0 5 10 15 20 25 30
NPS 0.00
0.02 0.04 0.06 0.08
probability
Experminent Markov Single Step Gaussian
Markov Model
Fig. 6.4 Probability for observing a certain NPS for the same episode shown in Figure 6.1, consisting of a 30 s video of VF without Pinacidil. Additionally to the experimentally ob-served distribution (black squares), different models are shown. The Markov models distri-bution is derived from the curves fitted to the rates in Figure 6.1. Markov single step denotes a Markov model which does not include pair creation and annihilation, but otherwise repro-duces the same total rates. The Gaussian model is a fit of the normal distribution to the data.
other words, half of the observed PS appear to be closely associated with a paired PS for their entire livespan. This PS is the one they were created with as well as the one they annihilate with.
6.2 Results for the Probability Distribution of the NPS
As detailed in Section 3.2.1 the NPS rates of change can be used to create a Markov model. This Markov model can predict the probability distribution of the NPS. To achieve this, the rates are fitted as introduced previously. Then, the detailed balance of the Markov model is solved to find the model’s stationary probability distribution 𝒫(NPS)(compare Sec. 3.2.1). For comparison the NPS distribution is also estimated using:
1. A Gaussian model, derived from the mean and standard deviation of the exper-imental data.
2. A single step Markov model, which assumes that only events affecting a single PS exist and no pair creations and annihilations occur. Such a model thus views
Markov Gaussian Single Step 10−30
10−26 10−22 10−18 10−14 10−10 10−6 10−2
χ2 -Test–p-Value
N= 15
Fig. 6.5 𝜒2 test comparing the hypothesis that the PS distribution can be described using the different models. The boxplot summarizes the median of 100 repetitions over all 15 time spans with multiple short VF episodes. Each repetition uses half of the episodes to estimate the model, and the other half for the𝜒2-test. Higher values of the𝜒2 test indicate that the given model matches the experimental results better. They thus correspond to a better fit for example in Figure 6.4.
6.2 Results for the Probability Distribution of the NPS 71 a pair creation as two individual events.
An example for the results of this analysis for all three models can be found in Fig-ure 6.4. It shows the three models as well as the experimentally observed distribution of the NPS for the same thirty second VF episode given in Figure 6.1.
To compare the Markov model’s stationary distribution with the other alternative descriptions for the NPS distribution, this is tested in more depth for all time spans composed of many individual episodes. One approach to compare how well the models describe the experimentally observed distribution is the use of a𝜒2 goodness of fit test.
Three problems arise when computing the test statistics:
1. Samples should be uncorrelated, but the NPS are correlated especially on the time scale of about one CL (compare Sec. 6.4).
2. The 𝜒2 test is not reliable when bins contain very few events.
3. The 𝜒2 test requires knowledge of the degrees of freedoms of the model fitted to the data.
To reduce issues due to the first point, the NPS time series used for the test is sub-sampled at 100 ms intervals removing strong correlations2.
The second issue can be solved by combining multiple NPS probabilities/bins with very few data points into a single probability bin including adjacent NPS values with a higher number of events. Here this is achieved by combining multiple NPS levels until each contains at least five events. Since few events occur for either large or small NPS values, this combining of bins is done starting from the largest NPS bin going down, and then starting from zero NPS going up.
The third issue is avoided here by using only part of the data for the model esti-mation step and the rest of the data for the 𝜒2 test.3 Thus, the rates are estimated using only half of the VF episodes within a time span (rounded up) and the test is then performed on the remaining half. Using such a training (model estimation) and validation (𝜒2 test) strategy ensures that model estimation uses a separate data set.
For the test this effectively means that there are no additional degrees of freedom con-tributed by the model parameters. This procedure is repeated 100 times with different splits of the data and the median over all 100 test results is reported.
The result of this procedure is summarized in Figure 6.5 shown as a boxplot over all 15 time spans. The higher value of the 𝜒2-test for the Markov model shows that it performs better than for example a Gaussian one. While the 𝑝-Value for the proba-bility of the Markov model matching the observed distribution is not very high, given enough data this is expected even for small model deviations.
To summarize the 𝜒2-test results, the Markov model works well when looking at an average over multiple episodes and this appears to be true even for unseen data.4
2Although this should mostly affect the absolute values and these should be interpreted with care in any case.
3Such as training and validation/testing sets in machine learning, see for example [97].
4Although not necessarily for predicting an individual episode.
0 5 10 15 20 25 30 NPS
0.00 0.02 0.04 0.06 0.08 0.10 0.12
probability
Experminent Markov Single Step Gaussian
Markov Model
Fig. 6.6Probability of observing a certain NPS for the episode in the experiment 2017-11-23 at 15:28. This video of a VF is 20 s long and no Pinacidil was used. From all 31 Markov models, it is the only exception showing a visually significant derivation of the Markov model from the experimental observation. Lines indicate the probability distributions of the different models and black squares the experimental data, as also shown in Figure 6.1 for a different episode.
