Spatial reconstruction of the human motion based on images of a single camera

The inverse dynamic analysis procedures used in the study of the human gait require that the kinematics of the supporting biomechanical model is known beforehand. The first step to obtain the kinematic data is the reconstruction of human spatial motion, i.e., the evaluation of the anatomic points positions that enables to uniquely define the position of all anatomical segments. In photogrammetry, the projection of each anatomical point is described by two linear equations relating its three spatial coordinates with the two coordinates of the projected point. The need for the image of two cameras arises from the fact that three equations are necessary to find the original spatial position of the anatomical point. It is shown here that the kinematic constraint equations associated with a biomechanical model can be used as the extra set of equations required for the reconstruction process, instead of the equations associated with the second camera. With this methodology, the system of equations arising from the point projections and biomechanical model kinematic constraints are solved simultaneously. Since the system of equations has multiple solutions for each image, a strategy based on the minimization of the cost function associated to the smoothness of the reconstructed motion is devised, leading to an automated computer procedure enabling a unique reconstruction.     

The appropriate use of regression equations for the estimation of segmental inertia parameters

Linear regression equations are commonly used in conjunction with experimental data to provide linear relationships between quantities which are dimensionally distinct. In many cases theoretical relationships between such quantities are known and can be used as a basis for non-linear regression equations. This study compares linear and non-linear approaches for estimating the segmental moments of inertia from anthropometric measurements using the data of Chandler et al. [Chandler et al. (1975) Investigation of inertial properties of the human body. AMRL Technical Report 74-137, Wright Patterson Air Force Base. OH.] Right limb data were used to derive the equations while left limb data were used as a cross-validation sample to evaluate the inertia estimates calculated from the equations. For the limb segments the standard error estimates had average values of 21% for the linear equations and 13% for the non-linear equations. Data on a 10 yr-old boy was used to compare the two approaches outside the sample range. The mean percentage residuals were 286% for the linear equations and 20% for the non-linear equations. A set of non-linear equations is provided.     

Effects of elastic property of the wall on flow characteristics through arterial stenoses

Hemodynamic characteristics of blood flow through arterial stenoses are numerically investigated in this work. The blood is assumed as a Newtonian fluid and the pulsatile nature of flow is modeled by using measured values of the flowrate and pressure for the canine femoral artery. An isotropic elastic and incompressible material is assumed for the wall at each axial section, but a non-uniform distribution of the shear modulus in axial direction is used to model the high stiffness of the wall at the stenosis location. Full Navier equations for a thick wall are used as the governing equations for the wall displacements. A continuous grid extending over the flow field and the wall is considered and governing equations are transformed for use in the computational domain. Discretized forms of the transformed wall and flow equations, which are coupled through the boundary conditions at their interface, are obtained by control volume method and simultaneously solved using the well-known SIMPLER algorithm. To study the effects of wall deformability, solutions are obtained for both rigid and elastic walls. The results indicate that deformability of the wall causes an increase in the time average of pressure drop, but a decrease in the maximum wall shear stress. Displacement and stress distributions in the wall are presented.     

Analytical characterization of microvascular resistance distribution

The bat wing is used as an experimental preparation and as a self-contained vascular bed. The number, dimensions, and distribution of the vessels of the real vascular bed are included into an analyzable, representative geometric configuration. Based on theoretical analysis and experimental data, equations are developed and utilized to characterize the pressure-flow relationships for each branching order of the vascular field. The geometric configuration and associated describing equations are used to determine the resistance distribution of the microvascular bed. The predicted values are compared with experimentally determined quantities in normal and hypertensive animals.     

Parameter identification of the calvin photosynthesis cycle

Summary This article is concerned with the determination of kinetic parameters of the Calvin photosynthesis cycle which is described by seventeen nonlinear ordinary differential equations. It is shown that the task requires dynamic data for several sets of initial conditions. The numerical technique is based upon an algorithm for non-linear optimization and Gear's numerical integration scheme for stiff systems of differential equations. The sensitivity of the parameters to noise in the data is tested with a method adapted from Rosenbrook and Storey. A preliminary set of parameters has been obtained from a preliminary set of experimental data. The numerical methods are then tested with synthetic data derived from these parameters. The mathematical model and the results obtained in the simulation are used as an aid in designing new experiments.     

