Triangular Mechanical Structure of the Proximal Femur

Objective The mechanical high modulus structure of the proximal femur could guide clinical surgical treatment and instrument design of proximal femoral fractures. The purpose of this study is to analyze and verify the mechanical structure of the proximal femur. Methods A total of 375 patients with intertrochanteric fractures were imaged using computed tomography (CT) scans. Patients were grouped according to age and sex. Cortical and medullary cavity parameters (cortical thickness [CTh], cortical mean density [CM], upper‐lower diameter length [ULL], and medial‐lateral diameter length [MLL]) were measured at eight planes. Six proximal femoral finite element models of different sexes and ages were constructed. To verify the measurement results, Abaqus was used to implement the force load to describe the von Mises stress distribution, and the maximum von Mises stress values of each wall of the proximal femur were compared. Results The CTh values of the lower and upper walls were higher than those of the anterior and posterior walls of the femoral neck (p < 0.05). The CM values of the lower and upper walls were higher than those of the anterior and posterior walls of the subcephalic and middle femoral neck (p < 0.05). The ULL value gradually increased from the subcephalic region to the bottom (p < 0.05). The CTh and CM values of the medial and lateral walls were higher than those of the anterior and posterior walls in the femoral trochanteric region (p < 0.05). The MLL value decreased gradually from the plane 20 mm above the upper edge to that 20 mm below the vertex of the femoral lesser trochanter (p < 0.05). The von Mises stress was concentrated on the upper and lower walls of the femoral neck and on the medial and lateral walls of the femoral trochanteric region. The maximum von Mises stress values of the upper and lower walls were higher than those of the anterior and posterior walls of the femoral neck. The maximum von Mises stress values of the medial and lateral walls were higher than those of the anterior and posterior walls in the femoral trochanteric region, except for the plane 20 mm above the upper edge of the femoral lesser trochanter. Conclusion The bone mass of the proximal femur presented a triangular high‐modulus distribution, which bore the main stress of the proximal femur. The triangular mechanical structure provides a guideline for the surgical strategy and instrument design of the proximal femur.


Introduction
W ith the increasing age of the population, the number of patients with hip fractures will reach 4.5 million by the year 2050, and proximal femoral fractures account for approximately 90% of hip fractures 1 . Surgery should be performed as soon as possible to avoid complications caused by long-term bed rest 2 . However, studies have shown that loss of fracture reduction occurs in 40% of patients postoperatively 3 . Thus, ensuring the success of an operation has attracted considerable attention.
The AO (Arbeitsgemeinschaftfür Osteosynthesefragen) principle of fracture management emphasizes that the primary principle of surgical treatment is to realize the mechanical integrity of the fracture site through internal fixation instruments in line with the local mechanical structure 4 . An accurate understanding of the mechanical structure is very important for the treatment of proximal femoral fractures. Through analogy and theoretical analysis, Ward 5 , Culmann 6 , and Koch 7 regarded the cantilever structure as the standard mechanical structure that was applied to guide the treatment of proximal femoral fractures. Based on this mechanical structure, an extramedullary fixation system, such as a Jewett-angle steel plate 8 or sliding screw plate 9 , has been developed and utilized in clinical treatment. However, with a further understanding of the mechanical structure of the proximal femur, it was found that the extramedullary fixation system, as an eccentric design, has a long force arm, resulting in a high stress concentration at the tip of the head nail. The junction between the head nail and the main nail exceeds the yield stress of the bone and fixation system, which is the cause of frequent complications including nail breakage, cut-out, and varus collapse 10 . Therefore, intramedullary fixation systems, such as gamma nail 11 , Proximal Femoral Nail Antirotation (PFNA) 12 , and InterTan 13 , have been developed to improve the surgical effect. However, most existing intramedullary fixation systems cannot reconstruct the medial support structure, which is important for the treatment of proximal femoral fractures, especially unstable fractures 14 . A postoperative complication rate of up to 20.5% was reported for the existing intramedullary fixation systems 15 .
Therefore, we propose that the characteristics of the mechanical structure of the proximal femur have not been clarified. Through retrospective analysis of 53 failed cases of fracture treatment, we found that the failure of any unilateral reconstruction of the medial, lateral, and upper walls of the proximal femur will lead to the failure of hip fracture fixation 16 . For the treatment of femoral intertrochanteric fractures, we designed a medial sustainable nail with a triangular structure, exhibiting a better biomechanical performance when compared with that of PFNA in reducing displacement and anti-varus 17 . In addition, for displaced femoral neck fractures, a medial anatomical buttress plate capable of forming a triangular structure was designed with more stable properties with respect to the stress distributions, stress peaks, and Z-axis displacements, and all healing was achieved in the treatment of 15 patients with femoral neck fracture nonunion 18,19 . Thus, we speculated whether there was a triangular mechanical structure in the proximal femur, so that any strategy realizing complete reconstruction of the triangular structure could be successful.
In this study, the anatomical structural parameters of the proximal femoral cortex bearing 47%-80% of the overall stress 20,21 were measured in 375 patients with intertrochanteric fractures, and the normal stress distribution of the proximal femur in six patients was analyzed. The aims of this study were: (i) to investigate the high-modulus structural morphology of bone mass in the proximal femur and (ii) to investigate the mechanical effect of the above structure.

