## 1 INTRODUCTION

Blood pressure and flow waveforms in systemic arteries carry valuable information for the diagnosis and treatment of cardiovascular disease and play a significant role in clinical conditions such as hypertension. The waveforms result from a complex ventricular-vascular interaction involving cardiac contraction, impedance of large and medium-sized distensible arteries and resistance of smaller arteries and arterioles. Blood behaves as an incompressible fluid in arteries, which distend to accommodate the sudden increase in blood volume delivered by cardiac contraction. When elastic energy stored in the distended arterial walls is released, arteries contract. The regular expansion and contraction of arteries (the *pulse*) that follows cardiac contraction propagates in the form of *pulse waves*. These produce continuous changes in blood pressure and flow that can be studied as pressure and flow *wavefronts* (infinitesimal changes in pressure and flow) 1 running forwards and backwards (away from and towards the heart, respectively), with backward wavefronts originating from reflected forward wavefronts at sites of vascular impedance mismatch.

Figure 1 shows typical blood pressure waveforms measured *in vivo* along human (left) and rabbit (right) aortas, from the root to the aorto-iliac bifurcation, under normal conditions. The slope of the line joining the feet of these waveforms shows clearly that the pressure wavefront originated at the start of cardiac contraction propagates away from the heart; the measured space-averaged speed is 6.9 m s ^{ − 1} in the human and 6.1 m s ^{ − 1} in the rabbit. Thus, during a typical cardiac cycle, which takes about 1 s in the human and 0.25 s in the rabbit, a pulse wave has sufficient time to travel from the heart to the arterial vasculature and back multiple times.

Several studies have used wave intensity analysis (WIA) to investigate the role of wave reflections in shaping *in vivo* pressure and flow waveforms in systemic arteries [2-11], including the coronary circulation [12-14]. Given simultaneous measurements of blood pressure and flow velocity with time at an arbitrary location in the arterial network, we can calculate the local pulse wave velocity (PWV) and apply WIA to quantify the timing, direction and magnitude of the predominant waves that shape the pressure and velocity waveforms [15, 16]. Accurate estimation of PWV is not only important for WIA but is also clinically relevant, because PWV is an important predictor of cardiovascular events [17].

Numerical modelling has been used to assess the following: (i) the ability of WIA to quantify reflection coefficients [18]; (ii) haemodynamic information provided by WIA in a model of aortic coarctation [19] and the fetal circulation [20]; (iii) a modified WIA based on the reservoir-wave separation [21]; and (iv) the performance of several methods for PWV calculation [22-25]. Numerically generated pressure and flow waveforms are free of measurement errors, and the theoretical values of haemodynamic properties that affect waveforms (e.g. PWV, location of reflection sites, and magnitude of reflected waves produced) are available for comparison with corresponding estimates given by WIA and methods of calculating PWV.

In the present work, we used pressure and flow velocity waveforms measured *in vivo* in the rabbit or generated numerically in several models of human compliant vessels to (i) show the inability of traditional WIA to identify the important role of peripheral reflections in shaping the pressure waveform; (ii) test the accuracy of the modified *PU*–loop method of calculating PWV proposed by Mynard *et al.* [24], which accounts for peripheral reflections originating in previous cardiac cycles; and (iii) propose a new analysis of arterial pulse wave propagation to study the predominant waves that shape pressure and flow waveforms during systole and the contribution to the pressure waveform, over the whole cardiac cycle, of wave reflections originating in previous cardiac cycles, vessel compliances, peripheral resistances, outflow pressures and the flow at the root. We used our new analysis to study the effects of vessel stiffness and peripheral resistance on numerically-generated aortic pressure and flow waveforms.

We generated all numerical data using the nonlinear one-dimensional (1-D) formulation of blood flow in compliant vessels, because WIA is derived from this formulation and, hence, 1-D model pressure and velocity waveforms provide an ideal mathematical framework for our study. Several comparisons against *in vivo* [26-29], *in vitro* [30-34] and 3-D numerical [35] data have shown the ability of the 1-D formulation to capture the main features of pressure and flow waveforms in large human arteries. The nomenclature and abbreviations used in this paper are listed in the supplementary material.