Original Article
Uncertainty optimization of dental implant based on finite element method, global sensitivity analysis and support vector regression
Proc IMechE Part H: J Engineering in Medicine 1–12 Ó IMechE 2018 Article reuse guidelines: sagepub.com/journals-permissions DOI: 10.1177/0954411918819116 journals.sagepub.com/home/pih
Hongyou Li1, Maolin Shi1,2, Xiaomei Liu1 and Yuying Shi1
Abstract In this work, an uncertainty optimization approach for dental implant is proposed to reduce the stress at implant–bone interface. Finite element method is utilized to calculate the stress at implant–bone interface, and support vector regression is used to replace finite element method to ease the computational cost. Deterministic optimization based on support vector regression is conducted, which demonstrates that the method using support vector regression replacing finite element method in dental implant optimization is efficient and reliable. Global sensitivity analysis based on support vector regression is used to assign different uncertainties (manufacturing errors) to different design variables to save the manufacturing cost. Two popular uncertainty optimization methods, k-sigma method and interval method, are used for the uncertainty optimization of dental implant. The results indicate that the stress at implant–bone interface is reduced greatly considering the uncertainties in design variables with the manufacturing cost increasing a little. This approach can be promoted to other types of bio-implants.
Keywords Dental implant, uncertainty optimization, finite element method, global sensitivity analysis, support vector regression
Date received: 4 June 2018; accepted: 19 November 2018
Introduction Dental implant is an important branch of dentistry that focuses on oral and maxillofacial prosthesis. As reported in the literature,1 90% of dental implants can service over 15 years with scrupulous care. However, there is still a risk failure rate (3%–5%) of dental implant, which is mainly because of too high stress at the implant–bone interface as reported by many researchers and dentists.2,3 It is necessary to find a tool to analyze, reduce and optimize the stress at implant– bone interface to decrease the failure rate of dental implant, and finite element method (FEM) is a natural tool to solve this problem and widely utilized for the mechanical analysis of bio-implants.2 FEM is developed from 1960s and first used in mechanical engineering to analyze the stress and strain of the truss structure. In recent decades, researchers introduce FEM to the design, analysis and optimization of bio-implants, and the attention is focused on the following three areas. The first one is model building. Generic anatomy was first used to build the model, but they cannot be used on the patients or volunteers.
Thus, computed tomography (CT) data are used to generate the model, and researchers developed a lot of methods such as hexahedral mesh generator based on a grid projection algorithm to improve the transformation accuracy from CT data to the FEM model. More details can be found in Viceconti et al.4 The second topic is improving the material parameters or model of biological tissues to better describe their material properties. In the past, biological tissues were considered as an isotropic material, but the view that biological tissues are anisotropic material is more accepted now.5,6 Thus, more recent works are moving to develop new devices or techniques to obtain more accurate material parameters including elastic modulus and Poisson’s 1
College of Mechanical Engineering and Automation, Huaqiao University, Xiamen, China 2 School of Mechanical Engineering, Dalian University of Technology, Dalian, China Corresponding author: Maolin Shi, School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, China. Email:
[email protected]
2 ratio.7,8 The last area is the analysis and optimization of bio-implants through FEM. The main purpose is to study the interactions between biological tissues and implants, and the effect of design variables on the mechanical performance of bio-implants such as the stress at implant–bone interface,9 the volume of implant10 and the fatigue probability of implant auxiliary components.11 In addition, some researchers paid their attention on model meshing, loading conditions and also boundary conditions.12–14 FEM has been successfully applied for the mechanical analysis of prosthodontics, especially dental implant. Researchers studied the effect of implant geometry, implant materials and misfit in implant on the stress on the implant and its surrounding bone by the help of FEM. The effect of implant geometry attracts people’s attention first. Stellingsma et al.15 concluded the studies about implant length before 2000 and pointed out that short implants can reduce the stress. Holmgren et al.16 and Mailath et al.17 compared cylindrical and conical implant shapes exposed to physiologic stresses. They both reported that cylindrical implants are preferable to conical implant shapes. The implant material properties have important effect on the stress at implant–bone interface. Stegariou et al.18 used threedimensional (3D) FEM to assess stress distribution in bone, implant and abutment when a gold alloy, porcelain or resin (acrylic or composite) was used for a 3unit prosthesis. Li et al.9 compared stress at implant– bone interface when implant is made of different Ti alloys. The results indicate that the Ti alloy with low elastic modulus can reduce the stress at implant–bone interface. However, Malaith’s study pointed out that the implant materials with too low elastic moduli also need to be avoided and suggested designers of dental implant and dentists to choose the implant materials based on the bone support conditions. The misfit in implant also has effect on the stress at implant–bone interface, and many corresponding studies have been conducted.19–25 The above works give insights on the relationships between the design variables such as implant geometry and the stress at implant–bone interface and provide a number of methods to reduce the stress at implant–bone interface, the optimization of implant geometry, better material selection and better operation parameters during implantation.4–6,9,16–25 However, the computational cost of FEM is too high for the designers to test every possible design scheme, thus the optimization based on FEM is mostly a discrete optimization, which means the results are not much likely to be the real optimum results. It is necessary to find a new method to replace FEM to realize the continuous optimization and to find the real optimum design variables of dental implant. Manufacturing errors cannot be avoided in the manufacturing process of dental implant, in other words, the uncertainties objectively exist in the design variables. The uncertainties in design variables can easily make the optimum results deteriorate. An obvious
Proc IMechE Part H: J Engineering in Medicine 00(0)
Figure 1. 3D model of dental implant and mandible segment.
