Influence of Different Boundary Conditions in Finite Element Analysis on Pelvic Biomechanical Load Transmission

Abstract

Objective

To observe the effects of boundary conditions and connect conditions on biomechanics predictions in finite element (FE) pelvic models.

Methods

Three FE pelvic models were constructed to analyze the effect of boundary conditions and connect conditions in the hip joint: an intact pelvic model assumed contact of the hip joint on both sides (Model I); and a pelvic model assumed the hip joint connecting surfaces fused together with (Model II) or without proximal femurs (Model III). The model was validated by bone surface strains obtained from strain gauges in an in vitro pelvic experiment. Vertical load was applied to the pelvic specimen, and the same load was simulated in the FE model.

Results

There was a strong correlation between the FE analysis results of Model I and the experimental results (R ² = 0.979); meanwhile, the correlation coefficient and the linear regression function increased slightly with increasing load force. Comparing the three models, the stress values in the point near the pubic symphysis in Model III were 48.52 and 39.1% lower, respectively, in comparison with Models I and II. Furthermore, the stress values on the dome region of the acetabulum in Models II and III were 103.61 and 390.53% less than those of Model I. Besides, the posterior acetabular wall stress values of Model II were 197.15 and 305.17% higher than those of Models I and III, respectively.

Conclusions

These findings suggest that the effect of the connect condition in the hip joint should not be neglected, especially in studies related to clinical applications.

Introduction

The pelvis, as the center of a musculoskeletal biomechanical system, facilitates transfer of weight from the upper part of the body to the lower extremities. The resultant force through the hip joint during walking is around three times greater than body weight, and may be up to six times greater than body weight during running and stair climbing1, 2. Thus, the reconstruction of pelvic comprehensive biomechanics after an injury is the primary consideration of clinical management. Various implants have been utilized in the management of pelvic injuries, such as reconstructive plates, dynamic compression plates, locking plates, lag screws, and sacral rods3. Generally, the clinical efficacy and biomechanical features of the implants used for pelvic injuries should be evaluated through biomechanical experiments in vitro. However, irregular geometry and material heterogeneity of the pelvis often make mechanical experiments challenging4. In the past decade, the finite element (FE) analysis method has been frequently utilized in the research of human pelvic biomechanics5, 6, 7, 8. Compared to clinical trials and mechanical experiments in animal models, FE analysis provides full field data more easily and may be used to investigate the effects of many variables quickly and economically.

The FE model simulation results could be affected by several factors, such as boundary conditions and anatomical structures9. Currently, there is not a universal standard on boundary conditions in pelvic FE development10, 11, 12. Zhao et al. compared the stability of screws for the treatment of pelvic fractures, and the boundary condition in their model was applied to the rotation center of the acetabulum, which could present limitations while observing the biomechanical behavior of pelvis10. Furthermore, the functional unit integrity of the anatomical structures in the pelvis is another crucial factor influencing the accuracy of the prediction13, 14. Phillips et al. explored the impacts of ligaments and muscles in stress distribution15; however, the fixed restraints in their model were applied to the articular surface at each of the sacroiliac joints, and the sacrum was not included. Eichenseer et al. analyzed the role of sacroiliac ligaments in joint motion and stress transmission, and their model did not consider the biomechanical impacts of the femur11. Overall, although all the investigations mentioned above provide fundamental theoretical guidance on clinical treatment, their models were all simplified with no consideration of the influence of the femur and/or the sacrum, both of which have significant effects on stress distribution and displacement in numerical simulations.

The present study aimed to: (i) construct an intact pelvic FE model including the ilium, sacrum, proximal femurs, and main ligaments; (ii) validate the FE model with mechanical experiments to facilitate further clinical research; and (iii) analyze the influence of different boundary conditions and connect conditions in the hip joint on the load transmission in the pelvic system.

