Mechanical and Fatigue Behavior of Cellular Structure Ti-6Al-4V Alloy Femoral Stems: A Finite Element Analysis

: Repetitive loads acting on the hip joint ﬂuctuate according to the type of activities produced by the human body. Repetitive loading is one of the factors that leads to fatigue failure of the implanted stems. The objective of this study is to develop lightweight femoral stems with cubic porous structures that will survive under fatigue loading. Cubic porous structures with different volumetric porosities were designed and subjected to compressive loading using ﬁnite element analysis (FEA) to measure the elastic moduli, yield strength, and ultimate tensile strength. These porous structures were employed to design femoral stems containing mechanical properties under compressive loading close to the intact bone. Several arrangements of radial geometrical porous functionally graded (FG) and homogenous Ti-6Al-4V porous femoral stems were designed and grouped under three average porosities of 30%, 50%, and 70% respectively. The designed stems were simulated inside the femoral bone with physiological loads demonstrating three walking speeds of 1, 3, and 5 km/h using ABAQUS. Stresses at the layers of the functionally graded stem were measured and compared with the yield strength of the relevant porous structure to check the possibility of yielding under the subjected load. The Soderberg approach is employed to compute the safety factor (N f > 1.0) for each design under each loading condition. Several designs were shortlisted as potential candidates for orthopedic implants.


Introduction
Total hip arthroplasty (THA) is a successful treatment procedure that helps patients achieve pain free movement by hip replacement [1,2]. The cementless femoral stems are introduced in the femur medullary canal and securely fixed to the bone to create a direct press-fit contact to transfer part of the applied loads developed during human daily activities to the adjacent bone [2][3][4]. The femoral stems were designed to live for a minimum of 20 years. However, the life span of the hip implants is affected by many limitation factors leading to stem failure [2]. Fatigue fracture is one of the major problems associated with femoral stem failures [5,6]. In addition, other factors include micromotion (that may lead to poor bone tissue ingrowth), stress shielding (where the stem doesn't transfer the applied load to the adjacent bone and, as a consequence, may cause aseptic loosening, stem migration), and failure, surrounding bone fractures and requiring revision surgery [3,[7][8][9][10]. The proposed solution is to use the mechanical properties of the stem material that can be controlled by introducing porous cellular structure to reduce the modulus of elasticity and yield strength as concluded in our previous study [11,12]. THA revision surgery is more complex compared with the primary one due to the higher risk and unsatisfactory outcomes in terms of further loss of the host bone, perforation, fracture, pain, time, and cost aspects [2,[13][14][15].
Total hip arthroplasty is keep raising drastically over the next 30 years. THA in USA is expected to increase from 930,575 in 2020 to 1,537,422 in 2050 with an annual growth rate of 1.7% [15,16]. The revision hip replacement surgery after 20 years varies from 40% for young of less than 50 years old, 30% for ages between 50 and 59 years old, and 15% for ages between 60 and 75 years old [16]. Early hip arthroplasty revisions with a diagnosis of infection and inflammatory reaction, fracture, instability, other mechanical complications, and aseptic loosening over the years 2012 to 2019 are 32.5%, 24.3%, 21,7%, 4.8%, and 4.2%, respectively [15,17].
The femoral stem's prefabricated design allows using different biocompatible materials with properties fitting the functional requirements [2,18] and increasing the expectations to last longer than before. Studies have proven that the Ti-6Al-4V (Ti alloy) is a perfect biomedical material for hip implants in terms of strength, biocompatibility, corrosion resistance, lightweight, and flexibility [19][20][21]. The stiffness of the Ti-6Al-4V alloy is 5-10 stiffer than femur bone [11,20]. Thus, stems with porous structure designs were developed to reduce the stiffness, enhance the trabecular bone tissue ingrowth that facilitates the integration of the implanted stem with the hosting bone [22] and significantly reduce the stress shielding [11]. The applied forces on the implanted stem that depend on the different human activities such as running, stair climbing, jogging, and walking at various speeds produce dynamic stresses leading to fatigue fracture [23]. Many researchers focused on transferring part of the applied stresses on the stem to the bone to reduce the stress shielding and micromotion without considering the fatigue failure due to the dynamic stresses in their studies. Most of the previous studies were found predicting the fatigue limit using FEA for bulk stems under concentrated compression loads, but few studies were found carried out for stems with porous structure [24,25]. No literature was discovered on computing the fatigue life for porous stems using different physiological loads.
In our previous study [11], mechanical performance, stress shielding, and micromotion of the designed femoral stems were studied and investigated. To complete the investigations to have the most ideal design that can be recommended for the surgent based on patients' conditions, it is vital to study the fatigue performance of the proposed designs. Hence, this current study aims to investigate the performance of the designed functionally graded porous structures femoral stem under high cyclic loads, focusing on the stresses obtained on each of the stem's layers and fatigue factor of safety for each stem. The hypothesis in this study is that such radial functionally graded porous structure will provide optimum designs that can reduce the stress shielding, promote the stability and survive under high cyclic loading focusing on the level of porosity in the stems required to achieve good biomechanical performance. Hence, in this study, Fifteen Ti-6Al-4V functionally and homogeneous fully porous stems were designed and grouped under three average volumetric porosities, namely, 30%, 50%, and 70%. The porosities were integrated into the stem and geometrically arranged in the radial direction using cubic porous structures and tested using 3D Finite Element model (ABAQUS) under physiological loads based on slow (1 km/h), medium (3 km/h), and fast walking speeds (5 km/h) [11]. The stresses within the stem's porous layers were measured to verify and ensure the designs are safe under certain loading conditions based on the yield strength of the porous structure and fatigue factor of safety limit (N f > 1.0) [24]. The factor of safety is calculated for each design using the Soderberg approach. This approach is efficient in terms of estimating the stem fatigue limit under high cyclic loads [24].

