Scapular kinematics.

The International Society of Biomechanics (ISB) recommended standard for describing the motion of the upper-extremity including the scapula [18] has been adopted by several models [10–12, 16, 17]. This standard defines the kinematics of the scapula in terms of a body-fixed Euler rotation sequence for a frame fixed to the acromion (Anglus Acromialis) of the scapula relative to a frame fixed to the clavicle, assuming a ball joint between these segments. Although ISB-recommended angles have been the standard description for scapular kinematics for nearly a decade, a consistent set of “normal” scapular kinematics for a standardized set of tasks has yet to be established. In contrast, clinical gait analysis routinely utilizes deviation from normative data [19] as a tool to assess the severity of gait pathology and to evaluate treatment outcomes. Normative data for shoulder motions would provide a stronger scientific basis for evaluation of shoulder function.

The rotations (Euler angles) and translations of the scapula are coupled [20]. As the scapula moves on the thoracic surface, it rotates to maintain congruity with the thoracic surface while under load. Several models kinematically constrain the scapula to glide on the thorax surface (e.g., [10, 11, 14]), in agreement with the motion described by Dvir and Berme [21] from X-ray photographs. This gliding plane motion has been represented by one or two fixed scapula points constrained to an ellipsoid surface [10], leading to 5 or 4 degrees of freedom (dof) of the scapula, respectively. These models enable the full motion of the scapula, but have shown to result in unrealistic scapula motion (e.g., [14]) and are computationally costly due to the simultaneous solution of constraint forces. Recently, Bolsterlee et al. [22] demonstrated that the constrained kinematic optimization solution implemented in the Delft Shoulder Elbow Model [10] can lead to nonphysical positions and may cause inaccuracies in model predictions when the scapulothoracic constraints must be satisfied, especially at higher humeral elevation angles. Another approach specifies the orientation of the scapula and clavicle bones by means of regression equations as a function of the humeral angles [23] for example [12, 13, 24, 25] which is often referred to as the “scapulohumeral” or shoulder rhythm. These models can emulate healthy flexion and abduction movements, but fail to represent independent scapula and humerus motions like shrugging, and by definition cannot discriminate between normal and pathological scapular kinematics. Therefore, neither the rigidly constrained scapula nor the coupled-regression solutions have adequately modeled scapular kinematics.

Scapula Dynamics.

The motion of the scapula is influenced by muscle forces and joint reaction forces, which arise from the thoracic surface as well as the acromioclavicular and glenohumeral joints. Muscle forces are dependent on scapular kinematics, which determine the length, velocity and effective moment arms of the large muscles that attach to the scapula. Joint reaction forces, especially between the scapula and thorax, are primarily a consequence of muscle loads. The more the structure of the scapulothoracic contact restricts the motion of the scapula in the direction of muscle forces, the greater the reaction forces will be.

To solve for the acceleration of a constrained multibody system, both applied forces and constraint reaction forces must be accounted for and the mass matrix must be inverted. Here a fundamental numerical problem arises since applied forces are large and the mass of segments are small, thus the solution for system accelerations approaches a singularity. This problem has been widely ignored. Several models of the upper extremity (e.g., [12]) were not built to solve the dynamics problem, while others struggle with poor performance [10] or eliminate the complex motion of the scapula [26]. Chadwick and colleagues recently achieved real-time performance during forward dynamics by implicit numerical integration methods that address the stiff dynamics due to scapulothoracic contact forces [16]; however, the model cannot be used for purely kinematics analyses and the accuracy of simulated motion has not been verified against experimental measures.

The inability to accurately and reliably describe the kinematics of the scapula and/or to enforce physiological scapula motion during dynamic simulations (Table 1) necessitates a new approach to modeling and simulating scapula mechanics.

Scapulothoracic Joint Definition.

