MSc Project Plan Template

Title

Finite element modelling of a linear chain of human cardiac cells

Abstract (maximum 100 words)

In order to understand the normal and diseased state of the human heart, modelling the activation and propagation of action potentials, the electrical signals responsible for muscle contraction, is critical. In this project we will use finite element modelling of a linear chain of cardiac cells coupled by gap junctions modelled as simple resistors. Instead of biophysically accurate models of the membrane currents, we will use the phenomenological Fitzhugh Nagumo model in order to investigate whether we can improve the simulation time over other studies and construct larger models, whilst still obtaining biologically accurate tissue properties.

Aims of the Project (maximum 100 words).

We aim to construct a model of an array of cardiomyocytes using the phenomenological Fitzhugh Nagumo (FHN) model [1], for transmembrane current rather than the commonly used Luo and Rudy (LR) model, which incorporates the all of the ion conductances separately [2]. These models will incorporate gap junctions, which are the resistive channels via which ion currents are transferred between cells. We will compare the results to state of the art simulations of realistic tissue structures [3], in order to investigate whether we can obtain similar conduction velocities, action potential shapes and gap junction resitivities to those reported.

Significance of topic: (maximum 100 words).

Cardiovascular disease is currently the largest cause of premature death globally and results in 17.3 million deaths per year [4] Computational models of the electrical activity in the heart have the potential to allow cardiac defects to be modelled in order to investigate their effects on the ECG [5]. Realistic human heart models use the bidomain model, where the cardiomyocytes are much smaller than the size of the heart, but these do not incorporate the role of gap junctions [ 6 ]which are only included in models of tissue segments. The use of the FHN model may allow models of larger heart structures to be developed.

Science/Engineering context: (maximum 150 words).

It has not so far been possible to model an entire heart because the organ consists of 109 cardiomyocytes [3] and the transmembrane ion currents transfer process is complex. The computational load involved in processing such a model is currently intractable. Smaller models of discrete cells connected by gap junctions have been developed [3] and these use the LR model (reference) to simulate small structures. In some of these models the cardiomyocytes have realistic shapes and dimensions and are represented by conducting regions which obey the Laplace equation. Gap junctions are assumed to be distributed over the interfaces between cells and all membrane currents are determined by the ion current model. The electrical properties of cardiac tissue are well known [3] including gap junction resistivities enabling detailed comparison of simulation results with biological behaviour.

Relevant literature: (maximum 150 words).

The LR model [2] treats the membrane as a capacitor in parallel with six of voltage dependent and independent conductances, giving rise to eight first order non-linear differential equations. In contrast the FHN model uses two differential equations to represent the ion current [1]. The FHN model has been successfully used to model human and zebrafish embryo(ZE) hearts and chains of ZE heart cells.[7] [8][9]. Roberts et al [10], using the LR model to describe ionic currents, constructed cardiomyocyte fibre model consisting of rectangular cells connected by gap junctions in a conducting extracellular space and found agreement with observed tissue behaviour. This was extended by Stinstra et al [3] who constructed a three dimensional model of a realistic cardiac fibre consisting of several hundred cells.

Design of solution (maximum 200 words).

This project will use the same approach as Roberts et al [10], employing a linear chain of rectangular cells in a conducting medium. The dimensions of the cells will match those of human cardiovascular cells and the size of the external domain be determined by computational constraints. A diagram of the proposed model is shown in Figure 1 in the appendix. The membrane currents will be determined by the FHN model and the gap junction resistances spread uniformly over the interfaces between the cells. Measurements of membrane voltage Vm will be made at intervals of 1mm or 10 cells along the length of the model but could be made at any point along the length. The parameters of the FHN model will be adjusted to obtain action potentials with similar properties to those observed in real tissue and reported by Stinstra et al [3]. The gap junction resistivities will be adjusted to ensure that the action potential conduction velocity of similar to tissue values. The aim of the simulation will be to investigate, how gap junctions affect the variation of Vm along the chain and the effect of the length of the chain on the simulation run time.

Method of solution (maximum 200 words).

This project will use the finite element method to solve the HVF models with COMSOL Multiphysics on a computer with an Intel Core i5-4590 CPU @3.3GHz and 16 GB of RAM. The model geometry will be defined in the COMSOL environment and appropriate toolboxes used to define the electrical properties of the intracellular and extracellular regions and the associated boundary conditions. The details of the electrical model are included in the appendix. Briefly, the transmembrane current crossing the interfaces between the intracellular regions will be determined by equations 1 to 7 which incorporate the FHN model. Current flow between cells along the chain will be determined by the difference between the intracellular voltages determined by the total gap junction resistance. The parameters of the model are shown in Figure 1 and Table 1. Table 2 shows a set of real tissue parameters to which the results will be compared. The conduction velocity will be measured by defining the time of activation as the time at which Vm has exceeded 10% above the resting for the reference cells shown in Figure 1 and measuring the slope of a distance versus time graph. The appendix contains a diagrammatic plan for the project.