implant ing process [32–34]. In the numerical computation aspect, H4 RFA can be powerfully implemented by a ﬁnite element Abutment method (FEM). FEM is capable of simulating puzzling geom- H3 W1 etry characteristics, material properties, and also boundary conditions, which are often tough to deal with in the labora- H2 tory [35–37]. Furthermore, it is feasible to allow any indepen- dent control of parameters in the implant system by the FEM. Then, performing systematic evaluation for each parameter corresponding to implant stability becomes convenient. H1 Cancellous bone Combining the Latin Hypercube (LH) sampling method with the FEM develops a competent and appropriate sto- chastic ﬁnite element method. The widespread popularity Cortical bone of LH has led to the technical development in various ﬁelds, such as the improvement of space ﬁlling [38, 39], optimization of projective properties [40–42], minimization of least square error, maximization of entropy [43, 44], and W3 reducing spurious correlations . Meanwhile, LH has been deeply explored in probabilistic analysis, ranging from Figure 1: Geometry characteristics of the dental implant. the assessment of reliability [46–49] to coeﬃcient evalua- tion for polynomial chaos and merging with the surrogate Table 1: Parameters of geometrical properties in FEM. models [50, 51]. Therefore, the combination of LH and the FEM is a promising method for the RFA of the dental Deﬁnition mm implantation research. H1 The height of the bottom part in dental implants 14.2 Since the exact relationship between the design variables of dental implants and RF of the system is correlated, compli- H2 The height of the middle part in dental implants 2 cated, and dependent, traditional regression calculation is H3 The height of the top part in dental implants 5 diﬃcult to reach a satisﬁed accuracy. The Kriging model is The height of the cortical bone 26 an interpolation method which ﬁnds its roots in geostatis- The height (in the front end) of the thread in tics . As one of the most promising spatial correlation h1 0.1 dental implants models, the Kriging model is more ﬂexible than the regres- The distance (in the front end) between the edges sion model and not as complicated and time-consuming as 0.9 of the thread in dental implants other metamodels . Recently, there is an increasing inter- est in applying the Kriging model in the industry, mechanical The width of whole FEM 16 engineering, and related ﬁelds [54–58]. The popularity of W1 The thickness of the cortical bone 1.3 Kriging consists in that Kriging is an accurate interpolating W2 The diameter of the ceramic crown 5.1 approximation model . Kriging model is attractive for W3 The diameter of dental implants 3.1 its interpolating characteristic, providing predictions with the same values as the observations and reducing the time w The width of teeth in the thread part of dental implants 0.4 for the expensive analysis. When there are highly nonlinear- H4 The height of the ceramic crown 2 ity in a great number of factors, polynomial regression Besides, d represents the degree of the included angle in the front end of modeling becomes insuﬃcient while the Kriging modeling thread and is settled as50 . is an alternative choice in spite of the added complexity . The goal of this study is to propose an appropriate math- short summary is concluded in the last section. The methods ematical model to evaluate and predict the implant stability developed in this paper provide useful information and and performance. The model is explored to precisely describe helpful guidance for implant designers and prosthetic pro- the exact relationship between RF and design parameters of fessions in the process of implant design and corresponding dental implants. LH-FEM can reduce the cost of physical surgical plans. experiments, which are expensive and time-consuming. The proposed model in this study has the advantages not only in the prediction reliability and accuracy but also in the com- 2. LH-FEM for Dental Implants patibility and ﬂexibility in experimental data and numerical simulation results. 