Hill Function

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Hill Function

The sequential steps for converting blood measurements to an input curve are explained in a dedicated section. The current parent fraction model serves for converting a plasma activity curve into an input curve (authentic, unchanged tracer in plasma).

The Hill model assumes that the concentration ratio of authentic tracer to total tracer in plasma has been determined at certain times during the acquisition and loaded with the Load Fraction entry in the menu.


Operational Model Curve

The Hill model applied by Gunn et al [1] for modeling of the parent fraction has the following functional form


whereby the calculated fraction is multiplied with fp, the free fraction of authentic tracer in plasma, i.e. the fraction of tracer not bound to plasma proteins.

The fraction calculation assumes negligible metabolites before a Begin time, and therefore remains constant at 1. From Begin on it decreases smoothly.


Parameter Fitting

The free fraction fp is an input parameter which has no impact on fitting the Hill curve to the fraction data. The default for fp is 1 as it is experimentally difficult to measure.

The model supports the fitting of the parameters Begin, A, B and C. All data samples are considered in the fitting process irrespective of Begin. The scaling parameter A is restricted to the range [0,1] per default.

Use without Measured Parent Fraction

The Hill parent fraction model can be applied even if no parent fraction measurements were loaded. In this case the user has to specify a parameter set which establishes a representative metabolite correction for the used tracer.


1.Gunn RN, Sargent PA, Bench CJ, Rabiner EA, Osman S, Pike VW, Hume SP, Grasby PM, Lammertsma AA: Tracer kinetic modeling of the 5-HT1A receptor ligand [carbonyl-11C]WAY-100635 for PET. Neuroimage 1998, 8(4):426-440. DOI