# Pharmacokinetics from PET plasma curves

This page covers the pharmacokinetics (PK) of PET radiopharmaceutical in the plasma, not in tissue, and not the pharmacokinetics or pharmacodynamics (PD) of the drug that is studied with PET radiopharmaceutical.

## AUC - Area-under-curve

AUC is the area under the plot (integral) of radioactivity concentration of authentic (metabolite corrected) PET tracer against time after tracer infusion. The unit of AUC is the unit of time multiplied by the unit of radioactivity concentration, usually min*kBq/mL. The AUC from 0 to infinite time, AUC0-∞, can be used to estimate the total clearance of radiopharmaceuticals, CLT.

AUC can be determined by "trapezoidal rule": the data points are connected by straight line segments, perpendiculars are erected from the abscissa to each data point, and the sum of the areas of the trapezoids is computed. This can be done e.g. using program interpol with option -i. However, the last measured concentration is usually not zero, sometimes not even close to zero. Then the AUC from 0 to infinite time, AUC0-∞, must be estimated using the slope of the end-part of the plot of the natural logarithm of tracer concentration against time. This can be done e.g. using program paucinf, or with Excel (Zhang et al., 2010), if dedicated software for pharmacokinetic analysis is not available.

If blood sampling is not started right after i.v. injection, the plasma peak will be missed, leading to uncertainties in estimation of the AUC. This has traditionally been the case in (non-PET) PK studies and measurement of GFR. In that case the initial concentration, C0, can be estimated for instance by back-extrapolating to t=0 from log-linear regression of the two first concentration values collected after the tracer or indicator has distributed in the total blood volume of the body. Small-molecular compounds usually have two phases (fast and slow) in the decrease of the plasma concentration curve (after the initial mixing phase), and bi-exponential decay model is then used to describe the plasma concentration as a function of time:

, where C0 = A+B. This function can be fitted to the concentration data for example using program fit_dexp. C0 has been used to estimate the initial distribution volume of the tracer (Iozzo et al., 2006), or, with suitable tracer or indicator, the total body plasma volume (PV) (Polidori & Rowley, 2014):

As an example, back-extrapolation using exponential function fitting method is applied to a plasma curve from a [18F]FDG study in Figure 1. The bias introduced by extrapolation to zero time in PK bolus studies ("mixing error") was noted by Sapirstein et al (1952, 1955). PK studies are traditionally also based on venous blood sampling instead of arterial sampling, which introduces another error. Since the arteriovenous difference is often small at later time points, the back-extrapolation method based on those late samples, the mixing error (or "bolus error") tends to become negligible (Russell & Dubovsky, 1999).

As another example, back-extrapolation is applied to blood curves from [18F]FTHA studies to estimate the total body blood volume Figure 2.

Based on the fitted parameters of the bi-exponential decay model, AUC0-∞ can be calculated as

, based on the definite integrals of decaying exponential functions. Alternatively, if initial part of AUC is calculated from the measured data points, the late part of AUC (after time t) can be calculated from equation

(assuming that α > 0 and β > 0 ). The AUC0-∞ or AUCt-∞ can be computed using program fit2auc from parameters fitted to data with program fit_dexp.

## CLT - Total clearance

The total clearance of PET radiopharmaceutical after a single intravenous dose can be calculated as

The estimation of AUC0-∞ was explained above.

The unit of clearance depends on the unit of dose, but in principle the unit of radioactivity is cancelled out, and thus the unit of clearance will be volume/unit time. Both of the measures, patient dose and AUC0-∞, must be specified either as radioactivity (Bq, Ci) or as moles or mass (mol, g). The specific radioactivity (SA, mCi/µmol or MBq/µmol) can be used to radioactivity-mass conversion, if necessary.

In mice and rat studies each animal is often sampled only once, and plasma curve must be constructed from separate animals. Variable injected dose and animal weight must then be taken into account by using SUV or %ID/g as concentration. When AUC0-∞ is calculated from the SUV curve, then CLT is calculated as

, and the outcome has unit (mL of plasma)*min-1*(animal weight (g))-1. If AUC0-∞ is calculated from %ID/g curves, then

, and its unit is normal (g or mL of plasma)*min-1.

Another possibility is to fit a sum of three exponentials to the (protein-free) plasma data and calculate the clearance from the fitted parameters (Abi-Dargham et al., 1994).

## kel - Elimination rate constant

When the natural logarithm of tracer concentration is plotted against time from bolus infusion, the plot becomes linear in the end phase, as the tracer is eliminated according to the laws of first-order reaction kinetics; linearity should be verified from the plot. The slope of the linear part of the plot equals -kel. The kel can be estimated e.g. using program paucinf, or with Excel (Zhang et al., 2010)

## t1/2 - Half-life of the PET radiopharmaceutical

This is the amount of time required for the concentration of the PET radiopharmaceutical in plasma to be halved. If the radiopharmaceutical is eliminated from the plasma according to the laws of first-order kinetics, it can be calculated as

Notice that this t1/2 has nothing in common with the physical half-life of positron emitting isotope labels (T1/2), except the name. For pharmacokinetic calculations all curves are corrected for decay, so that the curves represent the concentrations of PET radiopharmaceuticals, and not concentrations of the radioactive label.

## Cmax and Tmax

In principle, the maximal concentration of PET tracer in plasma (Cmax) and the time of maximum concentration (Tmax) could be determined from plasma TAC directly, or after fitting a function to plasma TAC. However, these parameters would be dependent on the injected dose and the bolus infusion protocol, and thus less useful than kel or t1/2.

