A correction for axis misalignment in the joint angle curves representing knee movement in gait analysis

This paper derives a simple mathematical model relating changes in the orientations of the two Cartesian coordinate systems involved in recording knee movement and the varus-valgus and the internal-external rotation angles for describing the knee's motion. Rotation matrices are given for changing the orientations of the two Cartesian coordinate systems in such a way that the quadratic variations in the varus-valgus and in the external-internal angles are minimal. These estimated rotation matrices are used to correct for axis misalignment. The correction is calibrated by considering the impact of the new orientation of the thigh Cartesian coordinate system on the hip joint angles. The procedure is applied to kinematic data collected on normal subjects. The uncertainty about the specification of the thigh Cartesian coordinate system is shown to explain some of the between subject variability in the varus-valgus and in the internal-external rotation angles curves.     

Muscle forces predicted using optimization methods are coordinate system dependent

Optimization methods are widely used to predict in vivo muscle forces in musculoskeletal joints. Moment equilibrium at the joint center (usually chosen as the origin of the joint coordinate system) has been used as a constraint condition for optimization procedures and the joint reaction moments were assumed zero. This study, through the use of a three-dimensional elbow model, investigated the effect of coordinate system origin (joint center) location on muscle forces predicted using a nonlinear static optimization method. The results demonstrated that moving the origin of the coordinate system medially and laterally along the flexion-extension axis caused dramatic variations in the predicted muscle forces. For example, moving the origin of the coordinate system from a position 5mm medial to 5mm lateral of the geometric elbow center caused the predicted biceps force to vary from 12% to 46% and the brachialis force to vary from 80% to 34% of the total muscle loading. The joint reaction force reduced by 24% with this medial to lateral variation of the coordinate system origin location. This data revealed that the muscle forces predicted using the optimization method are sensitive to the coordinate system origin location due to the zero joint reaction moment assumption in the moment constraint condition. For accurate prediction of muscle load distributions using optimization methods, it is necessary to determine the accurate coordinate system origin location where the condition of a zero joint reaction moment is satisfied.     

Comparison of normal mode analyses on a small globular protein in dihedral angle space and Cartesian coordinate space

Normal mode analyses on the protein, bovine pancreatic trypsin inhibitor, in dihedral angle space and Cartesian coordinate space are compared. In Cartesian coordinate space it is found that modes of frequencies lower than 30 cm(-1) contribute 80% of the total mean-square fluctuation and are represented almost completely by motions in the dihedral angles. Bond angle and length fluctuations dominate in modes above 200 cm(-1), but contribute less than 2% to the total mean-square fluctuation. In the low-frequency modes a good correspondence between patterns of atomic displacements was found, but on average the root-mean-square fluctuations of the Cartesian coordinate modes are 13% greater than their dihedral angle counterparts. The main effect of fluctuations in the bond angles and lengths, therefore, is to allow the dihedral angles to become more flexible. As the important subspaces determined from the two methods overlap considerably, dihedral angle space analysis can be applied to proteins too large for Cartesian coordinate space analysis.     

A new non-orthogonal decomposition method to determine effective torques for three-dimensional joint rotation

This paper describes a new non-orthogonal decomposition method to determine effective torques for three-dimensional (3D) joint rotation. A rotation about a joint coordinate axis (e.g. shoulder internal/external rotation) cannot be explained only by the torque about the joint coordinate axis because the joint coordinate axes usually deviate from the principal axes of inertia of the entire kinematic chain distal to the joint. Instead of decomposing torques into three orthogonal joint coordinate axes, our new method decomposes torques into three "non-orthogonal effective axes" that are determined in such a way that a torque about each effective axis produces a joint rotation only about one of the joint coordinate axes. To demonstrate the validity of this new method, a simple internal/external rotation of the upper arm with the elbow flexed at 90 degrees was analyzed by both orthogonal and non-orthogonal decomposition methods. The results showed that only the non-orthogonal decomposition method could explain the cause-effect mechanism whereby three angular accelerations at the shoulder joint are produced by the gravity torque, resultant joint torque, and interaction torque. The proposed method would be helpful for biomechanics and motor control researchers to investigate the manner in which the central nervous system coordinates the gravity torque, resultant joint torque, and interaction torque to control 3D joint rotations.     

