Balancing Complexity and Simplicity in Mathematical Models of Fontan Circulations

Characterizing heart‐arterial‐venous interaction in the healthy adult is quite complicated. Blood pressures and flows depend on a large number of critical parameters, including compliances and contractilities of two ventricles, as well as arterial compliances, venous compliances and resistances of both systemic and pulmonary circulations. The additional complexity of the staged Fontan to palliate single ventricle physiology has made characterization insurmountable—the single ventricle and vasculature adapt not only to large changes in pressure after each of three surgeries, but also to large changes in blood volume in a rapidly growing infant. Mathematical models promising to elucidate the physiology of Fontan circulations have been too simple, lacking adaptive responses to altered hemodynamics. They also have been too complex, requiring numerical solutions that are exquisitely sensitive to assumed parameter values and are notoriously difficult to interpret. We therefore modified standard minimal models of each of the stages of the Fontan procedure to simplify the model equations. We then added select adaptive processes that modify cardiac and vascular properties. Because the resulting model solutions are algebraic, mean pressures and volumes are expressed explicitly in terms of critical cardiovascular parameters. The result is a novel tool that is complex enough to trust, yet simple enough to use.