# 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, *C _{S}(t)* is calculated accordingly:

*C _{B}(t)* is the radioactivity concentration in the blood,

*V*is the vascular volume fraction inside the region of interest, and

_{B}*C*is the radioactivity concentration in the

_{i}(t)*i*th tissue compartment.

When we have the simulated tissue TAC, we can compare that curve to the tissue TAC that we have
measured with PET (*C _{PET}(t)*):
At each time point (

*t*, 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 Χ

_{i}^{2}):

, where p̂ is the set of compartmental model parameters that were used to simulate
the tissue TAC, and *w _{i}* are the weights for each data sample.
If measurement variance is known,

, otherwise *w _{i}*=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
Χ^{2}, 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 unrealizable 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.

## See also:

- 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

## References:

Dai X, Chen Z, Tian J. Performance evaluation of kinetic parameter estimation
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Floudas CA, Pardalos PM (eds.): *Encyclopedia of Optimization*. 2nd ed.,
Springer, 2009.

Huang S-C. Detection of measurement outliers in tracer kinetics.
*IFAC Proceedings Volumes* 2008; 41(2): 6654-6657.

Huang C-K, Wang W, Tzen K-Y, Lin W-L, Chou C-Y. FDOPA kinetics analysis in PET images for
Parkinson’s disease diagnosis by use of particle swarm optimization.
*2012 9th IEEE International Symposium on Biomedical Imaging (ISBI)*, 586-589.

Huesman RH, Coxson PG. Consolidation of common parameters from multiple fits in dynamic PET data
analysis. *IEEE Trans Med Imaging* 1997; 16(5): 675-683.

Kadrmas DJ, Oktay MB. Generalized separable parameter space techniques for fitting 1K-5K
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doi: 10.1096/fasebj.1.5.3315805.

Muzic RF Jr, Christian BT. Evaluation of objective functions for estimation of kinetic
parameters. *Med Phys.* 2006; 32(2): 342-353.
doi: 10.1118/1.2135907.

Raylman RR, Hutchins GD, Beanlands RSB, Schwaiger M. Modeling of carbon-11-acetate kinetics by
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*J Nucl Med.* 1994; 35: 1286-1291.

Sederholm K. Globaali optimointi
positroniemissiotomografia-kuvantamiseen liittyvässä mallintamisessa.
*Pro gradu, 2003*.

Yaqub M, Boellaard R, Kropholler MA, Lammertsma AA. Optimization algorithms and weighting factors
for analysis of dynamic PET studies. *Phys Med Biol.* 2006; 51: 4217-4232.

Yaqub M, Boellaard R, Kropholler MA, Lubberink M, Lammertsma AA.
Simulated annealing in pharmacokinetic modeling of PET neuroreceptor studies: accuracy and precision
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*Nuclear Science Symposium Conference Record, 2004 IEEE.* 5: 3222-3225.
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Young P: *Everything You Wanted to Know About Data Analysis and Fitting but Were Afraid
to Ask.* Springer, 2005.
ISBN 978-3-319-19051-8.

Tags: Modeling, Compartmental model, Fitting, Optimization algorithm, NLLS, Rate constant

Created at: 2014-01-23

Updated at: 2018-02-06

Written by: Vesa Oikonen