Parameter Uncertainty Analysis of a Mathematical Ion Channel Model.

In mathematical modeling of cell physiological processes, measurements required for parameter determination are often available only as aggregated data in the literature. Physiological measurements contain relatively large observation errors due to intrinsic variations in physiological processes, and the errors cause uncertainties in parameter values. This paper reports analyses of the uncertainty in parameter estimates of a simple mathematical model of an ion channel from a set of published experimental data. A conventional approach for estimating model parameters from aggregated data is applying the method of least squares to a series of the mean values of measurements. The parameter estimates by the conventional method significantly differed from those by a statistical approach, maximum likelihood estimation considering the standard errors of the means. Exhaustive analyses on the likelihood of parameter values show high parameter uncertainties and wide distribution of parameter values with no significant differences in the likelihood. These results imply the importance of considering variances of observations and uncertainties in parameter estimates.

A Preliminary Computational Model for Hypoxic Acidosis in Cardiac Myocytes.

Intracellular acidosis induced by hypoxia resulted from myocardial ischemia damages myocardium. However, the detailed mechanisms of hypoxic acidosis are not quantitatively explained. The purpose of this study is to create a novel computational model which can reproduce intracellular acidosis caused by myocardial ischemia. We constructed a computational model of myocardium, by using a mathematical ventricular cell model which includes pH regulation and a computational model of myocardial microcirculation for calculating extracellular conditions. The present model reproduced cellular hypoxia in an ischemic condition simulated by a reduced blood flow, and intracellular pH reduction in response to the hypoxia.

A computational model of myocardial microcirculation including interstitial flow.

Contributions of interstitial fluid (ISF) flow within the myocardial microcirculation is not well understood despite its importance due to difficulties in measurements. For analysing a contribution of interstitial fluid flow within myocardial microcirculation, we developed a computational model of myocardial microcirculation by introducing convection by the ISF flow into an existing myocardial microcirculation model, and performed simulations with varied ISF flows in normal and hypoperfusion conditions. Simulation results show that the ISF flow has a contribution only with low capillary flow. This might suggest partial comensation of oxygen supply by the ISF flow under ischemic conditions.

Modeling analysis of inositol 1,4,5-trisphosphate receptor-mediated Ca2+ mobilization under the control of glucagon-like peptide-1 in mouse pancreatic ß-cells.

Glucagon-like peptide-1 (GLP-1) is an intestinally derived blood glucose-lowering hormone that potentiates glucose-stimulated insulin secretion from pancreatic ß-cells. The secretagogue action of GLP-1 is explained, at least in part, by its ability to stimulate cAMP production so that cAMP may facilitate the release of Ca(2+) from inositol trisphosphate receptor (IP3R)-regulated Ca(2+) stores. However, a quantitative model has yet to be provided that explains the molecular mechanisms and dynamic processes linking GLP-1-stimulated cAMP production to Ca(2+) mobilization. Here, we performed simulation studies to investigate how GLP-1 alters the abilities of Ca(2+) and IP3 to act as coagonists at IP3R Ca(2+) release channels. A new dynamic model was constructed based on the Kaftan model, which demonstrates dual steady-state allosteric regulation of the IP3R by Ca(2+) and IP3. Data obtained from ß-cells were then analyzed to understand how GLP-1 facilitates IP3R-mediated Ca(2+) mobilization when UV flash photolysis is used to uncage Ca(2+) and IP3 intracellularly. When the dynamic model for IP3R activation was incorporated into a minimal cell model, the Ca(2+) transients and oscillations induced by GLP-1 were successfully reconstructed. Simulation studies indicated that transient and oscillatory responses to GLP-1 were produced by sequential positive and negative feedback regulation due to fast activation and slow inhibition of the IP3R by Ca(2+). The slow rate of Ca(2+)-dependent inhibition was revealed to provide a remarkable contribution to the time course of the decay of cytosolic Ca(2+) transients. It also served to drive and pace Ca(2+) oscillations that are significant when evaluating how GLP-1 stimulates insulin secretion.

Application of the Kalman Filter for Faster Strong Coupling of Cardiovascular Simulations.

In this paper, we propose a method for reducing the computational cost of strong coupling for multiscale cardiovascular simulation models. In such a model, individual model modules of myocardial cell, left ventricular structural dynamics, and circulatory hemodynamics are coupled. The strong coupling method enables stable and accurate calculation, but requires iterative calculations which are computationally expensive. The iterative calculations can be reduced, if accurate initial approximations are made available by predictors. The proposed method uses the Kalman filter to estimate accurate predictions by filtering out noise included in past values. The performance of the proposed method was assessed with an application to a previously published multiscale cardiovascular model. The proposed method reduced the number of iterations by 90% and 62% compared with no prediction and Lagrange extrapolation, respectively. Even when the parameters were varied and number of elements of the left ventricular finite-element model increased, the number of iterations required by the proposed method was significantly lower than that without prediction. These results indicate the robustness, scalability, and validity of the proposed method.

