# Comparison of Algorithms to Compute Relaxation Time Maps in Magnetic Resonance Imaging
## 4 Discussion  
A critical aspect of assessing the performance of a relaxation time fitting algorithm is the accuracy of its predictions, which can be quantified using the error between predicted and measured signal values. All fitting algorithms aim to minimize this error to achieve accurate and reliable relaxation time maps. Our results demonstrate that the RD-NLS algorithm outperforms the comparison algorithms regarding fit error, although the magnitude of improvement varies depending on the specific MRI dataset and signal model. It is noteworthy that we did not observe any instances where the RD-NLS algorithm was  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Test Number** & **Bias (ms)** & **Limits of Agreement (ms)** \\ \hline
1 & 14.07 & \(-\)1.63 to 29.76 \\ \hline
2 & 0.88 & \(-\)13.11 to 14.86 \\ \hline
3 & 0.07 & \(-\)5.22 to 5.36 \\ \hline
4 & 0.92 & \(-\)13.19 to 15.03 \\ \hline
5 & \(-\)1.87 & \(-\)27.21 to 23.47 \\ \hline
6 & \(-\)0.61 & \(-\)19.67 to 18.45 \\ \hline
7 & \(-\)1.32 & \(-\)22.08 to 19.45 \\ \hline
8 & 18.20 & \(-\)497.54 to 533.95 \\ \hline
9 & 17.34 & \(-\)472.89 to 507.57 \\ \hline
10 & \(-\)0.63 & \(-\)449.79 to 448.53 \\ \hline
11 & \(-\)0.01 & \(-\)3.32 to 3.31 \\ \hline
12 & 0.01 & \(-\)2.76 to 2.78 \\ \hline
13 & 0.01 & \(-\)2.75 to 2.78 \\ \hline
14 & 0.02 & \(-\)2.82 to 2.85 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Bias and limits of agreement from the tests. Bland-Altman plots corresponding to this table are given in Figures 1 and 2, and in Figures S1–S14 in the supplementary materials.  
outperformed by the comparison algorithms. As shown in Figure 1, the difference in relaxation time values between algorithms can have a significant impact on tissue characterization. Therefore, the bias introduced by inaccurate relaxation time mapping could have implications for clinical diagnosis and treatment planning, underscoring the importance of developing and validating robust algorithms such as RD-NLS for accurate and reliable relaxation time mapping in MRI.  
Figure 3 provides insight into the superior performance of the RD-NLS algorithm over the other tested algorithms and highlights the issue of bias in Trel values obtained from the fit. The \"W\" shape of the fitting error vs. Trel curve with multiple local minima can be observed for some pixels, as shown in Figure 3. The RD-NLS algorithm can find all such minima and choose the one that leads to the lowest error. In contrast, a local search algorithm may get stuck in a local minimum, depending on the starting conditions, which are known to be a critical issue when using this type of algorithm. In fairness, the authors attempted to find good starting conditions, but it is not always an easy task. For example, Figure 3 illustrates that for a particular pixel, the region between 680-760 ms is a good starting point for parameter T1. If the starting point is inappropriate, a local search algorithm may converge to a suboptimal solution and overestimate T1. For example, if the starting point is T1 = 730 ms, a local search algorithm will converge to T1 = 740 ms, and this local search algorithm will, in this case, overestimate T1. However, if the starting point is T1 = 710 ms, a local search algorithm will converge to T1 = 696 ms, and this local search algorithm will find the correct T1 value. This example illustrates the importance of using appropriate algorithms and techniques in fitting image data to models, as the choice of algorithm can greatly impact the accuracy of the results. The RD-NLS algorithm provided a more accurate estimate of the relaxation time, while the Matlab algorithm reached a suboptimal solution. The choice of algorithm can greatly impact the accuracy of the results, and the RD-NLS algorithm eliminates the risk of failure to find the global optimum. Although the running time was not measured in this paper, high-speed implementations of the RD-NLS algorithm are available [30].