Three-Dimensional Finite Element Analysis of the Knee Ligaments During Cycling in Normal Young Subjects
This was a three years group project for my master degree and also was my master thesis. The project aimed to tackle the challenging problem by developing novel techniques that are then integrated with the core knowledge of biomechanical engineering, sports science, sports medicine and rehabilitation medicine. My contribution to the project included in vitro robotic test, medical image processing, motion analysis and finite element analysis.
Goemetric Model of the Knee and Ligaments
The geometric data of the knee joint including the bones (femur, tibia and fibula), and ligaments (anterior cruciate ligament (ACL), posterior cruciate ligament (PCL), medial collateral ligament(MCL) and lateral collateral ligament (LCL)) will be obtained using the CT(Fig. 1) and MRI(Fig. 2). With the data of CT and MRI, the corresponding surface models will be reconstructed using the commercial-available software package (AMIRA 5.3.3, TSG, Inc., CA; Geomagic Studio 12, Raindrop Geomagic, Inc., USA) after the necessary segmentation and 3D reconstruction. The decimation of the vertices on all models will be carried out for the reduction of the computational effort, and the surfaces of the ligaments will be smoothed for reducing the artifacts of sampling and segmentation. The bone surface models will be reconstructed using the CT and MR images, respectively, denoted as the CT-derived and MR-derived models. The ligament models will be reconstructed only using the MRI. A point-based surface registration between the CT-derived and MR-derived bones models will be performed for the spatial transformation between the CT and MR coordinate systems. With the spatial transformation, the models of the ligaments and cartilage will be attached to the CT-derived models of the femur and tibia under the same coordinate system.
Figure 1. Finite Element Model of Knee Bone From CT Scan
Figure 2. Finite Element Model of Knee Ligaments From MRI
Constitutive Modeling of Ligaments
Ligaments are dense connective tissues consisting of mainly parallel-fibre collagenous tissues embedded in a highly compliant solid matrix of proteoglycans. The degree of anisotropy can vary substantially between different types of ligaments and the fiber orientation generally represents an adaptation to the mechanical environment. Therefore, the accurate description of the three-dimensional mechanical behavior of ligaments by constitutive equations is quite complicated and difficult. In the current study, the constitutive modeling (mechanical properties) of the ligaments used in the FE model will be defined based on the strain-energy function developed by Limbert as follows.
where (i=1~5) are subject-specific material parameters and have to be determined from the subject-specific experimental data. Parameters I1 and I3 are the first and third principal invariants of the modified right Cauchy-Green deformation tensor, respectively, and the I4 is an invariant characterizing the anisotropic property of ligament fibers. In the current study, data from the drawer test of the cadaveric knee at flexion angles 30-degree using RJTS will be used for in vitro study.
Constitutive Modeling of Ligaments
The use of anisotropic constitutive models to describe the material behavior of ligaments requires the specification of values for some number of material coefficients. Five material parameters, i.e. ci (i=1~5) in Eqs. (1) and (2), have to be determined using the data obtained from subject-specific experiments in the current project. For in vitro study, the force-displacement relationships of the cadaveric knee specimen during the anterior/posterior joint laxity test at 30° knee flexion will be obtained and used to customize the model to the specimen. For in vivo study, the force-displacement relationships of the knee of the subject was examined using knee arthrometer (KT-2000) during the anterior/posterior joint laxity test at 30° knee flexion. For this purpose, an optimization procedure will be used to determine the values of the 5 subject-specific parameters by minimizing the differences between the force-displacement relationships simulated from FE model and those measured by the RJTS and the KT-2000 at 30° knee flexion. In the FE model, the neutral position of the knee model is defined as the extended passive position. The ligament resting lengths are determined at the neutral passive position.
The positions and orientations of the femur and tibia imported from the poses obtained from the 3D fluoroscopy method as the boundary conditions(Fig. 3). In the FE analysis of the knee, movement of the knee is described in terms of the rotations and translations of the femur with respect to the tibia. The movement of the femur is controlled by a reference point on the femur and the tibia is fixed in an invariant position. All nodes on the femur will be moved relative to the reference point. Since insertions of the ligaments are fixed to the bones with the attached node, the movement of the bone will displace and deform the ligaments.
Figure 3. Boundary Conditions of Cycling From 3D Fluoroscopy
Construction of Subject-Specific 3D Finite Element Model of the Knee
Three-dimensional finite element (FE) modeling of the knee and the ligaments will be used to investigate non-invasively the stress and strain distributions and the forces of the ligaments in the knee joint during functional activities. As shown in Fig. 4, the construction of the subject-specific FE model of the knee consists of four main parts of information, namely the geometric model of the knee and ligaments, constitutive modeling of the ligaments, model parameters and the boundary conditions. It is clear that the results of the FE analysis will depend heavily on the accuracy of the information of the four main parts. In the current study, the effort is made to obtain accurate, subject-specific data for the four parts of the model, in order to ensure accurate results. This is in contrast to previous FE analysis in which generic models are often used, mainly because of lack of subject-specific data. After the validation with the in vitro data, the current framework for constructing subject-specific FE models of the bones and ligaments will be useful for in vivo modeling during the functional activities.
