Mathematical Model Shows How Sleep May Affect Amyloid-β Fibrillization

Mathematical models on Alzheimer's disease and its treatment: A review

Alzheimer's disease is a gradually advancing neurodegenerative disease. According to the report by "World Health Organization (WHO)", there are over 55 million individuals currently living with Alzheimer's disease and other dementia globally, and the number of sufferers is increasing every day. In absence of effective cures and preventive measures, this number is predicted to triple by 2050. The disease's origin is still unclear, and also no such treatment is available for eradicating the disease. Based on the crucial factors that are connected to the disease's progression, the authors developed several types of mathematical models. We review such mathematical models that are utilized to better understand the pathophysiology of Alzheimer's disease. Section-wise, we categorize the mathematical models in terms of different components that might be responsible for Alzheimer's disease. We explain the mathematical models with their descriptions and respective conclusions. In addition to mathematical models, we concentrate on biological aspects of the disease and possible therapeutic targets. We explore the disease's biological basis primarily to understand how proteins, glial cells, cytokines, genes, calcium signaling and oxidative stress contribute to the disease. We go through several treatment targets that might stop the progression of the disease or at least slow it down. We present a table that summarizes the mathematical models in terms of their formalisms, highlighting key components and important remarks.

Proteins clump: Mechanics and transport during neurodegeneration

This summary of recent contributions in the Biophysical Journal from 2020 to 2023 highlights new mechanistic insights into key biomechanical and biophysical aspects of neurodegeneration. Neurodegeneration encompasses complex diseases characterized by the progressive loss of neuronal function, often linked to protein accumulation and aggregation. Several factors, including mechanical properties and structural composition of brain tissue, formation of proteinaceous condensates within cells, and protein transport between cells, impact the loss of neural function.

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 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.

How Can We Use Mathematical Modeling of Amyloid-β in Alzheimer's Disease Research and Clinical Practices?

The accumulation of amyloid-β (Aβ) plaques in the brain is considered a hallmark of Alzheimer's disease (AD). Mathematical modeling, capable of predicting the motion and accumulation of Aβ, has obtained increasing interest as a potential alternative to aid the diagnosis of AD and predict disease prognosis. These mathematical models have provided insights into the pathogenesis and progression of AD that are difficult to obtain through experimental studies alone. Mathematical modeling can also simulate the effects of therapeutics on brain Aβ levels, thereby holding potential for drug efficacy simulation and the optimization of personalized treatment approaches. In this review, we provide an overview of the mathematical models that have been used to simulate brain levels of Aβ (oligomers, protofibrils, and/or plaques). We classify the models into five categories: the general ordinary differential equation models, the general partial differential equation models, the network models, the linear optimal ordinary differential equation models, and the modified partial differential equation models (i.e., Smoluchowski equation models). The assumptions, advantages and limitations of these models are discussed. Given the popularity of using the Smoluchowski equation models to simulate brain levels of Aβ, our review summarizes the history and major advancements in these models (e.g., their application to predict the onset of AD and their combined use with network models). This review is intended to bring mathematical modeling to the attention of more scientists and clinical researchers working on AD to promote cross-disciplinary research.

Personalized Computational Causal Modeling of the Alzheimer Disease Biomarker Cascade

BACKGROUND

Mathematical models of complex diseases, such as Alzheimer's disease, have the potential to play a significant role in personalized medicine. Specifically, models can be personalized by fitting parameters with individual data for the purpose of discovering primary underlying disease drivers, predicting natural history, and assessing the effects of theoretical interventions. Previous work in causal/mechanistic modeling of Alzheimer's Disease progression has modeled the disease at the cellular level and on a short time scale, such as minutes to hours. No previous studies have addressed mechanistic modeling on a personalized level using clinically validated biomarkers in individual subjects.

OBJECTIVES

This study aimed to investigate the feasibility of personalizing a causal model of Alzheimer's Disease progression using longitudinal biomarker data.

DESIGN/SETTING/PARTICIPANTS/MEASUREMENTS

We chose the Alzheimer Disease Biomarker Cascade model, a widely-referenced hypothetical model of Alzheimer's Disease based on the amyloid cascade hypothesis, which we had previously implemented mathematically as a mechanistic model. We used available longitudinal demographic and serial biomarker data in over 800 subjects across the cognitive spectrum from the Alzheimer's Disease Neuroimaging Initiative. The data included participants that were cognitively normal, had mild cognitive impairment, or were diagnosed with dementia (probable Alzheimer's Disease). The model consisted of a sparse system of differential equations involving four measurable biomarkers based on cerebrospinal fluid proteins, imaging, and cognitive testing data.