Solvers for the cardiac bidomain equations

The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue. They are especially important for accurately modeling extracellular stimulation, as evidenced by their prediction of virtual electrode polarization before experimental verification. However, solution of the equations is computationally expensive due to the fine spatial and temporal discretization needed. This limits the size and duration of the problem which can be modeled. Regardless of the specific form into which they are cast, the computational bottleneck becomes the repeated solution of a large, linear system. The purpose of this review is to give an overview of the equations and the methods by which they have been solved. Of particular note are recent developments in multigrid methods, which have proven to be the most efficient.     

Modelling the Effect of Gap Junctions on Tissue-Level Cardiac Electrophysiology

When modelling tissue-level cardiac electrophysiology, a continuum approximation to the discrete cell-level equations, known as the bidomain equations, is often used to maintain computational tractability. Analysing the derivation of the bidomain equations allows us to investigate how microstructure, in particular gap junctions that electrically connect cells, affect tissue-level conductivity properties. Using a one-dimensional cable model, we derive a modified form of the bidomain equations that take gap junctions into account, and compare results of simulations using both the discrete and continuum models, finding that the underlying conduction velocity of the action potential ceases to match up between models when gap junctions are introduced at physiologically realistic coupling levels. We show that this effect is magnified by: (i) modelling gap junctions with reduced conductivity; (ii) increasing the conductance of the fast sodium channel; and (iii) an increase in myocyte length. From this, we conclude that the conduction velocity arising from the bidomain equations may not be an accurate representation of the underlying discrete system. In particular, the bidomain equations are less likely to be valid when modelling certain diseased states whose symptoms include a reduction in gap junction coupling or an increase in myocyte length.     

Modeling of human reactions to whole-body vibration

A computer-automated approach for studying the human body vibration is presented. This includes vertical, horizontal, and torsional vibration. The procedure used is based on Finite Segment Modeling (FSM) of the human body, thus treating it as a mechanical structure. Kane's equations as developed by Huston et al. are used to formulate the governing equations of motion. The connective tissues are modeled by springs and dampers. In addition, the paper presents the transient response of different parts of the body due to a sinusoidal forcing function as well as an impulse function applied to the lower torso in the vertical direction.     

Regression equations for the estimation of stature and body mass using a Greek documented skeletal collection

Body size is an important variable in bioarchaeological and forensic studies, making the accurate calculation of stature and body mass imperative. Given that anatomical and morphometric approaches offer accurate results but require a particularly good preservation of the skeletal material, whereas mathematical and mechanical methods are more easily applicable but they are largely population-specific, the present paper uses a ‘hybrid’ approach in order to generate regression equations for the prediction of stature and body mass in a modern Greek sample. Specifically, anatomical and morphometric methods were used to calculate the stature and body mass of the individuals and regression equations using the Ordinary Least Squares and Reduced Major Axis methods were generated with long bone lengths and femoral head breadth as predictors. The obtained equations exhibit low random and directional error and perform better than existing equations designed using different samples from the United States, Europe, and the Balkans. Therefore, these equations are more appropriate for modern Greek material.     

Tumor Growth Modeling from the Perspective of Multiphase Porous Media Mechanics

Multiphase porous media mechanics is used for modeling tumor growth, using governing equations obtained via the Thermodynamically Constrained Averaging Theory (TCAT). This approach incorporates the interaction of more phases than legacy tumor growth models. The tumor is treated as a multiphase system composed of an extracellular matrix, tumor cells which may become necrotic depending on nutrient level and pressure, healthy cells and an interstitial fluid which transports nutrients. The governing equations are numerically solved within a Finite Element framework for predicting the growth rate of the tumor mass, and of its individual components, as a function of the initial tumor-to-healthy cell ratio, nutrient concentration, and mechanical strain. Preliminary results are shown.     

Methods for Fitting the Right-Truncated Weibull Distribution to Life-Test and Survival Data

This paper develops mathematical and computational methods for fitting, by the method of maximum likelihood (ML), the two-parameter, right-truncated Weibull distribution (RTWD) to life-test or survival data. Some important statistical properties of the RTWD are derived and ML estimating equations for the scale and shape parameters of the RTWD are developed. The ML equations are used to express the scale parameter as an analytic function of the shape parameter and to establish a computationally useful lower bound on the ML estimate of the shape parameter. This bound is a function only of the sample observations and the (known) truncation point T. The ML equations are reducible to a single nonlinear, transcendental equation in the shape parameter, and a computationally efficient algorithm is described for solving this equation. The practical use of the methods is illustrated in two numerical examples.     

