Back to Bernoulli: a simple formula for trans-stenotic pressure gradients and retrospective estimation of flow rates in cerebral venous disease

Background Venous sinus stenosis can be associated with cerebrovascular disorders. Understanding the role of blood flow disturbances in these disorders is often hampered by the lack of patient-specific flow rates. Our goal was to demonstrate the impact of this by predicting individual flow rates retrospectively from routine manometry and angiography. Methods Ten cases, spanning a range of stenosis severities and pressure gradients, were selected from a cohort of patients who had undergone venous stenting for pulsatile tinnitus. Lumen geometries were digitally segmented from CT venograms. A simplified Bernoulli formula was derived to estimate individual cycle-average flow rates from clinical pressure gradients and minimum lumen cross-section areas. High-fidelity pulsatile computational fluid dynamics (CFD) simulations were performed to compare predictions of flow disturbances using generic versus individual flow rates, and to validate the Bernoulli formula. Results Individual flow rates derived from the Bernoulli formula deviated by up to 47% from the assumed generic flow rate, resulting in substantial differences in CFD predictions of post-stenotic flow instabilities. Pressure gradients estimated by the simplified Bernoulli formula were, however, highly predictive of pressure gradients from the full CFD simulations (R 2 =0.95; slope=0.98, 95% CI 0.88 to 1.09). Conclusions A simple Bernoulli formula can predict CFD-estimated trans-stenotic pressure gradients in realistic venous geometries. As demonstrated here, this may be used to recover individual flow rates from routine-but-invasive clinical measurements; however, it also suggests a simpler path towards non-invasive estimation of trans-stenotic pressure gradients that may avoid some of the challenges associated with 4D flow MRI approaches.