Response surface modelsHill and median-effect functions

The Hill function (also known as the Emax model) is widely used to describe single-agent input–response relationships. It is defined as:

$$\begin E = E_ \frac}\right) ^\gamma }}\right) ^\gamma } , \end$$

(1)

where c is the input level of agent, e.g. drug concentration, \(C_\) is the median effective level (the input producing 50% of the maximum effect), \(E_\) denotes the maximum achievable effect, and \(\gamma \) is the Hill coefficient that determines the slope of the curve.

A frequently used special case is obtained by normalizing the maximum effect, i.e. setting \(E_=1\). This yields the median-effect equation:

$$\begin E = \frac}\right) ^\gamma }}\right) ^\gamma } \end$$

(2)

The median-effect equation can be further transformed into a linear form using the median-effect linearization, i.e. taking \(\log (E/(1-E))\) against \(\log (c)\):

$$\begin Y=\log \left( \frac\right) =\gamma (\log (c) - \log (C_))= \beta _0 + \beta _1\log (c) \end$$

(3)

with \(\beta _0 = -\gamma \log (C_)\) and \(\beta _1= \gamma \). This linearization establishes a direct relationship between the transformed response Y and the logarithm of the input level c, which is convenient for both parameter estimation and model comparison.

This linearization allows to introduce the Loewe model to characterize the interaction between two agents:

$$\begin \frac} + \frac} = 1 & \text \\ < 1 & \text \\ > 1 & \text \end\right. } \end$$

(4)

where \(c_1\) and \(c_2\) are the input levels (e.g., concentrations or application intensities) of agent 1 and agent 2 that in combination produce the effect level y, while \(C_\) and \(C_\) are the corresponding input levels of each agent that yield the same effect y when applied individually. It is defined as:

$$\begin C_ = C__i} \left( \frac\right) ^\frac, \quad i =1,2 \end$$

(5)

where \(C__i}\) denotes the median effective level for each agent, and \(\gamma _i\) represents the Hill coefficient for each agent.

The concept is commonly illustrated with an isobologram, where the additivity condition is represented by a straight line connecting the single-agent effect doses, as depicted in Fig. 1. Deviations below this line correspond to synergy, while deviations above it correspond to antagonism.

Fig. 1

Isobologram of the Loewe additivity model (\(y=0.5\)), with parameters \(C__1}=C__2}=3, \gamma _1 = \gamma _2 = 5\)

This representation provides the foundation for modeling multi-drug interactions under the principle of Loewe additivity, which in clinical medicine is applied to drugs within a therapy and is further extended through response surface models in the next subsection.

The Greco model

Assuming that the input–response curves of both agents follow the median-effect equation in Eq. 2, Greco et al. [6] proposed the following isobole equation corresponding to an effect y:

$$\begin 1 = \frac_1}\left( \frac\right) ^\frac}+\frac_2}\left( \frac\right) ^\frac}+\frac_1 C__2}}\left( \frac\right) ^+\frac}} \end$$

(6)

where the parameter \(\beta \) quantifies the degree of interaction, i.e. synergism, additivity, or antagonism. When \(\gamma _1=\gamma _2=\gamma \), this formula is reorganized and can be generalized as in [26]

$$\begin E = \frac \end$$

(7)

where the normalized index is

$$\begin I(c_1,c_2)=\frac_1}} + \frac_2}} + \beta \frac_1}} \frac_2}}. \end$$

The interaction parameter is \(\beta \), which captures the magnitude of synergy, additivity, or antagonism. For \( \beta = 0\), the model represents additivity; for \(\beta >0\), it indicates synergistic interaction; and for \(\beta <0\), it corresponds to antagonistic interaction. The three situations along with their respective isoboles at effect level \(y=0.5\), are illustrated in Fig. 2.

Fig. 2

Greco model representation of additivity, synergy, and antagonism, with isoboles shown at effect level \(y=0.5\). Parameters: \(C__1}=C__2}=3, \gamma _1 = \gamma _2 = 5 \)

The Minto model

Minto et al. [7] introduced an alternative formulation by normalizing the input levels of each agent to their respective potencies. Their key idea was to treat any fixed ratio of the two agents as a “new entity,” which is characterized by its own sigmoidal \(E_\). Accordingly, a normalized index is defined as:

$$\begin I(c_1,c_2)=\frac(\theta )} \end$$

(8)

where

$$\begin U_(\theta )= 1- \beta \theta +\beta \theta ^2, \quad I_1(c_1)=\frac}_1}}, \quad I_2(c_2)=\frac}_2}} \end$$

(9)

Here, \(I_1(c_1)\) and \(I_2 (c_2) \) are the normalized ratios of input levels of agent, \(c_\) and agent 2, \(c_\) with respect to their median effective levels \(C__1}\) and \(C__2}\), respectively. The term \(\beta \) indicates the interaction as in the Greco model. In addition to the features of the Greco model, the potency of the drug mixture \(\theta \) that ranges to 1 (only agent 1) from 0 (only agent 2) can be obtained as follows:

$$\begin \theta (c_1, c_2) = \frac \end$$

(10)

The Finney model

The model proposed by Finney [9] is expressed as:

$$\begin Y = \beta _0 + \beta _1 \log \left( c_1 + \rho c_2 + \beta _4 (\rho c_1 c_2)^\frac\right) , \end$$

(11)

where Y denotes the transformed effect, \(c_1\) and \(c_2\) are the amounts of agents 1 and 2 in the combination, respectively, and \(\beta _4\) represents the interaction term. This model assumes a fixed relative potency \(\rho \), defined as \(\rho = \frac_1}}_2}}\). Later, Plummer and Short [8] generalized this model to allow the relative potency \(\rho \) to vary, extending its applicability for more flexible interaction assessments.

