The simplified reference tissue model (SRTM, Lammertsma 1996) has been implemented in numerous forms. In differential form, the time derivative of a tissue time-activity curve (CT) can be written in terms of a reference region concentration (CR), a relative index of the extraction-flow product (R1), a time constant for outflow from the reference region into plasma (k'2), and the binding potential (BP) as:
Because this equation is linear in parameters, it can be solved by standard GLM methods. To convert to tissue concentrations rather than derivatives (and also regularize the equation), an integral form generally is used:
This equation now can be solved by GLM for 3 parameters: R1, k2, & k2a, with the binding potential (BP) equal to BP = k2/k2a - 1.
Fixing k'2 as a global constant
The parameter k'2 is really a global constant that should not be fit for every voxel (Ichise 2003). From Eq. 1, it's clear that this value will be biased toward zero in regions without specific binding in order to reduce noise from the last two terms. This bias on k'2 then can effect other parameters, so a good strategy is to first determine k'2 using a 3-parameter fit and then fix k'2 as a global constant and fit only 2 parameters per voxel:
Dynamic binding potentials
A binding potential, by definition, can only be defined at equilibrium, or for a physiological steady state. This can be a problem for challenge data, so it is often better to turn k2a into a dynamic quantity under such circumstances (Alpert 2003). For ligands with a low single-pass extraction fraction (so that flow doesn't change R1 very much), this strategy essentially turns BP into a dynamic quantity. However, it's important to remember that while this strategy enables kinetic modeling of challenge data, a "dynamic binding potential" is only an approximation of real changes in receptor availability in the specific binding compartment, because that compartment cannot be isolated from the free & non-specifically bound compartment within SRTM. If k2a is parameterized as a gamma function, for instance, then Eq. 3 is modified as
To determine whether or not this is a good thing to do, or whether a gamma function or some other function is appropriate, one can inspect the chi-squared per degree of freedom of fits.