Ola Skavhaug

A Second-Order Algorithm for Solving Dynamic Cell Membrane Equations

This paper describes an extension of the so-called Rush-Larsen scheme, which is a widely used numerical method for solving dynamic models of cardiac cell electrophysiology. The proposed method applies a local linearization of nonlinear terms in combination with the analytical solution of linear ordinary differential equations to obtain a second-order accurate numerical scheme. We compare the error and computational load of the second-order scheme to the original Rush-Larsen method and a second-order Runge-Kutta (RK) method. The numerical results indicate that the new method outperforms the original Rush-Larsen scheme for all the test cases. The comparison with the RK solver reveals that the new method is more efficient for stiff problems.

Unified framework for finite element assembly

At the heart of any finite element simulation is the assembly of matrices and vectors from discrete variational forms. We propose a general interface between problem-specific and general-purpose components of finite element programs. This interface is called Unified Form-assembly Code (UFC). A wide range of finite element problems is covered, including mixed finite elements and discontinuous Galerkin methods. We discuss how the UFC interface enables implementations of variational form evaluation to be independent of mesh and linear algebra components. UFC does not depend on any external libraries, and is released into the public domain.

Using Python to Solve Partial Differential Equations

This article describes two Python modules for solving partial differential equations (PDEs): PyCC is designed as a Matlab-like environment for writing algorithms for solving PDEs, and SyFi creates matrices based on symbolic mathematics, code generation, and the finite element method.

Using B SP and Python to simplify parallel programming

Scientific computing is usually associated with compiled languages for maximum efficiency. However, in a typical application program, only a small part of the code is time-critical and requires the efficiency of a compiled language. It is often advantageous to use interpreted high-level languages for the remaining tasks, adopting a mixed-language approach. This will be demonstrated for Python, an interpreted object-oriented high-level language that is well suited for scientific computing. Particular attention is paid to high-level parallel programming using Python and the BSP model. We explain the basics of BSP and how it differs from other parallel programming tools like MPI. Thereafter we present an application of Python and BSP for solving a partial differential equation from computational science, utilizing high-level design of libraries and mixed-language (Python-C or Python-Fortran) programming.

Existence of excitation waves for a collection of cardiomyocytes electrically coupled to fibroblasts.

We consider mathematical models of a collection of cardiomyocytes (myocardial tissue) coupled to a varying number of fibroblasts. Our aim is to understand how conductivity (δ) and fibroblast density (η) affect the stability of the collection. We provide mathematical and computational arguments indicating that there is a region of instability in the η-δ space. Mathematical arguments, based on a simplified model of the coupled myocyte-fibroblast system, show that for certain parameter choices, a stationary solution cannot exist. Numerical experiments (1D,2D) are based on a recently developed model of electro-chemical coupling between a human atrial myocyte and a number of associated atrial fibroblasts. The numerical experiments demonstrate that there is a region of instability of the form observed in the simplified model analysis.

Simple T-Wave Metrics May Better Predict Early Ischemia as Compared to ST Segment

There is pressing clinical need to identify developing heart attack (infarction) in patients as early as possible. However, current state-of-the-art tools in clinical practice, underpinned by the evaluation of elevation of the ST segment of the 12-lead electrocardiogram (ECG), do not identify all patients suffering from lack of blood flow to the heart muscle (cardiac ischemia), worsening the risk for further adverse events and patient outcome overall. In this study, we aimed to explore and compare the portions of cardiac repolarization in the ECG that best capture the electrophysiological changes associated with ischemia. We developed three-dimensional electrophysiological models of the human ventricles and torso, incorporating biophysically-based membrane kinetics and realistic activation sequence, to compute simulated ECGs and their alteration with the application of simulated ischemia of differing severity in diverse regions of the heart. Results suggest that metrics based on the T-wave in addition to the ST segment may be more sensitive to detecting ischemia than those using the ST segment alone. Further research into how such simulation-aided risk assessment methods may aid workflows in extant clinical practice, with the ultimate goal of multimodality clinical support, is warranted.

Computing the stability of steady-state solutions of mathematical models of the electrical activity in the heart

Instabilities in the electro-chemical resting state of the heart can generate ectopic waves that in turn can initiate arrhythmias. We derive methods for computing the resting state for mathematical models of the electro-chemical process underpinning a heartbeat, and we estimate the stability of the resting state by invoking the largest real part of the eigenvalues of a linearized model. The implementation of the methods is described and a number of numerical experiments illustrate the feasibility of the methods. In particular, we test the methods for problems where we can compare the solutions with analytical results, and problems where we have solutions computed by independent software. The software is also tested for a fairly realistic 3D model.

Unstable eigenmodes are possible drivers for cardiac arrhythmias

The well-organized contraction of each heartbeat is enabled by an electrical wave traversing and exciting the myocardium in a regular manner. Perturbations to this wave, referred to as arrhythmias, can lead to lethal fibrillation if not treated within minutes. One manner in which arrhythmias originate is an ill-fated interaction of the regular electrical signal controlling the heartbeat, the sinus wave, with an ectopic stimulus. It is not fully understood how and when ectopic waves are generated. Based on mathematical models, we show that ectopic beats can be characterized in terms of unstable eigenmodes of the resting state.

Mathematical Models of Financial Derivatives

In this chapter, we derive several mathematical models of financial derivatives, such as futures and options. The methodology used is commonly known as risk-neutral pricing, and was first presented by Merton, Black and Scholes in the 1970s. We start by presenting the basics of the Black-Scholes analysis, which leads to the Black-Scholes equation. Several option contracts such as plain European and American option contracts are derived. We also give an overview of some exotic option contracts. At last, we present mathematical models of the so-called Greeks, i.e., the partial derivatives of the value of the option contracts with respect to important model parameters.

Numerical Methods for Financial Derivatives

Analytical solutions of the mathematical equations modeling the behavior of financial derivatives, like the price of option contracts, are seldom available. Only in the simplest cases, e.g., vanilla European put and call options, do analytical solutions exist. For most other option models, numerical techniques must be applied to compute solutions of the mathematical models. For exotic option contracts, computing the option prices numerically may be the only pricing mechanism available.