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).

Graham's compartmental model for input function
Figure 2. Graham’s compartmental model for input function is a catenary model, in contrast to PK three-compartment model. VP is the volume fraction of plasma, VIF is the volume fraction of interstitial fluid, and VTF the volume fraction of tissue fluid. PS1 and PS2 are permeability-surface area products for exchange between VP and VIF, and between VIF and VTF, respectively. GFR is glomerular filtration rate, representing elimination rate of radiotracer from the circulation.

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 ki:

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

Inputs for Graham model
Figure 3. Example inputs for Graham’s CM plasma model. Bolus injection (red) is simulated with parameters H=1 and ki=5 min-1. Short infusion (green) is simulated with H=1, ki=10 min-1, and duration of infusion Tdur=0.25 min (15 s). Long infusion (blue) is simulated with H=1, ki=5 min-1, and Tdur=2 min. To use traditional boxcar (rectangular) function, set ki=0.

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 [15O]H2O. 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 [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).

See also:


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.

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Updated at: 2019-02-03
Created at: 2016-08-08
Written by: Vesa Oikonen