Quantitative Decomposition of Dynamics of Mathematical Cell Models: Method and Application to Ventricular Myocyte Models.

Mathematical cell models are effective tools to understand cellular physiological functions precisely. For detailed analysis of model dynamics in order to investigate how much each component affects cellular behaviour, mathematical approaches are essential. This article presents a numerical analysis technique, which is applicable to any complicated cell model formulated as a system of ordinary differential equations, to quantitatively evaluate contributions of respective model components to the model dynamics in the intact situation. The present technique employs a novel mathematical index for decomposed dynamics with respect to each differential variable, along with a concept named instantaneous equilibrium point, which represents the trend of a model variable at some instant. This article also illustrates applications of the method to comprehensive myocardial cell models for analysing insights into the mechanisms of action potential generation and calcium transient. The analysis results exhibit quantitative contributions of individual channel gating mechanisms and ion exchanger activities to membrane repolarization and of calcium fluxes and buffers to raising and descending of the cytosolic calcium level. These analyses quantitatively explicate principle of the model, which leads to a better understanding of cellular dynamics.

Nonlinear multiscale circulation model reproducable linear end-systolic pressure-volume relationship.

As a well-known property of the heart, many studies has reported that the left ventricular end-systolic pressure-volume relationship (ESPVR) is linear. However, the reason of the linearity is poorly understood. This article presents a multiscale circulation model to be a tool for theoretical analyses on the mechanism of the linearity of ESPVR. The model is composed of three sub-models; a detailed closed-loop lumped-parameter model for cardiovascular system, geometric left ventricle model, a comprehensive ventricular myocyte model. Although the present model integrates nonlinear sub-models, the model can successfully reproduce highly linear ESPVR without any arbitrary modifications.

Code generator for distributed parameter biological model simulation with PDE numerical schemes.

The physiological simulation at the tissue and organ level typically involves the handling of partial differential equations (PDEs). Boundary conditions and in cases like pharmacokinetics, distributed parameters add to the complexity of the PDE solution. These factors make most PDE solutions and their corresponding program codes tailored for specific problems. We propose a general approach for handling PDEs in computational models using a replacement scheme for discretization. This method allows for the handling of the different PDE types. The replacement scheme involves substituting all the partial differential terms with the numerical solution equations. Once the model 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 determine the simulation loop structure and generate the program code.

A study on the relationship between electrical transmural heterogeneity and ventricular energetics.

In this study, we use cardiovascular simulation to gain new insights on the correlation between electrical heterogeneity and ventricular energetics. Although there are numerous in vivo and in vitro studies on the electrical heterogeneity within the ventricular myocardium, not much attention has been directed to its correlation to cardiovascular mechanics, because of difficulties in simultaneously observing and analyzing multiple spatial scales (the cell, the organ, and the system). We performed simulations with two cardiovascular simulation models, one which uses different myocardial cell models for the epicardial, endocardial, and mid-myocardial cells, and another which uses a homogeneous model throughout the entire myocardium. The epicardial, endocardial, and midmyocardial cell models were created by parametrically tuning a homogenous cell model. From the cardiovascular simulation we obtained pressure-volume loops which were used to calculate cardiovascular energetic efficiency and myocardial contractility. We found that energetic efficiency is higher in the electrically heterogeneous model.

Theoretical analysis on the relationship between left ventricular energetic efficiency and acute infarct size.

Energetic efficiency is an important indicator of cardiac function in acute myocardial infarction. However, the relationship between cardiac energetic efficiency and infarct size is not perfectly elucidated. In this study, the relationship is analysed by means of simulation using a theoretical model of the guinea pig left ventricle. In simulation with varied ratios of infarct area, pressure-volume area (PVA), which is an index of total mechanical energy by ventricular contraction, and myocardial oxygen consumption (MVO2) are calculated for each infarct ratio. Then, change of PVA when MVO2 alters (PVA/MVO2) as a well-known index of energy conversion efficiency is evaluated. In addition, PVA/VO2, which represents a ratio of PVA change to alteration of mean oxygen consumption of myocytes except for infarct myocytes, is introduced as an index for real energetic efficiency. In simulation results, PVA/MVO2 increases but PVA/VO2 decreases as infarct area expands, because with expansion of infarct area PVA decreases but VO2 remains almost unchanged because of larger shortening of myocytes. This implies that the enlargement of shortening of noninfarcted myocyte to compensate for depression of cardiac output is a potential cause of myocardial remodelling.

