# Comparison of Algorithms to Compute Relaxation Time Maps in Magnetic Resonance Imaging
## 3 Results  
A summary of the comparisons is presented in Table 3, along with Figures 1 and 2 and Supporting Figures S1-S14. According to the results:  
* The T1 fits (Tests 1-10) show many pixels in which error vs. T1 has more than one local minimum, in the 9-70% range. Having more than one local minimum means that \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Test** & **Test Model** & **Algorithm to Compare** &
\begin{tabular}{c} **Software to** \\ **Compare** \\ \end{tabular} & **Data** \\ \hline
1 & I & Levenberg-Marquardt & ISA & mouse \\ \hline
2 & I & Levenberg-Marquardt & Matlab & mouse \\ \hline
3 & I & [PERSON] & Matlab & mouse \\ \hline
4 & I & Split & Matlab & mouse \\ \hline
5 & II & [PERSON] & Matlab & mouse \\ \hline
6 & II & Nelder-Mead & Matlab & mouse \\ \hline
7 & II & Split & Matlab & mouse \\ \hline
8 & II & [PERSON] & Matlab & simulated \\ \hline
9 & II & [PERSON] & Matlab & simulated \\ \hline
10 & II & Split & Matlab & simulated \\ \hline
11 & III & [PERSON] & ISA & phantom \\ \hline
12 & III & [PERSON] & Matlab & phantom \\ \hline
13 & III & Nelder-Mead & Matlab & phantom \\ \hline
14 & III & Split & Matlab & phantom \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of tests performed in this article. The RD-NLS algorithm has been compared to the reported algorithm in all cases. Inversion-recovery mouse images, inversion-recovery simulated human images, and multi-slice multi-echo images were used to fit T1 and T2 maps. a local search algorithm could get trapped in a local minimum which is not a global minimum (see Figure 3). This is not the case for the T2 fits (Tests 11-14), where error vs. T2 has only one local minimum. * The T\({}_{\rm rel}\) value obtained by the RD-NLS algorithm always corresponds to the global optimum. * Comparing [PERSON] and Matlab fit versus RD-NLS for the T1 maps (Tests 1-10), the fit is different in a significant number of pixels, in the range of 5-97%. However, this is not true for the T2 maps (Tests 11-14). Figure 1: Bland-Altman plot corresponding to Test 1 (inversion recovery, mouse data). It is evident that the T1 obtained by [PERSON]’s fit is greater than that derived from RD-NLS. Figure 2: Bland-Altman plot corresponding to Test 8 (inversion recovery, simulated human data). T1 values show a significant dispersion. Moreover, T1 obtained by Matlab’s fit (Lebenberg-Marquardt) is greater than the one attained by RD-NLS. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test Number** & **Minima (\%)** & **Pixels** & **Relaxation Time (ms)** \\ \hline
1 & 9.2\% & 97.0\% & 622/636 \\ \hline
2 & 9.2\% & 4.1\% & 622/623 \\ \hline
3 & 9.2\% & 0.6\% & 622/622 \\ \hline
4 & 9.2\% & 4.1\% & 622/623 \\ \hline
5 & 18.6\% & 4.8\% & 807/805 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results from the tests. Notice that the T2 fits (Tests 11–14) are much more similar for the compared algorithms than the T1 fits (Tests 1–10). Also, the T2 fits (Tests 11–14) show no multiple minima, whereas multiple minima are common in the T1 fits (Tests 1–10). of agreement for all tests. In all cases, the differences refer to the relaxation time obtained using the algorithm used as a comparison minus the relaxation time obtained using the RD-NLS algorithm. As it is obvious from the results of these plots, the T1 values obtained in mouse data fitted with Matlab versus RD-NLS are quite similar for Tests 2-7. For Tests 8-10, the T1 values obtained in simulated data fitted with Matlab and RD-NLS show much more dispersion than in Tests 2-7. Besides, in Tests 8 and 9, Levenberg-Marquardt and Nelder-Mead fit gives a higher T1 value than RD-NLS fit, whereas no such difference appears in Test 10 (Split versus RD-NLS). Test 1 (mouse data) shows a clear distinction, with ISA fit giving a higher T1 value than RD-NLS fit. However, T2 values obtained in both fits are very similar for Tests 11-14 (phantom data).