Alzheimer's disease: analysis of a mathematical model incorporating the role of prions

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.

Roles of astrocytes and prions in Alzheimer's disease: insights from mathematical modeling

We present a mathematical model that explores the progression of Alzheimer's disease, with a particular focus on the involvement of disease-related proteins and astrocytes. Our model consists of a coupled system of differential equations that delineates the dynamics of amyloid beta plaques, amyloid beta protein, tau protein, and astrocytes. Amyloid beta plaques can be considered fibrils that depend on both the plaque size and time. We change our mathematical model to a temporal system by applying an integration operation with respect to the plaque size. Theoretical analysis including existence, uniqueness, positivity, and boundedness is performed in our model. We extend our mathematical model by adding two populations, namely prion protein and amyloid beta-prion complex. We characterize the system dynamics by locating biologically feasible steady states and their local stability analysis for both models. The characterization of the proposed model can help inform in advancing our understanding of the development of Alzheimer's disease as well as its complicated dynamics. We investigate the global stability analysis around the interior equilibrium point by constructing a suitable Lyapunov function. We validate our theoretical analysis with the aid of extensive numerical illustrations.

A mathematical model on the propagation of tau pathology in neurodegenerative diseases

A system of partial differential equations is developed to study the spreading of tau pathology in the brain for Alzheimer's and other neurodegenerative diseases. Two cases are considered with one assuming intracellular diffusion through synaptic activities or the nanotubes that connect the adjacent cells. The other, in addition to intracellular spreading, takes into account of the secretion of the tau species which are able to diffuse, move with the interstitial fluid flow and subsequently taken up by the surrounding cells providing an alternative pathway for disease spreading. Cross membrane transport of the tau species are considered enabling us to examine the role of extracellular clearance of tau protein on the disease status. Bifurcation analysis is carried out for the steady states of the spatially homogeneous system yielding the results that fast cross-membrane transport combined with effective extracellular clearance is key to maintain the brain's healthy status. Numerical simulations of the first case exhibit solutions of travelling wave form describing the gradual outward spreading of the pathology; whereas the second case shows faster spreading with the buildup of neurofibrillary tangles quickly elevated throughout. Our investigation thus indicates that the gradual progression of the intracellular spreading case is more consistent with the clinical observations of the development of Alzheimer's disease.

Generating PET scan patterns in Alzheimer's by a mathematical model

Alzheimer disease (AD) is the most common form of dementia. The cause of the disease is unknown, and it has no cure. Symptoms include cognitive decline, memory loss, and impairment of daily functioning. The pathological hallmarks of the disease are aggregation of plaques of amyloid-β (Aβ) and neurofibrillary tangles of tau proteins (τ), which can be detected in PET scans of the brain. The disease can remain asymptomatic for decades, while the densities of Aβ and τ continue to grow. Inflammation is considered an early event that drives the disease. In this paper, we develop a mathematical model that can produce simulated patterns of (Aβ,τ) seen in PET scans of AD patients. The model is based on the assumption that early inflammations, R and [Formula: see text], drive the growth of Aβ and τ, respectively. Recently approved drugs can slow the progression of AD in patients, provided treatment begins early, before significant damage to the brain has occurred. In line with current longitudinal studies, we used the model to demonstrate how to assess the efficacy of such drugs when given years before the disease becomes symptomatic.

Analysis of fractal-fractional Alzheimer's disease mathematical model in sense of Caputo derivative

Alzheimer's disease stands as one of the most widespread neurodegenerative conditions associated with aging, giving rise to dementia and posing significant public health challenges. Mathematical models are considered as valuable tools to gain insights into the mechanisms underlying the onset, progression, and potential therapeutic approaches for AD. In this paper, we introduce a mathematical model for AD that employs the fractal fractional operator in the Caputo sense to characterize the temporal dynamics of key cell populations. This model encompasses essential elements, including amyloid-β ($\mathbb{ A_\beta }$), neurons, astroglia and microglia. Using the fractal fractional operator, we have established the existence and uniqueness of solutions for the model under consideration, employing Leray-Schaefer's theorem and the Banach fixed-point methods. Utilizing functional techniques, we have analyzed the proposed model stability under the Ulam-Hyers condition. The suggested model has been numerically simulated by using a fractional Adams-Bashforth approach, which involves a two-step Lagrange polynomial. For numerical simulations, different ranges of fractional order values and fractal dimensions are considered. This new fractal fractional operator in the form of the Caputo derivative was determined to yield better results than an ordinary integer order. Various outcomes are shown graphically by for different fractal dimensions and arbitrary orders.

