## 2 Methodology  
Successful PIV analysis requires optically transparent, physiological relevant phantoms. Physiological relevance requires a suitable global and local geometry and representative compliance. To match the experimental pulsatile flow characteristics to in vivo flow characteristics, the compliance of the phantoms should ensure that the Womersley number of the flows are equal. Compliance affects the arterial diameter change which is a key parameter in the determination of the Womersley number for time oscillating flow. Thus, if the compliance is correct (i.e., the relative change in vessel diameter during the cardiac cycle is identical in the phantom and physiological system), the Womersley number has been matched. In addition, compliance affects the pressure pulse peak and wave travelling speed. A complete description of phantom dynamic similarity matching is reported by [PERSON] et al. [24].  
The dimensions of the phantoms in this study were obtained from a meta-analysis of healthy aortic geometry reported by [PERSON] et al. [37]. Figure 1 shows the phantom dimensions. The model was designed at 1.5\(\times\) scale of the life-size for ease of fabrication and experimentation. Aortic arch geometry varies across the population. Using an idealized geometry enables modulation of certain characteristics to allow a generalizable understanding of certain flow phenomena. Thus, an idealized geometry of the aortic arch with the brachioccephalic artery (BCA) was designed using SolidWorks (SolidWorks, Concord, MA, USA) (Figure 1). A wall thickness of 1 mm is used to match experimentally determined in vivo compliance of the aortic arch [30].  
Two manufacturing strategies were used and compared in this study. The initial phantoms used the investment casting process described in [PERSON] et al. [24]. They fabricated a compliant model of the carotid artery with stenosis using a Sylgard-184 elastomer. The female mould was computer numerical control (CNC) machined from aluminium and the male mould was 3D printed from plaster. The second method was an innovation of the [PERSON] et al. method. The initial phantom utilised a wall thickness of 0.5 mm in the brachioccephalic artery to mimic compliance. From the initial casting, it was concluded that a 0.5 mm wall thickness demanded too much precision from the casting process as a minor dislocation may bridge the cavity across male and female moulds. Therefore, the second phantom utilised a 1 mm wall thickness in the brachioccephalic artery as the compliant behaviour in this artery was not considered clinically important. In addition, small wall thickness required significantly greater manufacturing precision, which seemed preclusive. This led to a compliance mismatch between the brachioccephalic artery in vivo and the phantom. However, since haemodynamics in the aortic arch and not the brachioccephalic artery was the object of this study, the trade-off was deemed acceptable.  
Compliance is defined as the ratio of volume gained over a pressure change. However, to achieve dynamic matching a lumen cross-sectional area normalised compliance that was a function of unit length was used [40]. This normalised compliance can be defined using Equation (1) [41]:  
\[C\ =\ \frac{1}{A_{min}}\frac{dA}{dP}\ =\ \frac{2\pi r^{3}}{A_{min}Eh} \tag{1}\]  
where \(A\) is the area, \(P\) is the pressure, \(r\) is the radius, \(A_{min}\) is the cross-sectional area of the lumen at zero transmural pressure, \(E\) is the modulus of elasticity, and \(h\) is the wall thickness of the phantom. Typical aorta structures have a radius of 0.011 m, a Young's modulus of \(0.526\times 10^{6}\) Pa and a wall thickness of 0.00132 m, leading to a compliance of \(31.25\times 10^{-6}\) Pa\({}^{-1}\)[37]. The phantom has a radius of 0.0206 m, a [PERSON]'s modulus of \(1.32\times 106\) Pa and a wall thickness of 0.001 m, leading to a compliance of \(31.25\times 10^{-6}\) Pa\({}^{-1}\). Polydimethylsiloxane (PDMS) Sylgard-184 Young's modulus was measured for a 1:10 mixing ratio and room temperature curing [24; 42].