2.1. Parameters of Dental Implants. The geometrical parame- This paper consists of four parts. Section 1 provides a ters of dental implants are expressed in Figure 1 and Table 1. brief overview of the Kriging surrogate model and reso- They include height, length, and width of dental implants, nance frequency analysis. Parameters of dental implants cortical bone, cancellous bone, and ceramic dent. There are are presented in Section 2. Besides, the implementation of 8 parameters (H1, H2, H3, W3, h1, h, w, and d) for dental LH-FEM is also performed in this section. In Section 3, implants and 2 parameters (W2 and H4) for ceramic dent. results are demonstrated and discussed comprehensively. A H, W, and W1 for the cortical bone are 26 mm, 16 mm, Applied Bionics and Biomechanics 3 Table 2: Physical properties of materials for FEM. Material Young’s modulus range (GPa) Poisson’s ratio range Physical density (kg/m ) Dental implant E1 (100–200) R1 (0.25–0.35) D1 (4000–8000) Ceramic crown E2 (5–12) R2 (0.25–0.35) D2 (2000–3000) Cortical bone E3 (10.0–20.0) R3 (0.25–0.35) D3 (1600–2000) Cancellous bone E4 (0.8–1.5) R4 (0.25–0.35) D4 (1600–2000) Figure 2: Finite element model of dental implants. and 3.1 mm, respectively. With the development of material objective surface and contact surface are supposed to be bonded. Perfect bonding makes the computation process science, stainless steel, titanium, gold, and even ﬁber can reach a biomedical grade and can be applied in dental avoid the problem of convergence, speeds up the calculation, implants. For material properties, Young’s modulus, Pois- and is proper to have the analysis in large deformation and son’s ratio, and physical density of dental implants (E1, R1, nonlinear problems; (4) there are no ﬂaws in any compo- and D1), ceramic crown (E2, R2, and D2), cortical bone nents; and (5) for the boundary condition in the FEM of the dental implant system, three surfaces (in the bottom (E3, R3, and D3), and cancellous bone (E4, R4, and D4) are 12 input variables in the ﬁnite element model. In order to and left and right sides) of the cortical bone surface are cho- perform the Latin Hypercube-based ﬁnite element method, sen and completely ﬁxed; the displacement in X, Y, and Z the speciﬁc intervals of parameters corresponding with mate- directions of these three surfaces is constrained to be zero. rial properties are given in Table 2. Besides, in the previous work of Chu et al. in reference , it has been proven that a suﬃcient number of samples In terms of the material properties, on the one hand, the work in this study provides the appropriate and wide inter- can guarantee a satisﬁed accuracy of LH and MCS. Enlarging val ranges for the material parameters instead of the certain the sampling pool can eﬀectively improve the result accuracy and speciﬁc values. The speciﬁc values of material parame- when the number of samples is small; however, when the ters are included in the interval ranges. On the other hand, amount of samples reaches a certain number, the improve- ment is not evident. Therefore, the number of samples for the material properties are analyzed and discussed in the aspects of stiﬀness and mass matrix, which are more com- each variable is supposed to be 500 to obtain an appropriate prehensive to analyze the eﬀects of material properties in accuracy while reducing computation costs. RFs of dental implants. According to the FEM, (1) all materials are homoge- 2.2. Latin Hypercube-Based Finite Element Method. A ﬁnite neous, isotropic, and linear elastic; (2) the diﬀerent compo- element model of the dental implant system is created as nents exhibit diﬀerent physical properties; the Young’s shown in Figure 2 by ANSYS (Mechanical Parameter modulus and Poisson’s ratio are given as input variables Design Language, Version 14.5, USA). Solid 285 is the cho- according to common values in the certain range; (3) perfect sen ﬁnite element, which is a tetrahedron solid element bonding is involved between implant-abutment-screw, abut- with 4 nodes in each element, and each node has 3 degrees ment-crown, and implant-bone. In the contact pair, the of freedom (displacement in X, Y, and Z directions). The 4 Applied Bionics and Biomechanics Start Latin Hypercube Initial configuration sampling simulation i=1 Fixing material Capturing the ith i=1+1 property parameters sample of each input variable Meshing the finite element model Creating finite element model of dental implant system Applying the boundary conditions Performing finite Performing finite element computation element method RF anaysisl Block Lanczos eigenvalue extraction Capturing natural Yes No frequencies of vibration Verified Results output No i> n Yes End Figure 3: The ﬂowchart of LH-FEM. tetrahedron shape of Solid 285 is ﬂexible and