Patients
From September 2009 to March 2017, 375 patients with intertrochanteric fractures who underwent surgery at the Chinese PLA General Hospital were included in our study to assess cortical structural characteristics of the proximal femurs. This study was approved by the institutional review board of our hospital (S2020-114-04).
Inclusion criteria were patients (1) older than 40 years, (2) with low-energy external injuries such as a simple fall from standing height or lower and twisting injuries 22 , (3) with CT scanning range from the upper edge of the first lumbar vertebra to 8 cm below the lesser trochanter of the femur, and (4) with imaging follow-up for at least 1 year. The exclusion criteria were as follows: (1) femoral head necrosis, (2) severe hip osteoarthritis or rheumatoid arthritis, (3) hip joint or femur deformity, and (4) history of contralateral hip fracture.
All CT data were collected using one CT machine (Siemens AG, Erlangen, Germany) with the same scanning parameters (120 KV, 210 mA; collimation, 4 mm; table speed, 3-5 mm/s; slice thickness 1.2 mm with 1.5 mm interval). In Mimics 20.0 (Materialize Inc., Leuven, Belgium), the osseous material was isolated via thresholding at 0-350 Hounsfield units, which could clearly display the outline of bone.

Anatomical Measurement of the Proximal Femur
In the femoral neck region, cortical bone was defined as the upper, lower, anterior, and posterior walls. The bottom of the femoral neck was determined using the method described by Zhang 23 , the subcephalic part of the femoral neck was determined according to Sparks 24 , and then the middle part of the femoral neck was determined ( Figure 1). In planes 1-3, blue lines (the longest line between the upper and lower walls) 112 were drawn with lengths measured as ULL and green lines (the longest line between the anterior and posterior walls) were drawn. The regions of the circle tangent to the cortical bone (where the blue and green lines intersect with the cortical bone) were identified as the measurement regions (Figure 2, P1-3).
In the trochanteric region, the cortical bone was defined by the medial, lateral, anterior, and posterior walls. Using the method described by Zhang 25 , Planes 4-8 were determined ( Figure 1). In each plane, the red line (the longest line between the medial and lateral walls) was drawn with the length measured as MLL, and the yellow line (the longest line between the anterior and posterior walls) was drawn. The measurement regions of the medial, anterior, lateral, and posterior walls in planes 4-8 were defined to extract the relevant anatomical parameters (Figure 2, P4-8). The diameter and mean Hounsfield unit values of these regions were regarded as the cortical thickness (CTh) and cortical mean density (CM) values, respectively, of each wall.