## Mean residence time (MRT) and Mean transit time (MTT)

Mean residence time (MRT) and mean transit time (MTT) are frequently used in pharmacokinetic literature, including system moment and system matrix MRTs, and MRTs of individual compartments such as central/plasma compartment (which is discussed here), tissue compartments, and the MRT of an absorption site (Wagner, 1988). By definition, MRT of drug molecules in a kinetic space is the average total time the drug molecules injected into a kinetic system at a given point spend in the kinetic space; MTT of drug molecules in a kinetic space is the average time taken by drug molecules injected into the kinetic system at a given point to leave the kinetic space after first and possible subsequent entries into that space (Veng-Pedersen, 1989a). In PET studies, MTT is sometimes estimated in tissue perfusion studies, for example as the inverse of cerebral perfusion pressure.

Based on a number of (often overlooked) assumptions, MRT, in the case of intravenous bolus injection, is often estimated using formula

, where AUMC is the area under the moment curve, and AUC is the area under the concentration-time curve. The limitations and assumptions behind this formula are reviewed by Kasuya et al. (1987) and Veng-Pedersen (1989a). An additional problem is that MRT is often calculated from peripheral venous data, and is then dependent on the sampling site (Chiou, 1989a).

If the blood curve is corrected for the recirculation, then the equation for MRT can be used to calculate the mean arrival time of the peak. For example, if these are calculated from blood pools in heart RV and LV cavities, then cardiopulmonary transit time can be calculated as their difference (Peters & Myers, 1998).

In PET studies MTT is often calculated from tissue concentration curves, which can be measured only limited time, and input function is delayed. Corrections for these can be included in the equations (Choi et al., 1993).

MTT is related to the clearance. For the central (plasma) compartment,

, where Vc is the distribution volume of the central compartment (Veng-Pedersen, 1989b).

## Literature

Bourne DWA: Mathematical Modeling of Pharmacokinetic Data. CRC Press, 1995. ISBN 1-56676-204-9.

Debruyne D, Abadie P, Barke L, Albessard F, Moulin M, Zarifian E, Baron JC. Plasma pharmacokinetics and metabolism of the benzodiazepine antagonist [11C] Ro 15-1788 (flumazenil) in baboon and human during positron emission tomography studies. Eur J Drug Metab Pharmacokinetics. 1991; 16: 141-152. doi: 10.1007/BF03189951.

Nosslin B. Determination of clearance and distribution volume with the single injection technique. Acta Med Scand. 1965; 442(suppl.): 97-101. doi: 10.1111/j.0954-6820.1965.tb02318.x.

Peters M, Myers MJ: Physiological Measurements With Radionuclides in Clinical Practice, 3rd ed., Oxford University Press, 1998. ISBN: 978-0192619945.

Qin S, Fite BZ, Gagnon KJ, Seo JW, Curry F-R, Thorsen F, Ferrara KW. A physiological perspective on the use of imaging to assess the in vivo delivery of therapeutics. Ann Biomed Eng. 2014; 42(2): 280-298. doi: 10.1007/s10439-013-0895-2.

Rescigno A. Clearance, turnover time, and volume of distribution. Pharmacol Res. 1997a; 35(3): 189-193. doi: 10.1006/phrs.1997.0131.

Rescigno A. Fundamental concepts in pharmacokinetics. Pharmacol Res. 1997b; 35(5): 363-390. doi: 10.1006/phrs.1997.0175.

van Rij CM, Huitema ADR, Swart EL, Greuter HNJM, Lammertsma AA, van Loenen AC, Franssen EJF. Population plasma pharmacokinetics of 11C-flumazenil at tracer concentrations. Br J Clin Pharmacol. 2005; 60(5): 477-485. doi: 10.1111/j.1365-2125.2005.02487.x.

Saleem A, Aboagye EO, Matthews JC, Price PM. Plasma pharmacokinetic evaluation of cytotoxic agents radiolabelled with positron emitting radioisotopes. Cancer Chemother Pharmacol. 2008; 61: 865-873. doi: 10.1007/s00280-007-0552-2.

Sestini S, Halldin C, Mansi L, Castagnoli A, Farde L. Pharmacokinetic analysis of plasma curves obtained after i.v. injection of the PET radioligand [11C] raclopride provides likely explanation for rapid radioligand metabolism. J Cell Physiol. 2012; 227: 1663-1669. doi: 10.1002/jcp.22890.

Strand SE, Zanzonico P, Johnson TK. Pharmacokinetic modeling. Med Phys. 1993; 20(2 Pt 2): 515-527. doi: 10.1118/1.597047.

Toutain PL, Mélou-Bousquet A. Plasma terminal half-life. J vet Pharmacol Therap. 2004; 27: 427-439. doi: 10.1111/j.1365-2885.2004.00600.x.

Urso R, Blardi P, Giorgi G. A short introduction to pharmacokinetics. Eur Rev Med Pharmacol Sci. 2002; 6: 33-44. PMID: 12708608.

van der Veldt AAM, Lammertsma AA. In vivo imaging as a pharmacodynamic marker. Clin Cancer Res. 2014; 20(10): 2569-2577. doi: 10.1158/1078-0432.CCR-13-2666.

Zhang Y, Huo M, Zhou J, Xie S. PKSolver: An add-in program for pharmacokinetic and pharmacodynamic data analysis in Microsoft Excel. Comput Methods Programs Biomed. 2010; 99: 306-314. doi: 10.1016/j.cmpb.2010.01.007.

Updated at: 2023-01-13
Created at: 2006-03-21
Written by: Vesa Oikonen, Anne Roivainen