Implementation of the Gaussian theorem with digital three-dimensional Doppler: A versatile flow quantitation method for laminar flows

Calculation of flow volumes through the great vessels or ventricular outflow tracts by conventional pulsed Doppler methods usually requires an assumption of a flat or predictably shaped cross-sectional flow profile. This may be invalid when applied to vessels which are non-circular and display dynamic changes in shape over the cardiac cycle. The Gaussian theorem states that for any arbitrarily shaped control surface system, the flow rate passing through it is equal to the sum of all the velocity components which are normal to the system surfaces. We tested this concept for the quantitation of flow in an oval, curved tube (major axis diameter 16mm, minor 12mm). A series of pulsatile flows (peak flow rates 60-205ml/s) were studied with reference flow data obtained by ultrasonic flow meter. Colour Doppler imaging was performed parallel to flow using an ATL HDI 3000 ultrasound system and a 7 MHz multiplane transoesophageal probe. 180° rotational acquisition at 6° step increments was used to generate a digital 3D dataset of tomographic Doppler velocities along the tube. The velocities at a 3D ovoid surface (equivalent to the vectors normal to a control surface) covering the cross-section of the tube lumen were integrated over time to derive actual flow rates. The time-surface velocity integration derived flow rates showed excellent correlation and agreement with actual peak flow rates (y = 1.02x + 4.9, r = 0.97, p<0.01). This method yielded reliable volume flow measurements with minimal angle dependency for laminar flows. It therefore provides a robust flow measurement that should be readily applicable to vessels and outflow tracts with varying flow profiles.