FE analysis of stress and displacements occurring in the bony chain of leg

Abstract

Aims

The aim of this study was to assess how the stress shielding can influence the integrity and resistance of bones.

Methods

With this purpose a complete FE model of the human leg was realised. A load of 700 N has been applied at the top of pelvis and the feet, at the tip, was rigidly fixed.

Results

Obtained results reveal interesting consequences deriving by taking into account the complete bony chain.

Conclusion

A comparison among the literature data and our models can furnish a complete vision of the global spreading of the forces along the various bony components.

1. Introduction

The introduction of finite element analysis (FEA) into orthopaedic biomechanics allowed continuum structural analysis of bone and bone-implant composites of complicated shapes. However, besides having complicated shapes, musculoskeletal tissues are hierarchical composites with multiple structural levels that adapt to their mechanical environment. Mechanical adaptation influences the success of many orthopaedic treatments, especially total joint replacements. Recent advances in FEA applications have begun to address questions concerning the optimality of bone structure, the processes of bone remodelling, the mechanics of soft hydrated tissues, and the mechanics of tissues down to the micro structural and cell levels, but still have deeply difficulties to analysed complied skeletal chains involving different bony parts, because of the model size, and above all the boundary conditions to impose. Many different works in literature have investigated the single bony part such as femur, knee, tibia, or feet, fixing the base and loading with more or less detailed systems of forces. In this paper this question is faced by adopting a simplified model of the human leg, intended to assess how the stress shielding can influence the integrity and resistance of bones, if loaded with a vertical force of 700 N, and by comparing the obtained results with the ones present in literature. D.J. Rapperport¹ et al considered the pressure distributions across the articular surface for a resultant femoral load magnitude of 1.000 N angled of 40° medial of vertical obtaining an equivalent stress of 10 MPa on the hip joint. Breuckmann optical scanning and Metris laser scanning devices are used for scanning any object and producing a 3D form, while the CT device is used to visualize hard tissue of the human body. As a consequence of statistical evaluations, Breuckmann, CT and Metris models can be also compared with finite element analysis for the human upright stance position.² Montanini et al³ addresses the question of evaluating, by combining both experimental and numerical methods, the stress/strain distribution within a standardized composite femur. Two different loading conditions of the femur were considered. In the first one (LC-1) loads and boundary conditions exactly replicate those used in the experimental set-up: a rigid plate was modelled to transfer the axial load to the femoral head by imposing a vertical incremental displacement. In addition of providing information about the mechanical response of the implant/bone assembly under simplified yet physiological loads,⁴ LC-1 furnished a sound basis for the validation of the numerical model by direct comparison with in vitro testing. The effect of muscle action was then taken into account by considering a more complex loading case (LC-2). This load case has the same force resultant (980 N) as LC-1 but it considers the interdependence of muscles and joint forces as proposed by Bergmann et al,^5,6 who developed a simplified computer model of the physiological loading of the femur under walking activity by grouping functionally similar hip muscles. This load profile, which basically consists of four distinct muscle groups in addition to the hip joint force, is based on a validated musculoskeletal analysis and provides consistent torsion loading of the femur in addition to bending, loaded on the head with a force of 980 N. The human knee is the largest joint in the musculoskeletal system, which supports the body weight and facilitates locomotion. Gardiner and Weiss⁷ developed a finite element model of the MCL to study its 3D stress–strain behaviour under valgus loading. Gabriel et al⁸ determined ‘‘in situ’’ forces between the two bundles of the ACL with the knee subjected to anterior tibial and rotational loads. Hirokawa and Tsuruno⁹ developed a 3D model of the ACL that they used to study the strain and stress distributions in the ACL during knee flexion. Limbert et al¹⁰ proposed a 3D finite element model of the human ACL. This model was used to simulate clinical procedures such as the Lachman and drawer tests. In all these studies, the finite element model incorporated only one ligament without considering menisci and articular cartilages. Other authors analysed the distribution of contact pressures and compression stresses in menisci and articular cartilages, Bendjaballah et al¹¹ only considered a compressive load and modelled the ligaments as nonlinear springs. Pena et al¹² presented a 3D model of the knee considering the ligaments as isotropic. The results obtained in ligaments showed high stresses at full extension. This is essentially due to the large sagittal plane rotation of the femoral insertion of the ACL. This was also observed experimentally by Yamamoto et al¹³ that used photo elasticity to track the strains at the surface of the ACL. The anterior load produced a stress distribution in the MCL similar to a shear problem, with tension in the anterior–distal and posterior–proximal parts. Similar results were obtained by Hull et al¹⁴ in their work, were they measured the strain distribution in the MCL to determine the single and combined external loads most likely to cause injury. Due to the location of the tibial insertion of the PCL, during an anterior displacement of the knee, the tibia pushes the PCL and provokes bending. Different authors have used numerical techniques to analyse the distribution of contact pressures and compression stress in menisci and articular cartilage in the healthy knee joint^14,15 considered a compression load of 1300 N. The ankle mortise is composed of the distal articular surfaces of the tibia and fibula that are connected through a ligamentous complex known as the ankle syndesmosis. Together, these structures provide stability to the ankle joint. The distal tibio-fibular syndesmosis is composed of four ligaments: the anterior inferior tibio-fibular ligament (AITFL), the interosseous ligament, the posterior inferior tibio-fibular ligament (PITFL), and the posterior transverse tibio-fibular ligament. Proximally, the interosseous membrane joins the tibia and fibula. The fibres of the syndesmosis and interosseous membrane course inferolaterally and insert into the medial border of the fibula. These structures also maintain the integrity and stability of the ankle mortise and are important for balance loading of the foot through the fibula. Li and Anderson,¹⁶ carried on an FE analyses included several provisional loading steps (to bring the joint into a seated apposition as governed by the articular surfaces), followed by 13 steps spanning the stance phase of gait. Prevailing joint contact stresses following surgical fracture reduction were quantified in this study using patient specific contact finite element (FE) analysis. FE models were created for 11 ankle pairs from tibial plafond fracture patients. Both (reduced) fractured ankles and their intact contra laterals were modelled. A sequence of 13 loading instances was used to simulate the stance phase of gait. Contact stresses were summed across loadings in the simulation, weighted by resident time in the gait cycle. The maximum applied load was scaled to 320% body weight, rather than the normal 470% body weight. Information on the internal stresses/strains in the human foot and the pressure distribution at the plantar support interface under loadings useful in enhancing knowledge on the biomechanics of the ankle–foot complex. While techniques for plantar pressure measurements are well established, direct measurement of the internal stresses/strains is difficult. A three-dimensional (3D) finite element model of the human foot and ankle was developed by Cheunga, et al¹⁷ using the geometry of the foot skeleton and soft tissues, which were obtained from 3D reconstruction of MR images.

