Numerical assessment of time-domain methods for the estimation of local arterial pulse wave speed

Abstract

A local estimation of pulse wave speed c, an important predictor of cardiovascular events, can be obtained at arterial locations where simultaneous measurements of blood pressure (P) and velocity (U), arterial diameter (D) and U, flow rate (Q) and cross-sectional area (A), or P and D are available, using the PU-loop, sum-of-squares (∑(2)), lnDU-loop, QA-loop or new D(2)P-loop methods. Here, these methods were applied to estimate c from numerically generated P, U, D, Q and A waveforms using a visco-elastic one-dimensional model of the 55 larger human systemic arteries in normal conditions. Theoretical c were calculated from the parameters of the model. Estimates of c given by the loop methods were closer to theoretical values and more uniform within each arterial segment than those obtained using the ∑(2). The smaller differences between estimates and theoretical values were obtained using the D(2)P-loop method, with root-mean-square errors (RMSE) smaller than 0.18 ms(-1), followed by averaging the two c given by the PU- and lnDU-loops (RMSE <2.99 ms(-1)). In general, the errors of the PU-, lnDU- and QA-loops decreased at locations where visco-elastic effects were small and nearby junctions were well-matched for forward-travelling waves. The ∑(2) performed better at proximal locations.

Fig. 1

Connectivity of the 55 larger systemic arteries in the human. Their names and properties were taken from Alastruey (in press) and are shown in Tables 1 and 2 of the supplementary material. A periodic flow rate with a mean flow of 5.6 l min⁻¹ was prescribed at the aortic root for the first 10 s, followed by zero flow for $t>10\mathrm{s}$.

Fig. 2

PU-loop (a), $\ln \mathit{DU}$-loop (b), QA-loop (c) and D²P-loop (d) in the midpoint of the thoracic aorta II (Segment 27) in the purely elastic (dashed lines) and visco-elastic (solid lines) 55-artery models. Wave speeds c calculated using the linear least-squares fit highlighted in red. The theoretical c is 5.28 m s⁻¹. Pressure (e) and c (f) with time at the same location for the elastic (dashed lines) and visco-elastic (solid lines) models. The linear regions used in the loops are highlighted in red. (The complete pressure waveform was used for the D²P-loop in the purely elastic model.) (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3

Pressure (a,c) and velocity (b,d) versus time and distance in 2 cm increments from the aortic root (x=0, Segment 1) to the end of the left anterior tibial artery (Segment 49), through Segments 2, 14, 18, 27, 28, 35, 37, 39, 41, 42, 44 and 46. The aortic bifurcation is at x=44.4 cm. Pulse waveforms generated using the purely elastic (a,b) and visco-elastic (c,d) formulations in the normal 55-artery model. Continuous pressure and velocity magnitudes are shown using coloured surfaces interpolated from the simulated pulse waveforms every 1 cm. After t=10 s, a zero flow rate was prescribed at the aortic root to show the relaxation of the models. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4

Theoretical (red lines) and estimated pulse wave speed with distance in 1 cm increments from the aortic root (x=0) to the end of the left anterior tibial artery, through the same segments as in Fig. 3. Estimated values calculated using the PU-loop, $\ln \mathit{DU}$-loop, QA-loop and sum-of-squares (${\sum }^2$) methods with the pulse waveforms from the (a,b) purely elastic and (c,d) visco-elastic normal models and the (e,f) well-matched model. Estimated values using the D²P-loop method are not shown, since they cannot be distinguished from the theoretical values in the scale of the figures. Note the different scales of c in the left and right figures, and the different theoretical values for the normal and well-matched models. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)