Electrocardiographic cancellation: Mathematical basis

Abstract

1. 1. The concept of electrocardiographic cancellation can be generalized by imposing no limitation on the number of leads utilized in the formation of a synthetic lead for the purpose of cancelling an actual lead. 2. 2. Mathematical methods are given in detail for the computation of two types of cancellation patterns: (a) the minimum root mean square cancellation in which the sum of the squares of the deviations of the potentials from zero is minimized; (b) the minimum range cancellation in which the maximum peak-to-nadir spread of the potential points is minimized. 3. 3. If a synthetic lead is formed by n leads, a cancellation pattern having zero potential at n separate instants of time can always be produced. Therefore the maximum peak and maximum nadir of an actual lead can always be brought to the zero base line by a synthetic lead formed by two or more leads. However, the production of complete cancellation of the upper and lower limiting potentials of an actual lead does not constitute the technique for obtaining either a minimum range or a minimum root mean square type of pattern. 4. 4. Although cancellation patterns usually show multiple zero crossing points, it is mathematically possible for all of the potential points of both a minimum root mean square cancellation and a minimum range cancellation to be grouped on either the positive side or the negative side of the zero base line. Likewise, either type of cancellation pattern may possess only one zero crossing point. 5. 5. The upper and lower limits of a calculated minimum range cancellation are formed by a group of maximally positive and maximally negative potential points, the total number of which exceeds by two the number of leads which form the synthetic lead. If the number of such leads is represented by n, the number of equal and maximally positive potential points may vary from 1 to (n + 1). If the number of these limiting positive potential points is represented by m, the number of equal and maximally negative potential points is (n + 2)-m. If a minimum range cancellation consists entirely of either positive or negative points, (n + 1) of the points are both equal and maximally positive or maximally negative. The respective lower or upper limit of the cancellation pattern is then formed by the two zero end points. 6. 6. Suitable means of expressing quantitatively in per cent the results of both minimum root mean square and minimum range cancellations are presented and discussed. It is pointed out that the denominators of such expressions should contain terms referable only to the actual lead rather than to both the actual and synthetic leads.