Linearization of 2TCM

Reversible 2TCM

The tracer concentrations in the two tissue compartments (C₁ and C₂) of the reversible three-compartment model (3CM), or two-tissue compartment model (2TCM), can be calculated as

$C_{1}(t) = K_1 \int_{0}^{t} C_{0}(t)dt - \left( k_2 + k_3 \right) \int_{0}^{t} C_{1}(t)dt + k_4 \int_{0}^{t} C_{2}(t)dt$ $C_{2}(t) = k_3 \int_{0}^{t} C_{1}(t)dt - k_4 \int_{0}^{t} C_{2}(t)dt$

, where C₀(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:

$C_{PET}(t) = V_{0} C_{0}(t) + C_{1}(t) + C_{2}(t)$

Substituting C₁ and C₂ in Eq 3 with equations 1 and 2, gives

$C_{PET}(t) = V_{0} C_{0}(t) + K_1 \int_{0}^{t} C_{0}(t)dt - k_2 \int_{0}^{t} C_{1}(t)dt$

, which can be rearranged, and integrated:

$k_2 \int_{0}^{t} C_{1}(t)dt = V_{0} C_{0}(t) + K_1 \int_{0}^{t} C_{0}(t)dt - C_{PET}(t)$ $k_2 \int_{0}^{t} \int_{0}^{t} C_{1}(t)dt dt = V_{0} \int_{0}^{t} C_{0}(t)dt + K_1 \int_{0}^{t} \int_{0}^{t} C_{0}(t)dt dt - \int_{0}^{t} C_{PET}(t)dt$

Integration and rearrangement of Eq 3 gives

$\int_{0}^{t} C_{2}(t)dt = \int_{0}^{t} C_{PET}(t)dt - V_{0} \int_{0}^{t} C_{0}(t)dt - \int_{0}^{t} C_{1}(t)dt$

, which is substituted into Eq 1, giving equation

$C_{1}(t) = \left( K_1 - V_0 k_4 \right) \int_{0}^{t} C_{0}(t)dt + k_4 \int_{0}^{t} C_{PET}(t)dt - \left( k_2 + k_3 + k_4 \right) \int_{0}^{t} C_{1}(t)dt$

and after integration, and multiplying both sides with k₂,

$k_2 \int_{0}^{t} C_{1}(t) = \left( K_1 k_2 - V_0 k_2 k_4 \right) \int_{0}^{t} \int_{0}^{t} C_{0}(t)dt dt + k_2 k_4 \int_{0}^{t} \int_{0}^{t} C_{PET}(t)dt dt \\ \quad \quad \quad \quad \quad - \left( k_2 + k_3 + k_4 \right) k_2 \int_{0}^{t} \int_{0}^{t} C_{1}(t)dt dt$

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

$C_{PET}(t) = V_{0} C_{0}(t) + \left( K_1 + V_0 ( k_2 + k_3 + k_4 ) \right) \int_{0}^{t} C_{0}(t)dt - \left( k_2 + k_3 + k_4 \right) \int_{0}^{t} C_{PET}(t)dt \\ \quad \quad \quad \quad + \left( K_1 ( k_3 + k_4 ) + V_0 k_2 k_4 \right) \int_{0}^{t} \int_{0}^{t} C_{0}(t)dt dt - k_2 k_4 \int_{0}^{t} \int_{0}^{t} C_{PET}(t)dt dt$

Irreversible 2TCM

Irreversible three-compartment model (3CM), or two-tissue compartment model (2TCM) is similar to the reversible 2TCM above, except that k₄=0. The tracer concentrations in the two tissue compartments (C₁ and C₂) of the irreversible 2TCM can be calculated as

$C_{1}(t) = K_1 \int_{0}^{t} C_{0}(t)dt - \left( k_2 + k_3 \right) \int_{0}^{t} C_{1}(t)dt$ $C_{2}(t) = k_3 \int_{0}^{t} C_{1}(t)dt$

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

$C_{PET}(t) = V_{0} C_{0}(t) + C_{1}(t)dt + k_3 \int_{0}^{t} C_{1}(t)dt$

which can be solved for C₁. Integration and multiplication of both sides with k₂ gives

$k_2 \int_{0}^{t} C_{1}(t) = k_2 \int_{0}^{t} C_{PET}(t)dt - V_0 k_2 \int_{0}^{t} C_{0}(t)dt - k_3 k_2 \int_{0}^{t} \int_{0}^{t} C_{1}(t)dt dt$

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

$C_{PET}(t) = V_{0} C_{0}(t) + \left( K_1 + V_0 ( k_2 + k_3 ) \right) \int_{0}^{t} C_{0}(t)dt - \left( k_2 + k_3 \right) \int_{0}^{t} C_{PET}(t)dt + K_1 k_3 \int_{0}^{t} \int_{0}^{t} C_{0}(t)dt dt$

Literature

Blomqvist G. On the construction of functional maps in positron emission tomography. J Cereb Blood Flow Metab. 1984; 4:629-632. doi: 10.1038/jcbfm.1984.89.

Evans AC. A double integral form of the three-compartmental, four-rate-constant model for faster generation of parametric maps. J Cereb Blood Flow Metab. 1987; 7:S453.

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 neurotransmitter functions in brain, heart, and tumors. (Ed. Kuhl DE). ACNP Publication 91-2, USA, 1991, 157-179.

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, Eckelman WC, Bahk Y-W, Wagner HN Jr), Springer-Verlag, 2003, pp 121–169. doi: 10.1007/978-3-642-55539-8_7.

Tags: Compartmental model, 2TCM

Updated at: 2023-06-16
Created at: 2023-06-07
Written by: Vesa Oikonen

Linearization of 2TCM

Reversible 2TCM

Irreversible 2TCM

See also:

Literature