2. Materials and methods

The geometrical data of the model developed herein were obtained by matching a nuclear magnetic resonance (MRI) for soft tissues, and a computerized tomography (CT) for bones, with images taken from a normal adult patient, separated at intervals of 1.5 mm in the sagittal, coronal and axial planes with the knee at 0° flexion (accuracy 0.5 mm). These lines were transferred into the commercial code Hypermesh by Altair^® where the main surfaces and solid version of the model were reconstructed; in particular 5.758 elements and 1.837 nodes were used for pelvis, 20.096 elements and 1.012 nodes in femur, 2.567 elements and 687 nodes in patella, 2.480 elements and 849 nodes in fibula, 17.831 elements and 2.032 nodes in tibia and 1.120 elements and 412 nodes for the foot. On the upper zone, the ilio-femoral ligament, the ligament of the hip joint, which extends from the ileum to the femur in front of the joint, was modelled with 252 tetrahedral elements and 151 nodes. The knee joint constituted by the medial collateral ligament, which extends from the medial femoral epicondyle to the tibia, the lateral collateral ligament, which extends from the lateral femoral epicondyle to the head of the fibula, the anterior cruciate ligament which extends postero-laterally from the tibia and inserts on the lateral femoral condyle, and the posterior cruciate ligament which extends anteromedially from the tibia posterior to the medial femoral condyles were modelled with 529 tetrahedral elements and 296 nodes. On the lower zone, the foot joint, constituted by the plantar fascia, the medial and lateral ligaments were modelled with 366 tetrahedral elements and 187 nodes. Nonlinear finite element analysis of the models were performed with Abaqus version 5.4 (Hibbitt, Karlsson and Sorensen, Inc., Pawtucket, RI) using the geometric nonlinearity and automatic time stepping options.