Numerical simulation for heat transfer in tissues during thermal therapy

In this paper, a mathematical model describing the process of heat transfer in biological tissues for different coordinate system during thermal therapy by electromagnetic radiation is studied. The boundary value problem governing this process has been solved using Galerkin’s method taking B-polynomial as basis function. The system of ordinary differential equation in unknown time variable, thus obtained, is solved by homotopy perturbation method. The effect of thermal conductivity, antenna power constant, surface temperature, and blood perfusion rate on temperature for different coordinates are discussed. It has been observed that the process is faster in spherical symmetric coordinates in comparison to axisymmetric coordinate and faster in axisymmetric in comparison to Cartesian coordinate.     

Why protein R-factors are so large: a self-consistent analysis

The R-factor and R-free are commonly used to measure the quality of protein models obtained in X-ray crystallography. Well-refined protein structures usually have R-factors in the range of 20-25%, whereas intrinsic errors in the experimental data are usually around 5%. We use molecular dynamics simulations to perform a self-consistent analysis by which we determine the major factors contributing to large values of protein R-factors. The analysis shows that significant R-factor values can arise from the use of isotropic B-factors to model anisotropic protein motions and from coordinate errors. Even in the absence of coordinate errors, the use of isotropic B-factors can cause the R-factors to be around 10%; for coordinate errors smaller than 0.2 A, the two errors types make similar contributions. The inaccuracy of the energy function used and multistate protein dynamics are unlikely to make significant contributions to the large R-factors.     

The effect of using different coordinate systems on in-vivo hip angles can be estimated from computed tomography images

Measurements of hip kinematics inherently depend on the coordinate system in which they are derived, yet the effect of the coordinate system definition on calculations of hip angles is not well-understood. Herein, hip angles calculated during dynamic activities were compared using coordinate systems described in the literature. In-vivo kinematic data of 24 participants (13 males) were analyzed during gait and the anterior impingement test using dual fluoroscopy and model-based tracking. Two coordinate systems for the pelvis (anterior pelvic plane, International Society of Biomechanics [ISB]) and three coordinate systems for the femur (table top plane with two definitions of the superior-inferior axis, ISB) were evaluated. Bony landmarks visible on computed tomography (CT) images were identified to establish each coordinate system and used as the basis to calculate differences in hip angles between coordinate system pairs. In the analysis during gait, the maximum differences derived from various coordinate system definitions were 6.7° ± 5.5° for flexion, 7.7° ± 2.1° for rotation, and 5.5° ± 0.7° for adduction. For the anterior impingement test, the differences were 8.1° ± 5.9°, 7.1° ± 1.2°, and 5.3° ± 0.7°, respectively. Landmark-based analysis using CT images could estimate these dynamic differences with errors less than 1.0°. Our results indicate that hip angles can be accurately transformed to angles calculated in different coordinate systems by accounting for the inherent bony anatomy. This information may aid in the interpretation of results across biomechanical studies of the hip.     

Visuo-motor control: giving the brain a hand

Sensory information is acquired in spatial coordinate systems linked to sense organs, yet movement must be executed in coordinate systems linked to motor effector organs. Neurophysiological experiments are yielding new insights into how the brain transforms coordinate systems to facilitate movement.     

The role of computed tomographic and digital radiographic techniques in stereotactic procedures for electrode implantation and mapping, and lesion localization

This paper describes a computer-based system for analyzing stereotactic CT scans and angiograms. Simple Plexiglas frames containing metallic marker pellets and rods are affixed to the sides of the frame during CT scanning and angiography. The images of these markers are recognized by the computer program and used to compute the frame coordinate system. Coordinates of a lesion on CT and angiogram images may be readily computed and coordinate sets specified by the operator may be displayed on the imager, along with positions of implanted electrodes and a numerical indication of recorded activity for each site.     

Retinotopic mapping of categorical and coordinate spatial relation processing in early visual cortex

Spatial relations are commonly divided in two global classes. Categorical relations concern abstract relations which define areas of spatial equivalence, whereas coordinate relations are metric and concern exact distances. Categorical and coordinate relation processing are thought to rely on at least partially separate neurocognitive mechanisms, as reflected by differential lateralization patterns, in particular in the parietal cortex. In this study we address this textbook principle from a new angle. We studied retinotopic activation in early visual cortex, as a reflection of attentional distribution, in a spatial working memory task with either a categorical or a coordinate instruction. Participants were asked to memorize a dot position, with regard to a central cross, and to indicate whether a subsequent dot position matched the first dot position, either categorically (opposite quadrant of the cross) or coordinately (same distance to the centre of the cross). BOLD responses across the retinotopic maps of V1, V2, and V3 indicate that the spatial distribution of cortical activity was different for categorical and coordinate instructions throughout the retention interval; a more local focus was found during categorical processing, whereas focus was more global for coordinate processing. This effect was strongest for V3, approached significance in V2 and was absent in V1. Furthermore, during stimulus presentation the two instructions led to different levels of activation in V3 during stimulus encoding; a stronger increase in activity was found for categorical processing. Together this is the first demonstration that instructions for specific types of spatial relations may yield distinct attentional patterns which are already reflected in activity early in the visual cortex.     

