From Graphical Analysis to Multilinear Regression Analysis of Reversible Radioligand Binding

To the Editor:

The graphical method of Logan et al. (1990) allows calculation of binding potential (BP = B_max/K _d ) for radioligands that reversibly bind to receptors. This is accomplished by graphically analyzing regional time–activity measurements made with positron emission tomography (PET) or single photon emission computed tomography (SPECT) after a bolus injection of the ligand according to the two linear equations shown here:

$\frac{{\int }_0^t{C}_{\text{RC}}(t)dt}{C_{\text{RC}}(t)}=a\frac{{\int }_0^{t}f{C}_{\mathrm{p}}(t)dt}{C_{\text{RC}}(t)}+b$

(1)

$\frac{{\int }_0^t{C}_{\text{RF}}(t)dt}{C_{\text{RF}}(t)}=a^{\prime }\frac{{\int }_0^{t}f{C}_{\mathrm{p}}(t)dt}{C_{\text{RF}}(t)}+b^{\prime }$

(2)

for times in which secular equilibrium can be assumed between the local precursor pool and the receptor pool, where C_RC(t) and C_RF(t) represent time–activity measurements in the receptor-rich and reference (receptor-devoid) regions, respectively; C_p(t) is the total plasma ligand concentrations, with f representing the unmetabolized free fraction in plasma; and a,a′,b, and b′ are constants. Assuming that nonspecific binding of the ligand is negligible and applying three- and two-compartment models to the receptor-rich and reference regions, respectively, the slopes a and a′ are given by V₂ + V₃ + V_P and V*₂ + V_P, respectively, where V₂ and V₃ are the equilibrium distribution volumes of the ligand in the nondisplaceable and receptor compartments of the receptor-rich region, respectively; whereas V*₂ is the equilibrium distribution volume of the nondisplaceable compartment of the reference region and V_P is the plasma volume within the tissue. If we assume V₂ = and that V_P is negligible, then V₃ is given by a — a′. In experiments using tracer doses of ligand, BP or B_max/K_d is identical to V₃, which can also be expressed using kinetic parameters as K₁k₃/fk₂k₄.

Now, Eqs. 1 and 2 may be combined by eliminating ∫ ^t ₀ fC_p(T)dt as shown herein. First, rearrange Eq. 2:

${\int }_0^{t}f{C}_{\mathrm{p}}(t)dt=\frac{C_{\text{RF}}(t)}{a^{\prime }}\frac{{\int }_0^t{C}_{\text{RF}}(t)dt}{C_{\text{RF}}(t)}$

(3)

Insert Eq. 3 into 1:

$\begin{array}{l}\frac{{\int }_0^t{C}_{\text{RC}}(t)dt}{C_{\text{RC}}(t)}=\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(\frac{a}{a^{\prime }}\right)\frac{C_{\text{RF}}(t)\left[\frac{{\int }_0^t{C}_{\text{RF}}(t)dt}{C_{\text{RF}}(t)}-b^{\prime }\right]}{C_{\text{RC}}(t)}+b\hfill \end{array}$

(4)

Rearrange Eq. 4:

$\begin{array}{l}\frac{{\int }_0^t{C}_{\text{RC}}(t)dt}{C_{\text{RC}}(t)}=\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(\frac{a}{a^{\prime }}\right)\frac{{\int }_0^t{C}_{\text{RF}}(t)dt}{C_{\text{RC}}(t)}+\left(-\frac{a{b}^{\prime }}{a^{\prime }}\right)\frac{C_{\text{RF}}(t)}{C_{\text{RC}}(t)}+b\hfill \end{array}$

(5)

Equation 5 is a multilinear equation with partial regression coefficients a/a′, — ab′/a′, and b, and a/a′ is given by (V₂ + V₃ + V_P)/(V*₂ + V_P). Although this linear relationship cannot be graphically analyzed by using conventional graph plotting techniques, these coefficients can be obtained by multiple regression analysis. If we assume V₂ = V*₂ and neglect V_P as shown, then the ratio of V₃ to V₂ (volume ratio, R_v) is as follows:

$R_{\mathrm{V}}=\frac{V_3}{V_2}=\frac{BP}{V_2}=\frac{a}{a^{\prime }}-1$

(6)

R_w is also identical to k₃/k₄ because V₂ is given by K₁/fk₂. Thus, this variation of the graphical method that no longer employs a graphical analysis but a multilinear regression analysis allows calculation of the receptor parameter k₃/k₄ without requiring the knowledge of ligand concentrations in plasma. We recently employed this technique to obtain noninvasively, i.e., without blood sampling, the dopamine D2 receptor parameter using [¹²³I]iodobenzofuran and SPECT (Ichise et al., 1995). This technique may be of general interest because it should be applicable to other reversibly binding ligands in use with PET or SPECT for noninvasive quantification of this receptor parameter.