Method for the design and analysis of a non-linear anisotropic lower limb prosthetic socket

A simplified computer-based structural analysis procedure has been developed for evaluating the stresses in a lower limb prosthetic socket when subjected to external loads. This technique has been tested by studying a simple linear axisymmetric shell subjected to uniform loads. It was found that the stress and strain distributions obtained are close to those determined by a finite element technique. The method that has been proposed involves an incremental procedure, which can be used to analyse the behaviour of a prosthetic socket of irregular shape made from non-linear anisotropic materials. By employing the proposed procedure to study an example, in which a simplified lower limb prosthetic socket is subject to internal pressure and frictional forces, the effects of the material non-linearity have been found to be significant.     

Asymmetric spindle positioning

When a spindle is positioned asymmetrically in a dividing cell, the resulting daughter cells are unequal in size. Asymmetric spindle positioning is driven by regulated forces that can pull or push a spindle. The physical and molecular mechanisms that can position spindles asymmetrically have been studied in several systems, and some themes have begun to emerge from recent research. Recent work in budding yeast has presented a model for how cytoskeletal motors and cortical capture molecules can function in orienting and positioning a spindle. The temporal regulation of microtubule-based pulling forces that move a spindle has been examined in one animal system. Although the spindle positioning force generators have not been identified in most animal systems, the forces have been found to be regulated by both PAR polarity proteins and G-protein signaling pathways in more than one animal system.     

Analytic analysis of the force sharing among synergistic muscles in one- and two-degree-of-freedom models

Mathematical optimization of specific cost functions has been used in theoretical models to calculate individual muscle forces. Measurements of individual muscle forces and force sharing among individual muscles show an intensity-dependent, non-linear behavior. It has been demonstrated that the force sharing between the cat Gastrocnemius, Plantaris and Soleus shows distinct loops that change orientation systematically depending on the intensity of the movement. The purpose of this study was to prove whether or not static, non-linear optimization could inherently predict force sharing loops between agonistic muscles. Using joint moment data from a step cycle of cat locomotion, the forces in three cat ankle plantar flexors (Gastrocnemius, Plantaris and Soleus) were calculated using two popular optimization algorithms and two musculo-skeletal models. The two musculo-skeletal models included a one-degree-of-freedom model that considered the ankle joint exclusively and a two-degree-of-freedom model that included the ankle and the knee joint. The main conclusion of this study was that solutions of the one-degree-of-freedom model do not guarantee force-sharing loops, but the two-degree-of-freedom model predicts force-sharing loops independent of the specific values of the input parameters for the muscles and the musculo-skeletal geometry. The predicted force-sharing loops were found to be a direct result of the loops formed by the knee and ankle moments in a moment-moment graph.     

A task-specific validation of homogeneous non-linear optimisation approaches

In biomechanics, musculoskeletal models are typically redundant. This situation is referred to as the distribution problem. Often, static, non-linear optimisation methods of the form “min: φ(f) subject to mechanical and muscular constraints” have been used to extract a unique set of muscle forces. Here, we present a method for validating this class of non-linear optimisation approaches where the homogeneous cost function, φ(f), is used to solve the distribution problem. We show that the predicted muscle forces for different loading conditions are scaled versions of each other if the joint loading conditions are just scaled versions. Therefore, we can calculate the theoretical muscle forces for different experimental conditions based on the measured muscle forces and joint loadings taken from one experimental condition and assuming that all input into the optimisation (e.g., moment arms, muscle attachment sites, size, fibre type distribution) and the optimisation approach are perfectly correct. Thus predictions of muscle force for other experimental conditions are accurate if the optimisation approach is appropriate, independent of the musculoskeletal geometry and other input required for the optimisation procedure. By comparing the muscle forces predicted in this way to the actual muscle forces obtained experimentally, we conclude that convex homogeneous non-linear optimisation approaches cannot predict individual muscle forces properly, as force-sharing among synergistic muscles obtained experimentally are not just scaled versions of joint loading, not even in a first approximation.     

Stoichiometric modeling of Clostridium acetobutylicum fermentations with non-linear constraints.

A stoichiometric model of Clostridium acetobutylicum and related strains has been previously derived. The stoichiometric matrix of the model contains a singularity which has prevented the calculation of a unique set of fluxes which describe the primary metabolic activity. To resolve the singularity, we have developed a non-linear constraint relating the acetate and butyrate uptake fluxes. Subsequently, we developed a software package utilizing a model independent heuristic global optimization approach to solve the resultant non-linear problem. We have validated the use of the non-linear constraint by correlating calculated butyrate production pathway flux profiles with measured intracellular pH profiles. Finally, we examined a controlled batch fermentation to determine that the acid formation pathways play critical roles throughout solventogenesis. The broader usefulness of reformulating the stoichiometric model as a constrained minimization problem is discussed.     

