# Fitting the fractions of unchanged tracer 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 labeled tracers. 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* tracer metabolism
during blood sample preparation before the chromatographic methods (Oikonen, 2014), but this
approach must be validated for each tracer and metabolite analysis protocol.

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

## How to fit the fraction curves?

### Functions

Different functions can be applied to different tracers
(Tonietto et al., 2016).
For most tracers fitting the sigmoidal
“Hill type” or
power functions can be recommended: in practise,
the curves of unchanged tracer fractions often do show a sigmoid shape, and could not be described
by declining exponential functions. This may be caused by slow injection of tracer, or a
redistribution phase of tracer from an initial deposition to a highly perfused tissue, e.g. lungs
(Suhara et al., 1998).
Metabolites should not appear in blood samples before the circulation time (normally 1 min) is
passed, unless tracer is metabolized in the lungs or by enzymes in blood.
Parent tracer fraction at *t=0* may be <1.0 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).

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

, where *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 tracer fractions, and plasma-to-blood ratio data with certain tracers.

#### 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 (tracers for κ opioid receptors) PET studies, but the function
would be applicable to most PET tracers.
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 tracers, the fraction of non-metabolized parent tracer 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 tracer ([^{11}C]DASB) binding to 5-HT transporters,
possibly caused by transient trapping of parent tracer in the lungs, 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 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 tracer in plasma or blood are very variable or measurements are missing for a few subjects, then a population based method should be considered.

## See also:

## References

Carson RE, Breier A, de Bartolomeis A, Saunders RC, Su TP, Schmall B, Der MG, Pickar D,
Eckelman WC. Quantification of amphetamine-induced changes in [^{11}C]raclopride binding
with continuous infusion. *J Cereb Blood Flow Metab.* 1997; 17(4): 437–447. doi:
10.1097/00004647-199704000-00009.

Meyer PT, Bier D, Holschbach MH, Boy C, Olsson RA, Coenen HH, Zilles K, Bauer A.
Quantification of cerebral A_{1} adenosine receptors in humans using [^{18}F]CPFPX
and PET. *J Cereb Blood Flow Metab.* 2004; 24(3): 323-333. doi: 10.1097/01.WCB.0000110531.48786.9D.

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.
Metabolite considerations in the in vivo quantification of serotonin transporters using
^{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.
Improved models for plasma radiometabolite correction and their impact on kinetic quantification
in PET studies. *J Cereb Blood Flow Metab.* 2015; 35: 1462-1469.
doi: 10.1038/jcbfm.2015.61.

Tonietto M, Rizzo G, Veronese M, Fujita M, Zoghbi SS, Zanotti-Fregonara P, Bertoldo A.
Plasma radiometabolite correction in dynamic PET studies: insights on the available modeling
approaches. *J Cereb Blood Flow Metab.* 2016; 36(2): 326-339. doi:
10.1177/0271678X15610585.

Watabe H, Channing MA, Der MG, Adams HR, Jagoda E, Herscovitch P, Eckelman WC, Carson RE.
Kinetic analysis of the 5-HT_{2A} ligand ([^{11}C]MDL 100,907.
*J Cereb Blood Flow Metab.* 2000; 20: 899-909. doi:
10.1097/00004647-200006000-00002.

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

Created at: 2007-07-18

Updated at: 2018-10-10

Written by: Vesa Oikonen