Benchmark solution for the prediction of temperature distributions during radiofrequency ablation of cardiac tissue

Several studies on radiofrequency (RF) ablation are aimed at accurately predicting tissue temperature distributions by numerical solution of the bioheat equation. This paper describes the development of a solution that can serve as a benchmark for subsequent numerical solutions. The solution was obtained using integral transforms and evaluated using a C program. Temperature profiles were generated at various times and for different convection coefficients. In addition, a numerical model was developed using the same assumptions made in obtaining the benchmark solution. Comparison of surface and axial temperature profiles shows that the two solutions match very closely, cross validating the numerical methods used in evaluating both solutions.     

Numerical simulation of propagating concentration profiles in renal tubules

Method-dependent mechanisms that may affect dynamic numerical solutions of a hyperbolic partial differential equation that models concentration profiles in renal tubules are described. Some numerical methods that have been applied to the equation are summarized, and ways by which the methods may misrepresent true solutions are analysed. Comparison of these methods demonstrates the need for thoughtful application of computational mathematics when simulating complicated time-dependent phenomena.     

Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments

In many semi-arid environments, vegetation is self-organised into spatial patterns. The most striking examples of this are on gentle slopes, where striped patterns are typical, running parallel to the contours. Previously, Klausmeier [1999. Regular and irregular patterns in semiarid vegetation. Science 284, 1826-1828.] has proposed a model for vegetation stripes based on competition for water. Here, we present a detailed study of the patterned solutions in the full nonlinear model, using numerical bifurcation analysis of both the pattern odes and the model pdes. We show that patterns exist for a wide range of rainfall levels, and in particular for much lower rainfall than have been considered by previous authors. Moreover, we show that for many rainfall levels, patterns with a variety of different wavelengths are stable, with mode selection dependent on initial conditions. This raises the possibility of hysteresis, and in numerical solutions of the model we show that pattern selection depends on rainfall history in a relatively simple way.     

On Comparison of Series and Numerical Solutions for Flow of Eyring-Powell Fluid with Newtonian Heating And Internal Heat Generation/Absorption

In this paper, we have investigated the combined effects of Newtonian heating and internal heat generation/absorption in the two-dimensional flow of Eyring-Powell fluid over a stretching surface. The governing non-linear analysis of partial differential equations is reduced into the ordinary differential equations using similarity transformations. The resulting problems are computed for both series and numerical solutions. Series solution is constructed using homotopy analysis method (HAM) whereas numerical solution is presented by two different techniques namely shooting method and bvp4c. A comparison of homotopy solution with numerical solution is also tabulated. Both solutions are found in an excellent agreement. Dimensionless velocity and temperature profiles are plotted and discussed for various emerging physical parameters.     

Cooperative polymerization reactions. Analytical approximations, numerical examples, and experimental strategy

How does one obtain kinetic rate constants from the time course of a reversible and cooperative polymerization reaction? We examine a simple version of the homogeneous nucleation-elongation model with both analytical and numerical techniques to test some common assumptions and develop an experimental strategy. The assumption of irreversible polymer formation is found to be a useful and adequate approximation for the numerical determination of monomer disappearance. The assumption of early "pre-equilibrium" between monomer and seed, however, is shown numerically and analytically to produce significant errors over a wide range of parameters, particularly for small seed lengths. We exhibit numerical solutions for many different parameters, and also discuss analytical techniques that allow approximate solutions for several conditions: the high-concentration limit; the long-time limit; and the long-seed-length, lows concentration limit. The overall reaction simplifies when the monomer concentration is large. An experimental strategy for elucidating the seed size and the rate constants for polymerization and depolymerization is presented.     

Adiabatic model of field reversal by fast ions in an axisymmetric open trap

A model of field reversal by fast ions has been developed under the assumption of preservation of fast-ion adiabatic invariants. Analytical solutions obtained in the approximation of a narrow fast-ion layer and numerical solutions to the evolutionary problem are presented. The solutions demonstrate the process of formation of a field reversed configuration with parameters close to those of the planned experiment.     