Materials and Methods

Biomechanical Experiments

A formalin‐preserved cadaveric pelvis (female, age 40 years, height 164 cm, weight 65 kg) that included the fifth lumbar vertebra and one‐third of the proximal femoral shaft was obtained from the Department of Anatomy of Hebei Medical University. A standard anteroposterior X‐ray was used to confirm the absence of any bone abnormalities or tumors, and dual‐energy X‐ray absorptiometry (Parc de la Mediterranée; Medilink, Madison, France) was used to examine the lumbar spine to ensure there was no osteoporosis. All soft tissue was dissected from the specimen, except the hip joint capsule, sacroiliac ligament, sacrospinous ligament, sacrotuberous ligament, and arcuate pubic ligaments. The pelvis with both hips was mounted on a BOSE Electroforce 3250‐AT (BOSE, Eden Prairie, USA) with the proximal femoral stem and was fixed in all degrees of freedom to emulate a person in a standing position. The study design and protocol were approved by the institutional review board.

According to previous research, six points on the surface of the pelvis were chosen as the measuring points16; these points were located along the pelvic biomechanical load transferring path (Fig. 1): the point on the sacral first vertebrae; the point on the sacral first vertebrae near the sacroiliac joint; the point on the sacral second vertebrae; the point on the ilium sacroiliac joint as high as the sacral second vertebrae; the midpoint of the iliopectineal line; and the point near the ischial notch. Resistance strain gauges were pasted on the surface of the pelvis to obtain the strain values of these chosen points. A digital static resistance strain indicator (type YE2539; Lianneng Electric Technology, Jiangsu, China) was used to obtain the strain at each level of resistance. The specimen was treated repeatedly with a vertical force preloaded from 0 to 500 N, and this was repeated three times to eliminate the impact of pelvic relaxation. Then, a pure vertical compressive load of 500 N was gradually loaded on the top of the S₁ vertebral body (loading strain rate 10 N/s). At the six chosen points, the strain was recorded when the load force was 100–500 N (with an interval of 100 N). The load was repeated three times, and an average of the strains was taken.

Figure 1.

The six points on the surface of the pelvis were selected as the measuring points, which were located along the pelvic biomechanical load transferring path.

Geometric Model Reconstruction

A geometrical model of the pelvis and femur was developed from computed tomography (CT; Philips Brilliance 64 CT; Philips Healthcare, the Netherlands) data of a healthy woman (40 years old, 160 cm, 63 kg, no known history of bone disease or surgical interventions), with slice thickness of 0.3 mm (947 slices). The volunteer underwent a CT scan with both lower extremities in a neutral position, and then the CT data were imported into Mimics 10.0 medical image processing software (Materialise, Belgium) to construct the 3D surface mesh of the intact pelvis and the proximal third of the femurs; data were exported as an ASCII stereo lithography (STL) file. The file was imported into Geomagic Studio 12 (Geomagic, Research Triangle Park NC, USA), reverse engineering software, which input the data into the FE analysis software Abaqus v.6.11 (Dassault Systemes Simulia, Providence, RI, USA). The ligaments and cartilage on the sacroiliac joint and hip joint could not be detected by CT, so they were not included.

Finite Element Model Construction

We selected C3D4, a type of 4‐nodetetrahedral element, to mesh the trabecular bone, and the cortical bone was simulated on both the sacrum and the iliac bones by adding a 2.0‐mm thick shell element. Young's modulus and Poisson's ratio were 400 MPa and 0.2 for trabecular bone, and 18,000 MPa and 0.3 for cortical bone, respectively. For the sake of simplifying the femoral part of the model, the femoral bony component was set as homogeneous cortical bone, and Young's modulus and Poisson's ratio were 18,200 MPa and 0.38, respectively. All material properties of bone were chosen in accordance with a previous study and are listed in Table 1 10, 12.

Table 1.

The material properties of pelvic bone in models

Material of bone	Properties		Element number	Element type
	Young's modulus (MPa)	Poisson's ration
Cortical bone	18,000	0.3	3210320	Shell element
Trabecular bone	400	0.2	102356	Tetrahedral element
Interpubic disc	5	0.45	693	Tetrahedral element

The main ligaments, namely the sacroiliac ligament, the sacroiliac interosseous ligament, the sacrospinous ligament, the sacrotuberous ligament, and the arcuate pubic ligaments, were incorporated based on the significance of the pelvic ring ligaments on pelvic biomechanics determined in previous studies. Because the main ligaments of the pelvis were too complex for constructing 3D models, each of the ligaments was modeled as a Truss element, a type of element that exhibit stiffness only with tension. The origin and insertion of each bundle of ligaments were chosen in accordance with a combination of anatomic data available in the literature. The sacroiliac interosseous ligament, which acted as the tension member in a suspension bridge, was covered by the posterior sacroiliac ligament and connected the ilium tuberosity and sacral tuberosity17; as a consequence, the long and short posterior sacroiliac ligaments played an important role in maintaining the normal position of the sacrum in the pelvic ring. The sacrospinous ligaments originated from the lateral edge of the sacrum to the ischial spine and resisted external rotation of the hemi‐pelvis, whereas the sacrotuberous ligaments resisted both rotational forces and shearing forces in the vertical plane17, 18. The material properties of the selected ligaments were assigned based on previous research and are listed in Table 2 15, 19.