Material properties were defined as nonlinear elastic materials for the structures, as reported in Fig. 1. Contact interfaces were imposed at the ilio-femoral (femur-pelvis), knee (femur-patella, patella-tibia, and fibula-tibia) and foot (tibia-feet) joints; defined using a penalty-based method with a weight factor, a coefficient of friction of 0.04 was chosen to be consistent. A distributed, on 70 nodes, vertical (y axes) load of 700 N was applied on the left side of the pelvis as depicted on Fig. 1, while a fixed constrain, of 200 nodes, was imposed at the lower extremity of the foot, see Fig. 1.

Fig. 1.

a) FE model of the whole structure: pelvis, femur, patella, fibula, tibia, foot, and ligaments; and Young modulus adopted for the various components. a) Particular of the ilio-femoral joint; b) Particular of the knee joint; c) Particular of the foot joint.

3. Results

A geometrical accurate 3D FE model of the human complete model of the leg was realised. The analysis of the entire chain allows having a complete picture of the stress distribution and of the most stressed bones and soft tissues, but, more importantly can overcome problems connected with boundary conditions imposed at single bony components. In Table 1 are reported the obtained results detailed for each part of the skeletal chain. As it is possible to notice by observing Fig. 2 and Table 1, the maximum total displacement is localised on the upper zone of the leg, concerning the pelvis and the femur, with values ranging from 12 to 9 mm. The lower part undergoes a minor effect producing displacements of the order of 3 mm localised in knee, tibia and fibula. Different consideration must be done for the Equivalent Von Mises Stress which don't follow a continuum trend like displacements, see Fig. 4. Maximum values of about 18 MPa are reached on the knee ligaments, and pelvis see Fig. 3, while values of about 15 MPa were found in fibula. Femur, femoral ligaments, and tibia register values of about 13 MPa, while patella, feet ligaments and feet are solicited with less than 8 MPa. The curves depicted in Fig. 4, total displacements and Equivalent Von Mises stress vs. position, have been calculated by choosing the average value for displacements and Eq. Von Mises stress localised along perpendicular intercepting lines, drown every 5 mm starting from the pelvis and finishing to the feet. As it is possible to notice the first curve, related to the displacements, exhibits a continuum trend regularly decreasing from the pelvis, with a maximum value of about 12 mm, to the feet. On the contrary the second curve, related to the Eq. Von Mises stress, exhibits a sinusoidal trend, with two high peaks on the knee ligaments and pelvis (18 MPa), then, in the area of tibia it reaches a value of about 15 MPa, while femur, femoral ligaments and tibia are solicited with about 13 MPa. Finally in Fig. 5 are reported the Eq. Von Mises Stress contour maps evidencing the femoral ligaments a); the knee ligaments b); tibias c); and the feet ligaments d). Moreover Table 1 shows also the equivalent total strain which reaches, as expected, its maximum value of 2.55e⁻³ on the knee ligaments.

Table 1.

Obtained results in terms of displacements and Von Mises Stress.

		X disp. [mm]	Y disp. [mm]	Z disp. [mm]	Tot. disp. [mm]	Eq. V. Mises Str. [MPa]	Eq. total strain
	Pelvis	0.76	−2.14	−11.68	11.74	17.08	1.65e−3
	Femural ligaments	0.21	0.57	−9.43	9.44	13.00	2.13e−3
	Femur	0.34	0.53	−9.28	9.29	13.38	1.29e−3
	Patella	0.25	0.47	−2.69	2.72	1.35	1.64e−4
	Knee ligaments	0.33	0.37	−2.75	2.76	18.00	2.55e−3
	Fibula	0.33	0.24	0.06	2.43	15.83	1.15e−3
	Tibia	0.21	−0.42	−2.63	2.64	13.02	7.73e−4
	Foot ligaments	0.05	−0.02	−0.08	0.09	7.75	1.27e−3
	Foot	0.05	0.01	0.01	0.01	4.12	4.51e−4
	Complete model	0.76	−2.14	−11.68	11.74	18.00	2.55e−3

Fig. 2.