solution is reducing the uncertainties as much as possible, in other words, increasing the manufacturing accuracy as high as possible. However, the two key components of dental implant, implant and abutment (Figure 1), are made of pure titanium or titanium alloys. Their manufacturing cost will increase exponentially with the increasing manufacturing accuracy.26 The uncertainties cannot be averted, but the manufacturing accuracy can be ensured. Thus, the manufacturing cost of dental implant can be saved through assigning different manufacturing accuracies to different design variables according to their importance rank affecting the stress at implant–bone interface. Thus, the importance rank of all the design variables needs to be known, and global sensitivity analysis (GSA) is a natural method to accomplish this work.27,28 In past decades, a number of GSA methods have been developed, including Morris method, iterated fractional factorial design, Fourier amplitude sensitivity test (FAST) and Sobol’s method.29,30 Sobol’s30 method is most popular and used in this work, but it needs a lot of repeated calculations (generally more than 10,000) to guarantee its accuracy and reliability.28 In addition, the uncertainty optimization also needs numerous iteration calculations. Thus, it is also necessary to find a new method to replace FEM to ease the computational cost for the uncertainty optimization of dental implant. In this work, we introduce a surrogate model, support vector regression (SVR), to replace FEM to ease the computational cost of GSA and uncertainty optimization. SVR is constructed based on a limited number of FEMs, which enables to formulate an explicit relationship between the objective and design variables with relatively simple expression.31 Thus, it is possible to predict the stress at implant–bone interface with relatively low computational cost, to analyze the effect of different design variables on the stress at implant–bone interface and to conduct the uncertainty optimization of dental implant. This work aims to reduce the stress at implant–bone interface through the uncertainty optimization of dental implant. Finite element model of dental implant is introduced in section ‘‘Finite element modeling.’’ The uncertainty approach based on GSA
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Table 1. Material properties of jawbone. Elastic modulus (GPa)
Cortical bone Cancellous bone
Shear modulus (GPa)
Poisson’s ratio
Ex
Ey
Ez
Gxy
Gyz
Gxz
Vxy
Vyz
Vxz
17.90 1.148
12.70 0.210
22.80 1.148
5.00 0.068
5.50 0.068
7.40 0.434
0.18 0.055
0.31 0.055
0.28 0.322
and SVR is given in section ‘‘Uncertainty optimization approach.’’ The prediction results are given in section ‘‘Results,’’ followed by the conclusions in section ‘‘Conclusion.’’
Table 2. Material properties of dental implant.
