The parameters of plasma input TACs and compartmental models are estimated (fitted) using non-linear least-squares optimization algorithms (NLLSs), such as Particle swarm optimization (PSO), artificial immune network (AIN), and gravitational search algorithm (GSA).

In times of very limited computing resources the traditional methods such as Newton-Gaussian or Levenberg-Marquardt algorithms were also used; being very fast, those methods are very dependent on the initial parameter guesses. These methods converge to the local optimum, but could be used as a part of global optimization routines. For example, the commonly used Nelder-Mead algorithm (Downhill Simplex) (Nelder & Mead, 1965; Price et al., 2002) forms the basis of globalized bounded Nelder-Mead (GBNM) algorithm (Luersen et al., 2004a and 2004b). Local optimization method can be used as part of simulated annealing (SA) algorithm (Wong et al., 2002; Yaqub et al., 2006). Topographical global optimization (TGO) is one of the methods that can be used to get good initial parameter estimates for local optimization routines (Sederholm, 2003; Henderson et al., 2017).

Population-based heuristic optimization methods are affected by the selection of the initial population of parameters. Initial population is usually set randomly using pseudo-random number generators such as Mersenne Twister. However, it is more important that the initial population is evenly distributed than random, and therefore quasi-random low-discrepancy sequences are preferred over pseudo-random numbers (Maaranen et al., 2004).

Several stopping rules for terminating global optimization methods have been proposed (Betrò and Schoen, 1986; Boender and Kan, 1991; Hart, 1998; Lagaris and Tsoulos, 2008).

While simple models with few parameters can be fitted with any algorithm, more effort is needed to select and refine the algorithm as the complexity of the model increases; no algorithm works well for all optimization problems (Wolpert & Macready, 1997).


Alternatively, some non-linear models can be linearised to estimate the macroparameter of interest using linear regression, for example the multiple-time graphical analyses, or individual model parameters using for example NNLS or GLLS (Feng et al., 1995, 1996, 1999). Based on simulations, Muzic and Christian (2006) have shown that iteratively re-weighted least squares (IRLS) and variations of extended least squares (ELS0, ELS1, ELS3) perform better than methods (WLS, GLLS, PWLS) that determine weights based directly on the measured data.

See also:


Dai X, Chen Z, Tian J. Performance evaluation of kinetic parameter estimation methods in dynamic FDG-PET studies. Nucl Med Commun. 2011; 32: 4-16. doi: 10.1097/MNM.0b013e32833f6c05.

Feng D, Huang S-C, Wang Z, Ho D. An unbiased parametric imaging algorithm for nonuniformly sampled biomedical system parameter estimation. IEEE Trans Med Imaging 1996; 15(4): 512-518. doi: 10.1109/42.511754.

Floudas CA, Pardalos PM (eds.): Encyclopedia of Optimization. 2nd ed., Springer, 2009.

Gill PE, Murray W, Wright MH: Practical Optimization. Academic Press, 1981. ISBN 0-12-283952-8.

He J, Wang T, Li Y, Deng Y, Wang S. Dynamic chaotic gravitational search algorithm-based kinetic parameter estimation of hepatocellular carcinoma on 18F-FDG PET/CT. BMC Medical Imaging 2022; 22:20. doi: 10.1186/s12880-022-00742-4.

Huang S-C, Wu L-C, Lin W-C, Lin K-P, Liu R-S. Adaptive weighted nonlinear least squares method for fluorodeoxyglucose positron emission tomography quantification. J Med Biol Eng. 2018: 38(1): 63-75. doi: 10.1007/s40846-017-0313-6.

Hughes IG, Hase TPA: Measurements and their Uncertainties - A Practical Guide to Modern Error Analysis. Oxford University Press, 2010. ISBN 978-0-19-956632-7.

Levy AB: The Basics of Practical Optimization. SIAM, 2009. ISBN 978-0-898716-79-5.

Liu L, Ding H, Huang HB. Improved simultaneous estimation of tracer kinetic models with artificial immune network based optimization method. Appl Radiat Isot. 2016; 107: 71-76. doi: 10.1016/j.apradiso.2015.09.012.

Lu R, Liu L, Shen L. A distributed artificial immune network for optimizing tracer kinetic models with MATLAB distributed computing engine. J Algorithms Computational Technol. 2013; 7(2): 173-185. doi: 10.1260/1748-3018.7.2.173.

Motulsky HJ, Ransnas LA. Fitting curves to data using nonlinear regression: a practical and nonmathematical review. FASEB J. 1987; 1: 365-374. doi: 10.1096/fasebj.1.5.3315805.

Murase K, Mochizuki T, Kikuchi T, Ikezoe J. Kinetic parameter estimation from compartment models using a genetic algorithm. Nucl Med Commun. 1999; 20(10): 925–932. doi: 10.1097/00006231-199910000-00010.

Muzic RF Jr, Christian BT. Evaluation of objective functions for estimation of kinetic parameters. Med Phys. 2006; 33(2): 342-352. doi: 10.1118/1.2135907.

Nelder JA, Mead R. A simplex method for function minimization. Comput J. 1965; 7(4): 308-314. doi: 10.1093/comjnl/7.4.308.

Nocedal J, Wright SJ: Numerical Optimization, 2nd ed., Springer, 2006. ISBN 978-0387-30303-1.

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

Wahde M: Biologically Inspired Optimization Methods - An Introduction. WIT Press, 2008. ISBN: 978-1-84564-148-1.

Zhu W, Ouyang J, Rakvongthai Y, Guehl NJ, Wooten DW, El Fakhri G, Normandin MD, Fan Y. A Bayesian spatial temporal mixtures approach to kinetic parametric images in dynamic positron emission tomography. Med Phys. 2016; 43(3): 1222-1234. doi: 10.1118/1.4941010.

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. doi: 10.1088/0031-9155/51/17/007.

Yaqub M, Boellaard R, Kropholler MA, Lubberink M, Lammertsma AA. Simulated annealing in pharmacokinetic modeling of PET neuroreceptor studies: accuracy and precision compared with other optimization algorithms. Nuclear Science Symposium Conference Record, 2004 IEEE. 5: 3222-3225. doi: 10.1109/NSSMIC.2004.1466368.

Tags: , ,

Updated at: 2022-03-26
Created at: 2017-08-07
Written by: Vesa Oikonen