Background— In the present study, we extended previous mathematical modeling work on patients with bidirectional cavopulmonary (“bidirectional Glenn”) anastomosis to assess the potential utility of several descriptors of oxygen status. We set out to determine which of these descriptors best represents the overall tissue oxygenation. We also introduce a new descriptor, S o ₂ min, defined as the lower of the superior and inferior vena cava oxygen saturations. Methods and Results— The application of differential calculus to a model of oxygen physiology of patients with bidirectional Glenn allowed simultaneous assessment of all possible distributions of blood flow and metabolic rate between upper and lower body, across all cardiac outputs, total metabolic rates, and oxygen-carrying capacities. When total cardiac output is fixed, although it may intuitively seem best to distribute flow to maximize oxygen delivery (total, upper body, or lower body), we found that for each variable, there are situations in which its maximization seriously deprives flow to the upper or lower circulation. In contrast, maximizing S o ₂ min always gives physiologically sensible results. If the majority of metabolism is in the upper body (typical of infancy), then oxygenation is optimized when flow distribution matches metabolic distribution. In contrast, if the majority of metabolism is in the lower body (typical of older children and during exercise), oxygenation is optimal when flows are equal. Conclusions— In patients with bidirectional cavopulmonary anastomosis, because there is a tradeoff between flow distribution and saturation, it is unwise to concentrate on maximizing oxygen delivery. Maximizing systemic venous saturations (especially S o ₂ min) is conceptually different and physiologically preferable for tissue oxygenation.