Human bone consists of rigid and dense connected tissues that are capable of repairing itself and adapt to stimuli such as various physical activities (walking, swimming, running, etc.). The strengthening or weakening of the bone tissues under such stimuli depends on weight, muscle strength, fitness, etc. . A damaged bone tissue heals over time by body’s natural healing process, which can be accelerated by external interferences such as mechanical loading under controlled conditions. The largest synovial joint in the human body is the knee joint that carries high loads. During the sitting, standing and climbing, it bears the weight of the body. Accordingly, it is highly susceptible to osteoarthritis and is one of the most injured parts of the human body. A lateral application of force on the knee joint was found effective in protecting bone tissue in the areas of distal femur and proximal tibia of the knee bone. The effects of the stimulation of bone regeneration are not limited to the applied areas but are seen along the length of the long bone . When a specific loading force is applied to the epiphyses of the femur and tibia, the trabecular bone tissue, which is characterized by axial stress resistance, resists this force from the opposite direction. A human lower body bone structure is shown in Figure 1 . This results in reversable deformations in that area. These deformations create a variation of the fluid pressure in the intramedullary cavity. This pressure gradient allows the flow of fluids that carry essential nutrients to the bone cortex initiating osteoblast differentiation and osteogenesis thus helping in repair and regeneration of the bone tissue . Thus such a knee loading regiment can be used as an effective treatment for bone rehabilitation as well as reducing the healing time of bone fractures and injuries. The lateral stress application is also less strenuous to the knee bone and reduces the amount of force that needs to be applied to get this result.
To our knowledge, few computational analyses are reported, which include not only cartilages but also the prominent ligaments such as anterior cruciate (ACL), posterior cruciate (PCL), medial collateral (MCL) and lateral collateral (LCL) as non-linear materials. Most previous studies used one-dimensional representations of the knee ligaments . This simplified approach was proved to be useful for predicting joint kinematics but nonuniform three dimensional (3D) stresses and strains could not be predicted. Other researchers developed 3D finite element models of individual human ligaments such as the ACL  or the MCL . Some papers presented specific computational models of parts of the human knee to discuss different aspects of its biomechanical behavior . For instance, Heegard et al.  developed a 3D model to analyze the human biomechanics during passive knee flexion. In all these latter models, ligaments were modelled as nonlinear springs.
A prototype device was designed and built in the lab whose effectiveness still needs to be clinically tested on human subjects [9 , 10]. Figure 2 shows such a prototype device. This device has been validated to produce force levels of up to 40 N with a frequency of application of 5 Hz, which is the recommended value based on mice studies . These forces are applied laterally on the soft tissues (skin, muscle, etc.) surrounding the knee joint, whereas the recommended force magnitudes are based on forces applied to the bone itself. Thus there is a need for analysis of forces transmitted from the knee loading device pads through the soft tissues to the knee joint bone and cartilage.
Figure 1. Brief anatomy of lower body bone structure of human body .
Figure 2. (a) Design details of the knee loading device; (b) Prototype of the design knee loading device.
Considerable efforts have been put into biomechanical modeling and computational analysis over the last few years [12 - 15]. Computational models present a standardized scheme for parametric studies, such as stress distributions for different geometries and kinematics. A finite element method is one of the computational techniques that has been extensively adopted to study biomechanics of the knee joint. Such analyses of the human knee joint can help in determining the underlying causes of cartilage degeneration that in turn results in osteoarthritis. The focus was to determine the effect of changes in the properties of bone joint on the stress and strain distribution in the knee joint. The soft tissues of the knee joint, namely the cartilages and ligaments, are particularly susceptible to wear owing to high levels of contact stresses. The main objective of this study is to determine the contact stresses in the knee joint due to the forces applied by a knee loading device.
We present here a complete 3D model of the healthy human knee joint. This included all the relevant ligaments and cartilage. Different experimental and numerical results were used to validate it. Once sufficiently validated, our main goal was to analyze the combined role of cartilage and ligaments in load transmission and stability.
