An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso

In this paper we present a numerical method for the bidomain model, which describes the electrical activity in the heart. The model consists of two partial differential equations (PDEs), which are coupled to systems of ordinary differential equations (ODEs) describing electrochemical reactions in the cardiac cells. Many applications require coupling these equations to a third PDE, describing the electrical fields in the torso surrounding the heart. The resulting system is challenging to solve numerically, because of its complexity and very strict resolution requirements in time and space. We propose a method based on operator splitting and a fully coupled discretization of the three PDEs. Numerical experiments show that for simple simulation cases and fine discretizations, the algorithm is second-order accurate in space and time.     

Efficient simulations of tubulin-driven axonal growth

This work concerns efficient and reliable numerical simulations of the dynamic behaviour of a moving-boundary model for tubulin-driven axonal growth. The model is nonlinear and consists of a coupled set of a partial differential equation (PDE) and two ordinary differential equations. The PDE is defined on a computational domain with a moving boundary, which is part of the solution. Numerical simulations based on standard explicit time-stepping methods are too time consuming due to the small time steps required for numerical stability. On the other hand standard implicit schemes are too complex due to the nonlinear equations that needs to be solved in each step. Instead, we propose to use the Peaceman–Rachford splitting scheme combined with temporal and spatial scalings of the model. Simulations based on this scheme have shown to be efficient, accurate, and reliable which makes it possible to evaluate the model, e.g. its dependency on biological and physical model parameters. These evaluations show among other things that the initial axon growth is very fast, that the active transport is the dominant reason over diffusion for the growth velocity, and that the polymerization rate in the growth cone does not affect the final axon length.     

The micromechanics of fluid–solid interactions during growth in porous soft biological tissue

In this paper, we address some modelling issues related to biological growth. Our treatment is based on a formulation for growth that was proposed within the context of mixture theory (J Mech Phys Solids 52:1595–1625, 2004). We aim to make this treatment more appropriate for the physics of porous soft tissues, paying particular attention to the nature of fluid transport, and mechanics of fluid and solid phases. The interactions between transport and mechanics have significant implications for growth and swelling. We also reformulate the governing differential equations for reaction-transport of solutes to represent the incompressibility constraint on the fluid phase of the tissue. This revision enables a straightforward implementation of numerical stabilisation for the advection-dominated limit of these equations. A finite element implementation with operator splitting is used to solve the coupled, non-linear partial differential equations that arise from the theory. We carry out a numerical and analytic study of the convergence of the operator splitting scheme subject to strain- and stress-homogenisation of the mechanics of fluid–solid interactions. A few computations are presented to demonstrate aspects of the physical mechanisms, and the numerical performance of the formulation.     

Analysis of limit-cycle conditions in intercoupled relay oscillators with reference to gastrointestinal modelling

This paper presents an exact analysis of a mutually coupled relay oscillator based on a method orignated by Tsypkin. Limit-cycle frequencies and phases can be determined exactly using this method, unlike other approximate methods based on describing functions and harmonic balance techniques. A new method of exact determination of limit-cycle stability is also shown to give excellent agreement with simulation studies. Different types of intercoupling are shown to give different stability conditions, and these are discussed in relation to gastrointestinal (GI) smooth muscle modelling. GI tract electrical activity has previously been modelled using bidirectionally coupled nonlinear oscillators, and the results of the present analysis of relay oscillators is compared with other studies using van der Pol dynamics.     

