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:

$\begin{equation} f_{p}(t) = \left\{ \begin{array}{l l} d & \quad \text{if } t \leq e\\ [d^{-\frac{1}{c}} + (a(t-e))^b ]^{-c} & \quad \text{if } t > e \end{array} \right. \end{equation}$

, 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.

$Power function for parent fraction$

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:

$\begin{equation} t = e \: + \: \frac{1}{a} \: \sqrt[b]{ {f_p}^{-\frac{1}{c}} - {d}^{-\frac{1}{c}} } \end{equation}$

The derivative of the extended power function is

$\begin{equation} \frac{d}{dt} f_{p}(t) = \left\{ \begin{array}{l l} 0 & \quad \text{if } t \leq e\\ - \frac{b c (a (t-e))^{b}}{t-e} \: (d^{-\frac{1}{c}} + (a (t-e))^{b})^{-c-1} & \quad \text{if } t > e \end{array} \right. \end{equation}$

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 [¹¹C]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 A₁ adenosine receptors in humans using [¹⁸F]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 ([¹¹C]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

Power functions in plasma metabolite correction

See also:

References:

Acknowledgements: