VI. CONCLUSION ABOUT LOCAL FIBRILLATION
1 Introduction
A repetition frequency in the range of 4.5 to 6.5 Hz (equivalent to 270
to 390 beats per minute or an interval range of 150 to 220 ms) fits the
known physiological properties of the myocardial cells of the dog much
better than the twice higher frequency found in signal analysis, see
chapter IV, par. 4. In the previous chapter (par. 3.5) was shown the tendency of the modelcells to get organized into two groups of more or less
synchronized cells. In this chapter the output of the topological model
will be treated as an electrogram during fibrillation in order to throw
some light upon the results of chapter 4.
2 Simulated potentials
The measurable electrical activity of the model was simulated by assuming
that the membrane potential could be described by the sum of 2 integrals
of Gauss-curves:
where
- P(t) stands for the membrane potential at time t after activation,
- m1, s1 for the steepness of the upstroke and
- m2 and s2 for the steepness of the downstroke of the membrane action potential.
The length of the
action potential is determined by m1+m2.
fig.6.1: simulated membrane action potential filtered at 30 Hz
 |
P(t) is drawn for m1=30, s1=10, m2=65 and s2=15.
|
The extracellular action potential after stimulation is reasonably
well approximated by the second derivative of the membrane potential in
course of time, (Scher 1979).
Thus:
fig.6.2: simulated membrane action potential filtered at 30 Hz
 |
A(t) with m1=30, s1=10, m2=65 and s2=15.
|
These formula's were used to transform the output of the topological
model into something that appeared like an electrogram, see the next figure
for an example.
fig.6.3: simulated actionpotentials of a group of 1000 modelcells and the central cell
 |
The apparently twice higher frequency of the conglomerate
of 1000 model cells - compared to the 'membrane potential' of the central
cell - is clearly seen. (The model is described in chapter V, par. 3.5).
3 Histogram
The amplitude histogram of the 'membrane potential' of the central model
cell is depicted in the next figure.
Such a histogram is typical for a
(almost) sinusoidal distribution. The histogram of the
'extracellular electrogram' of the above mentioned conglomerate of
1000 cells is shown in the next figure.
This histogram is almost identical
to that in fig4.7 in chapter IV.
Maybe the most interesting type of histogram is this two-peaked
form. If the signal from one electrode can be considered as the sum of n
independent sinusoidal distributions, other types of histograms will be
found and according to the central limit theorem in statistics even a
normal distribution when the number n grows to infinite. To see how fast
the sum of such a histogram would look like a normal distribution, the
amplitude histogram of 1000 pseudo-random drawings from such distributions has been calculated.
fig 6.6: comparison of 1000 pseudo-random
drawings from a sinusoidal distribution
and of the sum of 2 and 8 such distributions
 |
In the figure above the results are depicted
for the sinusoidal distribution and the sum of 2 or 8 independent
sinusoidal distributions. If the number n equals 8 the amplitude histogram is already unimodal symmetric, so instead of assuming different
types of ventricular fibrillation in different parts of the same heart,
one could also - restricting oneself to histogram analysis - assume one
type of ventricular fibrillation giving rise to a sinusoidal signal in
such small, independent islands of heart tissue, that in most cases the
electrodes used will record the summed electrical activity of several of
these islands.
4 Power spectrum
4.1 Peaks at regular intervals
Repetition of a function ad infinitum at regular intervals t removes all
of the Fourier transform except delta function samples at
f=+n/t (n=0,+1,+2,...).
The derivation of these results is seen simply if the process of repetition is regarded as the convolution of a function with an infinite set of
regularly spaced delta functions (Champeney 1973). The theoretical
power spectrum of A(t) (see par. 1) plus its convolution with a train of delta functions is sketched in the next figure.
fig.6.7: auto power spectrum of A(t)
added are deltafunctions spaced 5 Hz
 |
see fig. 6.2 |
In other words: the Fourier
transform of a repeated function is equal to zero except at the repetition frequency and its higher harmonics, where the transform equals the
original Fourier transform of that function.
The power spectra of the 'potentials' shown in fig. 6.3 are
depicted in the next 2 figures.
fig 6.8: power spectrum of simulated membrane potential
4.5 Hz corresponds to a repetition interval of 225 ms
 |
see fig. 6.3 |
fig 6.9: power spectrum of simulated external potential
9 Hz reflects apparent high repetition rate
 |
see fig. 6.3 |
The horizontal
axes are redrawn in such a way, that one timestep stands for 2 msec's.