Implant Abutment Screw
Elastic modulus (GPa)
Poisson’s ratio
105 105 96
0.345 0.345 0.360
Finite element modeling ANSYS is used for the finite element analysis of dental implant. The 3D model is built as follows. The computerized image is created through CT scanning data of an edentulous mandible first molar area and then extruded to create a 3D model in the direction of the mandible with the length of 8 mm. The implant is inserted into the middle of the mandible model, and the abutment is applied into the implant through the screw as shown in Figure 1. Since the dental crown is different for each tooth of each patient and many publications report that the results with loading on the crown and that on the abutment are similar, the crown is admitted.2,3 The diameter of implanting cylinder is 4 mm. The implanting length is 8 mm. The length and the pitch of the thread on the implant are 6 and 1.2 mm, respectively, and the cross-section of the thread is a regular triangle with a side length of 0.6 mm. The cancellous bone and cortical bone are regarded as anisotropic materials and the parameters are listed in Table 1.5,6 The material properties of implant and abutment are decided from the optimization results by our previous study as shown in Table 2. The screw is the common titanium alloy whose data are from the database of ANSYS. Contacts among cancellous bone, cortical bone and implant are defined as bonded, and those among implant, abutment and screw are fictional with a friction coefficient of 0.3. The automatic meshing partition function of ANSYS is used for the meshing of the model, and the thread contacts between bone and segment are refined (Figure 2). The bottom surface of the mandible is the fixed support. Both the flanks of mandible in the direction of the mandible are fixed as frictionless support. As a loading condition, 150 N of occlusal force is applied on the end upper surface of abutment vertically (Figure 3), and the bolt pretension of screw is 150 N. The index of stress at implant–bone interface is vonMises stress, and the results are similar to those in other studies as shown in Figure 4.32–36 Stress concentrations on the thread surface can be found, especially the first thread facing the buccolingual direction. The maximum value is 67.149 MPa. The stress decreases along with
Figure 2. Meshing of implant.
Figure 3. Loading on the end upper surface of abutment.
the implanting direction and reaches 0.755 MPa at the bottom of the implant. The time cost of one-time calculation of FEM is about 40 min. It can be seen that the
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Proc IMechE Part H: J Engineering in Medicine 00(0) generalization performance. The second summation polynomials measure the flatness. Introducing relaxation factors z, z , the constrained problem can be expressed as follows min J = C
N X i=1
1 zi + zi + v2 2
ð3Þ
Subjected to 8 < yi ðv, uðxÞÞ b4e + zi ðv, uðxÞÞ + b yi 4e + zi : zi , zi ø 0 where v is the linear combination of training data v=
l X ai ai uðxi Þ
ð4Þ
i=1
where ai and ai are the Lagrange multipliers. The decision equation (1) has become the explicit form Figure 4. Stress at implant–bone interface.
fðxÞ =
N X
ai ai Kðxi , xÞ + b
ð5Þ
i=1
GSA and uncertainty optimization of dental implant are difficult to conduct based on FEM. The maximum of the stress at implant–bone interface is considered as the optimization objective in this work.
Uncertainty optimization approach SVR SVR is used to replace FEM in this work. SVR originates from support vector machine (SVM). In SVM, the basic idea is to map the initial data into a higher dimensional feature space via a nonlinear mapping u, then an optimal separating hyperplane is created which maps the nonlinear problems in the lower dimensional feature space to the linear problems in the higher dimensional feature space, and finally, the classification of the initial space is accomplished. The difference between SVR and SVM is SVR maps the data into a higher dimensional feature space, but then does regression in this space. SVR function is formed as follows31 fðx, aÞ = ðv uðxÞÞ + b, u : Rn ! F, v 2 F
ð1Þ
fui (x)gN i=1
where are the data in feature space, and v and b are the coefficients which can be estimated by minimizing the regularized risk function RðCÞ = C
N 1X 1 Le ðyi , fðxi ÞÞ + v2 l i=1 2
ð2Þ
where L is the loss function which measures the approximate errors between the expected output yi and the calculated output f(xi), and C is a constant determining the trade-off between the training error and the
where K(xi , x) is the kernel function which replaces the nonlinear function u, and the advantage of using it is that one could deal with feature spaces of arbitrary dimensionality without computing the map u(x) explicitly. Finally, the predicting value of x could be approximated by equation (5), and the kernel function used in this article is the radial basis kernel K xi, xj = exp gxi x2j ð6Þ Therefore, C and g are the two key parameters in SVR.
