A biomechanical model of the lumbosacral joint during lifting activities

A biomechanical model of the lumbosacral region was constructed for the purpose of systematically studying the combined stresses and strains on the local ligaments, muscles and disc tissue during sagittal plane two-handed lifting. The model was validated in two ways. The first validation was a comparison of experimental study results with model predictions. In general predictions compared very reasonably with observed values of several authors with the exception of strain predictions on the articular ligaments. Second, a sensitivity analysis was performed over a wide range of lifting tasks. The predicted stress/strain values followed anticipated patterns and were of reasonable magnitudes. On the basis of the results of the sensitivity analysis it was concluded that typical lifting tasks can lead to excessive disc compressive forces, muscle moment generation requirements, and possibly lumbodorsal fascia strains. Conversely, annulus rupture of a healthy disc due to overstrain appears very unlikely.     

Interpolation of segment Euler angles can provide a robust estimation of segment angular trajectories during asymmetric lifting tasks

Video-based field methods that estimate L5/S1 net joint moments from kinematics based on interpolation in the sagittal plane of joint angles alone can introduce a significant error on the interpolated joint angular trajectory when applied to asymmetric dynamic lifts. Our goal was to evaluate interpolation of segment Euler angles for a wide range of dynamic asymmetric lifting tasks using cubic spline methods by comparing the interpolated values with the continuous measured ones. For most body segments, the estimated trajectories of segment Euler angles have less than 5° RMSE (in each dimension) with 5-point cubic spline interpolation when there is no measurement error of interpolation points. Sensitivity analysis indicates that when the measurement error exists, the root mean square error (RMSE) of estimated trajectories increases. Comparison among different lifting conditions showed that lifting a load from a high initial position yielded a smaller RMSE than lifting from a low initial position. In conclusion, interpolation of segment Euler angles can provide a robust estimation of segment angular trajectories during asymmetric lifting when measurement error of interpolation points can be controlled at a low level.     

Existence of a limiting pattern for a system of nonlinear equations describing interpopulation competition

A system of arbitrarily many nonlinear ordinary differential equations which can be interpreted as describing competition between populations is studied here. It is found that limits exist for the system given specified constraints on the “signal function” which describes input-output relations at the level of single populations. Existence of limits is shown by means of a function which records which variable in the system is growing fastest at a given time i.e. who is winning the competition. Generalizations are discussed to sigmoid signal functions which appear in models of pattern discrimination by neuron populations.     

Bottom-up estimation of joint moments during manual lifting using orientation sensors instead of position sensors

L5/S1, hip and knee moments during manual lifting tasks are, in a laboratory environment, frequently established by bottom-up inverse dynamics, using force plates to measure ground reaction forces (GRFs) and an optoelectronic system to measure segment positions and orientations. For field measurements, alternative measurement systems are being developed. One alternative is the use of small body-mounted inertial/magnetic sensors (IMSs) and instrumented force shoes to measure segment orientation and GRFs, respectively. However, because IMSs measure segment orientations only, the positions of segments relative to each other and relative to the GRFs have to be determined by linking them, assuming fixed segment lengths and zero joint translation. This will affect the estimated joint positions and joint moments. This study investigated the effect of using segment orientations only (orientation-based method) instead of using orientations and positions (reference method) on three-dimensional joint moments. To compare analysis methods (and not measurement methods), GRFs were measured with a force plate and segment positions and/or orientations were measured using optoelectronic marker clusters for both analysis methods. Eleven male subjects lifted a box from floor level using three lifting techniques: a stoop, a semi-squat and a squat technique. The difference between the two analysis methods remained small for the knee moments: <4%. For the hip and L5/S1 moments, the differences were more substantial: up to 8% for the stoop and semi-squat techniques and up to 14% for the squat technique. In conclusion, joint moments during lifting can be estimated with good accuracy at the knee joint and with reasonable accuracy at the hip and L5/S1 joints using segment orientation and GRF data only.     

Reconstruction of equilibrium trajectories and joint stiffness patterns during single-joint voluntary movements under different instructions

