Influence of design features of tibial stems in total knee arthroplasty on tibial bone remodeling behaviors

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Influence of design features of tibial stems in total knee arthroplasty on tibial bone remodeling behaviors Zhengbin Jia, He Gong∗, Shimin Hu, Juan Fang, Ruoxun Fan Department of Engineering Mechanics, Nanling Campus, Jilin University, Changchun130025, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 30 September 2016 Revised 30 April 2017 Accepted 2 June 2017 Available online xxx Keywords: Total knee arthroplasty Functionally graded material Bone remodeling Finite element analysis

a b s t r a c t In total knee arthroplasty, the optimal length and material of tibial stem remain controversial. This study aimed to evaluate influences of lengths and materials of cementless stems on tibial remodeling behaviors. Three groups of lengths were investigated (i.e., 110, 60, and 30 mm), and four materials (i.e., titanium, flexible ‘iso-elastic’ material, and two functionally graded materials [FGMs]) were selected for each group. FGM is a kind of material whose composition gradually varies in space. In this study, the compositions of two FGMs were Ti and hydroxyapatite (FGM I), and Ti and bioglass (FGM II), respectively. Tibial models were incorporated with finite element analysis to simulate bone remodeling. Distributions of bone mineral density, von Mises stress, and interface shear stress were obtained. For the length, the long stem produced more serious stress shielding and stress concentration than the short stem, but it could provide better mechanical stability. For the material, FGM I could reduce stress shielding and stress concentration and reduce the risk of loosening. Compared with the length, the material had a pronounced effect on remodeling. This study provided theoretical basis for optimal design of stem to improve service life of tibial components and to reduce pain of patients. © 2017 IPEM. Published by Elsevier Ltd. All rights reserved.

1. Introduction Total knee arthroplasty (TKA), which is an effective treatment of advanced arthritis of the knee, is expected with a consecutive distinct increasing applications [1,2]. However, 13% and 30% of TKA patients experience pain in short term and long term respectively [3], which may be caused by multifactorial etiology [4]. In addition, it was found that mechanical loosening was the most common reason for tibial component revision (24.6%) [5]. Several authors indicated that inadequate bone stock caused by stress shielding and stress concentration increases the risk of prosthesis loosening and periprosthetic fracture, and also triggers pain [6–8]. Stress shielding occurs when an implant, whose material property is usually stiffer than that of a bone, carries a part of the load originally carried by the host bone after implantation [7,9]. Thus, this implant forms a “shield” against the mechanical stimulus for bone tissues and causes bone resorption, thereby eventually leading to the possibility of aseptic loosening [6,10]. The prosthesis may increase the local stress level in the host bone, and this phenomenon is commonly known as stress concentration [9]. Studies showed that this phenomenon can stimulate the growth of the

∗

Corresponding author. E-mail addresses: [email protected], [email protected] (H. Gong).

surrounding bone tissue, and can also be associated with pain and can increase the risk of periprosthetic fracture [7,8,11–13]. Therefore, the mechanical environment of the host bone must be improved by investigating the design of tibial components. In recent years, studies regarding tibial components mainly focused on the materials [14,15], fixation [16,17], installation [18], and design of stems [7,19]. In the aspect of tibial tray materials, compared with the traditional titanium (Ti) and CoCrMo, functionally graded materials (FGMs) can reduce stress shielding of the host bone [14]. In the aspect of tibial component materials, a clinical study showed that in comparison with metal-backed tibial components, all-polyethylene counterparts significantly improved implant survival [15]. In the aspect of fixation ways, two kinds of techniques for primary TKA (i.e., cementless, and cemented fixations) were widely investigated [16,17]. In the aspect of final component alignment and tray position, the anatomic conflict between the tibial mechanical axis and intramedullary canal should not be ignored [18]. The optimal geometric length and material of stems, both affecting stress shielding and stress concentration directly, remain controversial particularly in the aspect of stem design [7,19]. Accordingly, the geometric length and the material of stems need further investigation and improvement. Bone has functional adaptation, that is, bone adjusts its mass and architecture in response to the change of mechanical environment. Functional adaptation results in the change of bone material

http://dx.doi.org/10.1016/j.medengphy.2017.06.046 1350-4533/© 2017 IPEM. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Z. Jia et al., Influence of design features of tibial stems in total knee arthroplasty on tibial bone remodeling behaviors, Medical Engineering and Physics (2017), http://dx.doi.org/10.1016/j.medengphy.2017.06.046