Numerical bifurcation analysis of delay differential equations arising from physiological modeling

This paper has a dual purpose. First, we describe numerical methods for continuation and bifurcation analysis of steady state solutions and periodic solutions of systems of delay differential equations with an arbitrary number of fixed, discrete delays. Second, we demonstrate how these methods can be used to obtain insight into complex biological regulatory systems in which interactions occur with time delays: for this, we consider a system of two equations for the plasma glucose and insulin concentrations in a diabetic patient subject to a system of external assistance. The model has two delays: the technological delay of the external system, and the physiological delay of the patient's liver. We compute stability of the steady state solution as a function of two parameters, compare with analytical results and compute several branches of periodic solutions and their stability. These numerical results allow to infer two categories of diabetic patients for which the external system has different efficiency.     

A compact model to predict quantized sub-band energy levels and inversion layer centroids of MOSFETs with a parabolic potential well approximation

A compact model to predict sub-band energy levels and inversion charge centroids in the MOSFET surface inversion layer has been presented in this paper for parabolic potential well approximation. Based on a coupled solution of the Schrödinger equation and the Poisson equation following the WKB method, one transcendental equation of the sub-band energy level has been rigorously derived and then the approximate analytical solutions for the sub-band energy levels and the inversion charge centroids have been obtained. The analytical results are compared with the numerical data and a good agreement between the analytical and numerical is found.     

Study of models of single populations: development of equations used in microorganism cultures

This article derives a number of equations which can be used in both continuous and the semicontinuous cultures of microorganism populations in chemostat systems. Using these equations, some phenomena which have been known for many years can be explained reasonably in terms of chemical kinetics, and a number of analytical solutions can be obtained instead of numerical solutions previously published.     

Steady states and outbreaks of two-phase nonlinear age-structured model of population dynamics with discrete time delay

In this paper we study the nonlinear age-structured model of a polycyclic two-phase population dynamics including delayed effect of population density growth on the mortality. Both phases are modelled as a system of initial boundary values problem for semi-linear transport equation with delay and initial problem for nonlinear delay ODE. The obtained system is studied both theoretically and numerically. Three different regimes of population dynamics for asymptotically stable states of autonomous systems are obtained in numerical experiments for the different initial values of population density. The quasi-periodical travelling wave solutions are studied numerically for the autonomous system with the different values of time delays and for the system with oscillating death rate and birth modulus. In both cases it is observed three types of travelling wave solutions: harmonic oscillations, pulse sequence and single pulse.     

根表面养分吸收通量和根围溶质浓度的近似解析解

该文用Nye-Tinker-Barber模型来研究植物根系表面的养分吸收通量和根围溶质浓度的近似解析解。将根围区域分为远场区域和近场区域, 在远场用相似变量, 在近场用尺度变换, 将远场解在根表面展开并与近场解进行待定函数的匹配, 从而获得对流扩散方程根表面通量和浓度的一阶近似解析解, 该解能够简化到扩散方程的解的形式。对氮、钾、硫、磷、镁、钙的养分吸收通量和氮、钾的浓度分别进行数值模拟, 比较模型的数值解、Roose的近似解析解和该文的近似解析解。结果表明: 在扩散方程中, 6种元素通量的解析解与Roose解析解相近, 但均高于数值解, 钾和磷的通量在短时间内迅速衰减; 钾和氮浓度的全局近似解析解与Roose解析解接近, 并与数值解的变化趋势一致。在对流扩散方程中, 除氮外的5种元素通量的近似解较Roose的解析解更接近于数值解, 且没有奇性。     

Standing-gradient osmotic flow Examination of its validity using an analytical method

The solutions to the non-linear differential equations governing solute-solvent coupling in the intercellular spaces of epithelial layers have been obtained by using an analytical method, rather than the usual numerical ones. When the present series solution includes second-order correction terms, the concentration and velocity profiles obtained by the analytical method agree very well with those coming from numerical solutions. This method has further allowed us to examine the standing-gradient hypothesis when applied to the backwards fluid transport system of the corneal endothelium. With the information presently available for the relevant parameters (osmotic permeability, rate of transport, radius and length of the spaces, and location of the pumping sites), near-isotonicity of the transported fluid would not be explained by the standing-gradient model.     

