# 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.

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).

Instead of boxcar function, Graham simulated the bolus injection and bolus infusion with
functions that include exponential terms, representing the initial dispersion that distorts the bolus shape before it enters the arterial network for the first
time. Bolus injection was simulated with simple decaying exponential with time constant
*k _{i}*:

, where *H* represents the amount (radioactivity) of injected radiotracer (Figure 3),
with units of radioactivity/time.
Longer bolus with infusion duration *T _{dur}* was simulated with boxcar function
including exponential terms on both leading and trailing edges:

Differential equations for the Graham’s model are (Graham, 1997):

This model depicts slowly varying plasma activity over several minutes, but is not intended
to model the first-pass kinetics and recirculation effects with tracers 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 tracers 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:

- Fitting input function
- Blood sampling
- Input function
- Input for simulations
- Modelling A-V difference
- Compartmental models for metabolite correction
- Fitting compartmental models
- Fitting TTACs
- Plasma pharmacokinetics
- Area under curve (AUC)

## 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.

Tags: Input function, Fitting, Compartmental model, Biexponential

Updated at: 2019-02-03

Created at: 2016-08-08

Written by: Vesa Oikonen