Bjørn Fredrik Nielsen

On the Computational Complexity of the Bidomain and the Monodomain Models of Electrophysiology

The bidomain model, coupled with accurate models of cell membrane kinetics, is generally believed to provide a reasonable basis for numerical simulations of cardiac electrophysiology. Because of changes occurring in very short time intervals and over small spatial domains, discretized versions of these models must be solved on fine computational grids, and small time-steps must be applied. This leads to huge computational challenges that have been addressed by several authors. One popular way of reducing the CPU demands is to approximate the bidomain model by the monodomain model, and thus reducing a two by two set of partial differential equations to one scalar partial differential equation; both of which are coupled to a set of ordinary differential equations modeling the cell membrane kinetics. A reduction in CPU time of two orders of magnitude has been reported. It is the purpose of the present paper to provide arguments that such a reduction is not present when order-optimal numerical methods are applied. Theoretical considerations and numerical experiments indicate that the reduction factor of the CPU requirements from bidomain to monodomain computations, using order-optimal methods, typically is about 10 for simple cell models and less than two for more complex cell models.

Global Upscaling of Permeability in Heterogeneous Reservoirs; The Output Least Squares (OLS) Method

This paper presents a new technique for computing the effective permeability on a coarse scale. It is assumed that the permeability is given at a fine scale and that it is necessary to reduce the number of blocks in the reservoir model. Traditional upscaling methods depend on local boundary conditions. It is well known that the permeability may depend heavily on the local boundary condition chosen. Hence the estimate is not stable. We propose to compute a coarse scale permeability field that minimises the error, measured in a global norm, in the velocity and pressure fields. This leads to stable problems for a large number of reservoirs. We present several algorithms for finding the effective permeability values. It turns out that these algorithms are not significantly more computational expensive than traditional local methods. Finally, the method is illustrated by several numerical experiments.

Stochastic Structural Modeling

A consistent stochastic model for faults and horizons is described. The faults are represented as a parametric invertible deformation operator. The faults may truncate each other. The horizons are modeled as correlated Gaussian fields and are represented in a grid. Petrophysical variables may be modeled in a reservoir before faulting in order to describe the juxtaposition effect of the faulting. It is possible to condition the realization on petrophysics, horizons, and fault plane observations in wells in addition to seismic data. The transmissibility in the fault plane may also be included in the model. Four different methods to integrate the fault and horizon models in a common model is described. The method is illustrated on an example from a real petroleum field with 18 interpreted faults that are handled stochastically.

Optimal monodomain approximations of the bidomain equations

Abstract The bidomain equations are widely accepted to model the spatial distribution of the electrical potential in the heart. Although order optimal methods have been devised for discrete versions of these equations, it is still very CPU demanding to solve the equations numerically on a sufficiently fine mesh in 3D. Furthermore, the equations are hard to analyze; from a mathematical point of view, very little is known about the qualitative behavior of the solutions generated by these equations. It is well known that upon appealing to a certain relation between the extracellular and intracellular conductivities, the bidomain model can be rewritten in terms of a scalar reaction diffusion equation referred to as the monodomain model. This model is of course much easier to solve, and also the qualitative properties of the solutions are well known; such equations have been studied intensively for decades. It is the purpose of the present paper to show how the bidomain equations can be approximated in an optimal manner by the solution of a monodomain model. The key feature here is that this optimal solution can be computed without solving the bidomain model itself. The solution is obtained by putting the problem into a framework of parameter identification problems. Our results are illuminated by a series of numerical experiments.

An order optimal solver for the discretized bidomain equations

The electrical activity in the heart is governed by the bidomain equations. In this paper, we analyse an order optimal method for the algebraic equations arising from the discretization of this model. Our scheme is defined in terms of block Jacobi or block symmetric Gauss–Seidel preconditioners. Furthermore, each block in these methods is based on standard preconditioners for scalar elliptic or parabolic partial differential equations (PDEs). Such preconditioners can be realized in terms of multigrid or domain decomposition schemes, and are thus readily available by applying ‘off-the-shelves’ software. Finally, our theoretical findings are illuminated by a series of numerical experiments. Copyright

On the use of the resting potential and level set methods for identifying ischemic heart disease: An inverse problem