Table 2.

The material properties of main ligaments in models

Material of ligament	Stiffness coefficient (N/mm)	Element number	Element type
Anterior sacroiliac ligament	700	20 × 2	Truss
Sacroiliac interosseous ligament	2800	20 × 2	Truss
Long posterior sacroiliac ligament	1000	20 × 2	Truss
Short posterior sacroiliac ligament	400	16 × 2	Truss
Sacrospinous ligament	1400	20 × 2	Truss
Sacrotuberous ligament	1500	15 × 2	Truss
Superior pubic ligaments	500	10	Truss
Arcuate pubic ligaments	500	10	Truss

In the anterior part of the pelvis, the symphysis pubis was covered with superior and arcuate pubic ligaments; its inter‐space was occupied by the interpubic disc, which was represented as continuity and connected both sides of the ilium. The interpubic disc meshed into a tetrahedral element, and Young's modulus and Poisson's ratio were 5 MPa and 0.45, respectively.

Loading and Boundary Conditions

The present study evaluated the effects of the boundary condition of the hip joint on load transmission from the sacrum to the lower extremities while in a standing position on both feet. Thus, three different boundary conditions of the hip joint were defined for the FE analysis: Models I, II, and III. (Fig. 2)

• Model I: Two connecting pairs were constructed on the interfaces of the femoral head and acetabulum with contact condition on both sides to observe the effects of the synovial condition in the hip joint on load transmission. Meanwhile, the condition in the sacroiliac joint was the same in all three models. In the sacroiliac joint, the friction coefficient of the connecting pair was set at 0.015, and the initial penetration was set at 0.01 mm20. However, in the hip joint, two frictionless finite sliding contact pairs were defined on both sides, and the ends of the proximal femur were fixed.

• Model II: The model included the sacrum, ilium, proximal femur, and main ligaments, which was similar to Model I; the sacroiliac joint interfaces connected with each other as well, and the parameter of the friction coefficient at initial penetration was also the same as in Model I. The difference between Model II and Model I was that the hip joints of both sides were assumed to be fused rather than under synovial conditions.

• Model III: This model was constructed under the assumption that the femur may reduce local stress concentrated in the acetabulum; thus, the femur was eliminated and acetabular cups were fixed in all directions to simulate a standing position. All the other structures were kept the same as those in the other two models. The contact condition of the sacroiliac joint interfaces was also the same as in Model I.

Figure 2.

The finite element (FE) model of human pelvic with main bone structures (proximal femur, sacrum, and iliac bone) and main ligaments (the sacroiliac ligament, sacroiliac interosseous ligament, sacrospinous ligament, sacrotuberous ligament, and arcuate pubic ligament). In Model III, the proximal femur was removed.

Model I contained all main bone structures and ligaments, and had a similar boundary condition to the biomechanical experiment of the pelvic specimen. Therefore, Model I was used for the validation of the specimen biomechanical experiment.

In each model, six vertical forces (100–600 N with an interval of 100 N) were applied on the top surface of the first sacral vertebral body to simulate body weight in the case of standing on both feet.

Results

Validation of the Finite Element Model

The average strain values in the biomechanical experiments of the pelvic specimen are listed in Table 3. Based on the data of the six measuring points under a 500 N vertical load, it was found that the minimal strain value of those selected points was 13.8 × 10⁻⁴, which was near the iliosciatic notch, and maximum strain value was 54.13 × 10⁻⁴, which was at the first sacral vertebrae; the difference in the strain values between the two points was 40.33 × 10⁻⁴.

Table 3.

