# 2TCM - compartments in series and in parallel

Three-compartment model (two-tissue compartment
model, 2TCM) is commonly used in analysis of PET data (Fig. 1a).
First compartment is for the input function, the concentration
of tracer in plasma or blood curve as a function of time (*C _{0}(t)*).
The following two compartments are for the two distinct kinetic compartments in the tissue
(

*C*and

_{1}(t)*C*), representing, for instance, the free and receptor-bound tracer concentrations. Compartments are connected with four rate constants,

_{2}(t)*K*,

_{1}*k*,

_{2}*k*, and

_{3}*k*.

_{4}**Figure 1a.** 2TCM with compartments in series, representing free and bound
tracer, or tracer and its metabolized form, in tissue.

Alternatively, the two tissue compartments could be in parallel (Fig. 1b), that is, two parallel two-compartment models (one-tissue compartment models) both sharing the common input function. The two parallel tissue compartments 1 and 2 are kinetically indistinguishable. This model could be used to describe tissue heterogeneity or partial volume effect when region-of-interest contains two distinct tissue types, for instance brain white and grey matter.

These two-tissue compartmental models are kinetically indistinguishable from each other. This is shown in Figures 2 and 3, where the same tissue time-activity curve is simulated using the two models, with tissue compartments in series (left plot) and in parallel (right plot). If PET data can be fitted using one of the models, it could be as well fitted with the other. Fits to both models would even provide the same AIC values. Other information on the properties of the tissue and tracer must be known to decide which 2TCM setting is correct.

**Figure 2.** The same TTAC can be simulated from the two 2TCM models
(Fig. 1a and 1b), using the same input function.
On the left side, total tissue concentration curve (red line) is simulated using the model with
tissue compartments (green and blue lines) in series.
On the right side, total tissue concentration curve (red line) is simulated using the model with
tissue compartments (green and blue lines) in parallel.
For simplicity, contribution of blood radioactivity inside tissue vasculature is not considered
in these simulations.

**Figure 3.** Also in the case of irreversible tissue uptake kinetics
the same TTAC can be simulated from the two 2TCM models.
On the left side, total tissue concentration curve (red line) is simulated using the model with
tissue compartments (green and blue lines) in series, and with *k _{4}=0*.
On the right side, total tissue concentration curve (red line) is simulated using the model with
tissue compartments (green and blue lines) in parallel, and with

*k*.

_{2b}=0In TCM the total tissue radioactivity is the sum of the two tissue compartments (ignoring the contribution from blood in tissue vasculature):

For the 2TCM with tissue compartments *in series* (Fig. 1a) the concentration curves
*C _{1}(t)* and

*C*can be calculated from the following differential equations:

_{2}(t)For the 2TCM with tissue compartments *in parallel* (Fig. 1b) the concentration curves
*C _{1}(t)* and

*C*can be calculated from the following (identical) differential equations:

_{2}(t)The parameters of the model with the tissue compartments in series can be transformed into parameters of the parallel compartments model (Phelps et al., 1979; Feng et al., 1995):

And vice versa, the parameters of the parallel model can be transformed into parameters of the model with the tissue compartments in series (Feng et al., 1995):

The interchangeability of these two representations of 2TCM has been used in analysis of PET
data, especially in the early days, and in computation of parametric
images, because the solution of ODEs for two one-tissue
compartmental models is easier than for the 2TCM with compartments in parallel, and the methods are
computationally lighter. **Basis functions method** is a robust and fast method when
applied to the 1TCM (Koeppe et al., 1985),
and it can be extended to 2TCM to estimate the parameters
*K _{1}*-

*k*and

_{4}*V*(Tomasi et al., 2009; Hong & Fryer, 2010; Kudomi et al., 2017). The same principle can be applied to models with an arbitrary number of compartments, which allows the use of spectral analysis.

_{B}## See also:

- Compartmental models
- Reference tissue input compartmental models
- Compartmental model ODEs
- Compartmental model fitting
- Computation of parametric images
- Tissue heterogeneity
- Partial volume effect

## Literature

Feng D, Ho D, Chen K, Wu L-C, Wang J-K, Liu R-S, Yeh S-H.
An evaluation of the algorithms for determining local cerebral metabolic rates of glucose using
positron emission tomography dynamic data. *IEEE Trans Med Imaging* 1995; 14(4): 697-710.
doi: 10.1109/42.476111.

Hong YT, Fryer TD.
Kinetic modelling using basis functions derived from two-tissue compartmental models with a plasma
input function: General principle and application to [^{18}F]fluorodeoxyglucose positron
emission tomography. *NeuroImage* 2010; 51: 164-172. doi:
10.1016/j.neuroimage.2010.02.013.

Koeppe RA, Holden JE, Ip WR. Performance comparison of parameter estimation techniques for the
quantification of local cerebral blood flow by dynamic positron emission tomography.
*J Cereb Blood Flow Metab.* 1985; 5: 224-234.
doi: 10.1038/jcbfm.1985.29.

Phelps ME, Huang S-C, Hoffman EJ, Selin C, Sokoloff L, Kuhl DE.
Tomographic measurement of local cerebral glucose metabolic rate in humans with
[F-18]2-fluoro-2-deoxy-D-glucose: validation of method. *Ann Neurol.* 1979; 6: 371-388.
doi: 10.1002/ana.410060502.

Tags: Modeling, Compartmental model, Parameters, 2TCM, PVE, Simulation, Basis functions

Updated at: 2019-02-05

Created at: 2018-02-07

Written by: Vesa Oikonen