Finite Element Analysis of the Proximal Femur
Finite element analysis was used to describe the stress distribution of the proximal femur in the different age and sex groups. Using the RAND function in Microsoft Excel 2016, the same-sex patients in the three age groups were sorted by the value received from the random number generation CT data were imported into Mimics 20.0, and threedimensional models of their proximal femurs were constructed through the protocols of thresholding, region growing, mask editing, polyline calculation, cavity filling from polylines, etc. The threshold units ranged from 226 to 3071. The "region growing" tool was utilized to split the segmentation as desired and to remove floating pixels. The polyline calculation was used to edit the details of the proximal femur. After the procedures above, six three-dimensional reconstruction models of the proximal femur were generated and further imported into Geomagic Studio 14.0 (Geomagic, Inc., Research Triangle Park, NC, USA) in STL format for modification. Using the material function of the FEA module in the Mimics software, the density and elastic modulus of the proximal femoral bone structure were automatically defined according to the CT value of the bone structure 26 , and the Poisson's ratio of all bone structures was defined as 0.3 27 . Finally, these models were imported into Abaqus (Simulia, France) to generate the finite element models. Subsequently, a mesh sensitivity analysis was performed to determine the correct size of the elements with the lowest computational cost 28 . For all the FE models, element sizes (mm) of 0.65-7.00 (with an interval of 0.05) were used. We used the bisection method to perform a difference analysis of the calculated maximum von Mises stress. The formula is as follows: where S is the difference ratio, X is the maximum von Mises stress value, and n is the mesh size. In total, 17 difference ratios were obtained. We performed linear regression analysis on the difference ratio results and obtained the regression curve Y = 0.0242X + 0.0017, where Y represents the difference ratio, and therefore the FE accuracy; X represents the mesh size. When Y is set as 0.05, we obtain X = 2, which means that a 2 mm mesh size guarantees an accurate result; therefore, we set the mesh size to 2 mm. Warpage, aspect ratio, and Jacobian elements were used to assess the mesh quality. In all models used in this study, the element Jacobian was above 0.7, warpage was less than 5, and the aspect ratio was less than 5, which indicates a good mesh quality. Linear tetrahedral elements (C3D4) were applied to the finite element models. All nodes on the surface of the distal femoral shaft were constrained with 0 of freedom to avoid rigidbody motion during the analysis. This study simulated the force loading on the hip during the stance phase of walking. According to the body weight of the volunteers and the literature 29 , the weight-bearing forces of the femoral head of models 1-6 were set to 1920N, 1800N, 1350N, 1980N, 1800N, and 1800N respectively, corresponding to 300% body weight. The net force was applied to the femoral head at 9 posteriorly on the sagittal plane and 10 laterally in the coronal plane 30 . The von Mises stress distribution in the proximal femur is also shown. As in the measurement sites of anatomical parameters of each section of the proximal femur, the maximum von Mises stress value of the same region in eight sections of models 1-6 was recorded to reflect the stress in different walls.

Reliability Study
The intraclass correlation coefficient (ICC) was used to assess the reliability of the measurement methods established in this study. PASS 11.0 (NCSS LLC. Kaysville, UT) was used to estimate the sample size for the intraclass correlation coefficient. The design solved for sample size: A random sample of several subjects who were each measured three times produced a two-sided 95% confidence interval with a width of 0.1 when the estimated intraclass correlation was 0.950. Data were analyzed using a two-way random effects ANOVA model. Given these settings, the minimum sample size was calculated to be 14. To increase the reliability, 20 femurs were selected for the reliability study. One independent physician (GXX) repeated the measurements of 70 proximal femoral parameters three times in a randomized order with a minimum of 24-h intervals to assess intraobserver reliability. Three other physicians (JTL, HZ, and CX) independently measured the same parameters on the same samples in a random order to assess interobserver reliability. Two-way random and two-way fixed models were selected to assess inter-and intra-observer reliability.     Tables 1, 2). The ULL values increased gradually from plane 1 to plane 3, regardless of sex and age (p < 0.05) ( Figure 5A, C; Table 5).
In the femoral trochanteric region, the CTh values of the medial and lateral walls were higher than those of the anterior and posterior walls in planes 5-8 regardless of sex and age (p < 0.05), and the highest CTh value was found in the lateral wall in plane 4, regardless of sex and age (p < 0.05) ( Figures 3D-H and 4D-H; Tables 3, 4). The spatial distribution of CM values was similar to that of CTh, regardless of sex and age. (Figures 3L-P and 4L-P; Tables 3,  4). The MLL values decreased gradually from plane 4 to plane 8, regardless of sex and age (p < 0.05) (Figure 5B, D; Table 6).