convenient to speciﬁc interval range as shown in Table 2. The stochastic mesh nonlinear and complicated geometry components, sampling process of the input variable creates a reliable data- such as thread of dental implants. There are a total of base for FEM computation. The ﬂowchart of LH-FEM in 133961 elements and 22168 nodes. The thread components Figure 3 presents the programing process. The process of dental implants have been ﬁne meshed. By performing a marked by blue color represents the deterministic ﬁnite ele- ﬁnite element (FE) procedure, the RF can be obtained by ment model of the dental implant system. The convergence solving the eigenvalue problem in the govern equation and accuracy of the deterministic ﬁnite element model for through the block Lanczos method. dental implants are veriﬁed before the next steps. After the The study of the relationship between the various cor- validation of the deterministic ﬁnite element model, the orig- responding parameters of the dental implant system and inal codes are assigned to LH-FEM. The loop of LH-FEM the values of RF requires fabrication of a huge sample set does not stop until all of the samples are computed by the for experiments, which is expensive and time-consuming. ﬁnite element model. Then, the results of RF are captured LH as an advanced Monte Carlo method is especially eﬃ- and stored in the output database, as shown in the right side cient, which divides the sample space into a number of of the ﬂowchart in Figure 3 which are marked by red color. subspaces, then samples from subspaces, thereby perfectly In order to ﬁnd the availability of LH-FEM, a param- avoiding sample clustering or variation in the boundary eter sampling record (E1) of input variables is presented in [49, 50]. Combing the traditional ﬁnite element model of Figure 4. The diﬀerence between samples is not regular dental implants with LH is a sophisticated method to over- because of the stochastic sampling process, which makes come the disadvantages of physical experiments. the sampling set including diﬀerent situations as far as The input variables of the ﬁnite element model were possible. The probability distribution in statistical mathe- assigned using LH sampling to eﬀectively avoid sample repe- matics is following a uniform distribution as in Figure 4. tition or clustering. For each variable, there are ﬁve hundred The LH sampling method is successfully implemented in randomly samples, which are uniformly distributed in the the whole sampling spaces. Applied Bionics and Biomechanics 5 x 10 11 1 x 10 0.8 1.8 1.6 0.6 1.4 0.4 1.2 0.2 0 50 100 150 200 250 300 350 400 450 500 Number of samples 1 1.2 1.4 1.6 1.8 2 (a) 11 x 10 E1 (Pa) (b) Figure 4: Sample records of Young’s modulus of dental implants by LH ((a) for stochastic sampling record and (b) for probability density distribution result, respectively). 3. Results and Discussion In this study, not only the resonant frequencies of den- tal implants are calculated by the FEM but also the vibra- tion modes of the dental implants are provided. As the 3.1. Results of the Deterministic Finite Element Model. RFA (resonance frequency analysis) as a noninvasive and nonde- direct measurement of the vibration performance of dental structive technique is proposed to assess the implant stability. implants higher than the third mode is diﬃcult, the displace- A deterministic ﬁnite element model of dental implants is ments of dental implants under the vibration mode in this performed to evaluate the feasibility of this method. The res- numerical study are an important supplement of the experi- onant displacements of the ﬁrst ﬁve order resonance frequen- mental measurements. Furthermore, the high mode vibra- cies are plotted in Figure 5, for the X, Y, and Z directions and tion is an appropriate reference for the safety and reliability vector sums, respectively. The axes in Figure 5 are the same as of the dental implants in the real operating environment. that in Figure 2. The contour results allow visualizing the dis- placement and mode shape of the dental implant system. The 3.2. Probabilistic Results of LH-FEM. The results of LH-FEM contours present that the displacement happened in the direc- are a large database, and the eﬀective and useful information tion of Y axis in RFA is a more dominant vector that inﬂu- is extracted and presented in Table 3. The maximum, mini- ences the whole system in the ﬁrst-order RFA than