**2. Modelling techniques**

**1. Introduction**

76 Numerical Simulation - From Brain Imaging to Turbulent Flows

In this chapter, the research on the electrical conduction system of the human heart is concerned. The parts of this system are pacemaker centres made of special cells similar to embryonic cells which form the following concentrations: sino-atrial node (SA), atrioven‐ tricular node (AV) and His-Purkinje system [1]. The key elements of the conduction system considered include SA node and AV node. Modelling the formation and conduction of electrical impulses in the heart is one of the most developed areas of mathematical biolo‐ gy. For years, the most popular models of action potentials which occur in the heart have included the Hodgkin-Huxley model and the Purkinje cells model. These models, even though very effective, are not very suitable for studying the dynamics of the system, which is described in view of high dimensionality of the phase space and very broad spatial parameters. In this chapter, we propose ordinary differential system which is based on the van der Pol models and which makes it possible to reconstruct pathological behaviours in the system of the heart, such as the WPW syndrome and different types of AVNRT. The motivation for writing this chapter was that there is a problem with making the appropri‐ ate diagnosis and therefore with treating the disease effectively. This kind of problem is observed mainly in different types of AVNRT. The reasons of those problems are not fully known and the mechanisms of these pathologies are not fully understood. Also, the symptoms are often mistakenly taken for other heart diseases. It should be emphasized that the clinical pictures of these disease are also non-specific. In the past few years, we understood that the structure of the AV node has a multi-level architecture in which there may be many pathways (slow and fast) at different locations in the AV node [2, 3]. This helped to recognize many types of AVNRT, which previously were understood as one, although the mechanisms of action were different. In literature, there are no mathematical models that would specifically model the various types of AVNRT. There were attempts to model AVNRT only as a single pathology having the slow and fast pathways [4, 5]. Part of the population has abnormal accessory pathways: fast and slow, cf. [1, 6]. The pathways in the AV node are anatomical and functional contributions of the most popular supraventric‐ ular tachycardia, which is a re-entry tachycardia from the AV node. The atrioventricular nodal re-entrant tachycardia is caused by re-entries. A condition for AVNRT to occur is that two electric pathways occur in and around the AV node (for example, slow and fast pathways). This gives way to the occurrence of re-entry. We can distinguish five different forms of the AVNRT (typical: slow/fast, atypical: fast/slow and other forms: slow/slow, more than two re-entries waves, one fast pathway with depolarization of slow pathway) [7, 8, 9]. This depends on the multi-level architecture of the AV node. The AVNRT circuit involves larger areas including atrioventricular junction, adjacent atrial structures and in particular so-called atrial inputs including at least antero-superior and postero-inferior entries, and sometimes also the left atrial entry. Based on the van der Pol equation, we study the influence of feedback which occurs in the normal heart action mode as well as in patholog‐ ical modes. Particularly, it is important to introduce the time delay into this feedback. Delay values used in our research correspond to those which occur in the electrical conduction system, for example, in the case of an accessory conducting pathway, so-called Wolff-

The van der Pol model was used in this research because it is a two-dimensional model with small number of parameters and it is a relaxation oscillator. The van der Pol equation is often used to describe an action potential occurring in the heart. The model with a delay and often with a term including a coupling coefficient has been a topic of many articles, e.g. [15–17]. However, in these articles there are no exact examples of application of this model for recreating a pathological behaviour of the electrical conduction system of the human heart such as the AVNRT and the WPW, and therefore the range of parameters considered by them is wider than that which can be used in medical applications. Below, the author will present one of the most important existing mathematical models of the action potential. Basing on the knowledge of van der Pol model given below will be carried out construction of the proposed models.

#### **2.1. Van der Pol model**

Since every node is a self-exciting pacemaker, it can be described using the van der Pol oscillator which is the relaxation oscillator. It was first introduced by van der Pol and van der Mark as a model in the electronic circuit theory in 1927, [19]:

$$
\ddot{\mathbf{x}} + a \left(\mathbf{x}^2 - 1\right) \dot{\mathbf{x}} + \mathbf{x} = \mathbf{0} \tag{1}
$$

where

$$f\left(\mathbf{x}\right) = 0.5a\left(\mathbf{x}^2 - \mathbf{l}\right) \tag{2}$$

*f* (*x*) is a damping coefficient, which is negative for | *x* | <1 and positive for | *x* | >1. Never‐ theless, the van der Pol model requires some changes to be introduced in order to reproduce features of heart's action potential. In paper [20], Postnov substituted a linear term by a nonlinear cubic force called the Duffing term

$$\ddot{\mathbf{x}} + a \left( \mathbf{x}^2 - \mu \right) \dot{\mathbf{x}} + \frac{\mathbf{x} \left( \mathbf{x} + d \right) \left( \mathbf{x} + 2d \right)}{d^2} = \mathbf{0} \tag{3}$$

where *a*, *μ*, *d* are positive control parameters.

In this way, he introduced modifications which allowed maintaining the required structure of the phase space. Such model can be applied to model either SA or AV node. The main property of a modified relaxation oscillator is the mutual interaction of a limit cycle which is presented around an unstable focus with a saddle and a stable node. This allows reproducing correctly the refraction period and non-linear phase sensitivity of an action potential of node cells.

