A description of the vascular input function is needed to obtain tissue kinetic parameter estimates from dynamic contrast enhanced MRI (DCE-MRI) data. This paper describes a general modelling framework for defining compact functional forms to describe vascular input functions. By appropriately specifying the components of this model it is possible to generate models that are realistic, and that ensure that the tissue concentration curves can be analytically calculated. This means that the computations necessary to estimate parameters from measured data are relatively efficient, which is important if such methods are to become of use in clinical practice. Three models defined by four parameters, using exponential, gamma-variate and cosine descriptions of the bolus, are described and their properties investigated using simulations. The results indicate that if there is no plasma fraction, then the proposed models are indistinguishable. When a small plasma fraction is present the exponential model gives parameter estimates that are biassed by up to 50%, while the other two models give very little bias; up to 10% but less than 5% in most cases. With a larger plasma fraction the exponential model is again biassed, the gamma-variate model has a small bias, but the cosine model has a very little bias and is indistinguishable from the model used to generate the data. The computational speed of the analytic approaches is compared with a fast-Fourier-transform-based numerical convolution approach. The analytic methods are nearly 10 times faster than the numerical methods for the isolated computation of the convolution, and around 4-5 times faster when used in an optimization routine to obtain parameter estimates. These results were obtained from five example data sets, one of which was examined in more detail to compare the estimates obtained using the different models, and with literature values.