Bmax and KD

Bmax is the total density (concentration) of receptors in a sample of tissue. To refer to the concentration of available (free) receptors, terms Bavail or B'max can be used.

KD is the (radioligand) equilibrium dissociation constant. Affinity of ligand binding is the inverse of KD, and it equals the ratio of rate constants for association (kon) and dissociation (koff) of the radioligand to and from the receptors.

The constants kon [1/(min*mM)] and koff [1/min] determine the rate of change in the concentration of the ligand-receptor complex, [LR]:

, where [L] and [R] are the concentrations of the radioligand and available receptors, respectively. [R] + [LR] = Bmax, and [R] = Bavail.

Binding potential (BP) is the ratio of Bmax (receptor density) to KD (radioligand equilibrium dissociation constant), as defined by Mintun et al. (1984):

In vitro receptor studies the equilibrium is easily achieved. At equilibrium, the concentration of specifically bound ligand, Cbound, is:

, where Cfree is the concentration of the free ligand. At tracer doses, the amount of injected ligand (labeled and unlabelled) is so low (low injected mass and high molar activity) that the number of available receptors (Bavail = Bmax - Cbound) is practically unaffected (BavailBmax). Thus, at equilibrium, and with tracer dose, BP equals the ratio of specifically bound ligand to its free concentration:

Bolus+infusion protocol has been used to achieve a true equilibrium during PET studies for calculation of the BP.

After a single bolus administration of radioligand the true equilibrium cannot be achieved, but a transient equilibrium that is formed after bolus injection when the Cbound(t) reaches its peak (dCbound(t)/dt=0) can be used to get an estimate of BP. After bolus injection, compartmental models can be used to estimate BP from dynamic PET data without assumption of equilibrium. First-order kinetics can be assumed, when the study is performed with tracer dose (Mintun et al., 1984).

Estimation of Bmax and KD

Farde et al 1986 applied this transient equilibrium approach for the quantification of dopamine D2 receptors using PET imaging with [11C]raclopride. Assumption was that the concentration of free radioligand, Cfree can be obtained from a reference region, devoid of receptors, and the Cbound can be calculated by by simple subtraction from the total radioligand concentration inside the region of interest Croi (Cbound = Croi - Cfree). Several PET studies are then performed, with different ligand mass (range of specific activities). Cbound, estimated at the time of transient equilibrium, is plotted as function of the Cfree, measured at the same time point. This plot is the saturation curve, from which the estimates for Bmax and KD can be estimated either with nonlinear fitting (Farde et al., 1986), or with linear regression fitting of data transformed into Scatchard plot (Sedvall et al., 1986). Scatchard plot is considered outdated and Woolf-Hanes plot should be used instead, but nowadays nonlinear methods are preferred over linearization. In practise, usually only two PET studies were conducted, one with high and the other with low specific activity (low and high ligand mass) (Blomqvist et al., 1989; Persson et al., 1989). Assumption of equilibrium or tracer dose is not necessary, if dynamic data from the two studies is analyzed using kinetic model presented by Huang et al. (1986), and transformed into multilinear form allowing parameter estimation pixel-by-pixel (Blomqvist et al., 1989).

These methods are only possible with ligands that have reversible and relatively fast kinetics, and that can be safely administered at high masses, such as [11C]flumazenil and [11C]raclopride, but not with ligands that would show potentially dangerous pharmacological activity, such as [11C]carfentanil.

If binding is irreversible during PET scan, k3 can applied instead (Wong et al., 1986, 1997).

By the development of radiochemistry the specific activities of the produced radioligands are improved, and therefore calculation of binding potential directly from one PET study with high molar radioactivity has replaced the more elaborate Bmax/KD method in human studies. However, in small-animal studies the currently available molar activity levels may not be high enough to provide correct BP from a single study, and in those cases conducting the study with two injections with different specific radioactivities may be the only possibility to reliably estimate the binding parameters.

Scatchard plot

Scatchard plot has been used to transform the data into linear representation

to which line can be fitted and Bmax and KD estimated from the regression line parameters (Sedvall et al., 1986). Cbound/Cfree values are plotted on the y axis, and Cbound on the x axis. Bmax is estimated as the x axis intercept, and KD is -1/slope. Y axis intercept is Bmax/KD

However, the transformation leads to violation of the assumption of independence in the linear regression model (concentration of bound ligand is used in both x and y axes of the Scatchard plot). This distorts experimental error and can lead to misleading results. Therefore Scatchard plot should be replaced by Woolf-Hanes plot.

Woolf-Hanes plot

When k3/k4, or an estimate of it (usually BPND or Bound/free ratio at pseudo-equilibrium), is measured under (at least) two different ligand concentrations, the data can be plotted against the concentration of free ligand (radioligand + possible drug). The high ligand concentration can be achieved either by injecting the labeled ligand with large amount of cold ligand (low specific activity), or by giving another inhibitor, for instance an oral dose of a drug compound that binds to the same receptor.

Bmax can be calculated as 1/slope of the Woolf-Hanes plot, and KD as -(x intercept) or Bmax × (y intercept). Y axis intercept is 1/(binding potential) at zero occupancy. If binding potential on y axis is unitless (as usually is the case in PET), the units of both Bmax and KD are the same as the unit of free concentration on x axis.

Woolf-Hanes plot

The concentration of the free ligand in the tissue can be estimated from the radioactivity concentration in a reference region (receptor free tissue), divided by specific activity, when the labeled ligand and cold ligand (inhibitor) are injected together (Farde et al., 1986, 1987). If the first PET study is performed with tracer dose, the concentration of the free ligand can be assumed to be zero. In that case,

If the second PET scan is performed with tracer dose of PET radioligand and high dose of another inhibitor, the concentration of free ligand in scan 2 can be estimated based on the plasma concentration of the drug and the free tissue-to-plasma ratio of the drug. This ratio can be estimated from the same PET study as the distribution volume (VT) of the drug in a reference region (arterial plasma sampling is required), or if a receptor free region does not exist, as the distribution volume of an enantiomer of the drug that does not bind to the receptor. Estimates of the ratio can also be measured in ex vivo or in vitro human and animal studies (Wong et al., 1986, 1997). When the free ligand concentration is calculated as VT × ligand concentration in plasma, it must be noticed that both measures either have to be corrected for binding to plasma proteins, or if binding of PET radioligand and the drug is similar, both can be left uncorrected.

See also:


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Updated at: 2018-12-07
Created at: 2013-12-11
Written by: Vesa Oikonen