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 (C₀(t)). The following two compartments are for the two distinct kinetic compartments in the tissue (C₁(t) and C₂(t)), representing, for instance, the free and receptor-bound tracer concentrations. Compartments are connected with four rate constants, K₁, k₂, k₃, and k₄.

Two-tissue compartmental model with compartments in series

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.

Two-tissue compartmental model with compartments in parallel

Figure 1b. 2TCM with compartments in parallel.

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.

TTAC simulated using 2TCM with compartments in series

TTAC simulated using 2TCM with compartments in parallel

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.

TTAC simulated using irreversible 2TCM with compartments in series

TTAC simulated using irreversible 2TCM with compartments in parallel

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 k₄=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=0.

In TCM the total tissue radioactivity is the sum of the two tissue compartments (ignoring the contribution from blood in tissue vasculature):

$C_{T}(t) = C_{1}(t) + C_{2}(t)$

For the 2TCM with tissue compartments in series (Fig. 1a) the concentration curves C₁(t) and C₂(t) can be calculated from the following differential equations:

$\frac{dC_{1}(t)}{dt} = K_{1} C_{0}(t) - (k_{2} + k_{3}) C_{1}(t) + k_{4} C_{2}(t)$ $\frac{dC_{2}(t)}{dt} = k_{3} C_{1}(t) - k_{4} C_{2}(t)$

For the 2TCM with tissue compartments in parallel (Fig. 1b) the concentration curves C₁(t) and C₂(t) can be calculated from the following (identical) differential equations:

$\frac{dC_{1}(t)}{dt} = K_{1a} C_{0}(t) - k_{2a} C_{1}(t)$ $\frac{dC_{2}(t)}{dt} = K_{1b} C_{0}(t) - k_{2b} C_{2}(t)$

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):

$K_{1a} = \frac{K_1}{k_{2b} - k_{2a}} \times (k_{3} + k_{4} - k_{2a})$ $k_{2a} = \frac{k_2 + k_3 + k_4 - \sqrt{ (k_2 + k_3 + k_4)^2 - 4 k_2 k_4 }}{2}$ $K_{1b} = \frac{K_1}{k_{2b} - k_{2a}} \times ( k_{2b} - k_{3} - k_{4})$ $k_{2b} = \frac{k_2 + k_3 + k_4 + \sqrt{ (k_2 + k_3 + k_4)^2 - 4 k_2 k_4 }}{2}$

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):

$K_1 = {K_{1a} + K_{1b}}$ $k_2 = \frac{K_{1a} k_{2a} + K_{1b} k_{2b}}{K_{1a} + K_{1b}}$ $k_3 = \frac{K_{1a} k_{2b} + K_{1b} k_{2a}}{K_{1a} + K_{1b}} - \frac{ k_{2a} k_{2b} ( K_{1a} + K_{1b} ) }{K_{1a} k_{2a} + K_{1b} k_{2b}}$ $k_4 = \frac{ k_{2a} k_{2b} ( K_{1a} + K_{1b} ) }{K_{1a} k_{2a} + K_{1b} k_{2b}}$

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 K₁-k₄ and V_B (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.

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 [¹⁸F]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

2TCM - compartments in series and in parallel

See also:

Literature