# Comparison of Algorithms to Compute Relaxation Time Maps in Magnetic Resonance Imaging
## 2 Materials and Methods  
Table 1 lists three distinct MR signal models that can be effectively fitted using the RD-NLS algorithm. **Model I. ** Inversion Recovery: In this case, \(T_{rel}=T1\), \(T=T_{I}\) and \(S=A+B\cdot|1-2\cdot\exp(-T/T_{rel})|\) where \(A\geq 0\). To fit this model, let \(S=y\) and \(x=|1-2\cdot\exp(-T/T_{rel})|\). The linear fit is carried out according to: \(y=A+B\cdot x\). If the result is \(A<0\), the algorithm lets \(A=0\) and repeats the fit according to: \(y=B\cdot x\). **Model II. ** Inversion Recovery [11; 13] or MOLLI [25]: In this case, \(T_{rel}=T1\) (Inversion Recovery) or \(T_{rel}=T1^{\prime}\) (MOLLI), \(T=T_{I}\) and \(S=|A-B\cdot\exp(-T/T_{rel})|\). To fit this model, let \(S=|y|\) and \(x=-\exp(-T/T_{rel})\). The linear fit is performed according to \(y=A+B\cdot x\). This case is more difficult since the absolute value of \(y\) is known but its sign is unknown. In principle, with \(n\) data points, \(2^{n}\) sign combinations are possible. However, since the signal is recovering from inversion, \(y\) increases as \(T\) increases, so only \(n\) sign combinations need to be considered [24; 26]. Therefore, \(n\) linear fits are considered corresponding to the sign of \(y\) and the one for which the error is lowest is taken. **Model III. ** Simplified Spin Echo (T2) or Simplified Gradient Echo (T2'): In this case, \(T=T_{E}\) and \(T_{rel}=T2\) for spin-echo or \(T_{rel}=T2^{*}\) for gradient echo. To fit this model, let \(S=y\) and \(x=\exp(-T/T_{rel})\). The linear fit is performed according to:\(y=A\cdot x\). For validating purposes, three sets of images were used:  
* Inversion recovery (IR) images from a healthy mouse. * A series of MRI human brain images, simulated using a web-based software available at [[http://brainweb.bic.muni.mcgill.ca/brainweb](http://brainweb.bic.muni.mcgill.ca/brainweb)]([http://brainweb.bic.muni.mcgill.ca/brainweb](http://brainweb.bic.muni.mcgill.ca/brainweb)) (accessed on 10 March 2018) [27; 28]. * Multi-slice multi-echo spin-echo images of a phantom consisting of scaffolds in which cells were grown, filled with phosphate-buffered saline. The first two datasets were used to fit T1 relaxation times, and the third dataset was used to obtain T2 relaxation times. **Imaging:** Healthy mouse images were taken using an IR sequence in a Bruker 1T benchtop MRI scanner (ICON 1-T MRI; Burker BioSpin GmbH, Ettlingen, Germany). The main sequence parameters were as follows: T\({}_{\rm R}\) = 7500 ms; Acquisition matrix = 80 \(\times\) 80; FOV = 20 \(\times\) 20 mm\({}^{2}\), giving a pixel resolution of 0.25 \(\times\) 0.25 mm\({}^{2}\); Number of Slices = 3; Slice Thickness = 1.25 mm; Slice Gap = 1 mm; T\({}_{\rm E}\) = 5 ms; T\({}_{\rm I}\) = 35, 50, 75, 100, 200, 250, 350, 500, 650, 800, 1000, 1500, 2000, 3000, 5000, 7000 ms; flip angle = 90\({}^{\circ}\). Human brain images were custom simulated with a FOV of 181 \(\times\) 217 \(\times\) 181 mm\({}^{3}\) and a 181 \(\times\) 217 \(\times\) 181 acquisition matrix, giving a voxel resolution of 1 \(\times\) 1 \(\times\) 1 mm\({}^{3}\). An IR sequence with T\({}_{\rm R}\) = 10 s was used. Echo time was T\({}_{\rm E}\) = 10 ms, and inversion times were T\({}_{\rm I}\) = 35, 50, 75, 100, 200, 250, 350, 500, 650, 800, 1000, 1500, 2000, 3000, 5000, and 7000 ms. Other simulation parameters were flip angle = 90\({}^{\circ}\), INU field = \"Field A\", INU level = 20%, and noise level = 10%. For simplicity, only 11 slices in the center were selected for fitting. Phantom images were taken using an MSME sequence in a T Bruker Biospec 70/30 USR MRI system (Bruker Biospin, GmbH, Ettlingen, Germany). Main sequence parameters were as follows: T\({}_{\rm R}\) = 4000 ms; Acquisition matrix = 256 \(\times\) 256; FOV = 32 \(\times\) 32 mm\({}^{2}\), giving a pixel resolution of 0.125 \(\times\) 0.125 mm\({}^{2}\); Number of Slices = 5; Number of echoes = 64; Slice Thickness = 1 mm; Slice Gap = 0.25 mm; T\({}_{\rm E}\) = 7 ms; flip angle = 90\({}^{\circ}\).