Kinematic analysis of over-determinate biomechanical systems

In this paper, we introduce a new general method for kinematic analysis of rigid multi body systems subject to holonomic constraints. The method extends the standard analysis of kinematically determinate rigid multi body systems to the over-determinate case. This is accomplished by introducing a constrained optimisation problem with the objective function given as a function of the set of system equations that are allowed to be violated while the remaining equations define the feasible set.

We show that exact velocity and acceleration analysis can also be performed by solving linear sets of equations, originating from differentiation of the Karush–Kuhn–Tucker optimality conditions.

The method is applied to the analysis of an 18 degrees-of-freedom gait model where the kinematical drivers are prescribed with data from a motion capture experiment.

The results show that significant differences are obtained between applying standard kinematic analysis or minimising the least-square errors on the two fully equivalent 3D gait models with only the way the experimental data is processed being different.     

Equivalent Voltage and Current Waves for TM SPP in Graphene Sheets

We present a theoretical model to describe the propagation of a transverse magnetic surface plasmon polariton in graphene based on equivalent voltage and charge current waves, which includes the spatial dispersion effect. Electrons and holes in graphene are governed by the Boltzmann equation in the particle conserving relaxation time approximation. First, we deduce expressions for the non-equilibrium distributions when there are charge oscillations in graphene as a response to the electromagnetic field applied to it. These distribution functions are used in the Boltzmann equations to derive other equations for the following four local macroscopic averages: the oscillating electron and hole densities, and the electron and hole current densities. Then, for a specific structure, we solve the wave equations for the electric and vector magnetic potentials to obtain the relations between the charge oscillations and the potentials. So, we reach a homogeneous system of four coupled equations relating the amplitudes of the voltage and the current waves. The non-trivial solutions of the system allow us to compute the dispersion and loss curves for such waves. As it is already known, for a given frequency, we can see that the higher the Fermi level is, the lesser the spatial-dispersion effect is. Following the analysis, a distributed-element circuit for the equivalent transmission line in which would propagate the waves, is developed. Finally, we analyze the dependence of these circuit elements and the impedance on both the frequency and Fermi level.

     

Sensitivity analysis of the Poisson Nernst–Planck equations: a finite element approximation for the sensitive analysis of an electrodiffusion model

Biological structures exhibiting electric potential fluctuations such as neuron and neural structures with complex geometries are modelled using an electrodiffusion or Poisson Nernst–Planck system of equations. These structures typically depend upon several parameters displaying a large degree of variation or that cannot be precisely inferred experimentally. It is crucial to understand how the mathematical model (and resulting simulations) depend on specific values of these parameters. Here we develop a rigorous approach based on the sensitivity equation for the electrodiffusion model. To illustrate the proposed methodology, we investigate the sensitivity of the electrical response of a node of Ranvier with respect to ionic diffusion coefficients and the membrane dielectric permittivity.

     

Multiscale bio-chemo-mechanical model of intimal hyperplasia

We consider a computational multiscale framework of a bio-chemo-mechanical model for intimal hyperplasia. With respect to existing models, we investigate the interactions between hemodynamics, cellular dynamics and biochemistry on the development of the pathology. Within the arterial wall, we propose a mathematical model consisting of kinetic differential equations for key vascular cell types, collagen and growth factors. The luminal hemodynamics is modeled with the Navier–Stokes equations. Coupling hypothesis among time and space scales are proposed to build a tractable modeling of such a complex multifactorial and multiscale pathology. A one-dimensional numerical test-case is presented for validation by comparing the results of the framework with experiments at short and long timescales. Our model permits to capture many cellular phenomena which have a central role in the physiopathology of intimal hyperplasia. Results are quantitatively and qualitatively consistent with experimental findings at both short and long timescales.

     

Multigrid Block Preconditioning for a Coupled System of Partial Differential Equations Modeling the Electrical Activity in the Heart

The electrical activity of the heart may be modeled with a system of partial differential equations (PDEs) known as the bidomain model. Computer simulations based on these equations may become a helpful tool to understand the relationship between changes in the electrical field and various heart diseases. Because of the rapid variations in the electrical field, sufficiently accurate simulations require a fine-scale discretization of the equations. For realistic geometries this leads to a large number of grid points and consequently large linear systems to be solved for each time step. In this paper, we present a fully coupled discretization of the bidomain model, leading to a block structured linear system. We take advantage of the block structure to construct an efficient preconditioner for the linear system, by combining multigrid with an operator splitting technique.     