Mechanical analysis of the sea-urchin lantern: the overall system in Paracentrotus lividus

The dental apparatus or Aristotle's lantern of sea-urchins is a complex system of interacting skeletal ossicles, joints, muscles and ligaments arranged in a rigorous geometry and involved in a variety of activities. In this paper we study the movement of the whole lantern system modelled as a rigid body. The model lantern is constrained at its apex by the peristomial membrane and its movement is controlled by five pairs of antagonistic forces (retractor and protractor muscles). The other main forces applied to the lantern are the elastic reactions of both muscles and ligamental structures (compass depressors and peristomial membrane). The lantern is allowed to perform vertical movements and lateral inclinations but cannot rotate around its main axis. The equilibrium conditions of the system have been found by means of a numerical iterative procedure for solving non-linear equations. The results of the present analysis allow simulation of the overall mechanical activity of the lantern taking into account the experimental data regarding active and passive muscular forces and the tensile constraints due to ligaments.     

Co-activation alters the linear versus non-linear impression of the EMG-torque relationship of trunk muscles

The use of electromyographic signals in the modeling of muscle forces and joint loads requires an assumption of the relationship between EMG and muscle force. This relationship has been studied for the trunk musculature and been shown to be predominantly non-linear, with more EMG producing less torque output at higher levels of activation. However, agonist-antagonist muscle co-activation is often substantial during trunk exertions, yet has not been adequately accounted for in determining such relationships. The purpose of this study was to revisit the EMG-moment relationship of the trunk recognizing the additional moment requirements necessitated due to antagonist muscle activity. Eight participants generated a series of isometric ramped trunk flexor and extensor moment contractions. EMG was recorded from 14 torso muscles, and the externally resisted moment was calculated. Agonist muscle moments (either flexor or extensor) were estimated from an anatomically detailed biomechanical model of the spine and fit to: the externally calculated moment alone; the externally calculated moment combined with the antagonist muscle moment. When antagonist activity was ignored, the EMG-moment relationship was found to be non-linear, similar to previous work. However, when accounting for the additional muscle torque generated by the antagonist muscle groups, the relationships became, in three of the four conditions, more linear. Therefore, it was concluded that antagonist muscle co-activation must be included when determining the EMG-moment relationship of trunk muscles and that previous impressions of non-linear EMG-force relationships should be revisited.     

Effect of dielectric constant of medium on protonation equilibria of ethylenediamine

Abstract

The effect of dielectric constant of medium on protonation equilibria has been studied by determining protonation constants of ethylenediamine pH metrically in various concentrations (0–60%v/v) of acetoni-trile– and ethylene glycol–water mixtures, at an ionic strength of 0.16mol L^?1 and at 303.0 K. MINIQUAD75 computer program has been used for the calculation of protonation constants. Linear and non-linear variations of step-wise protonation constants with reciprocal of dielectric constant of the solvent mixtures have been attributed to the dominance of the electrostatic and non-electrostatic forces, respectively. The trend is explained on the basis of solute–solute and solute–solvent interactions, solvation, proton transfer processes and dielectric constants of the media.     

A biomechanical model on muscle forces in the transfer of spinal load to the pelvis and legs.

Based on musculoskeletal anatomy of the lower back, abdominal wall, pelvis and upper legs, a biomechanical model has been developed on forces in the load transfer through the pelvis. The aim of this model is to obtain a tool for analyzing the relations between forces in muscles, ligaments and joints in the transfer of gravitational and external load from the upper body via the sacroiliac joints to the legs in normal situations and pathology. The study of the relation between muscle coordination patterns and forces in pelvic structures, in particular the sacroiliac joints, is relevant for a better understanding of the aetiology of low back pain and pelvic pain. The model comprises 94 muscle parts, 6 ligaments and 6 joints. It enables the calculation of forces in pelvic structures in various postures. The calculations are based on a linear/non-linear optimization scheme. To gain a better understanding of the function of individual muscles and ligaments, deviant properties of these structures can be preset. The model is validated by comparing calculations with EMG data from the literature. For agonistic muscles, good agreement is found between model calculations and EMG data. Antagonistic muscle activity is underestimated by the model. Imposed activity of modelled antagonistic muscles has a minor effect on the mutual proportions of agonistic muscle activities. Simulation of asymmetric muscle weakness shows higher activity of especially abdominal muscles.     