A solution of Eq. (3) in time presents the action potential whereas a solution in velocity makes it possible to obtain very important phase portrait. We can easily see that the main qualitative difference between Eqs. (1) and (3) is the appearance of two additional steady states, *x*<sup>2</sup> = −*d* and *x*<sup>3</sup> = −2*d*. As before, *x*<sup>1</sup> =0 is an unstable node or a focus surrounded by a stable unique limit cycle, *x*<sup>2</sup> = −*d* is a saddle and *x*<sup>3</sup> = −2*d* forms a focus or a node and can be either stable or unstable, depending on the sign of 4*d* <sup>2</sup> −*μ*. In the case considered by Postnov [20], the first steady state is an unstable focus, the third one is a stable node. In the considered model (5), the problem is the adjustment of the position of stationary states in the phase space. In order to reproduce behaviour of the heart, a new parameter *e* is added. For better regulation of the frequency, the *e* ⋅*d* factor is substituted with an independent coefficient *f* corresponding to harmonic oscillator's frequency, [20]. We present the final version of the model in its two variable firstorder form that reads [20]:

$$\begin{aligned} \dot{\mathbf{x}} &= \mathbf{y}, \\ \dot{\mathbf{y}} &= -a \left( \mathbf{x}^2 - \mathbf{l} \right) \mathbf{y} - f \mathbf{x} \left( \mathbf{x} + d \right) \left( \mathbf{x} + e \right) \end{aligned} \tag{4}$$

The system that we use as a reference one is given in following form:

one of the most important existing mathematical models of the action potential. Basing on the knowledge of van der Pol model given below will be carried out construction of the proposed

Since every node is a self-exciting pacemaker, it can be described using the van der Pol oscillator which is the relaxation oscillator. It was first introduced by van der Pol and van der

*f* (*x*) is a damping coefficient, which is negative for | *x* | <1 and positive for | *x* | >1. Never‐ theless, the van der Pol model requires some changes to be introduced in order to reproduce features of heart's action potential. In paper [20], Postnov substituted a linear term by a non-

( ) ( )( ) <sup>2</sup>

*xx d x d x ax x <sup>d</sup>* m

2

In this way, he introduced modifications which allowed maintaining the required structure of the phase space. Such model can be applied to model either SA or AV node. The main property of a modified relaxation oscillator is the mutual interaction of a limit cycle which is presented around an unstable focus with a saddle and a stable node. This allows reproducing correctly the refraction period and non-linear phase sensitivity of an action potential of node cells.

A solution of Eq. (3) in time presents the action potential whereas a solution in velocity makes it possible to obtain very important phase portrait. We can easily see that the main qualitative difference between Eqs. (1) and (3) is the appearance of two additional steady states, *x*<sup>2</sup> = −*d* and *x*<sup>3</sup> = −2*d*. As before, *x*<sup>1</sup> =0 is an unstable node or a focus surrounded by a stable unique limit cycle, *x*<sup>2</sup> = −*d* is a saddle and *x*<sup>3</sup> = −2*d* forms a focus or a node and can be either stable or unstable, depending on the sign of 4*d* <sup>2</sup> −*μ*. In the case considered by Postnov [20], the first steady state is an unstable focus, the third one is a stable node. In the considered model (5), the problem is the adjustment of the position of stationary states in the phase space. In order to reproduce behaviour of the heart, a new parameter *e* is added. For better regulation of the frequency, the

2

0

+ + &&+ -+& <sup>=</sup> (3)

*x ax x x* + - += 1 0 (1)

( ) ( ) <sup>2</sup> *f x ax* = - 0.5 1 (2)

( ) <sup>2</sup> && &

Mark as a model in the electronic circuit theory in 1927, [19]:

78 Numerical Simulation - From Brain Imaging to Turbulent Flows

linear cubic force called the Duffing term

where *a*, *μ*, *d* are positive control parameters.

models.

where

**2.1. Van der Pol model**

$$\dot{\mathbf{x}}\_1 = \mathbf{y}\_1$$

$$\dot{\mathbf{y}}\_1 = -a\_1 \left(\mathbf{x}\_1^2 - 1\right) \mathbf{y}\_1 - f\_1 \mathbf{x}\_1 \left(\mathbf{x}\_1 + d\_1\right) \left(\mathbf{x}\_1 + e\_1\right)$$

$$\dot{\mathbf{x}}\_2 = \mathbf{y}\_2$$

$$\dot{\mathbf{y}}\_2 = -a\_2 \left(\mathbf{x}\_2^2 - 1\right) \mathbf{y}\_2 - f\_2 \mathbf{x}\_2 \left(\mathbf{x}\_2 + d\_2\right) \left(\mathbf{x}\_2 + e\_2\right)$$

where *a*<sup>1</sup> =*a*<sup>2</sup> =5; *f* <sup>1</sup> = *f* <sup>2</sup> =3; *d*<sup>1</sup> =*d*<sup>2</sup> =3; *e*1=7; *e*2=4.5 are control parameters. In the next part of this paper, a modified van der Pol system with delayed feedback and couplings describes various pathologies observed in heart action, for example, WPW syndrome and different types of AVNRT pathology.