 A method for reconstructing joint compliant characteristics during voluntary movements was applied to the analysis of oscillatory and unidirectional elbow flexion movements. In different series, the subjects were given one of the following instructions: (1) do not intervene voluntarily; (2) keep the trajectory; (3) in cases of perturbations, return back to the starting position as quickly as possible (only during unidirectional movements). Under the instruction ‘keep trajectory’, the apparent joint stiffness increased by 50% to 250%. During oscillatory movements, this was accompanied by a decrease in the maximal difference between the actual and equilibrium joint trajectories and, in several cases, led to a change in the phase relation between the two trajectories. The coefficients of correlation between joint torque and angle were very high (commonly, over 0.9) under the ‘do not intervene’ instruction. They dropped to about 0.6 under the ‘keep trajectory’ and to about 0.3 under the ‘return back’ instructions. Under these two instructions, the low values of the coefficients of correlation did not allow reconstruction of segments of equilibrium trajectories and joint stiffness values in all the subjects. The results provide further support for the λ-version of the equilibrium-point hypothesis and for using the instruction ‘do not intervene voluntarily’ to obtain reproducible time patterns of the central motor command. Received: 14 December 1993/Accepted in revised form: 16 April 1994     

Application of the describing function to the Danziger-Elmegreen equations

The describing function method is used as a guide to the behaviour of the solutions of the equations of Danziger and Elmergreen, proposed as a model of periodic catatonia. The method suggests that whenever the equilibrium point is unstable it is surrounded by a stable closed periodic orbit. This is confirmed in specific cases by computation.     

Lumbar loading during lifting: a comparative study of three measurement techniques.

Low back loading during occupational lifting is thought to be an important causative factor in the development of low back pain. In order to regulate spinal loading in the workplace, it is necessary to measure it accurately. Various methods have been developed to do this, but each has its own limitations, and none can be considered a "gold standard". The purpose of the current study was to compare the results of three contrasting techniques in order to gain insight into possible sources of error to which each is susceptible. The three techniques were a linked segment model (LSM), an electromyographic (EMG)-based model, and a neural network (NN) that used both EMG and inertial sensing techniques. All three techniques were applied simultaneously to calculate spinal loading when eight volunteers performed a total of eight lifts in a laboratory setting. Averaged results showed that, in comparison with the LSM, the EMG technique calculated a 25.5+/-33.4% higher peak torque and the NN technique a 17.3+/-10.5% lower peak torque. Differences between the techniques varied with lifting speed and method of lifting, and could be attributed to differences in anthropometric assumptions, antagonistic muscle activity, damping of transient force peaks by body tissues, and, specific to the NN, underestimation of trunk flexion. The results of the current study urge to reconsider the validity of other models by independent comparisons.     

A stochastic flowering model describing an asynchronically flowering set of trees

A general stochastic model is presented that simulates the time course of flowering of individual trees and populations, integrating the synchronization of flowering both between and within trees. Making some hypotheses, a simplified expression of the model, called the 'shoot' model, is proposed, in which the synchronization of flowering both between and within trees is characterized by specific parameters. Two derived models, the 'tree' model and the 'population' model, are presented. They neglect the asynchrony of flowering, respectively, within trees, and between and within trees. Models were fitted and tested using data on flowering of Psidium cattleianum observed at study sites at elevations of 200, 520 and 890 m in Reunion Island. The 'shoot' model fitted the data best and reproduced the strong irregularities in flowering shown by empirical data. The asynchrony of flowering in P. cattleianum was more pronounced within than between trees. Simulations showed that various flowering patterns can be reproduced by the 'shoot' model. The use of different levels of organization of the general model is discussed.     

Sudden loading during a dynamic lifting task: a simulation study

It is believed that nurses risk the development of back pain as a consequence of sudden loadings during tasks in which they are handling patients. Forward dynamics simulations of sudden loads (applied to the arms) during dynamic lifting tasks were performed on a two-dimensional whole-body model. Loads were in the range of -80 kg to 80 kg, with the initial load being 20 kg. Loading the arm downwards with less than that which equals a mass of 20 kg did not change the compressive forces on the spine when compared to a normal lifting motion with a 20 kg mass in the hands. However when larger loads (40 kg to 80 kg extra in the hands) were simulated, the compressive forces exceeded 13,000 N (above 3400 N is generally considered a risk factor). Loading upwards led to a decrease in the compressive forces but to a larger backwards velocity at the end of the movement. In the present study, it was possible to simulate a fast lifting motion. The results showed that when loading the arms downwards with a force that equals 40 kg or more, the spine was severely compressed. When loading in the opposite direction (unloading), the spine was not compressed more than during a normal lifting motion. In practical terms, this indicates that if a nursing aide tries to catch a patient who is falling, large compressive forces are applied to the spine.     

Solution of a set of Maxwell-Lorentz equations for a ring relativistic electron beam

Nonlinear solutions to a set of Maxwell’s equations and the relativistic equations of electron motion are obtained that describe the equilibrium of a high-power ring relativistic electron beam against the background of immobile ions. By transforming the basic equations, a set of equations for a three-component vortex vector field is derived that describes ring beam configurations for plasma confinement. An example of a numerical calculation of the steady state of a compact beam torus of immobile ions and relativistic electrons is presented.     