Finite Element Analysis of the Proximal Femur
In the simulation of models 1-6, the von Mises stress was concentrated on the upper and lower walls in planes 1-3 and concentrated on the medial and lateral walls in planes 5-8 ( Figure 6).
In the femoral neck, the maximum von Mises stress values of the upper and lower walls were higher than those of the anterior and posterior walls in planes 1-3 ( Figure 7A-C and Table 7). In the femoral trochanteric region, the maximum von Mises stress values of the medial and lateral walls were higher than those of the anterior and posterior walls in planes 5-8 ( Figure 7E-H and Table 7).
The number of elements in models 1-6 is shown in Supplementary Table 1.

Intraobserver and Interobserver Reliability
The ICC showed values of >0.80 for interobserver and intraobserver reliability (Supplementary Table 2).

Discussion
I n this study, we verified the high-modulus mechanical structure of the proximal femur. Our major findings were as follows: (1) the high-modulus structure of cortical bone varied dramatically across cross-sections independent of age and sex, with prominent cortical thickening, densification, and stress concentrating at the upper, medial, and lateral walls; (2) the above three walls converged to a triangle in the coronal plane, independent of age and sex.

The High-Modulus Structural Morphology of Bone Mass in the Proximal Femur
The proximal femur consists of cancellous and cortical bone, bearing 15%-53% and 47%-85% of the stress load, respectively 20,21 . However, the previous understanding of the mechanical structure was mainly based on the morphology of the cancellous bone 31 , without investigation of different sex and age groups. In this study, the anatomical parameters of the proximal femoral cortical bone in patients of different ages and genders were measured, and heterogeneity of the cortical bone structure was found. In the femoral neck region, the CTh and CM values of the upper and lower walls were higher than those of the anterior and posterior walls, regardless of sex and age. The only exception was that the upper walls exhibited the lowest CM value in plane 3, which might be caused by the anatomical structure of the junction between the upper wall and greater trochanter, where the principal tensile trabecula is separated from the upper cortical bone 32 . In the trochanteric region, the medial and lateral walls had higher CTh and CM values than those of the anterior and posterior walls, regardless of sex and age.
Previous studies have demonstrated that the cortical geometry and microarchitecture of the proximal femoral structure display regional heterogeneity. Khoo 33 found that the mineral masses of the upper and lower segments were larger than those of the anterior and posterior femoral necks. Through a study of the histomorphometric properties of the subtrochanteric femoral region, Tong 34 found that the cortical widths in the medial and lateral quadrants were significantly higher than those detected in the anterior and posterior quadrants. However, the cortical bone density was not measured in these studies.
According to Wolff's law 35,36 , local bone tissue normally adapts its density and structure to the current loads, so it can be inferred that the medial, lateral, and upper walls bear the main stress of the proximal femur regardless of sex and age. Concurrently, we found that ULL values increased from the inside to the outside, and the MLL values decreased gradually from top to bottom regardless of sex and age; therefore, it can be inferred that the medial, lateral, and upper walls of the proximal femur converge in a triangular structure in the coronal plane and bear the main stress.

