Program Code Generator for Cardiac Electrophysiology Simulation with Automatic PDE Boundary Condition Handling

Clinical and experimental studies involving human hearts can have certain limitations. Methods such as computer simulations can be an important alternative or supplemental tool. Physiological simulation at the tissue or organ level typically involves the handling of partial differential equations (PDEs). Boundary conditions and distributed parameters, such as those used in pharmacokinetics simulation, add to the complexity of the PDE solution. These factors can tailor PDE solutions and their corresponding program code to specific problems. Boundary condition and parameter changes in the customized code are usually prone to errors and time-consuming. We propose a general approach for handling PDEs and boundary conditions in computational models using a replacement scheme for discretization. This study is an extension of a program generator that we introduced in a previous publication. The program generator can generate code for multi-cell simulations of cardiac electrophysiology. Improvements to the system allow it to handle simultaneous equations in the biological function model as well as implicit PDE numerical schemes. The replacement scheme involves substituting all partial differential terms with numerical solution equations. Once the model and boundary equations are discretized with the numerical solution scheme, instances of the equations are generated to undergo dependency analysis. The result of the dependency analysis is then used to generate the program code. The resulting program code are in Java or C programming language. To validate the automatic handling of boundary conditions in the program code generator, we generated simulation code using the FHN, Luo-Rudy 1, and Hund-Rudy cell models and run cell-to-cell coupling and action potential propagation simulations. One of the simulations is based on a published experiment and simulation results are compared with the experimental data. We conclude that the proposed program code generator can be used to generate code for physiological simulations and provides a tool for studying cardiac electrophysiology.     

A versatile approach for modeling and simulating the tacticity of polymers

We are introducing a versatile computerized approach to model and simulate polymer tacticities using seven single-stage statistical models. The theory behind the models, e.g., Bovey’s versus Price’s, Bernoullian, 1st or 2nd order Markovian, enantiomeric types, and combinations thereof is explained. One of the models, "E-B gen", which can be used to produce four types of enantiomorphically controlled tacticities, and the pentad distribution for the model "E-M1" are reported here for the first time. The relations of chain-end controlled models to binary copolymerizations are discussed in detail, and equations for the conversion of tacticity based probabilities to reactivity ratios to obtain related n-ad distributions are presented. The models were applied to 20 polymers with exemplary tacticities found in the literature. A related software program (“Polytact”) based on Microsoft’s Excel has been designed to calculate all relevant characteristics of the polymer tacticity and to present them in graphical form in a user-friendly manner. The program can be used to produce graphs of the triad, pentad and sequence length distributions and a simulation of 50 monomer repeat units in the polymer for each of the seven models. One of the main intended uses of the program is to compare the computed n-ad distributions to those of experimental polymers obtained from NMR spectroscopy and to gain insight into the polymerization mechanisms.     

Parameter estimation in Hodgkin-Huxley-Type equations for membrane action potentials in nerve and heart muscle

A method is presented which renders parameter estimation possible in systems of non-linear differential equations where normally no solution exists in terms of analytic functions and which have to be solved numerically. The method uses the concept of sensitivity equations. Two examples are given, taking mathematical models for membrane action potentials in nerve and heart muscle by Hodgkin and Huxley and by Beeler and Reuter. The model equations together with the corresponding system of sensitivity equations are given, which are necessary to estimate maximum conductivity coefficients defining the interactions of different ionic current components. A computer program is described and results of action potential numerical analysis are presented using simulated data. It can be seen, that even with superimposed simulated noise the real parameter values are estimated in an excellent manner. The method can be used to interpret observed changes in action potential time courses under physiological and pharmacological conditions.     

Multidimensional curve fitting program for biological data

A program written for use with the IBM-PC can be used to find least squares solutions to linearized multidimensional equations. The program is 'user-friendly' by requiring little from the user except to make decisions; most responses can be entered by a single keystroke. Once data are entered by the user, they can be repeatedly manipulated, graphed, and correlated. Many models relating data variables can be tried relatively easily, and best fit results found. Examples using respiratory mechanical data illustrate the ease of model comparisons.     