The average strain values in the biomechanical experiments of the pelvic specimen

	100 N	200 N	300 N	400 N	500 N
1	10.73 × 10−4	21.9 × 10−4	32.2 × 10−4	43.8 × 10−4	54.13 × 10−4
2	1.4 × 10−4	4.6 × 10−4	9 × 10−4	14.2 × 10−4	19.6 × 10−4
3	8.5 × 10−4	15.4 × 10−4	21.6 × 10−4	27.3 × 10−4	32.7 × 10−4
4	4.4 × 10−4	10 × 10−4	14.5 × 10−4	17.8 × 10−4	19.2 × 10−4
5	3.1 × 10−4	6.3 × 10−4	9.9 × 10−4	13.5 × 10−4	15.4 × 10−4
6	2.4 × 10−4	5.2 × 10−4	8.3 × 10−4	11.3 × 10−4	13.8 × 10−4

To validate the developed FE model, we compared the strain values of specimen experiments with that of the pelvic FE model for each corresponding point by linear regression analysis. The results of the linear regression analysis are shown in Fig. 3. The regression equation and correlation coefficient were obtained as follows:

Figure 3.

Results of the linear regression analyses of the finite element analysis and mechanical experiment. The x‐axis represents finite element‐simulated equilibrium strains, and the y‐axis represents the strain values in the mechanical experiment. The regression equation showed that the finite element analysis results had a strong correlation with the experimental results (R ² = 0.979).

$$Y=1.037X-1.114,R^2=0.979.$$

The R ² represents the correlation coefficient of the regression equation.

Furthermore, the strain values under different loads in each point of both the biomechanical experiment and the pelvic FE model were compared by linear regression analysis. The regression equation and correlation coefficient are displayed in Table 4. In the linear regression analysis, all P‐values in the analysis of variance of five regression equations were less than 0.05, which meant all regression equations had statistical significance; thus, there was a linear correlation between each of the load conditions and the biomechanical experiment and pelvic FE. Meanwhile, with the increase of load levels, the correlation coefficient of the linear regression function increased from R ² = 0.854 in 100 N to R ² = 0.988 in 500 N, indicating that a better fitting degree of linear regression function was obtained with the higher load than with the lower load. In addition, we found that the theoretical slope of the regression equation had a tendency to decrease with the load level, indicating that the FE analysis was similar to the experiment with the load increase.

Table 4.

The regression equation and correlation coefficient under different vertical load

100 N	200 N	300 N	400 N	500 N
Y = 0.847X + 0.674	Y = 0.950X − 0.072	Y = 1.055X − 1.717	Y = 1.074X − 2.626	Y = 1.098X − 2.592
R 2 = 0.854	R 2 = 0.931	R 2 = 0.963	R 2 = 0.979	R 2 = 0.988

Stress Analysis under Different Boundary Conditions

To explore more accurate load transmission of the pelvis under different boundary conditions, we added another three anatomical locations as measuring points on the surface of the pelvis: the dome region of the acetabulum, the point on the posterior acetabular wall, and the point on the anterior of the pelvis near the pubic symphysis. To better compare the biomechanical transfer, in addition to the six measuring points chosen in the validation test, those additional points were marked as points 7, 8, and 9, respectively, resulting in a total of nine measuring points in the FE analysis.

For each point, the von Mises stress was recorded. Unlike in the biomechanical experiment, the recorded stress value could not be statistically analyzed because the means and standard deviations were unavailable. Thus, we compared the change in absolute value between any two models to evaluate the differences in stress results among the three models.

The three FE models demonstrated diverse stress results after application of 100–500 N at 100‐N intervals (Fig. 4).

Figure 4.

The comparison of von Mises stress values of the nine measurement points among the three finite element models with 100 N loading (A), 300 N loading (B), and 500 N loading (C); the stress values in the point near the pubic symphysis in Model III were 48.52 and 39.1% lower, respectively, in comparison with Models I and Model II; on the dome region of the acetabulum in Model II and Model III were 103.61 and 390.53% less than those of Model I; the posterior acetabular wall stress values of Model II were 197.15 and 305.17% higher than those of Model I and Model III, respectively. *The points are represented as follows: (i) the point on the sacral first vertebrae; (ii) the point on the sacral first vertebrae near the sacroiliac joint; (iii) the point on the sacral second vertebrae; (iv) the point on the ilium sacroiliac joint as high as the sacral second vertebrae; (v) the midpoint of the iliopectineal line; (vi) the point near the ischial notch; (vii) the point on the dome region of the acetabulum; (viii) the point on the posterior acetabular wall; and (ix) the point on the anterior of the pelvis near the pubic symphysis.

