The main focus of the research in the experiments shown here is to study cardiac arrhythmia. For this, first initiation of the ventricular fibrillation is necessary. Then the arrhythmia and finally the defibrillation can be studied.
2.6.1 Induction Protocol
Induction of ventricular fibrillation was done using the local pacing electrode placed typically on the left ventricle. A biphasic rectangular stimulus with 3 ms total pulse duration and inverted polarity after 1.5 ms was used. Induction was performed well above the minimum required strength necessary for pacing the heart at a slow fre-quency (such as with a 300 ms period) and usually at 50 Hz for a duration of 3 s to 10 s.
In some cases, repeated or longer inductions were necessary or a lower frequency in the range of 20 Hz to 50 Hz was used in order to induce fibrillation.
2.6.2 Defibrillation
In general defibrillation was achieved using 7 ms biphasic rectangular shocks with an inverted polarity after 5 ms (5ms+2ms). The voltage for both shock parts is identical and adjusted for each defibrillation attempt. Three different strategies were employed in the experiments shown in this thesis:
• 1 Shock: A single, bi-phasic defibrillation shock.
• 5 Shocks: Five identical shocks given at a period that is a factor of 1.2 slower than the frequency of the arrhythmia as determined from the ECG.
• 50 Shocks: 50 shocks using overdrive pacing about 10% faster than the frequency of the arrhythmia.
After any successful defibrillation, the heart was allowed to recover for a minimum of about five minutes.
Various defibrillation techniques have been studied intensively in many contexts.
For example overdrive [4, 55–58] and underdrive [33, 59–61] pacing as well as shock forms [62]. All of the experiments done here were to some degree designed to com-pare such methods. However, most of the analysis presented in this thesis will be independent of that.
Chapter 3
Theoretical Background
In this chapter, the theoretical background with respect to the analysis performed in the result chapters is given. Since the main focus of the thesis is the analysis of experimental data the chapter is limited to two parts:
1. A brief review of the theory for activation time analysis with the expectation of a power law, which is used in Chapter 5.
2. An introduction and overview over the literature of defect mediated turbulence.
This introduces the stochastic Markov model used in Chapter 6.
3.1 Activation Maps and Time
In this section, I will briefly review the theory of activation times – i.e. the time after a shock until the whole tissue has been excited – as laid out in [4]. This theory motivates the calculation of power law exponents shown in Chapter 5.
3.1.1 Theoretical Foundation of Wave Emission at Boundaries
While a full understanding of boundary effects cannot be derived analytically and is only approachable using numerical methods using the bidomain model, it is possi-ble to derive approximate solutions for simple boundary conditions such as circular non-conducting heterogeneities in isotropic media [4, 18, 20, 63]. These theoretical considerations rely on a few assumptions and approximations:
1. Isotropic, homogeneous medium
2. Monodomain approximation of the cardiac bidomain 3. Low field strength approximation.
However, the qualitative results of wave emission site recruitment and general intuition can be confirmed in complex simulations [19] and in cell culture experiments [18].
3.1.2 From Circular Boundaries to Activation Times
Assuming a certain field strength, there is a minimum size at which non conducting circular enclosures will be activated. As laid out by Luther and Fenton et al. in [4]
these considerations may be applied to the vasculature. If one assumes a homogeneous distribution of vasculature in the tissue and a constant propagation velocity, it is possible to make a prediction for the time after a shock that is required to activate the whole volume or surface of a three or two dimensional tissue.
To approximate this activation time it is necessary to know the size distribution of the wave emitting sites. These are assumed to follow a size probability distribution for their radius given by:
P(𝑟) ∝ 𝑟𝛼 (3.1)
with the constant 𝛼 < −1. This formula can describe the vasculature for sufficiently small blood vessel sizes as shown in references [4, 33] and references therein.
Then the density of wave emitting sites with a radius or𝑟min or larger is given by its number𝑁 (𝑟 > 𝑟min) and the volume V:
𝜌(𝑟) = 𝑁 (𝑟 > 𝑟min)
𝑉 = ∫
∞ 𝑟min
P(𝑟)d𝑟 (3.2)
∝ ∫
∞ 𝑟min
𝑟𝛼 = [ 1
𝛼 + 1𝑟𝛼+1]
∞ 𝑟min
(3.3)
and with the assumption of 𝛼 < −1:
𝜌(𝑟) ∝ 0 − 1
𝛼 + 1𝑟𝛼+1min (3.4)
substituting the approximation 𝑟min ∝ 𝐸1 which was derived in [4, 63] gives:
𝜌(𝐸) ∝ 𝐸−(𝛼−1). (3.5)
Equation 3.5 thus gives an estimation of the density of new waves created by a shock of electric field strength 𝐸. These waves will travel in all directions from the emission site. For a given density 𝜌, the distance that has to be covered by a single site can be estimated by:
𝑑(𝜌) ∝ 𝜌𝐷1 (3.6)
3.1 Activation Maps and Time 25
Fig. 3.1 Numerical model of activation on a rabbit LV geometry (micro-CT scan) with increasing field strengths from left to right (0.2 V/cm, 0.4 V/cm, and 1.0 V/cm). The colours indicate the membrane potential from −80mV to +20mV (blue to red). Figure modified from [18].
with the dimension 𝐷. Thus, the time 𝜏 necessary to cover this distance, based on the wave propagation speed 𝑣 is:
𝜏 (𝜌) ∝ 𝜌𝐷1
𝑣 . (3.7)
Substituting Equation (3.5) gives:
𝜏 (𝐸) ∝ 𝐸−(𝛼−1)𝐷 = 𝐸−𝛽 (3.8)
with 𝛽 = (𝛼 − 1)/𝐷.
This gives a theoretical foundation for the expectation that, for sufficiently high field strengths, a power law for the activation time𝜏 with respect to the field strength 𝐸 is expected. This relationship is based on the main assumptions that the underlying heterogeneities are circular and follow a size distribution well described by a power law.
3.1.3 Typically Activated Areas
As mentioned in the previous section more sites get activated by an external shock when the shock voltage is increased. To give a better intuition, Figure 3.1 shows an example from a simulation with increasing field strengths. In this study the first site activated is a protuberance (left panel). Then the endocardial wall is activated in many areas and finally the larger blood vessels are activated. More detailed simulations for the rabbit heart can be found in [19].
NPS +1 +2
−1
−2
Enter
Creation Leave
Annihilation
…
0
…
Fig. 3.2 Illustration of the Markov process. For a given number of PS (NPS) the four possible transitions are shown. The only exception is for zero and one NPS where further removal of PS is either not possible at all or not possible through pair annihilation.