Graduate Research Training Program PharMetrX: Pharmacometrics & Computational Disease Modeling, Freie Universität Berlin and Universität Potsdam, Potsdam, Germany.
Institute of Mathematics, Universität Potsdam, Karl-Liebknecht-Str. 24-25, Golm, 14476, Potsdam, Germany.
J Pharmacokinet Pharmacodyn. 2018 Feb;45(1):139-157. doi: 10.1007/s10928-017-9561-x. Epub 2017 Dec 14.
A growing understanding of complex processes in biology has led to large-scale mechanistic models of pharmacologically relevant processes. These models are increasingly used to study the response of the system to a given input or stimulus, e.g., after drug administration. Understanding the input-response relationship, however, is often a challenging task due to the complexity of the interactions between its constituents as well as the size of the models. An approach that quantifies the importance of the different constituents for a given input-output relationship and allows to reduce the dynamics to its essential features is therefore highly desirable. In this article, we present a novel state- and time-dependent quantity called the input-response index that quantifies the importance of state variables for a given input-response relationship at a particular time. It is based on the concept of time-bounded controllability and observability, and defined with respect to a reference dynamics. In application to the brown snake venom-fibrinogen (Fg) network, the input-response indices give insight into the coordinated action of specific coagulation factors and about those factors that contribute only little to the response. We demonstrate how the indices can be used to reduce large-scale models in a two-step procedure: (i) elimination of states whose dynamics have only minor impact on the input-response relationship, and (ii) proper lumping of the remaining (lower order) model. In application to the brown snake venom-fibrinogen network, this resulted in a reduction from 62 to 8 state variables in the first step, and a further reduction to 5 state variables in the second step. We further illustrate that the sequence, in which a recursive algorithm eliminates and/or lumps state variables, has an impact on the final reduced model. The input-response indices are particularly suited to determine an informed sequence, since they are based on the dynamics of the original system. In summary, the novel measure of importance provides a powerful tool for analysing the complex dynamics of large-scale systems and a means for very efficient model order reduction of nonlinear systems.
对生物学中复杂过程的认识不断深入，导致了针对药理学相关过程的大规模机制模型的出现。这些模型越来越多地被用于研究系统对给定输入或刺激的反应，例如在药物给药后。然而，由于其组成部分之间相互作用的复杂性以及模型的规模，理解输入-响应关系通常是一项具有挑战性的任务。因此，一种能够量化不同组成部分对特定输入-输出关系的重要性并能够将动力学简化为其基本特征的方法是非常需要的。在本文中，我们提出了一种新的状态和时间相关的量，称为输入-响应指数，它可以量化在特定时间给定输入-响应关系中状态变量的重要性。它基于时间有界可控性和可观性的概念，并相对于参考动力学来定义。在应用于棕色蛇毒-纤维蛋白原（Fg）网络的实例中，输入-响应指数深入了解了特定凝血因子的协调作用以及那些对响应贡献不大的因子。我们展示了如何使用这些指数通过两步过程来简化大规模模型：（i）消除对输入-响应关系影响较小的状态，以及（ii）对剩余（低阶）模型进行适当的合并。在应用于棕色蛇毒-纤维蛋白原网络的实例中，这导致在第一步中将状态变量从 62 个减少到 8 个，在第二步中进一步减少到 5 个。我们进一步说明，递归算法消除和/或合并状态变量的顺序会对最终简化模型产生影响。输入-响应指数特别适合确定一个明智的顺序，因为它们基于原始系统的动力学。总之，这种新的重要性度量为分析大规模系统的复杂动力学提供了一个强大的工具，并为非线性系统的非常有效的模型降阶提供了一种手段。
