In the case of living tissues, the spectral width of the electrical bioimpedance dispersions (closely related with the alpha parameter in the Cole equation) evolves during the ischemic periods. This parameter is often ignored in favor of other bioimpedance parameters such as the central frequency or the resistivity at low frequencies. The object of this paper is to analyze the significance of this parameter through computer simulations (in the alpha and beta dispersion regions) and to demonstrate its practical importance through experimental studies performed in rat kidneys during cold preservation. The simulations indicate that the dispersion width could be determined by the morphology of the extra-cellular spaces. The experimental studies show that it is a unique parameter able to detect certain conditions such as a warm ischemia period prior to cold preservation or the effect of a drug (Swinholide A) able to disrupt the cytoskeleton. The main conclusion is that, thanks to the alpha parameter in the Cole equation, the bioimpedance is not only useful to monitor the intra/extra-cellular volume imbalances or the inter-cellular junctions resistance but also to detect tissue structural alterations.