Calculation of the Velocity Profile, Flow and Wall Shear Stress in Arteries From the Pressure Gradient: Importance of Distensibility and Taper

Abstract

Several studies have shown that an acute increase in blood flow causes an increase in arterial diameter. It is generally assumed that this flow dependent dilatation results from the increased shear stress acting on the wall. To test this hypothesis accurate assessment of the hemodynamic environment in the artery is necessary. Womersley [1] developed a linear theory for the calculation of the velocity profile, flow, and wall shear stress in arteries from the pressure gradient. This theory was validated by McDonald [2] who found good agreement between observed and calculated average velociites for the canine femoral artery. In contrast, Ling et al [3,4] who developed a nonlinear theory to calculate the velocity field from the pressure gradient, found the nonlinearities are important for the accurate prediction of flow and centerline velocity in the canine thoracic aorta. To evaluate the importance of nonlinearities we made a rigorous comparison of the linear theory of Womersley and the nonlinear theory of Ling and co-workers using data given by McDonald [5] for the canine femoral artery and by Ling et al [4] for the canine thoracic aorta. In the throacic aorta the velocity profile, flow and wall shear stress are significnatly different in the following cases: Distensible vessel with tper, which corresponds to the nonlinear solution, rigid vessel with taper, distensible vessel with no taper, and rigid vessel with no taper, which corresponds to the linear solution For exmplaes in these four cases the stroke volume is 18.2, 4.5 134.0, and 1176.6 cm3 respectively and the mean wall shear stress is 13.6, 6.5, 47.9, and 56.3 dyne/cm2, respectively. Assuming a physiological taper of 0.2° for the femoral artery [6], the velocity field is very similar for a distensible vessel with taper and a rigid vessel with taper, but it is different from those in the cases in which there is no tape. For example, in the cases of a distensible vessel with taper, a distensible vessel with no taper, and a rigid vessel with no taper, the stroke volume is 0.30, 0.54, ang 0.52 ml, respectively and the mean wall shear stress is 20.1, 29.1 and 29.0 dyne/cm2, respectively. In conclusion, nonlinearities arising from distensibility and taper of the artery can be important in the calculation of the velocity field from the pressure gradient.