Binding potentials have become important indicators of receptor availability in PET maps of receptors in the
living human brain. Current methods fail to determine true steady-state values of the binding potential (pB) and
the corresponding steady-state magnitude of bound tracer. The tracer flux into the tissue is driven by the
concentration difference between the tracer concentration in the capillary circulation and in the tissue with which
the capillary is directly exchanging (precursor compartment). When tracers are introduced into living tissue, the
tracer precursor and its products distribute in separate compartments, which cannot be probed individually. The
solution may therefore be to determine the individual time courses of the precursor and the products. In the
cases of more than two compartments, however, the number of parameters is so great that regression analysis
fails to identify a meaningful and unique set of estimates. Is it possible instead to use model-independent
regression analysis to identify the precursor compartment that determines the influx to all of the subsequent
compartments? In the following discussion, m is a PET record of the total contents of the precursor and product
compartments, ca is the arterial concentration, and Vo the volume of the capillary compartment. The total flux of
labeled substance into the system of compartments is dm/dt = Vo dca/dt + K1 ca - k2 me. The equation can be
solved for the tracer content of the precursor (exchange) compartment, me = Ve (ca – 1/K1 (dm/dt – Vo dca/dt))
where Ve = K1/k2 is the partition volume where Ve is known from separate measurement of the tracer’s
partition volume in a non-binding reference region. Hence, to obtain an estimate of the quantity of exchangeable
tracer, it is necessary to determine the derivatives of m and ca, and the magnitudes of Vo, Ve, and K1. To
differentiate the observed m, the record could first be
optimized by a formula that both minimizes the sum of
squares but also has the property that the observed values
maintain normal distribution about the optimized values.
Both analytic differentiation with known derivatives, and
simple numerical differentiation, is considered. In the end,
the precursor–product ratio (i.e., the binding potential) is
obtained by subtracting the precursor from the total, such
that mb = m – me – Vo ca where at the peak of mb, (mb(p)),
dmb/dt is zero and mb(p) therefore is at steady-state (transient
equilibrium), and the binding potential is the ratio between mb
and me at this time, such that pB = (m(p)-Vo ca(p)/me(p))-1
where the bound tracer at the peak is the ratio between mb(p)
and the specific activity of the tracer, as shown in the graphs
of actual uptakes of the dopamine receptor tracer raclopride.