Material and Methods
In this section, various essential aspects are described in detail, including the finite element model details for both porous microstructure and femoral stems, mechanical behavior of the designed stems under physiological loading conditions aiming to predict the fatigue life of the designed functionally graded porous stems.

Porous Microstructure
Square pores microstructure three-dimensional (3D) models with various volumetric porosities ranging from 10% to 90% were designed and simulated under compression loading using ABAQUS 6.17 as illustrated in Figure 1 and presented in Table 1. The square pores' structure was selected based on the results of a previous fatigue study carried out by Amin Yavari et al. [26] on different Ti-4Al-6V alloy porous structures. The experimental results show the superiority of the cubic porous structures over the other structures such as diamond shape structures. None of the cubic porous structure specimens failed under fatigue testing even at a maximum load of 80% of the relevant porous structure yield strength (0.8σ Ys ). These porous cubic structures reached 10 6 loading cycles without signs of failure. This is attributed to the vertical struts performing better than the diagonal and horizontal ones in compression loading conditions [26,27]. Different sizes of square pores with minimum strut thicknesses of 3.0 mm [26] were fitted within 15 mm × 15 mm × 15 mm 3D cubes. The volumetric porosity percentages were calculated based on the volume of the square pores introduced to the 3D cubes based on Equation (1) generated in our previous study [11]. The top and bottom sides of the cubes were fitted with plates; a displacement boundary condition was assigned to the top plate; and a encastre boundary condition was used to tie up the bottom plate. The designed structures were assigned with bulk Ti-4Al-6V material properties. C3D10 quadratic tetrahedral meshing elements had been used to mesh the assembly using a mesh size of 0.2 mm based on a sensitivity study determined from stiffness convergence of dense cube [11]. where: V p : Volumetric porosity L 2 : cubic pore inner dimension N: Number of pores in a single cube face x: Full cube dimension t: Strut thickness The upper plate was assigned with axial displacement to attain the yield points using 0.05 increment size to generate more accurate values of the yield points, modulus of elasticity, and ultimate tensile strength. Mass properties in ABAQUS were used to compare the designed volumetric porosity for each designed cube.

