# 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 tracer,
the models can be extended to dual-tracer PET studies, where two tracers 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
*K _{1}/k_{2}* is similar in all studied regions. If

*K*and

_{1}’*2*are used to represent these rate constants in the reference tissue, we can write:

_{2}’, and introduce *R _{1}* (or

*R*):

_{influx}## 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, butBPis the one that can be estimated with reference region input compartmental models._{ND}

### 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.

**Figure 1.** The original reference tissue compartmental model.
The compartments for free tracer (FT) and non-specifically bound tracer
(NS) in tissue
are combined into a single compartment, called non-displaceable
(ND) tracer in tissue,
that is, *C _{ND}* =

*C*+

_{FT}*C*.

_{NS}The four parameters *R _{1}* (ratio of the

*K*values of regions of interest and reference tissue),

_{1}*k*,

_{2}*k*, and

_{3}*BP*(

_{ND}*k*/

_{3}*k*) can be estimated using nonlinear fitting (Cunningham et al., 1991).

_{4}Assumptions:

- Reference region has no specific binding (devoid of receptors)
K/_{1}kis same in the regions of interest and in the reference region_{2}

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
tracer to the tissue (*R _{1}*).

Differential equations for the FRTM:

, 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:

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)

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 tracer in tissue (Lammertsma and Hume, 1996). Differential equation for SRTM:

Analytical solution using Laplace transformation for SRTM gives:

The three parameters of simplified model (*R _{1}*,

*k*and

_{2}*BP*) 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.

_{ND}Assumptions:

- Reference region has no specific binding (devoid of receptors)
K/_{1}k(=_{2}R) is same in the regions of interest and in the reference region_{1}- 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 tracer are simple enough to fulfill 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 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

*R*images, but leads to negative bias in low

_{1}*BP*values (Schuitemaker et al., 2007).

_{ND}#### SRTM with parameter coupling

Coupling of *k’ _{2}* (

*k*in the reference region) to a common value across brain regions or to a first-pass estimate reduces the variance of parameter estimates (Endres et al., 2011).

_{2}#### 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).

### Ichise’s multilinear regression techniques

Ichise et al (1996) presented a method in which the equations of Logan graphical analysis are solved with multilinear regression. This 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).

## 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 (*k _{3}*), when

*k*: the (reduced) reference tissue compartment model (RRTM), and the transport limited reference tissue model (TRTM).

_{4}=0### Reduced reference tissue model (RRTM)

If tracer uptake is irreversible during the PET experiment, i.e. *k _{4}=0*,
the full reference tissue model is reduced into a model where three parameters

*R*,

_{1}*k*, and

_{2}*k*can be estimated with nonlinear fitting. The

_{3}*k*value provided by the RRTM model will be proportional to the concentration of unoccupied receptors (Wong et al., 1986).

_{3}**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)
K/_{1}kis same in the regions of interest and in the reference region_{2}kin the regions of interest_{4}=0

In reference region the rate constant *k _{3}* is assumed to be negligible.
Differential equations for RRTM:

As an alternative, if a region exists, where the irreversible binding or metabolism is very rapid
(*k _{3}‘>>k_{2}’*), it can be used as the reference region in
TRTM (see below).

### Transport-limited reference tissue model (TRTM)

If tracer uptake is irreversible during PET experiment, i.e. *k _{4}=0*, and
there is no traditional reference region where

*k*=0, but there exists a region where irreversible binding or metabolism is very rapid (

_{3}*k*), 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

_{3}‘>>k_{2}’*R*,

_{1}*k*, and

_{2}*k*in regions where

_{3}*k*is not too high.

_{3}**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:

kin the reference region_{3}‘>>k_{2}’K/_{1}kis same in the regions of interest and in the reference region_{2}kin all regions_{4}=0

Differential equation for TRTM:

## Units

The parameters *R _{1}* and

*BP*are unitless. The units of

_{ND}*k*and

_{2}*k*are min

_{3}^{-1}.

## See also:

- Analysis of regional TAC data
- Analysis of image data
- Calculation of
*BP*images_{ND} - Compartmental models
- Model fitting
- PET data
- Input function
- Multiple-time graphical analysis (MTGA)
- Binding potential
- Receptor occupancy

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

Created at: 2005-11-30

Updated at: 2018-12-13

Written by: Vesa Oikonen