The two-compartment cerebral blood flow model can be characterized by the following three equations:

In these equations, *A*(*t*) is the PET data collected over a set of
frames, *C*_{a}(*t*) is the delay and dispersion corrected arterial
blood sample data, and *M*(*t*) is the radioactive tracer activity
present in cerebral tissue. We know both *A*(*t*) and *C*_{a}(*t*), but
cannot know *M*(*t*) without knowing *V*_{0}.

One additional point to consider is that these equations are written for continuous functions. However, the PET data that is actually available is average activity across each frame. Therefore, in order to solve the equations, we must average the non PET data across frames. By combining equations (2) and (3), and averaging the non PET data over frames, we get:

where *A*^{*}(*t*) is the PET data that is actually collected (by
definition, averaged over frames), and
performs an averaging over frames (*T*_{1} is the frame start
time, *T*_{2} is the frame stop time, and the integral is evaluated for
each frame).

We wish to solve equation (4) for *K*_{1}, *k*_{2}, and
*V*_{0}. Of course, one approach would be to try to perform an
explicit least squares curve fitting. However, this approach would be
quite computationally intensive since the fitting would need to be
performed for every pixel of a
pixel image. Fortunately,
there is a method of solution that gives good results with reduced
computational difficulty.

Greg Ward <greg@bic.mni.mcgill.ca>

Sean Marrett <sean@bic.mni.mcgill.ca>