# 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 *T _{a}* and

*T*, respectively; and ⊗ is the convolution operator, with decaying biexponential as the response function. Fourier transform of the boxcar function can be calculated using

_{a}*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.

Instead of assuming a fixed number of compartments (exponential terms), spectral analysis could be used. There a large number of exponentials (basis functions), covering the full range of feasible kinetics, are convolved with the boxcar function, and NNLS is used to solve suitable weights for each exponential in the sum that represents the fitted input function. This method has been used for example in dynamic NMR studies (Reynolds et al., 2022).

For AIFs from PET studies, compartmental model has been developed for
[^{15}O]H_{2}O 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 [^{15}O]H_{2}O.
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 [^{18}F]FLT study.

Comprehensive whole-body pharmacokinetic models have been developed especially for
[^{15}O]H_{2}O and [^{18}F]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).

## See also:

- Graham's compartmental model for AIF
- Fitting input function
- Blood sampling
- Input function
- Input for simulations
- Modelling A-V difference
- Compartmental models for metabolite correction
- Whole-body physiology-based pharmacokinetic model
- Fitting compartmental models
- Fitting TTACs
- Plasma pharmacokinetics
- Area under curve (AUC)

## Literature

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.

Tags: Input function, Plasma, PTAC, Fitting, Compartmental model, Biexponential

Updated at: 2023-02-06

Created at: 2016-08-08

Written by: Vesa Oikonen