Verification of a one-dimensional finite element method for modeling blood flow in the cardiovascular system incorporating a viscoelastic wall model

In this study we present the implementation and verification of a space–time finite element method for solving the nonlinear one-dimensional (1-D) equations of blood flow incorporating a viscoelastic arterial wall model. The viscoelastic model used is based on the generalized Maxwell model and thin-walled tube theory assumptions. Verification of the implementation was conducted using two different methods: (1) the analytic solution to the linearized 1-D equations of blood flow and (2) the Method of Manufactured Solutions (MMS). MMS enables verification for conditions such as tapered vessel geometries with spatially varying wall properties which cannot be verified with the analytic solution. RMS error values for all variables of interest (blood flow rate, pressure and vessel cross-sectional area) were calculated for simulations that utilized different mesh and time step sizes. The error values decreased with each mesh and time step size refinement. A theoretical analysis of numerical dissipation was also carried out for the linearized 1-D equations of blood flow. Dissipation in simulations due to different time step sizes matched theoretical estimations for given frequencies. This dissipation analysis can provide one way to estimate mesh and time step sizes to control numerical dissipation in the nonlinear simulations. The presence of wall viscoelasticity in all simulations is clearly seen in the hysteresis (pressure–area) loops.