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property, and then the bone adapts to the new mechanical environment [20,21]. Since the 1970s, some authors have quantified functional adaptation through bone remodeling governing equations, and these equations were improved by Huiskes et al. [13], Weinans et al. [22], and Mullender et al. [23], etc. “Lazy zone,” nonlinear remodeling coefficient, and other concepts were also developed, which increased the accuracy of the bone remodeling algorithm in describing bone functional adaptation [24,25]. Given the functional adaptation, the investigations only examine the effects on bone tissue just only after the implantation provides an incomplete appreciation when describing the influence of prosthesis on the host bone [26]. However, a follow-up (long-term) quantitative simulation study can be implemented with bone remodeling governing equations [27–29], and many applications in the study of TKA have emerged. Nyman et al. [30,31] successively used this method to compare the effects of three kinds of fixation techniques (i.e., interlocking screws, cementless, and cement fixations) on bone loss and to assess the ability of bisphosphonates in reducing proximal bone loss in TKA. Chong et al. [32] focused on the influence of three methods of fixation (i.e., cement, cementless, and hybrid) on bone resorption, and they concluded that the cementless fixation with partial ingrowth or hybrid cementing fixation is preferred to maintain tibial bone stock. It was shown that the effects of tibial components on the host tibia can be simulated quantitatively and effectively with bone remodeling equation. Accordingly, the present study aimed to evaluate the influence of design features (i.e., geometric length and materials) of tibial stems in TKA on tibial bone tissues, and to improve the mechanical environment of the host bone by seeking favorable material and length of stem. 2. Methods The method incorporating finite element analysis (FEA) with quantitative bone remodeling algorithm was used in this study. The bone mineral density (BMD) distribution of proximal tibia was initially simulated, and then the three groups of cementless prostheses with different stem lengths, which included four types of materials in each group, were built to simulate the influences of different lengths and materials of the stem on tibial bone remodeling behaviors. The following three aspects were investigated in this study: BMD distribution of the proximal tibia before and after implantation, von Mises stress of the periprosthetic bone tissue, and shear stress on the interface between a stem and a bone. Fig. 1 illustrated the simulating steps. 2.1. Models Due to the complexity of bone remodeling algorithm and the simplified structures used in previous studies [10,31,33], a simplified model of tibia was established in SolidWorks (Dassault Systèmes, SolidWorks Corp.) according to the morphology of tibial plateau and the coronal dimensions of tibia in computed tomography (CT) scan images (0.73 × 0.73 mm/pixel resolution, and 0.6 mm slice thickness). The CT data were from the left tibia of a male patient, who was 35 years old, 177 cm tall, and weighed 70 kg. The right tibia was fractured, and the CT scan data of the healthy left tibia were used. Considering that morphology of the tibial plateau was important for selection and design of the tibial component [34], and no differences in the anatomical shape of tibial plateau were found when comparing female and male or younger and older patients [35], as a result, using a young male tibial shape would not have a significant influence on the simulation results in comparison with using a tibial shape of an older osteoporotic patient. Then, the simplified model was imported into ANSYS (ANSYS Inc.) to mesh (Fig. 2a). Three groups of cementless