Design of experiments The design variables of structural parameters of the implant are considered in this work. According to the references and our previous studies,1–5,32–36 the implanting length (LC), the length of thread (LT) and the pitch of thread (P) are considered as design variables. The tooth profile is an isosceles triangle, and the bottom width (L) and the base angle (b) are used to control the form of the tooth profile, which are also considered as design variables. In recent decades, many studies report that the material parameters have significant effect on the stress on implant–bone interface.3–5,9,10 Four material parameters, the elastic modulus and Poisson’s ratio (Ei, Ea, Pi and Pa) of the implant and abutment, are also considered as design variables. To build SVR, some sampling points need to be generated first to explore the design space. Then, these sampling points and their responses calculated by FEM are used to build the SVR. Design of experiments (DOE) using Latin hypercube design (LHD) is adopted
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to acquire the sampling points in this work, which can distribute the sample points evenly in the design space.37 This method can maximize the minimum distance between the experimental points, and is expressed as follows ns ns X X 1 d i = 1 j = i + 1 ij
ð7Þ
where dij means the distance between the sample points i and j. In this work, 180 sampling points are generated using LHD.
Error analysis To assess the accuracy of SVR, additional 20 validation points are generated. Three error indexes, R-square (R2), root mean square error (RMSE) and maximum absolute percentage error (MAPE), are used as follows, respectively m P 1 R2 = 1 i = m P
3 3 2 0 x 11 x0 1i x0 1k x11 x1i x1k 6 x21 x2i x2k 7 6 x0 21 x0 2i x0 2k 7 7 7 6 6 7 7 6 6 : : 7 7 6 6 7 7 6 6 : : A=6 7 B=6 7 7 7 6 6 : : 7 7 6 6 5 5 4 4 : : 0 0 0 xn1 xni xnk x n1 x ni x nk 2
ð11Þ
where k is the number of parameters. Exchanging the ith row between matrices A and B with other rows’ invariant, and let the new matrices A and B be noted as Ai and Bi 3 3 2 2 0 x11 x0 1i x1k x 11 x1i x0 1k 6 x21 x0 2i x2k 7 6 x0 21 x2i x0 2k 7 7 7 6 6 7 7 6 6 : : 7 7 6 6 7 Bi = 6 7 : : Ai = 6 7 7 6 6 7 7 6 6 : : 7 7 6 6 5 5 4 4 : : xn1 x0 ni xnk x0 n1 xni x0 nk ð12Þ
ðyi y^i Þ ð8Þ ðyi yi Þ
i=1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 1X ðyi y^i Þ2 RSME = m i=1 jyi y^i j MAPE = max yi
ð9Þ ð10Þ
where m = 20 is the number of newly created validation points, yi is the true FEM results on the validation points and y^i is the corresponding approximate results given by SVR. R2 and RMSE measure the overall accuracy of an approximation model, whereas the MAPE measures the local error. The closer the value of R2 to 1 and the smaller the value of RMSE and MAPE, the better the accuracy.
GSA based on Sobol’s method GSA is a method analyzing the effect of each parameter on the output of the model with all the design variables changing simultaneously. Sobol’s method is a popular GSA method which has been successfully applied in many areas of engineering. The basic idea of Sobol’s sensitivity analysis is to decouple the model to a number of functions where each function is the combination of each parameter and the other ones, and then use Mote-Carlo estimation to compute the first-, secondand higher order sensitivity of parameters. The details are as follows. First, two sets of n sampling points are generated, and two matrixes A (n*k) and B (n*k) can be achieved
Every group of parameters (every line in Ai and Bi) is inputted into the model or function, and the responses are recorded. Then, the vibration of responses of the model and the sensitivity index are calculated as equations (13)–(16) n 1X f^2o = fðxr1 , xr2 , . . . , xrk Þ f ðx0 r1 , x0 r2 , . . . , x0 rk Þ n r=1
ð13Þ V^ðyÞ =
n 1X
n r=1
f2 ðxr1 , xr2 , . . . , xrk Þ f^2o
n X ^i = 1 f2 ðxr1 , . . . , xrk Þ U n r=1 f x0 r1 , . . . , x0 rði1Þ , xri , x0 rði + 1Þ , . . . , x0 rk n X ^i = 1 U f2 ðxr1 , . . . , xrk Þ n r=1 f xr1 , . . . , xrði1Þ , x0 ri , xrði + 1Þ , . . . , xrk
ð14Þ
ð15Þ
ð16Þ
^ is the where f^2o is the estimation of matrix A or B; V(y) ^i estimate of variance of the output of the model; and U ^ and Ui are the mean estimators of matrices A and Ai, and B and Bi, respectively. The main effect of the parameter xi, S^xi , can be estimated as ^i f^2 U o S^xi = V^ðyÞ
ð17Þ
The total effect of the parameter xi, S^Txi , can be estimated as ^i f^2 U o S^Txi = 1 V^ðyÞ
ð18Þ
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In this work, S^Txi is used to evaluate the effect of design variables on the stress at implant–bone interface. The sum, S^Txi , of all design variables is 1. The higher the S^Txi , the higher the effect of the design variable.
