Analytical solution for steady flow in a nonlinearly elastic vessel: prediction of negative resistance for positive transmural pressures

The Poiseuille equation describes steady flow in a rigid tube, and has been extensively used to relate pressure gradients to flow in blood vessels. However, blood vessel are distensible, and thus the true pressure-flow relationship is more complicated. Starting with the nonlinear Navier-Stokes equation, steady flow is derived for a cylindrical vessel with a nonlinear compliance. Although this derivation applies to blood vessels stretched with a positive transmural pressure, the nonlinear pressure-flow characteristics are similar to those of collapsible vessels. Thus the interesting behavior associated with collapsible vessels in the negative transmural pressure range (such as negative resistance) are predicted for cylindrical vessels stretched with positive transmural pressures.