Modelling of non-linear elastic tissues for surgical simulation

Realistic modelling of the interaction between surgical instruments and human organs has been recognised as a key requirement in the development of high-fidelity surgical simulators. Primarily due to computational considerations, most of the past real-time surgical simulation research has assumed linear elastic behaviour for modelling tissues, even though human soft tissues generally possess non-linear properties. For a non-linear model, the well-known Poynting effect developed during shearing of the tissue results in normal forces not seen in a linear elastic model. Using constitutive equations of non-linear tissue models together with experiments, we show that the Poynting effect results in differences in force magnitude larger than the absolute human perception threshold for force discrimination in some tissues (e.g. myocardial tissues) but not in others (e.g. brain tissue simulants).     

Computer simulation of model lipid membrane dynamics

The motions of individual lipid molecules in a model membrane have been studied by computer simulation using the molecular dynamics technique. The intermolecular forces were of both the Lennard-Jones and the Coulomb type. The influence of temperature and electrical screening on the order-disorder transition was examined, and it was also found that this transition is initiated by the spontaneous generation of disclinations.     

The effects of extra chances to live on life table functions

A decline in mortality can be described as a process according to which the earlier forces of mortality become what has been defined here as the forces of debility. The main thesis is that individuals contract debilitating conditions now with the same force that used to result in deaths in the past. The difference in the two situations is that in the latter period, a certain proportion of the population can recover and thus escape death. Based on these assumptions, the relationships between the initial and the resulting life table functions have been developed in this paper. Among other things, the relative gain in life expectancy has been found to be inversely related to its initial value. Further, interesting results are obtained when a fixed proportion of the population can manage to survive a debilitating condition only once, only twice and so on. The limiting situation has been found to be equivalent to one in which the initial forces of mortality at all ages are reduced by a constant proportion.     

A quasi-static three-dimensional, mathematical, three-body segment model of the canine knee

A mathematical, three-dimensional, anatomically accurate model of the canine knee was created to determine the forces in the knee ligaments and the knee joint reaction forces during the stance phase of a slow walk. This quasi-static model considered both the tibio-femoral and patello-femoral articulations. The geometric and morphometric data of the hind limb were obtained from cadaver data. Muscle forces acting on the femur and the hip joint reaction force were determined by numerical optimization. Ligaments were modeled as non-linear-springs. Ligament material properties were obtained from the literature pertaining to the human knee. The model consists of-28 non-linear algebraic equations describing equilibrium of the femur and the patella, and geometric constraints. This system of equations was solved by a non-linear least-squares method. Results are presented for a knee with an intact cranial cruciate ligament (CCL) and for a knee with a ruptured CCL. Forces predicted to occur in the CCL by analysis of the model were found to be very similar to reported results of CCL forces measured in vivo in goats.     

Antagonistic activity of one-joint muscles in three-dimensions using non-linear optimisation

Non-linear optimisation, such as the type presented by R.D. Crowninshield and R.A. Brand [The prediction of forces in joint structures: Distribution of intersegmental resultants, Exercise Sports Sci. Rev. 9 (1981) 159], has been frequently used to obtain a unique set of muscle forces during human or animal movements. In the past, analytical solutions of this optimisation problem have been presented for single degree-of-freedom models, and planar models with a specific number of muscles and a defined musculoskeletal geometry. Results of these studies have been generalised to three-dimensional problems and for general formulations of the musculoskeletal geometry without corresponding proofs. Here, we extend the general solution of the above non-linear, constrained, planar optimisation problem to three-dimensional systems of arbitrary geometry. We show that there always exists a set of intersegmental moments for which the given static optimisation formulation will predict co-contraction of a pair of antagonistic muscles unless they are exact antagonists. Furthermore, we provide, for a given three-dimensional system consisting of single joint muscles, a method that describes all the possible joint moments that give co-contraction for a given pair of antagonistic muscles.     

Biomechanical aspects of the repair of intertrochanteric fractures

A two-dimensional model of a repaired intertrochanteric fracture has been proposed in order to estimate the forces transmitted by a sliding screw implant and at the fracture site. These forces have been estimated from the radiographs of 55 repairs which were previously graded as satisfactory or unsatisfactory. The unsuccessful group was found to have significantly higher fracture angles as well as increased forces and moments transmitted by the implant. The results support the view that increased loading of implant and therefore high stresses in the surrounding cancellous bone contribute to the failure of repairs.     

The steady-state kinetics of yeast phosphoglycerate kinase. Anomalous kinetic plots and the effects of salts on activity.

