Compartmental model fitting
Compartmental model equations can be used to do simulations of tissue TAC, and simulations are used to
- test simplified analysis methods and software
- fit (estimate) compartmental model parameters from measured PET data
Simulation of tissue curve is based on
- input function (usually arterial plasma TAC)
- compartmental model (equations)
- physiological model parameters
PET is used to measure the tissue TAC, which is the sum of the isotope label concentrations in all tissue compartments and blood in tissue vasculature. Simulated tissue TAC, CS(t) is calculated accordingly:
CB(t) is the radioactivity concentration in the blood, VB is the vascular volume fraction inside the region of interest, and Ci(t) is the radioactivity concentration in the ith tissue compartment.
When we have the simulated tissue TAC, we can compare that curve to the tissue TAC that we have measured with PET (CPET(t)): At each time point (ti, we calculate the difference between measured and simulated tissue radioactivity concentration, square it, and sum all these together. That is the so called Sum-of-Squares (SS or Χ2):
, where p̂ is the set of compartmental model parameters that were used to simulate the tissue TAC, and wi are the weights for each data sample. If measurement variance is known,
, otherwise wi=1.
The smaller that is, the better match we have between the measured and simulated curves.
Optimization algorithm is used for iteratively moving from one set of compartmental model parameters (p̂) to a set of parameters, which provides smaller Χ2, until progress is stalled or until a fixed maximum number of iterations has passed. As a result of this non-linear least-squares (NLLS) method, we have estimates of the rate constants.
Outlier detection can be implemented in the optimization algorithm (Huang, 2008).
If the criterion function has multiple local minima, the iterative search may end up at any one of these, leading to more or less flawed model parameter estimates. Therefore we are often applying global optimization algorithms, although they take much more computation time.
If no constraints are imposed on the parameters, the minimum could correspond to a physically unrealisable set of parameters. Noise and inadequate handling of the vascular volume fraction in the initial phase of the tissue TAC may lead to “impulse phenomenon”, where an initial peak in the data that actually is just noise is interpreted as representing very fast kinetics of the first tissue compartment, leading to improper rate constant estimates (Huang et al., 2018). At least negative parameter values would be non-physiological and should be ruled out.
Certain compartmental models can be linearized, and parameters solved using e.g. non-negative least-squares (NNLS) method.
One dynamic PET image contains several regions of interest, which usually are analyzed individually. Usually, however, we can assume that certain model parameters are common to all regions. If the TACs from several regions are fitted simultaneously, keeping the assumption of one or more common parameters, all of the parameters can in theory be estimated more reliably (Huesman & Coxson, 1997). Also the input function can be assumed to be the same for all regions, which is utilized in model-based input function estimation. However, the number of parameters to fit in SIME methods can be very large, and many usual optimization methods do not work reliably.
- Compartmental models
- PET data
- Input function
- Calculation of reference tissue input models for regional TAC data
- Plasma input compartmental model analysis of regional TACs
- Model calculations for PET images
- AIC in model selection
- Plotting TACs
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Updated at: 2019-01-18
Created at: 2014-01-23
Written by: Vesa Oikonen