Reference region input compartmental models

Reference tissue (input) compartmental models do not require or use plasma sampling as the model input function, but instead are using time-activity curve of a reference region with non-existent (or very low) specific uptake (Cunningham et al., 1991; Lammertsma & Hume, 1996).

Usually reference tissue models are used to estimate binding potential (BP_ND) from reversible ligand-receptor PET studies, but there are also modified models that can be applied to irreversible binding. While models are usually applied to dynamic PET data collected after injection of one radiopharmaceutical, the models can be extended to dual-radiopharmaceutical (dual-tracer) PET studies, where two radiopharmaceuticals targeting different transmitter systems are injected ∼20-30 min apart (Joshi et al., 2009).

The advantage of all reference tissue models is that since blood sampling and plasma metabolite analysis are not needed, the errors caused by the uncertainties in the measured plasma metabolite fractions are avoided.

One assumption common to all of reference tissue models is that K₁/k₂ is similar in all studied regions. If K₁' and k₂' are used to represent these rate constants in the reference tissue, we can write:

$\frac{K_1}{k_2} = \frac{K^\prime_1}{k^\prime_2}$

, and introduce R₁ (or R_influx):

$R_1 = \frac{K_1}{K^\prime_1} = \frac{k_2}{k^\prime_2}$

Reversible uptake

Two reference tissue compartment models are available for quantitation of binding potential (BP_ND): the (original) full reference tissue compartment model (FRTM or RTCM), and the simplified reference tissue model (SRTM).

There are many binding potentials, but BP_ND is the one that can be estimated with reference region input compartmental models.

Full reference tissue compartment model (RTCM)

The (full) reference tissue compartmental model (Cunningham et al., 1991) is the original and "gold standard" of reference tissue input methods for the estimation of BP_ND from reversible ligand receptor studies, and is based on the two-tissue compartmental model.

Reference tissue compartmental model (original)

Figure 1. The original reference tissue compartmental model. The compartments for free radiotracer (FT) and non-specifically bound radiotracer (NS) in tissue are combined into a single compartment, called non-displaceable (ND) radiotracer in tissue, that is, C_ND = C_FT + C_NS.

The four parameters R₁ (ratio of the K₁ values of regions of interest and reference tissue), k₂, k₃, and BP_ND (k₃/k₄) can be estimated using nonlinear fitting (Cunningham et al., 1991).

Assumptions:

Reference region has no specific binding (devoid of receptors)

K₁/k₂ is same in the regions of interest and in the reference region

This model has some advantages over the Logan plot: dynamic study can be used from the beginning with no need to wait for any equilibrium or search for linear phase. Reference tissue model also provides an index for the perfusion and transport of radioligand to the tissue (R₁).

Differential equations for the FRTM:

$\frac{dC_{ND}(t)}{dt} = R_1 \frac{dC_{R}(t)}{dt} + k_2 C_{R}(t) - (k_2 + k_3) C_{ND}(t) + k_4 C_{S}(t)$ $\frac{dC_{S}(t)}{dt} = k_3 C_{ND}(t) - k_4 C_{S}(t)$

, where C_R(t) represents the radioactivity concentration as a function of time in the reference tissue.

Analytical solution for FRTM using Laplace transformation is:

$\begin{align*} C_T(t) &= R_1 C_{R}(t) + [A_1 e^{-\alpha_1 t} + A_2 e^{-\alpha_2 t} ] \otimes C_{R}(t) \\ &, where \\ &\alpha_1 = \frac{k_2 + k_3 + k_4 - \sqrt{(k_2 + k_3 + k_4)^2 - 4 k_2 k_4}}{2} \\ &\alpha_2 = \frac{k_2 + k_3 + k_4 + \sqrt{(k_2 + k_3 + k_4)^2 - 4 k_2 k_4}}{2} \\ &A_1 = (\frac{k_3 + k_4 - \alpha_2}{\alpha_1 - \alpha_2}) (\frac{k_2}{R_1} - \alpha_2) \\ &A_2 = (\frac{\alpha_1 - k_3 - k_4}{\alpha_1 - \alpha_2}) (\frac{k_2}{R_1} - \alpha_1) \end{align*}$

The full reference tissue compartmental model has four free parameters, and often the model is too complex for the noisy PET data. In practise it is usually replaced by the simplified reference tissue model, although that may introduce some bias into the BP_ND estimates.

Simplified reference tissue model (SRTM)

Figure 2. Simplified reference tissue compartmental model (SRTM).

Simplified reference tissue (or region) model (SRTM or SRRM) can be used when two-compartmental model (one-tissue compartmental model) could reasonable describe the kinetics of the radiopharmaceutical in tissue (Lammertsma and Hume, 1996). Differential equation for SRTM:

$\frac{dC_{T}(t)}{dt} = R_{1} \frac{dC_{R}(t)}{dt} + k_2 C_{R}(t) - \frac{k_2}{1+BP_{ND}} C_{T}(t)$

Analytical solution using Laplace transformation for SRTM gives:

$C_T(t) = R_1 C_{R}(t) + [k_2 - \frac{R_1 k_2}{1 + BP_{ND}}] C_{R}(t) \otimes \ e^{-\frac{k_2}{1 + BP_{ND}} t}$

The differential equation can be written with parameter k'₂ instead of k₂ ; k₂ is solved from equation R₁ = k₂/k₂' , giving:

$\frac{dC_{T}(t)}{dt} = R_{1} \frac{dC_{R}(t)}{dt} + R_{1} k_2^\prime C_{R}(t) - \ \frac{R_{1} k_2^\prime}{1+BP_{ND}} C_{T}(t)$

The three parameters of simplified model (R₁ , BP_ND , and k₂ or k'₂) can be solved not only using nonlinear fitting but also using linearized methods (Blomqvist, 1984), spectral analysis, or with basis function approach (Gunn et al. 1997). This makes it possible to produce parametric images of model parameters, and, when linearized, also to do the calculations at the sinogram level.