tibial prostheses of the P.F.C Sigma Modular Knee System (DePuy International Inc.) in TKA were built in SolidWorks (Fig. 2b–d) directly according to the geometric parameters of tibial trays and stems listed in Table 1. And then the prosthetic models were imported into ANSYS. Overlap operation was performed on the tibia and on the three groups of prostheses in ANSYS. Thus, the three groups of tibial models containing prosthesis could be obtained. Moreover, in consideration of the complicated contact conditions of osseointegration during bone remodeling process and hypotheses in previous studies [29,30,36], the interface between tibia and prosthesis was assumed to be tied and a perfect fit, simulating complete osseointegration [17,37]. Free meshing was conducted for the three groups of models, and the internal shape of prosthesis in each group should be considered to assign the material properties of the prosthesis before simulating the effects of prosthesis on the tibial BMD distributions. A 3D 10-node solid element (i.e., tetrahedron Solid 95) was selected for these models. This element type could simulate irregular shapes appropriately. It was also suitable for the models with plastic deformation or large strain [38]. To verify the FE models, mesh patterns with different element sizes were generated. The same material properties and the same loads were applied to the FE models with different mesh sizes. The material properties were E = 2039 MPa and v = 0.3, and the loads were applied according to Fang et al [39]. When variations of average strain energy density (SED) were less than 5% [39], the mesh was considered to be convergent. After convergence tests, the mesh size used in this study was 2 mm. Fig. 3 shows finite elements of the prosthesis and the tibia. The numbers of elements and nodes of the tibial models containing the prosthetic shapes are listed in Table 2. The loads were applied according to Fang et al. [39] including joint contact force, shear force, medial collateral ligament force, and anterior cruciate ligament force acting on the proximal tibia with the magnitudes of 1233.3, 102.04, 6.4, and 139.3 N, respectively (Fig. 2e). These loads were the mean values of gait cycle loading. Of the joint contact force, 55% was applied on the medial side of the tibial plateau, and 45% was applied on the lateral side [42]. All the nodes at the bottom surface of the tibial models were fully constrained in all degrees of freedom. 2.2. Materials of the stem In this study, three groups of stem lengths were investigated, and four kinds of materials (i.e., two homogeneous materials and two FGMs) were selected for comparisons in each group. The two homogeneous materials were Ti and the flexible ‘iso-elastic’ material, whereas the two different functionally graded composites were the mixture of Ti and bioactive hydroxyapatite/collagen (HAP/Col) (FGM I) and the mixture of Ti and bioglass (FGM II). The material properties of Ti were E = 110 GPa and v = 0.27 [43], whereas those of the flexible ‘iso-elastic’ material were E = 14.22 GPa and v = 0.3, which were close to those of cortical bone [10]. For the tibial trays, the material in all the groups was Ti. The design of FGMs was based on the studies of Lin et al. [44] and Hedia et al. [9]. The material composition of the stem gradually transformed from Ti rich under the tibial tray to bioglass or HAP/Col rich toward the bottom of the stem, and the Young’s modulus gradually decreased. The volume fraction distributed over z-direction (vertical distribution) is as follows:

Vt = (z/l )m

(1)

Vc = 1 − Vt

(2)

where Vt and Vc were the fractions of Ti and HAP/Col (or bioglass), respectively; l was the total length of the stem; and m was the

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Fig. 1. Flow chart for the simulation of the influence of each prosthetic group on bone remodeling.

Fig. 2. Finite element meshes of the tibia and stems and the loading conditions. (a) Intact tibia, (b) 110 mm stem group, (c) 60 mm stem group, (d) 30 mm stem group, (e) Loading conditions. Blue lines and arrows indicated directions and positions of the loads (joint contact force, shear force, medial collateral ligament force, and anterior cruciate ligament force). Red lines and arrows indicated the loads applied on the FE model. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

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Z. Jia et al. / Medical Engineering and Physics 000 (2017) 1–11 Table 1 Geometric characteristics of the tibial knee prostheses. Group

PFC sigma knee system

Stem

110 mm stem group [8] 60 mm stem group [40] 30 mm stem group [19]

Tibial plate-size 5-Ti-6Al-4 V 83 mm ML-55 mm AP Tibial plate-size 5-Ti-6Al-4 V 83 mm ML-55 mm AP Tibial plate-size 5-Ti-6Al-4 V 83 mm ML-55 mm AP

∅13 mm × 110 mm, Ti-6Al-4V ∅13 mm × 60 mm, Ti-6Al-4V ∅13 mm × 30 mm, Ti-6Al-4V

Fig. 4. The flow chart of bone remodeling algorithm in combination with FEA [48].

Et = 110 GPa, vc = 0.35, and vt = 0.27 [9,10]; for FGM II, Ec = 30 GPa and vc = 0.35 [43]. 2.3. Bone remodeling simulation

Fig. 3. The cross-sectional view of meshes of the tibial model and the partition for comparing BMDs. (a) 110 mm stem group, (b) 60 mm stem group, (c) 30 mm stem group. When prosthesis was included in the model, the red elements were elements of prosthesis, and the yellow ones were the elements of tibia. ROIs (regions of interests) I and II were the selected region according to the measurements in clinics [41]. M was the medial side of the tibial model, L was the lateral side, and Region D was the tip region of the stem. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

Number of elements 141,229 141,427 158,446

Number of nodes 196,863 197,135 220,060

parameter controlling the composition variation through the stem. In the present investigation, m = 0.1 [44]. The Young’s modulus and Poisson’s ratio of the material composition are as follows:



E = Ec

Ec + (Et − Ec )Vt2/3

  Ec + (Et − Ec ) Vt2/3 − Vt

v = vt Vt + vcVc





dρ U =B − k , 0 < ρ ≤ ρmax dt ρ U=

n 1 Ui n

(5)

(6)

i=1

where B was a constant, k was the mechanical setpoint, and ρ max was the maximum density of cortical bone. U was the SED, n was the total number of loading cases, and Ui was the SED for loading case i. The parameters of Eqs. (5) and (6) are: B = 0.05 (g cm−3 )2 ·(M Pa·time-unit)−1 , k = 0.14 J·g−1 [39], and ρ max = 1.74 g/cm3 [22]. The relationship between the Young’s modulus E and the apparent density ρ is as follows [45]:

Table 2 Numbers of nodes and elements in the tibial models (including prosthesis meshes). Group 110 mm stem group 60 mm stem group 30 mm stem group

Weinans et al. [22] described bone internal structure with apparent density ρ as the control variable. The rate changing with time is as follows:



(3)

(4)

where E was the Young’s modulus of the FGM; Ec and Et were the Young’s moduli of bioglass (or HAP/Col) and titanium, respectively; and vc and vt were the Poisson’s ratios of bioglass (or HAP/Col) and titanium, respectively. In this paper, for FGM I, Ec = 1 GPa,

⎧ 2 ⎪ ⎨1007 × ρ 255 × ρ E= ⎪2972 × ρ 2 − 933 × ρ ⎩ 1763 × ρ 3.25

ρ ≤ 0.25 g/cm3 0.25 g/cm3 < ρ ≤ 0.40 g/cm3 (7) 0.40 g/cm3 < ρ ≤ 1.20 g/cm3 3 ρ > 1.20 g/cm

where the unit of the Young’s modulus was MPa, and the Poisson’s ratio was 0.3. The segmented equation was specific in describing the relationship between the Young’s modulus and the apparent density caused by the heterogeneity of bone tissue. Bone tissue was assumed to be isotropic and had linear elastic behavior. The initial density had no effect on the result of bone remodeling simulation [46]. Therefore, our simulation started from a uniform density of 1 g/cm3 [47]. The flowchart of the quantitative simulation of the influence of prosthesis on the proximal tibia is shown in Fig. 4.

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The iterative process stopped when the equilibrium condition was reached. The equilibrium condition is as follows [39]:





dρ U =0 = k or ρ =ρmin or ρ = ρmax dt ρ

(8)

where ρ min = 0.01 g/cm3 and ρ max = 1.74 g/cm3 . After 310 iterations, the BMD distribution of the proximal tibia model was obtained in this study. 2.4. BMD measurement ROIs, the same as those measured in clinics by Hudson et al. [41], were selected to verify the simulation results by comparing the BMDs (as shown in Fig. 3). The ROIs were placed distal to the cortical endplate region and throughout the tibial plateau from back to front. The width of the ROIs was one-half of the proximal tibia, their height was one-half the distance from the superior border of the cortical plate to the fibular head, and this distance was 12 mm in the model. Like in the previous clinical study focusing on the changes of tibial BMDs at the medial and lateral sides after implantation, in this study, medial and lateral sides were divided into several regions, similar to the method by Gruen et al. [49], to compare the effects of prosthesis on the BMDs of the surrounding bone tissue (Fig. 3). Fig. 3 shows the partitioning methods of the three groups, (e.g., for the 110 mm stem group, medial and lateral sides were divided into seven regions and tip region D, respectively). All the regions were through the entire tibia from back to front, and the height of region D was 5 mm. The group of 60 mm stem only retained the first four regions for both sides and region D because of the length of the stem. For the 30 mm stem group, only the first two regions for both sides and region D were retained. For different lengths of the stem model, the dimension of region D would change, but it was still located under the stem. 3. Results 3.1. BMD distribution of the proximal tibia The simulated proximal tibial structure shown in Fig. 5 was in line with the real tibial structure, which could be observed from the following aspects: medullary cavity was surrounded by cortical bone at distal tibia, typical cancellous bone was beneath the tibial plateau, and the BMD of cancellous bone in the medial side was larger than that in the lateral side. The average BMDs in ROIs I and II (shown in Fig. 3) of the three groups of tibias (i.e., 110 mm stem group, 60 mm stem group, and 30 mm stem group) were 0.803 and 0.715 g/cm3 , 0.803 and 0.718 g/cm3 , and 0.804 and 0.719 g/cm3 , respectively. Hence, the M:L BMD ratios of the three groups (M was the medial side of the tibial model, and L was the lateral side) were all 1.12. 3.2. BMD distribution of the implanted tibia The changes of BMD distribution in the coronal section view are illustrated in Fig. 6. These changes are caused by stems with different lengths and materials. Fig. 7 shows the comparisons of the average BMD distribution among medial side, lateral side, and tip region D (as shown in Fig. 3). It is found that, except the tip region, the decreases in BMDs on both sides of the stem were different. For the length of the stem, the greatest bone resorption on both sides occurred on the 110 mm stem group, whereas the least bone resorption on both sides occurred on the 30 mm stem group. For the material of the stem, the BMDs on both sides of the model were the greatest with the flexible ‘iso-elastic’ stem. However, the model with Ti prosthesis had the minimum BMDs on both sides. The BMD of the model with FGM I stem was similar to that of