We characterized the scapulothoracic joint by the translation and rotation of the scapula on the surface of the thorax modeled as an ellipsoid [9, 10]. The scapulothoracic joint defines the kinematics of a joint frame on the scapula with respect to a joint frame on the thorax body. We parameterized scapulothoracic motion (Fig 1) by four position coordinates: 1) abduction-adduction [10], 2) elevation-depression [10], 3) upward rotation [27], and 4) internal rotation or “winging” [10, 27, 28]. The first two coordinates, abduction and elevation, locate the origin of the joint reference frame on the thoracic (ellipsoid) surface and are analogous to how longitude and latitude are used to locate a point on the Earth’s surface. The third coordinate, upward rotation, rotates the scapula about the normal to the thoracic surface at the origin of the joint frame. The fourth coordinate, winging, rotates the scapula about a longitudinal axis in the plane of the scapula (which remains tangent to the thoracic surface) and enables the medial border and the Angulus Inferior of the scapula to lift off the thoracic surface. The origin of the joint frame on the scapula is specified as the centroid of the Angulus Acromialis (AA), Trigonum Spinae (TS) and Angulus Inferior (AI) markers, which are the anatomical landmarks recommended by the ISB [18]. The orientation of the joint frame on the scapula is also computed according to the ISB recommendations; however, the scapulothoracic joint frame is rotated -90° about the scapular Y-axis to define upward rotation as a positive rotation about the scapulothoracic joint frame’s Z-axis (Fig 1).

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Fig 1. The four modeled degrees of freedom of the scapulothoracic joint.

The joint reference frame on the scapula (axes X,Y,Z) is used to locate the scapula with respect to the thorax. The joint reference frame on the scapula is computed according to the ISB recommendations [18] (shown as X_S, Y_S, Z_S), however our joint origin is located at the centroid of the anatomical markers used to define the joint frame instead of the Angulus Acromialis and its axes are rotated -90° about Y (to enable positive upward rotation about Z). The joint frame on the thorax defines the center of the scapulothoracic surface modeled as an ellipsoid (red shaded surface). Abduction (adduction) followed by elevation (depression) locate the joint frame origin of the scapula (blue) on the ellipsoid fixed to the parent thorax body (green). The scapula rotates upward (downward) about the normal to the surface (scapula Z-axis). Internal rotation or “winging” is a positive rotation about the Y-axis of the joint frame in the scapular plane, which remains tangent to the thoracic surface.

https://doi.org/10.1371/journal.pone.0141028.g001

The Mobilizer Formulation.

Internally, the scapulothoracic joint is composed of two mobilizers: an ellipsoid mobilizer [9] and a pin mobilizer [29]. A mobilizer can be thought of as the mathematical dual of a constraint—while a constraint removes dofs from a model, a mobilizer grants dofs to a body, which we term the body’s “mobilities”. Unlike typical engineering joints, which can be reconstituted by a series of ideal 1-dof joints [14], basically pins and sliders, the mobilizer enables smooth and continuous spatial motion between two bodies parameterized by 1–6 internal coordinate speeds or mobilities. This formulation is utilized by Simbody [29], an open source multibody solver available from Simtk.org, and serves as the computational foundation of OpenSim [30, 31].

In Simbody, the kinematics of a body, B, with respect to its parent, P, are fully described by the following four mobilizer equations: (1) (2) (3) (4)

Eq (1) describes the position transform, ^PX^B, composed of the rotation matrix, R, and translation vector, p, of a mobilizer frame, B, fixed in the body (B_o frame) with respect to a parent mobilizer frame, P, fixed in the parent body (P_o) (e.g., scapula body and thorax as parent in Fig 1). The spatial velocity, ^PV^B (composed of angular velocity vector ω and linear velocity vector v) in Eq (2), and acceleration, ^PA^B in Eq (3), of B with respect to P, are specified by the mobilizer matrix, H, and its time derivative, . The evolution of the internal coordinates, q, is governed by the differential relationship Eq (4) with the mobilities, u, according to the kinematic coupling matrix, N. While constant H matrix captures the kinematics of most typical engineering joints (e.g., pin or ball-and-socket), if we do not assume H(q) to be constant, H(q) can represent curvilinear paths in the 6-dimensional spatial kinematics basis of a body with respect to its parent. For instance, H(q) can represent movement about a helix or slider path that is curved in space (curvy slot), or other coupled motions, which are typical of biological joints [29].