Assumptions:

Reference region has no specific binding (devoid of receptors)

K₁/k₂ (=R₁) is same in the regions of interest and in the reference region

kinetics in all brain regions is fast and simple: if we had an arterial plasma input function, we could fit one-tissue compartmental model to tissue curves fairly well.

If kinetics of the radiopharmaceutical are simple enough to fulfil the requirements of SRTM, then the more complicated RTCM would produce results with high variance. If this requirement is not fulfilled, then results will be biased.

Basis function implementations

The widely used basis function (BF) implementation of SRTM (Gunn et al., 1997) produces unbiased BP_ND maps presuming that the range of basis functions is carefully optimized. The selection of a specific range of basis functions has been criticized of being slow and inefficient as it is based on a compromise between accuracy and precision (Cselényi et al., 2006; Schuitemaker et al., 2007).

Wu and Carson (2002) proposed that the washout rate constant of the reference tissue, k'₂ , could be first estimated using RPM (the original software implementation of BF-SRTM), and the median value (including voxels where BP_ND>0 ) would be fixed for all voxels during the second basis function evaluation. This approach mostly improves the quality of R₁ images, but leads to negative bias in low BP_ND values (Schuitemaker et al., 2007).

SRTM and MRTM with parameter coupling

Ichise et al (1996) presented a method in which the equations of Logan graphical analysis are solved with multilinear regression. This MRTM approach may lead to marked negative bias with noisy data (Schuitemaker et al., 2007). The modified method (Ichise et al., 2002 and 2003) may lead to higher variance with slightly reduced bias (Schuitemaker et al., 2007).

Coupling of the k'₂ (k₂ in the reference region) to a common value across brain regions, or to a first-pass estimate (as proposed for the BF implementations), reduces the variance of parameter estimates (Wu & Carson, 2002; Ichise et al., 2003; Endres et al., 2011). The method has been referred to as SRTM2 or MRTM2 (two-parameter version of multilinear reference tissue model), depending on the calculation method.

Modified SRTM to detect neurotransmitter release

SRTM assumes a steady physiological state throughout the PET experiment, from radiotracer injection to the end of scanning. Steady-state can be intentionally perturbed to study the effect of task- or drug-induced changes in neurotransmission (Friston et al., 1997). To analyze this kind of data the reference tissue model had to be modified to account for the changes in the dissociation rate of the radioligand (LSSRM, Alpert et al., 2003).

Irreversible uptake

Reference tissue compartmental models are usually applied to quantification of reversible binding, but two reference tissue compartmental models are available for quantitation of irreversible binding or metabolism (k₃), when k₄=0: the (reduced) reference tissue compartment model (RRTM), and the transport limited reference tissue model (TRTM).

Reduced reference tissue model (RRTM)

If radiopharmaceutical uptake is irreversible during the PET experiment, i.e. k₄=0, the full reference tissue model is reduced into a model where three parameters R₁, k₂, and k₃ can be estimated with nonlinear fitting. The k₃ value provided by the RRTM model will be proportional to the concentration of unoccupied receptors (Wong et al., 1986).

Reduced reference tissue compartmental model

Figure 3. Reference tissue compartmental model for situations where binding is irreversible in the region of interest, and reversible in the reference region.

Assumptions:

Reference region has no specific uptake (devoid of the target enzyme or receptor)

K₁/k₂ is same in the regions of interest and in the reference region

k₄=0 in the regions of interest

In reference region the rate constant k₃ is assumed to be negligible. Differential equations for RRTM:

$\frac{dC_{ND}(t)}{dt} = R_1 \frac{dC_{R}(t)}{dt} + k_2 C_{R}(t) - (k_2 + k_3) C_{ND}(t)$ $\frac{dC_{S}(t)}{dt} = k_3 C_{ND}(t)$

As an alternative, if a region exists, where the irreversible binding or metabolism is very rapid (k₃'>>k₂'), it can be used as the reference region in TRTM (see below).

Transport-limited reference tissue model (TRTM)

If radiopharmaceutical uptake is irreversible during PET experiment, i.e. k₄=0, and there is no traditional reference region where k₃=0, but there exists a region where irreversible binding or metabolism is very rapid (k₃'>>k₂'), then this region can be used as a "positive" reference region in transport-limited reference tissue model (Herholz et al., 2001; Nagatsuka et al., 2001) to estimate parameters R₁, k₂, and k₃ in regions where k₃ is not too high.

Transport-limited reference tissue compartmental model

Figure 4. Reference tissue compartmental model for situations where binding is irreversible in the region of interest and reference region, and binding in reference region is so fast that uptake there is only limited by transport.

Assumptions:

k₃'>>k₂' in the reference region

K₁/k₂ is same in the regions of interest and in the reference region

k₄=0 in all regions

Differential equation for TRTM:

$\frac{dC_{T}(t)}{dt} = R_{1} \frac{dC_{R}(t)}{dt} + (R_1 k_3) C_{R}(t) - (R_1 k_{2}^\prime + k_3) C_{T}(t)$

Units

The parameters R₁ and BP_ND are unitless. The units of k₂ and k₃ are min^-1.

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Tags: Modeling, Compartmental model, Reference tissue, SRTM, Rate constant, Convolution, Basis functions

Updated at: 2020-02-03
Created at: 2005-11-30
Written by: Vesa Oikonen