Comparison of the functional and numerical responses of resistant versus non-resistant populations of the copepod Acartia hudsonica fed the toxic dinoflagellate Alexandrium tamarense

The functional and numerical responses of grazers are key pieces of information in predicting and modeling predator–prey interactions. It has been demonstrated that exposure to toxic algae can lead to evolved resistance in grazer populations. However, the influence of resistance on the functional and numerical response of grazers has not been studied to date. Here, we compared the functional and numerical responses of populations of the copepod Acartia hudsonica that vary in their degree of resistance to the toxic dinoflagellate Alexandrium tamarense. In common environment experiments carried out after populations had been grown under identical conditions for several generations, female copepods were offered solutions containing different concentrations of either toxic A. tamarense or the non-toxic green flagellate Tetraselmis sp. ranging from 25 to 500 μgC L⁻¹, and ingestion and egg production rates were measured. Throughout most of the range of concentrations of the toxic diet, copepod populations that had been historically exposed to toxic blooms of Alexandrium exhibited significantly higher ingestion and egg production rates than populations that had little or no exposure to these blooms. In contrast, there were no significant differences between populations in ingestion or egg production for the non-toxic diet. Hence, the between population differences in functional and numerical response to A. tamarense were indeed related to resistance. We suggest that the effect of grazer toxin resistance should be incorporated in models of predator and toxic prey interactions. The potential effects of grazer toxin resistance in the development and control of Alexandrium blooms are illustrated here with a simple simulation exercise.     

Accurate modelling of unsteady flows in collapsible tubes

The context of this paper is the development of a general and efficient numerical haemodynamic tool to help clinicians and researchers in understanding of physiological flow phenomena. We propose an accurate one-dimensional Runge–Kutta discontinuous Galerkin (RK-DG) method coupled with lumped parameter models for the boundary conditions. The suggested model has already been successfully applied to haemodynamics in arteries and is now extended for the flow in collapsible tubes such as veins. The main difference with cardiovascular simulations is that the flow may become supercritical and elastic jumps may appear with the numerical consequence that scheme may not remain monotone if no limiting procedure is introduced. We show that our second-order RK-DG method equipped with an approximate Roe's Riemann solver and a slope-limiting procedure allows us to capture elastic jumps accurately. Moreover, this paper demonstrates that the complex physics associated with such flows is more accurately modelled than with traditional methods such as finite difference methods or finite volumes. We present various benchmark problems that show the flexibility and applicability of the numerical method. Our solutions are compared with analytical solutions when they are available and with solutions obtained using other numerical methods. Finally, to illustrate the clinical interest, we study the emptying process in a calf vein squeezed by contracting skeletal muscle in a normal and pathological subject. We compare our results with experimental simulations and discuss the sensitivity to parameters of our model.     

Algorithms for ultrasound elastography: a survey

Elastography is a useful and interesting technique that can be used to infer stiffness information from ultrasound medical images. In one decade of activity, the scientific community has developed this technique to a more and more mature stage, such that it has evolved into a fruitful application for clinical practice. During this decade, the evolution has proceeded from qualitative stiffness information to numerical quantification using different algorithms and approaches, based on both iterative and direct approaches. Moreover, different post- and pre-processing techniques as well as evaluation methods have been implemented. This work presents a survey on the methods developed by giving a short overview of algorithms and mathematical solutions and by analysing results and comparing methodologies.     