Prediction of the power of detection of marker-quantitative trait locus linkages using analysis of variance

Analysis of variance can be used to detect the linkage of segregating quantitative trait loci (QTLs) to molecular markers in outbred populations. Using independent full-sib families and assuming linkage equilibrium, equations to predict the power of detection of a QTL are described. These equations are based on an hierarchical analysis of variance assuming either a completely random model or a mixed model, in which the QTL effect is fixed. A simple prediction of power from the mean squares is used that assumes a random model so that in the mixed-model situation this is an approximation. Simulation is used to illustrate the failure of the random model to predict mean squares and, hence, the power. The mixed model is shown to provide accurate prediction of the mean squares and, using the approximation, of power.     

Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations.

Modifications based on experimental results reported in the literature are made to the Hodgkin-Huxley equations to describe the electrophysiological behavior of the Aplysia abdominal ganglion R15 cell. The system is then further modified to describe the effects with the application of the drug tetrodotoxin (TTX) to the cells' bathing medium. Methods of the qualitative theory of differential equations are used to determine the conditions necessary for such a system of equations to have an oscillatory solution. A model satisfying these conditions is shown to preduct many experimental observations of R15 cell behavior. Numerical solutions are obtained for differential equations satisfying the conditions of the model. These solutions are shown to have a form similar to that of the bursting which is characteristic of this cell, and to preduct many results of experiments conducted on this cell. The physiological implications of the model are discussed.     

Mechanical analysis of the development of left ventricular aneurysms

The left ventricle is modelled as a spherical shell with an infarcted wall segment. The mechanics of the circumstances causing this infarcted segment to develop into an aneurysm is presented. Both the wall stresses and deformations are worked out for aneurysms developing from infarcts of different sizes and percentages of wall damage. The governing equations consist of incompressibility relations, force-equilibrium relations and stress-strain relations. Newton Raphson technique is used to solve these nonlinear simultaneous algebraic equations, for the values of the myocardial stresses in the infarcted segment and the bulge values, in terms of the ventricular geometry and the damage extent (expressed in terms of the damage angle and percentage of wall damage). The results indicate that in general it is innermost layer which is severely stressed and that in the rupture of the ventricle the critical factor involved is the percentage of infarct thickness rather than the angle of damage.     

A direct substitution, equation error technique for solving the thermographic tomography problem

A new technique for solving the combined state and parameter estimation problem in thermographic tomography is presented. The technique involves the direct substitution of known skin temperatures into the finite difference form of the bio-heat transfer equation as formulated for solving an initial value problem with a convection boundary condition at the skin surface. These equations are then used to solve the inverse bio-heat transfer problem for the unknown subcutaneous tissue temperatures and physiological parameters. For a small number of nodal points, closed form algebraic solutions are obtained. For larger sets of equations, a hybrid technique is used in which the problem is initially posed as an unconstrained optimization problem in which the model equation error is minimized using the conjugate gradient descent technique to get close to a solution. Then a generalized Newton-Raphson technique was used to solve the equations. A numerical simulation of a one-dimensional problem is investigated both with and without noise superimposed on the input (transient) skin temperature data. The results show that the technique gives very accurate results if the skin temperature data contains little noise. It is also shown that if the physical properties of the tissue and the metabolism are known, that a given set of proper transient skin temperature inputs yields a unique solution for the unknown internal temperatures and blood perfusion rates. However, the similar problem with known blood perfusion rates and unknown metabolisms does not yield a unique solution for the internal temperatures and metabolisms.     

Boundary Layer Flow Past a Stretching Surface in a Porous Medium Saturated by a Nanofluid: Brinkman-Forchheimer Model

In this study, the steady forced convection flow and heat transfer due to an impermeable stretching surface in a porous medium saturated with a nanofluid are investigated numerically. The Brinkman-Forchheimer model is used for the momentum equations (porous medium), whereas, Bongiorno’s model is used for the nanofluid. Uniform temperature and nanofluid volume fraction are assumed at the surface. The boundary layer equations are transformed to ordinary differential equations in terms of the governing parameters including Prandtl and Lewis numbers, viscosity ratio, porous medium, Brownian motion and thermophoresis parameters. Numerical results for the velocity, temperature and concentration profiles, as well as for the reduced Nusselt and Sherwood numbers are obtained and presented graphically.     