4.2 Subharmonics and alternans
The spectrum of the total activity would lead to the conclusion of a
repetition frequency of ventricular fibrillation of 9 Hz like in our
previous publications (Herbschleb 1979, Herbschleb 1980a and
Herbschleb 1980b). Analysis of individual cells (the central cell
is shown as an example) however shows that the cells have a repetition
frequency of 4.5 Hz. The two groups of model cells are not quite equal in
size, so a clear alternans in amplitude is present. As will be proven in
appendix E, this means that the low repetition rate of the individual
cells is present in the spectrum as a low peak at a frequency half of the
overall repetition frequency. In chapter IV - par. 4.1 - has already been mentioned that 25% of the auto power spectra during ventricular fibrillation
exhibit this low frequency peak. An example of a subharmonic peak has
already been given in fig. 4.25 and another one is given in the next figure.
fig 6.10: intramural unipolar electrogram during VF
main peaks: 12.5, 25 & 37.5 Hz
minor peaks: ±6.5 & ±18.5 Hz
 |
Another indication is seen in fig. 11.8 and
fig. 11.9 showing the cardiogram of an intramural needle electrode
(unipolar) during the transition from a very fast ventricular rhythm
(stimulation by electrical pulses of 5 msec duration at 20 msec intervals) to ventricular fibrillation. Not only is the repetiton rate during
ventricular fibrillation immediately twice the rate during regular contractions, but clearly the fibrillation waves alternate in height. The
same phenomenon is present in figure 11 of Ideker et al (Ideker 1980)
and in figure 2A of Sano et al (Sano 1958) - see figures 6.14 and 6.13 -
and the doubling of frequency in figure 3 of Josephson
et al (Josephson 1980) - see fig. 6.15 - and less clear in figure 8
of Vanremoortene (Vanremoortene 1968). None of these authors mentions
these features of their figures. In chapter IX more evidence will be
presented to support the claim, that ventricular fibrillation is a
tachycardia with a splitting of the phases of the cells in 2 opposite
groups.
5 Regularity
If the repetition is not exactly regular, but some irregularity like e.g.
a regular modulation of the intervals is present, the delta functions
mentioned will be accompanied by sidebands, which will result in practical signal analysis in broadening of the peaks. The more irregular the
repetition, the less peaks are discernible. As shown in appendix E one
could use the figures of that chapter as a kind of nomogram to express
the regularity of ventricular fibrillation as the coefficient of
variation - cv=s/m - (standard deviation divided by the mean) of the
repetition interval. Assuming a normal distribution of the intervals,
95 % of the intervals will lie in between m-1.96·s and m+1.96·s; in terms
of coefficient of variation: 95 % lies in between m(1-1.96·cv) and
m(1+1.96·cv).
tabel 6.1: Classification of spectra in this study, using the width of the second spectral peak at -20 dB
| figures | cv | frequency | m | 95 % interval |
|---|
| fig. 4.12 | 0.025 |
11.5 Hz | 87 ms | 79 - 95 ms |
| fig. 4.13 | 0.156 |
10.5 Hz | 95 ms | 66 - 124 ms |
| fig. 4.14 | 0.25 |
10.5 Hz | 95 ms | 48 - 142 ms |
| fig. 4.25 | 0.075 |
11.5 Hz | 87 ms | 74 - 100 ms |
| fig. 4.26 | 0.125 |
10.5 Hz | 95 ms | 73 - 118 ms |
| fig. 6.10 | 0.025 |
12.5 Hz | 80 ms | 76 - 84 ms |
6 Reconstructed action potential
The spectrum in fig. 6.10 contains very sharp peaks at the
frequencies 12.5, 25 and 37.5 Hz and less predominant peaks at 6.0 - 6.5,
18.5 - 19.0 and 31.0 Hz. Assuming a completely regular repetition (cv <
0.025, see previous paragraph) and considering the estimated spectral
values at 12.5, 25 and 37.5 Hz as the best estimation of the height of
the delta functions mentioned earlier, one could reconstruct an idealized
function with such a spectrum by inverse Fourier transformation.
fig 6.11: a function with the same complex spectral
properties as the spectrum in fig. 6.10
at 12.5, 25 & 37.5 Hz
 |
Taking into account the spectral values at the half
frequencies, inverse Fourier transformation yields a function like in the next figure.
fig 6.12:
a function with the same complex spectral
properties as the electrogram in fig. 6.10
at 6.5, 12.5, 18.5, 25 & 31 Hz
 |
This function differs in shape from the one in fig. 6.11
and also displays a clear alternans. The spectrum of fig. 6.10
belongs to an unipolar, intramural electrogram from
an electrode with a surface of 1 square mm. Such an electrode collects
the electrical output of a great number of cells and if all these cells
fire at random the result is something akin to white noise. If the cells
that contribute most to the potential field of the electrode are in
synchrony and fire regularly, a spectrum with sharp peaks at the repetition frequency and its higher harmonics will arise. Should there however
be two equal groups of cells firing very regularly in synchrony, but with
half a repetition period shift between the two groups, then the odd peaks
would disappear from the spectrum, suggesting a twice higher repetition
frequency than really present; see appendix E. If these two groups are not
exactly equal, whether in size or contribution to the potential field of
the electrode, the odd frequencies will still be present in a diminished
form like in fig. 6.10.