Molecular Dynamics as a Mathematical Mapping. II. Partial Derivatives in the Microcanonical Ensemble

Abstract

Performing molecular dynamics in a fully continuous and differentiable framework can be viewed as a deterministic mathematical mapping between, on one side, the force field parameters that describe the potential energy interactions and input macroscopic conditions, and, on the other, the calculated corresponding macroscopic properties of the bulk molecular system.

Within this framework, it is possible to apply standard methods of variational calculus for the computation of the partial derivatives of the molecular dynamics mapping based on the integration of either the adjoint equations or the sensitivity equations of the classical Newtonian equations of motion. We present procedures for these computations in the standard microcanonical (N, V, E) ensemble, and compare the computational efficiency of the two approaches. The general formulations developed are applied to the specific example of bulk ethane fluid.

With these procedures in place, it is now possible to compute the partial derivatives of any property determined by molecular dynamics with respect to any input property and any potential parameter. Moreover, these derivatives are computed to essentially the same level of numerical accuracy as the output properties themselves.     

Theoretical evaluation of the parameters of glucose metabolism on the basis of continuous glycemia monitoring data using mathematical modeling

The aim of this paper is to construct a mathematical model that takes the main physiological parameters of blood-glucose regulation into account, in order to identify these parameters for an individual patient according to continuous glucose-monitoring data. The constructed mathematical model consists of six ordinary differential equations that describe the dynamics of changes in glucose concentrations, as well as insulin and anti-insulin factors in the blood. Estimation of the parameters of the equations was performed using an evolutionary programming method. The model predictions were fitted to the continuous glucosemonitoring data. As a result of the identification of the model parameters for two patients with type 1 diabetes mellitus, the estimated insulin secretion was close to zero and the estimated glucose utilization and insulin clearance were increased in comparison with the data for healthy donors. Here, we present a personalized model of the regulation of blood glucose, which can be used to predict the results of continuous glucose monitoring depending on modification of the prescribed glucose-lowering therapy. This approach can significantly reduce the number of iterations of the selection of medical hypoglycemic therapy and therefore increase the effectiveness of treatment according to glucose-monitoring data.

     

A novel,FFT-based one-dimensional blood flow solution method for arterial network

In the present work, we propose an FFT-based method for solving blood flow equations in an arterial network with variable properties and geometrical changes. An essential advantage of this approach is in correctly accounting for the vessel skin friction through the use of Womersley solution. To incorporate nonlinear effects, a novel approximation method is proposed to enable calculation of nonlinear corrections. Unlike similar methods available in the literature, the set of algebraic equations required for every harmonic is constructed automatically. The result is a generalized, robust and fast method to accurately capture the increasing pulse wave velocity downstream as well as steepening of the pulse front. The proposed method is shown to be appropriate for incorporating correct convection and diffusion coefficients. We show that the proposed method is fast and accurate and it can be an effective tool for 1D modelling of blood flow in human arterial networks.

     

Modelling dendritic spines with the finite element method,investigating the impact of geometry on electric and calcic responses

Understanding the relationship between shape and function of dendritic spines is an elusive topic. Several modelling approaches have been used to investigate the interplay between spine geometry, calcium diffusion and electric signalling. We here use a second order finite element method to solve the Poisson–Nernst–Planck equations and describe electrodiffusion in dendritic spines. With this, we obtain relationships between dendritic geometry and calcic as well as electric responses to synaptic events. Our findings support the hypothesis that spine geometry plays a role shaping the electrical responses to synaptic events. Our method was also able to reveal the fine scale distribution of calcium in spines with irregular shapes.

     

Impact of the Error Structure on the Design and Analysis of Enzyme Kinetic Models

The statistical analysis of enzyme kinetic reactions usually involves models of the response functions which are well defined on the basis of Michaelis–Menten type equations. The error structure, however, is often without good reason assumed as additive Gaussian noise. This simple assumption may lead to undesired properties of the analysis, particularly when simulations are involved and consequently negative simulated reaction rates may occur. In this study, we investigate the effect of assuming multiplicative log normal errors instead. While there is typically little impact on the estimates, the experimental designs and their efficiencies are decisively affected, particularly when it comes to model discrimination problems.

     

Myocardial material parameter estimation: a comparison of invariant based orthotropic constitutive equations

This study investigated a number of invariant based orthotropic and transversely isotropic constitutive equations for their suitability to fit three-dimensional simple shear mechanics data of passive myocardial tissue.

A number of orthotropic laws based on Green strain components and one microstructurally based law have previously been investigated to fit experimental measurements of stress-strain behaviour. Here we extend this investigation to include several recently proposed functional forms, i.e. invariant based orthotropic and transversely isotropic constitutive relations.

These laws were compared on the basis of (i) ‘goodness of fit’: how well they fit a set of six shear deformation tests, (ii) ‘variability’: how well determined the material parameters are over the range of experiments. These criteria were utilised to discuss the advantages and disadvantages of the constitutive laws.

