DYNAMIC ANALYSIS OF THE HUMAN KNEE

Transcription

1 122 Vol. 14 No. 3 June 2002 DYNAMIC ANALYSIS OF THE HUMAN KNEE R. TARLOCHAN,' S. RAMESH,' AND B. M. HILLBERRY 2 ' Department of Mechanical Engineering, University Tenaga Nasional, Selangor, Malaysia. 2Department of Mechanical Engineering, Purdue University, IN, USA. ABSTRACT Biomed. Eng. Appl. Basis Commun : Downloaded from The objective of this paper is to present a dynamic analysis of the human knee for various activities such as walking and stair ascending/descending. The results of this work is significant as it can used as inputs to joint simulators for artificial Fttjoint NWsf study, Y for rehabilitation purposes and to some extend applied to sports biomechanics. In the present esrt work an experimental study consisting of a motion sensor and a force plate Wivi was used c?r^ to capture t the r3teinor#o motion of subjects afst and to provide ground force reactions data that will serve su as inptis inputs to rt the tyre model. wndel Subjects.i. j bis in the years old bracket performed activities such as walking, stairs ascending nt'1ing and descending. descerdm,g..7h This age bracket was chosen because most of patients with knee-related c-rc1atedproblerrts problems are found in 'n this age r ' group. Some of the results from the present work correlated with that r Q of l# published literurur. literature. However, ffa# the present analysis was based on a more comprehensive model of f the 7 knee, ; which makes m<tk?s flee the results more realistic to the actual mechanical behavior of the lower limb. Biomed Eng Appl Basis Comm, 2002 (June); 14: INTRODUCTION The functions of the lower limb are to support body weight, provide movement and to maintain stability of the entire body. The knee joint is the largest and the most complex joint in the body. The importance of this joint to the mobility and stability of the human body cannot be denied. In addition, many sports injuries and arthritis are related to the knee joint. Hence this joint has been studied extensively for decades. Numerous models have been developed to gain a better understanding of the knee joint and motion [1]. Several knee simulators were designed to further understand the behavior of the knee and to evaluate knee replacement components [2]. In general the knee joint has been modeled as a spatial movement with rotations and translations in two and three-dimension [1]. In the force analysis, very few models were developed that takes into ac- Received: April 26, 2002; Accepted: June 10, 2002 Correspondence: Dr. S. Ramesh Department of Mechanical Engineering, University Tenaga Nasional, Km-7 Jalan Kajang-Puchong, Kajang, Selangor, Malaysia. rameshjit@uniten.edu.my count all vital structures of the joint, such as the patello-femoral and tibio-femoral joints, menisci, articular-surfaces and the cruciate ligaments structures [1]. Lacking of these vital structures in model predictions can lead to inaccurate results. In addition, a single dynamic model that consists of both the tibio-femoral and patello-femoral joints and anatomical structures like menisci and ligaments has yet to be developed with proper contact conditions and surface geometry. The understanding of the behavior of the knee not only highlight the importance of this joint but also assist the prosthetic industry in designing, developing and evaluating artificial knee replacement joints and rehabilitation centers in developing and correcting locomotion deficiency in patients that have abnormal motion patterns. The objective of this work was to study the dynamic motion of the knee by adopting and improving existing models. The analysis incorporates most of the vital structures of the knee joint and a multi rigid open chain system [3] was employed to determine the inverse dynamics of the system. 2. METHODS 2.1 Modeling -26-

2 BIOMEDICAL ENGINEERING- APPLICATIONS, BASIS & COMMUNICATIONS 123 Biomed. Eng. Appl. Basis Commun : Downloaded from V. Fa. FEMORAL FRAME Fig. 1. Open chain system to describe the motion of the knee joint [3]. To describe the motion of the tibio-femoral, modifications of the inverse kinematics of a three-cylindric robot developed by Pennock and Vierstra [4, 5] was used (Fig. 1). Since a cylindric joint permits rotation and translation, combinations of three of these joints describes the motion of the knee in three-dimension. Basically, a cylindric joint is the kinematic equivalent of a revolute joint and a prismatic joint sharing the same axis and has two degrees of freedom (Fig. 1). For a typical link, there are two Cartesian reference frames attached to the link namely the body frame denoted by F; and the joint frame denoted by Ci. The frames F0 and F3 are referred to as the base frame and the end effector frame, respectively (Fig. 1). The equations used to derive the displacements are well documented [4, 5] and expressed in terms of dual-number algebra and (3x3) dual-number transformation matrices. An example of a dual algebra representation is the dual unit vector U that can be written as: U, U= U2 U3 U1 +U,, s U2 + Uz s U3 +U3 s where the subscript number 1, 2 and 3 are the reference system directions analogous to the x, y and z Cartesian reference system. The real parts of the dual algebra are direction cosines and are re?resented by U1, U2 and U3, whereas the dual parts U,, U2 and U3 (1) are the components of the moment of the line about the origin. The symbol, s, designates exclusively the dual unit having the properties s = 0. Details of the formulation and mathematical derivation employed in the present work are described elsewhere [6]. For the patello-femoral joint, the model developed by Rullkoeter [7] was used. In this model, the open chain system used is similar to that proposed by Grood and Suntay [8]. The prediction of external moments and forces were based on the analytical technique proposed by Pennock and Yang [3] to derive and solve for the dynamic equations of a multi rigid body open chain system. The analytical technique is based on Newtonian mechanics with screw calculus. This technique provides very concise expression for the description of the forces and moments of a rigid body in general motion and allow the freedom of choosing a body reference frame with any point as its origin on the geometric constraints imposed on the body [3]. The derivation and solutions of this technique are based on dual number algebra and (3x3) dual matrix, consistent with that used for the kinematics analysis. The external moments and forces at the patello-femoral joints were assumed zero [9]. In the present study, the knee joint analysis was extended to the formulation of a three-dimensional mathematical dynamic model of a general three body segmented articulating knee joint [6]. This formulation was applied to investigate the relative dynamic motions between the femur-tibia-patella as well as the -27-

