An analytical approach to the determination of optimal phasing for external counterpulsation

An inhomogeneous linear one-dimensional mathematical model is constructed as a conceptual approach to the study of the effects of External Counterpulsation (ECP) on the pressure and flow at the root of the aorta. The optimal operation of ECP is defined by two conditions: (1) minimization of the mean systolic pressure; and (b) maximization of the ratio of diastolic area over systolic area under the total pressure curve. The phase shift of the external pressure is determined so as to satisfy these two requirements. It is demonstrated within our approximation that with a given magnitude of external pressure, the phase shifts that satisfy these two requirements are the same. These phase shifts are linear functions of the systolic fraction of the total cardiac period, and depend on the time for the external wave to travel from the site of application up the vascular bed to the root of the aorta, plus the reflection contributions. Even though these results are derived from a simple model far from the complexity of the actual vasculature, the basic concepts would remain valid even if more complex mathematical treatments would have been used.