A lipid core that occupies a high proportion of the plaque volume in addition to a thin fibrous cap is a predominant indicator of plaque vulnerability. Nowadays, noninvasive imaging modalities can identify such structural components, however, morphological criteria alone cannot reliably identify high-risk plaques. Information, such as stresses in the lesion's components, seems to be essential. This work presents a methodology able to analyze the effect of changes in the lipid core and calcification on the wall stresses, in particular, on the fibrous cap vulnerability. Using high-resolution magnetic resonance imaging and histology of an ex vivo human atherosclerotic carotid bifurcation, a patient-specific three-dimensional geometric model, consisting of four tissue components, is generated. The adopted constitutive model accounts for the nonlinear and anisotropic tissue behavior incorporating the collagen fiber orientation by means of a novel and robust algorithm. The material parameters are identified from experimental data. A novel stress-based computational cap vulnerability index is proposed to assess quantitatively the rupture-risk of fibrous caps. Nonlinear finite element analyses identify that the highest stress regions are located at the vicinity of the shoulders of the fibrous cap and in the stiff calcified tissue. A parametric analysis reveals a positive correlation between the increase in lipid core portion and the mechanical stress in the fibrous cap and, hence, the risk for cap rupture. The highest values of the vulnerability index, which correlate to more vulnerable caps, are obtained for morphologies for which the lipid cores were severe; heavily loaded fibrous caps were thus detected. The proposed multidisciplinary methodology is able to investigate quantitatively the mechanical behavior of atherosclerotic plaques in patient-specific stenoses. The introduced vulnerability index may serve as a more quantitative tool for diagnosis, treatment and prevention.