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, fp, 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
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 (t½) 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.
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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 18F-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 11C-WAY-100635. J Nucl Med. 2007; 48: 926-931.
Created at: 2007-07-18
Updated at: 2016-04-08
Written by: Vesa Oikonen