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.

A glance through the effects of CD4+ T cells, CD8+ T cells, and cytokines on Alzheimer's disease

Alzheimer's disease (AD) is the most common form of dementia. Unfortunately, despite numerous studies, an effective treatment for AD has not yet been established. There is remarkable evidence indicating that the innate immune mechanism and adaptive immune response play significant roles in the pathogenesis of AD. Several studies have reported changes in CD8+ and CD4+ T cells in AD patients. This mini-review article discusses the potential contribution of CD4+ and CD8+ T cells reactivity to amyloid β (Aβ) protein in individuals with AD. Moreover, this mini-review examines the potential associations between T cells, heme oxygenase (HO), and impaired mitochondria in the context of AD. While current mathematical models of AD have not extensively addressed the inclusion of CD4+ and CD8+ T cells, there exist models that can be extended to consider AD as an autoimmune disease involving these T cell types. Additionally, the mini-review covers recent research that has investigated the utilization of machine learning models, considering the impact of CD4+ and CD8+ T cells.

Recent Development in the Understanding of Molecular and Cellular Mechanisms Underlying the Etiopathogenesis of Alzheimer's Disease

Alzheimer's disease (AD) is a progressive neurodegenerative disorder that leads to dementia and patient death. AD is characterized by intracellular neurofibrillary tangles, extracellular amyloid beta (Aβ) plaque deposition, and neurodegeneration. Diverse alterations have been associated with AD progression, including genetic mutations, neuroinflammation, blood-brain barrier (BBB) impairment, mitochondrial dysfunction, oxidative stress, and metal ion imbalance.Additionally, recent studies have shown an association between altered heme metabolism and AD. Unfortunately, decades of research and drug development have not produced any effective treatments for AD. Therefore, understanding the cellular and molecular mechanisms underlying AD pathology and identifying potential therapeutic targets are crucial for AD drug development. This review discusses the most common alterations associated with AD and promising therapeutic targets for AD drug discovery. Furthermore, it highlights the role of heme in AD development and summarizes mathematical models of AD, including a stochastic mathematical model of AD and mathematical models of the effect of Aβ on AD. We also summarize the potential treatment strategies that these models can offer in clinical trials.

A continuous-time mathematical model and discrete approximations for the aggregation of β-Amyloid

Alzheimer's disease is a degenerative disorder characterized by the loss of synapses and neurons from the brain, as well as the accumulation of amyloid-based neuritic plaques. While it remains a matter of contention whether β-amyloid causes the neurodegeneration, β-amyloid aggregation is associated with the disease progression. Therefore, gaining a clearer understanding of this aggregation may help to better understand the disease. We develop a continuous-time model for β-amyloid aggregation using concepts from chemical kinetics and population dynamics. We show the model conserves mass and establish conditions for the existence and stability of equilibria. We also develop two discrete-time approximations to the model that are dynamically consistent. We show numerically that the continuous-time model produces sigmoidal growth, while the discrete-time approximations may exhibit oscillatory dynamics. Finally, sensitivity analysis reveals that aggregate concentration is most sensitive to parameters involved in monomer production and nucleation, suggesting the need for good estimates of such parameters.

A reaction-diffusion model of spatial propagation of A[Formula: see text] oligomers in early stage Alzheimer's disease

The misconformation and aggregation of the protein Amyloid-Beta (A[Formula: see text]) is a key event in the propagation of Alzheimer's Disease (AD). Different types of assemblies are identified, with long fibrils and plaques deposing during the late stages of AD. In the earlier stages, the disease spread is driven by the formation and the spatial propagation of small amorphous assemblies called oligomers. We propose a model dedicated to studying those early stages, in the vicinity of a few neurons and after a polymer seed has been formed. We build a reaction-diffusion model, with a Becker-Döring-like system that includes fragmentation and size-dependent diffusion. We hereby establish the theoretical framework necessary for the proper use of this model, by proving the existence of solutions using a fixed point method.