3 124 ligaments, muscles and contact forces developed in the joint while taking into account the geometry of the articulating surfaces and the appropriate constitutive behavior of the ligaments. The articulating surfaces are modeled as rigid and non-deformable. On the other hand, the femoral articulating surfaces are modeled as spheres whereas the articulating surfaces of the tibia and patellar facets are modeled as planes. By assuming the above geometrical shapes for the surfaces, point contacts are assumed on each medial and lateral articulating surfaces. The locations of the contact points are assumed based from studies conducted by Rahman [10]. In the present study, twelve nonlinear spring elements are modeled as ligaments to represent the different ligamentous and the capsular tissue posterior of the knee joint. For the kinematic and force analysis of the lower extremity, transformation matrices that provide information on the relative orientation of one segment with respect to another are important. In human motion studies, the most frequently used method for measuring human movement involves certain markers being placed on the subject's skin surface. These markers are generally used to represent the underlying bone movement although doubts exist on the accuracy of this technique. However in the present work, the method developed by Andriacchi et al. [I I ] was employed to determine the motion of the underlying bone. In general, this method uses a set of cluster points uniformly distributed on the limb segment. Each point or marker is assigned a mass, and the center of mass and inertia tensor of this system of cluster points is determined. From the inertia tensor, eigenvalues and eigenvectors are used to define a coordinate frame as well as providing means of studying the contribution of the skin motion. 2.2 Experimental Study Experimental study was carried out to provide inputs into the models. Four subjects were chosen in the age bracket of years. This age bracket was chosen since this age bracket has a high percentage of patients with related lower limb problem [6]. The devices used in this study were an Optotrak and a force plate. The Optotrak is a three-dimensional motion system that measures the motion in space by using infra red light emitting markers. The force plate provides the reaction forces, moments and center of pressure when a load is applied to the plate. The data from the motion sensor was used in determining the joint motion, velocity and acceleration of the limb segment via numerical differentiation. The data from the force plate along with the data from the motion sensor was used in determining the external forces and moments. The types of activities performed in this study were level walking, ascending and descending stairs. Vol. 14 No. 3 June RESULTS AND DISCUSSION The important parameters in describing the knee are kinematic and force distribution in three dimension. Nonetheless, the knee flexion extension angles (Fig. 2), adduction - abduction moments (Fig. 3) and the contact forces at the condyles (Fig. 4) are the important parameters worth discussing. The knee flexionextension is the main motion of the knee joint. The adduction-abduction moment is the main moment used in stabilizing the knee in the support phase of the gait, whereas the contact forces are the actual forces that contribute to the stress distribution on the articulating surfaces. The data presented here is for level walking since it is the most significant activity in gait analysis. For the knee flexion and extension angles (Fig. 2), the waveform is very similar to that reported in literature [12]. Slight variations however do exist and are attributed to variability in parameters such as age, sex, walking speed and natural responses of subjects. The average pattern of the flexion/extension is biphasic where a slight flexion followed by an extension during the stance phase and a large flexion accompanied by extension in the swing phase. As the heel strikes the ground, the knee flexes slightly before the quadriceps muscles contract to extent the knee to achieve support and stability of the limb during the stance phase. As the limb approaches the end of the stance phase the activation of the flexor muscles become dominant in flexing the legs and hence providing the forward propulsion of the body. At this time the limb is entering the swing phase where the flexion of the knee joint reaches its maximum value before the quadriceps muscles activate to extend the knee preparing for the next stance phase. Therefore this completes one gait cycle (Fig. 2) and the cycle continues for the entire gait. The external moments when compared with that of reported by other researchers [12] differs in magnitude but not in the general waveform as depicted in Fig. 3. Some shifts in the curve were expected due to different muscular activation in the subjects. This is because the external moments are balanced in a dynamic % Gait Fig. 2. Knee flexion-extension angle for a complete gait cycle. -28-