Solution of the equations describing the flow of Na in a semi-infinite composite system containing a membrane

Solutions to the flow equations in a semi-infinite composite system consisting of protein solution and a phospholipid sol with a membrane at the interface are described in this paper. The composite system was contained in a length of precision bore capillary tubing and the flow rate in the system was followed by using labelled NaCl and a continuous monitoring apparatus.     

System of kinetic equations describing large-scale processes in collisionless space plasma

A system of kinetic equations describing relatively slow large-scale processes in collisionless magnetoplasma structures with a spatial resolution of about the characteristic gyroradius is derived. Plasma is assumed to be quasineutral, while the magnetic and electric fields are determined by the instantaneous distributions of the particle and current densities and the stress tensor of all plasma components in the longrange instantaneous interaction approximation. A special version of equations is derived for the case of magnetized electrons described by the Vlasov equation in the drift approximation. The obtained system of equations can be used to develop a global numerical kinetic model of the Earth’s magnetosphere with a spatial resolution of about 100 km, as well as local models of certain regions of the Earth’s magnetosphere with a higher resolution.     

The kinematics of the scapulae and spine during a lifting task

Simultaneous motion of the scapula and humerus is widely accepted as a feature of normal upper limb movement, however this has usually been investigated under conditions in which purposeful, functional tasks were not considered. The aim of this study was to investigate the synchrony and coordination of the constituent 3D movements of the shoulder girdle and trunk, during a functional activity. 45 healthy women, aged between 20 and 80 years, performed a simple lifting task, moving a loaded box from a shelf at waist level to one at shoulder level and then reversed the movement, during which the linear and angular motions of the scapulae, upper and lower thoracic spine and upper limbs were monitored and analysed using cross-correlation techniques. Results indicated a close and consistent set of coordinated movement patterns, which suggest biomechanical invariance in the responses of the structures adjacent to the upper limb during such a lifting task. These scapulohumeral relationships were, however, more constant and phase-locked when there was a specific purpose to the movement than during periods in which the arm was lowered without load. There were no age-related differences in any movement responses.     

Verification of empirical equations describing subsidence rate of peatland in Central Poland

Wetlands Ecology and Management - Currently, due to prolonged soil drought, dehydrated peat soils are particularly exposed to subsidence and, as a consequence, even to disappearance from the...     

Reconstruction of equilibrium trajectories during whole-body movements

The framework of the equilibrium-point hypothesis was used to reconstruct equilibrium trajectories (ETs) of the ankle, hip and body center of mass during quick voluntary hip flexions (`Japanese courtesy bow') by standing subjects. Different spring loads applied to the subject's back were used to introduce smooth perturbations that are necessary to reconstruct ETs based on a series of trials at the same task. Time patterns of muscle torques were calculated using inverse dynamics techniques. A second-order linear model was employed to calculate the instantaneous position of the spring-like joint or center of mass characteristic at different times during the movement. ETs of the joints and of the center of mass had significantly different shapes from the actual trajectories. Integral measures of electromyographic bursts of activity in postural muscles demonstrated a relation to muscle length corresponding to the equilibrium-point hypothesis. Received: 3 March 1997 / Accepted in revised form: 2 November 1998     

Approximate analytic solution of a simple system of kinetic equations describing the enzymatic synthesis of nucleic acids

Macroscopic kinetic models describing the process of polymerase chain reaction (PCR) are currently solved only by numerical methods, which hampers the development of effective software algorithms for processing the results of the reaction. This paper considers the application of the homotopy perturbation method for obtaining approximate analytical solution of the simplest system of enzymatic kinetic equations describing the synthesis of nucleic acid molecules during PCR. The resulting approximate analytic solution with high accuracy reproduces the results of a numerical solution of the system in a wide range of ratios of enzyme and substrate concentrations both for the case of a large excess of the substrate over the enzyme and vice versa.     

Homeostatic control of thyroxin concentration expressed by a set of linear differential equations

The purpose of this work is to express current concepts on the relationship between the rates of secretion of thyroxin and of thyroid stimulating hormone (TSH) by a set of linear differential equations (two attempts have been made previously in this direction; cf. Roston,Bull. Math. Biophysics,21, 271–282, 1959; Danziger and Elmergreen,Bull. Math. Biophysics,16, 15–21, 1954), and to show that the solutions to these equations fulfill two criteria: that they correctly express the previously observed behavior of thyroxin and TSH, and that they allow certain predictions to be made which are amenable to experimental verification or disproval by currently existing techniques. This mathematical model is necessarily only an approximation of reality.