others. mum, mean value, and variance of the ﬁrst ﬁve order RFs are all shown in Table 3. The results of the ﬁrst-order and While in the second-order RFA, the more important displace- ment is observed in the direction of X axis. In addition, the second-order RFs are very close, which is because of the geo- displacement vector sums in the third-order RFA are more metrical symmetry in the ﬁnite element model. However, the approximated to the displacement in the direction of Z axis. vibration modes of the ﬁrst and second orders are diﬀerent in For the fourth- and ﬁfth- order RFA, the displacements in Figure 5. In the ﬁrst-order vibration mode, the dominant dis- placement happens in the direction of Y axis, while in the vibration modes are more complicated and the natural fre- quencies are larger than the lower-order vibration modes. second-order vibration mode, the important displacement Besides, the largest deformation in the ﬁrst- and second- occurs in the direction of Z axis. order vibration modes happens in the top of the system, which Diﬀerent with the probability distribution of input well agreed with the results in the work of Li et al. . variables, the output results of the ﬁrst ﬁve order RFs for den- tal implants do not have uniform distributions as shown in In order to be more convenient, the results of dental implants are demonstrated independently. Figure 6 presents Figures 7 and 8. From Figure 7, the results of ﬁrst- and the displacement vector sum and Von Mises stress of dental second-order RFs are more concentrated in the smaller inter- implants in the ﬁrst ﬁve order RFA. The associated displace- val than the third, fourth, and ﬁfth orders, which means the ment corresponding to the ﬁrst bending modes of a cantile- diﬀerence in material property of dental implants causes a large deviation of RF in higher-order vibration modes. ver beam has been experimentally observed in the literature . The results of the deterministic ﬁnite element model Besides, in Figures 7 and 8, the probability distribution and are consistent with those in experiments of the literature. cumulative probability of the ﬁrst- and second-order RFs The accordance can be conﬁrmed in both the displacement are approximated, but when compared with the third-order vector sums and Von Mises stress. The failures and risks RF, the diﬀerence is very evident. These characteristics of th most likely occur at the beginning of the threads and the e probabilistic results provide helpful guidance in experi- junction of implant collar. This fact is in a good agreement mental designs and research. with numerical and experimental investigation [37, 62]. Furthermore, the primary (ﬁrst order) RF of LH-FEM is Therefore, the results in this study are validated. The ﬁnite ﬂuctuated in the interval range from 6100 to 10000 Hz element model of dental implants is feasible and appropriate approximately, and the average is 7800 Hz. In regard to RFA, for further studies. tremendous substantial studies have been accomplished in E1 (Pa) Destiny 6 Applied Bionics and Biomechanics 12 3 4 5 −.695E−04 −.387E−04 −.792E−05 .229E−04 .537E−04 −.541E−04 −.233E−04 .747E−05 .383E−04 .690E−04 −.337E−03 −.143E−03 .505E−04 .244E−03 .438E−03 −.240E−03 −.463E−04 .147E−03 .341E−03 .535E−03 −.682E−03 −.379E−03 −.755E−04 .228E−03 .531E−03 −.530E−03 −.227E−03 .761E−04 .379E−03 .682E−03 0 .154E−03 .307E−03 .461E−03 .614E−03 768E−04 .230E−03 .384E−03 .537E−03 .691E−03 Figure 5: Displacement in the ﬁrst ﬁve order RFA (X, Y, and Z axis directions and sum vector). experiments and clinical research. In the work of Glauser During a 12-month interval, the average RF ranges from et al. , the implant stability based on the RFA of dental 6100 Hz at the 1st month to approximately 6600 Hz at the implants under an early functional loading is explored. 