First, comparing Model II with Model I, it was found that the stress values of points on the posterior acetabular wall in Model II were 1.97 times higher than those of the latter; furthermore, the von Mises stress values were less concentrated on the dome region of the acetabulum in Model II; stress values were approximately 103.61% less than those of Model I.

Second, comparing Model III with Model I, it was found that the stress values in the point near the pubic symphysis were 48.52% less than those of Model I; on the dome region of the acetabulum and the posterior acetabular wall, the stress values of Model I were 390.53% and 105.17% higher than those of Model III, respectively.

Third, comparing Model III with Model II, it was found that on the dome region of acetabulum and the posterior acetabular wall, the stress values of Model II were 286.92 and 305.17% higher than that of Model III; however, in the point near the pubic symphysis, the stress values was 39.1% less than that of Model II.

Finally, in addition to the abovementioned measuring points, there were no significant differences among the other points, indicating differences of less than 20%.

With regard to the load transmission, the stress distribution was transmitted along the iliopectineal line in all three models, and the maximum stress, which was located on the sacroiliac interosseous ligament, was similar among the three models (Fig. 5).

Figure 5.

The distribution of von Mises stress in three finite element (FE) models under 500 N vertical load. The stress distribution along the iliopectineal line, on the posterior acetabular wall, and on the dome region of the acetabulum changed significantly in the comparison of three FE models: (A, B) Model I (anterior and posterior views); (C, D) Model II (anterior and posterior views); and (E, F) Model III (anterior and posterior views).

Discussion

The pelvis is one of the major anatomical components that transmit body weight to the lower extremities and exhibits complicated biomechanical features. The pelvis consists of complex bony material and has an irregular geometric structure, which is cumbersome to the model calculation and often results in large variation of strain across the structure in FE analysis21. As a result, to simplify analyses, some previously established models did not include the sacrum or femur, and their articular surfaces were assumed to be rigidly connected; this could impact prediction accuracy. Additional issues need to be addressed in pelvis models, such as mesh generation, the assigning of boundary conditions, and material properties; therefore, any evaluation of parameters in the pelvis may contribute to the improvement of FE model simulation. Meanwhile, the FE model boundary condition is a crucial factor in exploring the biomechanical behavior of the pelvis16. However, there is currently no uniform standard for establishing boundary conditions in pelvic FE models. The objective of this study was to study the effect of boundary conditions on predictions of pelvic biomechanics and to establish an intact pelvic FE that included the sacrum, ilium, femur, and major ligaments.

Validation of the Finite Element Model

With the purpose of predicting the simulation of stresses and strains more accurately in FE analysis, the validation of FE predictions is mandatory, especially while the FE model is subjected to clinically relevant investigations. In previous reports, several validation methods were used to validate FE models of the pelvis. Eichenseer et al. adopted load displacement data from cadaveric studies reported by Miller et al. to validate their FE model11, 22; however, the pelvic or anatomical structure force‐displacement curve was too global to be suitable for validation of the FE model, and the strain distributions were not validated. Hao et al. designed the equilibrium strain as the validating criterion to verify the FE model, although the assumption of the density–elasticity relationships might not be accurate in the material assignment16. Furthermore, Shi et al. validated their FE model by comparing their results with those reported in other studies, and this might have been affected by the different assignment of bone material properties in each model6.

In the present study, we validated the FE model by utilizing a linear regression analysis to compare the strain of the specimen mechanical experiment from multiple points with the two‐phase bone material FE model. Based on the linear regression function, the slope was 1.062, which meant the strain result in the FE model was 1.062 times higher than that in the specimen experiment. This shows that the results of the FE model and biomechanical model are basically equivalent. Meanwhile, the high correlation coefficient of 0.979 obtained in our model was better than those in many other reports, and indicated a strong correlation between the FE analysis and the experimental results23, 24. Our model validation revealed that prediction accuracy was partly dependent on the boundary conditions in the FE model; the high correlation coefficient demonstrated that the model with contact conditions and the proximal femur could realistically simulate the mechanical transmission of the pelvis. In the present study, strain values corresponding to each load in both the mechanical experiments and the pelvic FE model were compared by linear regression analysis. As further analysis showed, we found that the correlation coefficient increased slightly from 0.854 to 0.988 with increasing load force, and the slope of the linear regression function increased from 0.847 to 1.098. The differences in the correlation coefficients and the slopes might have been caused by the poor linearity of the pelvic specimen with low loading levels; thus, when the pelvic bones and ligaments were compressed or tensed completely under a higher load force, the result was more similar to the FE prediction.

