# Power functions in plasma metabolite correction

Power functions have been used to fit the fractions of parent radiopharmaceutical in plasma.

An extended power function can be used when the parent radiopharmaceutical fraction,
*f _{p}*, is initially 1 or less then 1 (in some cases radiopharmaceutical is
metabolized already in vasculature before reaching sampling site), and then approaches 0:

, where *0 < a ≤ 1*, *b > 1*, *c > 0*,
*0 < d ≤ 1*, and *e ≥ 0*.

Parameter *d* represents the initial level of parent fraction, and parameter *e*
is the time after which the fraction starts to decrease.

Unchanged (parent) radiopharmaceutical fraction curves can be fitted with
fit_ppf with option `-model=PF`

.

If parameter *d* is set to 1, we have essentially the same function as was proposed by
Meyer et al. (2004), and
if additionally parameter *b* is set to 2 and delay parameter *e = 0*,
the function is the same as used by
Watabe et al. (2000).

If you prefer a function that approaches a steady level above 0, then use the Hill functions.

The x axis value *t* for a known *f _{p}* can be solved from the extended
power function, giving equation:

The derivative of the extended power function is

## See also:

- Hill function in plasma metabolite correction
- Metabolite correction
- Fractions of unchanged radiotracer in plasma
- Fitting PET input curves

## References:

Hinz R, Bhagwagar Z, Cowen PJ, Cunningham VJ, Grasby PM. Validation of a tracer kinetic model for
the quantification of 5-HT_{2A} receptors in human brain with [^{11}C]MDL 100,907.
*J Cereb Blood Flow Metab.* 2007; 27: 161-172.
doi: 10.1038/sj.jcbfm.9600323.

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.

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.

## Acknowledgements:

Thanks to Matteo Tonietto for notifying of a typo in the equation.

Tags: Metabolite correction, Parent fraction, Fitting

Updated at: 2018-12-18

Created at: 2007-07-18

Written by: Vesa Oikonen