12th month. Besides, the results of the experiment in the tibia Applied Bionics and Biomechanics 7 (a) (b) Figure 6: Results of dental implants in the ﬁrst ﬁve order RFA ((a) displacement vector sums, (b) Von Mises stress of the dental implant). Table 3: Statistical results of LH-FEM for RFA. 1 0.9 Maximum Minimum Mean Variance (kHz) (kHz) (kHz) (kHz) 0.8 F1 0.7 10.0 6.1 7.8 0.7 F2 0.6 10.1 6.1 7.9 0.7 F3 28.2 17.2 21.9 4.4 0.5 F4 35.7 21.5 27.9 7.9 0.4 F5 37.0 22.4 28.8 8.3 0.3 0.2 0.1 0.45 0.4 10 15 20 25 30 35 0.35 Resonance frequency (KHz) 0.3 F1 F4 F2 F5 0.25 F3 0.2 Figure 8: Cumulative probability of the ﬁrst ﬁve order RFs. 0.15 0.05 range of LH-FEM, which strongly proves the feasibility of the proposed model in this study. Besides, the probability 0.1 density distribution results of the ﬁrst ﬁve order RFs for den- tal implants in Figures 7 and 8 provide important informa- 10 15 20 25 30 35 tion corresponding to the safety and reliability of dental Resonance frequency (kHz) implants’ stability. F1 F4 F2 F5 3.3. Prediction of the Kriging Surrogate Model. The resonance F3 frequency is directly attributed to the stiﬀness matrix and mass matrix. The material stiﬀness and the stiﬀness of Figure 7: Probability density of the ﬁrst ﬁve order RFs. implant-bone interfaces and surrounding tissues have posi- tive eﬀects on RF, while RF usually has the inverse proportion of three groups of guinea pigs in a period of 4 weeks  to the mass matrix. However, the Young’s modulus of the also indicated that the average RF is roughly 5650 Hz cancellous bone is crucially related with the physical density, amongst the three groups. Compared with these literature and the increase of physical density of cancellous bones data, the results of LH-FEM are in a reasonable range. There- (mass matrix) can contribute to the improvement of the fore, the database of LH-FEM for RFA of dental implants is Young’s modulus (stiﬀness matrix). On the other hand, can- available and feasible. cellous bones have a much larger contact area to the dental The work of this study provides the maximum, mini- implant and can hugely aﬀect the whole dental implant sys- mum, mean values, and the variances of the resonant fre- tem. Therefore, the relationship between RF and parameters quencies of dental implants. The primary (ﬁrst order) of material properties in the dental implant system is compli- resonant frequencies of dental implants corresponding to cated and correlated. It is diﬃcult to be described by a simple the implant stability in the literatures fall within the interval linear interpolation or regression function. Density Cumulative probability 8 Applied Bionics and Biomechanics (a) (b) (c) (d) Figure 9: Prediction results of the Kriging surrogate model ((a), (b), (c), and (d) represent the relationships between E1, E2, and F1; E1, E3, and F1; E1, E4, and F1; and E1, P2, and F1, respectively). 1.05 1 0.95 0.95 0.9 0.9 0.85 0.85 0.8 0.8 0.75 0.75 0.7 17.5 8 18.5 19 19.5 20 20.5 21 0.7 Young's modulus (GPa) in cortical bone 1.8 2 2.2 2.4 2.6 2.8 1 order RF 1 order prediction Young's modulus (GPa) in cancellous region 2 order RF 2 order prediction 1 order RF 1 order prediction 3 order RF 3 order prediction 2 order RF 2 order prediction 3 order RF 3 order prediction Figure 10: Comparison between deterministic results and prediction results of RF in the cortical bone. Figure 11: Comparison between deterministic results and prediction results of RF in the cancellous region. Based on the reliable database of LH-FEM, a huge amount of samples for parameters of material properties in is created. The accuracy and convergence of the prediction by the dental implant system and corresponding RF is provided. the Kriging surrogate model are demonstrated in Figure 9. By performing the Kriging surrogate model, the numerical The black spots in Figure 9 are prediction results of the Kri- relationship between RF and parameters of material property ging surrogate model; the mesh surface is the results obtained Applied Bionics and Biomechanics 9 1 10 0.9 0.8 8 0.7 0.6 6 0.5 0.4 4 0.3 0.2 2 0.1 01 24680 0 0.2 0.4 0.6 0.8 1 (a) (b) Figure 12: Comparison between the LH sampling method and MCS ((a) for MCS, (b) for the LH sampling method). from LH-FEM. Satisﬁed level of accuracy is reached by the the cancellous region is large. The reasons causing this phe- Kriging surrogate model. Figure 9 well conﬁrms the pre- nomenon can be the original database of LH-FEM, the low diction accuracy of the proposed Kriging surrogate model sensitivity of the Kriging surrogate model for the cancellous in this study. region, or computational relative errors of the method applied in this study. On the other hand, the Kriging surro- In order to have quantitative comparison, the prediction results of the Kriging surrogate model and the results in the gate model provides continuous result prediction of RF for literature  are presented in Figures 10 and 11. In clinical the dental implant system, which is more convenient, com- and experimental observation, RF increases and changes in prehensive, and