Porous Stem Design
The designed porous cellular structures were utilized to design twelve different functionally graded porous stems and three homogeneous porous stems with average volumetric porosities of 30%, 50%, and 70%. Additional porous structures were designed that are needed for designing these 15 stems as shown in Figure 2. The fifteen designed stems were divided into three groups representing these aforementioned average volumetric porosities, each group contains four functionally graded designs and one homogeneous design The functionally graded porous structures were created within the stem's layers in the radial direction, the layers cross-sectional areas are 15.9 mm 2 , 17.3 mm 2 , and 23.6 mm 2 for L1 (the stem's core), layer 2 (L2), and layer 3 (L3), respectively. Figure 2 illustrates the porosities configurations within the stem's layers to achieve the required overall average volumetric porosity of the pertinent group. The average porosities of the designed stems in a radial direction were calculated based on Equation (2).  These fifteen designs (D1 to D15) were integrated into three-dimensional (3D) finite element models in ABAQUS 6.17 software to study the mechanical behavior of the designed stems under physiological loading conditions obtained from walking at speeds of 1 km/h, 3 km/h, and 5 km/h. The stem-bone assembly as shown in Figure 3 was built based on ISO 7206-4 standard [11,28] using surface to surface contact between the stem and the bone interface surfaces with a 0.4 friction coefficient. The assembly was fixed into cylindrical shape epoxy (3.7 GPa) with specific orientations shown in Figure 3, the bone was tied with the epoxy and constrained to move or revolve in any direction [7]. The 3D model of the femur was constructed using computed tomography (CT) scanned images of a patient femur as explained in previous studies [7,29]. Cortical and trabecular bone were considered homogeneous and linear elastic isotropic material, respectively, with material properties as listed in Table 2 [24,30,31].   The femur was subjected to three physiological loads, specifically hip contact force (P 0 ) and muscular forces (P 1 and P 2 ) [11,32,33], as shown in Figure 3. The applied forces in their components F x , F y , and F z were calculated based on an average bodyweight of 900 N at different walking speeds (i.e., 1 km/h, 3 km/h, and 5 km/h) as shown in Table 3. The highest hip contact forces used in percent of bodyweight (%BW) were 293%, 352%, and 471% for walking speeds of 1 km/h, 3 km/h, and 5 km/h, respectively [32,33]. The assembly was meshed with a mesh size of 3 mm using C3D10 quadratic tetrahedral elements as illustrated in Figure 3, which was used based on a previous sensitivity study [11,34].

Fatigue Analysis of the Designed Porous Stems
A fatigue study was performed on the designed stems implanted inside bone by performing finite element analysis based on physiological loading conditions obtained from the three walking speeds as presented in Table 3. Von Mises stress on the stem/stem layers was measured using ABAQUS FEA software and the results were used to calculate the factor of safety as illustrated in Figure 4. Designs D12 and D14 denote that no results were obtained as the FEA simulation does not converge due to the high volumetric porosity at the outer layer of these stems (Figure 4) [11]. The Soderberg approach was used for calculating the factor of safety [24] as shown in Equations (3)-(6): The minimum stresses and the maximum stresses were measured at the loads associated with a stress ratio of 0.1 (R = 0.1) during the test: where; (σ m ) and (σ a ) denote mean and alternating stresses generated in the porous stems and (σ min ) and (σ max ) are refereeing to minimum stresses at 0.1 of the applied physio-logical load and maximum stresses associated with the full load generated from each walking speed. σ a S e + σ m S ys = 1 N Hence, the fatigue factor of safety N f is as follows: Whereas S e , S ys , and N denote the endurance limit of the porous material calculated using Equation (7) [36], yield strength of the porous material, and factor of safety, respectively.
K a : Surface quality factor. K b : Size factor. K c : Reliability factor. K d : Temperature factor. K e : Stress concentration factor. K f : Miscellaneous factor. For Ti6Al4V [37]: S e =0.54 × S u S u : The Ultimate tensile stress of the porous material, (the S u for the outer stem's porous layer for each design was used where it's associated with the maximum stresses obtained from each walking speed), the reduction percentage in S u and the calculated endurance limit S e are shown in Table 4 below: Table 4. The obtained elastic moduli, yield strength, ultimate tensile strength, and the associated calculated of the endurance limit S e values based on the percentage of reduction in S u due to the introduced porosities.

