Logan plot

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 ${\displaystyle C_{p}(t)}$ can be described by:

${\displaystyle {\frac {d\mathbf {A} }{dt}}=\mathbf {KA} +\mathbf {Q} C_{p}(t)}$

where ${\displaystyle \mathbf {A} }$ is a column vector of activity concentration for each compartment at time ${\displaystyle t}$, ${\displaystyle \mathbf {K} }$ is the matrix of the transfer constants between compartments, and ${\displaystyle \mathbf {Q} }$ is the vector of plasma-to-tissue transfer constants. Patlak and Blasberg^[3] showed that the above equation can be written as:

${\displaystyle \int _{0}^{t}A(\tau )\,d\tau =-\mathbf {U} _{n}^{T}\mathbf {K} ^{-1}\mathbf {Q} \int _{0}^{t}C_{p}(\tau )\,d\tau +\mathbf {U} _{n}^{T}\mathbf {K} ^{-1}\mathbf {A} }$

where ${\displaystyle \mathbf {U} _{n}^{T}}$ represents a row vector of 1s and ${\displaystyle A(t)=\mathbf {U} _{n}^{T}\mathbf {A} }$. The total activity in the region of interest, ${\displaystyle \mathrm {ROI} (t)}$, is a combination of radioactivities from all compartments plus a plasma volume fraction (${\displaystyle V_{p}}$)^[2] and thus:

${\displaystyle \int _{0}^{t}\mathrm {ROI} (\tau )\,d\tau =(-\mathbf {U} _{n}^{T}\mathbf {K} ^{-1}\mathbf {Q} +V_{p})\int _{0}^{t}C_{p}(\tau )\,d\tau +\mathbf {U} _{n}^{T}\mathbf {K} ^{-1}\mathbf {A} }$

By dividing both sides by ${\displaystyle \mathrm {ROI} (t)}$, one obtains the following linear equation:

${\displaystyle {{\int _{0}^{t}\mathrm {ROI} (\tau )\,d\tau } \over \mathrm {ROI} (t)}=(-\mathbf {U} _{n}^{T}\mathbf {K} ^{-1}\mathbf {Q} +V_{p}){{\int _{0}^{t}C_{p}(\tau )\,d\tau } \over \mathrm {ROI} (t)}+{{\mathbf {U} _{n}^{T}\mathbf {K} ^{-1}\mathbf {A} } \over {\mathbf {U} _{n}^{T}\mathbf {A} +V_{p}C_{p}}}}$

For ${\displaystyle t>t'}$, Patlak and his colleagues^[2] showed that ${\displaystyle \mathbf {A} =-\mathbf {K} ^{-1}\mathbf {Q} C_{p}(t)}$, 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 ${\displaystyle {{\int _{0}^{t}C_{p}(\tau )d\tau } \over \mathrm {ROI} (t)}}$ versus ${\displaystyle {{\int _{0}^{t}\mathrm {ROI} (\tau )d\tau } \over \mathrm {ROI} (t)}}$ becomes a straight line with slope ${\displaystyle (-\mathbf {U} _{n}^{T}\mathbf {K} ^{-1}\mathbf {Q} +V_{p})}$ and intercept ${\displaystyle {{\mathbf {U} _{n}^{T}\mathbf {K} ^{-1}\mathbf {A} } \over {\mathbf {U} _{n}^{T}\mathbf {A} +V_{p}C_{p}}}}$.^[1]

For a catenary two-tissue compartment model with transfer constants ${\displaystyle K_{1}}$ (forward transport from plasma to tissue), ${\displaystyle k_{2}}$ (reverse transport from tissue to plasma), ${\displaystyle k_{3}}$ (binding parameter proportional to ${\displaystyle B_{\max }k_{\mathrm {on} }}$), and ${\displaystyle k_{4}}$ (dissociation constant) to analyze enzyme or receptor system, the slope represents the total distribution volume (${\displaystyle V_{d}}$) and is given by ${\displaystyle {\frac {K_{1}}{k_{2}}}(1+{\frac {k_{3}}{k_{4}}})+V_{p}}$,^[1] where ${\displaystyle k_{3}=B_{\max }k_{\mathrm {on} }}$, ${\displaystyle k_{4}=k_{\mathrm {off} }}$, ${\displaystyle {\frac {k_{3}}{k_{4}}}={\frac {B_{\max }}{K_{d}}}}$, and ${\displaystyle K_{d}=k_{\mathrm {off} }/k_{\mathrm {on} }}$, in which ${\displaystyle B_{\max }}$ is the concentration of ligand binding sites, ${\displaystyle K_{d}}$ is the equilibrium dissociation constant for the ligand-binding site complex, ${\displaystyle k_{\mathrm {on} }}$ is the ligand-binding association constant, ${\displaystyle k_{\mathrm {off} }}$ is the ligand-binding dissociation constant. For a one-tissue compartment model with transfer constants ${\displaystyle K_{1}}$ and ${\displaystyle k_{2}}$, the slope is ${\displaystyle \lambda +V_{p}}$, where ${\displaystyle \lambda }$ is the partition coefficient (${\displaystyle {K_{1}}/{k_{2}}}$) and the intercept is ${\displaystyle {\frac {-1}{k_{2}(1+V_{p}/\lambda )}}}$.^[1]

• Patlak plot
• Multi-compartment model
• Positron emission tomography
• Binding potential
• Distribution volume