4 BIOMEDICAL ENGINEERING- APPLICATIONS, BASIS & COMMUNICATIONS sense by an equal an opposite internal moments. Muscles, ligaments and contact forces generate these internal moments, although the muscles are the primary structure contributing to these moments. The adductor moment is the largest compared to the flexion and rotation moment since it is used to stabilize the knee during the support phase. This compresses the medial compartment of the knee and stretches the lateral soft tissue. Internal-external moments in the transverse plane are also important in providing stability to the knee joint. This stability is due to the contribution of the ligaments. The kinematic and external forces and moments calculated from this study were very consistent with that reported in the literature [12]. The model of the knee develop in the present work has provided a satisfactory prediction of the motions and the moments. The model predictions are not intended as a substitute for accurate studies of the biomechanical aspect of the knee due to variations exist between subjects. However, the current model proved to be effective in providing reliable predictions of the knee kinematics and external moments with regards to normal and abnormal gait. On the other hand the waveform obtained for the contact forces for the tibio-femoral joint were not in agreement with published values [13]. This discrepancy in the results could be attributed to the way the Time (seq Fig. 3 Knee adduction - abduction moment (Nm) against time. 7 4 Force to Body Weight 3 Ratio Time (sec( Fig. 4 Tibio-femoral contact forces. Rmedial Rlateral knee joint was modeled, the assumptions made and the solution method used. As mentioned earlier the knee joint was modeled taking into consideration the twelve ligaments with different stiffness. In addition, the muscular activation assumptions used [6] can also result in different outputs. Nonetheless, the present work affirmed that the medial compartment of the tibial plateau experiences a larger contact force than the lateral compartment (Fig. 4). This could be attributed to the adduction moment that compresses the medial compartment of the knee joint as reflected in Fig. 3. In this study several assumptions were made to reduce the indeterminacy of the system. Future recommendations to the model include developing a more comprehensive and complex model that takes into account the menisci and the deformability of the articular cartilage. Besides this optimization algorithm can be used to converge to a solution due to the highly nonlinear behavior of the system. Optimization is also important to some extend to better predict muscle and bone attachment behaviour. 4. CONCLUSIONS In the present work, the model developed and approach taken was to best mimic the anatomical behavior of the knee. This was done by inclusion of vital structures and better analytical and experimental techniques. The current model proved to be effective in providing reliable predictions of the knee kinematics and external moments with regards to normal and abnormal gait. Insight information of the knee behaviour was gained and the results obtained from this study could serve as a source for clinical and component development usage. It is envisage that this model can be expanded to the area of sports biomechanics where ways to reduce athlete's injuries can be studied. ACKNOWLEDGEMENT Rameshjit Tarlochan would like to thank the Biomedical Engineering Department of Purdue University USA for providing the financial support in carrying out this research. REFERENCES 1. M. S. Hefzy and T. D. V. Cooke, "Review of Knee Models: 1996 Update," Applied Mechanics Review, 49 (1996) S. W. Peter, "A Knee Simulating Machine for Performance Evaluation of Total Knee Replacements," J ofbiomechanics, 30 (1997) G. R. Pennock and A. T. Yang, "Dynamic Analysis -29-

5 126 of a Multi-Rigid Body Open Chain System," Journal of Mechanism, Transmission, and Automation in Design, 105 (1983 ) G. R. Pennock and B. C. Vierstra, "The Inverse Kinematics of a Three Cylindric Robot," International Journal of Robotics and Research, 9 (1990) G. R. Pennock and K. J. Clark, "An Anatomical Based Coordinate System for the Description of the Kinematics Displacement in the Human Knee," Journal ofbiomechanics, 23 (1990) R. Tarlochan, "Lower Limb Biomechanical Analysis," Master Thesis, Purdue University (2001). 7. P. J. Rullkoeter, "A Graphical and Quantitative Method for Determining Joint Surface Contact under Dynamic Loading Conditions," Master's Thesis, Purdue University, (1992) pp E. S. Grood and W. I. Suntay, "A Joint Coordinate System for the Clinical Description of Three- Dimensional Motions: Application to the Knee," Journal of Biomechanical Engineering, 105 (1983) Vol. 14 No. 3 June M. S. Hefzy and H. Yang, "Three-Dimensional Anatomical Model of the Human Patello-Femoral.Joint to Determine Motions and Contact Characteristics," Journal of Biomedical Engineering, 15 (1993) E. M. A. Rahmen and M. S. Hefzy, "Three - Dimensional Dynamic Behavior of the Human Knee Joint under Impact Loading," Journal of Medical Engineering and Physics, 20 (1998) T. P. Andriachi, E. J. Alexander, M. K. Toney, C. Dyrby and J. Sum, "A Point Cluster Method for In- Vivo Motion Analysis: Applied to a Study of Knee Kinematics," Journal of Biomechanical Engineering, 120 (1999) J. Apkarian, S. Naumann and B. Cairns, "A Three - Dimensional Kinematic and Dynamic Model of the Lower Limb," Journal of Biomechanics, 22 (1989) A. Seireg and R. J. Arvikar, "The Prediction of Muscular Load Sharing and Joint Forces in the Lower Extremities During Walking," Journal of Biomechanics, 8 (1975)