An In-Vitro Flow Study Using an Artificial Circle of Willis Model for Validation of an Existing One-Dimensional Numerical Model

Abstract

A one-dimensional (1D) numerical model has been previously developed to investigate the hemodynamics of blood flow in the entire human vascular network. In the current work, an experimental study of water–glycerin mixture flow in a 3D-printed silicone model of an anatomically accurate, complete circle of Willis (CoW) was conducted to investigate the flow characteristics in comparison with the simulated results by the 1D numerical model. In the experiment, the transient flow and pressure waveforms were measured at 13 selected segments within the flow network for comparisons. In the 1D simulation, the initial parameters of the vessel network were obtained by a direct measurement of the tubes in the experimental setup. The results verified that the 1D numerical model is able to capture the main features of the experimental pressure and flow waveforms with good reliability. The mean flow rates measurement results agree with the predictions of the 1D model with an overall difference of less than 1%. Further experiment might be needed to validate the 1D model in capturing pressure waveforms.

Appendices

Appendix 1: The Measurement of the Viscosity of the Working Fluid

To measure the viscosity of the working fluid, a falling sphere method was used. The relationship between the viscosity of the working fluid and the terminal velocity of the sphere can be determined as $\mu = 2gr_{\text{s}}^{2} (\rho_{\text{s}} - \rho_{\text{l}} )/9u_{\text{t}}$, where $\mu$ is the viscosity of the working fluid, $g$ is acceleration due to gravity (a fixed value of 9.8 m s⁻²), $r_{\text{s}}$ is the radius of the sphere, $\rho_{\text{s}}$ is the density of the sphere, $\rho_{\text{l}}$ is the density of the working fluid, and $u_{\text{t}}$ is the velocity of the sphere. Briefly, in a graduated cylinder, a sphere was allowed to fall between a certain distance with marked positions at the top and bottom through the working fluid. A stopwatch was used to record the time during the sphere falling through the marked distance and its velocity was determined. The density of the sphere and working fluid was measured by dividing the mass by the volume. With having all the parameter values, the viscosity of the working fluid can be computed by the above equation.

Appendix 2: Phase-Averaging Method

The 150 measurements were split into individual waveforms by defining a reference point to distinguish each complete cardiac cycle. The reference point was chosen at the middle point of the rising edge of waveform to ensure the accuracy of the alignment of waveforms, because less fluctuation was observed in this region. The time scale for each waveform was normalized by dividing by its time interval for phase-based averaging. For pressure curves, the individual pressure waveform was end-diastolic aligned in the time domain for averaging.