Biomedical Engineering Reference
In-Depth Information
predict the three integers. Clinical studies demonstrate the applicability of these
results to patients with an artificial parasystolic pacemaker produced by periodi-
cally stimulating the ventricles with an intracardiac catheter at a fixed rate (12).
The mathematics for this problem is related to the "gaps and steps" problem in
number theory (13).
Just as with Wenckebach rhythms, careful examination of data in patients
shows ways in which the simple model of pure parasystole is not followed ex-
actly. In some patients, the parasystolic pacemaker is reset, or modulated by the
beat originating at the sinus node (14-16). In addition, although the model
makes predictions of the exact timing of ectopic beats, in some cases there are
unexpected missed ectopic beats. To account for such observations, the presence
of stochastic mechanisms that affect the timing of the parasystolic beats has
been hypothesized (17).
Both Wenckebach rhythms and parasystole can be used to illustrate the im-
portant mathematical concept of universal bifurcations (6). Thus, for both these
rhythms, changes in the parameters in the model (e.g., the frequency of the sinus
beat, the frequency of the ectopic beat, or the parameters specifying the quantita-
tive properties of the AV node or resetting of the parasystolic focus by the sinus
beat) will nevertheless lead to the same sequences of rhythms, but the exact val-
ues at which any rhythm appears will be different from person to person. How-
ever, the observation that the sequence of dynamic behaviors will be the same is
a triumph of mathematics and also provides a fundamental mathematical expla-
nation why the same similar rhythms appear in different people, in whom the
anatomical and physiological properties of the heart must necessarily differ.
However, both Wenckebach rhythms and parasystole are diagnosed and treated
when necessary, by physicians who have no knowledge of the underlying
mathematics. Further, it is not clear how knowledge of the mathematics could
lead to improvements in the therapy. Thus, at the current time the mathematical
analyses are not relevant to the practice of medicine.
3.
REENTRANT ARRHYTHMIAS
From a medical perspective, the most important class of arrhythmias are
called reentrant arrhythmias. In these arrhythmias, the period of the oscillation is
set by the time an excitation takes to travel in a circuitous path, rather than the
period of oscillation of a pacemaker (18). In some cases the reentrant circuit can
be found in a single chamber of the heart. For example, in typical atrial flutter,
there is a wave circulating around the tricuspid valve in the right atrium, and in
some patients who have had a heart attack there is reentrant circuit entirely con-
tained entirely in the ventricles. In contrast, in Wolf-Parkinson-White syndrome,
there can be excitation following the normal excitation pathways from the atria
to the AV node to the ventricles, but then the excitation is conducted retro-
