The objective of this study is to propose a method for preliminary processing of the experimental data from an inflation-extension test on tubular arterial specimens. The method is based on the condition for existence of a strain energy function (SEF) and can be used to verify whether the data from a certain experiment validate the assumption that the tissue can be considered as an elastic solid. As an illustrative example of the proposed method, experimental data for a porcine renal artery are used and the sources of the error in satisfying the condition of elasticity are analyzed. The results lead to the conclusion that the experimental data for a renal artery validate that the artery exhibits an elastic mechanical response and a constitutive formulation based on the existence of the SEF is justified. A modification of the proposed method for the case of an in-plane biaxial stretching test of mechanically isotropic and orthotropic tissues is considered.