A Scoping Review of Mathematical Models Covering Alzheimer’s Disease Progression

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 of AD with a search strategy applied to the PubMed database which yielded 846 entries. After applying our exclusion criteria, only 17 studies remained from which we extracted data, focusing on three aspects of mathematical modeling: how authors addressed continuous time, how models were solved, and how the high dimensionality and non-linearity of models were managed. Most articles modeled AD at the cellular range of the disease process, operating on a short time scale (e.g., minutes; hours), i.e., the micro view (12/17); the rest considered regional or brain-level processes, with longer timescales (e.g., years, decades) (the macro view). Most papers were concerned primarily with Aβ (n = 8), few modeled with both Aβ and tau proteins (n = 3), and some considered more than these two factors in the model (n = 6). Models used partial differential equations (PDEs; n = 3), ordinary differential equations (ODEs; n = 7), both PDEs and ODEs (n = 3). Some didn’t specify the 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. The majority (n = 13) either was invalid or there was insufficient information to ascertain their status. While mathematical models are powerful and useful tools for the study of AD, closer attention to reporting is necessary to gauge the quality of published studies to replicate or continue with their contributions.