Manual inspection of the NPS distributions from all time spans and VF episodes shows a visually good fit in all but one event. This exception is the VF episode 2017-11-23 at 15:28 and shown in Figure 6.6. Two overlapping reasons may explain the strong deviation:
• When the NPS drops too low on average, the model may not perform well due to hidden dynamics, for example within the septum, having a large influence. One may hypothesize that a hidden Markov model with additional unseen states may capture this to some degree.5
• This video visually shows repetitive patterns that quickly activate most of the heart akin to a VT. This could indicate that, more than in most events, the dynamics are driven by unseen dynamics not captured by the Markov model.
Above, this was interpreted as a general shortcoming of the model when few PS are present. However, it could also be interpreted as the existence of a dominating part of the dynamics that is unseen and cannot be captured by
5A simple extension, by allowing negative NPS, indeed shows a large improvement. Such an extension may be reasonable even in other cases where the results do not change much since the probability to reach no NPS is low. However, the choice of the hidden states and their emission probabilities in the hidden Markov model add too much complexity to be discussed or analyzed in depth here. For a formal definition of hidden Markov models see for example [98, Ch. 3.2]
6.2 Results for the Probability Distribution of the NPS 73
10−7 10−6 10−5 10−4 10−3 10−2 pMarkov(NPS = 0)
2018-03-28 2017-11-30 2017-11-23 2017-11-16 2017-11-02 2017-10-17 2017-10-12 2017-10-05 2017-09-28 2017-09-21
Experiment
no self-term.
self-term.
a)
10 20 30
hNPSi b)
Fig. 6.7For each of the experiment on the𝑦-Axis each dot represents on time span with the colour indicating whether or not self-termination was observed. a) gives the probability of finding no PS in the Markov model on the𝑥-Axis, whileb)shows the average NPS. Both (a) and (b) give similar results with the average NPS being anti-correlated with the probability of no phase singularities𝑝Markov(NPS= 0).
the stochastic Markov model. One possibility for this would for example be a relatively stable dominating spiral wave. In cardiac literature such a dominating rotor is called a “mother rotor” and its importance to sustaining the arrhythmia is contested. This is discussed for example in [2] and references therein. If this interpretation would be substantiated for example by simulations, it would suggest that in most cases a dominating mother rotor does not exist in the rabbit heart.
In general, the Markov model works very well to describe the probability distribu-tion of the NPS for the analyzed time spans and videos. This is a behaviour shared with other pattern forming systems as well as the simulations mentioned in the theory Section 3.2. It further provides the basis for the following section where the knowledge of the probability distribution is used.
6.2.1 Relation to Self Termination
All data analyzed here was recorded from fully developed fibrillation that did not stop during the recording time. However, in some of these experiments self-termination, the spontaneous end of fibrillation, is observed. For example it happens that after inducing a VF, it stops within a short time6. Alternatively, a VF may stop even after a long time of fibrillation.
6If it stops too fast the induction would be considered unsuccessful itself.
For the previously analyzed time spans it is now possible to fit the rates to find the Markov model and its stationary probability distribution. Since termination is char-acterized by the fact that no spirals remain, a necessary condition for it is that no PS is observed. Thus, the probability of finding no PS in the Markov model 𝑝(NPS= 0) may be an indicator for whether or not self-termination was indeed observed. This is similar to the approach taken by Sugimura et al. for simulations [76].
Furthermore, it is possible to analyze all experiments and determine whether self-terminations occurred at or around a given time span. Here, this was done using two sources of information:
1. Often it was noted during the experiment when self-termination occurred.
2. When induction events were repeated within a short time, although the first one was noted as a successful induction, a self-termination has most likely occurred between the two.
While these records do not provide a quantification of how commonly self-termination occurred, they do provide an approximate categorization into whether or not it was observed at all.
The result of this analysis is given in Figure 6.7 where blue dots show episodes with observed self-termination and orange ones identify those without.
The correlation between 𝑝(NPS = 0) and the average NPS is very high so that both provide similar information. However, the plot indicates that self-terminations may indeed be more likely when there are fewer PS observed and 𝑝(NPS= 0)is large.
The results shown in Figure 6.7 are based on a fairly small set of data and depend more on the heart than on whether or not Pinacidil was given. Further, the identification of whether or not self-termination occurred may be prone to errors. Nevertheless, the results motivate that such an approach to understand self-termination may not be limited to simulations and could be applied successfully to experimental observations.