Stochastic models of kleptoparasitism

In this paper, we consider a model of kleptoparasitism amongst a small group of individuals, where the state of the population is described by the distribution of its individuals over three specific types of behaviour (handling, searching for or fighting over, food). The model used is based upon earlier work which considered an equivalent deterministic model relating to large, effectively infinite, populations. We find explicit equations for the probability of the population being in each state. For any reasonably sized population, the number of possible states, and hence the number of equations, is large. These equations are used to find a set of equations for the means, variances, covariances and higher moments for the number of individuals performing each type of behaviour. Given the fixed population size, there are five moments of order one or two (two means, two variances and a covariance). A normal approximation is used to find a set of equations for these five principal moments. The results of our model are then analysed numerically, with the exact solutions, the normal approximation and the deterministic infinite population model compared. It is found that the original deterministic models approximate the stochastic model well in most situations, but that the normal approximations are better, proving to be good approximations to the exact distribution, which can greatly reduce computing time.     

Focus

A solid background in population ecology is valuable for anyone concerned with natural resources management. However, without the opportunity of field experience, it may be difficult for students to become familiar with some of the principles of population biology. Microcomputers and programmable pocket calculators allow the construction of simple simulation models, making it possible to carry out some ‘numerical experiments’ in the classroom which may play the same role in the learning process as in vivo experimentation

This work introduces a mathematical model based on the Lotka—Volterra equations and a computer program developed for a programmable calculator. They can be used to assess the influence of environmental heterogeneity and disturbances on the results of competitive relationships between two plant populations. As an example, the interaction between two tussock grass species of the Patagonian steppe, Stipa speciosa Trin. et Rupr. and Festuca pallescens (St Yves) Parodi (the second being less xerophytic than the first one), is analysed.     

The Routine Fitting of Kinetic Data to Models: A Mathematical Formalism for Digital Computers

A mathematical formalism is presented for use with digital computers to permit the routine fitting of data to physical and mathematical models. Given a set of data, the mathematical equations describing a model, initial conditions for an experiment, and initial estimates for the values of model parameters, the computer program automatically proceeds to obtain a least squares fit of the data by an iterative adjustment of the values of the parameters. When the experimental measures are linear combinations of functions, the linear coefficients for a least squares fit may also be calculated. The values of both the parameters of the model and the coefficients for the sum of functions may be unknown independent variables, unknown dependent variables, or known constants. In the case of dependence, only linear dependencies are provided for in routine use. The computer program includes a number of subroutines, each one of which performs a special task. This permits flexibility in choosing various types of solutions and procedures. One subroutine, for example, handles linear differential equations, another, special non-linear functions, etc. The use of analytic or numerical solutions of equations is possible.     

Diffusion models for chemotaxis: a statistical analysis of noninteractive unicellular movement.

A program is developed for applying stochastic differential equations to models for chemotaxis. First a few of the experimental and theoretical models for chemotaxis both for swimming bacteria and for cells migrating along a substrate are reviewed. In physical and biological models of deterministic systems, finite difference equations are often replaced by a limiting differential equation in order to take advantage of the ease in the use of calculus. A similar but more intricate methodology is developed here for stochastic models for chemotaxis. This exposition is possible because recent work in probability theory gives ease in the use of the stochastic calculus for diffusions and broad applicability in the convergence of stochastic difference equations to a stochastic differential equation. Stochastic differential equations suggest useful data for the model and provide statistical tests. We begin with phenomenological considerations as we analyze a one-dimensional model proposed by Boyarsky, Noble, and Peterson in their study of human granulocytes. In this context, a theoretical model consists in identifying which diffusion best approximates a model for cell movement based upon theoretical considerations of cell physiology. Such a diffusion approximation theorem is presented along with discussion of the relationship between autocovariance and persistence. Both the stochastic calculus and the diffusion approximation theorem are described in one dimension. Finally, these tools are extended to multidimensional models and applied to a three-dimensional experimental setup of spherical symmetry.     

