When applying pharmacokinetic (PK) models to dynamic contrast enhanced MRI (DCE-MRI) data it is important to appropriately deal with the enhancement onset time, because errors in the onset time will affect the PK parameter estimates. This paper presents a Bayesian approach to the estimation of the PK parameters k_ep and K^trans that robustly treats the onset time. This approach involves the computation of an analytically intractable integral, so two approximate methods are developed. The first uses adaptive numerical quadrature, which produces results accurate to a given tolerance, and the other a simple approximation with a summation. These approaches are compared with each other, and with the standard least-squares (LS) approach. The results of a Monte Carlo experiment show that the LS approach produces biased estimates when k_ep is large and K^trans is small, whereas both the Bayesian methods are unbiased. The two Bayesian methods produce very similar estimates, but the simple summation method requires less than half the computation time of either the LS, or the quadrature approximation. The standard deviation of the LS estimates is shown to be larger than either of the Bayesian estimates, while uncertainty estimates based around a Hessian approximation are shown to be too small for all three methods. A more detailed method of assessing the uncertainty of the Bayesian approach is described, and the results show that this is a more accurate description of the estimation uncertainty.