Global displacements contours maps registered in the skeletal chain of the human leg.

Fig. 4.

Eq. Von Mises Stress and Global displacements trends vs. position (0–90 mm) calculated for the FE model.

Fig. 3.

Eq. Von Mises contours maps registered in the skeletal chain of the human leg.

Fig. 5.

Eq. Von Mises Stress evidenced on the femoral ligaments a); on the knee ligaments b); on tibias c); and on the feet ligaments d).

4. Discussion

The obtained results reveal interesting consequences deriving by taking into account the complete bony chain. A comparison among the literature data and our models, carried on the works studying femur, knee, and tibia show less stressing conditions ageing on structure, and moreover can furnish a complete vision of the global spreading of the forces along the various bony components. D.J. Rapperport¹ et al considered the pressure distributions across the articular surface for a resultant femoral load magnitude of 1.000 N angled of 40° medial of vertical obtaining an equivalent stress of 10 MPa on the hip joint, see Table 2. Breuckmann optical scanning, Metris laser scanning, and CT devices, were compared with finite element analysis for the human upright stance position.^2,7 As a result of this process, the equivalent maximum stress values obtained on the femur were estimated in 25 MPa.

Table 2.

Comparison of the obtained data, in terms of displacements and Eq. Von Mises Stress, on hip joint and femur, among different authors.

Femur and hip joint
		D.J. Rapperport (hip joint),1	Verim et al,2	Montanini et al (LC-1),3	Montanini et al (LC-2),3	Our model
Femur disp.	[mm]	/	5	6	21	9
Femur stress	[MPa]	10	37	31–47	25–49	13

Montanini et al³ addressed the question by evaluating the stress/strain distribution within a standardized composite femur for two different loading conditions. In the first one (LC-1) loads and boundary conditions exactly replicate those used in the experimental set-up: a rigid plate was modelled to transfer the axial load to the femoral head by imposing a vertical incremental displacement. In addition of providing information about the mechanical response of the implant/bone assembly under simplified yet physiological loads,⁴ LC-1 furnished a sound basis for the validation of the numerical model by direct comparison with in vitro testing. The effect of muscle action was then taken into account by considering a more complex loading case (LC-2). This load case has the same force resultant (980 N) as LC-1 but it considers the interdependence of muscles and joint forces as proposed by Bergmann et al,^5,6 who developed a simplified computer model of the physiological loading of the femur under walking activity by grouping functionally similar hip muscles. This load profile, which basically consists of four distinct muscle groups in addition to the hip joint force, is based on a validated musculoskeletal analysis and provides consistent torsion loading of the femur in addition to bending, loaded on the head with a force of 980 N. The mentioned results are compared to ours in Table 2. As it is possible to notice, in terms of displacements, results are quite similar to ours in the first two cases, Verim et al² and Montanini et al LC-1,³ while in the third case, Montanini et al LC-2, they are much more elevated. Different considerations must be done for the Equivalent Von Mises stress which result higher, except for the first author D.J. Rapperport (Hip Joint),¹ because of the boundary condition imposed, femur fixed on the condilar region. The human knee is the largest joint in the musculoskeletal system, which supports the body weight and facilitates locomotion. Pena et al¹² presented a 3D model of the knee considering a combined load of 1150 N in compression and 134 N anterior–posterior was applied to the femur. The results obtained in ligaments showed high stresses at full extension. Different authors have used numerical techniques to analyse the distribution of contact pressures and compression stress in menisci and articular cartilage in the healthy knee joint, Song et al¹⁸ imposed an initial in situ strain of the AM and PL bundles of 3% and the intact knee was tested at full extension in response to an incremental anterior tibial load ranging from 0.0 to 134 N while the forces and moments in all other directions were minimized. The maximum equivalent Cauchy stresses of both bundles were localized on the lateral portion of the bundles near the femoral insertion site where the stresses in the PL bundle ranged from 7 to 24 MPa, and, in the AM bundle, the stresses ranged from 6 to 17 MPa. Different authors have used numerical techniques to analyse the distribution of contact pressures and compression stress in menisci and articular cartilage in the healthy knee joint. Bendjaballah et al¹¹ considered a compression load of 1.300 N over the femur and found a compression stress on the meniscus of 4 MPa in the external and internal periphery, respectively, for the healthy joint. Donahue et al¹⁹ obtained similar results with a maximum stress of 4 MPa in the medial meniscus for the healthy joint coincident with experimental results. As it is possible to see by observing Table 3, results obtained by several authors in knee ligaments, except in the case of the author Pena et al¹² which adopt similar loading conditions, are quite lower than ours. This could be due to the different loading conditions analysed and to the simplifications adopted in the various FE models. In human anatomy the tibia is the second largest bone next to the femur. As in other vertebrates the tibia is one of two bones in the lower leg, the other being the fibula, and is a component of the knee and ankle joints. Gabriel et al⁸ obtained a tibial displacement of 5 mm, while Song et al¹⁸ obtained about 4.5 mm, under an anterior tibial load of 134 N, at full extension. With the inclusion of the posterior capsule that decreases the global displacement, Yagi et al²⁰ obtained a value of 4 mm. A peak stress value of 12 MPa was measured by Vrahas et al,²¹ in cadaveric tibia statically loaded with 1360 N, while Wendy Li et al¹⁶ obtained a peak stress of 13 MPa under the same load. Cheung et al¹⁷ developed a geometrical accurate 3D FE model of the human foot and ankle, and for a subject with body mass of 70 kg, a vertical force of approximately 350 N is applied on each foot during balanced standing. The results confirm an Equivalent Von Mises stress of 5 MPa, under a load of 700 N, not too much different by the one obtained in this work of 3 MPa. The described results are compared with ours in Table 4. As shown in table, displacements of tibia, found in this paper, result lower than literature ones, while the stresses are comparable, the same considerations can be done for the stress calculated on the feet.

