# Compartmental model analysis of regional TACs

This page handles the general compartmental models with arterial plasma input that are applicable to most tracers; tracer specific models are described elsewhere, as well as reference tissue input models.

## Model fitting software

In TPC, Carimas is available for all researchers, and it can be used to fit one- and two-tissue compartmental models to regional data. ROIs can either be drawn and regional TACs computed inside Carimas, or regional TACs can be imported to Carimas.

Certain research groups in TPC have also acquired PMOD licenses.

Alternatively, in-house developed command-line programs (Open Source) can be used directly in Windows computers that are connected to TPC network, or can be downloaded for Windows, Linux, and macOS computers.

### Nonlinear least-squares (NLLS) fitting

Programs
fitk2,
fitk3,
fitk4,
fitk5, and
fitkloss
can be used to fit two, three, or four compartment
models to the PET TACs;
fitkloss
can be used to fit a three-compartment model where the last rate constant
*k _{4}* or

*k*represents the efflux of labelled metabolite or radioligand directly to the venous plasma.

_{Loss}These programs do use the weighting information if that is included in the tissue datafile.

#### Constrained *K*_{1}/k_{2}

_{1}/k

_{2}

In many cases all of the model parameters of two-tissue model can not be reliably fitted, but
some of them need to be constrained to a pre-determined value. A commonly used method is to
constrain the *K _{1}/k_{2}*, representing the distribution volume of
nonspecifically bound and free radioligand in the tissue, to a value that is derived from
a reference region with no specific binding or uptake
(

*k*). Not only does this method allow lower variances of model parameters, but it also enables us to calculate the binding potential in case of receptor studies.

_{3}=0As in all models applying reference tissue, also here it is assumed that
*K _{1}/k_{2}* is the same in all (brain) regions.
First, two- or three-compartment model is fitted to the reference tissue curve, providing
the values for

*K*or

_{1}/k_{2}*K*and

_{1}/k_{2}*k*. These are used as constraints for these parameters when fitting the compartmental model to the regions of interest.

_{5}/k_{6}Constraining *K _{1}/k_{2}* can be optionally done with programs
fitk3,
fitk4, and
fitk5,
when the name of reference region in the TAC file is given as command-line argument.

Constraining *K _{1}/k_{2}* to zero with program
fitk2 actually constrains

*k*, allowing the use of an extremely simple irreversible model with only two parameters,

_{2}=0*K*and

_{1}*V*.

_{B}#### Akaike information criteria (AIC)

Various compartmental models can be constructed and used to analyze PET data. The more complicated the model is, the better is the achieved fit to the data. However, at the same time, also the variance of the fitted parameters is increased. To find the optimum model, the programs compute also Akaike information criteria values: the smaller the AIC values are, the better the model is, considering the degrees of freedom of the fit. However, the physiological interpretation of the fitted parameters is on the responsibility of the user.

### General linear least squares method

Most compartmental models can be transformed into general linear least squares functions (Blomqvist, 1984), which can be solved using very fast linear methods, and are therefore suitable for computing parametric images. For regional data, program lhsol can be used to estimate the model parameters using Lawson-Hanson nonnegative least squares (NNLS) algorithm. The compartmental model can be selected with options -k1, -k2, -k3, -k4 for models excluding vascular volume fraction, or options -vk1, -vk2, -vk3, -vk4 for models including vascular volume fraction.

Note that if *V _{A}* is fitted using this methods, the vascular blood TAC is
assumed to be similar to the model input, i.e. metabolite corrected arterial plasma curve.
This is close to truth for a few tracers only, e.g. [

^{18}F]FDG. For other studies, a fixed amount of blood background can be subtracted before the model fit using taccbv.

The fitted parameters from these programs may have non-physiological values, because there is no other constraints than non-negativity.

#### V_{T} and K_{i} using NNLS method

In receptor binding studies distribution volume
(*V _{T}*) is usually the only model parameter of interest.
Instead of solving separate model rate constants and calculating

*V*from those afterwards, more reliable estimates of

_{T}*V*can be obtained by solving

_{T}*V*directly without division (Zhou et al., 2002; Hagelberg et al., 2004). Program lhsoldv uses Lawson-Hanson non-negative least squares (NNLS) algorithm and one or two tissue compartment models (with options -1 and -2) to solve the

_{T}*V*without division. The noise in regional TACs does not cause bias when using this method. Two tissue compartment model is recommended, since one tissue compartment model may lead to biases with more complex tissue kinetics. By default (or with option -A), the model can be selected automatically, based on lower AIC. With option -0 the AIC weighted average (Turkheimer et al. 2002) of

_{T}*V*from 1- and 2-tissue compartment models is calculated.

_{T}For irreversible tracer uptake models, the influx rate constant *K _{i}*
can be calculated accordingly with program lhsolki.
The two tissue model is applied by default.

## Units

In-house analysis programs, including Carimas, automatically convert the sample time and radioactivity concentrations units of plasma, blood and tissue data, if the units are specified in the data files. This is not always the case, and therefore it would be safest that researcher verifies that the units are the same in all data files before proceeding to the modelling.

In TPC, the units of radioactivity concentrations in plasma and blood are by default given per
volume (mL), not per mass (g). Therefore the unit of model parameter *K _{1}* is
(mL plasma)*(mL tissue)

^{-1}*min

^{-1}, and the units of other rate constants

*k*, ...

_{2}*k*are min

_{6}^{-1}. The units of the vascular blood or arterial plasma volume fractions

*V*and

_{B}*V*are mL/mL.

_{A}## Steps of calculation using command-line tools

All of of the following steps can be done in Linux terminal window or MS Windows command prompt window (preferably using scripts):

- Preparation of arterial plasma curve
- Adding weights to regional tissue TAC data
- Computing the parameter estimates: execute one of the model fit programs
with the command line parameters that are specified in the programs
user help information (with option
`--help`

).

## See also:

- Instruction by tracer
- Analysis models for regional TACs
- Input function
- Model calculations for PET images
- Regional result files
- Processing and further analysis for regional results

## 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.

Lawson CL, Hanson RJ. *Solving least squares problems.* Prentice-Hall, 1974.

Turkheimer FE, Hinz R, Cunningham VJ. On the undecidability among kinetic models: from model
selection to model averaging. *J Cereb Blood Flow Metab.* 2003; 23: 490-498. doi: 10.1097/01.WCB.0000050065.57184.BB.

Zhou Y, Brasic J, Endres CJ, Kuwabara H, Kimes A, Contoreggi C, Maini A, Ernst M, Wong DF.
Binding potential image based statistical mapping for detection of dopamine release by
[^{11}C]raclopride dynamic PET. *NeuroImage* 2002; 16: S91.

Tags: Modeling, Compartmental model, Rate constant, Fitting, NLLS, NNLS, Analysis

Updated at: 2020-05-01

Created at: 2013-05-17

Written by: Vesa Oikonen