Institute of Theoretical Chemistry, Ruhr-University Bochum, 44780, Bochum, Germany.
Sci Rep. 2022 Jun 22;12(1):10524. doi: 10.1038/s41598-022-13469-7.
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 to describe co-operative effects are currently in use, the Additive Dose (ADM) or concentration addition (CA) and the Multiplicative Survival (MSM) or independent action (IA) models. Recently we proposed an approach which describes 'zero-interaction' surfaces based on the only requirement that simultaneous administration of different drugs leads to Hill-type response surfaces, which are solutions of the underlying logistic differential equations. No further assumptions, neither on mechanisms of action nor on limitations of parameter combinations are required. This defines-and limits-the application range of our approach. Resting on the same principle, we extend this ansatz in the present paper in order to describe deviations from the reference surface by generalized Hill-type functions. To this end we introduce two types of parameters, perturbations of the pure drug Hill-parameters and interaction parameters that account for n-tuple interactions between all components of a mixture. The resulting 'full-interaction' response surface is a valid solution of the basic partial differential equation (PDE), satisfying appropriate boundary conditions. This is true irrespective of its actual functional form, as within our framework the number of parameters is not fixed. We start by fitting the experimental data to the 'full-interaction' model with the maximum possible number of parameters. Guided by the fit-statistics, we then gradually remove insignificant parameters until the optimum response surface model is obtained. The 'full-interaction' Hill response surface ansatz can be applied to mixtures of n compounds with arbitrary Hill parameters including those describing baseline effects. Synergy surfaces, i.e., differences between full- and null-interaction models, are used to identify dose-combinations showing peak synergies. We apply our approach to binary and ternary examples from the literature, which range from mixtures behaving according to the null-interaction model to those showing strong synergistic or antagonistic effects. By comparing 'null-' and 'full-response' surfaces we identify those dose-combinations that lead to maximum synergistic or antagonistic effects. In one example we identify both synergistic and antagonistic effects simlutaneously, depending on the dose-ratio of the components. In addition we show that often the number of parameters necessary to describe the response can be reduced without significantly affecting the accuracy. This facilitates an analysis of the synergistic effects by focussing on the main factors causing the deviations from 'null-interaction'.
混合药物作用的协同、拮抗或相加效应的分类，严重依赖于“零相互作用”参考模型。目前，有两种主要的方法来描述协同作用，即加和剂量（ADM）或浓度加和（CA）和乘法生存（MSM）或独立作用（IA）模型。最近，我们提出了一种方法，该方法基于以下唯一要求来描述“零相互作用”表面，即同时给予不同药物会导致 Hill 型反应表面，这是潜在逻辑微分方程的解。不需要对作用机制或参数组合的限制进行进一步的假设。这定义并限制了我们方法的应用范围。基于相同的原理，我们在本文中扩展了这个假设，以便用广义 Hill 型函数来描述偏离参考表面的情况。为此，我们引入了两种类型的参数，即纯药物 Hill 参数的扰动和考虑混合物中所有成分之间 n 重相互作用的相互作用参数。由此产生的“全相互作用”响应表面是基本偏微分方程（PDE）的有效解，满足适当的边界条件。这是正确的，无论其实际功能形式如何，因为在我们的框架内，参数的数量不是固定的。我们首先用最大可能数量的参数拟合实验数据到“全相互作用”模型。根据拟合统计数据，我们逐渐删除不重要的参数，直到获得最佳的响应曲面模型。“全相互作用”Hill 响应曲面假设可以应用于具有任意 Hill 参数的 n 种化合物的混合物，包括描述基线效应的参数。协同作用表面，即全相互作用和零相互作用模型之间的差异，用于识别表现出协同作用峰值的剂量组合。我们将我们的方法应用于文献中的二元和三元实例，这些实例从符合零相互作用模型的混合物到表现出强烈协同或拮抗作用的混合物。通过比较“零响应”和“全响应”表面，我们确定了导致最大协同或拮抗作用的剂量组合。在一个实例中，我们同时识别出协同和拮抗作用，这取决于成分的剂量比。此外，我们还表明，通常可以减少描述响应所需的参数数量，而不会显著影响准确性。这便于通过聚焦于导致与“零相互作用”偏离的主要因素来分析协同作用。