time-saving for RFA than the traditional the beginning of the ﬁrst several months and reaches a ﬁnite element method calculation and clinical test. In addition, based on the accuracy and reliability of the certain steady state after a period. In the healing process, the Young’s modulus of the cortical region and cancellous Kriging surrogate model prediction, it is also useful in the region increases. Therefore, the other input variables are analysis process of dental implant stability. However, the settled as certain values in the Kriging surrogate model; dental stability cannot be directly measured or simulated the Young’s modulus of cortical (E3) and cancellous (E4) with an in vivo or vitro model under speciﬁc clinical situa- tions. The Kriging surrogate model proposed in this study bones are input variables. From Figure 10, it can be found that in general, according can play important roles and provide believable prediction to the increase of the Young’s modulus cortical region, the results of RF corresponding with the dental stability. Besides, ﬁrst three order RFs are all ampliﬁed in the beginning period the Kriging surrogate model is compatible to combine the in the results of the literature  and the prediction of the numerical simulation results with clinical and experimental results in the original database, which makes it a more Kriging surrogate model, where β is the ratio of the RF related to the speciﬁc Young’s modulus with the steady RF sophisticated surrogate model for the dental implant system. in the ﬁnal phase. Due to the database of LH-FEM, the Through LH-FEM proposed in this paper, a series of ﬁrst-order and the second-order prediction results are very meaningful results are obtained for RFA of dental implants. close, while the prediction results of the Kriging surrogate However, there are still some limitations caused by simpliﬁ- cation and assumptions involving in the numerical simula- model in the third RF are approximated with the results in the work of Glauser et al. , especially in the beginning tion. Firstly, material properties of each component in the and end parts of the result curve. dental implant system are supposed to be homogenous and isotropic. Secondly, the damping eﬀect is totally neglected Furthermore, the prediction results of RF in the cancel- lous region by the Kriging surrogate model are also com- during the RFA, while damping may have a certain extent inﬂuence on the accuracy of RF. Thirdly, the interfaces in pared with the results in the literature  as shown in Figure 11. A good agreement is achieved when the Young’s the system are all treated as perfect bonding without consid- modulus in the cancellous region is large. However, the pre- ering the local specialty. In the real operation situation, the diction results of the Kriging surrogate model are larger than bone-implant interface is a dynamic living surface that the deterministic results  when the Young’s modulus in evolves from a debonded interface to a bonded interface. Li 10 Applied Bionics and Biomechanics et al.  applied the bonded model to calculate the RF of the In the ﬁrst-order linear Kriging surrogate model, the pre- bone-implant interface in a dental implant. The results were dictor can be expressed as follows : in a quantitative agreement with some experimental mea- surements, because the bone-implant interface was fully ̂yx = c Y, A 4 osseointegrated after several weeks [4, 61]. Despite such sim- pliﬁcation in the simulation process, the computational where c = c x ∈ R . The relative error is results in RFA are fairly reasonable, which could be useful and helpful for implant designers and prosthetic researchers. T T T ̂yx −yx = c Y −yx = c Fβ + Z − fx β + z Furthermore, the prediction results of the Kriging surrogate model for RFA of dental implants by LH-FEM are reliable T T = c Z − z + F c −fx β and believable. A 5 4. Conclusion In order to make sure the predictor is unbiased, F c is In conclusion, the computational model proposed in this equal to f x . The mean squared error of the predictor can paper is a successful numerical tool for noninvasive RFA in be written as follows: dental implant research. LH-FEM is appropriate and feasible to study the inﬂuence of the material properties of the 2 T implant medical components on RF. The Kriging surrogate φ x =Ey∧ x −yx =Ec Z − z A 6 model is an eﬀective model to precisely describe the relation- 2 T T = σ 1+ c Rc − 2c r ship