### Introduction

_{max}model (Hill equation); let's discuss drug interactions and response surface focusing on this model.

### Reference models for zero interaction

### Bliss independence

_{max}relationship [14,15] is an appropriate pharmacodynamic model for each drug, are as follows:

_{1}, fu

_{2}, and fu

_{12}are the fractions of response for drug 1, drug 2, and combination unaffected [16]. In Eq. (2), is a fraction of the effect resulting from the mixture of d

_{1}and d

_{2}; d

_{1}and d

_{2}are doses of each drug in the mixture that yield the effect E; ED

_{50,1 or 2}is a dose of drugs to produce 50% of the maximal effect (E

_{max}) for each drug acting alone; and γ

_{1}and γ

_{2}are Hill coefficients (slope of dose-response curve). When Eq. (1) and (2) are recast in terms of the fraction of effect affected (fa

_{1 or 2}) (if 1 - fa exchanges fu in Eq. (1), because fa + fu = 1), then Eq. (3) and (4) are the results.

### Loewe additivity

_{max}model) of two full agonist drugs are different, each case has a varying potency ratio. For a full agonist and a partial agonist, the isobole of no interaction is no longer straight but curved [22]. For two full agonists with a varying potency, termed "heterodynamic" by Loewe [12], the application of dose equivalence leads to not just one, but to two nonlinear additive isoboles [21,23]. However, Berenbaum [9] maintained that a straight-line isobole is appropriate for a zero interactive combination, for it is not derived from the knowledge of the shapes of the dose-response curves or of their mechanisms of action, and he concluded that it is valid irrespective of the shapes of these curves, similar or dissimilar, and also, irrespective of their mechanisms of action. Although the shape of the isobole for Loewe additivity remains to be further studied, a straight line of additivity is commonly employed to distinguish synergistic and antagonistic from additive interactions. Although there are many reasons to prefer the Berenbaum method, the most predominate reason to use it may be because of easy calculations.

_{max}relationship is appropriate for each drug, is defined as

_{1}and d

_{2}are doses of each drug in the mixture that yield an effect, E, while D

_{1 or 2}is the dose of each drug to produce the same effect of E when given alone. The term is called the interaction index at the combination dose (d

_{1}, d

_{2}). The dose (D

_{1}, or D

_{2}) for each single drug producing the same effect of E is expressed as

_{1}, d

_{2}) is equal to, less than, or greater than 1, the combination dose is asserted to be additive, synergistic, or antagonistic, respectively.

### Median-effect method from the law of mass action

_{a}is the fraction affected by d; f

_{u}is the fraction unaffected by d (i.e., f

_{u}= 1 - f

_{a}, and f

_{a}/ f

_{u}= odds ratio); D

_{m}is the median effect dose (e.g. EC

_{50}, or ED

_{50}); and m is the slope coefficient of dose-response curve. When m is greater than 1, the dose-response curve becomes S-shaped, so m is equivalent to the Hill coefficient of sigmoid E

_{max}equation.

_{1}and d

_{2}are doses of each drug in the mixture that yield an effect, f

_{a}, while D

_{1 or 2}is the dose of each drug to produce the same effect of f

_{a}when given alone. D

_{m,1 or 2}is the median effect dose of drug 1 or drug 2, and m

_{1 or 2}is the slope coefficient of dose-response curve of drug 1 or drug 2. In the view of interaction, CI < 1, = 1, and > 1 indicate synergy, additivity, and antagonism, respectively.

_{max}model is 1. Therefore, this model under-estimates synergistic effects.

### Response surface

_{5}, ED

_{25}, ED

_{50}, ED

_{75}, etc.) in order to classify all possible interactions. To depict the entire set of levels of effect (from ED

_{1}to ED

_{99}), one may build all the isoboles in three dimensions, thus, creating a surface of isoboles, the so-called response surface. It illustrates the drug effect (Z axis) versus two-drug doses (X and Y axes) [10] and presents an entire drug interaction at all dose pairs.

### Response surface models with a single interaction parameter

### Greco model [34]

_{max}, the formula of this model is:

### Machado model

### Plummer model

_{1 or 2}is the dose of each drug to produce the same effect of E when given alone. The Newton-Raphson method is used to find D'

_{2}in the Plummer model. For this model, the plots of Y versus log (d) for two drugs should be linear but need not be parallel. This model contains five parameters (β

_{0}, β

_{1}, β

_{2}, β

_{3}, and β

_{4}). When you set d

_{2}= 0 or d

_{1}= 0 for Eq. (14), you will get , which are the doses of drug 1 and drug 2 producing the effect E. You can rearrange Eq. (14) to be

_{4}is the interaction parameter which captures synergism (β

_{4}> 0), additivity (β

_{4}= 0), and antagonism (β

_{4}< 0). The larger positive is β

_{4}, the smaller the interaction index, and the stronger the synergy.

### Carter model

_{1}= d

_{2}= 0, you will get the dose (D

_{1}, and D

_{2}) of each drug 1 or drug 2 alone eliciting an effect E: , . After moving β

_{0}to the left side of Eq. (17), and then dividing both sides of the equation with , you will obtain the interaction index:

_{12}is the interaction parameter which captures synergism (β

_{12}> 0), additivity (β

_{12}= 0), or antagonism (β

_{12}< 0).

### Response surface models with an interaction function

### Minto model

_{max}curve. The sigmoid E

_{max}model for a single drug is extended to a model that takes each ratio of two drugs as a drug in its own right.

_{50}, and the following forms are obtained.

