# 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, *AUC _{0-∞}*, can be used to estimate
the total clearance of radiopharmaceuticals,

*CL*.

_{T}*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,
*AUC _{0-∞}*, 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, *C _{0}*, 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 *C _{0} = A+B*. This function can be
fitted to the concentration data for example using program
fit_dexp.

*C*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 (

_{0}*PV*) (Polidori & Rowley, 2014):

As an example, back-extrapolation using exponential function fitting method is applied to
a plasma curve from a [^{18}F]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 [^{18}F]FTHA
studies to estimate the total body blood volume Figure 2.

Based on the fitted parameters of the bi-exponential decay model, *AUC _{0-∞}*
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 *AUC _{0-∞}* or

*AUC*can be computed using program fit2auc from parameters fitted to data with program fit_dexp.

_{t-∞}##
*CL*_{T} - Total clearance

_{T}

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

The estimation of *AUC _{0-∞}* 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 *AUC _{0-∞}*, 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 *AUC _{0-∞}* is calculated from the
SUV curve, then

*CL*is calculated as

_{T}, and the outcome has unit (mL of plasma)*min^{-1}*(animal weight (g))^{-1}.
If *AUC _{0-∞}* 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).

##
*k*_{el} - Elimination rate constant

_{el}

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 *-k _{el}*.
The

*k*can be estimated e.g. using program paucinf, or with Excel (Zhang et al., 2010)

_{el}##
*t*_{1/2} - Half-life of the PET
radiopharmaceutical

_{1/2}

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 *t _{1/2}* has nothing in common with the physical half-life of
positron emitting isotope labels (

*T*), 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.

_{1/2}##
*C*_{max} and *T*_{max}

_{max}

_{max}

In principle, the maximal concentration of PET tracer in plasma (*C _{max}*) and
the time of maximum concentration (

*T*) 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

_{max}*k*or

_{el}*t*.

_{1/2}## 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 *V _{c}* is the distribution volume of the central compartment
(Veng-Pedersen, 1989b).

## See also:

- Input function preprocessing
- Metabolite correction
- Fitting the input TAC
- Area under curve (AUC)
- Pharmacokinetic one-, two-, and three-compartment model
- Urinary clearance
- Renal clearance
- Receptor occupancy

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Tags: Input function, Plasma, Pharmacokinetics, Fitting, AUC, Clearance, Extrapolation, Exponential function, Biexponential, Transit time

Updated at: 2023-01-13

Created at: 2006-03-21

Written by: Vesa Oikonen, Anne Roivainen