Identi®cation of optimal strategies for increasing whole arm strength using Karush±Kuhn±Tucker multipliers Richard E. Hughes, Michael G. Rock, Kai-Nan An * Department of Orthopedic Research, Mayo Clinic/Mayo Foundation, Biomechanics Lab, 200 First West Street SW, Rochester, MN 55905, USA Received 28 July 1998; accepted 22 March 1999 Abstract Objective. The purpose of this study was to develop a computer model for identifying muscles critical to improving functional upper extremity strength. Design. A three-dimensional biomechanical model of the upper extremity was developed, and the predictions were compared to maximal arm strength data collected from healthy volunteers. Background. Although several optimization-based mathematical models of the shoulder have been developed, none have utilized the mathematical properties of the Karush±Kuhn±Tucker multipliers to eciently estimate the eect of strengthening individual muscles on functional strength of the whole arm. Methods. A static three-dimensional biomechanical model of the glenohumeral, radio-humeral, ulno-humeral and wrist joints was developed for predicting maximal hand exertion forces. The model was formulated as a linear program. Constraints consisted of moment equilibrium conditions and limits on maximum and minimum allowable muscle forces. Predicted arm strengths were compared to maximal pull strength measurements made on 10 subjects (5 male; 5 female). The task involved pulling toward the mid- sagittal plane of the body with the arm ¯exed 45 degrees. The Karush±Kuhn±Tucker variables associated with the maximal limits on muscle force were computed to estimate the eect of altering the strength of individual muscles on functional arm strength. Results. Maximum pull strengths were predicted well by the model. Karush±Kuhn±Tucker values ranged from 0 (for muscles not at their upper force limits) to 0.11 for the ¯exor carpi radialis and pectoralis major muscles. Karush±Kuhn±Tucker multipliers were found to be insensitive to the assumed speci®c tension of muscle. Conclusions. Upper extremity strength can be predicted from musculoskeletal geometry and physiology using linear program- ming. Relevance Karush±Kuhn±Tucker multipliers associated with the muscle force upper limits give insight into the eect of strengthening in- dividual muscles on whole arm exertion strength. Such an analysis may provide insight into the development of optimal rehabil- itation protocols. Ó 1999 Elsevier Science Ltd. All rights reserved. Keywords: Strength; Optimization; Arm; Shoulder; Rehabilitation 1. Introduction Functional arm strength is necessary for activities of daily living. Muscular contraction acts to generate tor- ques about joints, and the skeleton acts as a lever system to convert the muscular torques into functional strength (e.g. force that can be exerted against objects in the environment). Although it is easy to clinically assess strength de®cits of individual joints resulting from pa- thology, the ability to generate forces against objects in the environment is more important. The relationship between individual muscle insuciency and functional strength is complex, because of the three-dimensional nature of the musculoskeletal system. Biomechanical models can provide insight into the relationship between musculoskeletal parameters and functional strength. A detailed three-dimensional bio- mechanical model of the arm that includes muscle physiology and musculoskeletal geometry could be used to derive the relationship between changes in strength of individual muscles and changes in functional strength. Such a model could provide information on which muscles are most critical to restoration of arm function. www.elsevier.com/locate/clinbiomech Clinical Biomechanics 14 (1999) 628±634 * Corresponding author. E-mail: an@mayo.edu. 0268-0033/99/$ - see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 8 - 0 0 3 3 ( 9 9 ) 0 0 0 2 2 - 4