according to its values and standard deviations; m ^ (x) and s ^ (x) of objective and constraints are calculated by MC-LHS and maximum likelihood estimate (MLE) in this work.
k-sigma method
Interval method
In the previous studies of optimization of dental implant, most concentrate on the deterministic approaches as formulated follows
Another popular uncertainty optimization method is interval method,40 and equation (19) is translated as follows
minimize f(x)
Subjected to ( gi ðxÞ40 xL 4x4xU
minimize½mðfðx, pÞÞ, wðfðx, pÞÞ
ð19Þ
where f(x) is the optimization objective, x = (x1, x2, x3, ., xn)T is the vector of the design variables, gi is the ith constraint, and xL and xU are the lower and upper bounds of the design variables, respectively. In this work, the optimization objective f(x) is the maximum von-Mises stress at the implant–bone interface Sv. Deterministic optimization trends to excessively push the final optimum results at or close to one or more constraints such that tiny uncertainties of design variables would likely make the deterministic optimization result deteriorate dramatically. Uncertainty objectively exists in the design variables of dental implant due the machining and manufacturing errors. Therefore, the uncertainty optimization of dental implant is necessary, and a popular uncertainty optimization method based on the stochastic variables (usually the mean value and standard deviation) of design variables, constraints and optimization objectives, named as ‘‘k-sigma-method,’’38 is used here, which can be formulated as minimize m ^ f ðxÞ + n^ sf ðxÞ
Subjected to 8 ^ gm ðxÞ + k^ sgm ðxÞ4bm , m = 1, . . . , Ns > < mðfðx, pÞÞ = 1 ðf L ðxÞ + f U ðxÞÞ 2
1 U L > > : wðfðx, pÞÞ = 2 ðf ðxÞ f ðxÞÞ xL 4x4xU
ð21Þ
where p is the uncertainty vector of the design variables, m(f(x, p)) is the midpoint of the interval of f and w(f(x, p)) is the radius of the interval of f(x). fL (x) and fU (x) are calculated as follows f L ðxÞp2t = min fðx, pÞ
ð22Þ
f U ðxÞp2t = max fðx, pÞ
p 2 t = pjpL 4p4pU
ð23Þ ð24Þ
The advantage of this method is that it does not need the probability distribution assumption. The disadvantage is that it takes much higher computational cost compared to probability method since it needs solving a nested loops problem during its optimization. The outer optimization seeks the most optimum design variables, and the inner optimization determines the maximum and/or minimum response for each design variable considering its uncertainties. In this work, f U Sv (x) (the maximum of Sv) is considered as the optimization objective, which can be formulated as minimize f SUv ðxÞ
ð25Þ
Subjected to 8 gi ðx, pÞ40 < xL 4x4xU : p 2 t = pjpL 4p4pU where p is determined by the results of GSA and Chinese national manufacturing standards GB3935.1-96.
Results Deterministic optimization The deterministic optimization process is shown in Figure 5. Genetic algorithm (GA) and cross-validation are used to seek the optimum parameters of the SVR model, as listed in Table 3. The MAPE and RMSE are
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Figure 5. Flow chart of deterministic optimization. Figure 6. GSA results based on SVR.
Table 3. Optimum parameters of SVR. C
g
84.449
0.007
Table 4. Error indexes of SVR. R2
MAPE
RMSE
0.932
0.074
2.986
variables depend on our investigation and the advices from the manufactures. The optimum results are listed in Table 5. It can be found that the stress at implant– bone interface is reduced greatly with 36.529%. The stress calculated by SVR and FEM are almost equal with a percent error of 2.464%, which indicates that the method using SVR replacing FEM is effective and reliable. It is noted that b, Ei and LC are located near their boundaries, which suggests that uncertainty optimization is necessary.