1. A re-investigation of the kinetics of yeast phosphoglycerate kinase in the direction of 1,3-bisphosphoglycerate formation has been carried out, covering a 1000-fold range in substrate concentrations. A variety of improved spectrophotometric and fluorimetric assay procedures have been used. 2. Kinetic plots proved to be non-linear for each variable substrate. A variety of checks have been carried out to show that this is not due to artifacts in the assay procedures or heterogeneity of the enzyme preparation. 3. The effects of a variety of salts on the activity of the enzyme have been examined. Most salts, especially those with multivalent anions, can cause activation of the enzyme, but inhibit at high concentration. 4. The salt effect is shown to be principally due to anions rather than cations, and not to ionic strength changes. Sulphate, as one of the most effective anions has been used in most comparisons. 5. Salt activation is steepest when the substrate concentrations are low; maximum activation has been about 5-fold with 0.2 mM MgATP and 0.2 mM 3-phosphoglycerate. Inhibition at the higher salt concentrations is strongest at the same substrate concentrations as when activation is steepest, indicating a link between the two effects. 6. The presence of 20 mM or more Na2SO4 converted non-linear kinetic plots to linear ones. A study of the kinetics in the presence of 40 mM Na2SO4 was interpreted in terms of a random sequential binding mechanism, with sulphate acting as a competitive inhibitor. 7. Possible explanations for these anomalous results are discussed in terms of several mechanisms which have been shown to apply in other systems.     

Ultraviolet microbeam irradiations of mitotic diatoms: investigation of spindle elongation

Our simple instrumentation for generating a UV-microbeam is described UV microbeam irradiations of the central spindle in the pennate diatom Hantzschia amphioxys have been examined through correlated birefringence light microscopy and TEM. A precise correlation between the region of reduced birefringence and the UV-induced lesion in the microtubules (MTs) of the central spindle is demonstrated. The UV beam appears to dissociate MTs, as MT fragments were rarely encountered. The forces associated with metaphase and anaphase spindles have been studied via localized UV-microbeam irradiation of the central spindle. These spindles were found to be subjected to compressional forces, presumably exerted by stretched or contracting chromosomes. Comparisons are made with the results of other writers. These compressional forces caused the poles of a severed anaphase spindle to move toward each other and the center of the cell. As these poles moved centrally, the larger of the two postirradiational central spindle remnants elongated with a concomitant decrease in the length of the overlap. Metaphase spindles, in contrast, did not elongate nor lose their overlap region. Our interpretation is that the force for anaphase spindle elongation in Hantzschia is generated between half-spindles in the region of MT overlap.     

Simulation of deformable models with the Poisson equation

In this paper, we present a new methodology for the deformation of soft objects by drawing an analogy between the Poisson equation and elastic deformation from the viewpoint of energy propagation. The potential energy stored due to a deformation caused by an external force is calculated and treated as the source injected into the Poisson system, as described by the law of conservation of energy. An improved Poisson model is developed for propagating the energy generated by the external force in a natural manner. An autonomous cellular neural network (CNN) model is established by using the analogy between the Poisson equation and CNN to solve the Poisson model for the real-time requirement of soft object deformation. A method is presented to derive the internal forces from the potential energy distribution. The proposed methodology models non-linear materials with the non-linear Poisson equation and thus non-linear CNN, rather than geometric non-linearity. It not only deals with large-range deformations, but also accommodates isotropic, anisotropic and inhomogeneous materials by simply modifying constitutive coefficients. A haptic virtual reality system has been developed for deformation simulation with force feedback. Examples are presented to demonstrate the efficiency of the proposed methodology.     

Simulation of deformable models with the Poisson equation

In this paper, we present a new methodology for the deformation of soft objects by drawing an analogy between the Poisson equation and elastic deformation from the viewpoint of energy propagation. The potential energy stored due to a deformation caused by an external force is calculated and treated as the source injected into the Poisson system, as described by the law of conservation of energy. An improved Poisson model is developed for propagating the energy generated by the external force in a natural manner. An autonomous cellular neural network (CNN) model is established by using the analogy between the Poisson equation and CNN to solve the Poisson model for the real-time requirement of soft object deformation. A method is presented to derive the internal forces from the potential energy distribution. The proposed methodology models non-linear materials with the non-linear Poisson equation and thus non-linear CNN, rather than geometric non-linearity. It not only deals with large-range deformations, but also accommodates isotropic, anisotropic and inhomogeneous materials by simply modifying constitutive coefficients. A haptic virtual reality system has been developed for deformation simulation with force feedback. Examples are presented to demonstrate the efficiency of the proposed methodology.     

Formation of three-dimensional structures in supported lipid bilayers

The creation of three-dimensional structures in supported lipid bilayers has been examined. In bilayers, shape transformations can be triggered by adjusting a variety of parameters. Here, it is shown that bilayers composed of phosphatidylcholine and phosphatidic acid can be induced to reversibly form cap structures when exposed to an asymmetry in ionic strength. The structures that form depend on the asymmetry in the ionic strength and the amount of anionic lipid. Other factors that may be of importance in the creation of the structures, expansion forces, osmotic forces, and the bilayer-support interaction are discussed. The cap structures have the potential to be of considerable utility in examining the effect that curvature has on membrane processes.