Neuroscience Vol. 40, No. 3, pp. 793-804, 1991 Printed
in Great
03064522/91 $3.00 + 0.00 Pergamon Press plc 0 1991 IBRO
Britain
PHYSIOLOGICAL PREDICTION OF MUSCLE FORCES-II. APPLICATION TO ISOKINETIC EXERCISE K. R. KAUFMAN,* K.-N. AN,~ W. J. LITCHY~ and E. Y. S. CrrAot$ *Motion Analysis Laboratory, Children’s Hospital, San Diego, CA 92123, U.S.A. TBiomechanics Laboratory, Department of Orthopedics and ISection of Electromyography, Division of Clinical Nemophysiology, Department of Neurology, Mayo Clinic/Mayo Foundation, Rochester, MN 55905, U.S.A. Ahstraet-The successful application of a physiological model of the musculoskeletal system capable of accounting for nonequilibrium dynamic loading and predicting individual muscle forces in the knee is presented. The model incorporates rigid-body mechanics and musculoskeletal physiology. Unknown muscle and joint contact forces outnumber the equilibrium equations resulting in an indeterminant
problem. Mathematical optimization is utilized to resolve the indeterminacy. The model is used to estimate individual muscle forces during isokinetic exercise. Five subjects were tested at speeds of 6o”/s and lSO”/s.A newly proposed optimal criterion, minimizing muscular activation, results in muscle force predictions which have significantly higher correlations with myoelectric activity than other linear and nonlinear optimal criteria. The results demonstrate that properly constrained linear programming methods do not limit the number of active muscles and allow for uniform recruitment of the active muscles.
For the analysis of human movements, relationships between the unknown internal (muscle and joint) forces and the known external forces can be obtained from force and moment equilibrium equations. The number of unknown forces will in general exceed the number of equations, so a unique solution cannot be found. This mechanical redundancy of the musculoskeletal system provides the body with versatility in selecting the muscle to perform a given movement. Usually the total required effort is shared among several muscles (synergistic muscle action). The distribution is not necessarily unique: intra- and interindividual differences can be observed. Despite these differences, general patterns of load sharing between the muscles can still be expected during normal activities. For example, the average temporal pattern of muscle action during the human gait cycle is well defined.” This suggests that the control of muscle action is governed by certain physiological criteria, which result in more or less unique load sharing patterns. A mathematical formulation of the physiological criteria to supplement the equations of equilibrium provides the starting point for solving a redundant biomechanical system. The hypothesis that efficiency principles are inherent to neuromuscular control is the rationale for using optimization techniques. With this approach the indeterminate problem can be solved, often uniquely. The objective function to be optimized represents the physiological principles governing the muscular load sharing. The minimization §To whom correspondence should be addressed. Abbreuiutions: EMG, electromyograph; PCSA, physiological cross-sectional area.
of this function, subject to the constraints of the force and moment equations and limits on muscle force, results in a solution for the muscular load sharing. The previous article2r presented the theoretical basis for an optimization method which incorporates physiological constraints into the prediction of muscular forces. In the present paper, this method is used to develop an analytical model of the knee joint to predict muscle forces during isokinetic exercise. The features of this model are presented and comparisons are made between several optimal criteria. EXPERIMENTAL PROCEDURES Subjects The right knee of five male subjects was used for this study. The age of the participants was 27 + 2 (mean & S.D.) years. They weighed 81 + 13 kg, and had a height of 178 + 5 cm. To be included in the study, the subjects had to have no previous surgery and no history of knee disorders. Examinations revealed stable ligaments with a full range of knee motion symmetric with the other side, no effusions, and no generalized ligamentous laxity in any of the subjects. Radiographic examinations confirmed the clinical findings. Equipment Triaxial electrogoniometer. Angular displacement of the knee was measured with a triaxial electrogoniometer (Fig. 1). The triaxial electrogoniometer has been used in the measurement of three-dimensional rotations of the lower extremity joints during gait. lo Experimental and theoretical justifications have been performed with satisfactory results.9v’0The device consists of three precision, miniature, rotational potentiometers mounted in a gimbal mechanism, joined to record the angular excursions in three planes according to the gyroscopic mechanism. This is the special property of the triaxial goniometer which proves threedimensional angular motion in terms of the unique Eulerian angles without sequence dependence. The potentiometer cluster was aligned with the knee axis of rotation by a