Accurate computation of the stable solitary wave for the FitzHugh-Nagumo equations

Comparisons are made between three different methods for computing the stable solitary wave solution for the FitzHugh-Nagumo equations which consist of a nonlinear diffusion equation coupled to an ordinary differential equation in time. They model the Hodgkin-Huxley equations which describe the propagation of the nerve impulse down the axon. Two of the methods involve the travelling wave equations. Previous accurate numerical computations of these equations as an initial-value problem using a shooting method lead to inaccurate values for the wave speed; however, nonlinear corrections to the initial values are shown to yield accurate values. A boundary-value method applies asymptotic boundary conditions and uses a spline-collocation code called COLSYS for numerical solution of boundary-value problems which leads to accurate wave profiles and speeds. The third method is to solve an initial-boundary-value problem with an adaptive outgoing wave condition for the partial differential equations where the solitary wave emerges as the stable long time solution. The concept of a wave integral is introduced and they are derived to determine the wave speed used in the adaptive boundary condition and to measure the closeness of the computed solutions to the exact solitary wave solution.This work was supported in part by the Natural Sciences and Engineering Research Council Canada under Grant A4559 and by the John Simon Guggenheim Memorial Foundation     

Predictive algorithms for neuromuscular control of human locomotion.

The problem of quantifying muscular activity of the human body can be formulated as an optimal control problem. The current methods used with large-scale biomechanical systems are non-derivative techniques. These methods are costly, as they require numerous integrations of the equations of motion. Additionally, the convergence is slow, making them impractical for use with large systems. We apply an efficient numerical algorithm to the biomechanical optimal control problem. Using direct collocation with a trapezoidal discretization, the equations of motion are converted into a set of algebraic constraint equations. An augmented Lagrangian formulation is used for the optimization problem to handle both equality and inequality constraints. The resulting min-max problem is solved with a generalized Newton method. In contrast to the prevalent optimal control implementations, we calculate analytical first- and second-derivative information and obtain local quadratic convergence. To demonstrate the efficacy of the method, we solve a steady-state pedaling problem with 7 segments and 18 independent muscle groups. The computed muscle activations compare well with experimental EMG data. The computational effort is significantly reduced and solution times are a fraction of those of the non-derivative techniques.     

A mathematical simulation of the ureter: effects of the model parameters on ureteral pressure/flow relations

Ureteral peristaltic mechanism facilitates urine transport from the kidney to the bladder. Numerical analysis of the peristaltic flow in the ureter aims to further our understanding of the reflux phenomenon and other ureteral abnormalities. Fluid-structure interaction (FSI) plays an important role in accuracy of this approach and the arbitrary Lagrangian-Eulerian (ALE) formulation is a strong method to analyze the coupled fluid-structure interaction between the compliant wall and the surrounding fluid. This formulation, however, was not used in previous studies of peristalsis in living organisms. In the present investigation, a numerical simulation is introduced and solved through ALE formulation to perform the ureteral flow and stress analysis. The incompressible Navier-Stokes equations are used as the governing equations for the fluid, and a linear elastic model is utilized for the compliant wall. The wall stimulation is modeled by nonlinear contact analysis using a rigid contact surface since an appropriate model for simulation of ureteral peristalsis needs to contain cell-to-cell wall stimulation. In contrast to previous studies, the wall displacements are not predetermined in the presented model of this finite-length compliant tube, neither the peristalsis needs to be periodic. Moreover, the temporal changes of ureteral wall intraluminal shear stress during peristalsis are included in our study. Iterative computing of two-way coupling is used to solve the governing equations. Two phases of nonperistaltic and peristaltic transport of urine in the ureter are discussed. Results are obtained following an analysis of the effects of the ureteral wall compliance, the pressure difference between the ureteral inlet and outlet, the maximum height of the contraction wave, the contraction wave velocity, and the number of contraction waves on the ureteral outlet flow. The results indicate that the proximal part of the ureter is prone to a higher shear stress during peristalsis compared with its middle and distal parts. It is also shown that the peristalsis is more efficient as the maximum height of the contraction wave increases. Finally, it is concluded that improper function of ureteropelvic junction results in the passage of part of urine back flow even in the case of slow start-up of the peristaltic contraction wave.     

Bifurcation of the Hodgkin and Huxley equations: A new twist

The Hodgkin and Huxley equations model action potentials in squid giant axons. Variants of these equations are used in most models for electrial activity of excitable membranes. Computational tools based upon the theory of nonlinear dynamical systems are used here to illustrate how the dynamical behavior of the Hodgkin Huxley model changes as functions of two of the system parameters.