A CellML simulation compiler and code generator using ODE solving schemes.

: Models written in description languages such as CellML are becoming a popular solution to the handling of complex cellular physiological models in biological function simulations. However, in order to fully simulate a model, boundary conditions and ordinary differential equation (ODE) solving schemes have to be combined with it. Though boundary conditions can be described in CellML, it is difficult to explicitly specify ODE solving schemes using existing tools. In this study, we define an ODE solving scheme description language-based on XML and propose a code generation system for biological function simulations. In the proposed system, biological simulation programs using various ODE solving schemes can be easily generated. We designed a two-stage approach where the system generates the equation set associating the physiological model variable values at a certain time t with values at t + Δt in the first stage. The second stage generates the simulation code for the model. This approach enables the flexible construction of code generation modules that can support complex sets of formulas. We evaluate the relationship between models and their calculation accuracies by simulating complex biological models using various ODE solving schemes. Using the FHN model simulation, results showed good qualitative and quantitative correspondence with the theoretical predictions. Results for the Luo-Rudy 1991 model showed that only first order precision was achieved. In addition, running the generated code in parallel on a GPU made it possible to speed up the calculation time by a factor of 50. The CellML Compiler source code is available for download at http://sourceforge.net/projects/cellmlcompiler.

A coupling method for a cardiovascular simulation model which includes the Kalman filter.

Multi-scale models of the cardiovascular system provide new insight that was unavailable with in vivo and in vitro experiments. For the cardiovascular system, multi-scale simulations provide a valuable perspective in analyzing the interaction of three phenomenons occurring at different spatial scales: circulatory hemodynamics, ventricular structural dynamics, and myocardial excitation-contraction. In order to simulate these interactions, multiscale cardiovascular simulation systems couple models that simulate different phenomena. However, coupling methods require a significant amount of calculation, since a system of non-linear equations must be solved for each timestep. Therefore, we proposed a coupling method which decreases the amount of calculation by using the Kalman filter. In our method, the Kalman filter calculates approximations for the solution to the system of non-linear equations at each timestep. The approximations are then used as initial values for solving the system of non-linear equations. The proposed method decreases the number of iterations required by 94.0% compared to the conventional strong coupling method. When compared with a smoothing spline predictor, the proposed method required 49.4% fewer iterations.

A program code generator for multiphysics biological simulation using markup languages.

To cope with the complexity of the biological function simulation models, model representation with description language is becoming popular. However, simulation software itself becomes complex in these environment, thus, it is difficult to modify the simulation conditions, target computation resources or calculation methods. In the complex biological function simulation software, there are 1) model equations, 2) boundary conditions and 3) calculation schemes. Use of description model file is useful for first point and partly second point, however, third point is difficult to handle for various calculation schemes which is required for simulation models constructed from two or more elementary models. We introduce a simulation software generation system which use description language based description of coupling calculation scheme together with cell model description file. By using this software, we can easily generate biological simulation code with variety of coupling calculation schemes. To show the efficiency of our system, example of coupling calculation scheme with three elementary models are shown.

Time-dependent changes in membrane excitability during glucose-induced bursting activity in pancreatic ß cells.

In our companion paper, the physiological functions of pancreatic ß cells were analyzed with a new ß-cell model by time-based integration of a set of differential equations that describe individual reaction steps or functional components based on experimental studies. In this study, we calculate steady-state solutions of these differential equations to obtain the limit cycles (LCs) as well as the equilibrium points (EPs) to make all of the time derivatives equal to zero. The sequential transitions from quiescence to burst-interburst oscillations and then to continuous firing with an increasing glucose concentration were defined objectively by the EPs or LCs for the whole set of equations. We also demonstrated that membrane excitability changed between the extremes of a single action potential mode and a stable firing mode during one cycle of bursting rhythm. Membrane excitability was determined by the EPs or LCs of the membrane subsystem, with the slow variables fixed at each time point. Details of the mode changes were expressed as functions of slowly changing variables, such as intracellular [ATP], [Ca(2+)], and [Na(+)]. In conclusion, using our model, we could suggest quantitatively the mutual interactions among multiple membrane and cytosolic factors occurring in pancreatic ß cells.

A study on prediction methods for a cardiovascular strong-coupling simulation.