2. Knee Model
A practical model of the human knee generally consists of four bones, the femur, tibia, fibula, and patella, together with five ligaments that limit the movement between the bones. These ligaments, in addition, keep the bones at a specified distance, which is necessary for the correct function of the joint. A representative three dimensional geometric model of the knee structure was used for this work (Figure 3). The development of a knee joint three-dimensional geometric model is described in detail in [16 , 17]. This work also includes the design of the skin model to study the forces transmitted through the skin and soft tissues. The skin model was designed using a CAD software Solidworks and was studied for force transmission analysis. The CAD design of the skin model developed in this work is shown in Figure 4.
For the analysis of the model, software package SIMSOLID was used. The assembled geometric models were input to the analysis module, and material properties were assigned to them. The different materials properties were used for the bone, cartilage, skin, and ligaments (Table 1) [18 , 19]. All materials were assumed to be isotropic. The nonlinear stress-strain properties for these materials of the knee joint components are shown in Figures 5(a)-(d). Following the assignment of material properties to each individual geometric entity, appropriate constraints were applied to the assembly.
The stress-strain curves for the various materials are shown below:
Table 1. Material properties [18 , 19].
Figure 3. Knee modelwhere (1) Femur; (2) Lateral collateral ligament (LCL); (3) Medial collateral ligament (MCL); (4) Anterior cruciate ligament (ACL); (5) Posterior cruciate ligament (PCL); (6) Tibia; (7) Fibula; (8) Cartilage.
Figure 4. Skin model.
(a) (b) (c) (d)
Figure 5. (a) Bone mechanical properties; (b) Cartilage mechanical properties; (c) Ligament mechanical properties; (d) Skin mechanical properties.
3. Analysis for Skin Model
The analysis for the skin model was performed in order to assess how the applied forces are transmitted from the skin to the knee joint components. It is challenging to develop the integrated 3D models with both hard and soft tissues due to major differences in their mechanical properties. The top and bottom surfaces of the skin model are constrained according to force loading conditions as applied by the knee loading device. A force was applied to the left side of the model, while the right side of the model was constrained with slider movement in order to emulate the force reaction on the right side. Figure 9 shows the skin model with boundary conditions. The force magnitudes evaluated in this study are similar to the levels recommended for the portable knee loading device . The development of this device is described in detail in a previous work [9 , 10]. The values of different thicknesses and the different forces used in the analysis are shown in Table 2.
4. Analysis for Knee Components
The automatic connections were used to represent the contact between bone and cartilage, between bone and ligament, and between the medial collateral ligament and the medial meniscus. The top and bottom surfaces of the tibia and femur were constrained in order to analyze the knee joint. The right side of the knee joint was constrained. The right side of the knee was also constrained. The forces obtained from skin model was applied to the left side of the knee joint. Figure 6 also shows the model of the knee joint with boundary conditions. The analysis of the knee joint was performed with the given boundary conditions. The contact stresses were calculated for the entire model, but the focus was on the cartilage.
5. Results and Discussion
The knee joint was mainly analyzed for the Von Mises stress (Maximum Equivalent Stress) and displacement magnitudes. The maximum Von Mises stress of 2.8077e−2 [MPa] and the maximum Displacement of 2.4974e−3 [MM] were located on the cartilage and tibia for a force magnitude of 33.80 N. Figure 7(a) and Figure 7(b) show the displacement and the Von Mises stress distributions on the knee joint components, respectively. Figure 8(a) and Figure 8(b) show the displacement and the Von Mises stress distributions on the femur. Figure 9(a) and Figure 9(b) show the displacement and the Von Mises stress distributions on the tibia, respectively. Figure 10(a) and Figure 10(b) show the displacement and the Von Mises stress distributions on the cartilage, respectively. Figure 11(a) and Figure 11(b) show the displacement and the Von Mises stress distributions on the ACL, respectively. Figure 12(a) and Figure 12(b) show the displacement and the Von Mises stress distributions on the fibula, respectively. Figure 13(a) and Figure 13(b) show the displacement and the Von Mises stress distributions for on the LCL, respectively. Figure 14(a) and Figure 14(b) show the displacement and the Von Mises stress distributions on the MCL, respectively. Figure 15(a) and Figure 15(b) show the displacement and the Von Mises stress distributions on the PCL, respectively.