Table 3.

Comparison of the obtained data, in terms of Eq. Von Mises Stress, on the knee ligaments, among different authors.

Knee ligaments
		Song et al,18	Bendjaballah et al,11	Donahue et al,19	Pena et al,12	Our model
Knee lig. stress	[MPa]	7	4	4	20	18

Table 4.

Comparison of the obtained data, in terms of displacements and Eq. Von Mises Stress, on the tibia and feet, among different authors.

Tibia and feet
		Gabriel et al7	Song et al18	Our model
Tibial disp.	[mm]	5	5	3
		Vrahas et al21	Wendy Li et al16	Our model
Tibial stress	[MPa]	12	13	13
		Cheung et al17		Our model
Feet stress	[MPa]	4		3

5. Conclusions

The aim of this study was to assess how the stress shielding can influence the integrity and resistance of bones. With this purpose a complete model of the human leg was realised. The analysis of the entire chain allows having a complete picture of the stress distribution and of the most stressed bones and soft tissues, but, more importantly can overcome problems connected with boundary conditions imposed at single bony components. Several limitations of the here described model have in any case to be considered. First of all, the results were obtained for a static model of the full extension position of the knee joint. The use of FE modelling necessarily involves simplifying assumptions, and pre defined loading and boundary conditions which influence the final results. It would be important to study how different joint angles modify the contact areas, the stress shielding, and the total displacements. Second, the cartilages here modelled were assumed to be composed by a single-phase linear elastic and isotropic material. Finally, the visco-elastic properties of ligaments and meniscus were not considered, although this aspect does not seem to be much relevant in this analysis. Several authors have found that the cartilages are much stiffer in circumferential direction, which could have some influence in the results (Proctor et al,²² Spilker et al,²³ and Fithian et al²⁴). In spite these limitations, the obtained results resulted close to the literature ones or almost comparable, demonstrating that subject specific FE models can predict the complex, non uniform stress and deformations fields that occur in biological bony complex chains. The obtained results reveal interesting consequences deriving by taking into account the complete bony chain. A comparison among the literature data and our models, carried on the works studying femur, knee, and tibia show less stressing conditions ageing on structure, and moreover can furnish a complete vision of the global spreading of the forces along the various bony components.

Conflicts of interest

The author has none to declare.