Fig. 5. Simulated BMD distribution of 3D proximal tibia. (a) Top view, (b) Posterior view, (c) Coronal section view, (d) Coronal CT image of a real tibia, which was from CT scan data of the left tibia of a healthy elderly men (0.79 × 0.79 mm/pixel resolution, and 0.6 mm slice thickness).

the model with the flexible ‘iso-elastic’ material, and it was better than that of the model with FGM II. For Ti prosthesis group, the most serious resorption was observed under the tibial tray (i.e., L1 and M1) with an average reduction in BMD of 33% compared with that under the intact tibia. In addition, the medial side experienced a larger amount of bone resorption than the lateral side, and the BMD reduction of trabecular bone in the metaphysis was greater than that of cortical bone in the diaphysis. The smallest resorption was observed at the lowest L region for each group. In addition, bone densifications were observed at the distal stems (i.e., region D and the bottom region of the medial side) in all models. However, the increases in BMDs for different models were distinct. For the length of the stem, the degree of densification with the short stem was smaller than that with the long stem. For the materials, the BMDs at the distal stem for Ti and FGM II cases were the largest, the flexible ‘iso-elastic’ case had the smallest BMD, and the BMD for FGM I case was close to that for the flexible ‘iso-elastic’ case. 3.3. Von mises stress distribution of the proximal tibia The tibial mechanical environment was changed because of the tibial component implantation. Thus, von Mises stress distributions at both sides of tibia were investigated (Fig. 8). It shows that when the lengths of the stems were the same, excluding the distal region of the stem, the flexible ‘iso-elastic’ case had the largest von Mises stress at both sides, whereas the Ti case had the smallest one. On the contrary, at the distal tip of the stem, the Ti case had the largest von Mises stress, whereas the flexible ‘iso-elastic’ case had the smallest one. Except for the region next to tibial tray, the von

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Fig. 6. The coronal section view of the tibial BMD distribution after implantation. The first column was the distribution just implanted, and the last four columns were the simulated results of tibia implanted with Ti, flexible ‘iso-elastic’, FGM I, and FGM II stems, respectively. One to three lines were the distributions of 110 mm stem group, 60 mm stem group, 30 mm stem group, respectively.

Mises stress distribution of the model with FGM I stem was considerably closer to that with flexible ‘iso-elastic’ stem. The comparison of the uses of the same material with different stem lengths determined that the von Mises stress of 110 mm stem group was the smallest and that of 30 mm stem group was the largest in the same area on both sides excluding the distal region. 3.4. Medial and lateral interface shear stress distributions Fig. 9 shows the medial and lateral interface shear stress distributions between the tibia and the stem. The interface shear stress of the model with the same lengths of stems reached the maximum (absolute value) at the distal end of the stem on both sides. Furthermore, the interface shear stress was the greatest in the model with Ti stem compared with those in the models with the other three materials. For the material of the stem, the average value of the interface shear stress caused by Ti stem was the largest and the most uneven, whereas that caused by the flexible ‘iso-elastic’ case was the smallest and the most uniform. In addition, the FGM I case was close to the flexible ‘iso-elastic’ case. For the length of the stem, although the interface shear stress distribution of the 110 stem mm group was the most nonuniform, its average value was the smallest compared with those of other two

length stem groups. The average value of 30 stem mm group was the greatest.