It was found that a specific form of the polyconvex type as well as the exponential Fung-type law from the previous study were most suitable for modelling the orthotropic behaviour of myocardium under simple shear.     

Theory of the locomotion of nematodes: Dynamics of undulatory progression on a surface

We develop a model of the undulatory locomotion of nematodes, in particular that of Caenorhabditis elegans, based on mechanics. The model takes into account the most important forces acting on a moving worm and allows the computer simulation of a creeping nematode. These forces are produced by the interior pressure in the liquid-filled body cavity, the elasticity of the cuticle, the excitation of certain sets of muscles and the friction between the body and its support.

We propose that muscle excitation patterns can be generated by stretch receptor control. By solving numerically the equations of motion of the model of the nematode, we demonstrate that these muscle excitation patterns are suitable for the propulsion of the animal.

     

Numerical Study of the Structural and Thermal Properties of Vitreous Silica

Abstract

We use classical molecular dynamics simulations to study both the structural modifications through the glass transition and the thermal conductivity k of a model silica glass. The first part is based on the Voronoï tessellation and we show that the structural freezing following upon the glass transition is noticeable in all the geometric characteristics of the Voronoï cells and a possible interpretation in terms of geometrical frustration is proposed.

In the second part we calculate k directly in the simulation box by using the standard equations of heat transport. The calculations have been done between 10 and 1000 Kelvin and the results are in good agreement with the experimental data at temperatures above 20 K. The plateau observed around 10 K can be accounted for by correcting our results taking into account finite size effects in a phenomenological way.     

Thrombin Flux and Wall Shear Rate Regulate Fibrin Fiber Deposition State during Polymerization under Flow

Thrombin is released as a soluble enzyme from the surface of platelets and tissue-factor-bearing cells to trigger fibrin polymerization during thrombosis under flow conditions. Although isotropic fibrin polymerization under static conditions involves protofibril extension and lateral aggregation leading to a gel, factors regulating fiber growth are poorly quantified under hemodynamic flow due to the difficulty of setting thrombin fluxes. A membrane microfluidic device allowed combined control of both thrombin wall flux (10⁻¹³ to 10⁻¹¹ nmol/μm² s) and the wall shear rate (10-100 s⁻¹) of a flowing fibrinogen solution. At a thrombin flux of 10⁻¹² nmol/μm² s, both fibrin deposition and fiber thickness decreased as the wall shear rate increased from 10 to 100 s⁻¹. Direct measurement and transport-reaction simulations at 12 different thrombin flux-wall shear rate conditions demonstrated that two dimensionless numbers, the Peclet number (Pe) and the Damkohler number (Da), defined a state diagram to predict fibrin morphology. For Da < 10, we only observed thin films at all Pe. For 10 < Da < 900, we observed either mat fibers or gels, depending on the Pe. For Da > 900 and Pe < 100, we observed three-dimensional gels. These results indicate that increases in wall shear rate quench first lateral aggregation and then protofibril extension.     

A PDMP model of the epithelial cell turn-over in the intestinal crypt including microbiota-derived regulations

Mechanistic models are a powerful tool to gain insights into biological processes. The parameters of such models, e.g. kinetic rate constants, usually cannot be measured directly but need to be inferred from experimental data. In this article, we study dynamical models of the translation kinetics after mRNA transfection and analyze their parameter identifiability. That is, whether parameters can be uniquely determined from perfect or realistic data in theory and practice. Previous studies have considered ordinary differential equation (ODE) models of the process, and here we formulate a stochastic differential equation (SDE) model. For both model types, we consider structural identifiability based on the model equations and practical identifiability based on simulated as well as experimental data and find that the SDE model provides better parameter identifiability than the ODE model. Moreover, our analysis shows that even for those parameters of the ODE model that are considered to be identifiable, the obtained estimates are sometimes unreliable. Overall, our study clearly demonstrates the relevance of considering different modeling approaches and that stochastic models can provide more reliable and informative results.

     

Two enzyme active transport in vitro with ph induced asymmetrical functional structures: I. The model and its analytical treatment

A class of systems is characterized by the asymmetrical distribution of a sink and a source reaction, the asymmetry of the global chemical equation (energy liberation) and by an asymmetrical one-wave space profile. These systems belong to the family of primary chemical cells and can deplete and enrich the media they separate. A “ one way ” transport-reaction chain is needed for specific “ real ” active transport. A two enzyme model of this class is described in which the spatial asymmetry is due to a (diffusive) pH gradient; this distribution of “ potential ” enzyme activities is called the “ functional structure ”. Equal potential enzyme activities and absence of reactive back action on local pH are assumed in the “ square model ” version of the pump. Analytical expressions of the enzymatic diffusion reactions are derived for zero and first order kinetics, i.e. in function of substrate concentrations. Tables of equations are presented. The intrinsic properties of the pump are characterized by (dimensionless) transport reaction parameters, (membrane composition); the “ potential ” activity is controlled by the pH gradient; the “ effective ” pumping is also a function of the substrate concentrations on the boundaries.     

