Good predictions of the local mechanical environment of the tissue with known geometry and applied loads are fundamental to quantifying the biological response of tissues to mechanical stimuli. Whereas mean stresses in cylindrical sections of blood vessels may be calculated directly from measured loads and vessel geometry (e.g., Laplace’s law), predicting how these stresses are distributed across the wall requires knowledge of the constitutive behavior of the tissue. Previously, we reported biaxial biomechanical data for mouse carotid arteries before and after exposure to altered axial extension in organ culture. Here we considered phenomenological and microstructurally motivated constitutive models and identified material parameters for each via nonlinear regression. Specifically, we considered the model of Chuong and Fung, a four fiber-family model, and several new variants of a rule-of-mixtures model; in the latter, we modeled the artery as a mixture of collagen, elastin, muscle, and water. We found that the four fiber-family model fitted data significantly better than the model of Chuong and Fung. When identifying parameters for the rule-of-mixtures models, we imposed penalties that required each constituent to be structurally significant; e.g., elastin contributing significantly to the overall response over low loads and collagen dominating the response over high loads. Such constraints ascribe additional microstructural “meaning” to the constitutive model. Although imposing such penalties necessarily reduces the goodness of fit of model predictions to experimental data compared to regression without such penalties, the modest reduction in the goodness of fit observed in our results was off-set by the improved structural interpretation such models provide. Such microstructurally motivated models will be useful in characterizing vascular growth and remodeling in terms of the evolution of microstructural metrics that may be quantified experimentally.