4. Discussion In this study, the 3D internal structure of tibia and tibial bone remodeling behavior after implantation were simulated. The distributions of tibial BMD, von Mises stress, and interface shear stress between the tibia and the stem were analyzed to investigate the effects of the stem on stress shielding, stress concentration, and tibial component stability. The simulation of tibial structure shown in Fig. 5 was quite similar to the BMD distribution simulated by Fang et al. [39], and the structure agreed well with the clinical measurements [50]. In addition, for the same ROIs, the measured average M:L BMD ratio for postmenopausal females was 1.21 ± 0.10 (Hudson et al. [41]). Therefore, the M:L BMD ratios simulated in this study were consistent with the results measured by Hudson et al. [41]. The simulation result was in accordance with the BMD distribution of the real proximal tibia. Therefore, the simulation results of tibial BMD were suitable for investigating the influences of tibial components on the mechanical environment of tibias for older patients.

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Fig. 7. Comparison of average BMDs in different regions. Left columns: medial side and region D. Right columns: Lateral side. Top: 110 mm stem group. Middle: 60 mm stem group. Bottom: 30 mm stem group.

The influence of length and material of the stem on tibial bone remodeling was studied directly and quantitatively by comparing the distribution of tibial BMDs (Figs. 6 and 7). The simulation results of BMD reduction caused by stress shielding from clinical cementless Ti prosthesis were consistent with the clinical observations and the experimental results [32,51–53]. Bone densifications were observed at the distal stems (i.e., region D and the bottom region of the medial side) in Ti prosthesis group because of stress concentration [8,51]. Therefore, the model used in this study could be used to predict the effects of prosthesis with other materials on the tibial mechanical environment. The mechanical environment of the host tibia could be studied based on the comparison of von Mises stress distribution (Fig. 8); the mechanical environment was mutually verified with the conclusion from BMD distributions. Then, the length and material of the stem with a smaller effect on the host bone could also be selected. The long stem resulted in more pronounced stress concentration and stress shielding. This finding supported those of the previous studies [8,26]. For the stem material, compared with Ti, FGM I could improve the conditions of stress shielding and stress concentration. Specifically, FGM I could increase the von

Mises stresses on both sides of the stem (but they were still lower than those of intact tibia, and they did not cause stress concentration). The maximum differences of von Mises stresses for the models with FGM I and Ti stems were 3.694 and 1.598 MPa, respectively (at point A), thereby facilitating the reduction of stress shielding of both sides significantly. This scenario could also result in stress reduction at the tip region. The maximum differences of von Mises stresses for the cases of FGM I and Ti stems were 3.924 and 5.109 MPa, respectively (at point B). Therefore, FGM I effectively relieved the stress concentration at the tip region. The comparison of Figs. 8 and 9 indicated that both trends were consistent. With the model with 110 mm Ti stem as an example, the stem caused the minimum von Mises stress, and its interface shear stress (absolute value) was the smallest except for the region of distal tip. At the tip region, the maximum von Mises stress and interface shear stress were both observed for the model with 110 mm Ti stem. The consistency of trends could be explained by the following two aspects: On the one hand, shear stress could reflect stress shielding and stress concentration. On the other hand, it could also represent load transfer [14]. Further comparison of the

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Fig. 8. Von Mises stress distributions of the proximal tibia. Left columns: medial side. Right columns: Lateral side. Top: 110 mm stem group. Middle: 60 mm stem group. Bottom: 30 mm stem group. A was the point showing the maximum deviation between flexible ‘iso-elastic’ and Ti excluding the region of distal tip, and B was the one at the tip region.

shear stresses for different cases showed that for the stem length, the shear stress of the model with long stem case was the most nonuniformly distributed because this model had the minimum interface shear stress excluding the tip region and the maximum interface shear stress at the tip region. This scenario also reflected the most serious stress shielding and stress concentration. However, its average shear stress was the smallest, meaning that it was the least likely to loosen. This finding agreed with the conclusion that long stems could provide better mechanical stability [54,55], which are helpful for bone defects to reconstruct and recover. For the stem material, in comparison with Ti, FGM I caused more homogeneous and smaller shear stress. Thus, it not only relieved stress shielding and stress concentration effectively but also provided better stability to reduce the possibility of interface failure [10,26,56].