Advances in disinfection testing and modelling

AIMS: To develop a set of kinetic equations which more ably describe the disinfection process. METHODS AND RESULTS: A group of functions, the fat equations, based on the model used for the quantification of microbial inhibition, was produced. These functions introduce a limit to the numbers of micro-organisms capable of being disinfected. These new expressions were shown to be more general forms of currently-used (e.g. log-linear) disinfection models, and accommodate the lags and/or tails of non-linear log-survivor--time plots. An advance in the experimental procedures used to obtain disinfection data, using an optical density technique, was developed concomitantly. CONCLUSION: The methods of analyses (experimental and modelling) allow the researcher to examine, more ably, five-minute disinfection (or specific time disinfection tests) as well as the more important disinfection rate analyses. SIGNIFICANCE AND IMPACT OF THE STUDY: The fat equations are an improvement over commonly-used rate models of disinfection, which are shown to be special cases of these equations. This raises the question as to whether our current understanding of the kinetic basis of disinfection requires revision.     

Visco-hyperelastic law for finite deformations: a frequency analysis

Some biological tissues are repeatedly stimulated under cyclic loading, and this stimulation can be combined with large pressures, thus leading to large deformations. For such applications, visco-hyperelastic models have been proposed in the literature and used in finite-element studies. An extensively used quasi-linear model (QLVH), which assumes linear evolution equations, is compared with a nonlinear model (NLVH), which assumes a multiplicative split of the deformation gradient. The comparison is made here using sets of simulations covering a large frequency range. Lost and stored energies are computed, and the additional parameter of the NLVH model is set to two values found in the literature (NLVH-2 and NLVH-30 models). The predicted behaviour is very similar for all models at small strains, with each time constant (and corresponding viscous modulus) being associated with a damping peak and a stored-energy increase. When the strain amplitude is increased, the ratio of lost to stored energy increases for the QLVH model, but decreases for the NLVH models. The NLVH-30 model also displays a shift of the peak damping towards higher frequencies. Before reaching a steady state, all models display a decay of energy independent of the frequency, and the additional parameter of the NLVH model permits the modelling of complex types of evolution of the damping. In conclusion, this study compares the behaviour of two viscous hyper-elastic laws to allow an informed choice between them.     

Threshold virus dynamics with impulsive antiretroviral drug effects

The purposes of this paper are twofold: to develop a rigorous approach to analyze the threshold behaviors of nonlinear virus dynamics models with impulsive drug effects and to examine the feasibility of virus clearance following the Manuals of National AIDS Free Antiviral Treatment in China. An impulsive system of differential equations is developed to describe the within-host virus dynamics of both wild-type and drug-resistant strains when a combination of antiretroviral drugs is used to induce instantaneous drug effects at a sequence of dosing times equally spaced while drug concentrations decay exponentially after the dosing time. Threshold parameters are derived using the basic reproduction number of periodic epidemic models, and are used to depict virus clearance/persistence scenarios using the theory of asymptotic periodic systems and the persistence theory of discrete dynamical systems. Numerical simulations using model systems parametrized in terms of the antiretroviral therapy recommended in the aforementioned Manuals illustrate the theoretical threshold virus dynamics, and examine conditions under which the impulsive antiretroviral therapy leads to treatment success. In particular, our results show that only the drug-resistant strain can dominate (the first-line treatment program guided by the Manuals) or both strains may be rapidly eliminated (the second-line treatment program), thus the work indicates the importance of implementing the second-line treatment program as soon as possible.     

Modelling of biofilms

A mixed-culture biofilm (MCB) model is available which describes the progression of biofilm thickness and the spatial distribution and development in time of dissolved and paniculate components in the biofilm. The MCB model is able to predict the physico-chemical conditions at the interface between the biofilm and the solid surface, on which the biofilm grows, as a function of the conditions in the bulk fluid, the microbial composition of the biofilm, and the transport and transformation processes which take place in the biofilm. The mass balance equations of the MCB model are generally valid and can be applied to almost any microbial system if its kinetics and stoichiometry can be provided. AQUASIM is a new computer program for the identification and simulation of aquatic systems. The program solves the equations of the MCB model. It has a window-type user interface and includes routines for simulation, sensitivity analysis, automatic parameter estimation and data fitting. The MCB model has been developed and is primarily used in the field of waste water treatment. However, under certain conditions and with some additional simplifications this model can also be used for the investigation of biofouling and biocorrosion problems. The possibilities and limitations of the application of the MCB model and of AQUASIM to this type of problem are briefly discussed.     

