extended the method to tracers with reversible kinetics.
Description
The kinetics of radiolabeled compounds in a compartmental system can be described in terms of a set of first-order, constant-coefficient, ordinary differential equations.[4][5] The time course of the activity in the multicompartmental system driven by a metabolite-corrected plasma input function can be described by:
where is a column vector of activity concentration for each compartment at time , is the matrix of the transfer constants between compartments, and is the vector of plasma-to-tissue transfer constants. Patlak and Blasberg[3] showed that the above equation can be written as:
where represents a row vector of 1s and . The total activity in the region of interest, , is a combination of radioactivities from all compartments plus a plasma volume fraction ()[2] and thus:
By dividing both sides by , one obtains the following linear equation:
For , Patlak and his colleagues[2] showed that , i.e., the steady-state condition. When this condition is satisfied, the intercept has reached its constant value so that after some time a plot of versus becomes a straight line with slope and intercept .[1]
For a catenary two-tissue compartment model with transfer constants (forward transport from plasma to tissue), (reverse transport from tissue to plasma), (
binding
parameter proportional to ), and (dissociation constant) to analyze enzyme or receptor system, the slope represents the total
distribution volume
() and is given by ,[1] where , , , and , in which is the concentration of ligand binding sites, is the equilibrium dissociation constant for the ligand-binding site complex, is the ligand-binding association constant, is the ligand-binding dissociation constant. For a one-tissue compartment model with transfer constants and , the slope is , where is the partition coefficient () and the intercept is .[1]