The Tofts model (TM) and extended Tofts model (ETM) have become a standard for the analysis of dynamic contrast‐enhanced MRI. In this study, a mathematical analysis is used to identify exactly in which tissue types the Tofts models may be applied. The results show that the TM is accurate if and only if the tissue is weakly vascularised (small blood volume). The ETM is additionally accurate in highly perfused tissues (high blood flow). In tissues that are highly vascularised, or where tracer exchange is very fast or very slow, TM and ETM accurately fit the data but lead to a misinterpretation of the parameters. In tissue types with intermediate vascularity, perfusion and tracer exchange, neither model offers a good fit to the tissue concentrations. A good fit can be obtained with a measured input function by reducing the temporal resolution, but this does not improve the accuracy of the parameters. In conclusion, the Tofts models only produce reliable parameter values if the tissue is weakly vascularized (TM or ETM) or highly perfused (ETM). Without prior knowledge that at least one of these constraints is fulfilled, the physiological interpretation of the values produced by the Tofts models is unclear. Magn Reson Med, 2011. © 2011 Wiley‐Liss, Inc.