Mathematical model on Alzheimer’s disease

Hao, Wenrui; Friedman, Avner

doi:10.1186/s12918-016-0348-2

Cited by 106 publications

(135 citation statements)

References 95 publications

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“…Third, our CCM is a simplified causal model of biomarkers interacting with each other, an abstraction that does not model the actual underlying cellular and molecular processes. Prior CCM efforts in AD have modeled the disease at a molecular and cellular level (12,13) as well as at a whole-brain, systems level using MRI and EEG data (14,15). There have been few CCM approaches that have specifically focused on clinical AD biomarkers (14,16), and none have incorporated all three clinically available categories of biomarkers, amyloidopathy, tauopathy, and neurodegeneration.…”

Section: Discussionmentioning

confidence: 99%

Computational Causal Modeling of the Dynamic Biomarker Cascade in Alzheimer’s Disease

Petrella

Hao

Rao

et al. 2018

Preprint

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Background: Alzheimer's disease (AD) is a major public health concern and there is an urgent need to better understand its complex biology and develop effective therapies. AD progression can be tracked in patients though validated imaging and spinal fluid biomarkers of pathology and neuronal loss. We still, however, lack a coherent quantitative model that explains how these biomarkers interact and evolve over time. Such a model could potentially help identify the major drivers of disease in individual patients and simulate response to therapy prior to entry in clinical trials. A current theory of AD biomarker progression, known as the dynamic biomarker cascade model, hypothesizes AD biomarkers evolve in a sequential, but temporally overlapping manner. A computational model incorporating assumptions about the underlying biology of this theory and its variations would be useful to test and refine its accuracy with longitudinal biomarker data from clinical trials. Methods:We implemented a causal model to simulate time-dependent biomarker data under the descriptive assumptions of the dynamic biomarker cascade theory. We modeled pathologic biomarkers (beta-amyloid and tau), neuronal loss biomarkers and cognitive impairment as non-linear first order ordinary differential equations (ODEs) to include amyloid-dependent and non-dependent neurodegenerative cascades. We tested the feasibility of the model by adjusting its parameters to simulate three specific natural history scenarios in early-onset autosomal dominant AD and late-onset AD, and determine whether computed biomarker trajectories agreed with current assumptions of AD biomarker progression. We also simulated the effects of anti-amyloid therapy in late-onset AD.Results: The computational model of early-onset AD demonstrated the initial appearance of amyloid, followed by biomarkers of tau and neurodegeneration, followed by onset of cognitive decline based on cognitive reserve, as predicted by prior literature. Similarly, the late-onset AD computational models demonstrated the first appearance of amyloid or non-amyloid-related tauopathy, depending on the magnitude of comorbid pathology, and also closely matched the biomarker cascades predicted by prior literature. Forward simulation of anti-amyloid therapy in symptomatic late-onset AD failed to demonstrate any slowing in progression of cognitive decline, consistent with prior failed clinical trials in symptomatic patients. Conclusions:We have developed and computationally implemented a mathematical causal model of the dynamic biomarker cascade theory in AD. We demonstrate the feasibility of this model by simulating biomarker evolution and cognitive decline in early and late-onset natural history scenarios, as well as in a treatment scenario targeted at core AD pathology. Models resulting from this causal approach can be further developed and refined using patient data from longitudinal biomarker studies, and may in the future play a key role in personalizing approaches to treatment.