大兴安岭东部主要林分类型乔木层生物量估算模型

大尺度森林生物量的估算方法是人们目前关注的焦点,建立林分生物量模型成为一种趋势.本研究以大兴安岭东部6个主要林分类型为研究对象,构建了其总量及各分项一元、二元可加性林分生物量模型.采用似然分析法判断总量及各分项生物量异速生长模型的误差结构(可加型或相乘型),采用非线性似乎不相关回归模型方法估计模型参数.结果表明: 经似然分析法判断,大兴安岭东部6个主要林分类型总量及各分项生物量异速生长模型的误差结构都是相乘型的,对数转换的可加性生物量可以被选用.各林分类型可加性生物量模型的调整后确定系数为0.78～0.99,平均相对误差为-2.3%～6.9%,平均相对误差绝对值6.3%～43.3%.增加林分平均高可以提高绝大多数生物量模型的拟合效果和预测能力,而且总量、地上和树干生物量模型效果较好,树根、树枝、树叶和树冠生物量模型效果较差.为了使模型参数估计更有效,所建立的生物量模型应当考虑林分总生物量及各分项生物量的可加性.本研究建立的林分总量与各分项生物量模型都能对大兴安岭东部6个主要林分类型生物量进行较好的估计.     

Robust dynamic experiments for the precise estimation of respiration and fermentation parameters of fruit and vegetables

Dynamic models based on non-linear differential equations are increasingly being used in many biological applications. Highly informative dynamic experiments are valuable for the identification of these dynamic models. The storage of fresh fruit and vegetables is one such application where dynamic experimentation is gaining momentum. In this paper, we construct optimal O₂ and CO₂ gas input profiles to estimate the respiration and fermentation kinetics of pear fruit. The optimal input profiles, however, depend on the true values of the respiration and fermentation parameters. Locally optimal design of input profiles, which uses a single initial guess for the parameters, is the traditional method to deal with this issue. This method, however, is very sensitive to the initial values selected for the model parameters. Therefore, we present a robust experimental design approach that can handle uncertainty on the model parameters.     

On the uniqueness of the positive solution of an integral equation which appears in epidemiological models

 In this paper, we show that the positive solution of a non-linear integral equation which appears in classical SIR epidemiological models is unique. The demonstration of this fact is necessary to justify the correctness of any approximate or numerical solution. The SIR epidemiological model is used only for simplicity. In fact, the methods used can be easily extended to prove the existence and uniqueness of the more involved integral equations that appear when more biological realities are considered. Thus the inclusion of a latent class (SLIR models) and models incorporating variability in the infectiousness with duration of the infection and spatial distribution lead to integral equations to which the results derived in this paper apply immediately. Received: 7 May 1999     

A reliable simulator for dynamic flux balance analysis

Dynamic flux balance analysis (DFBA) provides a platform for detailed design, control and optimization of biochemical process technologies. It is a promising modeling framework that combines genome‐scale metabolic network analysis with dynamic simulation of the extracellular environment. Dynamic flux balance analysis assumes that the intracellular species concentrations are in equilibrium with the extracellular environment. The resulting underdetermined stoichiometric model is solved under the assumption of a biochemical objective such as growth rate maximization. The model of the metabolism is coupled with the dynamic mass balance equations of the extracellular environment via expressions for the rates of substrate uptake and product excretion, which imposes additional constraints on the linear program (LP) defined by growth rate maximization of the metabolism. The linear program is embedded into the dynamic model of the bioreactor, and together with the additional constraints this provides an accurate model of the substrate consumption, product secretion, and biomass production during operation. A DFBA model consists of a system of ordinary differential equations for which the evaluation of the right‐hand side requires not only function evaluations, but also the solution of one or more linear programs. The numerical tool presented here accurately and efficiently simulates large‐scale dynamic flux balance models. The main advantages that this approach has over existing implementation are that the integration scheme has a variable step size, that the linear program only has to be solved when qualitative changes in the optimal flux distribution of the metabolic network occur, and that it can reliably simulate behavior near the boundary of the domain where the model is defined. This is illustrated through large‐scale examples taken from the literature. Biotechnol. Bioeng. 2013; 110: 792–802. © 2012 Wiley Periodicals, Inc.     

