# Particle swarm optimization (PSO)

The parameters of plasma input TACs and compartmental models are estimated (fitted) using non-linear least-squares optimization algorithms, such as PSO.

Particle swarm optimization technique is a derivative-free stochastic optimization algorithm based on social interaction of bird flocks, developed by Kennedy and Eberhart. In PSO, the particles (birds) are placed in the search space of the objective function, each with their own location. The movement of each particle is determined by the past of its best (best-fitness) and current locations with the members of the swarm. The position and velocity of each particle are iteratively adjusted with previous best position and neighbouring particle in the swarm.

Each particle *P _{i}* (

*i=1,2,...,S*) in the swarm of size

*S*is represented by its position

*x*, velocity

_{i}(t)*v*, and the best position

_{i}(t)*x*throughout its trajectory so far. The best position of the particles is given by

_{bi}(t)*x*. The movement of the particles to the next positions is calculated using equations

_{g}(t), where *r _{1}(t)* and

*r*are randomly set values that enable the stochastic search. Acceleration coefficients

_{2}(t)*c*and

_{1}*c*regulate the velocities of the particle, considering the previous best position of the individual particle and other particles of the swarm. Coefficient

_{2}*w*(inertia weight) controls the local and global exploration degree of the search; high value of

*w*enables better global exploration, while small

*w*allows the search of local optimum. Iterations (movement of each particle) are carried on until until progress is stalled (fitness does not improve) or until a fixed maximum number of iterations has passed.

The initial population is usually set randomly. Since it is more important that the initial population is evenly distributed than that is random, quasi-random low-discrepancy sequences, such as Halton and Sobol sequences, are preferred (Maaranen et al., 2004; Weerasinghe et al., 2016).

## See also:

- Compartmental models
- Compartmental model fitting
- Input function
- Fitting PET input curves
- Fitting the fractions of unchanged tracer in plasma

## Literature

Clerc M: *Particle Swarm Optimization.* ISTE Ltd, 2006.
doi: 10.1002/9780470612163.

Eberhart RC, Kennedy J. A new optimizer using particle swarm theory.
*Proceedings of the Sixth International Symposium on Micromachine and Human Science*,
Nagoya, Japan, 1995: 39-43.

Floudas CA, Pardalos PM (eds.): *Encyclopedia of Optimization*.
2nd ed., Springer, 2009.
doi: 10.1007/978-0-387-74759-0.

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. *9th IEEE International Symposium on Biomedical Imaging (ISBI 2012)*, 586-589.
doi: 10.1109/ISBI.2012.6235616.

Kennedy J, Eberhart RC. Particle swarm optimization. *Proceedings of IEEE International
Conference on Neural Networks, Piscataway, NJ. 1995: 1942-1948.
doi: 10.1109/ICNN.1995.488968.*

Kennedy J, Eberhart RC, Shi Y: *Swarm Intelligence.* Academic Press, 2001.
eISBN: 9780080518268.

Meng A, Li Z, Yin H, Chen S, Guo Z. Accelerating particle swarm optimization using crisscross
search. *Information Sciences* 2016; 329: 52-72.
doi: 10.1016/j.ins.2015.08.018.

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.

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: Fitting, Optimization algorithm, NLLS

Updated at: 2022-03-27

Created at: 2017-01-25

Written by: Vesa Oikonen