Influence of Connecting Conditions in the Hip Joint

The hip joint in situ is a frictionless joint; the contact conditions of the hip joint in the FE analysis were similar to those of reality. However, most reports on pelvic FE analysis assumed the hip joint connecting surfaces were fused together with only a negligible displacement across the hip joint during processing, which was not a good representation of the load transmission of the pelvis as a transference center. Therefore, in this study, the influence of connecting conditions in the hip joint was analyzed in the FE model, and two connecting conditions in the hip joint were developed: contact condition (Model I) and tie condition (Model II). In Model I, the dome region of the acetabulum was increased significantly compared with that of Model II, while the stress in the posterior acetabular wall was reduced compared with that in Model II. The results revealed that articular surfaces assumed to be rigidly connected could cause stress concentration on the posterior acetabular wall and adjacent areas. In normal human acetabular biomechanical distribution, the dome region of the acetabulum is the main mechanical bearing part, and the posterior wall of the acetabulum is the weak part of the acetabulum, so the stress in simulation should be concentrated on the dome region of the acetabulum instead of the posterior wall of the acetabulum; this is more in line with the transfer of human biomechanics. Thus, we believe that Model I better simulated the reality of the human biomechanics.

Meanwhile, we found that the maximum stress among the nine measuring points was observed in the dome region of the acetabulum in Model I. This phenomenon corresponded to the dome region of the acetabulum bearing the load transfer from the upper limbs to the femur. In both models, the interosseous sacroiliac and sacrotuberous ligaments had higher stress than bone structures, which was similar to the results reported by Bohme et al.25.

Influence of the Femur in the Pelvic Finite Element Model

In previous research, stress and strain were found to be associated with changes in boundary conditions23. Li et al. investigated the influence of fixed and released boundary conditions on the pubic symphysis in simulating lateral pelvic impact injury26. Phillips et al. demonstrated that adding the ligaments and muscles in the boundary conditions of the FE model could reduce the stress concentration on the cortical bone surface15. The proximal femurs are actually involved in the transformation of the load‐transmitting pattern from the upper body through the pelvis to the lower extremities. However, to our knowledge, there have been few studies on the effects of boundary conditions on pelvic biomechanics, especially boundary conditions on the femur.

Therefore, in Model III, the acetabulum in all degrees of freedom was fixed as the boundary condition, and the proximal femur was removed. Although most of the stress distributions in Model III were similar to those in Models I and II, there were still differences in stress on the superior ramus, the posterior acetabular wall, and the dome region of the acetabulum; values in Model III were less than those in the other two models. Hao et al. found that in a model with the acetabulum fixed in all directions and the femur removed, the stress values on the acetabular inner plate were twice those obtain in a model that included the proximal femur, which were similar to our results16. Therefore, the assumptions of ignoring the femur and fixation of the acetabulum in all degrees of freedom instead of femur fixation can influence the stress distribution in the area around the hip joints and pubis bone. If the model concentrates on the acetabulum and pubis bone, which simulate the pelvic ring before surgery or prosthesis implantation, the model with its boundary conditions at the proximal femur could obtain more accurate prediction results.

The major potential limitation of this study was the lack of muscle elements due to the lack of sufficient muscle data, so muscle forces were not taken into consideration in the models. Furthermore, cartilage in the joint could not be detected clearly by CT; thus, the hip and sacroiliac joint cartilage was not included in the model, and this could have influenced the stress distribution around the acetabulum and sacroiliac joint.

In conclusion, we developed a 3D FE model of the pelvis including intact bone structures and main ligaments to explore the effects of boundary conditions, especially regarding the effects of the proximal femur structure on pelvic biomechanical predictions. We suggest that the determination of boundary conditions and connect conditions of pelvic FE models should be more closely aligned with reality, especially for studies related to clinical applications.