between RF and parameters of material property. The output results of RF from LH-FEM are in the reasonable The objective of the optimization program is to have the and believable range. The prediction results from the Kriging surrogate model have a good agreement with the published minimum value of φ; the constraint is paper. Based on the database of LH-FEM, the Kriging surro- 2 T T T T gate model is an appropriate and powerful method in RFA of Lc, λ = σ 1+ c Rc − 2c r − λ F c − f A 7 dental implants. The computation of the gradient of the constraint func- Appendix tion with respect to c can be expressed as follows: A.1. Kriging Surrogate Model L c, λ =2σ Rc − r − Fλ A 8 A surrogate model likes a black box, where the mathematical relationship between the parameters in the input database Suppose λ = −λ/2σ , the following system of equations is and parameters in the output results is expressed by approx- imated implicit methods. Usually, the surrogate model can be RF c r classiﬁed into two kinds of groups: local and global models. = A 9 The response surface method is a typical local model and F 0 λ f can be written in polynomial series as follows : The solution can be obtained: F β x = β + 〠 β x , A 1 0 i i −1 T −1 T −1 i=1 λ = F R F F R r − f , A 10 −1 c = R r − Fλ , n n n F β x = β + 〠 β x + 〠〠 β x x A 2 i i j 0 i ij i=1 i=1 j=1 −1 where R is the inverse matrix of the correlation matrix R. Equation (A.1) and equation (A.2) are the local ﬁrst- and −1 second-order response surface model, respectively, where F ̂yx = r − Fλ R Y A 11 β x is a deterministic regression model and β is the cor- i T −1 T −1 T −1 T −1 T −1 = r R Y − F R r − f F R F F R Y responding regression coeﬃcient. In addition, a global surrogate model generally has global searching space. Kriging models ﬁt a spatial correlation func- The generalized least squares solution is tion as follows: −1 ∗ T −1 T −1 β = F R F F R Y A 12 Gx = F β x +zx , A 3 Substitute β into the predictor of the Kriging surro- where z x is assumed to have mean zero and covariance. gate model, Applied Bionics and Biomechanics 11 T −1 T −1 ∗ T ∗ T −1 ∗ For the linear system, free vibrations will be harmonic of ̂yx = r R Y − F R r − f β = f β + r R Y − Fβ the form: T ∗ T ∗ =fx β +rx γ u = ϕ cos ω t A 21 A 13 i i In the above equation, ϕ is eigenvector representing Besides, the corresponding maximum likelihood esti- the mode shape of the ith natural frequency, ω is the ith mate of the variance is written as follows: i natural circular frequency, and t is time. Thus, the free vibration can be presented as follows: 2 ∗ ∗ σ = Y − Fβ Y − Fβ A 14 −ω M + K ϕ =0 A 22 i i If the errors are uncorrelated and have diﬀerent vari- 2 2 This equality is satisﬁed if either ϕ = 0 or −ω ances, E e e = σ and E e e =0 for i ≠ j. It is logical to i i i i i i j M + K equals to zero. Then, ﬁnd that R is the diagonal matrix, 2 2 K − ω M =0 A 23 σ σ 1 m R = diag ,⋯, A 15 2 2 σ σ This is an eigenvalue problem which may be solved for up to n values of ω and n eigenvectors ϕ , where n is the Besides, the weight matrix W is given as follows: number of degree of freedom. The eigenvalue and eigenvector problem needs to be σ σ 2 −1 W = diag , … , ⇔ W = R A 16 solved for mode-frequency and buckling analyses. It also σ σ 1 m has the form as follows: Therefore, K ϕ = λ M ϕ , A 24 i i Y = WY = WFβ + e A 17 where λ is an eigenvalue. The block Lanczos method uses the sparse direct solver to perform Lanczos iterations and And, the below equations are satisﬁed: extracts the requested eigenvalues. Rather than the natural circular frequencies ω , the E e =0, natural frequency f is A 18 T T T 2 E ee =WE ee W = σ I f = A 25 2π Replacing F and Y by weighted function, the results can be depicted as follows: Therefore, the natural frequency f is not only corre- sponding with structure stiﬀness matrix K but also be T 2 ∗ T 2 aﬀected by M structure mass matrix. It is an eﬀective F W F β = F W Y, index to observe the changes or deviation of the structure A 19 2 ∗ 2 ∗ geometry and material property. σ = Y − Fβ W Y − Fβ A.3. Latin Hypercube Sampling Method A.2. Resonance Frequency The Latin Hypercube sampling method is one kind of Modal analysis is eﬃcient to identify the RF of objects in advanced Monte Carlo simulation (MCS). This approach the ﬁelds of engineering, industry, and medicine. Resonance divides the range of each variable into disjoint intervals of happens when a structural system vibrates at its