# Compartmental models for input function

Biexponential arterial input function (AIF) model has been used in DCE-MRI studies and also in PET field. A rectangular (boxcar) function that represents the infusion of the contrast agent or radiotracer is convolved with an exponential function, which represents circulatory system (Pellerin et al., 2007; Poulin et al., 2013; Richard et al., 2017). In the case of biexponential AIF model a two-compartmental model is assumed. The biexponential AIF model can be written as

, where D is the infusion rate of the contrast agent or radiotracer, Π(t) is boxcar function with height 1 between the infusion start and end times Ta and Ta, respectively; and ⊗ is the convolution operator, with decaying biexponential as the response function. Fourier transform of the boxcar function can be calculated using sinc function. If we set the AUC of the biexponential response function to unity, and assume that its value initially is 1, that is, h(0)=1, then the response function can be given with just two parameters:

These models are based on the mammillary three-compartmental pharmacokinetics model of drug plasma concentration.

For AIFs from PET studies, compartmental model has been developed for [15O]H2O studies (Maguire et al., 2003). General AIF model for PET (Figure 2) was presented by Graham (1997).

The Graham's compartmental model for arterial plasma curves is explained in more detail on this page. This model is not intended to model the first-pass kinetics and recirculation effects with radiotracers such as [15O]H2O. As Graham points out, all compartmental models will have a simple exponential behaviour at late times, although many radiopharmaceuticals tend to have a slight upward convexity long times after injection. This model cannot account for this phenomenon, nor the appearance of label-carrying metabolites in the blood or plasma.

Compartmental model for metabolite correction was developed by Huang et al. (1991), and Graham (1997) suggested combining the compartmental models for plasma curve and metabolites. Graham et al (2000) used his compartmental model for smoothing and interpolating TACs from aorta in a FDG study to construct a population-based input function. Spence et al (2008) used the method to extrapolate metabolite-corrected blood TAC in a [18F]FLT study.

Comprehensive whole-body pharmacokinetic models have been developed especially for [15O]H2O and [18F]FDG. These model can be used for simulation, but fitting measured AIFs is not possible because of the large number of model parameter, unless a regularization/penalization method is used (O'Sullivan et al., 2009; Huang et al., 2014).

## References

Feng D, Huang S-C, Wang X. Models for computer simulation studies of input functions for tracer kinetic modeling with positron emission tomography. Int J Biomed Comput. 1993; 32: 95-110. doi: 10.1016/0020-7101(93)90049-C.

Graham MM. Physiologic smoothing of blood time-activity curves for PET data analysis. J Nucl Med. 1997; 38(7): 1161-1168. PMID: 9225813.

Updated at: 2019-04-03
Created at: 2016-08-08
Written by: Vesa Oikonen