Mathematical simplification of a PET blood flow model

The positron emission tomography (PET) H₂¹⁵O bolus injection model for cerebral blood flow (CBF) requires calculation of a certain double integral that, when calculated, provides the pixel values of a reconstructed image (PET number) in terms of the tissue flow, the arterial input function, a decay constant for ¹⁵O, the partition coefficient and a camera calibration constant that relates the flow-dependent integrated tissue activity to the measured PET number (cts/pixel). The tissue activity is assumed to be zero at the time of injection. A mathematical simplification, changing the order of integration, enabled the integration with respect to time to be performed analytically before the integration of the arterial input function. As a result of this simplification, only single integrals remain to be calculated numerically; cubic spline integration was used to calculate numerically these remaining integrals. This technique increases the accuracy and speed of evaluating blood flow without making simplifying assumptions. Similar simplifications may be applicable to other physiological models