_{V}and U

_{R}are units of potency, and the normalized concentrations of drugs V and R. A set of new drugs, each having a unique unit ratio (θ) of U

_{V}and U

_{R}, is defined in a set of terms of θ. The term of θ is defined as

_{V}+ U

_{R}. These terms can be combined with the sigmoid E

_{max}equation:

_{V}+ U

_{R}, then γ(θ) is the sigmoidicity of the concentration-response curve at a specific ratio (θ); U

_{50}(θ) is the number of units of potency associated with 50% of the maximum effect at a specific ratio θ; and E

_{max}(θ) is the maximum drug effect at a specific ratio θ. The term U

_{50}(θ) is the potency of drug combination at a specific ratio θ. Because one 50% of maximum effect is 1 unit of potency, when only drug V (θ = 0) or drug R (θ = 1) is present, the number of units associated with one 50% drug effect, U

_{50}(0) or U

_{50}(1), must be one.

_{max}(θ), U

_{50}(θ), and γ(θ). Coefficients β

_{x}are model parameters that are estimated by the data. The two terms, β

_{0}and β

_{1}in E

_{max}(θ), U

_{50}(θ), or γ(θ), are replaced by other terms already defined.

_{50}(θ) is 1 for all values of θ, the interaction will be additive. If U

_{50}(θ) is less than 1 for all values of θ, this amplifies the term in Eq. (21). This will create a synergistic effect. If U

_{50}(θ) is greater than 1 for all values of θ, this lessen the term in Eq. (21). This will create an antagonistic effect. This model can be used in investigating the drug interaction when the maximum effects of drugs V and R are not identical.

_{50,V}, C

_{50,R}, γ

_{V}, and γ

_{R}will be yielded after a given data is fitted to this model. The specific effect E is produced by the combination of [V] and [R].

### Fidler model

_{max}(θ) are given as functions of potency fraction:

_{A}θ + γ

_{B}(1-θ)], and [E

_{max,A}θ+E

_{max,B}(1-θ) are simple estimates for γ(θ) and E

_{max}(θ). The parameters β and ζ influence the type of change in the linear estimates of γ(θ) and E

_{max}(θ). Positive values of β and ζ indicate an increase in γ(θ) and E

_{max}(θ); negative indicates a decrease in them; and 0 means no change from the line of estimates of γ(θ) and E

_{max}(θ). The terms, β·f(m

_{β},w

_{β},θ) and ζ·f(m

_{β},w

_{β},θ), are restricted to be greater than -1 to keep each function positive. The parameters m

_{β}and m

_{ζ}are the symmetry parameters of the respective changes, where the maximum changes in γ(θ) and E

_{max}(θ) occur; the parameters w

_{β}and w

_{ζ}are the line shape parameters around m

_{β}and m

_{ζ}, and are greater than 0.

_{50,V}× m : C

_{50,R}× (1 - m); in a symmetric case, the best ratio is C

_{50,V}: C

_{50,R}.

### Kong model

^{-1}[V] + [R] in terms of drug R. That is, the combination doses on different isoboles may have different relative potencies. The form of ρ(z) is derived as

_{iso}and [R]

_{iso}are the respective single drug doses of drug V and drug R, and each of them creates the effect z. To describe the interspersion of interaction modes, Kong and Lee use the following form as an interaction function:

_{1}and γ

_{2}capturing the varying relative potency ρ and the coefficients δ's. The above polynomial function is substituted for the interaction parameter (β

_{4}) of the Plummer model, and the following equation is given as

### Data analysis

_{0}'s of two drugs are 0, and E

_{max}'s are 1 (or 100%). For interaction parameter and interaction index of the Greco, and Machado models, the values of Hill slope and potency of each drug acting alone were calculated (vecuronium: 6.5125, 2.98 µM, rocuronium: 7.8975, 5.34 µM, respectively) and then substituted for ED

_{50}'s and γ's of the two models. Interaction indices of the Berenbaum method (Eq. (5)) were directly calculated using the sigmoid E

_{max}equation substituted with the above calculated values. Plummer model (Eq. (14)) and Carter Model (Eq. (17)) were fitted with the data by using Tablecurve3D®.

_{50}(θ) and γ(θ) of the Minto model were calculated and illustrated (Fig. 6). The full model (10 parameters) for the Minto model was analyzed; the Fidler model (9 parameters) was analyzed when w = 1; and the Kong model (8 parameters) was analyzed when the relative potency ratio was constant. The isobole plot of raw data is rendered via Renka I (nonparametric algorithm) procedure, and it is shown in Fig. 4, panel A. Pharmacodynamic parameters of response surface models are listed at Table 2. The values are different depending on the model. In the Kong and Plummer models, because relative potency ratio is constant, the Hill slopes of each model are identical.

_{max}, U

_{50}(θ), and γ(θ), especially the value of U

_{50}(θ) (Fig. 6). The isobole plot of the Minto model is shown in Fig. 4, panel B. In this article, with the equation of the interaction index, the contour plot of the interaction index (Fig. 5, panel A) can be illustrated and help to make the quantitative evaluation. Chou and Hayball [45] proposed that an interaction index from 0.9 to 1.1 is designated as being additive. In the Minto model, the lower extreme effect region has greater synergistic interaction. Chou [13] said that for anticancer or antiviral agents, synergy at high effect levels (fa > 0.8) is more relevant to therapy than at low effect levels (fa < 0.2). On the contrary, in the anesthetic area, synergy at low effect levels may be more crucial than at high levels because the residual effects of anesthetic agents could threaten the safety of patients.