A functional form for the vascular concentration of MRI contrast agent after intravenous bolus injection was developed that can be used to model the concentration at any vascular site at which contrast concentration can be measured. The form is based on previous models of blood circulation, and is consistent with previously measured data at long post-injection times, when the contrast agent is fully and evenly dispersed in the blood. It allows the first-pass and recirculation peaks of contrast agent to be modelled, and measurement of the absolute concentration of contrast agent at a single time point allows the whole time course to be rescaled to give absolute contrast agent concentration values. This measure of absolute concentration could be performed at a long post-injection time using either MRI or blood-sampling methods. In order to provide a model that is consistent with measured data, it was necessary to include both rapid and slow extravasation, together with excretion via the kidneys. The model was tested on T1-weighted data from the descending aorta and hepatic portal vein, and on T*2-weighted data from the cerebral arteries. Fitting of the model was successful for all datasets, but there was a considerable variation in fit parameters between subjects, which suggests that the formation of a meaningful population-averaged vascular concentration function is precluded. ; This paper was published as Physics in Medicine and Biology, 2009, 54, pp. 2933-2949. It is available from http://www.iop.org/EJ/abstract/0031-9155/54/9/023/ Doi: 10.1088/0031-9155/54/9/023