# Linearization of 2TCM

## Reversible 2TCM

The tracer concentrations in the two tissue compartments (*C _{1}* and

*C*) of the reversible three-compartment model (3CM), or two-tissue compartment model (2TCM), can be calculated as

_{2}, where *C _{0}(t)* is the input function.
The radioactivity concentration in tissue, measured using PET, is contaminated by spillover
from adjacent blood vessels and vascular volume inside
the region-of-interest (or image voxel). We will assume that the radioactivity concentration in
blood can be represented by the model input function:

Substituting *C _{1}* and

*C*in Eq 3 with equations 1 and 2, gives

_{2}, which can be rearranged, and integrated:

Integration and rearrangement of Eq 3 gives

, which is substituted into Eq 1, giving equation

and after integration, and multiplying both sides with *k _{2}*,

Substituting equations 5 and 6 into this and rearrangement gives multilinear representation of the model:

## Irreversible 2TCM

Irreversible three-compartment model (3CM), or
two-tissue compartment model (2TCM) is similar to the reversible 2TCM above, except that
*k _{4}=0*.
The tracer concentrations in the two tissue compartments (

*C*and

_{1}*C*) of the irreversible 2TCM can be calculated as

_{2}The equations 3-6 of the reversible 2TCM apply to the irreversible model as well. Substitution of 12 into 3-6 gives

which can be solved for *C _{1}*. Integration and multiplication of both sides with

*k*gives

_{2}Substituting equations 5 and 6 into this and rearrangement gives multilinear representation of the irreversible 2TCM model:

## See also:

## Literature

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doi: 10.1038/jcbfm.1984.89.

Evans AC. A double integral form of the three-compartmental, four-rate-constant model for
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Gjedde A, Wong DF. Modeling neuroreceptor binding of radioligands *in vivo*.
In: Quantitative imaging, neuroreceptors, neurotransmitters, and enzymes. (Ed. JJ Frost and
HN Wagner Jr). Raven Press, New York, 1990, 51-79. ISBN: 978-0881676112.

Gjedde A. Modeling the dopamine system in vivo. *In:* In vivo imaging of
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Gjedde A. Modelling metabolite and tracer kinetics. *In:* Molecular Nuclear Medicine:
The Challenge of Genomics and Proteomics to Clinical Practice (Eds. Feinendegen LE, Shreeve WW,
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Tags: Compartmental model, 2TCM

Updated at: 2023-06-16

Created at: 2023-06-07

Written by: Vesa Oikonen