RESULTS

Personalization of the Alzheimer Disease Biomarker Cascade model with individual serial biomarker data yielded fourteen personalized parameters in each subject reflecting physiologically meaningful characteristics. These included growth rates, latency values, and carrying capacities of the various biomarkers, most of which demonstrated significant differences across clinical diagnostic groups. The model fits to training data across the entire cohort had a root mean squared error (RMSE) of 0.09 (SD 0.081) on a variable scale between zero and one, and were robust, with over 90% of subjects showing an RMSE of < 0.2. Similarly, in a subset of subjects with data on all four biomarkers in at least one test set, performance was high on the test sets, with a mean RMSE of 0.15 (SD 0.117), with 80% of subjects demonstrating an RMSE < 0.2 in the estimation of future biomarker points. Cluster analysis of parameters revealed two distinct endophenotypic groups, with distinct biomarker profiles and disease trajectories.

CONCLUSION

Results support the feasibility of personalizing mechanistic models based on individual biomarker trajectories and suggest that this approach may be useful for reclassifying subjects on the Alzheimer's clinical spectrum. This computational modeling approach is not limited to the Alzheimer Disease Biomarker Cascade hypothesis, and can be applied to any mechanistic hypothesis of disease progression in the Alzheimer's field that can be monitored with biomarkers. Thus, it offers a computational platform to compare and validate various disease hypotheses, personalize individual biomarker trajectories and predict individual response to theoretical prevention and therapeutic intervention strategies.

Importance of CSF-based Aβ clearance with age in humans increases with declining efficacy of blood-brain barrier/proteolytic pathways

The kinetics of amyloid beta turnover within human brain is still poorly understood. We previously found a dramatic decline in the turnover of Aβ peptides in normal aging. It was not known if brain interstitial fluid/cerebrospinal fluid (ISF/CSF) fluid exchange, CSF turnover, blood-brain barrier function or proteolysis were affected by aging or the presence of β amyloid plaques. Here, we describe a non-steady state physiological model developed to decouple CSF fluid transport from other processes. Kinetic parameters were estimated using: (1) MRI-derived brain volumes, (2) stable isotope labeling kinetics (SILK) of amyloid-β peptide (Aβ), and (3) lumbar CSF Aβ concentration during SILK. Here we show that changes in blood-brain barrier transport and/or proteolysis were largely responsible for the age-related decline in Aβ turnover rates. CSF-based clearance declined modestly in normal aging but became increasingly important due to the slowing of other processes. The magnitude of CSF-based clearance was also lower than that due to blood-brain barrier function plus proteolysis. These results suggest important roles for blood-brain barrier transport and proteolytic degradation of Aβ in the development Alzheimer’s Disease in humans.

To understand if brain interstitial fluid/cerebrospinal fluid (ISF/CSF) exchange, CSF turnover, blood-brain barrier function or proteolysis were affected by aging or the presence of β amyloid plaques, Elbert et al. develop a non-steady state physiological model using MRI-derived brain volumes, stable isotope labeling kinetics of Aβ, and lumbar CSF Aβ concentration. Their model suggests an important role for blood-brain barrier transport and proteolytic degradation of Aβ in the development Alzheimer’s Disease in humans.

Circadian rhythms in neurodegenerative disorders

Endogenous biological clocks, orchestrated by the suprachiasmatic nucleus, time the circadian rhythms that synchronize physiological and behavioural functions in humans. The circadian system influences most physiological processes, including sleep, alertness and cognitive performance. Disruption of circadian homeostasis has deleterious effects on human health. Neurodegenerative disorders involve a wide range of symptoms, many of which exhibit diurnal variations in frequency and intensity. These disorders also disrupt circadian homeostasis, which in turn has negative effects on symptoms and quality of life. Emerging evidence points to a bidirectional relationship between circadian homeostasis and neurodegeneration, suggesting that circadian function might have an important role in the progression of neurodegenerative disorders. Therefore, the circadian system has become an attractive target for research and clinical care innovations. Studying circadian disruption in neurodegenerative disorders could expand our understanding of the pathophysiology of neurodegeneration and facilitate the development of novel, circadian-based interventions for these disabling disorders. In this Review, we discuss the alterations to the circadian system that occur in movement (Parkinson disease and Huntington disease) and cognitive (Alzheimer disease and frontotemporal dementia) neurodegenerative disorders and provide directions for future investigations in this field.