Mathematical Model Shows How Sleep May Affect Amyloid-β Fibrillization

Deposition of amyloid-β (Aβ) fibers in the extracellular matrix of the brain is a ubiquitous feature associated with several neurodegenerative disorders, especially Alzheimer's disease (AD). Although many of the biological aspects that contribute to the formation of Aβ plaques are well addressed at the intra- and intercellular levels in short timescales, an understanding of how Aβ fibrillization usually starts to dominate at a longer timescale despite the presence of mechanisms dedicated to Aβ clearance is still lacking. Furthermore, no existing mathematical model integrates the impact of diurnal neural activity as emanated from circadian regulation to predict disease progression due to a disruption in the sleep-wake cycle. In this study, we develop a minimal model of Aβ fibrillization to investigate the onset of AD over a long timescale. Our results suggest that the diseased state is a manifestation of a phase change of the system from soluble Aβ (sAβ) to fibrillar Aβ (fAβ) domination upon surpassing a threshold in the production rate of sAβ. By incorporating the circadian rhythm into our model, we reveal that fAβ accumulation is crucially dependent on the regulation of the sleep-wake cycle, thereby indicating the importance of good sleep hygiene in averting AD onset. We also discuss potential intervention schemes to reduce fAβ accumulation in the brain by modification of the critical sAβ production rate.

A mathematical model demonstrating the role of interstitial fluid flow on the clearance and accumulation of amyloid β in the brain

A system of partial differential equations is developed to describe the formation and clearance of amyloid β (Aβ) and the subsequent buildup of Aβ plaques in the brain, which are associated with Alzheimer's disease. The Aβ related proteins are divided into five distinct categories depending on their size. In addition to enzymatic degradation, the clearance via diffusion and the outflow of interstitial fluid (ISF) into the surrounding cerebral spinal fluid (CSF) are considered. Treating the brain tissue as a porous medium, a simplified two-dimensional circular geometry is assumed for the transverse section of the brain leading to a nonlinear, coupled system of PDEs. Asymptotic analysis is carried out for the steady states of the spatially homogeneous system in the vanishingly small limit of Aβ clearance rate. The PDE model is studied numerically for two cases, a spherically symmetric case and a more realistic 2D asymmetric case, allowing for non-uniform boundary conditions. Our investigations demonstrate that ISF advection is a key component in reproducing the clinically observed accumulation of plaques on the outer boundaries. Furthermore, ISF circulation serves to enhance Aβ clearance over diffusion alone and that non-uniformities in ISF drainage into the CSF can lead to local clustering of plaques. Analysis of the model also demonstrates that plaque formation does not directly correspond to the high presence of toxic oligomers.

Amyloid β and tau are involved in sleep disorder in Alzheimer's disease by orexin A and adenosine A(1) receptor

Sleep disorder is confirmed as a core component of Alzheimer's disease (AD), while the accumulation of amyloid β (Aβ) in brain tissue is an important pathological feature of AD. However, how Aβ affects AD‑associated sleep disorder is not yet well understood. In the present study, experiments on animal and cell models were performed to detect the association between sleep disorder and Aβ. It was observed that Aβ25‑35 administration significantly decreased non‑rapid eye movement sleep, while it increased wakefulness in mice. In addition, reverse transcription‑quantitative polymerase chain reaction and western blot analysis revealed that the expression levels of tau, p‑tau, orexin A and orexin neurons express adenosine A1 receptor (A1R) were markedly upregulated in the brain tissue of AD mice compared with that in samples obtained from control mice. Furthermore, the in vitro study revealed that the expression levels of tau, p‑tau, orexin A and adenosine A1R were also significantly increased in human neuroblastoma SH‑SY5Y cells treated with Aβ25‑35 as compared with the control cells. In addition, the tau inhibitor TRx 0237 significantly reversed the promoting effects of Aβ25‑35 on tau, p‑tau, orexin A and adenosine A1R expression levels, and adenosine A1R or orexin A knockdown also inhibited tau and p‑tau expression levels mediated by Aβ25‑35 in AD. These results indicate that Aβ and tau may be considered as novel biomarkers of sleep disorder in AD pathology, and that they function by regulating the expression levels of orexin A and adenosine A1R.