RF with a equal probability, and one value is randomly selected from tendency to oscillate in higher amplitudes. Since the value each interval. It improves the stability of MCS and keeps of RF is associated with material stiﬀness, structural damp- satisﬁed accuracy and good convergence [49, 50]. Consider ing, physical density, and boundary condition, RF can be a statistic system described by the function, employed to symbolize a structural system. The calculation of RF of a system is essentially a general- Y = FX , X = X , X ,⋯,X , A 26 1 2 n ized eigenvalue problem. The free vibration without damping is governed by the following equation: where X is the random vector and represents the indepen- dent input random variables. F is the operator and performs M u + K u =0 , A 20 computer simulation, such as ﬁnite element computation. Traditional MCS relies on simple random sampling, where K is the structure stiﬀness matrix and M is the in which realization of X, denoted by x , k =1, … , N, where structure mass matrix. N is the amount of samples. In the sample space, 12 Applied Bionics and Biomechanics −1 element shakedown analyses,” Computer Methods in Biome- x = P U , i =1, … , n, A 27 ki X i chanics and Biomedical Engineering, vol. 6, no. 2, pp. 141– 152, 2003. where U are the uniformly distributed samples on [0, 1] and  M. A. Pérez, P. Moreo, J. M. García-Aznar, and M. Doblaré, P is the cumulative distribution function. Besides, Nataf or “Computational simulation of dental implant osseointegration Rosenblatt transformations can be used to produce a set of through resonance frequency analysis,” Journal of Biomechan- uncorrelated random variables from correlated variables. ics, vol. 41, no. 2, pp. 316–325, 2008. Latin Hypercube sampling method divides the range of  C. Rungsiyakull, Q. Li, G. Sun, W. Li, and M. V. Swain, each vector component into disjoint subsets of equal proba- “Surface morphology optimization for osseointegration of bility. Samples of each vector component are captured from coated implants,” Biomaterials, vol. 31, no. 27, pp. 7196– the respective subsets according to equation as follows: 7204, 2010.  W. Schulte and D. Lukas, “Periotest to monitor osseointe- −1 x = P U , A 28 gration and to check the occlusion in oral implantology,” ki X ij The Journal of Oral Implantology, vol. 19, no. 1, pp. 23– 32, 1993. where i =1, … , n and j =1, … , m, where n refers to the total  M. Bischof, R. Nedir, S. Szmukler-Moncler, J. P. Bernard, and number of vector components or dimensions of the vector J. Samson, “Implant stability measurement of delayed and and m is the number of the subset in a design. k is the sub- immediately loaded implants during healing,” Clinical Oral script which denotes a speciﬁc sample. Implants Research, vol. 15, no. 5, pp. 529–539, 2004. Besides, U are uniformly distributed samples on ξ , ξ , ij j j  I. Miyamoto, Y. Tsuboi, E. Wada, H. Suwa, and T. Iizuka, “Inﬂuence of cortical bone thickness and implant length on j − 1 implant stability at the time of surgery—clinical, prospective, ξ = , biomechanical, and imaging study,” Bone, vol. 37, no. 6, pp. 776–780, 2005. A 29 ξ = ,  J. T. Hsu, Y. W. Shen, C. W. Kuo, R. T. Wang, L. J. Fuh, and H. L. Huang, “Impacts of 3D bone-to-implant contact and −1 x = P U implant diameter on primary stability of dental implant,” Jour- X ij ki nal of the Formosan Medical Association, vol. 116, no. 8, pp. 582–590, 2017. By the Latin Hypercube sampling method, the ranges of  M. Atsumi, S. H. Park, and H. L. Wang, “Methods used to all random input variables are all divided into intervals with assess implant stability: current status,” International Journal equal probability, the eﬃciency of sampling is improved, of Oral & Maxillofacial Implants, vol. 22, no. 5, pp. 743–754, and the disadvantage of clustering points is well overcome, as presented in Figure 12.  E. Nkenke, M. Hahn, K. Weinzierl, M. Radespiel-Tröger, In Figure 12, the data are the examples to compare F. W. Neukam, and K. Engelke, “Implant stability and histo- the eﬃciency of the Latin Hypercube sampling method morphometry: a correlation study in human cadavers using and Monte Carlo method. Actually, it can be seen as the stepped cylinder implants,” Clinical Oral Implants Research, example of sampling space of parameters in the model of vol. 14, no. 5, pp. 601–609, 2003. dental implants.  N. Meredith, D. Alleyne, and P. 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