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# A lumped model for blood flow and pressure in the systemic arteries based on an approximate velocity profile function

• Previously, by assuming a viscous dominated flow in the boundary layer and an inertia dominated flow in the vessel core, a velocity profile function for a 1D-wave propagation model was derived. Because the time dependent shape of the velocity profile in this boundary layer model depends on the size of the inviscid core and the boundary layer, and thus on the Womersley number, it differs along the arterial tree. In this study we evaluated a lumped model for a vessel segment in which the element configuration is based on physical phenomena described by the boundary layer model and for which all parameters have a physically based quantitative value dependent on the Womersley number. The proposed electrical analog consists of a Womersley number dependent resistor and an inductor arranged in parallel, representing the flow impedance in respectively the vessel core and the boundary layer, in series with a second resistor. After incorporating a capacitor representing the vessel compliance in this rigid tube model, the element configuration resembles the configuration of the four-element windkessel model. For arbitrary Womersley numbers the relative impedance of Womersley theory is approximated with high accuracy. In the limits for small and large Womersley numbers the relative impedances of the proposed lumped model correspond exactly to Womersley theory.
Mathematics Subject Classification: 76Z05, 92C35.

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