Stability analysis of a steady state of a model describing Alzheimer's disease and interactions with prion proteins

Alzheimer's disease (AD) is a neuro-degenerative disease affecting more than 46 million people worldwide in 2015. AD is in part caused by the accumulation of A[Formula: see text] peptides inside the brain. These can aggregate to form insoluble oligomers or fibrils. Oligomers have the capacity to interact with neurons via membrane receptors such as prion proteins ([Formula: see text]). This interaction leads [Formula: see text] to be misfolded in oligomeric prion proteins ([Formula: see text]), transmitting a death signal to neurons. In the present work, we aim to describe the dynamics of A[Formula: see text] assemblies and the accumulation of toxic oligomeric species in the brain, by bringing together the fibrillation pathway of A[Formula: see text] peptides in one hand, and in the other hand A[Formula: see text] oligomerization process and their interaction with cellular prions, which has been reported to be involved in a cell-death signal transduction. The model is based on Becker-Döring equations for the polymerization process, with delayed differential equations accounting for structural rearrangement of the different reactants. We analyse the well-posedness of the model and show existence, uniqueness and non-negativity of solutions. Moreover, we demonstrate that this model admits a non-trivial steady state, which is found to be globally stable thanks to a Lyapunov function. We finally present numerical simulations and discuss the impact of model parameters on the whole dynamics, which could constitute the main targets for pharmaceutical industry.

The development of a stochastic mathematical model of Alzheimer's disease to help improve the design of clinical trials of potential treatments

Alzheimer's disease (AD) is a neurodegenerative disorder characterised by a slow progressive deterioration of cognitive capacity. Drugs are urgently needed for the treatment of AD and unfortunately almost all clinical trials of AD drug candidates have failed or been discontinued to date. Mathematical, computational and statistical tools can be employed in the construction of clinical trial simulators to assist in the improvement of trial design and enhance the chances of success of potential new therapies. Based on the analysis of a set of clinical data provided by the Alzheimer's Disease Neuroimaging Initiative (ADNI) we developed a simple stochastic mathematical model to simulate the development and progression of Alzheimer's in a longitudinal cohort study. We show how this modelling framework could be used to assess the effect and the chances of success of hypothetical treatments that are administered at different stages and delay disease development. We demonstrate that the detection of the true efficacy of an AD treatment can be very challenging, even if the treatment is highly effective. An important reason behind the inability to detect signals of efficacy in a clinical trial in this therapy area could be the high between- and within-individual variability in the measurement of diagnostic markers and endpoints, which consequently results in the misdiagnosis of an individual's disease state.

A physical model for dementia

Aging associated brain decline often result in some kind of dementia. Even when this is a complex brain disorder a physical model can be used in order to describe its general behavior. A probabilistic model for the development of dementia is obtained and fitted to some experimental data obtained from the Alzheimer's Disease Neuroimaging Initiative. It is explained how dementia appears as a consequence of aging and why it is irreversible.

Mathematical model on Alzheimer's disease

Background

Alzheimer disease (AD) is a progressive neurodegenerative disease that destroys memory and cognitive skills. AD is characterized by the presence of two types of neuropathological hallmarks: extracellular plaques consisting of amyloid β-peptides and intracellular neurofibrillary tangles of hyperphosphorylated tau proteins. The disease affects 5 million people in the United States and 44 million world-wide. Currently there is no drug that can cure, stop or even slow the progression of the disease. If no cure is found, by 2050 the number of alzheimer’s patients in the U.S. will reach 15 million and the cost of caring for them will exceed $ 1 trillion annually.

Results

The present paper develops a mathematical model of AD that includes neurons, astrocytes, microglias and peripheral macrophages, as well as amyloid β aggregation and hyperphosphorylated tau proteins. The model is represented by a system of partial differential equations. The model is used to simulate the effect of drugs that either failed in clinical trials, or are currently in clinical trials.

Conclusions

Based on these simulations it is suggested that combined therapy with TNF- α inhibitor and anti amyloid β could yield significant efficacy in slowing the progression of AD.

Unraveling the mechanistic complexity of Alzheimer's disease through systems biology

Alzheimer's disease (AD) is a complex, multifactorial disease that has reached global epidemic proportions. The challenge remains to fully identify its underlying molecular mechanisms that will enable development of accurate diagnostic tools and therapeutics. Conventional experimental approaches that target individual or small sets of genes or proteins may overlook important parts of the regulatory network, which limits the opportunity of identifying multitarget interventions. Our perspective is that a more complete insight into potential treatment options for AD will only be made possible through studying the disease as a system. We propose an integrative systems biology approach that we argue has been largely untapped in AD research. We present key publications to demonstrate the value of this approach and discuss the potential to intensify research efforts in AD through transdisciplinary collaboration. We highlight challenges and opportunities for significant breakthroughs that could be made if a systems biology approach is fully exploited.