## 3 Materials and Methods
### Optimization of the VOIs Selection Using Multi-Objective Optimization Evolutionary Algorithm (MOE)  
In this contribution, a Multi-Objective Optimization Evolutionary Algorithm (MOE) based on a controlled elitist algorithm variant of NSGA-II is used to select the most relevant region of the brain (VOIS), taking into account two objective functions: The classification accuracy and the complexity of the classifier (measure as the number of VOIs required for the system). This optimization tool is based on [PERSON]'s survival of the fittest principle, which states that an initial population of individuals evolves through natural selection. Therefore, a set of solution candidates is retained in MOE, performing selection processes (fitness assignment and sampling are performed on these potential solutions), and finally, several potential solutions can be mixed (in terms of recombination operators) to produce new solutions. In resemblance to [PERSON]'s theory on survival (from an initial population of individuals, and due to a process of natural selection, the population evolves towards better individuals in terms of the fitness function used), the potential solutions are called individuals, and the set of solutions is called the population. In this contribution, a binary coding of the individuals (the solutions) that make up the EMO population has been used. In the terminology of evolutionary algorithms, each individual is represented by a chromosome, which is made up of different genes. The genes in our case represent the VOIs used, indicated by a 1 if the VOI is selected and by a 0 when that VOI is not used. Therefore, the number of brain regions that are selected in a solution of the Multi-Objective Optimization Evolutionary Algorithm is equal to the number of genes that have a value of 1. In our case, as we worked with whole matter and gray matter, the total number of genes that can be activated in a solution of the MOE (equal to the number of VOIs that can be selected for the two volumetric images of the brain) is 3456. The upper part of the Figure 6 shows an example of codification for an individual-j of the population of the evolutionary algorithm. The total size of the population is 250 individuals in the simulations carried out, with a coding example of the entire population shown in the lower part of Figure 6.  
The objective of this research is twofold. The first objective is to deepen the knowledge of which the most relevant regions of the brain are (taking into account both gray and whole matter) that are related to [PERSON]'s disease and the different types of patients analyzed in this contribution. Obtaining these VOIs can be of great relevance for expert doctors and neurologists and specialists in [PERSON]'s disease. The second objective is the realization of an accurate multi-class classifier, whose inputs are the 3D-DWT coefficients of the brain regions that have been selected as most relevant. For the construction of the classifier, in this paper, a multi-class classification system based on SVM will be used.The algorithm has the following steps:  
1. Initialization of the Multi-Objective Optimization Evolutionary Algorithm (MOE), through an initial population \(P_{0}\), and the creation of an initial (empty) set of non-dominated solutions (\(ND_{0}\)). The population size that was used in the simulation was 250. 2. Calculate the fitness value for each individual. For this calculation, using the set of training patients, the mean precision value (10-k-fold is used) of the SVM classifier used in the MOE fitness function must be computed, together with the number of genes (complexity of the system) that are set to one. 3. In the NSGA-II algorithm, there are two components that are relevant in its elitist strategy: sorting the non-dominated solutions and calculating the crowding distance. With this methodology, individuals from the population will be selected based on their fitness function. 4. The evolutionary algorithm generates a new population of individuals.