Pharmfit-a Nonlinear Fitting Program for Pharmacology

A new program is presented for nonlinear fitting of data from pharmacological and chronobiological investigations. It contains functions for calculating data from ligand-binding studies and competition experiments, for the analysis of dose-response curves, for pharmacokinetic calculations, and for cosine analysis of harmonic and overlapping rhythms. In addition, it is possible to implement general equations by the user. The program allows data exchange with most spreadsheet, database, and graphics presentation programs, and accepts data from two widely used ambulatory 24-h blood-pressure monitoring systems. The fitting procedure uses the Marquardt-Levenberg algorithm. It calculates the weighted or the unweighted fit together with a great variety of statistics for estimation of goodness of fit. A graphics module permits graphical presentation of the fitted curve. Moreover, fitting of data to different models can be compared for the most likely fit and model discrimination statistics for improvement of further experiments are provided. To demonstrate the chronobiological application of the fitting program PHARMFIT, the analysis of telemetric heart rate data from rats is presented.     

Genomic predictions of breeding values in a cloned <Emphasis Type="Italic">Eucalyptus globulus</Emphasis> population in Chile

In Chile, an intensive Eucalyptus globulus clonal selection program is being carried out to increase forest productivity for pulp production. A breeding population was used to investigate the predicted ability of single nucleotide polymorphism (SNP) markers for genomic selection (GS). A total of 310 clones from 53 families were used. Stem volume and wood density were measured on all clones. Trees were genotyped at 12 K polymorphic markers using the EUChip60K genotype array. Genomic best linear unbiased prediction, Bayesian lasso regression, Bayes B, and Bayes C models were used to predict genomic estimated breeding values (GEBV). For cross-validation, 260 individuals were sampled for model training and 50 individuals for model validation, using 2 folds and 10 replications each. The average predictive ability estimates for wood density and stem volume across the models were 0.58 and 0.75, respectively. The average rank correlations were 0.59 and 0.71, respectively. Models produced very similar bias for both traits. When clones were ranked based on their GEBV, models had similar phenotypic mean for the top 10% of the clones. The predicted ability of markers will likely decrease if the models are used to predict GEBV of new material coming from the breeding program, because of a different marker–trait phase introduced by recombination. The results should be validated with larger populations and across two generations before routine applications of GS in E. globulus. We suggest that GS is a viable strategy to accelerate clonal selection program of E. globulus in Chile.     

Oscillations in a model of repression with external control

A mathematical model for control by repression by an extracellular substance is developed, including diffusion and time delays. The model examines how active transport of a nutrient can produce either oscillatory or stable responses depending on a variety of parameters, such as diffusivity, cell size, or nutrient concentration. The system of equations for the mathematical model is reduced to a system of delay differential equations and linear Volterra equations. After linearizing these equations and forming the limiting Volterra equations, the resulting linear system no longer has any spatial dependence. Local stability analysis of the radially symmetric model shows that the system of equations can undergo Hopf bifurcations for certain parameter values, while other ranges of the parameters guarantee asymptotic stability. One numerical study shows that the model can exhibit intracellular biochemical oscillations with increasing extracellular concentrations of the nutrient, which suggests a possible trigger mechanism for morphogenesis.The work of this author was supported in part by NSF grants DMS-8603787 and DMS-8807360The work of this author was supported under the REU program of NSF by grant DMS-8807360The work of this author was supported under the REU program of NSF by grant DMS-8807360