793
K. R. KAUFMANet ul.
Fig. I. Experimental setup. The subject was seated on a bench which was inclined backward 15” from the horizontal. A triaxial electrogoniometer was mounted on the subject’s knee to measure the three-dimensional motion of the lower limb. A Cybex II isokinetic dynamometer was used to provide a load. A load cell was placed on the Cybex arm to measure the three orthogonal force components acting on the shank. physical therapist familiar with the test equipment. Selfaligning mechanisms and instrument mounting guides were utilized to minimize error due to improper placement. Normal cross-talk due to external placement of the device and the non-colinear alignment of the axes of rotation was corrected mathematically.9 The goniometer was strapped to the upper and lower limb segments with elastic straps and, with a mass of only 85g, did not inhibit free movement. The decision was made to utilize a triaxial electrogoniometer affixed to the leg since the position of the knee cannot be assumed to remain constant relative to the rotational axis of the Cybex isokinetic dynamometer. Movement of the knee relative to the Cybex machine would result in measurement errors of joint orientation if the goniometer of the Cybex d~~ometer had been utilized in this study. The position of the electrogoniometer relative to the knee remained constant since it was attached directly to the leg by elastic straps (Fig. I). Accordingly, utilization of the three degrees of freedom electrogoniometer reduced motion measurement errors in this study. Three-component load cell. The torque measurements recorded by the Cybex dynamometer were not used for data analysis. The Cybex machine was simply used to provide an isokinetic movement. In order to increase the accuracy of the moment measurements, a custom-designed load cell was developed for this project (NK Biotechnical Engineering, Minneapolis, MN). This device was attached to the arm of the Cybex machine and accurately measured the applied force on the shank about three orthogonal axes (Fig. 1). The load cell was calibrated using a loading device traceable to
the National Bureau of Standards. The calibration scheme was linear with the output signal proportional to the applied load. The resolution of the load cell was 50.4N with a nonlinearity of ~0.3% F.S. and a hysterisis of
in Great
03064522/91 $3.00 + 0.00 Pergamon Press plc 0 1991 IBRO
Britain
PHYSIOLOGICAL PREDICTION OF MUSCLE FORCES-II. APPLICATION TO ISOKINETIC EXERCISE K. R. KAUFMAN,* K.-N. AN,~ W. J. LITCHY~ and E. Y. S. CrrAot$ *Motion Analysis Laboratory, Children’s Hospital, San Diego, CA 92123, U.S.A. TBiomechanics Laboratory, Department of Orthopedics and ISection of Electromyography, Division of Clinical Nemophysiology, Department of Neurology, Mayo Clinic/Mayo Foundation, Rochester, MN 55905, U.S.A. Ahstraet-The successful application of a physiological model of the musculoskeletal system capable of accounting for nonequilibrium dynamic loading and predicting individual muscle forces in the knee is presented. The model incorporates rigid-body mechanics and musculoskeletal physiology. Unknown muscle and joint contact forces outnumber the equilibrium equations resulting in an indeterminant
problem. Mathematical optimization is utilized to resolve the indeterminacy. The model is used to estimate individual muscle forces during isokinetic exercise. Five subjects were tested at speeds of 6o”/s and lSO”/s.A newly proposed optimal criterion, minimizing muscular activation, results in muscle force predictions which have significantly higher correlations with myoelectric activity than other linear and nonlinear optimal criteria. The results demonstrate that properly constrained linear programming methods do not limit the number of active muscles and allow for uniform recruitment of the active muscles.