A generating model of realistic synthetic heart sounds for performance assessment of phonocardiogram processing algorithms

A new model which is capable of generating realistic synthetic phonocardiogram (PCG) signals is introduced based on three coupled ordinary differential equations. The new PCG model takes into account the respiratory frequency, the heart rate variability and the time splitting of first and second heart sounds. This time splitting occurs with each cardiac cycle and varies with inhalation and exhalation. Clinical PCG statistics and the close temporal relationship between events in ECG and PCG are used to deduce values of PCG model parameters.In comparison with published PCG models, the proposed model allows a larger number of known PCG features to be taken into consideration. Moreover it is able to generate both normal and abnormal realistic synthetic heart sounds. Results show that these synthetic PCG signals have the closest features to those of a conventional heart sound in both time and frequency domains. Additionally, a sound quality test carried out by eight cardiologists demonstrates that the proposed model outperforms the existing models.This new PCG model is promising and useful in assessing signal processing techniques which are developed to help clinical diagnosis based on PCG.     

Nonlinear equations of reaction-diffusion type for neural populations

Several simplified differential equations are derived from the Wilson and Cowan model describing the dynamics of excitatory and inhibitory neurons. It is shown, by expansions of the convolution integrals and the input-output functions, that the basic integrodifferential equations can be reduced to two coupled nonlinear partial differential equations of reaction-diffusion type. Further simplification leads to the coupled partial differential equations mathematically equivalent to the FitzHugh-Nagumo equations for the nerve impulse. Through a brief stability analysis in relation to the existing investigations on the bifurcation phenomena, an attempt is made to clarify the consequence due to the approximations introduced in this paper.     

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.     

Soft Tissue Modelling of Cardiac Fibres for Use in Coupled Mechano-Electric Simulations

The numerical solution of the coupled system of partial differential and ordinary differential equations that model the whole heart in three dimensions is a considerable computational challenge. As a consequence, it is not computationally practical—either in terms of memory or time—to repeat simulations on a finer computational mesh to ensure that convergence of the solution has been attained. In an attempt to avoid this problem while retaining mathematical rigour, we derive a one dimensional model of a cardiac fibre that takes account of elasticity properties in three structurally defined axes within the myocardial tissue. This model of a cardiac fibre is then coupled with an electrophysiological cell model and a model of cellular electromechanics to allow us to simulate the coupling of the electrical and mechanical activity of the heart. We demonstrate that currently used numerical methods for coupling electrical and mechanical activity do not work in this case, and identify appropriate numerical techniques that may be used when solving the governing equations. This allows us to perform a series of simulations that: (i) investigate the effect of some of the assumptions inherent in other models; and (ii) reproduce qualitatively some experimental observations.     

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.     

Active muscle fiber as an equivalent cardiac current generator

Equations are derived describing potentials due to an active muscle fiber in an infinite medium in terms of two surface integrals—one of the propagated action potential and the other of the membrane current density, both integrals being taken over the surface of the muscle. These equations are incorporated into an equivalent cardiac current generator in which the left ventricle (i.e. the current source) is represented by a three-dimensional wedge and the thorax (i.e. the volume conductor), by a homogeneous circular cylinder. Since this current generator expresses the body surface potentials in terms of the membrane current density and the membrane potential at any point on the surface of the electrically active muscle fiber, the calculated ECG can be correlated with theactual sources within the heart. This equivalent cardiac generator possesses many of the physical and physiological properties of cardiac muscle. The equations were evaluated numerically on a digital computer. The results indicate that equivalent cardiac current generators of this type can yield clinically significant results and that further research is necessary to investigate their properties fully.     

A Bond Graph Model of the Cardiovascular System

The study of the autonomic nervous system (ANS) function has shown to provide useful indicators for risk stratification and early detection on a variety of cardiovascular pathologies. However, data gathered during different tests of the ANS are difficult to analyse, mainly due to the complex mechanisms involved in the autonomic regulation of the cardiovascular system (CVS). Although model-based analysis of ANS data has been already proposed as a way to cope with this complexity, only a few models coupling the main elements involved have been presented in the literature. In this paper, a new model of the CVS, representing the ventricles, the circulatory system and the regulation of the CVS activity by the ANS, is presented. The models of the vascular system and the ventricular activity have been developed using the Bond Graph formalism, as it proposes a unified representation for all energetic domains, facilitating the integration of mechanic and hydraulic phenomena. In order to take into account the electro-mechanical behaviour of both ventricles, an electrophysiologic model of the cardiac action potential, represented by a set of ordinary differential equations, has been integrated. The short-term ANS regulation of heart rate, cardiac contractility and peripheral vasoconstriction is represented by means of continuous transfer functions. These models, represented in different continuous formalisms, are coupled by using a multi-formalism simulation library. Results are presented for two different autonomic tests, namely the Tilt Test and the Valsalva Manoeuvre, by comparing real and simulated signals.     