Figure 4. A schematic representation of the framework of the finite element modeling based on a preliminary study. The establishment of the FE model can be separated into four parts namely (a) subject-specific parameters measured from the knee laxity test at 30 degrees of flexion as in KT2000, (b) constitutive modeling of ligaments, (c) surface geometry of the knee and (d) boundary conditions obtained from the 3D fluoroscopy method
Validation of the Finite Element Model
The FE analysis of the knee will be performed step by step according to the discrete kinematic data of the femur and tibia recorded by RJTS, and the forces at the terminal node of the cross-section of the ligaments will be output for each step for the subsequent validation purposes. The reaction forces at the level of the fixation of the UFS sensor will also be obtained. As shown in Fig. 5, the force-displacement relationships obtained from the FE simulation and from the joint laxity test using RJTS will be compared. Through the validation, the validity and the accuracy of this FE model will be evaluated.
Figure 5. The validation procedure for the resultant forces estimated by the FE model
Results and Discussion
In order to construct the subject-specific 3D finite element knee model, the root mean square error have to under the 5% of total force range, which is 193.6 newton. The kt-2000 finite element analysis result of the subject was 4.5 newton that below the 5% of 193.6 newtons(Fig. 6). Thus, we successfully obtained the personal ligaments parameters.
Figure 6. Finite Element Analysis of KT-2000
The resultant force of PCL was much greater than ACL during cycling.
The mean peak ACL strain values generated during bicycling were relatively low.
The resultant force of MCL was higher than LCL during cycling.
The resultant force of MCL and LCL during cycling were relatively low.
There was no difference between unloaded and loaded about the maximum principal stress and distribution during cycling.
Figure 7. Resultant Force of Knee Ligaments
Stress Distribution of the Knee Ligaments
The knee ligaments resultant force was decreased during power phase and increased during the recovery phase.
PCL was mainly stretched and stressed of the knee ligaments during cycling.
There was no effect on ligaments force, maximum principal stress value and stress distribution under unloaded and loaded.
Bicycling will appropriate as a knee rehabilitation exercise after ACL injury but not applicable to PCL injury patients.
J. D. Li, M. Y. Kuo, T. C. Lin, Y. H. Wu and T. W. Lu, “Skin Movement Artifacts Calculated Knee Kinematics and Kinetics During Cycling,” International Scientific Meeting on Biomechanics, Taipei, Taiwan, 2014.
J. D. Li, T. W. Lu, Y. H. Wu, M. Y. Kuo, T. C. Lin, C. C. Lin, Y. H. Liu and H. C. Hsu, “Effects of Soft Tissue Artifacts on the Calculated Kinematic and Kinetic Variables of the Knee During Cycling,” International Symposium on 3D Analysis of Human Movement, Lausanne, Switzerland, 2014.
H. C. Hsu, J. D. Li, Y. H. Wu, T. W. Lu, M. Y. Kuo, T. C. Lin, C. C. Lin and Y. H. Liu, “Comparisons of Soft Tissue Artifacts on the Thigh and Shank During Loaded and Unloaded Cycling Exercises Using 3D Fluoroscopy,” International Symposium on 3D Analysis of Human Movement, Lausanne, Switzerland, 2014.
J. D. Li, M. Y. Kuo, T. W. Lu, T. C. Lin, Y. H. Wu and H. C. Hsu, “Differences of Skin Movement Artifacts During Loaded and Unloaded Cycling Exercises on the Thigh and Shank Using 3D Fluoroscopy,” Asian-Pacific Conference on Medical and Biological Engineering, Tainan, Taiwan, 2014.
J. D. Li, T. W. Lu, M. Y. Kuo, Y. H. Wu, T. C. Lin, and H. C. Hsu, “Effects of Skin Movement Artifacts on Kinematics and Kinetics of the Knee During Cycling,” Asian-Pacific Conference on Medical and Biological Engineering, Tainan, Taiwan, 2014.
J. D. Li, Y. H. Wu, T. W. Lu, T. C. Lin, M. Y. Kuo, C. C. Lin, Y. H. Liu and H. C. Hsu, “Comparisons of Knee Joint Loading Between Forward and Backward Pedaling on an Instrumented Cycling Ergometer Using 3D Fluoroscopy Method,” World Congress of Biomechanics, Boston, USA, 2014.