In every case, the distribution of stress and strain on the model was similar. The maximum stress and strain appeared in the areas where the cartilage connected to the femur for every loading case. The results of this analysis for a soft tissue thickness of 1 mm and an applied force magnitude of 40 N are shown in Table 3. As can be seen from this analysis, the cartilage underwent the largest maximum desplacement and maximum equivalent strain while tibia exhibited the largest maximum Von Mieses stress. The stress-strain characteristics had linear response for both stress and strain analyses, where increased loads produce increased stresses and strains, for each type of load.
In order to study the effect of soft tissue thickness on the force transmission, three different soft tissue thicknesses were selected for finite element analysis. For each soft tissue thickness, three force magnitudes, as applied by the knee loading device, were considered: 20 N, 30 N, and 40 N. Finite element analysis of these force transmissions through the soft tissues with varying thickness are shown in Table 4.
A 3D model of the human knee was analyzed under a compressive force in this study. The main objective was to determine the contact stresses in the knee joint using analysis. Finite element analysis was used by other researchers to determine the distribution of compressive stresses in the healthy knee joint. Bendjaballah et al.  found that the compressive stress on the menisci varied between 1 MPa in the external periphery to 4 MPa in the internal periphery, under a compressive force of 1300 N. Dong et al.  obtained a compressive stress on the medial and lateral meniscus of 3.00 and 2.83 MPa, respectively. Pena et al.  found that the compressive stress on the medial meniscus was 3.31 MPa. The compressive stresses and displacement on the cartilage obtained in this study was 2.8077e−2 [MPa] and 2.4974e−3 [MM], respectively. The difference in the stress values may be due to the fact of low force values, and that frictionless nonlinear contact between the femoral and tibial cartilage, cartilage and menisci, and the femoral and patellar cartilage was not considered. The limitation of this study is that the knee joint components were assumed transversely isotropic. As it is shown in Figure 16, the model has a linear response, as it is expected, for both stress and strain analysis, where increased loads produce increased stresses and strains, for each type of load.
Figure 6. Skin and Knee Joint Model with boundary conditions.
Figure 7. (a) Displacement magnitude for 33.8 N force; (b) Von Mises stress for 33.8 N force.
Figure 8. (a) Displacement for 33.8 N force on Femur; (b) Von Mises Stress for 33.8 N force on Femur.
Figure 9. (a) Displacement for 33.8 N force on Tibia; (b) Von Mises Stress for 33.8 N force on Tibia.
Figure 10. (a) Displacement for 33.8 N on Cartilage; (b) Von Mises Stress for 33.8 N on Cartilage.
Figure 11. (a) Displacement for 33.8 N force on ACL; (b) Von Mises Stress for 33.8 N force on ACL.
Figure 12. (a) Displacement for 33.8 N on Fibula; (b) Von Mises Stress for 33.8 N on Fibula.
Figure 13. (a) Displacement for 33.8 N on LCL; (b) Von Mises Stress for 33.8 N on LCL.
Figure 14/span>. (a) Displacement for 33.8 N on MCL; (b) Von Mises Stress for 33.8 N on MCL.
Figure 15. (a) Displacement for 33.8 N on PCL; (b) Von Mises Stress for 33.8 N on PCL.
Figure 16. (a) Force vs. Displacement; (b) Force vs. Stress; (c) Force vs. Strain.
Table 2. Forces and thickness used for the skin model.
Table 3. Maximum stress and strain in each individual knee joint component.
Table 4. Maximumstress and strainon the cartilage foreach load case.
A 3D human knee joint model was built and analyzed under an axial compressive force, i.e. lateral force, which was generated with the knee loading device. Obviously, the model predicted that stresses, strains, and displacements elevated with increasing loads. The low and high load-bearing regions were estimated to be situated on the cartilage. According to several load case analyses for different values of direct loads, the solid model estimated stresses and strains within the elastic range. Furthermore, high stresses and strains were predicted to develop between the connections of soft and hard tissues. In particular, the results of this study demonstrated that the high load bearing areas were located on the posterior portion of the cartilages. It is expected that this prediction can be used to improve the design of the knee loading device.
This study was in part supported by a National Institute of Health (NIH) R01 grant number AR052144 (HY).
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