Modeling left ventricular dynamics with characteristic deformation modes

A computationally efficient method is described for simulating the dynamics of the left ventricle (LV) in three dimensions. LV motion is represented as a combination of a limited number of deformation modes, chosen to represent observed cardiac motions while conserving volume in the LV wall. The contribution of each mode to wall motion is determined by a corresponding time-dependent deformation variable. The principle of virtual work is applied to these deformation variables, yielding a system of ordinary differential equations for LV dynamics, including effects of muscle fiber orientations, active and passive stresses, and surface tractions. Passive stress is governed by a transversely isotropic elastic model. Active stress acts in the fiber direction and incorporates length–tension and force–velocity properties of cardiac muscle. Preload and afterload are represented by lumped vascular models. The variational equations and their numerical solutions are verified by comparison to analytic solutions of the strong form equations. Deformation modes are constructed using Fourier series with an arbitrary number of terms. Greater numbers of deformation modes increase deformable model resolution but at increased computational cost. Simulations of normal LV motion throughout the cardiac cycle are presented using models with 8, 23, or 46 deformation modes. Aggregate quantities that describe LV function vary little as the number of deformation modes is increased. Spatial distributions of stress and strain change as more deformation modes are included, but overall patterns are conserved. This approach yields three-dimensional simulations of the cardiac cycle on a clinically relevant time-scale.

     

Analytical threshold and stability results on age-structured epidemic models with vaccination

This paper examines mathematical models for common childhood diseases such as measles and rubella and in particular the use of such models to predict whether or not an epidemic pattern of regular recurrent disease incidence will occur. We use age-structured compartmental models which divide the population amongst whom the disease is spreading into classes and use partial differential equations to model the spread of the disease. This paper is particularly concerned with an analytical investigation of the effects of different types of vaccination schemes. We examine possible equilibria and determine the stability of small oscillations about these equilibria. The results are important in predicting the long-term overall level of incidence of disease, in designing immunisation programs and in describing the variations of the incidence of disease about this equilibrium level.     

由微分方程所描述的微生物连续培养动力系统(II)

该文为论文 [1 ]的继续 ,主要介绍具有时间滞后的微生物连续培养模型     

Traction patterns of tumor cells

The traction exerted by a cell on a planar deformable substrate can be indirectly obtained on the basis of the displacement field of the underlying layer. The usual methodology used to address this inverse problem is based on the exploitation of the Green tensor of the linear elasticity problem in a half space (Boussinesq problem), coupled with a minimization algorithm under force penalization. A possible alternative strategy is to exploit an adjoint equation, obtained on the basis of a suitable minimization requirement. The resulting system of coupled elliptic partial differential equations is applied here to determine the force field per unit surface generated by T24 tumor cells on a polyacrylamide substrate. The shear stress obtained by numerical integration provides quantitative insight of the traction field and is a promising tool to investigate the spatial pattern of force per unit surface generated in cell motion, particularly in the case of such cancer cells.     