The ellipsoid mobilizer of the scapulothoracic joint enables the abduction (adduction), elevation (depression) and upward (downward) rotation of the scapula with respect to the thorax. The scapulothoracic ellipsoid mobilizer has the same 3 mobilities, u, as a ball-and-socket, which are the components of the angular velocity of the child (scapula) joint frame with respect to the parent joint frame on the thorax (center of the ellipsoid). However, rather than grant 3 additional mobilities for spatial translations, which would then have to be constrained, the translations are coupled to the orientation of the scapula: (5) where q is the scapula-fixed Euler angle sequence relating to abduction (θ₁), elevation (θ₂) and upward rotation (θ₃, about the ellipsoid normal). The vector p(q) describes the translation of the scapula’s joint frame, S, on the surface of an ellipsoid, fixed in the thorax joint frame, T: (6) where n = n(q) is the surface normal expressed in the thorax, and h, w, d are the ellipsoid radii along the axes of T that correspond to height, width and depth, respectively. The angular velocity remains the same function of u as for a ball-and-socket joint, so N is (7)

Since there are coupled linear velocities as a consequence of the rotating normal vector, the ellipsoid mobilizer matrix has the following form: (8)

In this case is nonzero and it describes the angular velocity contributions to the body’s linear acceleration: (9)

The mobilizer matrix ^TH^S and its derivative span exactly and map only to the subspace of the permissible-motion manifold of an ellipsoid surface. Representing the same motion with conventional joints would require a free joint (6 dofs) and 3 constraint equations, for a total of 9 differential algebraic equations versus the mobilizer formulation’s 3 ordinary differential equations.

An additional pin mobilizer enables the 4^th dof of the scapulothoracic joint, which is internal rotation of the scapula about a winging axis in the plane of the scapula that enables the medial border and the inferior angle (Angulus inferior) to raise off the thoracic surface (Fig 1, winging axis is the scapula Y-axis). In the scapulothoracic joint, the pin’s axis is automatically aligned with the “winging” axis specified in the joint frame on the scapula and can be set by the user.

Multibody Dynamics.

Eqs (1–4) define the kinematics of a body with respect to its parent, thus the spatial kinematics of each body can be computed by recursively traversing the bodies from the ground or root body (e.g., thorax) out to the most distal bodies (e.g., hand). From the most distal bodies we apply forces to determine the spatial acceleration of the body in terms of the generalized coordinates and mobilities (q, u) and recurse inward towards the ground to form the complete set of system accelerations, . This approach describes a recursive Newton-Euler algorithm for solving multibody dynamics equations [32]. When position and velocity kinematics constraint equations are differentiated and expressed in terms of the system accelerations, the complete system dynamics can be written as a system of differential-algebraic equations [33] of this form [29]: (10.a) (10.b) where M is the system mass matrix in the mobility space; G is the Jacobian matrix of the kinematic constraints; b contains the coordinate and mobility (speed) terms of the differentiated constraints; λ are the constraint Lagrange multipliers corresponding to components of the constraint forces; f_applied is the net applied generalized force, and f_inertial is the net velocity-based generalized force due to rotating frames. Given system accelerations () we can solve for the required applied forces (inverse dynamics) or use the same set of equations to evaluate accelerations due to applied forces and integrate from initial coordinates and mobilities (q_o, u_o) to compute the trajectory of coordinates and mobilities forward in time (i.e., a forward dynamics simulation).

