PROBABILITY OF CARDIAC DISEASES AND CIRCLE MAP Aliyev A. Abstract For diagnosis of using electrocardiogram findings and cardiac markers (blood tests for heart muscle cell damage). However, due to some influences, like , it can occur some difficulties to get accurate results and to give a proper diagnosis. We study the probability of missing the correct diagnosis of myocardial infraction when the patient has also cardiac . Keywords. myocardial infarction, arrhythmias, pure parasystole, circle maps. Myocardial infarction is a dangerous disease caused by a blood clot in some part of the heart as a result of thrombosis in the blood vessels or narrowing blood vessels. To identify this disease used the Smith criteria which generalization of the Sragbossa criteria [1]. The Sgarbossa criteria have three rules.  Concordant ST-segment elevation more than 1 mm in a lead with positive QRS complex;  Concordant ST-segment depression 1 mm or more in lead V1, V2 or V3;  ST-segment elevation of 5 mm or more discordant with the QRS complex in any lead.

Then Smith modified the third rule of Sgarbossa criteria.  Proportionally excessively discordant ST elevation in V1-V4, as defined by an ST/S ratio of equal to or more than 0.2 and at least 2 mm of STE. But the sensitivity of these rules are various:  First criteria of Sgarbossa is unable to be calculated;  The second one has 19% sensitivity;  The third rule’s sensitivity is 10%; Unfortunately, if a patient has an arrhythmia, the ratio of sensitive these criteria decrease. Let the patent suffer from both myocardial infraction and pure parasystole. Pure parasystole was first studied by Leon Glass. [2] In this illness, there are rhythms which arise from the competition between two pacemakers. The normal rhythm is still generated by the normal pacemaker in the right atrium, but in addition, there is a second rhythm generated by an abnormal pacemaker is called the ectopic pacemaker. The two pacemakers compete for the control of the heart and result in an abnormal rhythm called ventricular parasystole. An example of an electrocardiogram displaying this rhythm is shown in the following picture.

The sinus beats labelled S, and the ectopic beats, labelled E, are identifiable by their morphology. A schematic interpretation of this rhythm is shown below the trace. The periods of the sinus and ectopic pacemaker are approximately constant and are designated 푡푆 and 푡퐸, respectively. Following each sinus beat is a period, called the refractory time designated 휃, during which an is blocked (this means it is not observed). But if the ectopic beat falls outside of this period it is observed, but the next sinus beat is blocked. As a result, there is a compensatory pause. When the sinus and ectopic beats fall at the same time there is a beat of different morphology, called a fusion beat (labelled F). But our patient suffers from both of these illnesses. Since the sensitive region is ST complex, we should investigate hitting the time of ectopic beat entry into ST complex. Let the length of ST-segment 휃1. If we divide this ECG curve function by

푡퐸 and take its fraction part, we can make a circle map. Let 푓 be orientation preserving homeomorphism of a circle, which represent fraction part of above function and 푥0 be first observed time of ectopic beat. We investigate hitting the time of [0, 휃1). Denote Φ(푛) is the normalized random value of n-th times in the interval [0, 휃1) of the point 푥0. Theorem. Let 푝 be the probability of sensitivity of Smith criteria. The probability of missing the correct diagnosis of myocardial infarction in conditions the patient suffer from cardiac arrhythmias is equal 푃 = 푝퐸Φ(푛), where 퐸Φ(푛)- mean of random value Φ(푛).

References 1. S. W. Smith, K. W. Dodd, T. D. Henry, D.M. Dvorak, L. A. Pearce, Diagnosis of ST-elevation myocardial infarction in the presence of left with the ST-elevation to S-wave ratio in a modified Sgarbossa rule 2. L. Glass, L. Goldberger, J. Belair, Dynamics of pure parasystole, Physiology, p.841-847, 1986