Dunster2016 - Nondimensional Coagulation Model

We undertake a mathematical investigation of a model for the generation of thrombin, an enzyme central to haemostatic blood coagulation, as well as to thrombotic disorders, that is the end product of a complicated protein cascade with multiple feedbacks that ensures its production in the right place at the right time. In a laboratory setting, its central role is reflected in thrombin evolution over time being used as a measure of the ability of a patient's blood to clot. Here, we present a model for the generation of thrombin (based on earlier work) and analyse it using the method of matched asymptotic expansions to derive a sequence of simplified models that characterize the roles of distinct interactions over various timescales. In particular, we are able through the asymptotic analysis to provide simplified models that are an excellent substitute for the full model (capturing the explosive growth and decay of thrombin) and approximations for the key experimental measurements used to describe thrombin's characteristic evolution over time. The asymptotic results are validated against numerical simulations.我们针对凝血酶（thrombin）的生成模型开展数学研究——凝血酶既是止血凝血（haemostatic blood coagulation）过程的核心酶类，同时也是血栓性疾病（thrombotic disorders）的关键酶类。凝血酶源自一套带有多重反馈的复杂蛋白质级联反应，该反应可确保凝血酶在正确的时间、正确的位置生成。在实验室研究场景中，凝血酶的核心作用体现为：其随时间的演化过程被用作评估患者血液凝血功能的指标。本文基于前期研究成果，提出了一套凝血酶生成模型，并采用匹配渐近展开法（matched asymptotic expansions）对其进行分析，推导出一系列简化模型，用以刻画不同时间尺度下各类特异性相互作用的功能角色。尤为关键的是，通过渐近分析，我们得以获得可完美替代完整模型的简化模型——该模型能够精准捕捉凝血酶的爆发式增长与衰减过程，同时可作为描述凝血酶特征性时间演化过程的关键实验测量指标的近似方案。本研究得到的渐近结果已通过数值模拟（numerical simulations）得到验证。