# Non-negative least squares (NNLS)

In the analysis of PET data, compartmental models can be used to simulate the concentration of radiotracer in tissue as a function of time, and compartment model parameters can be fitted by minimizing the difference between the simulated and measured tissue radioactivity concentration. Non-linear least-squares optimization algorithms, such as PSO, are commonly used methods for estimating model parameters from regional tissue curves, but they tend to be too slow for pixel-by-pixel calculation of PET images. Some compartmental models for PET data analysis can be represented in multilinear form which can be solved using very fast linear least squares methods. The NNLS method is a robust multilinear optimization methods for this purpose, because the parameters of compartmental models represent volumes and rate constants which cannot be negative. The non-negative least square problem is expressed as

which is suitable for use with PET data as explained in TPC modelling report 20. In addition, NNLS can be used in spectral analysis and basis function methods.

## Literature

Bro R, De Jong S. A fast non-negativity-constrained least squares algorithm. J Chemometr. 1997; 11(5): 393-401. doi: 10.1002/(SICI)1099-128X(199709/10)11:5<393::AID-CEM483>3.0.CO;2-L.

Franc V, Hlaváč V, Navara M. (2005). Sequential Coordinate-Wise Algorithm for the Non-negative Least Squares Problem. In: Gagalowicz A, Philips W (eds) Computer Analysis of Images and Patterns. CAIP 2005. Lecture Notes in Computer Science, vol 3691. Springer. doi: 10.1007/11556121_50.

Lawson CL, Hanson RJ: Solving Least Squares Problems. Prentice-Hall, 1974. doi: 10.1137/1.9781611971217.

Sederholm K: Using NNLS in multilinear PET problems. TPC modelling reports, 2003. tpcmod0020.pdf.

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.

Updated at: 2023-06-16
Created at: 2023-06-07
Written by: Vesa Oikonen