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.

阿尔茨海默氏病是一种退行性疾病，其特征是大脑中的突触和神经元丢失，以及淀粉样蛋白样神经斑的积累。虽然β-淀粉样蛋白是否引起神经变性仍是争论的问题，但β-淀粉样蛋白聚集与疾病进展有关。因此，更清楚地了解这种聚集可能有助于更好地了解该疾病。我们为β建立一个连续时间模型-从化学动力学和种群动力学的概念来看淀粉样蛋白的聚集。我们证明了该模型能节省质量并为平衡的存在和稳定性建立条件。我们还为模型开发了两个动态一致的离散时间近似值。我们用数字显示连续时间模型产生S形增长，而离散时间近似值可能显示出振荡动力学。最后，敏感性分析表明，聚集体浓度对单体生产和成核中涉及的参数最敏感，这表明需要对此类参数进行良好的估算。