Theory of synergistic effects: Hill-type response surfaces as 'null-interaction' models for mixtures.

Q1 Mathematics
Michael Schindler
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引用次数: 0

Abstract

Background: The classification of effects caused by mixtures of agents as synergistic, antagonistic or additive depends critically on the reference model of 'null interaction'. Two main approaches are currently in use, the Additive Dose (ADM) or concentration addition (CA) and the Multiplicative Survival (MSM) or independent action (IA) models. We compare several response surface models to a newly developed Hill response surface, obtained by solving a logistic partial differential equation (PDE). Assuming that a mixture of chemicals with individual Hill-type dose-response curves can be described by an n-dimensional logistic function, Hill's differential equation for pure agents is replaced by a PDE for mixtures whose solution provides Hill surfaces as 'null-interaction' models and relies neither on Bliss independence or Loewe additivity nor uses Chou's unified general theory.

Methods: An n-dimensional logistic PDE decribing the Hill-type response of n-component mixtures is solved. Appropriate boundary conditions ensure the correct asymptotic behaviour. Mathematica 11 (Wolfram, Mathematica Version 11.0, 2016) is used for the mathematics and graphics presented in this article.

Results: The Hill response surface ansatz can be applied to mixtures of compounds with arbitrary Hill parameters. Restrictions which are required when deriving analytical expressions for response surfaces from other principles, are unnecessary. Many approaches based on Loewe additivity turn out be special cases of the Hill approach whose increased flexibility permits a better description of 'null-effect' responses. Missing sham-compliance of Bliss IA, known as Colby's model in agrochemistry, leads to incompatibility with the Hill surface ansatz. Examples of binary and ternary mixtures illustrate the differences between the approaches. For Hill-slopes close to one and doses below the half-maximum effect doses MSM (Colby, Bliss, Finney, Abbott) predicts synergistic effects where the Hill model indicates 'null-interaction'. These differences increase considerably with increasing steepness of the individual dose-response curves.

Conclusion: The Hill response surface ansatz contains the Loewe additivity concept as a special case and is incompatible with Bliss independent action. Hence, when synergistic effects are claimed, those dose combinations deserve special attention where the differences between independent action approaches and Hill estimations are large.

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协同效应理论:作为混合物 "无相互作用 "模型的山丘型响应面。
背景:将混合制剂引起的效应分为协同效应、拮抗效应或相加效应,关键取决于 "无效相互作用 "的参考模型。目前使用的主要有两种方法,即加成剂量模型(ADM)或浓度加成模型(CA)和乘法生存模型(MSM)或独立作用模型(IA)。我们将几种响应面模型与新开发的希尔响应面进行了比较,后者是通过求解逻辑偏微分方程 (PDE) 得到的。假设具有独立希尔型剂量-反应曲线的化学混合物可以用 n 维 logistic 函数来描述,那么纯药剂的希尔微分方程就会被混合物的 PDE 所取代,后者的求解提供了作为 "无相互作用 "模型的希尔反应曲面,既不依赖于 Bliss 独立性或 Loewe 可加性,也不使用 Chou 的统一一般理论:方法:求解了描述 n 组分混合物希尔型响应的 n 维逻辑 PDE。适当的边界条件确保了正确的渐近行为。本文使用 Mathematica 11(Wolfram,Mathematica 11.0 版,2016 年)进行数学计算和图形绘制:希尔响应面公式可用于具有任意希尔参数的化合物混合物。从其他原理推导响应面分析表达式时所需要的限制是不必要的。许多基于 Loewe 可加性的方法都是希尔方法的特例,其灵活性的提高使其能够更好地描述 "无效应 "反应。Bliss IA 的假顺应性缺失(在农业化学中被称为 Colby 模型)导致与希尔表面公式不相容。二元和三元混合物的例子说明了两种方法之间的差异。对于希尔斜率接近 1 且剂量低于半最大效应剂量的情况,MSM(Colby、Bliss、Finney、Abbott)预测会产生协同效应,而希尔模型则显示 "无相互作用"。这些差异随着单个剂量-反应曲线陡度的增加而显著增大:结论:希尔反应曲面解析式包含了作为特例的卢瓦加成概念,与布利斯独立作用不相容。因此,当声称存在协同效应时,应特别注意独立作用法和希尔估计法之间差异较大的剂量组合。
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来源期刊
Theoretical Biology and Medical Modelling
Theoretical Biology and Medical Modelling MATHEMATICAL & COMPUTATIONAL BIOLOGY-
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0
审稿时长
6-12 weeks
期刊介绍: Theoretical Biology and Medical Modelling is an open access peer-reviewed journal adopting a broad definition of "biology" and focusing on theoretical ideas and models associated with developments in biology and medicine. Mathematicians, biologists and clinicians of various specialisms, philosophers and historians of science are all contributing to the emergence of novel concepts in an age of systems biology, bioinformatics and computer modelling. This is the field in which Theoretical Biology and Medical Modelling operates. We welcome submissions that are technically sound and offering either improved understanding in biology and medicine or progress in theory or method.
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