Biomechanical experiments will be more likely to improve our management of knee conditions if the mechanical performance of the human knee can be accurately reproduced in the laboratory. Since it is difficult to measure the muscle forces and joint reaction forces associated with functional knee activities in vivo, it is necessary to develop methods for predicting these forces from more easily measured quantities such as femoral and tibial motion, and external resistance forces. In mechanical analyses it is straightforward to calculate unknown parameters from known parameters, provided that there are as many consistent relationships - or equilibrium equations - as unknown variables. The equations can be used to solve for the unknown variables.

          These types of calculations are not possible in the study of the knee joint. Since there are far more muscles that act across the knee than degrees of freedom in the articulation there are not enough equilibrium equations to solve for all of the muscle forces and joint contact forces. When this situation arises, engineers use optimization methods to calculate the unknown values. Optimization is proving to be a useful tool in the study of the knee.

Background

          Ten unknown muscle forces are considered in models of the human knee, including the quadriceps group (QUAD), Tensor Fasciae Latae (TFL), Sartorius (SAR), Gracillis (GRA), Semimembranosus (SM), Semitendinosus (ST), Biceps Femoris Long Head (BFLH), Biceps Femoris Short Head (BFSH), Gastrocnemius Medial (GASM), and Gastrocnemius Lateral (GASL). In addition, the joint reaction forces have three unknown components, which represent the combined effect of joint contact forces and ligamentous tension.

          Since the knee has six degrees of freedom, only 6 equilibrium equations are available for the knee joint. The six equilibrium equations are derived from the assumption that the joint reaction moments are absorbed by the muscles in order to keep the knee stable in rotation. These six equilibrium equations are insufficient to solve for the 13 unknown forces. An optimization method is therefore used to calculate the muscle forces and knee joint reaction forces by minimizing a specific objective function (optimization procedure). The use of optimization methods to calculate muscle forces during specific activities has been referred to as inverse dynamic optimization(IDO).

Optimization

           Optimization methods use computer algorithms to determine the parameters that produce the minimum value for an objective function or optimization criteria, "J", within specific constraints. In other words, the values derived from optimization calculations represent the best fit for equations that are constructed on the basis of assumptions about knee mechanics. The success of the technique depends upon accurate formulation of both the constraints and the objective function to be minimized. The closer these equations represent the actual behavior of the system being studied, the more likely it is that the values derived will be accurate.

Prediction of Agonist Knee Muscle Forces Using Inverse Dynamic Optimization

             In the study of the knee, the values which can be measured include the external forces, femoral and tibial motion. Optimization methods are used to calculate the more difficult to measure muscle and joint contact forces from these data by determining the values for these parameters that will produce the minimum value for the objective function (J) within the constraints. For knee mechanics the constraints are defined by the six equilibrium equations; the requirement for non-negative muscle forces (i.e., the muscles can only generate tensile forces); and the upper bounds of muscle stresses.1-4

          The IDO method has been useful for predicting agonist muscle forces from the measured three-dimensional kinematics and kinetics of the joint segments, but has not been able to predict antagonistic muscle forces accurately. For example, Collins and co-workers found that inverse dynamic optimization did not predict the substantial antagonist muscle activity that is detected when electromyography is used to analyze normal gait.5 Much effort has been devoted to formulating a reasonable optimization criterion "J" which could reflect the physiological activities developed within the agonist and antagonist muscles which control the knee.3, 6-10 As a result, numerous linear, nonlinear, and physiological optimization criteria have been proposed. However, a realistic prediction of antagonistic muscle activity is still a challenge in biomechanics.

Attempts to Predict Antagonist Muscle Forces Using Inverse Dynamic Optimization

           We recently analyzed muscle recruitment and its effect on joint reaction forces of the knee using the IDO method.11, 12 We found that many previous studies had simplified the knee using sagittal plane models.2, 4, 5, 8, 9, 13-19 This produces only one equilibrium equation (i.e., flexion/extension motion only) for use in the optimization procedure if only sagittal plane rotation is considered. When the sagittal plane is considered in isolation, non-physiologic rigid constraints to knee motion in the coronal and transverse planes are introduced. Due to a failure to account for external loads outside of the sagittal plane, sagittal plane models cannot accurately reflect the physiological activities of the knee.

          In an attempt to improve the ability of optimization methods to accurately predict anatogonist as well as agonist muscle forces across the knee, we formulated a model of the knee joint that accounts for knee motion in three degrees of joint rotation--flexion/extension, varus/valgus and internal/external rotation.11, 12 Our work--and that of Glitsch and Baumann20 - has demonstrated that antagonist muscles can be accurately predicted when knee joint rotation is considered in three orthogonal planes.

Current Work

More recently, we tested the hypothesis that any optimization criterion satisfying the efficiency criteria of neuromuscular control (i.e., minimization of energy expenditure) could be used to predict antagonistic muscle forces during flexion/extension motion of the knee provided that the inverse dynamic optimization procedure was properly formulated by involving the knee joint rotation in three orthogonal planes. In order to test this hypothesis, four typical optimization criteria were selected:

          We analyzed the muscle forces and joint reaction forces of the knee during an isokinetic flexion/extension exercise performed at an flexion/extension velocity of 60 degrees per-second.11, 12 This model was able to accurately predict quadriceps muscle forces during flexion motion and semitendinosus muscle forces during extension equally well using each of the four optimization criteria (Figure 2). Axial joint compressive forces were also accurately predicted with each optimization criteria (Figure 3). A joint reaction force greater than two times body weight was predicted during flexion (between 10-70¡ of flexion). Maximum compressive forces as much as five times body weight were predicted at an angle of 46¡ during the extension portion of the exercise.

Conclusions

         All four optimization criteria adopted in this study predicted similar antagonistic muscle activities, axial joint compression, and anterior-posterior tibial shear forces. The key to accurate prediction of muscle and joint reaction forces using the IDO method appears to be formulation of the equations to account for knee joint motion in three dimensions. This finding may help simplify future analyses. If joint reaction forces are of major interest, a simple linear optimization criterion may be able to predict muscle force results similar to those predicted by complex nonlinear or physiological optimization procedures.

Guoan Li, PhD is a Researcher at the Orthopaedic Biomechanics Laboratory, Beth Israel Deaconess Medical Center and Assistant Professor of Orthopaedic Surgery at Harvard Medical School

Harry Rubash, MD is Chief of the Department of Orthopaedics at Massachusetts General Hospital and Professor of Orthopaedic Surgery at Harvard Medical School

Address correspondence to:
Guoan Li, PhD; Orthopaedic Biomechanics Laboratory; Beth Israel Deaconess Medical Center; RN 115, 330 Brookline Ave.; Boston, MA 02215
email: gli@obl.caregroup.harvard.edu