For the analysis of human movements, relationships between the unknown internal (muscle and joint) forces and the known external forces can be obtained from force and moment equilibrium equations. The number of unknown forces will in general exceed the number of equations, so a unique solution cannot be found. This mechanical redundancy of the musculoskeletal system provides the body with versatility in selecting the muscle to perform a given movement. Usually the total required effort is shared among several muscles (synergistic muscle action). The distribution is not necessarily unique: intra- and interindividual differences can be observed. Despite these differences, general patterns of load sharing between the muscles can still be expected during normal activities. For example, the average temporal pattern of muscle action during the human gait cycle is well defined.” This suggests that the control of muscle action is governed by certain physiological criteria, which result in more or less unique load sharing patterns. A mathematical formulation of the physiological criteria to supplement the equations of equilibrium provides the starting point for solving a redundant biomechanical system. The hypothesis that efficiency principles are inherent to neuromuscular control is the rationale for using optimization techniques. With this approach the indeterminate problem can be solved, often uniquely. The objective function to be optimized represents the physiological principles governing the muscular load sharing. The minimization §To whom correspondence should be addressed. Abbreuiutions: EMG, electromyograph; PCSA, physiological cross-sectional area.
of this function, subject to the constraints of the force and moment equations and limits on muscle force, results in a solution for the muscular load sharing. The previous article2r presented the theoretical basis for an optimization method which incorporates physiological constraints into the prediction of muscular forces. In the present paper, this method is used to develop an analytical model of the knee joint to predict muscle forces during isokinetic exercise. The features of this model are presented and comparisons are made between several optimal criteria. EXPERIMENTAL PROCEDURES Subjects The right knee of five male subjects was used for this study. The age of the participants was 27 + 2 (mean & S.D.) years. They weighed 81 + 13 kg, and had a height of 178 + 5 cm. To be included in the study, the subjects had to have no previous surgery and no history of knee disorders. Examinations revealed stable ligaments with a full range of knee motion symmetric with the other side, no effusions, and no generalized ligamentous laxity in any of the subjects. Radiographic examinations confirmed the clinical findings. Equipment Triaxial electrogoniometer. Angular displacement of the knee was measured with a triaxial electrogoniometer (Fig. 1). The triaxial electrogoniometer has been used in the measurement of three-dimensional rotations of the lower extremity joints during gait. lo Experimental and theoretical justifications have been performed with satisfactory results.9v’0The device consists of three precision, miniature, rotational potentiometers mounted in a gimbal mechanism, joined to record the angular excursions in three planes according to the gyroscopic mechanism. This is the special property of the triaxial goniometer which proves threedimensional angular motion in terms of the unique Eulerian angles without sequence dependence. The potentiometer cluster was aligned with the knee axis of rotation by a
793
K. R. KAUFMANet ul.
Fig. I. Experimental setup. The subject was seated on a bench which was inclined backward 15” from the horizontal. A triaxial electrogoniometer was mounted on the subject’s knee to measure the three-dimensional motion of the lower limb. A Cybex II isokinetic dynamometer was used to provide a load. A load cell was placed on the Cybex arm to measure the three orthogonal force components acting on the shank. physical therapist familiar with the test equipment. Selfaligning mechanisms and instrument mounting guides were utilized to minimize error due to improper placement. Normal cross-talk due to external placement of the device and the non-colinear alignment of the axes of rotation was corrected mathematically.9 The goniometer was strapped to the upper and lower limb segments with elastic straps and, with a mass of only 85g, did not inhibit free movement. The decision was made to utilize a triaxial electrogoniometer affixed to the leg since the position of the knee cannot be assumed to remain constant relative to the rotational axis of the Cybex isokinetic dynamometer. Movement of the knee relative to the Cybex machine would result in measurement errors of joint orientation if the goniometer of the Cybex d~~ometer had been utilized in this study. The position of the electrogoniometer relative to the knee remained constant since it was attached directly to the leg by elastic straps (Fig. I). Accordingly, utilization of the three degrees of freedom electrogoniometer reduced motion measurement errors in this study. Three-component load cell. The torque measurements recorded by the Cybex dynamometer were not used for data analysis. The Cybex machine was simply used to provide an isokinetic movement. In order to increase the accuracy of the moment measurements, a custom-designed load cell was developed for this project (NK Biotechnical Engineering, Minneapolis, MN). This device was attached to the arm of the Cybex machine and accurately measured the applied force on the shank about three orthogonal axes (Fig. 1). The load cell was calibrated using a loading device traceable to
the National Bureau of Standards. The calibration scheme was linear with the output signal proportional to the applied load. The resolution of the load cell was 50.4N with a nonlinearity of ~0.3% F.S. and a hysterisis of
