## 2 Materials and Methods
### Kullback-Leibler Divergence Measure  
The Kullback-Leibler (KL) divergence was first introduced in [25]. It is also known as the Kullback-Leibler distance, cross-entropy, information divergence, and information for discrimination [26]. The Kullback-Leibler (KL) divergence measures the difference between two probability distributions, i.e., it quantifies how one distribution, \(P\), diverges from another, \(Q\)[27; 28; 29]. The KL divergence is defined as shown in Equation (4).  
\[D_{\text{KL}}(P\|Q)=\sum_{i}P(i)\log\frac{P(i)}{Q(i)} \tag{4}\]  
where \(P(i)\) represents the probability of observing a specific feature (e.g., hippocampal volume) in the target distribution, and \(Q(i)\) denotes the corresponding probability in the reference distribution.  
The KL divergence quantifies the relative entropy between these distributions, with larger values indicating more significant divergence. The measure is non-negative (\(D_{\text{KL}}\geq 0\)), with a zero value when the distributions are identical. It is asymmetric, meaning that \(D_{\text{KL}}(P\|Q)\
eq D_{\text{KL}}(Q\|P)\), which emphasizes the importance of choosing an appropriate reference distribution. Unlike summary metrics such as the mean or variance, the KL divergence evaluates the entire shape of the distributions, capturing nuanced variations in atrophy patterns across groups. This comprehensive evaluation was particularly relevant for detecting the effects of physical activity on brain structures like the hippocampus, where the impact may manifest as small but significant differences in atrophy progression.  
The distributions were extracted from preprocessed MRI scans, where voxel intensities or regional volumes were normalized to create valid probability distributions that summed to one. In the context of this study, the KL divergence was applied to analyze the following:1. The intersubject variability;
2. The intrasubject variability.  
(A) The intersubject variability was analyzed to evaluate differences in hippocampal atrophy distributions between groups with varying physical activity levels. The target distribution, \(P\), represented individuals with higher physical activity levels, while the reference distribution, \(Q\), corresponded to sedentary individuals. By computing \(D_{\text{KL}}(P\|Q)\), the divergence highlighted the extent to which physical activity mitigated brain atrophy.  
(B) The intrasubject variability was analyzed to compare MRI images of the same patient to analyze dementia progression and track how the divergence between active and passive groups evolved. The target distribution, \(P\), represented the current state of the brain (the MRI taken at a later time point). In contrast, the reference distribution, \(Q\), corresponded to the baseline state of the brain (the MRI taken at an earlier time point). A higher \(D_{\text{KL}}\) value indicated greater divergence, suggesting more significant structural changes, which may correspond to disease progression. Conversely, a lower \(D_{\text{KL}}\) value indicated minimal divergence, suggesting stability in the brain structure over time.  
As a result, our study offers more profound insights into the temporal dynamics of dementia progression and neuroprotection.