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A '''Logan plot''' (or '''Logan graphical analysis''')<ref name="Logan1990">{{cite journal | author=J. Logan, J.S. Fowler, N.D. Volkow, A.P. Wolf, S.L. Dewey, D.J. Schlyer, R.R. MacGregor, R. Hitzemann, B. Bendriem, S.J. Gatley, D.R. Christman | title=Graphical analysis of reversible radioligand binding from time-activity measurements applied to [N-<sup>11</sup>C-methyl]-(-)-cocaine PET studies in human subjects | journal=[[Journal of Cerebral Blood Flow and Metabolism]] | volume=10 | issue=5 | pages=740–747 | year=1990 | month=September | doi=10.1038/jcbfm.1990.127 | pmid=2384545}}</ref> is a [[Graph of a function|graphical]] analysis technique based on the [[Compartment (pharmacokinetics)|compartment]] model that uses [[linear regression]] to analyze [[pharmacokinetics]] of tracers involving reversible uptake. It is mainly used for the evaluation of [[nuclear medicine]] [[medical imaging|imaging]] data after the injection of a labeled ligand that binds reversibly to specific [[Receptor (biochemistry)|receptor]] or [[enzyme]].

In conventional [[Multi-compartment model|compartmental analysis]], an [[iterative method]] is used to fit the individual model parameters in the solution of a compartmental model of specific configuration to the measurements with a measured plasma time-activity curve that serves as an forcing (input) function, and the binding of the tracer can then be described. Graphical analysis is a simplified method that transforms the model equations into a linear equation evaluated at multiple time points and provides fewer parameters (i.e., slope and intercept). Although the slope and the intercept can be interpreted in terms of a combination of model parameters if a compartmental model configuration is assumed, the graphical methods are independent of any specific model configuration. In case of irreversible tracers, certain fraction of the radioactivity is trapped in the tissue or the binding site during the course of the experiment, whereas reversible tracers show uptake and loss from all compartments throughout the study. The theoretical foundation of graphical analysis for irreversible tracers (also called [[Patlak plot|Patlak graphical analysis]] or [[Patlak plot]]) was laid by [[Clifford Patlak]] and his colleagues<ref name="Patlak1983">{{cite journal | author=C.S. Patlak, R.G. Blasberg, J.D. Fenstermacher | title=Graphical evaluation of blood-to-brain transfer constants from multiple-time uptake data | journal=[[Journal of Cerebral Blood Flow and Metabolism]] | volume=3 | issue=1 | pages=1–7 | year=1983 | month=March | doi=10.1038/jcbfm.1983.1 | pmid=6822610}}</ref><ref name="Patlak1985">{{cite journal | author=C.S. Patlak, R.G. Blasberg | title=Graphical evaluation of blood-to-brain transfer constants from multiple-time uptake data. Generalizations | journal=[[Journal of Cerebral Blood Flow and Metabolism]] | volume=5 | issue=4 | pages=584–590 | year=1985 | month=April | doi=10.1038/jcbfm.1985.87 | pmid=4055928}}</ref> at [[NIH]]. Based on the original work of Patlak, [[Jean Logan]] and her colleagues<ref name="Logan1990" /> from [[Brookhaven National Laboratory]] extended the method to tracers with reversible kinetics.

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.<ref>{{cite book | author=K. Godfrey | title=Compartmental Models and Their Application | publisher=Academic Press, New York | year=1983}}</ref><ref>{{cite book | author=J.A. Jacquez | title=Compartmental Analysis in Biology and Medicine | publisher=The University of Michigan Press, Ann Arbor | edition=2nd | year=1985}}</ref> The time course of the activity in the multicompartmental system driven by a metabolite-corrected plasma input function <math>C_p(t)</math> can be described by:

: <math>\frac{d\mathbf{A}}{dt} = \mathbf{KA} + \mathbf{Q}C_p(t)</math>

where <math>\mathbf{A}</math> is a column vector of activity concentration for each compartment at time <math>t</math>, <math>\mathbf{K}</math> is the matrix of the transfer constants between compartments, and <math>\mathbf{Q}</math> is the vector of plasma-to-tissue transfer constants. Patlak and Blasberg<ref name="Patlak1985" /> showed that the above equation can be written as:

: <math>\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}</math>

where <math>\mathbf{U}_n^T</math> represents a row vector of 1s and <math>A(t) = \mathbf{U}_n^T \mathbf{A}</math>. The total activity in the [[region of interest]], <math>\mathrm{ROI}(t)</math>, is a combination of radioactivities from all compartments plus a plasma volume fraction (<math>V_p</math>)<ref name="Patlak1983" /> and thus:

: <math>\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}</math>

By [[division (mathematics)|dividing]] both sides by <math>\mathrm{ROI}(t)</math>, one obtains the following linear equation:

: <math>{{\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}}</math>

For <math>t > t'</math>, Patlak and his colleagues<ref name="Patlak1983" /> showed that <math>\mathbf{A} = -\mathbf{K}^{-1} \mathbf{Q} C_p(t)</math>, 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 <math>{{\int_{0}^{t} C_p(\tau)d\tau} \over \mathrm{ROI}(t)}</math> versus <math>{{\int_{0}^{t} \mathrm{ROI}(\tau)d\tau} \over \mathrm{ROI}(t)}</math> becomes a straight line with slope <math>(-\mathbf{U}_n^T \mathbf{K}^{-1} \mathbf{Q} + V_p)</math> and intercept <math>{{\mathbf{U}_n^T \mathbf{K}^{-1} \mathbf{A}} \over {\mathbf{U}_n^T \mathbf{A} + V_p C_p}}</math>.<ref name="Logan1990" />

For a catenary two-tissue compartment model with transfer constants <math>K_1</math> (forward transport from plasma to tissue), <math>k_2</math> (reverse transport from tissue to plasma), <math>k_3</math> ([[Binding (molecular)|binding]] parameter proportional to <math>B_\max k_\mathrm{on}</math>), and <math>k_4</math> (dissociation constant) to analyze enzyme or receptor system, the slope represents the total [[distribution volume]] (<math>V_d</math>) and is given by <math>\frac{K_1}{k_2}(1+\frac{k_3}{k_4}) + V_p</math>,<ref name="Logan1990" /> where <math>k_3 = B_\max k_\mathrm{on}</math>, <math>k_4 = k_\mathrm{off}</math>, <math>\frac{k_3}{k_4} = \frac{B_\max}{K_d}</math>, and <math>K_d = k_\mathrm{off}/k_\mathrm{on}</math>, in which <math>B_\max</math> is the concentration of ligand binding sites, <math>K_d</math> is the equilibrium [[dissociation constant]] for the ligand-binding site complex, <math>k_\mathrm{on}</math> is the ligand-binding association constant, <math>k_\mathrm{off}</math> is the ligand-binding dissociation constant. For a one-tissue compartment model with transfer constants <math>K_1</math> and <math>k_2</math>, the slope is <math>\lambda + V_p</math>, where <math>\lambda</math> is the [[partition coefficient]] (<math>{K_1}/{k_2}</math>) and the intercept is <math>\frac{-1}{k_2(1+V_p/\lambda)}</math>.<ref name="Logan1990" />

== See also ==
* [[Patlak plot]]
* [[Multi-compartment model]]
* [[Positron emission tomography]]
* [[Binding potential]]
* [[Distribution volume]]

== References ==
{{Reflist}}

[[Category:Mathematical modeling]]
[[Category:Systems theory]]
[[Category:Plots (graphics)]]

Electrocommunication - Revision history

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