Alzheimer's disease is a complex, multi-factorial, and multi-parametric neurodegenerative etiology. Mathematical models can help understand such a complex problem by providing a way to explore and conceptualize principles, merging biological knowledge with experimental data into a model amenable to simulation and external validation, all without the need for extensive clinical trials. We performed a scoping review of mathematical models describing the onset and evolution of Alzheimer's disease as a result of biophysical factors following the PRISMA standard. Our search strategy applied to the PubMed database yielded 846 entries. After using our exclusion criteria, only 17 studies remained from which we extracted data, which focused on three aspects of mathematical modeling: how authors addressed continuous time (since even when the measurements are punctual, the biological processes underlying Alzheimer's disease evolve continuously), how models were solved, and how the high dimensionality and non-linearity of models were managed. Most articles modeled Alzheimer's disease at the cellular level, operating on a short time scale (e.g., minutes or hours), i.e., the micro view (12/17); the rest considered regional or brain-level processes with longer timescales (e.g., years or decades) (the macro view). Most papers were concerned primarily with amyloid beta (n = 8), few described both amyloid beta and tau proteins (n = 3), while some considered more than these two factors (n = 6). Models used partial differential equations (n = 3), ordinary differential equations (n = 7), and both partial differential equations and ordinary differential equations (n = 3). Some did not specify their mathematical formalism (n = 4). Sensitivity analyses were performed in only a small number of papers (4/17). Overall, we found that only two studies could be considered valid in terms of parameters and conclusions, and two more were partially valid. This puts the majority (n = 13) as being either invalid or with insufficient information to ascertain their status. This was the main finding of our paper, in that serious shortcomings make their results invalid or non-reproducible. These shortcomings come from insufficient methodological description, poor calibration, or the impossibility of experimentally validating or calibrating the model. Those shortcomings should be addressed by future authors to unlock the usefulness of mathematical models in Alzheimer's disease.

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涵盖阿尔茨海默病进展的数学模型的范围审查

阿尔茨海默病是一种复杂、多因素、多参数的神经退行性病因。数学模型可以通过提供一种探索和概念化原理的方法，将生物学知识与实验数据合并成一个易于模拟和外部验证的模型来帮助理解这样一个复杂的问题，所有这些都不需要进行广泛的临床试验。我们对数学模型进行了范围审查，描述了阿尔茨海默病的发病和演变，这是遵循 PRISMA 标准的生物物理因素的结果。我们应用于 PubMed 数据库的搜索策略产生了 846 个条目。使用我们的排除标准后，只剩下 17 项研究可供我们提取数据，这些研究重点关注数学建模的三个方面：作者如何处理连续时间（因为即使测量是准时的，阿尔茨海默病的生物过程也会不断演变），如何模型得到了解决，以及如何管理模型的高维和非线性。大多数文章在细胞水平上模拟阿尔茨海默病，在短时间尺度（例如分钟或小时）上进行操作，即微观视图（12/17）；其余的则考虑了具有较长时间尺度（例如，几年或几十年）的区域或大脑水平过程（宏观观点）。大多数论文主要关注β淀粉样蛋白（n= 8），很少有人描述了β淀粉样蛋白和tau蛋白（n= 3），而有些人考虑的不仅仅是这两个因素（n= 6）。模型使用偏微分方程（n= 3), 常微分方程 (n= 7), 偏微分方程和常微分方程 (n= 3）。有些没有具体说明他们的数学形式（n= 4）。仅在少数论文中进行了敏感性分析 (4/17)。总体而言，我们发现只有两项研究在参数和结论方面可以被认为是有效的，另外两项研究是部分有效的。这使得大多数（n= 13) 无效或没有足够的信息来确定其状态。这是我们论文的主要发现，因为严重的缺陷导致其结果无效或不可重现。这些缺点来自方法描述不足、校准不良或无法通过实验验证或校准模型。未来的作者应该解决这些缺点，以释放数学模型在阿尔茨海默病中的有用性。