It was recently submitted that the rupture risk of an ascending thoracic aortic aneurysm (ATAA) is strongly correlated with the aortic stiffness. To validate this assumption, we propose a non-invasive inverse method to identify the patient-specific local extensional stiffness of aortic walls based on gated CT scans. Using these images, the local strain distribution is reconstructed throughout the cardiac cycle. Subsequently, obtained strains are related to tensions, through local equilibrium equations, to estimate the local extensional stiffness at every position. We apply the approach on 11 patients who underwent a gated CT scan before surgical ATAA repair and whose ATAA tissue was tested after the surgical procedure to estimate the rupture risk criterion. We find a very good correlation between the rupture risk criterion and the local extensional stiffness. Finally it is shown that patients can be separated in two groups: a group of stiff and brittle ATAA with a rupture risk criterion above 0.9, and a group of relatively compliant ATAA with a rupture risk below 0.9. Although these results need to be repeated on larger cohorts to impact the clinical practice, they support the paradigm that local aortic stiffness is an important determinant of ATAA rupture risk.