Pulse wave propagation in a model human arterial network: Assessment of 1-D visco-elastic simulations against in vitro measurements

Abstract

The accuracy of the nonlinear one-dimensional (1-D) equations of pressure and flow wave propagation in Voigt-type visco-elastic arteries was tested against measurements in a well-defined experimental 1:1 replica of the 37 largest conduit arteries in the human systemic circulation. The parameters required by the numerical algorithm were directly measured in the in vitro setup and no data fitting was involved. The inclusion of wall visco-elasticity in the numerical model reduced the underdamped high-frequency oscillations obtained using a purely elastic tube law, especially in peripheral vessels, which was previously reported in this paper [Matthys et al., 2007. Pulse wave propagation in a model human arterial network: Assessment of 1-D numerical simulations against in vitro measurements. J. Biomech. 40, 3476-3486]. In comparison to the purely elastic model, visco-elasticity significantly reduced the average relative root-mean-square errors between numerical and experimental waveforms over the 70 locations measured in the in vitro model: from 3.0% to 2.5% (p<0.012) for pressure and from 15.7% to 10.8% (p<0.002) for the flow rate. In the frequency domain, average relative errors between numerical and experimental amplitudes from the 5th to the 20th harmonic decreased from 0.7% to 0.5% (p<0.107) for pressure and from 7.0% to 3.3% (p<10(-6)) for the flow rate. These results provide additional support for the use of 1-D reduced modelling to accurately simulate clinically relevant problems at a reasonable computational cost.

Fig. 1

(left) Planview schematic of the 1:1 hydraulic model of the 37 larger conduit arteries in the human systemic circulation. Arteries were simulated using silicone tubes. 1: Pump (left heart); 2: catheter access; 3: aortic valve; 4: peripheral resistance tube; 5: stiff plastic tubing (veins); 6: venous overflow; 7: venous return conduit; 8: buffering reservoir; 9: pulmonary veins. The arrows indicate the location of the results shown in Figs. 3 to 5. (right) Topology and reference labels of the arteries simulated, whose properties are given in Table 1. (Modified from Matthys et al., 2007.)

Fig. 2

(top) Experimental uniaxial load and extension with time for a sample of the silicone used in the experimental arterial network. (bottom) Load-extension loop for the first load–unload cycle, whose direction is indicated by the arrows. A cubic spline was fitted to the loading and unloading points measured in the extension test. Loads and extensions are shown normalised by their corresponding maximum values.

Fig. 3

Experimental (exp) and simulated elastic (elas) and visco-elastic (visc) pressure (left) and flow (right) waveforms in the midpoint of the thoracic aorta I, left renal artery, right iliac-femoral III artery and right carotid artery in the hydraulic model in Fig. 1. Note the different scales of flow rates.

Fig. 4

Spectrum of the flow harmonics $\widehat{Q}$ on a semi-logarithm scale in the midpoint of the left renal artery (Fig. 1) of the experimental (crosses) and simulated elastic (circles) and visco-elastic (triangles) models.

Fig. 5

Area–pressure curve in the midpoint of the thoracic aorta I (Fig. 1) simulated using the elastic (elas) and visco-elastic (visc) numerical models. We also show the curve obtained using the visco-elastic model with eight times the measured viscosity of silicone $(8\phi )$.