# Comparison of Algorithms to Compute Relaxation Time Maps in Magnetic Resonance Imaging
## 1 Introduction  
Magnetic Resonance Imaging (MRI) uses magnetic fields and radio waves to create detailed body images. The intensity of the signal in an MRI image is influenced by various factors, including relaxation times. These times are constants that indicate the magnetic properties of the tissue being imaged. Images that show the relaxation times for each pixel in the image are called relaxation time maps. Accurate relaxation time mapping in MRI is crucial for clinical diagnosis and treatment planning. T1 and T2 relaxation times are sensitive to tissue composition, microstructure, and other physiological properties. These maps have many uses in both research and clinical settings, such as determining the amount of Myocardial Extracellular Volume (ECV) in the heart [1], detecting edema in cases of acute myocardial infarction [2], identifying early tumor progression in the brain [3], differentiating between healthy and early degenerative cartilage [4], or providing helpful information to classify between normal and pathological brain aging [5]. Accurate relaxation time mapping is, therefore, critical for identifying and characterizing disease, monitoring its progression, and guiding interventions such as radiation therapy or surgical resection. Additionally, there is growing interest in using quantitative MRI for research in other areas, such as food products and processing [6]. Moreover, the choice of the algorithm used for relaxation time mapping can have a significant impact on the accuracy and precision of the resulting maps. To improve the accuracy of MRI data and reduce errors, it is important to continue developing new models and algorithms for processing MRI data, which will reduce variability and facilitate a more accurate diagnosis/classification [7]. Relaxation times that are routinely measured in MR are longitudinal or spin-lattice relaxation time (T1), transversal or spin-spin relaxation time (T2), and apparent transversal relaxation time (T2*) [8; 9; 10]. In what follows, this will be referred to as \"relaxation time\" (T\({}_{\text{rel}}\)). The type of pulse sequence and the technical parameters used during the acquisition determine how each time factor affects the final signal intensity. For a detailed description of MR signal formation, the reader is referred to the textbooks on this topic [11; 12; 13]. Creating relaxation time maps in MRI is a multi-step process because the MR signal is affected by various factors. A group of MR pulse sequences is used to generate the maps, with different techniques listed in Table 1. The basic concept behind most of these methods is to acquire a set of images with similar imaging parameters, except for one, such as repetition time (TR), echo time (TE), inversion time (TI), or saturation time (TS). This parameter is referred to as the \"time parameter\" (T). Once the images are acquired, a pixel-by-pixel curve fitting analysis is done according to a theoretical equation that describes the physical phenomena behind the process. This analysis produces the desired quantitative maps. As it is clear from the formulas in Table 1, nonlinear regression is needed to fit the data to obtain relaxation times. A widely used method to obtain the optimum fit is to minimize the sum of the squares of the differences between predicted and measured signals. In this context, the predicted signal is the value predicted by the mathematical model and varies as the fitting parameters are changed. The measured signal refers to the values measured in the MRI experiment (or simulated if the experiment is a simulation) and does not vary during the fitting process. The goal of the fitting process is to vary the model parameters in a way in which the predicted values are as close as possible to the measured values. The sum of the squares of the differences between predicted and measured values is the error, which is to be minimized. If two algorithms are fitting experimental data to the same model, we consider that the algorithm resulting in the lowest error outperforms the other algorithm.