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Chu C, Low YLC, Ma L, Wang Y, Cox T, Doré V, Masters CL, Goudey B, Jin L, Pan Y. How Can We Use Mathematical Modeling of Amyloid-β in Alzheimer's Disease Research and Clinical Practices? J Alzheimers Dis 2024; 97:89-100. [PMID: 38007665 DOI: 10.3233/jad-230938] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2023]
Abstract
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.
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Affiliation(s)
- Chenyin Chu
- Florey Institute of Neuroscience and Mental Health, University of Melbourne, Parkville, Victoria, Australia
| | - Yi Ling Clare Low
- Florey Institute of Neuroscience and Mental Health, University of Melbourne, Parkville, Victoria, Australia
| | - Liwei Ma
- Florey Institute of Neuroscience and Mental Health, University of Melbourne, Parkville, Victoria, Australia
| | - Yihan Wang
- Florey Institute of Neuroscience and Mental Health, University of Melbourne, Parkville, Victoria, Australia
| | - Timothy Cox
- The Australian e-Health Research Centre, CSIRO, Parkville, Victoria, Australia
| | - Vincent Doré
- The Australian e-Health Research Centre, CSIRO, Parkville, Victoria, Australia
| | - Colin L Masters
- Florey Institute of Neuroscience and Mental Health, University of Melbourne, Parkville, Victoria, Australia
| | - Benjamin Goudey
- Florey Institute of Neuroscience and Mental Health, University of Melbourne, Parkville, Victoria, Australia
- ARC Training Centre in Cognitive Computing for Medical Technologies, University of Melbourne, Carlton, Victoria, Australia
| | - Liang Jin
- Florey Institute of Neuroscience and Mental Health, University of Melbourne, Parkville, Victoria, Australia
- Drug Delivery, Disposition and Dynamics, Monash Institute of Pharmaceutical Sciences, Monash University, Parkville, Victoria, Australia
| | - Yijun Pan
- Florey Institute of Neuroscience and Mental Health, University of Melbourne, Parkville, Victoria, Australia
- Drug Delivery, Disposition and Dynamics, Monash Institute of Pharmaceutical Sciences, Monash University, Parkville, Victoria, Australia
- Department of Organ Anatomy, Graduate School of Medicine, Tohoku University, Sendai, Miyagi, Japan
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Bertsch M, Franchi B, Tesi MC, Tora V. The role of A[Formula: see text] and Tau proteins in Alzheimer's disease: a mathematical model on graphs. J Math Biol 2023; 87:49. [PMID: 37646953 PMCID: PMC10468937 DOI: 10.1007/s00285-023-01985-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/17/2023] [Revised: 06/25/2023] [Accepted: 08/13/2023] [Indexed: 09/01/2023]
Abstract
In this Note we study a mathematical model for the progression of Alzheimer's Disease in the human brain. The novelty of our approach consists in the representation of the brain as two superposed graphs where toxic proteins diffuse, the connectivity graph which represents the neural network, and the proximity graph which takes into account the extracellular space. Toxic proteins such as [Formula: see text] amyloid and Tau play in fact a crucial role in the development of Alzheimer's disease and, separately, have been targets of medical treatments. Recent biomedical literature stresses the potential impact of the synergetic action of these proteins. We numerically test various modelling hypotheses which confirm the relevance of this synergy.
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Affiliation(s)
- Michiel Bertsch
- Department of Mathematics, University of Roma “Tor Vergata”, Rome, Italy
- Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Rome, Italy
| | - Bruno Franchi
- Department of Mathematics, Alma Mater Studiorum University of Bologna, Bologna, Italy
| | - Maria Carla Tesi
- Department of Mathematics, Alma Mater Studiorum University of Bologna, Bologna, Italy
| | - Veronica Tora
- Department of Mathematics, University of Roma “Tor Vergata”, Rome, Italy
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Mohammadi S, Rafii-Tabar H, Sasanpour P. A modeling study of the effect of an alternating magnetic field on magnetite nanoparticles in proximity of the neuronal microtubules: A proposed mechanism for detachment of tau proteins. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2022; 222:106913. [PMID: 35738092 DOI: 10.1016/j.cmpb.2022.106913] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2022] [Revised: 05/17/2022] [Accepted: 05/24/2022] [Indexed: 06/15/2023]
Abstract
BACKGROUND AND OBJECTIVE It is known that the disintegration of microtubules in neurons occurs in response to the phosphorylation of the tau proteins that promotes the structural instability of the microtubules, as one of the factors underlying the onset of Alzheimer's disease (AD). METHODS In this study, the mechanical variations undergone by the tau protein's and microtubule's structures due to the action of intrinsic magnetite nanoparticles inside the brain tissue have been computationally modeled using the finite element (FEM) method. RESULTS The von Mises stress induced by magnetite nanoparticles, subject to an applied alternating magnetic field, leads to local heating and mechanical forces, prompting a corresponding deformation in, and displacement of, the microtubule and the tau protein. CONCLUSIONS The induction of these deformations would increase the probability of the microtubules' depolymerization, and hence their eventual structural disintegration.
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Affiliation(s)
- Simah Mohammadi
- Department of Medical Physics & Biomedical Engineering, School of Medicine, Shahid Beheshti University of Medical Sciences, Tehran, Iran
| | - Hashem Rafii-Tabar
- Department of Medical Physics & Biomedical Engineering, School of Medicine, Shahid Beheshti University of Medical Sciences, Tehran, Iran; The Physics Branch of Iran Academy of Sciences, Tehran, Iran.
| | - Pezhman Sasanpour
- Department of Medical Physics & Biomedical Engineering, School of Medicine, Shahid Beheshti University of Medical Sciences, Tehran, Iran.
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Mukherjee S, Tithof J. Model of glymphatic clearance of aggregating proteins from the brain interstitium. Phys Rev E 2022; 105:024405. [PMID: 35291186 DOI: 10.1103/physreve.105.024405] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/01/2021] [Accepted: 02/06/2022] [Indexed: 06/14/2023]
Abstract
A growing body of evidence suggests that cerebrospinal fluid circulates through the brain to sweep away high-molecular-weight solutes. Multiple studies demonstrate that flow through this pathway, often referred to as the glymphatic system, is most active during sleep. We numerically model the clearance of amyloid-β (a high-molecular-weight protein connected to Alzheimer's disease) from the brain interstitium by combined diffusion and glymphatic advection. We first compare the clearance for a range of different flow conditions and quantify the relation between the clearance rates and Péclet number Pe. We then simulate protein buildup using a reaction-advection-diffusion equation based on the Smoluchowski aggregation scheme and quantify the buildup for different Pe. We find that for flows with Pe≳1, the rate of accumulation of heavy aggregates decreases exponentially with Pe. We finally explore the effect of the sleep-wake cycle by incorporating a variation in the flow speed motivated by experimental measurements. We find that periods of sleep lead to better clearance of intermediate protein aggregates and deter the buildup of large aggregates in the brain. In a conservative estimate, for Pe≈1, we find a 32% reduction in the buildup rate of heavier protein aggregates compared to purely diffusive clearance.
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Affiliation(s)
- Saikat Mukherjee
- Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota 55455, USA
| | - Jeffrey Tithof
- Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota 55455, USA
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