Approximate analytical solutions of root surface nutrient uptake flux and rhizosphere solute concentrations

该文用Nye-Tinker-Barber模型来研究植物根系表面的养分吸收通量和根围溶质浓度的近似解析解。将根围区域分为远场区域和近场区域, 在远场用相似变量, 在近场用尺度变换, 将远场解在根表面展开并与近场解进行待定函数的匹配, 从而获得对流扩散方程根表面通量和浓度的一阶近似解析解, 该解能够简化到扩散方程的解的形式。对氮、钾、硫、磷、镁、钙的养分吸收通量和氮、钾的浓度分别进行数值模拟, 比较模型的数值解、Roose的近似解析解和该文的近似解析解。结果表明: 在扩散方程中, 6种元素通量的解析解与Roose解析解相近, 但均高于数值解, 钾和磷的通量在短时间内迅速衰减; 钾和氮浓度的全局近似解析解与Roose解析解接近, 并与数值解的变化趋势一致。在对流扩散方程中, 除氮外的5种元素通量的近似解较Roose的解析解更接近于数值解, 且没有奇性。     

A practical approach to significance assessment in alignment with gaps.

Current numerical methods for assessing the statistical significance of local alignments with gaps are time consuming. Analytical solutions thus far have been limited to specific cases. Here, we present a new line of attack to the problem of statistical significance assessment.We combine this new approach with known properties of the dynamics of the global alignment algorithm and high performance numerical techniques and present a novel method for assessing significance of gaps within practical time scales. The results and performance of these new methods test very well against tried methods with drastically less effort.     

Analysis of tracer experiments: VIII. Integro-differential equation treatment of partly accessible,partly injectable multicompartment systems

An integro-differential equation treatment of multi-compartment systems is developed which permits formal analysis of the incomplete data which is available from partly accessible, partly injectable systems. New transport functions are defined which can be obtained directly from the experimental data. These functions serve to characterize the communication and topology between different accessible compartments and also the reentrant contributions from inaccessible sites. The method gives solutions consistent with those of the differential equation approach when the system is uniformly contiguous and accessible, more complete solutions than those of the integral equation approach when all measured compartments are injectable, and in addition provides complete or partial solutions for certain otherwise analytically intractable systems. Detailed numerical illustrations of the method are given.     

A new numerical solution to traditional phytosociological tabular classification

Often, manually and numerically derived phytosociological classifications yield different results. Hitherto, a twostep procedure has been suggested in which numerical analysis of the data is followed by the revision of the resulting table (c.f. van der Maarel 1982). In this paper a new methodology is presented which makes manual refinements superfluous. Objectives are derived from phytosociological paradigms and conclusions drawn for the analytical process. The problems to be solved are: data transformation, detection of outliers, selection of clustering methods, checking within-group diversity, analysis of the resulting group structure, rearrangement of relevés and species within the groups, and finally the selection of differential species. The method has been derived using the well known example of Ellenberg (Mueller-Dombois & Ellenberg 1974). The results almost perfectly reproduce the intuitively widely accepted manual refinements in structure and presentation. A test with plant sociological data from Swiss forests (Ellenberg & Klötzli 1972) proves that the method can also classify complex gradient- and group systems and that the numerical result matches Landolt's (1977) system of indicator values. Since the solutions can be exactly reproduced, it is no longer necessary to combine numerical analysis with additional editing.     

Analytical SAR computation in a multilayer elliptic cylinder: the near-field line-current radiation case

In this paper, a previously proposed analytical procedure for the computation of the specific absorption rate (SAR) inside a biological elliptic cylinder model is extended to the case in which the body is illuminated under near-field conditions. The elliptic model is made up of layers of different biological tissues and the source is constituted by a line-current distribution. The recursive procedure in which the field is expressed in terms of Mathieu function is modified to express the incident electromagnetic wave produced by the line current. The new procedure makes it possible to check and validate numerical solutions obtained by accurate numerical techniques for SAR prediction, under more realistic illumination condition.