From Molecular to Process Simulation: Novel Approaches to the Prediction of Phase Equilibria and PVT Behavior Based on Molecular/Quantum Mechanics and Molecular Dynamics Simulations

Abstract

Actually, in modern process simulators, more than 75% of the code implemented is dedicated to physical properties estimation, calculation and predictions. Data banks storing pure component parameters and binary interaction parameters for phase equilibrium calculations are extensively used and continuously implemented in actual process simulators. This gives an idea of the important role physical properties availability plays in process simulation.

In this paper we propose a new way for coupling molecular and process simulation. The basic machinery is to resort to molecular/quantum mechanics and molecular dynamics simulation techniques for generating the parameters of some equations of state that will subsequently be used for the prediction of phase equilibria and PVT behavior of small and polymeric molecules as well. This information, in turn, will be used as input in the process simulator, thus creating a final and well-defined bridge between molecular and process simulations in chemical engineering.     

Time-dependent propagators for stochastic models of gene expression: an analytical method

The inherent stochasticity of gene expression in the context of regulatory networks profoundly influences the dynamics of the involved species. Mathematically speaking, the propagators which describe the evolution of such networks in time are typically defined as solutions of the corresponding chemical master equation (CME). However, it is not possible in general to obtain exact solutions to the CME in closed form, which is due largely to its high dimensionality. In the present article, we propose an analytical method for the efficient approximation of these propagators. We illustrate our method on the basis of two categories of stochastic models for gene expression that have been discussed in the literature. The requisite procedure consists of three steps: a probability-generating function is introduced which transforms the CME into (a system of) partial differential equations (PDEs); application of the method of characteristics then yields (a system of) ordinary differential equations (ODEs) which can be solved using dynamical systems techniques, giving closed-form expressions for the generating function; finally, propagator probabilities can be reconstructed numerically from these expressions via the Cauchy integral formula. The resulting ‘library’ of propagators lends itself naturally to implementation in a Bayesian parameter inference scheme, and can be generalised systematically to related categories of stochastic models beyond the ones considered here.

     

Terahertz Plasmonic Quantum Cascade Lasers: a Comprehensive Physics Study

Generation of terahertz (THz) radiation has been a hot research topic in recent years. Plasmonic quantum cascade lasers (QCLs) are among the most compact and efficient sources to generate THz radiation. In this paper, we comprehensively study plasmonic QCLs designed based on the antenna-feedback structure to generate efficient radiation about the center frequency of 3 THz. By changing the geometric structure of the plasmonic cavity and using two-dimensional simulation, a minimum loss less than 5.9 cm^?1 is achieved at the lasing frequency. It is also possible to control the orientation of the output beam either vertically or tilted by changing the geometry of the antenna design via chirped or non-chirped grating scheme. Moreover, the output characteristics of the QCL are simulated based on the three-level rate equations through which the dynamics of the laser, as well as the P-I curve, are investigated. Also, the gain spectra for two laser designs (with chirped and non-chirped gratings) are simulated and compared to each other. The results of this paper may provide deep insight into designing efficient laser sources in the THz region.

     

Association analyses of repeated measures on triglyceride and high-density lipoprotein levels: insights from GAW20

Background

The GAW20 group formed on the theme of methods for association analyses of repeated measures comprised 4sets of investigators. The provided “real” data set included genotypes obtained from a human whole-genome association study based on longitudinal measurements of triglycerides (TGs) and high-density lipoprotein in addition to methylation levels before and after administration of fenofibrate. The simulated data set contained 200 replications of methylation levels and posttreatment TGs, mimicking the real data set.

Results

The different investigators in the group focused on the statistical challenges unique to family-based association analyses of phenotypes measured longitudinally and applied a wide spectrum of statistical methods such as linear mixed models, generalized estimating equations, and quasi-likelihood–based regression models. This article discusses the varying strategies explored by the group’s investigators with the common goal of improving the power to detect association with repeated measures of a phenotype.

Conclusions

Although it is difficult to identify a common message emanating from the different contributions because of the diversity in the issues addressed, the unifying theme of the contributions lie in the search for novel analytic strategies to circumvent the limitations of existing methodologies to detect genetic association.