['The reliability of dialysis venous pressure (VP) in detecting stenosis is controversial. A mathematical model may help to resolve the controversy by providing insight into the factors that influence static VP.', 'This study used inflow artery and outflow vein luminal diameters from duplex ultrasound studies of 94 patients. These diameters were applied to a mathematical model, and how they affect the relation among VP, mean arterial pressure (MAP), blood flow, and stenosis was determined. Whether VP/MAP is a valid adjustment for the influence of MAP on VP, and whether the standard VP/MAP referral threshold of 0.50 is valid, were also determined.', 'It was found that there is an approximate one-to-one relation between MAP and VP, so VP/MAP is a valid adjustment. Also, the 0.50 threshold successfully identifies most grafts with stenosis of 65% or more. However, the ratio of artery/vein diameters varied widely between patients, and the ratio independently influences VP/MAP. When the inflow artery is relatively narrow, the VP/MAP increase is delayed followed by a more rapid increase as critical stenosis is reached.'] Does mathematical model demonstrate influence of luminal diameters on venous pressure surveillance?