All tracer-kinetic models currently employed with positron-emission tomography (PET) are based on compartmental assumptions. Our first indication that a compartmental model might suffer from severe limitations in certain circumstances when used with PET occurred when we implemented the Kety tissue-autoradiography technique CBF estimates, rather than remaining constant (to within predictable statistical uncertainty) as expected, fell with increasing scan duration T when T > 1 min. After ruling out other explanations, we concluded that a one-compartment model does not possess sufficient realism for adequately describing the movement of labeled water in brain. This article recounts our search for more realistic substitute models. We give our derivations and results for the residue-detection impulse responses for unit capillary-tissue systems of our two candidate distributed-parameter models. In a sequence of trials beginning with the simplest, we tested four progressively more detailed candidate models against data from appropriate residue-detection experiments. In these, we generated high-temporal-resolution counting-rate data reflecting the history of radiolabeled-water uptake and washout in the brains of rhesus monkeys. We describe our treatment of the data to yield model-independent empirical values of CBF and of other parameters. By substituting these into our trial-model functions, we were able to make direct comparisons of the model predictions with the experimental dynamic counting-rate histories, confirming that our reservations concerning the one-compartment model were well founded and obliging us to reject two others. We conclude that a two-barrier distributed-parameter model has the potential of serving as a substitute for the Kety model in PET measurements of CBF in patients, especially when scan durations for T > 1 min are desired.