## 3 Results and Discussion  
The following methodological steps (Figure 1) were carried out to examine the relationship between the body mass index (BMI) values and brain alterations observed in MRI scans:  
1. Initially, a dataset comprising patients who underwent body weight, height, and musculature measurements was curated. To be included, patients were required to have undergone multiple assessments, with the initial examination indicating minimal changes and subsequent assessments reflecting advanced stages of dementia or Alzheimer's disease. 2. Cluster analysis was performed using numerical data, including calculated BMI values and muscularity scores, forming up to five distinct clusters. 3. The Kullback-Leibler divergence was computed separately for each patient's first and final MRI images. 4. The mean Kullback-Leibler divergence value within each cluster was calculated to evaluate whether participants' physical characteristics corresponded to the progression of brain changes. This study employed a cluster analysis on numerical data, focusing on BMI and muscle test scores, to explore patterns across different cluster configurations. The elbow method and the silhouette score were utilized to determine the optimal number of clusters. The scores were computed for clusters of two, three, four, and five. The results of the silhouette score are displayed in Figure 2 and Table 1.  
When analyzing the silhouette score in relation to the overall average score of the entire dataset, the optimal number of clusters was two, as these clusters were the only ones with a maximum score exceeding the dataset's overall average. For three clusters, one cluster's score was equal to the average. In the case of four clusters, the scores for two were close enough to warrant consideration. However, when examining five clusters, the scores for two were too low to be significant. Figure 1: An overview of the experimental research process and methodology. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Cluster**} & \multicolumn{2}{c}{**Two Clusters**} & \multicolumn{2}{c}{**Three Clusters**} & \multicolumn{2}{c}{**Four Clusters**} & \multicolumn{2}{c}{**Five Clusters**} \\ \cline{2-7}  & **Avg. Score** & **Max Score** & **Avg. Score** & **Max Score** & **Avg. Score** & **Max Score** & **Avg. Score** & **Max Score** \\ \hline C1 & 0.6502 & 0.7357 & 0.5118 & 0.6569 & 0.4391 & 0.5397 & 0.7193 & 0.8126 \\ C2 & 0.3950 & 0.5613 & 0.3627 & 0.5378 & 0.5079 & 0.6799 & 0.3887 & 0.5460 \\ C3 & & & 0.6352 & 0.7596 & 0.3627 & 0.5378 & 0.5580 & 0.7100 \\ C4 & & & & 0.7369 & 0.8290 & 0.4152 & 0.6417 \\ C5 & & & & & 0.4391 & 0.5397 \\ \hline Schema Average & 0.5287 & 0.5382 & 0.5538 & & 0.5641 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Silhouette scores for clustering schemas with two, three, four, and five clusters. Figure 2: The results of the silhouette score in different clustering schemas ((**A**)—two clusters; (**B**)—three clusters; (**C**)—four clusters; (**D**)—five clusters). The red dashed line represents the average silhouette score for each schema. The comparison of patients grouped into two clusters revealed that individuals with a higher body mass index (BMI) exhibited fewer brain changes over time. Moreover, these individuals achieved higher scores on muscle strength assessments. However, the results from the two-cluster analysis provided limited insights, as dividing patients into only two groups did not allow for detailed conclusions. Nonetheless, a consistent subgroup of patients was identified from the beginning: this group exhibited the lowest average body mass index (BMI) and predominantly achieved a physical test score of 1.