GSA based on SVR 0.074 and 2.986, respectively (Table 4), and R2 is close to 1, which suggests an accurate approximation of SVR min Sv 8:000 mm4LC 49:000 mm 6:000 mm4LT 47:000 mm 1:000 mm4P41:400 mm 0:400 mm4L40:800 mm
ð26Þ
50:0008 4b470:0008 90:000 MPa4Ei 4120:000 MPa 0:3204Pi 40:370 90:000 MPa4Ea 4120:000 MPa 0:3204Pa 40:370
The deterministic optimization of dental implant is conducted based on SVR. The ranges of design
The GSA based on the SVR model built above is conducted as follows. Two sets of 100,000 sample points are generated first, and their responses are achieved by the SVR model built above. Then, Sobol’s method is used to compute the sensitivity indexes of each design variable. The results are exhibited in Figure 6 and Table 6. It can be seen that the S^Txi of L, b and Ei is much higher than the other design variables, which indicates that they have much more effect on the stress at implant–bone interface. For structural parameters, it can be found that the dimension of tooth mainly determines the stress at implant–bone interface, and the other parameters such as the implanting length and the thread length have little influence even less than some material parameters. For the material parameters, it is noted that the material property, Ei, has essential influence which claims that more attention need to be paid to the materials used for implant. The results of GSA provide advices to the manufacturers and designers
Table 5. Deterministic optimum result based on SVR. LC (mm)
LT (mm)
P (mm)
L (mm)
b (8)
Ei (MPa)
8.922
6.631
1.088
0.767
50.355
90.619
Pi
Ea (MPa)
Pa
Sv by SVR (MPa)
Sv by FEM (MPa)
0.329
100.796
0.346
42.620
41.595
SVR: support vector regression; FEM: finite element method.
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Table 6. ^ST of the design variables calculated by Sobol’s method. Design variables
LC
LT
P
L
b
Ei
Pi
Ea
Pa
^ST
0.023
0.010
0.011
0.059
0.786
0.092
0.011
0.006
0.001
Table 7. Uncertainties of the design variables. LC (mm)
LT (mm)
P (mm)
L (mm)
b (8)
Ei (MPa)
Pi
Ea (MPa)
Pa
0.150
0.150
0.100
0.030
0.660
2.000
0.020
5.000
0.020
which parameters need to be paid more attention, in other words, which structural parameters are set as higher machining accuracy and/or which material parameters need to be controlled more stably. In this work, the uncertainties of structural parameters are decided with the GSA results and Chinese national standards GB3935.1-96. L and b are with higher machining accuracy compared to other structural parameters. The uncertainties of material parameters are according to the titanium manufacturing report and some references about titanium alloys.26Ei is with smaller uncertainty. Finally, the uncertainties of design variables are listed in Table 7. It can be found that the optimum design variables are unfeasible when considering their uncertainties. Figure 7. Flow chart of uncertainty optimization.
Uncertainty optimization The uncertainty optimization is conducted as shown in Figure 7. The former four steps are as same as the deterministic optimization given in section ‘‘Deterministic optimization.’’ Step 5 is conducted based on the k-sigma method and the interval method. In step 6, 50 sampling points are generated using LHD according to the optimum design variables and their uncertainties to verify the effectiveness of uncertainty optimization. The uncertainty optimization using ksigma method is conducted first. k is 3 which implies that the probabilities of satisfying the constraints and design variables are 99.73%, and n is 6 which implies
that the probabilities of satisfying the objective is close to 100%. Based on the above GSA results, different machining accuracies are set to the design variables according to their effects on the stress at implant–bone interface. From Table 6, it can be seen that the base angle plays the most important role in the stress at implant–bone interface, but Poisson’s ratio of abutment has the least effect. Thus, the base angle gets the highest machining accuracy, and Poisson’s ratio of abutment has the lowest machining accuracy. Thus, the formulation in equation (20) is transformed as follows minimize m ^ sv ðxÞ + 6^ ssv ðxÞ
Subjected to 8 8:000 mm + 3sLC 4LC 49:000 mm 3sLC , sLC = 0:025 mm > > > > 6:000 mm + 3sLT 4LT 47:000 mm 3sLT , sLT = 0:025 mm > > > > 1:000 mm + 3sP 4P41:400 mm 3sP , sP = 0:017 mm > > > > < 0:400 mm + 3sL 4L40:800 mm 3sL , sL = 0:005 mm 50:0008 + 3sb 4b470:0008 3sb , sb = 0:1108 > > 90:000 MPa + 3sEi 4Ei 4120:000 MPa 3sEi , sEi = 0:333 MPa > > > > > 0:320 + 3sPi 4Pi 40:370 3sPi , sPi = 0:0033 > > > > 90:000 MPa + 3sEa 4Ea 4120:000 MPa 3sEa , sEa = 0:833 MPa > : 0:320 + 3sPa 4Pa 40:370 3sPa , sPa = 0:0033
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Table 8. Uncertainty optimization results using k-sigma method and interval method.