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Section: Discussionmentioning

confidence: 99%

Computational Causal Modeling of the Dynamic Biomarker Cascade in Alzheimer’s Disease

Petrella

Hao

Rao

et al. 2018

Preprint

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“…Let ϕ = ϕ(x, a) : Ω × [0, 1] → R be a test function representing any observable quantity that can be computed out of the microscopic state (X, A τ ) of a neuron. From (17) we get:…”

Section: 2mentioning

confidence: 99%

Microscopic and macroscopic models for the onset and progression of Alzheimer's disease

Bertsch¹,

Franchi²,

Tesi³

et al. 2017

J. Phys. A: Math. Theor.

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Abstract. In the first part of this paper we review a mathematical model for the onset and progression of Alzheimer's disease (AD) that was developed in subsequent steps over several years. The model is meant to describe the evolution of AD in vivo. In [1] we treated the problem at a microscopic scale, where the typical length scale is a multiple of the size of the soma of a single neuron. Subsequently, in [2] we concentrated on the macroscopic scale, where brain neurons are regarded as a continuous medium, structured by their degree of malfunctioning.In the second part of the paper we consider the relation between the microscopic and the macroscopic models. In particular we show under which assumptions the kinetic transport equation, which in the macroscopic model governs the evolution of the probability measure for the degree of malfunctioning of neurons, can be derived from a particle-based setting.The models are based on aggregation and diffusion equations for β Amyloid, a protein fragment that healthy brains regularly produce and eliminate. In case of dementia Aβ monomers are no longer properly washed out and begin to coalesce forming eventually plaques. Two different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble polymers of amyloid, produced by damaged neurons; ii) neuron-to-neuron prion-like transmission.In the microscopic model we consider basically mechanism i), modelling it by a system of Smoluchowski equations for the amyloid concentration (describing the agglomeration phenomenon), with the addition of a diffusion term as well as of a source term on the neuronal membrane. At the macroscopic level instead we model processes i) and ii) by a system of Smoluchowski equations for the amyloid concentration, coupled to a kinetic-type transport equation for the distribution function of the degree of malfunctioning of the neurons. The second equation contains an integral term describing the random onset of the disease as a jump process localized in particularly sensitive areas of the brain.Even though we deliberately neglected many aspects of the complexity of the brain and the disease, numerical simulations are in both cases (microscopic and macroscopic) in good qualitative agreement with clinical data.

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“…We propose a minimal model and show that the onset of AD is driven by a phase change from the soluble form of Aβ to its fibrillar form. The proposed model has the advantage of simplicity compared with the recent mathematical models [18,19,20,21]. A few other in-silico studies explained how the Aβ fibres form in the brain, using an agent-based model [22] or numerically solving the partial differential equations [21,23].…”

Section: Introductionmentioning

confidence: 99%

“…The proposed model has the advantage of simplicity compared with the recent mathematical models [18,19,20,21]. A few other in-silico studies explained how the Aβ fibres form in the brain, using an agent-based model [22] or numerically solving the partial differential equations [21,23]. Although many aspects of the disease are explained and understood from these models, we still lack a simple explanation of how different factors affect the brain in the course of AD.…”

Section: Introductionmentioning

confidence: 99%

Mathematical model shows how sleep may affect amyloid β fibrillization

Hoore

Khailaie

Montaseri

et al. 2019

Preprint

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Deposition of amyloid β (Aβ) fibers in extra-cellular matrix of the brain is a ubiquitous feature associated with several neurodegenerative disorders, especially Alzheimer's disease (AD). While many of the biological aspects that contribute to the formation of Aβ plaques are well addressed at the intra-and inter-cellular level in short timescales, an understanding of how Aβ fibrillization usually starts to dominate at a longer timescale in spite of 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 sleep-wake cycle. In this study, we develop a minimal model of Aβ fibrillization to investigate the onset of AD over a long time-scale. 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 soluble Aβ. By incorporating the circadian rhythm into our model, we reveal that fAβ accumulation is crucially dependent on the regulation of sleep-wake cycle, thereby indicating the importance of a 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.

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Mathematical model on Alzheimer’s disease

Cited by 106 publications

References 95 publications

Computational Causal Modeling of the Dynamic Biomarker Cascade in Alzheimer’s Disease

Computational Causal Modeling of the Dynamic Biomarker Cascade in Alzheimer’s Disease

Microscopic and macroscopic models for the onset and progression of Alzheimer's disease

Mathematical model shows how sleep may affect amyloid β fibrillization

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