We investigated numerical methods for predictors in a multiscale cardiovascular simulation model. The proposed method predicts initial approximations for the iterative convergence calculations of the strong coupling method using the smoothing spline to remove errors from values of past timesteps and using the linear and second-order extrapolation. The new coupling algorithm was used for coupling a left ventricular finite element model to a myocardial excitation-contraction model. We performed experiments with different values for the smoothing parameter λ and with linear and second-order extrapolations. λ = 1 with the linear extrapolation gave the best results. It reduced computation time by 91% compared to the strong coupling method. With the use of the smoothing spline, distance between the initial approximation and converged solution reduced by 62%, while the average number of iterations reduced by 32%. The smoothing spline can be used to improve the accuracy of predictors and reduce the number of iterations needed for the computation of the convergence procedure.

A general CellML simulation code generator using ODE solving scheme description.

To cope with the complexity of the biological function simulation models, model representation with description language is becoming popular. However, simulation software itself becomes complex in these environment, thus, it is difficult to modify target computation resources or numerical calculation methods or simulation conditions. Typical biological function simulation software consists of 1) model equation, 2) boundary conditions and 3) ODE solving scheme. Introducing the description model file such as CellML is useful for generalizing the first point and partly second point, however, third point is difficult to handle. We introduce a simulation software generation system which use markup language based description of ODE solving scheme together with cell model description file. By using this software, we can easily generate biological simulation program code with different ODE solving schemes. To show the efficiency of our system, experimental results of several simulation models with different ODE scheme and different computation resources are shown.

A novel method to quantify contribution of channels and transporters to membrane potential dynamics.

The action potential, once triggered in ventricular or atrial myocytes, automatically proceeds on its time course or is generated spontaneously in sinoatrial node pacemaker cells. It is induced by complex interactions among such cellular components as ion channels, transporters, intracellular ion concentrations, and signaling molecules. We have developed what is, to our knowledge, a new method using a mathematical model to quantify the contribution of each cellular component to the automatic time courses of the action potential. In this method, an equilibrium value, which the membrane potential is approaching at a given moment, is calculated along the time course of the membrane potential. The calculation itself is based on the time-varying conductance and the reversal potentials of individual ion channels and electrogenic ion transporters. Since the equilibrium potential moves in advance of the membrane potential change, we refer to it as the lead potential, V(L). The contribution of an individual current was successfully quantified by comparing dV(L)/dt before and after fixing the time-dependent change of a component of interest, such as the variations in the open probability of a channel or the turnover rate of an ion transporter. In addition to the action potential, the lead-potential analysis should also be applicable in all types of membrane excitation in many different kinds of cells.

Evaluation of cardiac oxygen consumption under hypoxia with tissue model integrating microcirculation model and cell model.

Analyzing the microscopic energy balance of cardiac tissue is very important for understanding heart diseases. However, such analysis is difficult with animal experiments. Therefore, the accurate simulation model is expected to be an important tool for such research. We propose a cardiac tissue model which can reproduce accurate distribution of oxygen consumption under hypoxia. The model includes blood tissue exchange model of capillary and oxygen consumption model of cells. The capillary model is based on the model proposed by Dash et al. 2006, and the cell model is based on the model proposed by Kuzumoto et al. 2007. By analyzing the oxygen consumption of the proposed model, the relation between the oxygen consumption and the arterial oxygen concentration was found to be largely different from that of single cell model. This implies that the animal experimental data should be carefully used for constructing a biological simulation model, depending on whether the experiment is performed within a cell or a tissue.

A method for analysis of simultaneous equations in cell models.

Some models of cellular physiological functions are formulated as ordinary differential equations that contain multiple systems of simultaneous nonlinear equations. Simulation of such a model described in a declarative representation format requires determination of equations to be simultaneously solved with specification of independent and parameter variables in the model. In this report, a method for extracting systems of simultaneous equations in a cell model is presented. The present method analyzes a graph representation of a model and extracts the subgraphs that represent equation systems to be simultaneously solved, by efficiently interactive selection of independent variables of the model.

An approximation model of myocardial crossbridge for weak coupling calculation of left ventricle model and circulation model.

It is necessary to use complicated myocardial cell model and heart model to evaluate the regional energy production and consumption which leads to the unrealistic computational time. In this research, a left ventricle (LV) simulation model was constructed which includes accurate myocardial cell model. In order to simulate the model in realistic time, we introduced an approximation model of the crossbridge model which can be calculated with weak coupling calculation. The LV model was combined with a circulation model to validate the proposed model by calculating the hemodynamics parameters and ventricular energetics indices. The ESPVR (End Systolic Pressure Volume Relation) showed linear relation, and also the PVA - ATP consumption relation showed linear relation which are widely known as the physiological characteristics of mammalian hearts. From these results, we can say that the model can be used as a model for physiological simulation experiments which are related to the ventricular energetics.