The electrical activity in the heart is modeled by a complex, nonlinear, fully coupled system of differential equations. Several scientists have studied how this model, referred to as the bidomain model, can be modified to incorporate the effect of heart infarctions on simulated ECG (electrocardiogram) recordings. We are concerned with the associated inverse problem; how can we use ECG recordings and mathematical models to identify the position, size and shape of heart infarctions? Due to the extreme CPU efforts needed to solve the bidomain equations, this model, in its full complexity, is not well-suited for this kind of problems. In this paper we show how biological knowledge about the resting potential in the heart and level set techniques can be combined to derive a suitable stationary model, expressed in terms of an elliptic PDE, for such applications. This approach leads to a nonlinear ill-posed minimization problem, which we propose to regularize and solve with a simple iterative scheme. Finally, our theoretical findings are illuminated through a series of computer simulations for an experimental setup involving a realistic heart in torso geometry. More specifically, experiments with synthetic ECG recordings, produced by solving the bidomain model, indicate that our method manages to identify the physical characteristics of the ischemic region(s) in the heart. Furthermore, the ill-posed nature of this inverse problem is explored, i.e. several quantitative issues of our scheme are explored.

Computing the size and location of myocardial ischemia using measurements of ST-segment shift

It is well known that the presence of myocardial ischemiastract can be observed as a shift in the ST segment of an electrocardiogram (ECG) recording. The question we address in this paper is whether or not ST shift can be used to compute approximations of the size and location of the ischemic region. We begin by investigating a cost functional (measuring the difference between synthetic recorded data and simulated values of ST shift) for a parameter identification problem to locate the ischemic region. We then formulate a more flexible representation of the ischemia using a level set framework and solve the associated minimization problem for the size and position of the ischemia. We apply this framework to a set of ECG data generated by the Bidomain model using the cell model of Winslow et al. Based on this data, we show that values of ST shift recorded at the body surface are capable of identifying the position and (roughly) the size of the ischemia.

Havana — a fault modeling tool

Improved knowledge on faults and hydrocarbon seal put pressure on geologists and reservoir engineers doing reservoir modeling. All geo-knowledge must be built into the reservoir models to assure that it is taken into account in the decision processes. The need for advanced modeling tools is increasing. This paper describes the development of a fault modeling tool, the methodology behind it and examples of fault modeling studies. The general focus is on the uncertainty related to faults. The tool can be used for sensitivity analysis of fault effects, including studies of the flow effects of all faults scales, adding faults to simulation grids, and studies of the geometric uncertainty of the faults. The work started out as a development of a tool for stochastic modeling of sub-seismic scale faults. The faults can be added to a flow simulation grid as both displacement and seal. The current tool has been designed to operate together with the Eclipse flow simulator and the IRAP RMS program package. IRAP RMS is the main tool for visualizing output and Eclipse is used to examine the effect of the faults on hydrocarbon recovery. The techniques for modeling of fault seal, outputting results in a format that Eclipse can directly utilize, and the possibility for displacing simulation grids has proved useful also to seismic scale faults. This has led to further development, more detailed fault models and improvements of the general fault modeling capabilities. Examples of fault modeling, including three field examples, Statfjord, Heidrun and Sleipner, are presented to illustrate ways of including fault modeling as part of the reservoir modeling workflow.

Computing Ischemic Regions in the Heart With the Bidomain Model—First Steps Towards Validation

We investigate whether it is possible to use the bidomain model and body surface potential maps (BSPMs) to compute the size and position of ischemic regions in the human heart. This leads to a severely ill posed inverse problem for a potential equation. We do not use the classical inverse problems of electrocardiography, in which the unknown sources are the epicardial potential distribution or the activation sequence. Instead we employ the bidomain theory to obtain a model that also enables identification of ischemic regions transmurally. This approach makes it possible to distinguish between subendocardial and transmural cases, only using the BSPM data. The main focus is on testing a previously published algorithm on clinical data, and the results are compared with images taken with perfusion scintigraphy. For the four patients involved in this study, the two modalities produce results that are rather similar: The relative differences between the center of mass and the size of the ischemic regions, suggested by the two modalities, are 10.8% ± 4.4% and 7.1% ± 4.6%, respectively. We also present some simulations which indicate that the methodology is robust with respect to uncertainties in important model parameters. However, in contrast to what has been observed in investigations only involving synthetic data, inequality constraints are needed to obtain sound results.

An upscaling method for one‐phase flow in heterogeneous reservoirs. A weighted output least squares (WOLS) approach

In this paper we study the problem of determining the effective permeability on a coarse scale level of problems with strongly varying and discontinuous coefficients defined on a fine scale. The upscaled permeability is defined as the solution of an optimization problem, where the difference between the fine scale and the coarse scale velocity field is minimized. We show that it is not necessary to solve the fine scale pressure equation in order to minimize the associated cost‐functional. Furthermore, we derive a simple technique for computing the derivatives of the cost‐functional needed in the fix‐point iteration used to compute the optimal permeability on the coarse mesh. Finally, the method is illustrated by several analytical examples and numerical experiments.