# Hill function in plasma metabolite correction

Hill type functions have been often used to fit the fractions of parent tracer in plasma.

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

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

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

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

.

If parameter *d* is set to 1 and *e* to 0, we have the
traditional Hill type equation, which always starts from 1, but can still stop
decreasing at a certain level (*1-a*) higher than 0:

Time when half of the isotope label carrying compounds in plasma are
metabolites
(*t _{½}*) can then be calculated from equation:

Both parent fractions and two metabolite fractions can be fitted simultaneously with fith2met, assuming that the a single Hill function shape can fit all the fractions.

Jucaite et al (2015) used a modified Hill function, mixture of the Hill and Richards equations (Giraldo, 2003; Bindslev, 2008). Guo et al (2013) used another modification:

If measured parent tracer fractions do not seem to reach a steady level during the measurement, or extrapolation is needed and you are not comfortable with the assumption of steady final level, then the power function should be applied instead.

## References:

Bindslev N. Hill in hell. In: *Drug-acceptor interactions.* Co-Action
Publishing; 2008. p. 257–82.

Giraldo J. Empirical models and Hill coefficients.
*Trend Pharmacol Sci.* 2003; 24(2): 63-65.

Guo Q, Colasanti A, Owen DR, Onega M, Kamalakaran A, Bennacef I, Matthews PM,
Rabiner EA, Turkheimer FE, Gunn RN. Quantification of the specific translocator
protein signal of ^{18}F-PBR111 in healthy humans: a genetic
polymorphism effect on in vivo binding.
*J Nucl Med.* 2013; 54: 1915-1923.

Jucaite A, Svenningsson P, Rinne JO, Cselényi Z, Varnäs K, Johnström P,
Amini N, Kirjavainen A, Helin S, Minkwitz M, Kugler AR, Posener JA, Budd S,
Halldin C, Varrone A, Farde L. Effect of the myeloperoxidase inhibitor AZD3241
on microglia: a PET study in Parkinson’s disease.
*Brain* 2015; 138(Pt 9): 2687-2700.

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.

Tags: Metabolite correction, Parent fraction, Fitting

Created at: 2007-07-18

Updated at: 2016-04-08

Written by: Vesa Oikonen