# Comparison of Algorithms to Compute Relaxation Time Maps in Magnetic Resonance Imaging
## 1 Introduction
In addition to the models listed in Table 1, it is also possible to obtain T1 maps using a technique called the \"variable flip angle approach\" (VFA). This approach involves acquiring several images with different flip angles and then using a linear fit to calculate the T1 values \begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Equation** \\ \hline I. Inversion Recovery & \(S=A+B\cdot|1-2\cdot\exp(-T/T_{rel})|\ A\geq 0\) \\ \hline II. Inversion Recovery or MOLLI & \(S=|A-B\cdot\exp(-T/T_{rel})|\) \\ \hline III. Simplified Spin Echo or Simplified & \\ Gradient Echo & \(S=A\cdot\exp(-T/T_{rel})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Some MRI signal models can be fitted using the algorithm described in this manuscript. The fit can be partially linearized in all cases, so only one parameter (relaxation time) needs to be varied to find the lowest error. from these images [14; 15; 16; 17]. The advantage of this technique is that it does not require the use of nonlinear regression. It is also possible to use specialized nonlinear algorithms, such as NOVIFAST, specifically tailored to VFA T1 mapping [18]. This approach has been widely used in the medical field for T1 mapping, but it is not the focus of this paper. Instead, this paper will discuss other models that are more complex and require nonlinear regression. This paper does not aim to study the effect of imaging conditions or mathematical models on relaxation time maps. This paper, given acquired data and a mathematical model, aims to compare how different fitting algorithms perform only in terms of fit error. Levenberg-Marquardt [19; 20] and Nelder-Mead [21] algorithms are frequently used for this purpose. These algorithms, widely used in relaxation time mapping software such as Segment or Matlab [7], implement a search in a two or three-dimensional space. Since the optimization problem is partly linear, it is possible to split the linear and nonlinear problems. One such approach is the variable projections algorithm (VARPRO) [22], which allows using an algorithm such as Levenberg-Marquardt for the nonlinear part of the problem. It is also possible to apply other search strategies for the nonlinear part of the problem [23]. In 2010, a new algorithm called the Reduced Dimension Nonlinear Least Squares (RD-NLS) [24] was presented as a method to compute relaxation times in MRI. This algorithm is a variation of the traditional nonlinear least squares algorithm. It addresses some of the limitations of the previous methods by splitting the linear and nonlinear problems and then optimizing the remaining nonlinear parameter, relaxation time, using a grid search instead of a local search. The main advantage of this algorithm is that it is more computationally efficient and has the potential to produce more accurate results. This article compares the performance of the RD-NLS algorithm to other existing algorithms in terms of fit error. A lower fit error indicates a better fit and more accurate results. By comparing the RD-NLS algorithm to other algorithms, the authors aim to demonstrate the superiority of this new method in terms of accuracy and efficiency.