The height of the even peaks is equal to
the sum of the corresponding spectral values for the separate groups and
the height of the odd peaks is equal to the difference of the corresponding spectral values. If one assumes that the spectra of both groups do not
differ in shape and that the recorded extracellular waveform is proportional to the second temporal derivative of the membrane action potential
(Scher 1979), then from the complex spectrum a reconstruction of the
membrane action potential would be feasible. This reconstruction with the
help of the complex spectrum belonging to the power spectrum of fig. 6.10
is sketched in the next figure.
fig 6.13: membrane potentials during VF
redrawn to the same timescale as the
'reconstruction' from the spectrum in fig. 6.10
 |
redrawn after Trautwein 1952 & Sano 1958
|
Trautwein and Zink
published repetition rates of single heart cells in frogs, dogs and cats
(Trautwein 1952). Contrary to Sano 1958 their recorded
membrane potentials became completely repolarized between contractions and
the repetition frequency during ventricular fibrillation was slightly
higher than the highest normal beating rate (250 beats/min versus 200
beats/min). Reasonably normal action potentials during ventricular fibrillation were also reported by Hoffman and Suckling (Hoffman 1954).
First of all nothing more has been stated than: if the above mentioned
function is repeated regularly at intervals of 160 ms (375 pm) and a
second identical function is formed with a 80 ms shift and the sum is
formed of proper weight factors times the temporal second derivatives of
these two repeated functions, then this sum will have a power spectrum
like the one in fig. 6.10, but many other functions will yield
the same spectrum, as only information of the frequency components of
6.25, 12.5, 18.75, 25, 31.25 and 37.5 Hz is available.
Furthermore, if one wishes to compare the reconstruction to the actual
action potentials as reported in the literature fig. 6.13, one
should keep in mind that the original signal had been filtered at 30 Hz.
7 Anatomy
In this paragraph a lot of loosely connected remarks are presented, that
throw some light on the reasoning behind the model of chapter V.
7.1 Nexus
Sperelakis and Hoshiko (Sperelakis 1960) supported the theory of
junctional transition between myocardial cells by showing there are no
low resistance pathways between cells. By phase-contrast microscopy was
shown that in living tissue the intercalated disks were clearly seen and
that they formed no fixation artifact (Yokoyama 1961). Moreover they
observed how these disks formed a definite partition between a segment of
the fiber which survived and contracted and an adjoining portion which
failed to contract.
The idea that one myocardial cell can excite another, if there are enough
nexuses between them, comes from the observation of Goshima 1975 in
tissue cultures that one single myocardial cell will synchronize another,
even if the current flows via a non-excitable cell in between.
7.2 Repetition frequency
From theoretical studies (Winfree 1980) and experimental work
(Jongsma 1975) it is known that an aggregate of coupled periodically
active elements can beat (much) faster than the isolated elements. All
myocardial cells are thus supposed to be potential pacemakers;
Goshima 1975, who reports that 70% of single fetal myocardial
cells in culture is spontaneously active.
Although thus theoretically an increase in spontaneous activity from 20-
40 b.p.m. (3000-1500 ms) in complete heart block to 6.5 Hz (154 ms)
during ventricular fibrillation in human patients cannot be excluded, the
sudden frequency doubling from ventricular tachycardia (150-220 b.p.m.;
400-273 ms) into fibrillation is not explained be these models.
7.3 Neighbouring cells
In fig. 5.6 one can clearly see how neighbouring 'cells' are in opposite phases. Although this might seem impossible in an actual heart to
physiologists who more or less tacitly assume a syncytial heart tissue,
Imchanitzky indicated already in 1905 and 1906 that adjacent cells in the
myocardium during ventricular fibrillation could be in quite different
states, i.e.: one was fully contracted and its neighbour relaxed
( Imchanitzky 1905 and Imchanitzky 1906).
The model thus exhibits the focal reexcitation phenomenon as defined by
Han: "due to the flow of current between adjacent fibers that are
repolarized at grossly disparate times." (Han 1971).
7.4 Size considerations
An aggregate of 10 by 10 by 10 cells of the dog myocard will have dimensions of circa 1.0 by 0.1 by 0.1 mm. Such a small piece of tissue would
stop fibrillation immediately or after several seconds, see
Garrey 1914, so the instability of the model is not surprising.
All cells in such a group are supposed to have the same characteristics,
as the tight electrotonic coupling prevents significant differences in
e.g. action potential duration, see Mendez 1969.
8 Some mathematical remarks
No attempt was made to describe the action potentials and membrane coupling in terms of the Hodgkin-Huxley equations, as it would cost too much
to resolve them simultaneously for 1000 cells.
Moreover some justification for the methods of chapter IV comes from the
statement by DeHaan and DeFelice (DeHaan 1978):
"The magnitude and kinetics of recorded electrical events may reflect
tissue geometry more than the characteristics of the cell membranes".
9 Conclusion
Keeping all this in mind the results of chapter IV could be explained as
being caused by measuring during ventricular fibrillation a fairly
regular electrical phenomenon or the sum of a number of such independent
phenomena.
This phenomenon consists out of action potentials of myocardial cells,
organized in 2 interwoven groups that constantly activate each other.
10 P.S.
After completion of the text I added 2 more examples of frequency-doubling. See par. 4.2 for an explanation.
fig 6.14: transition of ventricular tachycardia into fibrillation
 |
redrawn after fig. 11 in Ideker 1980 |
fig 6.15: transition of ventricular
tachycardia into fibrillation
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redrawn after Josephson 1980 |
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