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Our knowledge about the heart dates back more than two millenia. Already in the days of Aristotle (350 b.c.) the importance of the heart was recognized, and it was, in fact, considered to be the most important organ in the body. Other vital organs, such as the brain and lungs, were thought to exist merely to cool the blood. Over two thousand years later, the heart maintains its position as one of the most important, and also most studied, organs in the human body.

As described in the previous chapter, the human body consists of billions of cells, which may be connected by various coupling mechanisms depending on the type of tissue under consideration. When constructing mathematical models for electrical activity in the tissue, one possible approach would be to model each cell as a separate unit, and couple them together using mathematical models for the known coupling mechanisms. However, the large number of cells will prohibit using this type of model for anything but very small samples of tissue. When studying electrical phenomena on the level of complete organs or even organisms, the level of detail provided by such an approach also goes far beyond what it is necessary, or even possible, to utilize.

The mathematical models derived in the previous chapter give a quantitative description of the electrical activity in the heart, from the level of electrochemical reactions in the cells to body surface potentials that may be recorded as ECGs. However, the models are formulated as systems of nonlinear partial and ordinary differential equations, for which analytical solutions are not available. To be of any practical use, the equations of the models must therefore be solved with numerical methods. The choice of numerical methods that may be applied to the equations is large, see e.g. [83], but we have chosen to focus entirely on finite element methods (FEM). One reason for this is that the geometries of the heart and the body are irregular, and this is more conveniently handled by FEM than, for instance, by finite difference methods.

The physical relevance of computations based on the model problems arising from the electrical activity in the heart depends on high accuracy of the solution. High accuracy requires the solution of large linear or nonlinear systems of ODEs and PDEs. This chapter deals with solution algorithms for the discretization of (linear) PDEs, which is a huge research field around the world. Much of the research in this field has been centred around simple model problems such as the Poisson problem, where a solid theoretical framework has been developed. We will briefly review this theory in the simplest possible manner. Then, at the end of the chapter, we explain how the powerful concept of (block) preconditioning extends these algorithms to systems of PDEs that arise from the discretization of the Bidomain model.

The operator splitting algorithms introduced in Chapter 3 reduced the solution of the bidomain equations to solving linear PDE systems and nonlinear systems of ODEs. Techniques for discretizing the PDE system were presented in Chapter 3, while techniques for solving the resulting linear systems were discussed in Chapter 4. What remains to have a complete computational method for the bidomain model is to find an efficient method for solving the nonlinear ODE systems. Note that the spatial discretization of the bidomain equations results in one ODE system for each node in the finite element grid. Realistic simulations may require several millions of nodes, and it is therefore of the utmost importance to solve the ODE systems efficiently.

In the preceding chapters, we have discussed various numerical techniques for solving the different parts of our mathematical model problem. Now it is time to turn our attention to simulating the complete mathematical model. First, we will explain the diverse computational tasks that constitute an electrocardiac simulator. Then, we will estimate the computational resources needed to carry out high-resolution simulations. It will be shown that is an essential technique for largescale electrocardiac simulations. Thereafter, this chapter will focus on the general principles for parallelizing a sequential simulator, as well as on major software components needed to achieve this end. These software components will be described in a general setting, without reference to specific programming languages. On the other hand, a simple pseudo language will be used to explain certain details when necessary. Some performance measurements of electrocardiac simulations will also be presented.

We have seen how mathematical models, numerical methods, software and computers can be used to simulate the electrical activity in the human body. Provided that the physical characteristics of the involved tissues are known, such techniques can, in particular, be applied to compute the electrical potential along the surface of the body generated by the heart. This can be very useful for gaining a better insight into these biological processes, improving the interpretation of ECG recordings, and may serve as a starting point for developing suitable educational tools and training facilities.