Ultimate Tensile Strength (Su), Endurance Limit, Elastic Moduli and Yield Strength (Ys) for the Designed Porous Structures
Volumetric

Result and Discussion
Simple cubic microstructure (squire bore) with different volumetric porosities were designed and simulated under compression loading test to estimate the mechanical properties associated with each design. Stress-strain diagram for each sign was plotted (Figure 1) to obtain the yield strength, ultimate tensile stress, and elastic moduli, and the results were listed in Table 4. The results were used for designing and modeling the 15 stems which were grouped under three average volumetric porosities (namely 30%, 50%, and 70%) as shown in Figure 2.
Static finite element analysis (FEA) of the stem-bone assembly was performed for the fifteen designed functionally graded and homogeneous porous stems by applying the physiological loading conditions of the three walking speeds as illustrated in Table 3. Force-displacement data from the porous stem were used to validate the finite element models as explained in our previous study [11].

Mechanical Properties of the Porous Structures
Mechanical properties of the cubic porous structures were obtained to study the fatigue life and the factor of safety of the designed stems. It was observed that as the porosity increase the yield strength and elastic moduli decrease as presented in Table 4 and illustrated in Figure 5. A bulk Ti cube with the same dimensions was simulated using the same boundary conditions for validation and comparison purposes, the computed effective Young's Modulus was 123.6 GPa, which shows a good agreement with Hedayati et. al. results of 122.6 GPa [38].

Stem-Bone Model Validation
A bulk Ti stem was simulated inside the femur by applying a concentrated force of 3000 N in ten increments with increment size of 300 N at the stem head. Force-displacement data were plotted by tracking the displacement values at a certain point on the femur head as illustrated in Figure 6, the results were compared with Jette et al. finite element model results [19] for our model validation using the same boundary conditions test data, a strong agreement was noticed between both models as shown in Figure 6. It can be concluded from this validation result that the difference in implant geometry between Jette et al. [19] and our model has less impact on the results. The slight difference in the curves can be attributed to the difference in the exact position of the selected point on the bone head.

Stresses in Porous Stem's Section
The stresses within each porous layer were computed through the FEA results for the fifteen FG designs and for each of the three walking speeds (as illustrated in Figure 7). The stresses within the stem's layers are increasing radially up to reach the maximum stress at the outer stem's layer (L3). The stresses associated with layer 3 (L3) were considered for the fatigue calculations as this layer is associated with the maximum stress and on direct contact with the bone. Stresses at the outer layer (L3) of the porous stems were found to be less than the stresses generated on the bulk stem for all designs and walking speeds, which indicates that the applied loads/stresses are distributed between the femur and the stem (more stress transfer to the femur). This is attributed to the reduction of the stiffness associated with each porous stem designs in comparison with the bulk stem stiffness (114 GPa). Hence, the stress level is on the stem is affected by two factors: the applied loads associated with the walking speeds and the stiffness of the designed stems. Although the stress distribution along the FG stem's layers was varied, this is attributed to the difference in the Young's modulus of each porous layer. The yield strength was studied for each stem's porous layer and compared with the stress values computed via FEA of the stem-bone models and found that all designs are not yielding at walking speed of 1 km/h except designs 9, 12 which holds stress value at the stem's outer layer equal to the yield strength value related to 70% and 80% porosities, respectively, and design 14 which gave no results. At 3 km/h walking speed only designs 1, 3 and 8 were observed to be not yielding and safe. All designs found to be yielded at walking speed of 5 km/h as the stresses encountered are higher than the yield strength associated with the relevant porous structure of each design. stem's failure (SF) is calculated based on the stress ratio between the relevant stem layer yield strength (Y sl ) and the maximum stress value of that layer (σ Lm ) based on Equation (8): The design will not fail/yield if SF value > 1.0, the results are illustrated in Figure 8. Previous experimental study Amin Yavari et al. [26] shows the superiority of the cubic porous structures over the other structures such as diamond shape structures. None of their cubic structure tested samples were failed under cyclic loading at maximum load of 80% of the relevant porous structure yield strength (0.8σ Ys ) [26]. This finding is comparable with our computational analysis findings up to certain loading conditions as illustrated in Figure 8.

