Full compartmental models for regional analysis
Background
There are compartmental models that are specific to a certain tracer. This text handles only computation of the common models that are applicable to several tracers.
The input to these models is always the metabolite corrected plasma curve. The unit of model parameters K1,.., k6 is min-1 and for the vascular blood or arterial plasma volume fractions VB and VA mL/mL.
Model fit programs
General linear least squares method ("Blomqvist")
Compartmental models can be transformed into general linear least squares functions (Blomqvist 1984), which can be solved using different methods. The following programs estimate the model parameters using Lawson-Hanson non-negative least squares (NNLS) algorithm (therefore program names start with lh*):
| Fitted parameters | Lawson-Hanson |
|---|---|
| K1 | lhsol with option -k1 |
| K1, k2 | lhsol with option -k2 |
| K1, k2, k3 | lhsol with option -k3 |
| K1, k2, k3, k4 | lhsol with option -k4 |
| K1, VA | lhsol with option -vk1 |
| K1, k2, VA | lhsol with option -vk2 |
| K1, k2, k3, VA | lhsol with option -vk3 |
| K1, k2, k3, k4, VA | lhsol with option -vk4 |
Note that if VA is fitted using these these methods, the vascular blood 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. [18F]FDG. For other studies, a fixed amount of blood background can be subtracted before the model fit using the program dftcbv.
These programs do use the weighting information if that is included in the datafile. The fitted parameters from these programs may have non-physiological values, because there is no other constraints than non-negativity.
DV and Ki using general linear least squares method
In receptor binding studies distribution volume (DV) is usually the only model parameter of interest. Instead of solving separate parameter values and calculating DV from those afterwards, more reliable estimates of DV can be obtained by solving DV 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 DV. 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, the models are averagted using Akaike weights (Turkheimer et al. 2002) or selected based on smaller AIC.
For irreversible models, the infulx rate constant Ki can be calculated accordingly with program lhsolki. The two tissue model is applied by default.
Non-linear fit
Programs fitk2, fitk3 and fitk4 can be used to fit two or three compartment models to the PET TACs. In addition, program fitkloss can be used to fit a three-compartment model where the last rate constant k4 is replaced with kLoss which represents the efflux of labelled metabolite directly to the venous plasma. Program fitglob is an alternative, when different model structures are tested.
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, some of the programs compute also Akaike information criteria values (and/or Schwarz criteria values, SC): 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.
Steps of calculation:
Any or all of of the following steps can be done in Solaris terminal window or MS Windows command prompt window on either SUN or PC platform.
1. Preparation of arterial plasma curve
The procedure is dependent on the tracer and study protocol. Detailed instructions on the preparation of input curves in given elsewhere. In short, if on-line detector was used to collect the blood curve during the early phase of the study, the blood curve must be corrected and converted to plasma TAC, and then combined to the manually sampled plasma curve. Fractions of labeled metabolites in plasma must be corrected.
Usually the blood TAC is not measured separately. If it is needed for fitting the vascular volume fraction, it can be computed from the plasma TAC, which is not corrected for metabolites(!), using tracer specific programs.
Delay correction can be done for plasma and blood TACs at the same time using program fitdelay.
2. Preparation of regional tissue TAC data
This is explained in detail elsewhere. In short: draw ROIs and calculate regional TACs from dynamic images, and calculate averages over planes and regions if required. If you have regional TACs in old format (*.roi.kbq), convert those to DFT format using program nci2dft.
3. Adding weights to regional tissue TAC data
Weights can be added to tissue data file using program dftweigh. Weights can be extracted either from a SIF file (*dy1.img.sif), or automatically from the average tissue curves. The weights are not absolute, but only relational to each time frame in the TAC.
If weights are extracted from the SIF file, the command could be e.g.:
dftweigh ua0268.dft ua0268dy1.img.sif C-11
If SIF file is not used, the command would be:
dftweigh ua0268.dft
Note that weights need to be added to the DFT datafile only once, and they
may affect the results of other calculation programs. The weights can be
removed using the same program, dftweigh, with option -rm.
3. Computing the parameter estimates
Depending on the suitable model, run one of the model fit programs with the command line parameters that are given in the specific program information.