## 1 Introduction  
Cardiovascular disease is responsible for approximately one-third of deaths worldwide [1]. Atherosclerosis is the most common cardiovascular disease and involves the deposition of lipids in the arterial wall, leading to the thickening of the wall and the formation of a volume-lesion called a plaque, with the later stages of the disease involving the narrowing of the lumen of the artery, called a'stenosis' [2]. A common site for atherosclerosis is the carotid bifurcation, where plaque rupture can lead to stroke and death. The surgical treatment of atherosclerosis is considered if the diameter reduction is greater than 70% [3, 4]. Current clinical practice for the selection of at-risk patients is based on a minimum stenosis diameter as a measure of the risk of rupture and on patients' symptoms. It has become recognised that atherosclerosis develops as an interplay between the local mechanical environment and biology, where central to this interplay is wall shear stress (WSS). Early studies linked the development of atherosclerosis to regions of low and/or oscillating WSS. Early plaques are associated with the preservation of the lumen associated with outward remodelling, thought to be a response to a control mechanism in which the WSS is kept within a narrow range. For the established plaque, the upstream (proximal) side of the stenosis has high WSS [5, 6]. A theory developed by [PERSON] et al. [5] states that the high WSS leads to a series of biological events, leading to the thinning of the plaque wall and leaving the wall at risk of rupture. In addition, there is much experimental and clinical evidence that inflammation contributes to the development and destabilization of atherosclerosis but with a complex pathophysiology [7, 8]. The measurement of WSS in the individual patient is therefore of potential interest in understanding atherosclerosis and as a possible marker of the risk of the rupture of the plaque. Wall shear stress is estimated via the wall shear rate (WSR). The wall shear rate is the velocity gradient estimated at the vessel wall. Its measurement, therefore, involves the estimation of the velocities near the wall and the extrapolation of these velocities back to the measured location of the wall to obtain the WSR. Finally, the WSS is obtained by multiplying the WSR by the assumed viscosity of blood near the wall. Previous attempts to measure WSS have used magnetic resonance imaging (MRI) [9; 10] and patient-specific modelling involving a combination of medical imaging and computational fluid dynamics [11; 12]. The first reports of the ultrasound measurement of WSS were by [PERSON] et al. [13] using a single-beam Doppler approach. Recently, ultrafast Doppler [14] has been developed, which has the advantages of a very high frame rate and a large field of view [15; 16; 17; 18; 19]. Flow phantoms are used to investigate the performance of ultrasound imaging methods, including the validation of measurements of blood velocity and related quantities [20; 21; 22; 23]. The geometry of the flow phantom may come from the reconstruction of MRI or Computed Tomography (CT) scans [24] or from idealised geometries [25; 26]. Based on an in vitro experimental setup, [PERSON] et al. [27] discovered that plane wave (PW) vector Doppler has the potential to estimate WSS in a straight tube but provided few details about the used Doppler methodology. [PERSON] et al. [28] used PW vector Doppler to estimate WSS in a straight-vessel phantom and a 50% stenosed human femoral artery phantom under pulsatile flow conditions. In these phantom studies, the influence of tissue was not considered, and only one degree of stenosis was used. In terms of other aspects of WSS, such as increasing the accuracy of WSS estimation, some researchers have developed methods to improve the estimation accuracy of the near-wall velocity components [11; 29]. In this paper, our aim is to investigate the association of the WSS in carotid artery (CA) phantoms measured by PW vector Doppler with variations in the degree of stenosis.