Moment arms and lengths of human upper limb muscles as functions of joint angles

Modeling of musculoskeletal structures requires accurate data on anatomical parameters such as muscle lengths (MLs), moment arms (MAs) and those describing the upper limb position. Using a geometrical model of planar arm movements with three degrees of freedom, we present, in an analytical form, the available information on the relationship between MAs and MLs and joint angles for thirteen human upper limb muscles. The degrees of freedom included are shoulder flexion/extension, elbow flexion/extension, and either wrist flexion/extension (the forearm in supination) or radial/ulnar deviation (the forearm in mid-pronation). Previously published MA/angle curves were approximated by polynomials. ML/angle curves were obtained by combining the constant values of MLs (defined by the distance between the origin and insertion points for a specific upper limb position) with a variable part obtained by multiplying the MA (joint radius) and the joint angle. The MAs of the prime wrist movers in radial/ulnar deviation were linear functions of the joint angle (R2 ≥ 0.9904), while quadratic polynomials accurately described their MAs during wrist flexion/extensions. The relationship between MAs and the elbow angle was described by 2nd, 3rd or 5th-order polynomials (R2 ≥ 0.9904), with a lesser quality of fit for the anconeus (R2 = 0.9349). In the full range of angular displacements, the length of wrist, elbow and shoulder muscles can change by 8.5, 55 and 200%, respectively.