The Mechanical Effect of Triangular High-Modulus Structure in the Proximal Femur
The finite element analysis in this study found that the main stress in the proximal femur was concentrated in the upper, medial, and lateral walls, which confirmed that the triangular structure bears the main stress in the proximal femur.
Through finite element analysis of the femoral neck, Nawathe 20 found that stress was mainly concentrated in the upper and lower lateral walls. Endo 37 proposed that stress was mainly concentrated in the medial and lateral walls using finite element analysis of the femoral shaft. However, the above studies were based on the mechanical verification of individuals and did not consider the proximal femur as a whole for mechanical testing, nor did they investigate the mechanical structure of the bone cortex. According to the stress distribution law of the proximal femur, the upper, medial, and lateral edges of the proximal femoral coronal plane are defined as follows: (1) the upper edge is the high-stress area of bone along the upper part of the femoral neck, which runs from the circle point of the femoral head to the cortex of the vastus lateralis ridge; (2) the medial edge is the high-stress area of bone extending from the femoral head, along the medial cortex to the distal end; and (3) the lateral edge runs from the vastus lateralis ridge along the high-stress area of the lateral wall to the distal end. At the same sites, the cortical bone fusing with principal compressive, principal tensile, secondary compressive, secondary tensile, and greater trochanter trabeculae forms a triangular mechanical structure to maintain mechanical stability ( Figure 8). Meanwhile, as the junction of the upper and lateral walls, the greater trochanter realizes the effective transmission of tensile stress, which has not been mentioned in previous literature, to the best of our knowledge.
The structure has obvious mechanical advantages: (1) the medial part forms the oblique support of the proximal femoral cantilever structure, greatly reducing the bending stress and deflection of the structure; (2) the upper part acts as a connection between the medial and lateral edges of the proximal femur, resisting the bending moment caused by physiological loads; and (3) the lateral part can effectively reduce the sliding and deflection of the femoral neck under physiological loads. Under a physiological load, the complete mechanical triangular structure can effectively reduce the bending moment, balance the shear force of the physiological load, realize a balanced distribution of stress in the structure, and maintain balance and stability. Therefore, we propose a theory for triangular hip stabilization reconstruction. We believe that the treatment principle of hip fracture is to reconstruct the overall structure to achieve a low bending moment and self-balanced stress distribution bone on the fixation and skeletal system, and further obtain firm mechanical stability for patients to walk. During the operation, if any part of the triangular structure is not well reconstructed, it cannot form a strong mechanical stability and will lead to the collapse of the structure, resulting in treatment failure. When fractures occur, the integrity of the mechanical structure is destroyed, which leads to mechanical instability. Complete reconstruction of the triangular mechanical structure is key to a successful operation. Any technical means to achieve the overall reconstruction of the triangular mechanical structure can achieve mechanical stability in this region. Ye et al. 38 used a hollow screw combined with a medial support plate to treat patients with femoral neck fractures, and all patients healed without any postoperative complications. Through biomechanical experiments, Kunapuli 39 confirmed that the overall reconstruction of the three parts of the proximal femur had a significant advantage in terms of mechanical stability. Based on this mechanical structure, we designed a series of fixation instruments that have been used in the treatment of hip fractures with satisfactory clinical results 18,40 . Both theory and practice have proven the reliability of a triangular mechanical structure.

Strengths and Limitations
Through the combination of anatomical measurement and finite element analysis, our study confirmed the existence of a triangular mechanical structure of the proximal femur from the perspective of morphology and function and provided a new perspective for the exploration of the mechanical structure of human bones. This study had some limitations. First, the sample size of some age groups was limited. The inclusion of additional groups could strengthen the representation in the study. Second, the subjects in this study were patients with femoral intertrochanteric fractures. A normal population should be included to ascertain the universality of the results.

Conclusions
In this study, we identified a triangular structure of bone mass concentration in the proximal femur using anatomical measurements and further confirmed that this structure bears the main stress transmission in the proximal femur through finite element analysis. The triangular mechanical structure proposed in this study will help guide future improvements in the surgical strategy and instrument design of the proximal femur.

Etics Statement
T he ethics statement has been provided in Methods section: This study was approved by the institutional review board of our hospital (S2020-114-04).

Supporting Information
Additional Supporting Information may be found in the online version of this article on the publisher's web-site: Supplementary Table 1 Number of elements and nodes in the Models 1-6. Supplementary Table 2 Reliability study results.