A kinematic model of the human ankle

The spatial gross motion of the foot with respect to the shank is modelled as rotations about two fixed ankle axes: the upper ankle rotation axis (plantarflexion/dorsiflexion) and the subtalar rotation axis (inversion/eversion). The positions of the axes are determined by externally visible bony landmarks of the lower leg and are measured for a living subject. The model input data are the plantarflexion/dorsiflexion and inversion/eversion rotation angles; the model output is a 4 × 4 transformation matrix which quantitatively describes the relative position of a foot coordinate system with respect to a shank coordinate system.     

A motion-decomposition approach to address gimbal lock in the 3-cylinder open chain mechanism description of a joint coordinate system at the glenohumeral joint

In this study, the standard-sequence properties of a joint coordinate system were implemented for the glenohumeral joint by the use of a set of instantaneous geometrical planes. These are: a plane that is bound by the humeral long axis and an orthogonal axis that is the cross product of the scapular anterior axis and this long axis, and a plane that is bounded by the long axis of the humerus and the cross product of the scapular lateral axis and this long axis. The relevant axes are updated after every decomposition of a motion component of a humeral position. Flexion, abduction and rotation are then implemented upon three of these axes and are applied in a step-wise uncoupling of an acquired humeral motion to extract the joint coordinate system angles. This technique was numerically applied to physiological kinematics data from the literature to convert them to the joint coordinate system and to visually reconstruct the motion on a set of glenohumeral bones for validation.     

A procedure for determining angular positional data relative to the principal axes of the human body

This paper describes a simple computational procedure for determining angular displacement-time histories of human motion from three-dimensional cine data. The method is based on algebraic transformations of coordinates and coordinate axes. Through these coordinate transformations data was acquired for a multi-axial tumbling skill to illustrate angular displacement-time data relative to the moving coordinate system described by the human body through space.     

Correlated conformational fluctuations during enzymatic catalysis: Implications for catalytic rate enhancement

Correlated enzymatic conformational fluctuations are shown to contribute to the rate of enhancement achieved during catalysis. Cytidine deaminase serves as a model system. Crystallographic temperature factor data for this enzyme complexed with substrate analog, transition-state analog, and product are available, thereby establishing a measure of atomic scale conformational fluctuations along the (approximate) reaction coordinate. First, a neural network-based algorithm is used to visualize the decreased conformational fluctuations at the transition state. Second, a dynamic diffusion equation along the reaction coordinate is solved and shows that the flux velocity through the associated enzymatic conformation space is greatest at the transition state. These results suggest (1) that there are both dynamic and energetic restrictions to conformational fluctuations at the transition state, (2) that enzymatic catalysis occurs on a fluctuating potential energy surface, and (3) a form for the potential energy. The Michaelis-Menten equations are modified to describe catalysis on this fluctuating potential energy profile, leading to enhanced catalytic rates when fluctuations along the reaction coordinate are appropriately correlated. This represents a dynamic tuning of the enzyme for maximally effective transformation of the ES complex into EP.     

Ambivalence in modelling oblique saccades

Human oblique saccades might be made by synchronized but independent vertical and horizontal pulse generators; we call this the Cartesian theory. Another hypothesis is that the oblique amplitude and angle are coded centrally and trigonometricallyderived signals are sent to the horizontal and vertical muscles (the polar coordinate theory). We took a Cartesian model and cross-coupled the two generators to produce saccades identical to those of a polar coordinate model. This is disproof by counterexample: the experimental evidence claimed to support the polar coordinate model does not necessarily do so. Moreover, the behavior reported for oblique saccades is so variable, contradictory, idiosyncratic, and species-dependent that any model of the central organization of oblique saccades is probably premature.     