Fatigue Factor of Safety (Soderberg Approach)
The fatigue factor of safety is one of the important factors used for the evaluation of the femoral stem's durability [23]. Taking into consideration the other factors that need to be included in the porous femoral stem design such as stress shielding, micromotion for enhancing the implant's stability through improving bone tissue ingrowth in order to avoid premature implant failures. Stresses encountered within the stem during the daily activities shall be computed and compared with the yield strength of the used porous structure [39,40]. Most of the previous studies were predicting the fatigue limit using FEA for solid stems and under concentrated compression load, and less studies were found for porous stem [24,25]/No literature was discovered on computing the fatigue life for FG porous stems in the radial direction using different physiological loads. Hence, in this study, the factor of safety was calculated for each design using Soderberg approach. This approach is efficient in terms of estimating the stem fatigue limit under high cyclic loads [24] and this approach provide the worst-case scenario among the other approaches (namely, Goodman and Gerber [24]).
The maximum and the minimum stress were identified based on increments 1 and 10 of the FEA (ABAQUS) results, i.e., the cyclic stress ratio of R = 0.1 for calculating the minimum and maximum stresses as shown in Equations (3)- (6). The factors of safety (N f ) were calculated and the results for all designs with the three walking speeds are shown in Figure 9. The designs that have a factor of safety > 1 are considered safe under cyclic fatigue loading. None of the fifteen designs were found to be safe under a walking speed of 5 km/h. only Design 8 (D8) is safe to be used up to a walking speed of 3 km/h and the von Mises stress is less the associated yield strength at 1 km/h and 3 km/h. Designs D1, D2, D3, D5, D6, D8, D10, D11, and D15 are safe under walking speeds of 1 km/h all of these designs also had von Mises stresses less than their associated yield strength at 1 km/h walking speed. Most of the THA patients are normally have a maximum normal walking speed of 1.54 km/h and this level is less for the elderly patients [41,42]. The proposed femoral stem designs can be improved to enhance the fatigue factor of safety by adding an outer dense shell to the porous stem structure with different thicknesses [3] which will be part of our future study.

Conclusions
The aim of this study is to identify the fatigue durability of the cementless functionally graded and homogeneous porous structure femoral stems. Fifteen different volumetric porosities arrangements within the stems layers in a radial direction were grouped into three average porosities of 30%, 50%, and 70%. Different physiological loading conditions corresponding to three walking speeds (1 km/h, 3 km/h, and 5 km/h) were applied on the stem-femoral bone assembly. The von Mises stresses were computed within the stem's layers and studied against the yield strength associated with the relevant porous layer porosity. Fatigue factors of safety were computed for each design under the three loading conditions using Sodeberg approach. The outcomes obtained from this study lead to the following conclusions: The porous structure's mechanical properties varied as the volumetric porosities changed and this is affecting the stems performance and fatigue endurance under daily human physical activities; Changing the porosities arrangement within the stem's layers affects the stems durability and the fatigue performance; Designs D1, D2, D3, and D5 with volumetric percentage of 30%, D6, D8, and D10 with volumetric percentage of 50%, and D11 and D15 with volumetric percentage of 70% were found to be safe at walking speed 1 km/h in terms of von Mises stress values compared with the associated yield strength and factor of safety based on Soderberg approach; Design 8 (D8) is the best candidate that found not failed up to walking speed of 2 km/h. The performance of the proposed designs can be enhanced by adding outer dense shell to the porous stem structures with different thicknesses which will be part of our future study.