Neural network firing-rate models on integral form

Firing-rate models describing neural-network activity can be formulated in terms of differential equations for the synaptic drive from neurons. Such models are typically derived from more general models based on Volterra integral equations assuming exponentially decaying temporal coupling kernels describing the coupling of pre- and postsynaptic activities. Here we study models with other choices of temporal coupling kernels. In particular, we investigate the stability properties of constant solutions of two-population Volterra models by studying the equilibrium solutions of the corresponding autonomous dynamical systems, derived using the linear chain trick, by means of the Routh–Hurwitz criterion. In the four investigated synaptic-drive models with identical equilibrium points we find that the choice of temporal coupling kernels significantly affects the equilibrium-point stability properties. A model with an α-function replacing the standard exponentially decaying function in the inhibitory coupling kernel is in most of our examples found to be most prone to instability, while the opposite situation with an α-function describing the excitatory kernel is found to be least prone to instability. The standard model with exponentially decaying coupling kernels is typically found to be an intermediate case. We further find that stability is promoted by increasing the weight of self-inhibition or shortening the time constant of the inhibition.     

Left ventricular wall stress compendium

Left ventricular (LV) wall stress has intrigued scientists and cardiologists since the time of Lame and Laplace in 1800s. The left ventricle is an intriguing organ structure, whose intrinsic design enables it to fill and contract. The development of wall stress is intriguing to cardiologists and biomedical engineers. The role of left ventricle wall stress in cardiac perfusion and pumping as well as in cardiac pathophysiology is a relatively unexplored phenomenon. But even for us to assess this role, we first need accurate determination of in vivo wall stress. However, at this point, 150 years after Lame estimated left ventricle wall stress using the elasticity theory, we are still in the exploratory stage of (i) developing left ventricle models that properly represent left ventricle anatomy and physiology and (ii) obtaining data on left ventricle dynamics. In this paper, we are responding to the need for a comprehensive survey of left ventricle wall stress models, their mechanics, stress computation and results. We have provided herein a compendium of major type of wall stress models: thin-wall models based on the Laplace law, thick-wall shell models, elasticity theory model, thick-wall large deformation models and finite element models. We have compared the mean stress values of these models as well as the variation of stress across the wall. All of the thin-wall and thick-wall shell models are based on idealised ellipsoidal and spherical geometries. However, the elasticity model's shape can vary through the cycle, to simulate the more ellipsoidal shape of the left ventricle in the systolic phase. The finite element models have more representative geometries, but are generally based on animal data, which limits their medical relevance. This paper can enable readers to obtain a comprehensive perspective of left ventricle wall stress models, of how to employ them to determine wall stresses, and be cognizant of the assumptions involved in the use of specific models.     

Left ventricular wall stress compendium

Left ventricular (LV) wall stress has intrigued scientists and cardiologists since the time of Lame and Laplace in 1800s. The left ventricle is an intriguing organ structure, whose intrinsic design enables it to fill and contract. The development of wall stress is intriguing to cardiologists and biomedical engineers. The role of left ventricle wall stress in cardiac perfusion and pumping as well as in cardiac pathophysiology is a relatively unexplored phenomenon. But even for us to assess this role, we first need accurate determination of in vivo wall stress. However, at this point, 150 years after Lame estimated left ventricle wall stress using the elasticity theory, we are still in the exploratory stage of (i) developing left ventricle models that properly represent left ventricle anatomy and physiology and (ii) obtaining data on left ventricle dynamics. In this paper, we are responding to the need for a comprehensive survey of left ventricle wall stress models, their mechanics, stress computation and results. We have provided herein a compendium of major type of wall stress models: thin-wall models based on the Laplace law, thick-wall shell models, elasticity theory model, thick-wall large deformation models and finite element models. We have compared the mean stress values of these models as well as the variation of stress across the wall. All of the thin-wall and thick-wall shell models are based on idealised ellipsoidal and spherical geometries. However, the elasticity model's shape can vary through the cycle, to simulate the more ellipsoidal shape of the left ventricle in the systolic phase. The finite element models have more representative geometries, but are generally based on animal data, which limits their medical relevance. This paper can enable readers to obtain a comprehensive perspective of left ventricle wall stress models, of how to employ them to determine wall stresses, and be cognizant of the assumptions involved in the use of specific models.