In the case of the scapulothoracic joint, the motion of the scapula is fully captured by the mobilities of the ellipsoid and pin mobilizers and there are no constraints associated with the joint, however constraints do appear in the general form of the system dynamics (Eq 10) for a shoulder model (e.g. acromioclavicular ball-joint constrains 3 translations). Instead, the joint (mobilizer) reaction loads can be determined by solving the Newton-Euler dynamics per body, for example for the scapula: (11) where M_s is the spatial inertia matrix of the scapula and is the spatial acceleration of the scapula in ground (by expressing the acceleration from Eq 3 in ground); f_i is an applied spatial force on the scapula; ^TR_G^S is the unknown thoracic reaction load applied to the scapula expressed in ground; is the known glenohumerual joint reaction load on the scapula expressed in ground; and is the acromial (scapuloclavicular) constraint force (λ_ac) on the scapula expressed in ground. The system accelerations () and constraint forces (λ) are determined from Eq 10. Simbody determines the reaction loads as part of computing inter-body spatial forces [34].

The scapulothoracic joint was implemented as an OpenSim Joint and included as part of the OpenSim 3.2 application via a plugin library. Model scaling, inverse kinematics, inverse dynamics and forward dynamics simulations were performed in OpenSim 3.2.

Shoulder Model.

We constructed a shoulder model consisting of the thorax, scapula, clavicle, and humerus to evaluate the accuracy, robustness to noise and computational speed of the scapulothoracic joint. The sternoclavicular joint was modeled by a universal joint enabling protraction-retraction and elevation-depression of the clavicle, since axial rotation cannot be accurately measured and the conoid ligament limits axial rotation. The acromioclavicular joint was modeled as a ball joint. The glenohumeral joint was modeled as a custom gimbal joint using the ISB standard coordinates for angle of the elevation plane, elevation, and internal rotation [18]. Segment dimensions and joint locations were obtained from Holzbaur et al. [12] and the segment inertial properties were obtained from Breteler et al. [35].

Reconstruction of Experimental Kinematics.

We evaluated the kinematic accuracy of the scapulothoracic joint by comparing thorax, scapula and humerus model marker locations to measured marker data from bone-pin motion-capture experiments [28]. Ludewig et al. measured marker locations with magnetic sensors taped to the thorax on the anterior aspect of the sternum, rigidly attached into the scapula at the scapular spine at the acromial base, into the lateral third of the clavicle and distal to the deltoid insertion on the lateral aspect of the humerus, for three tasks: arm flexion (forward raise), arm abduction (to the side) and internal/external rotation of the upper arm when the humerus is abducted at the glenohumeral joint by 90° with respect to the thorax. The magnetic sensor device (Flock of Birds—Ascension Technology, Burlington, Vermont) had an accuracy of 1.8mm in location and 0.5° in orientation under static conditions at a recording frequency of 120frames/s. Ludewig et al. also used a digitizing stylus to register the anatomical landmarks of the Angulus inferior and Trigonum spinae scapulae to virtual markers (AI and TS) in the rigidly secured sensor frame, which had a measurement accuracy < 1mm [28]. Combining the orientation error with a mean distance of 10cm for two landmarks of interest with static location and stylus errors, yields a measurement accuracy of 1.85mm for the location of a scapula marker in the ground reference frame.

We scaled upper-extremity segment dimensions to obtain a match between model markers and corresponding experimental marker locations identifying the same bony landmarks. Scaling functionality was provided by the Scale Tool in OpenSim [30, 31].

We customized the location and orientation of the thoracic ellipsoid surface, with respect to the thorax origin and the location and orientation of the joint reference frame with respect to the scapula origin on the acromion according to the ISB recommendations [18], to construct a subject specific model. The origin of the scapulothoracic joint reference frame on the scapula was located at the centroid of the measured anatomical markers (AA, TS, AI) and its axes rotated 90° (see Fig 1). The remaining parameters were determined by minimizing the squared errors between model and subject marker locations from nine poses evenly selected from each task. For each task the ellipsoid orientation and radii were adjusted because marker errors were most sensitive to these parameters and account for rib-cage expansion and spinal bending/twisting differences between the tasks.

