### Water PET Model with Flow and Dispersion

This model implements the 1-tissue compartment model for H_{2}^{15}O water PET studies developed by E. Meyer [1].

Operational Model Curve

It is assumed that at by the arrival in tissue the true input curve C_{p}(t) has been dispersed. This dispersion is modeled by convolving C_{p}(t) with an exponential kernel

whereby the parameter t is called *Dispersion*. Incorporating the convolution into the 1-tissue compartment model and solving it using the Laplace transform yields the following analytic formula for the tissue concentration

K_{1} represents flow, and k_{2} flow divided by the partition coefficient p.

The operational equation is given by

Note that in this particular case there is no difference between the plasma and blood concentrations C_{p}(t) and C_{B}(t) because water is a freely diffusible tracer.

Parameter Fitting

The tissue model includes four fitable parameters **vB**, **K1 **, **k2**, (Flow/Part.Coeff), t (Dispersion). Per default, however, v_{B} is fixed at **vB**=0 as in the original model. For water studies with the blood activity sampled in a peripheral artery, the resulting input curve is delayed respect to the arrival in tissue. In this case, the model should be fitted using the **Fit region and blood delay** button.

Implementation Note

Unlike the standard compartment models in PKIN the model curve of the operational equation is not calculated by integrating a set of differential equations, but rather using the analytical solution given above. The convolution integral is approximated by summing rectangles of 0.1sec in length, assuming that the acquisition time of dynamic water studies is short.

Reference

1. Meyer E: Simultaneous correction for tracer arrival delay and dispersion in CBF measurements by the H215O autoradiographic method and dynamic PET. J Nucl Med 1989, 30(6):1069-1078. PDF