k-sigma method Interval method
k-sigma method Interval method
LC (mm)
LT (mm)
P (mm)
L (mm)
b (8)
Ei (MPa)
8.924 8.925
6.631 6.686
1.072 1.091
0.769 0.771
50.363 50.363
91.150 91.150
Pi
Ea (MPa)
Pa
Predicted max. Sv
Real max. Sv
0.331 0.331
100.102 102.602
0.344 0.345
45.411 45.802
48.267 47.712
Table 9. Computation cost of FEM and SVR.
Time
FEM
SVR
38.783 min
0.311 s
FEM: finite element method; SVR: support vector regression.
information to the designers to locate much better design variables compared to initial ones effectively. Interval method is used in this work, and the problem can be formulated as follows minimize f U SV ðxÞ
Subjected to Figure 8. Probability density distribution of Sv.
In this work, 2000 sample points are created through LHD according to the values and standard deviations ^ sv (x) are calof design variables, and then m ^ sv (x) and s culated by MLE of the responses of the SVR at sampling points. GA is used to find the optimum result which is [8.924, 6.631, 1.072, 0.769, 50.363, 91.150, 0.331, 100.102, 0.344]. The final value of optimization ^ sv is 0.063 (Table 8). objective is 45.411, m ^ sv is 45.033 and s It can be found that the mean and maximum of Sv after uncertainty optimization are little higher than the result of deterministic optimization, but the uncertainties of optimum design variables do not break through the boundaries of design variables. Another 50 sampling points of the uncertainty optimum results are generated using LHD, and their maximum stresses at the implant–bone interface are calculated by FEM. Figure 8 exhibits their probability density distribution. It can be found that the maximum value is higher than that given by k-sigma method. The mean of SVR predictions and that given by FEM are similar, but the variances of these two models are much different with 2.808 of FEM compared to 0.063 of SVR. It can be concluded that the error occurs mainly because the probability distribution of Sv does not strictly follow the normal distribution which is the premise assumption of k-sigma method. It needs to be noted that although the results of k-sigma method are with less accuracy, it can provide important
8 8:925 mm4LC 48:075 mm, 0:075 mm4pLC > > > > > 40:075 mm > > > > > 6:075 mm4LT 46:925 mm, 0:075 mm4pLT > > > > > > 40:075 mm > > > > > 1:050 mm4L41:350 mm, 0:050 mm4pL > > > > > 40:050 mm > > > > > > < 0:415 mm4P40:785 mm, 0:015 mm 4pP 40:015 mm > > > > 50:3308 4b469:6708 , 0:3308 4pb 40:3308 > > > > 91:000 MPa4E 4119:000 MPa, 1:000 MPa > i > > > > > 4p 41:000 MPa > Ei > > > > 0:3304Pi 40:360, 0:0104pPi 40:010 > > > > > 92:500 MPa4Ea 4117:5 MPa, 2:500 MPa > > > > > 4pEa 42:500 MPa > > > : 0:3304Pa 40:360, 0:0104pPa 40:010
GA is used to seek the optimum design variables. The optimum design variable vector is [8.925, 6.686, 1.091, 0.771, 50.363, 91.15024, 0.331, 102.60152, 0.345], and the finial f SUV (x) is 45.802 at [8.921, 6.613, 1.075, 0.770, 50.425, 91.114, 0.331, 100.213, 0.349] compared to 47.712 given by FEM. The stress is reduced significantly, and the optimum design variables are feasible considering the uncertainties existing in design variables. It can be found that the result of the interval method is more reliable, but the computational time is about 800 times the k-sigma method.
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Proc IMechE Part H: J Engineering in Medicine 00(0)
Table 10. Uncertainty optimization computation cost. FEM GSA k-sigma method Interval method Total
1.293 3 105 h 5.172 3 106 h – 5.301 3 106 h
SVR – 1.034 3 107 h 1.047 3 107 h
1.728 3 102 h 7.356 3 102 h – 9.084 3 102 h
– 1.382 3 103 h 1.555 3 103 h
FEM: finite element method; SVR: support vector regression; GSA: global sensitivity analysis.
