The Obstacle-Set Method for Representing Muscle Paths in Musculoskeletal Models

A computational method is introduced for modeling the paths of muscles in the human body. The method is based on the premise that the resultant muscle force acts along the locus of the transverse cross-sectional centroids of the muscle. The path of the muscle is calculated by idealizing its centroid path as a frictionless elastic band, which moves freely over neighboring anatomical constraints such as bones and other muscles. The anatomical constraints, referred to as obstacles, are represented in the model by regular-shaped, rigid bodies such as spheres and cylinders. The obstacles, together with the muscle path, define an obstacle set. It is proposed that the path of any muscle can be modeled using one or more of the following four obstacle sets: single sphere, single cylinder, double cylinder, and sphere-capped cylinder. Assuming that the locus of the muscle centroids is known for an arbitrary joint configuration, the obstacle-set method can be used to calculate the path of the muscle for all other joint configurations. The obstacle-set method accounts not only for the interaction between a muscle and a neighboring anatomical constraint, but also for the way in which this interaction changes with joint configuration. Consequently, it is the only feasible method for representing the paths of muscles which cross joints with multiple degrees of freedom such as the deltoid at the shoulder.     

A Three-Dimensional Musculoskeletal Model of the Human Knee Joint. Part 2: Analysis of Ligament Function

A three-dimensional model of the knee is used to study ligament function during anterior-posterior (a-p) draw, axial rotation, and isometric contractions of the extensor and flexor muscles. The geometry of the model bones is based on cadaver data. The contacting surfaces of the femur and tibia are modeled as deformable; those of the femur and patella are assumed to be rigid. Twelve elastic elements are used to describe the geometry and mechanical properties of the cruciate ligaments, the collateral ligaments, and the posterior capsule. The model is actuated by thirteen musculotendinous units, each unit represented as a three-element muscle in series with tendon. The calculations show that the forces applied during a-p draw are substantially different from those applied by the muscles during activity. Principles of knee-ligament function based on the results of in vitro experiments may therefore be overstated. Knee-ligament forces during straight a-p draw are determined solely by the changing geometry of the ligaments relative to the bones: ACL force decreases with increasing flexion during anterior draw because the angle between the ACL and the tibial plateau decreases as knee flexion increases; PCL force increases with increasing flexion during posterior draw because the angle between the PCL and the tibial plateau increases. The pattern of ligament loading during activity is governed by the geometry of the muscles spanning the knee: the resultant force in the ACL during isometric knee extension is determined mainly by the changing orientation of the patellar tendon relative to the tibia in the sagittal plane; the resultant force in the PCL during isometric knee flexion is dominated by the angle at which the hamstrings meet the tibia in the sagittal plane.     

A Dynamic Optimization Solution for Vertical Jumping in Three Dimensions

A three-dimensional model of the human body is used to simulate a maximal vertical jump. The body is modeled as a 10-segment, 23 degree-of-freedom (dof), mechanical linkage, actuated by 54 muscles. Six generalized coordinates describe the position and orientation of the pelvis relative to the ground; the remaining nine segments branch in an open chain from the pelvis. The head, arms, and torso (HAT) are modeled as a single rigid body. The HAT articulates with the pelvis via a 3 dof ball-and-socket joint. Each hip is modeled as a 3 dof ball-and-socket joint, and each knee is modeled as a 1 dof hinge joint. Each foot is represented by a hindfoot and toes segment. The hindfoot articulates with the shank via a 2 dof universal joint, and the toes articulate with the hindfoot via a 1 dof hinge joint. Interaction of the feet with the ground is modeled using a series of spring-damper units placed under the sole of each foot. The path of each muscle is represented by either a series of straight lines or a combination of straight lines and space curves. Each actuator is modeled as a three-element, Hill-type muscle in series with tendon. A first-order process is assumed to model muscle excitation-contraction dynamics. Dynamic optimization theory is used to calculate the pattern of muscle excitations that produces a maximal vertical jump. Quantitative comparisons between model and experiment indicate that the model reproduces the kinematic, kinetic, and muscle-coordination patterns evident when humans jump to their maximum achievable heights.     

A Three-Dimensional Musculoskeletal Model of the Human Knee Joint. Part 1: Theoretical Construction

A three-dimensional model of the knee is developed to study the interactions between the muscles, ligaments, and bones during activity. The geometry of the distal femur, proximal tibia, and patella is based on cadaver data reported for an average-size knee. The shapes of the femoral condyles are represented by high-order polynomials: the tibial plateaux and patellar facets are approximated as flat surfaces. The contacting surfaces of the femur and tibia are modeled as deformable, while those of the femur and patella are assumed to be rigid. Interpenetration of the femur and tibia is taken into account by modeling cartilage as a thin, linear, elastic layer, mounted on rigid bone. Twelve elastic elements describe the geometry and mechanical properties of the cruciate ligaments, the collateral ligaments, and the posterior capsule. The model is actuated by thirteen musculotendinous units, each unit modeled as a three-element muscle in series with tendon. The path of each muscle is approximated as a straight line, except where it contacts and wraps around bone and other muscles; changes in muscle paths are taken into account using data obtained from MRI. In the first part of this paper, the model is used to simulate passive knee flexion. Quantitative comparisons of the model results with experimental data reported in the literature indicate that the relative movements of the bones and the geometry of the ligaments and muscles in the model are similar to those evident in the real knee. In Part II, the model is used to describe knee-ligament function during anterior-posterior draw, axial rotation, and isometric knee-extension and knee-flexion exercises.