A Mathematical Model for the Pressure-Flow Relationship in a Segment of Vein

The cross section of veins changes both in area and in shape as the transmural pressure varies. This implies that the relationship between pressure and flow in such collapsible vessels is not a linear one. In this paper, the techniques which apply to arteries are extended to include the case where the tube may collapse, as in the case of veins, and a relationship between pressure and flow is determined. The veins were considered to be long, straight, unbranched, and flexible, and the function which gives the cross-sectional area in terms of the transmural pressure was assumed to be known from other sources. The blood was assumed to be homogeneous, incompressible, and viscous. The equations to be solved were the linearized form of the Navier-Stokes equation and the continuity equation. An operating point was chosen and all variables were restricted to small neighborhoods about these operating points. The resulting analysis was identical to that for the arterial case except for the one condition that the cross sections were not circular. In this case, closed-form solutions to the equations were not available. It was shown that, as long as the velocity profiles were not needed, explicit solutions to the equations were not required. The relationship between the flow and the pressure was expressed in terms of the boundary conditions, the angular frequency, and the complex wave velocity. An expression for the complex wave velocity could be obtained only from a solution to the equations of fluid flow. However, it was shown that, except for a multiplicative constant, or "shape factor," the complex wave velocity was approximately independent of the cross-sectional shape. This permitted the use of the expression for the complex wave velocity which applies to the circular case. With this information, the pressure-flow relationship was comIpletely specified. This relationship was modified so that expressions for the fluid longitudinal impedance and the fluid transverse aamittance were obtained. In addition, an electric circuit analog was developed.