Randomized controlled trials have not shown that surveillance of graft blood flow (Q) prolongs graft life. Because luminal diameters affect flow resistance, this study examined whether the influence of diameters on Q can explain the limitations of surveillance. Inflow artery and outflow vein diameters were determined from duplex ultrasound studies of 94 patients. These diameters were applied to a mathematical model for determination of how they affect the relation between Q and stenosis. Also determined was the correlation between Q (by ultrasound dilution) and diameters, stenosis, and mean arterial pressure in 88 patients. Artery and vein diameters varied widely between patients, but arteries generally were narrower than veins. The model predicts that the relation between Q and stenosis is sigmoid: as stenosis progresses, Q initially remains unchanged but then rapidly decreases. A narrower artery increases flow resistance, causing a longer delay followed by a more rapid reduction in Q. In a multiple regression analysis of data from patients, Q correlated with artery and vein diameters, sum of largest stenoses from each circuit segment, and mean arterial pressure (R = 0.689, P < 0.001). This study helps to explain why Q surveillance predicts thrombosis in some patients but not others. Luminal diameters control the relation between Q and stenosis, and these diameters vary widely. During progressive stenosis, the delay and then rapid reduction in Q may impair recognition of low Q before thrombosis occurs. Surveillance outcomes might be improved by taking frequent measurements so that there is no delay in discovering that Q has decreased.