# Fitting the fractions of unchanged radiopharmaceutical in plasma

## Why to fit the fraction curves?

The chromatographic methods used in the metabolite analysis are slow, and hampered by the fast decay of radioactivity, especially with C-11 labelled radiopharmaceuticals. The fractions can often be determined only from sparse samples with increased uncertainty with time. Fitting of an empirical mathematical function to the fraction curves reduces variation and enables extrapolation, and may thus be required to achieve an acceptable metabolite correction.

Parent fraction fitting could also enable simple correction of *ex vivo*
radiopharmaceutical metabolism during blood sample preparation before the chromatographic methods
(Oikonen, 2014), but this approach must be validated for each radiopharmaceutical and metabolite
analysis protocol.

Fitted fractions can be applied in the metabolite correction using metabcor as usual.

## How to fit the fraction curves?

Compartmental models can be used to model
the appearance of metabolites in the plasma or blood
(Huang et al., 1991).
A variety of mathematical functions can be applied to different radiopharmaceuticals
(Tonietto et al., 2016).
For most radiopharmaceuticals fitting the sigmoidal
"Hill type" or
power functions can be recommended: in practise,
the curves of unchanged radiopharmaceutical fractions often do show a sigmoid shape, and could not
be described by declining exponential functions. This may be caused by
slow injection of radiopharmaceutical, or a redistribution phase of radiopharmaceutical from
an initial deposition to a highly perfused tissue, mainly lungs,
which may include specific binding to for instance serotonin transporters
(Suhara et al., 1998).
Metabolites should not appear in blood samples before the circulation time (normally 1 min) is
passed, unless radiopharmaceutical is metabolised in the lungs or by enzymes in blood
(Hinz et al., 2007).
Parent radiopharmaceutical fraction at *t=0* may be <1.0 also if the administered
radioligand is not 100% pure, which is common in preclinical animal studies.

Hill function may even work better than a compartment model (Wu et al., 2007).

Sigmoidal fraction curve can also be fitted using a function suggested by Sorger et al (2007):

In this function *f _{0}* is the initial fraction, which can be markedly lower than
one in early preclinical studies with poor radiochemical purity.

### Sum of exponentials

Declining exponential functions may be preferred in some cases, especially in small animal studies where circulation is fast and the initial 'shoulder' cannot be observed. For example, this function

, where 0<*a*≤1, *b*>0, and *c*>0, was used for
[^{11}C]PK11195
(Kropholler et al., 2005).
Program fit_fexp can be used to fit this function to
the measured fractions.

Simple two-exponential function with background was used in [^{18}F]fallypride
radiometabolite analysis study (Peyronneau et al., 2013):

Two-exponential function could be presented so that the relative weights of the exponentials are easily seen:

In brain receptor studies where a reference region, for example cerebellum, is available, the terminal rate of radiotracer washout from the reference region and the smallest elimination rate of constant of the total plasma curve can be used to constrain the second exponential of the two-exponential function for the parent fraction (Abi-Dargham et al., 1999).

Two-exponential function can even be used to fit sigmoidal data, as proposed by Blomqvist et al. (1990):

, where the parameters are interchangeable but *b _{1}* ≠

*b*.

_{2}*One-phase exponential* (monoexponential) function

, where *A _{0}* and

*A*represent the level at time 0 and at infinity, respectively, and

_{i}*k*represents the decay constant, may also be useful in fitting parent radiopharmaceutical fractions, and plasma-to-blood ratio data with certain radiopharmaceuticals. Program fit_fexp with option

`-mono`

can be
used to fit this function to the measured fractions.### Cumulative gamma distribution function

Naganawa et al (2014a,
2014b) fitted a function based on cumulative
(regularized) gamma distribution to plasma parent fraction data from [^{11}C]GR103545 and
[^{11}C]LY2795050 (radiotracers for κ opioid receptors) PET studies, but the function
would be applicable to most PET radiopharmaceuticals.
The function has four parameters (*a-d*), where *a* and *b* define the overall
and end level of the parent fraction, and *c* and *d* affect the shape of the gamma
distribution function.

Standard gamma distribution function has two parameters *x* and *α*,
gammadist(*x*, *α*). The proposed function for the plasma parent tracer
fractions as a function of time, *t*, is:

### Other functions

Tonietto et al (2015) validated a method where the bolus injection is modelled as a boxcar function, convoluted with power, Hill, or exponential function.

For certain radiopharmaceuticals, the fraction of non-metabolized parent radiopharmaceutical in
plasma is not approaching 1.0 at the injection time, but may even be increasing during the first
few minutes of the study. This has been shown for a radiopharmaceutical ([^{11}C]DASB)
binding to 5-HT transporters, possibly caused by transient trapping of parent radiopharmaceutical in
the lungs or GI system, while
the radioactive metabolite has no affinity for the 5-HT transporter
(Parsey et al., 2006).
A power-function-damped 2-exponential function

was shown to fit the metabolite data better and improve test-retest reproducibility (Parsey et al., 2006).

Another extension to power function was used by Hinz et al. (2007).

### Weighting

Fraction data are usually either not weighted, or weighted by 1 / sampling frequency to prevent overfitting the initial part with more frequent sampling. However, fractions could also be weighted based on count statistics (Tsujikawa et al., 2014).

### Function parameters

Function parameters are saved into specific fit file format, which are ASCII text files.

Program fit2dat can be used to calculate the fitted fraction curve for other purposes, e.g. for drawing graphs.

#### Population average of fractions

If the fractions of unchanged radiopharmaceutical in plasma or blood are very variable or measurements are missing for a few subjects, then a population based method should be considered. It may be useful to constrain one or more function parameters to population mean to reduce the number of blood samples for metabolite analysis.

## See also:

- Metabolite correction
- Fractions of unchanged radiotracer in plasma
- Hill function in plasma metabolite correction
- Power function in metabolite correction
- Compartmental models for plasma metabolite correction
- Fitting PET input curves
- Processing input data

## References

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Lammertsma AA, Hume SP, Bench CJ, Luthra SK, Osman S, Jones T. Measurement of monoamine oxidase B
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Oikonen VJ. Effect of tracer metabolism during sample preparation.
*Poster presentation in XIII Turku PET Symposium*, 24-27 May, 2014.
figshare.

Parsey RV, Ojha A, Ogden RT, Erlandsson K, Kumar D, Landgrebe M, Van Heertum R, Mann JJ.
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^{11}C-DASB and PET in humans. *J Nucl Med* 2006; 47: 1796-1802.

Tonietto M, Veronese M, Rizzo G, Zanotti-Fregonara P, Lohith TG, Fujita M, Zoghbi SS, Bertoldo A.
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in PET studies. *J Cereb Blood Flow Metab.* 2015; 35: 1462-1469.
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Tonietto M, Rizzo G, Veronese M, Fujita M, Zoghbi SS, Zanotti-Fregonara P, Bertoldo A.
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Watabe H, Channing MA, Der MG, Adams HR, Jagoda E, Herscovitch P, Eckelman WC, Carson RE.
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Wu S, Ogden RT, Mann JJ, Parsey RV. Optimal metabolite curve fitting for kinetic modeling of
^{11}C-WAY-100635. *J Nucl Med.* 2007; 48: 926-931.
doi: 10.2967/jnumed.106.038075.

Tags: Input function, Metabolite correction, Parent fraction, Fitting, NLLS, Extrapolation, Interpolation, Exponential function

Updated at: 2019-12-20

Created at: 2007-07-18

Written by: Vesa Oikonen