Measurement of hepatic blood perfusion and of the exchange of substances between blood and cells is a challenge. Long before the development of PET, multiple indicator data were analysed using models based on elaborate capillary theories. Today, PET is a unique modality that allows external quantitative measurements of the regional distribution of intravenously injected radiotracers and their metabolites in tissues, but dynamic PET data are analysed using less physiologically based schemes. The standard compartment model is an inlet equilibration model that does not naturally incorporate blood flow, and a single-uptake model that does not allow substances to re-enter the capillaries. These deficiencies lead to paradoxes when modelling fast blood-cell exchange. We have combined compartmental and capillary theory and developed microvascular models that account for blood flow and concentration gradients in capillaries. The microvascular models can be regarded as revisions of the input function which include more physiological realism and provide a superior description and interpretation of dynamic PET data when compared to the standard compartmental scheme.