Nonlinear heart model predicts range of heart rates for 2:1 swinging in pericardial effusion

We analyze two mathematical models of Rigney and Goldberger (14) of heart swinging in large pericardial effusions. Both models represent the torques due to the outflow of blood from the heart. The first assumes that the duration of systole does not vary with heart rate (in beats/min), whereas the second assumes that it varies linearly with heart rate. We examine the motion of the heart for heart rates between 50 and 200 and for a range of initial positions and velocities. Both models predict that the heart swings once every other beat (2:1 swinging, giving rise to electrical alternans) in a discrete range of heart rates and swings once per beat otherwise; both models explain the appearance and disappearance of 2:1 swinging mathematically. The first model predicts a rate range from 105 to 116 for the occurrence of 2:1 swinging. The second model predicts the same qualitative behavior but with 2:1 swinging occurring at heart rates between 88 and 119, which agrees well with published clinical data showing 2:1 swinging at heart rates between 90 and 144. We describe an analysis program for ordinary differential equations that analyzed the models quickly and automatically.