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Elastic properties of arteries have long been recognized as playing a major role in the cardiovascular system. However, non-invasive in vivo assessment of local arterial stiffness remains challenging and imprecise as current techniques rely on indirect estimates such as wall deformation or pulse wave velocity. Recently, Shear Wave Elastography (SWE) has been proposed to non-invasively assess the intrinsic arterial stiffness. In this study, we applied SWE in the abdominal aortas of rats while increasing blood pressure (BP) to investigate the dependence of shear wave speed with invasive arterial pressure and non-invasive arterial diameter measurements. A 15MHz linear array connected to an ultrafast ultrasonic scanner, set non-invasively, on the abdominal aorta of anesthetized rats (N=5) was used. The SWE acquisition followed by an ultrafast (UF) acquisition was repeated at different moment of the cardiac cycle to assess shear wave speed and arterial diameter variations respectively. Invasive arterial BP catheter placed in the carotid, allowed the accurate measurement of pressure responses to increasing does of phenylephrine infused via a venous catheter. The SWE acquisition coupled to the UF acquisition was repeated for different range of pressure. For normal range of BP, the shear wave speed was found to follow the aortic BP variation during a cardiac cycle. A minimum of (5.06$\pm$0.82) m/s during diastole and a maximum of (5.97$\pm$0.90) m/s during systole was measured. After injection of phenylephrine, a strong increase of shear wave speed (13.85$\pm$5.51) m/s was observed for a peak systolic arterial pressure of (190$\pm$10) mmHg. A non-linear relationship between shear wave speed and arterial BP was found. A complete non-invasive method was proposed to characterize the artery with shear wave speed combined with arterial diameter variations. Finally, the results were validated against two parameters the incremental elastic modulus and the pressure elastic modulus derived from BP and arterial diameter variations.