# Compartmental models for metabolite correction

Compartmental models can be used to model the appearance of metabolites in the plasma or blood. While Hill function or power function can fit well the plasma parent radiotracer fractions in bolus studies, those are usually not applicable in metabolite correction for bolus+infusion studies.

Huang et al. (1991) introduced general models for metabolite correction, and specific models for [18F]FDOPA and [15O]O2. Compartmental model for metabolite correction of [15O]O2 blood data was simplified by Iida et al. (1993). Several models for [11C]CO2 have been presented, since it is a common metabolite of 11C-labelled radiotracers. Lammertsma et al (1993) used a multi-compartmental model in metabolite analysis of L-[11C]deprenyl, but did not disclose details of the model. Carson et al (1997) used a two-compartmental model in metabolite analysis of [11C]raclopride. In this model, the concentration of metabolites in plasma was calculated from total plasma activity using equation

, where λ1≥0, λ2≥0, t≥0, and λ1λ2. Here, the response function is one form of biexponential,

and its constant, λ1λ2/(λ21) normalises the AUC to unity. If one-compartmental model would suffice, then the biexponential response function can be replaced by surge function, with AUC=1:

Gillings et al (2001) linearised the fitting by applying Patlak plot, using parent plasma concentration as input. Patlak plot has negative curvature if the metabolite(s) are eliminated from the plasma, which was corrected by adding the elimination rate into the Patlak plot when necessary. The method was applied to [11C]NNC112, [11C]NS2214, and [11C]PK11195 in minipigs (Gillings et al., 2001). A different formulation of the linear plot was previously used by the same group for [18F]FDOPA, [11C]deprenyl, [11C]SCH23390, [11C](S)-nicotine, and [11C]raclopride (Cumming et al., 1999).

When the plasma-to-blood ratio and/or the rate of plasma-to-RBC transport is different between the parent radiotracer and its metabolites, a compartmental model can be used to model both fractions simultaneously, as shown in a 6-[18F]fluoro-L-m-tyrosine study (Asselin et al., 2002). Wu et al. (2007) applied compartmental model for [11C]WAY-100635 when testing the performance of different metabolite correction methods.

Model-based input methods rely on assumption that the input function is common to all tissue regions in the PET image, and can be solved from the data (simultaneous estimation method, SIME). Burger & Buck (1996) have proposed using SIME for metabolite correction, and method has been applied for instance to [11C]flumazenil (Sanabria-Bohórquez et al., 2000).

General arterial input function model for PET was presented by Graham (1997), and while the model does not account for the appearance of label-carrying metabolites in the blood or plasma, Graham (1997) suggested combining the compartmental models for the plasma curve and metabolites. The Graham's model can be extended to account for the disappearance of the parent radiopharmaceutical from plasma pool via metabolism (Figure 1):

The metabolites formed in this model setting can be assumed to appear first in the venous/pulmonary plasma, and could be used as input for compartmental models describing the plasma kinetics of the metabolites. For a single metabolite, the compartmental model could that be defined as (Figure 2):

In this model we assume that the metabolite that appears in the BV compartment (venous blood and pulmonary vascular system), possibly produced from the parent compound via different routes, can be modelled with similar model as the parent compound (but obviously with different rates). If there are more than one label-carrying metabolite, then each of them can have their own compartmental model. In addition, label-carrying metabolite typically metabolises further, and this can be modelled by using the first metabolite as the "parent" compound for the second metabolite. These kind of models typically contain too many parameters for reliable fitting of measured data (overdetermined), but could be useful for simulation purposes.

## References

Carson RE, Breier A, de Bartolomeis A, Saunders RC, Su TP, Schmall B, Der MG, Pickar D, Eckelman WC. Quantification of amphetamine-induced changes in [11C]raclopride binding with continuous infusion. J Cereb Blood Flow Metab. 1997; 17(4): 437-447. doi: 10.1097/00004647-199704000-00009.

Lammertsma AA, Hume SP, Bench CJ, Luthra SK, Osman S, Jones T. Measurement of monoamine oxidase B activity using L-[11C]deprenyl: inclusion of compartmental analysis of plasma metabolites and a new model not requiring measurement of plasma metabolites. In: Quantification of brain function: Tracer kinetics and image analysis in brain PET. Uemura K et al., (eds.) 1993, Elsevier, The Netherlands, p. 313-318.

Updated at: 2019-04-20
Created at: 2007-07-18
Written by: Vesa Oikonen