In the optimization design of the stem from stress shielding aspect, the stress shielding on the host bone for Ti and that on the host bone for FGM II cases were very similar. This observation agreed with the conclusion that the change of material stiffness had a slight effect on stress shielding [7,8]. The FGM I and flexible ‘iso-elastic’ materials used in this study could reduce stress shielding significantly. This finding was consistent with those of the previous studies indicating that the materials, such as PE, would facilitate reduction of stress shielding [7,26]. Therefore, the following conclusion was drawn. The host bone was considerably relieved only when the Young’s modulus of the stem was close to that of bone. In the optimization design from stress concentration aspect, FGM I could greatly reduce the stress level at distal tip. It could also avoid the load shifting from the tip to the interface between two materials when the distal tip of the Ti stem was

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Fig. 9. Interface shear stress distributions of the proximal tibia. Left columns: medial side. Right columns: Lateral side. Top: 110 mm stem group. Middle: 60 mm stem group. Bottom: 30 mm stem group.

replaced by PE stem, thereby resulting in the increase in the local stress level and uneven stress distribution [8]. In addition, its material property was also close to that of Ti at the proximal regions. Thus, the increase in the stress levels of cancellous bone caused by its low stiffness could be avoided particularly in PE and flexible ‘iso-elastic’ materials. Such increase in stress could trigger cancellous bone fatigue failure and could consequently lead to component loosening [26,57]. The material had significant effect on the bone when comparing the effects of both stem material and length on the host bone tissue. Given the view of clinical application, the selection of length was not usually determined with the mechanical environment of the host bone but with the intended fixation and anatomy of patients [55]. Thus, the improvement on materials would be more flexible to be applied in clinics. The mechanical environments were different at distinct locations. In consideration of bone functional adaptation process, the

bone structures were also different. As a result, the FGMs for optimizations on different bone tissues were different [29]. Therefore, two kinds of FGMs were investigated in the present study to analyze their effects on tibial bone remodeling. This study determined that FGM I could reduce stress shielding and stress concentration of the host bone. Some limitations must be acknowledged and addressed in the present study. First, in addition to geometric length and the material of stem, other variants existed. These variants, such as contact condition (including contact areas and coefficients of friction) of the interface, might have effects on the tibial bone remodeling but were not compared in this study. Second, four kinds of materials were investigated to study the influence of stem material on stress shielding at host bone. We concluded that stress shielding on the host bone was considerably relieved only when the Young’s modulus of the stem was close to that of bone. Nevertheless, only

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two materials (i.e., FGM I and flexible ‘iso-elastic’ ), whose Young’s moduli were close to that of bone, were investigated to evaluate the influence. Thus, the limited number of samples could not allow any statement regarding the range of Young’s modulus of the stem that could conspicuously relieve stress shielding. In our future study, a more accurate 3D tibial model of an older patient should be used to discuss the influences of more loads during activities of daily living [58] on bone remodeling behaviors. 5. Conclusions In summary, this study showed that the length and the material of the stem used in TKA could affect tibial remodeling behaviors. For the length of the stem, results supported that the long stem produced more serious stress shielding and stress concentration than the short stem, but it could provide better mechanical stability for tibial components. For the material of the stem, compared with Ti, FGM I could reduce stress shielding and stress concentration and could provide more uniform and smaller interface shear stress, thereby contributing to a subsequent reduction in the risk of loosening. In addition, the relationship between variation of the Young’s modulus of prosthesis and stress shielding of the tibia was concluded, i.e. the change of Young’s modulus had a significant influence on relieving stress shielding only when the material property of the stem was close to that of host bone. Finally, when comparing the effects of both lengths and materials of stems on tibia, it was concluded that the material had more pronounced effect than the length. This study provided a theoretical basis for the optimal design of the stem, which was helpful to improve the service life of tibial components and to reduce the pain of patients. Ethical approval This study was approved by the Medical Ethics Committee of the First Hospital of Jilin University (No. 2012-064). The patient gave written consent to participate in this study. Conflict of interest: None Acknowledgments Funding: This work was supported by the National Natural Science foundation of China (Nos. 11322223, 11432016, and 81471753), and Science and Technology Department of Jilin Province (20160101297JC). Appendix Table A.1. Table A.1 The detailed information and average SED results of finite element models with different meshes. Element sizes (mm)

Number of elements

Average SED (MPa)

Variations with 2 mm model (%)

3.2 2.8 2.4 2.0 1.6 1.2

38,091 53,019 83,130 141,229 214,586 393,960

0.0 0 016871 0.0 0 017474 0.0 0 017736 0.0 0 018751 0.0 0 018867 0.0 0 018835

10 6.8 5.4 – 0.62 0.45

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