To further analyze and discuss the computation cost of the proposed approach, the mean time of one-time SVR calculation and that of one-time FEM analysis for the 180 sampling points in DOE is listed in Table 9. It can be seen that SVR saves a lot of time than FEM but loses a bit of accuracy (Tables 4 and 9). The GSA and uncertainty optimization in this article need certain amount of calculations. To guarantee the accuracy and reliability of GSA results, 2.0 3 106 calculations are used. The settings of GA are 200 iterations and 20 individuals. Thus, the k-sigma method needs 200 3 20 3 2000 = 8 3 106 calculations, and the interval method needs 200 3 20 3 200 3 20 = 1.6 3 107 calculations. The computation cost based on SVR for GSA, k-sigma method and interval method is 6.220 3 105 s (1.728 3 102 h), 2.648 3 106 s (7.356 3 102 h) and 4.976 3 106 s (1.382 3 103 h), respectively. However, the computation cost based on FEM for GSA, k-sigma method and interval method is 7.757 3 106 min (1.293 3 105 h), 3.103 3 108 min (5.172 3 106 h) and 6.205 3 108 min (1.034 3 107 h) which is much higher than FEM. The total computation costs of GSA, k-sigma method and interval method based on FEM and SVR are listed in Table 10. It can be found that the computation cost of FEM is unacceptable, so the proposed approach based on SVR is suggested for the uncertainty optimization of dental implant. In addition, designers of dental implant can choose the uncertainty optimization method (ksigma method and interval method) according to their time or cost requirement referring to Tables 4, 9 and 10.
Conclusion In this work, an uncertainty optimization approach is proposed for dental implant. SVR is introduced to replace FEM to predict the stress at implant–bone interface with much less computational cost. The validation test indicates that it has higher accuracy with R2 of 0.932, MAPE 0.074 and RMSE 2.986. The stress at implant–bone interface is reduced by 36.529% after deterministic optimization, and the stress predicted by SVR is similar to that calculated by FEM with a percent error of 2.464%. GSA is used to analyze the effect of different design variables on the stress at implant– bone interface. The results indicate that the structural parameters of tooth on implant exhibits higher effect compared to other structural parameters. The material parameters of implant and abutment have important effect especially the elastic modulus of implant.
Different uncertainties are set to different design variables according to the importance rank given by GSA results. Both the k-sigma method and the interval method are used for the uncertainty optimization of dental implant. The stress at implant–bone interface slightly increases compared to that of deterministic optimization, but the results are feasible when considering the uncertainties existing in design variables. Between these two methods, the k-sigma method is faster but loses some accuracy. Designers and researchers can choose one depending on their special requirements. It needs to be noted that vertical loading condition is considered in this work which is not suitable to represent the real chewing cycle. In our current and future work, we are developing a special device to obtain the loading conditions of real chewing cycle from the volunteers, and a standard loading spectrum will be proposed and published to help the study of dental implant in the future. The main contribution of this work is to propose a new uncertainty optimization approach based on SVR, GSA and k-sigma method/ interval method, which can dramatically ease the computational cost and provide competitive accurate optimization results. The users can easily change the loading condition according to their conditions and requirements in the analysis of FEM, and the optimization objective can be easily set to satisfy their needs such as the strain at implant–bone interface, the area of implant–bone interface and the volume of implant. For example, if one user had a standard chewing loading spectrum and wanted to optimize the strain at implant– bone interface, he or she can load the spectrum on the dental implant in the FEM model and calculate the strain, and then use SVR to replace FEM to estimate the strain. After that, the stain can be optimized considering the uncertainties of design variables through k-sigma method or interval method. The proposed approach can be used in the uncertainty optimization of other bio-implants. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article. Funding The author(s) received no financial support for the research, authorship and/or publication of this article.
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Appendix 1 Notation Ea Ei L Lc LT P Pa Pi Sv b
elastic modulus of abutment elastic modulus of implant bottom width of tooth profile implanting length length of thread pitch of thread Poisson’s ratio of abutment Poisson’s ratio of implant maximum stress at implant–bone interface base angle of tooth profile