Loal three-dimensional MHD stability in equilibrium optimized coordinates

The linear equation for ideal magnetohydrodynamic ballooning modes in three-dimensional configurations is derived in the coordinate system that is optimal for the representation of the equilibrium state. The magnetic field lines in this coordinate system, however, are not straight. The form of the Mercier criterion that is currently in use is recovered from the asymptotic analysis of the ballooning equation. To determine the parallel-current density, a magnetic differential equation expressed in the optimal coordinates must be inverted.     

Perspectives in the coordinate regulation of cell cycle events in Synechococcus

The concepts of cell theory and the notions of coordinate regulation of the cell cycle have been known for centuries but the conundrum of coordinate regulation of the cell cycle remains to be resolved. The unique characteristics of the cell division cycle of Synechococcus, a photosynthetic bacterium, suggest the existence of a complex network of light/dark responsive gene regulatory factors that coordinate its cell cycle events. Evaluation of the highly ordered cell cycle of Synechococcus led to the construction of workable models that coordinate the cell cycle events. A central issue in bacterial cell growth is the elucidation of the genetic regulatory mechanisms that coordinate the cell cycle events. Synechococcus, a unicellular cyanobacterium, displays a peculiar cell growth cycle. In the light growth conditions, a highly ordered and sequentially coordinated appearances of r-protein synthesis, rRNA synthesis, DNA replication, chromosome segregation, and cell septum formation occur (Figs 1, 2A). Cell membrane syntheses occur predominantly during mid-cell cycle and cell division period. Synthesis of thylakoid (=photosynthetic apparatus) is thought to occur during mid-cell cycle and coincides with a period of peak phospholipid synthesis and oxygen production (Csatorday and Horvath, 1977; Asato, 1979). Cell wall syntheses occur in short discontinuous periods throughout the cell cycle and during cell division (Asato, 1984). Distinct D1 (=G1), C (S) and D2 (=G2) periods as defined by Cooper and Helmstetter (1968) are observed in synchronized cultures of Synechococcus (Asato, 1979). When light grown cultures are placed in the dark, the ongoing cell cycles are aborted in the dark (Fig. 3A) and cell divisions do not occur (Asato, 1983; Marino and Asato, 1986). Upon re-exposure of the cell cultures to the light growth conditions, about 14 h later, new cell cycles are re-initiated. These characteristics of cell growth are considered to be expressions of a unique strategy of obligate phototrophic mode of growth to perpetuate their species (Asato, 2003). Nevertheless, the intermediate metabolism, the synthesis of building block molecules, the genetics and molecular biology in the formation of major macromolecules are similarto heterotrophs such as E. coli. In any case, the genes that are involved in the formation of the cellular structures and the genes that control the orderly appearances of the cell cycle events must be coordinated by novel genetic mechanisms. Currently, there are no known physiological/physical mechanisms, growth rate dependent factors or traditional genetic regulatory mechanisms that could explain the coordinate regulation of the cell cycle events in bacteria (Newton and Ohta, 1992; Vinella and D'Ari, 1995; Donachie, 2001; Margolin and Bernander, 2004). Because the genetic mechanisms of coordinate regulation of cell cycle events in bacteria are largely unexplained, the questions on how Synechococcus coordinates the cell cycle events present a difficult problem to resolve. Nevertheless, the problems with regard to the coordinate regulation of the cell cycle events of Synechococcus must be considered. Possible solutions are developed and described in this article. The proposed schemes do not exclude the formation of other genetic mechanisms on the regulation of cell cycle events in Synechococcus. Although the cell cycle of Synechococcus is not widely known, the issues on the coordinate regulation of the cell cycle events are not trivial since similar regulatory mechanisms most likely occur in other prokaryotes.     

Enhancing backbone sampling in Monte Carlo simulations using internal coordinates normal mode analysis

Normal mode methods are becoming a popular alternative to sample the conformational landscape of proteins. In this study, we describe the implementation of an internal coordinate normal mode analysis method and its application in exploring protein flexibility by using the Monte Carlo method PELE. This new method alternates two different stages, a perturbation of the backbone through the application of torsional normal modes, and a resampling of the side chains. We have evaluated the new approach using two test systems, ubiquitin and c-Src kinase, and the differences to the original ANM method are assessed by comparing both results to reference molecular dynamics simulations. The results suggest that the sampled phase space in the internal coordinate approach is closer to the molecular dynamics phase space than the one coming from a Cartesian coordinate anisotropic network model. In addition, the new method shows a great speedup (∼5–7×), making it a good candidate for future normal mode implementations in Monte Carlo methods.