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Alzheimer’s disease (AD) is characterized in part by the accumulation and spread of amyloid beta proteins in the brain. Recent experiments have revealed that amyloid beta oligomers induce microvascular mural cells to contract, thereby constricting capillaries and increasing resistance to blood flow. Conversely, hypoperfusion promotes amyloid beta production and hinders its clearance, hence creating a pathogenic positive feedback loop. Here, we develop a mathematical model that combines protein–capillary interaction with the prion-like behaviour of amyloid beta. For sufficiently strong interaction, we find that healthy and diseased steady states, both stable, can exist simultaneously, implying that pathogenic protein seeds must exceed a critical threshold in order to trigger disease outbreak. We explore the consequences of this bistability for disease propagation through the brain’s structural connectome network. Finally, in a first attempt to model the AD two-hit vascular hypothesis mathematically, we describe how spatially localized deficits in blood supply, e.g. due to embolic stroke or atherosclerosis of the leptomeningeal vessels, may trigger disease outbreak and propagation. 

Throughout the 19th and 20th centuries, aided by advances in medical imaging, discoveries in physiology and medicine have added nearly 25 years to the average life expectancy. This resounding success brings with it a need to understand a broad range of age-related health conditions, such as dementia. Today, mathematics, neuroimaging and scientific computing are being combined with fresh insights, from animal models, to study the brain and to better understand the etiology and progression of Alzheimer's disease, the most common cause of age-related dementia in humans. In this manuscript, we offer a brief primer to the reader interested in engaging with the exciting field of mathematical modeling and scientific computing to advance the study of the brain and, in particular, human AD research. 

ABSTRACT Microfluidic devices (MDs) present a novel method for detecting circulating tumor cells (CTCs), enhancing the process through targeted techniques and visual inspection. However, current approaches often yield heterogeneous CTC populations, necessitating additional processing for comprehensive analysis and phenotype identification. These procedures are often expensive, time‐consuming, and need to be performed by skilled technicians. In this study, we investigate the potential of a cost‐effective and efficient hyperuniform micropost MD approach for CTC classification. Our approach combines mathematical modeling of fluid–structure interactions in a simulated microfluidic channel with machine learning techniques. Specifically, we developed a cell‐based modeling framework to assess CTC dynamics in erythrocyte‐laden plasma flow, generating a large dataset of CTC trajectories that account for two distinct CTC phenotypes. Convolutional neural network (CNN) and recurrent neural network (RNN) were then employed to analyze the dataset and classify these phenotypes. The results demonstrate the potential effectiveness of the hyperuniform micropost MD design and analysis approach in distinguishing between different CTC phenotypes based on cell trajectory, offering a promising avenue for early cancer detection.