Patlak plot

A Patlak plot (sometimes called Gjedde–Patlak plot, Patlak–Rutland plot, or Patlak analysis){{cite journal |author1=C. S. Patlak |author2=R. G. Blasberg |author3=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 |date=March 1983 | doi=10.1038/jcbfm.1983.1 | pmid=6822610| doi-access=free }}{{cite journal |author1=C.S. Patlak |author2=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 |date=April 1985 | doi=10.1038/jcbfm.1985.87 | pmid=4055928| doi-access=free }} is a graphical analysis technique based on the compartment model that uses linear regression to identify and analyze pharmacokinetics of tracers involving irreversible uptake, such as in the case of deoxyglucose.{{cite journal | author=A. Gjedde | title=High- and low-affinity transport of D-glucose from blood to brain | journal=Journal of Neurochemistry | volume=36 | issue=4 | pages=1463–1471 |date=April 1981 | doi=10.1111/j.1471-4159.1981.tb00587.x| pmid=7264642 }}{{cite journal | author=A. Gjedde | title=Calculation of glucose phosphorylation from brain uptake of glucose analogs in vivo: A re-examination | journal=Brain Research Reviews | volume=4 | issue=2 | pages=237–274 |date=June 1982 | doi=10.1016/0165-0173(82)90018-2| pmid=7104768 }} It is used for the evaluation of nuclear medicine imaging data after the injection of a radioopaque or radioactive tracer.

The method is model-independent because it does not depend on any specific compartmental model configuration for the tracer, and the minimal assumption is that the behavior of the tracer can be approximated by two compartments – a "central" (or reversible) compartment that is in rapid equilibrium with plasma, and a "peripheral" (or irreversible) compartment, where tracer enters without ever leaving during the time of the measurements. The amount of tracer in the region of interest is accumulating according to the equation:

: $R(t)\; =\; K\; \backslash int\_0^t\; C\_p(\backslash tau)\; \backslash ,\; d\backslash tau\; +\; V\_0\; C\_p(t)$

where $t$ represents time after tracer injection, $R(t)$ is the amount of tracer in region of interest, $C\_p(t)$ is the concentration of tracer in plasma or blood, $K$ is the clearance determining the rate of entry into the peripheral (irreversible) compartment, and $V\_0$ is the distribution volume of the tracer in the central compartment. The first term of the right-hand side represents tracer in the peripheral compartment, and the second term tracer in the central compartment.

By dividing both sides by $C\_p(t)$, one obtains:

: $\{R(t)\; \backslash over\; C\_p(t)\}\; =\; K\; \{\backslash int\_0^t\; C\_p(\backslash tau)\; \backslash ,\; d\backslash tau\; \backslash over\; C\_p(t)\}\; +\; V\_0$

The unknown constants $K$ and $V\_0$ can be obtained by linear regression from a graph of $\{R(t)\; \backslash over\; C\_p(t)\}$ against $\backslash int\_0^t\; C\_p(\backslash tau)\; \backslash ,\; d\backslash tau\; /\; C\_p(t)$.