Alzheimer's disease (AD) is a devastating neurodegenerative condition that precedes progressive and irreversible dementia; thus, predicting its progression over time is vital for clinical diagnosis and treatment. Numerous studies have implemented structural magnetic resonance imaging (MRI) to model AD progression, focusing on three integral aspects: (i) temporal variability, (ii) incomplete observations, and (iii) temporal geometric characteristics. However, deep learning-based approaches regarding data variability and sparsity have yet to consider inherent geometrical properties sufficiently. The ordinary differential equation-based geometric modeling method (ODE-RGRU) has recently emerged as a promising strategy for modeling time-series data by intertwining a recurrent neural network and an ODE in Riemannian space. Despite its achievements, ODE-RGRU encounters limitations when extrapolating positive definite symmetric metrics from incomplete samples, leading to feature reverse occurrences that are particularly problematic, especially within the clinical facet. Therefore, this study proposes a novel geometric learning approach that models longitudinal MRI biomarkers and cognitive scores by combining three modules: topological space shift, ODE-RGRU, and trajectory estimation. We have also developed a training algorithm that integrates manifold mapping with monotonicity constraints to reflect measurement transition irreversibility. We verify our proposed method's efficacy by predicting clinical labels and cognitive scores over time in regular and irregular settings. Furthermore, we thoroughly analyze our proposed framework through an ablation study.

翻译：阿尔茨海默病（AD）是一种毁灭性的神经退行性疾病，会导致进行性且不可逆的痴呆；因此，预测其随时间的变化对临床诊断和治疗至关重要。众多研究利用结构性磁共振成像（MRI）对AD进展进行建模，重点关注三个核心方面：（i）时间变异性，（ii）不完全观测，以及（iii）时间几何特征。然而，现有基于深度学习的方法在处理数据变异性和稀疏性时，尚未充分考虑到固有的几何特性。基于常微分方程的几何建模方法（ODE-RGRU）近期作为一种有前景的时序数据建模策略崭露头角，它通过在黎曼空间中将循环神经网络与ODE相结合。尽管取得了成就，ODE-RGRU在从不完全样本中外推正定对称度量时遇到限制，导致特征逆反现象，这在临床层面尤为棘手。因此，本研究提出了一种新颖的几何学习方法，通过结合三个模块（拓扑空间移位、ODE-RGRU和轨迹估计）对纵向MRI生物标志物和认知评分进行建模。我们还开发了一种训练算法，将流形映射与单调性约束相结合，以反映测量转换的不可逆性。我们通过预测常规和不规则设置下的临床标签及认知评分随时间的变化，验证了所提方法的有效性。此外，我们通过消融研究对所提框架进行了深入分析。