Once the upper extremity model was appropriately scaled, we performed inverse kinematics to determine the joint coordinates of the model that minimized the weighted-least-squares error between model marker and corresponding experimental marker locations at each frame of the movement trial [6, 30]. We applied uniform weights for bone-pin marker locations since these markers were all assumed to have the same accuracy. To evaluate the model’s kinematic accuracy, we calculated the root-mean-squared (RMS) errors for each marker across each trial.

Inverse Kinematics from Noisy Marker Data.

We evaluated the robustness of kinematics determined from the scapulothoracic joint model by introducing errors in the experimental bone-pin marker locations to simulate the effects of using surface markers attached to the skin. The purpose of the noise model was to test the effect of random error and bias in the markers movements relative to the bone, such as stretching, translation and rotational offsets, as well as warping. The noise model combines a systematic bias (as a function of the movement) and a random component (for the offset direction and white noise), and we tested a range of error magnitudes. We specified the mean and standard deviation of the noise distribution for each marker at each instant in the movement. We used skin marker errors (location on skin surface to corresponding fixed bony landmark location) measured by MRI [36] to specify the mean of the added noise. The mean of the noise over the task trial was modeled as a Gaussian function over time where the timing of the peak coincided with the time of maximum arm elevation (or rotation), which was the posture in which the maximum skin error was measured by Matsui and colleagues [36]. The Gaussian function gradually scaled the mean skin noise (i.e. the bias) to zero in the neutral static start and end postures. Mean skin noise levels were varied by 1mm increments from 1mm to the mean measured maximum skin surface error (41mm, [36]) at maximum elevation to represent varying amounts of skin movement artifact. We used the standard deviations of the skin surface error from experiments [36] in the simulated noise distribution to reflect random errors, which were included at all noise levels. Noise was added as offsets in marker locations at every instant in time with the direction of the offset also selected at random, sampled from a uniform distribution, for each marker and held constant for each trial. We synthesized 100 trials of “noisy” marker data for each noise level to generate a dataset containing a host of skin movement artifacts dictated by the offset directions of the markers. For example, markers whose noise directions were similar represented a shifting bias (translational offset); opposite directions represented stretching or compressing (depending on whether the directions were away from or towards one another), while other directions reflect rotational shifts and combinations of artifacts.

We computed Euler angles from the noisy marker data according to the ISB standard [18] after lowpass filtering the marker data at 5Hz to duplicate clinical estimates of scapular kinematics with respect to the thorax. We also determined Euler angles from the model’s markers affixed to the scapula after performing inverse kinematics from noisy marker data for each noise level (41) and each activity (3) for a total of 12,300 trials. We compared the mean error and standard deviations of Euler angles with and without the model, as well as those of the scapulothoracic joint coordinates, from trials with noise against those obtained from bone-pin data.

Computing Scapulothoracic Forces that Generate Shoulder Movements.

We next performed an inverse dynamics analysis to calculate the generalized forces necessary to produce the reconstructed shoulder motions (i.e., the joint (generalized) coordinates, velocities, and accelerations). Coordinates were determined from inverse kinematics of bone-pin measurements. A third order lowpass Butterworth digital filter with a 2Hz cut-off was used to filter the coordinate data and a generalized cross-validated quintic spline [37] was used to interpolate the filtered data. Spline functions for each coordinate were differentiated to obtain velocities and then again to obtain coordinate accelerations as inputs to inverse dynamics.

Forward Dynamics Simulation of Shoulder Motion.

The multibody dynamic equations of motion were used to determine the acceleration of the joint coordinates in response to applied forces. The velocities and accelerations were integrated forward in time to yield the position and velocity of the coordinates to generate a simulation of passive arm swing. Passive force elements prevented the scapula and humerus from reaching non-physiological positions. We initialized the simulation to be at rest at the maximum humerus elevation angles calculated from the bone-pin trial of each task, then allowed the arm to swing passively for a duration of 2s. The total system energy was computed and monitored during each simulation above. We used a variable step Runge-Kutta-Merson integration algorithm [38] with a relative accuracy of 1e-4 for all forward dynamics simulations, with constraints maintained to 0.01mm.