Analytical feasibility analysis

This section investigates the conditions under which response surface models yield real-valued and biologically interpretable responses under standard PKPD constraints. We begin by stating the basic assumptions on the input variables, model parameters, and transformations. These are followed by remarks that highlight immediate implications for the response and its transformed variables. Building on these observations, we establish a general feasibility condition as a proposition, which serves as the basis for the detailed analysis of each surface formulation in the subsequent sections.

Assumption 1

The input levels satisfy \(c_1, c_2 > 0\), and the median effect doses are positive, i.e. \(C__1}, C__2} > 0\).

Assumption 2

The Hill coefficients in Eq. 2 are strictly positive, i.e. \(\gamma _1, \gamma _2, \gamma > 0\).

In the PKPD context, Assumptions 1 and 2 are naturally satisfied, since drug concentrations, their effects and Hill coefficient are intrinsically non-negative. Therefore, within PKPD applications these conditions should be interpreted as remarks rather than restrictive modeling assumptions. They are stated explicitly here to preserve generality of the response surface modeling framework and to accommodate potential applications in other domains where such positivity cannot be taken for granted.

Based on these assumptions, several immediate mathematical consequences can be derived regarding the model parameters and the domains of response and their transformations.

Remark 1

Based on Assumption 2, the slope parameter \(\beta _1\) in Eq. 3 is strictly positive, i.e., \(\beta _1 > 0\).

Remark 2

Based on Assumptions 1 and 2, the intercept parameter \(\beta _0\) in Eq. 3 is strictly negative, i.e., \(\beta _0 < 0\).

Remark 3

The response variable \( E \) in Eq. 2 is bounded within the interval \([0,1)\).

Remark 4

The transformed variable \( Y \) in Eq. 3 is well-defined only when \( E \in (0,1) \).

These observations lead to a general feasibility condition, summarized in the following proposition.

Proposition 1

For all input pairs \((c_1,c_2)\) such that \(I(c_1,c_2)\) is non-negative, the Hill surface model yields real-valued responses that are interpretable under domain-specific constraints; in PKPD applications, this corresponds to biological interpretability.

This condition forms the foundation for all global and local feasibility analyses of the specific surface formulations presented hereafter.

Proof

Recall that the Hill surface response is given in Eq. 7. For \(I(c_1,c_2) \ge 0\) and \(\gamma > 0\), the numerator is non-negative and the denominator strictly positive. Hence, the response E is real-valued and bounded in [0, 1), which corresponds to biological interpretability in PKPD applications. If \(I(c_1,c_2)<0\), the exponentiation may become undefined or complex for non-integer \(\gamma \), yielding biologically implausible values. Thus, the condition \(I(c_1,c_2)\ge 0\) with \(\gamma >0\) guarantees real and interpretable responses, forming the basis for the feasibility analyses.\(\square \)

The subsequent subsections introduces a feasibility analysis that distinguishes between isobole-level feasibility (i.e. the existence of well-defined isoboles at a given effect level) and global feasibility (i.e. well-posedness across the entire domain).

Feasibility of the Greco model Theorem 1

(Global feasibility) Consider the Greco model in Eq. 7. The model admits global feasibility for all \((c_1,c_2)>0\) if and only if \(\beta \ge 0\).

Proof

We examine the sign of \(I(c_1,c_2)\) to assess global feasibility.

Case A (\(\beta \ge 0\)). For any \(c_1,c_2>0\), the normalized index satisfies \(I(c_1,c_2)>0\). Consequently, the effect level \(E \in (0,1)\) is well-defined for all \((c_1,c_2)>0\).

Case B (\(\beta < 0\)) Along the diagonal \(c_1=c_2\rightarrow \infty \), the normalized index becomes

$$ I(c_1,c_1) = 2c_1 + \beta c_1^2 \;\rightarrow \; -\infty . $$

Hence, I attains negative values. In this case, \(I^\gamma <0\) or a complex value if \(\gamma \) is different from even integer. In this case, E is no longer biologically interpretable. Therefore, no global feasibility holds when \(\beta <0\).

It follows that the Greco model is globally feasible for all \((c_1,c_2)>0\) if and only if \(\beta \ge 0\).\(\square \)

Theorem 2

(Isobole-level feasibility) The Greco model produces real, feasible values provided that the interaction parameter \(\beta \) satisfies

$$ \beta> -\frac_1}}} \quad \text \quad \beta > -\frac_2}}} . $$

Proof

Consider the isobole corresponding to a fixed effect level \(y\). The corresponding normalized index is

$$ I_y = \Big (\tfrac\Big )^} $$

Case A (fixing \(c_1\)). Solving for \(c_2\) gives

$$ c_2 = \frac_1}}}_2}} + \beta \tfrac_1}} \tfrac_2}}} . $$

The denominator vanishes when

$$ \frac_2}} + \beta \frac_1}} \frac_2}} = 0, $$

which yields \(\beta = -C__1}/c_\). Since \(c_1 \in [0, c_]\), \(\beta \in [-\frac_1}}}, -\infty ]\). It follows the feasible condition:

$$ \beta > -\frac_1}}} . $$

Case B (fixing \(c_2\)). Analogously, solving for \(c_1\) gives

$$ c_1 = \frac_2}}}_1}} + \beta \tfrac_2}} \tfrac_1}}} . $$

The denominator vanishes when

$$ \frac} + \beta \frac_2}} \frac_1}} = 0, $$

which yields \(\beta = -C__2}/c_\). Since \(c_2 \in [0, c_]\), \(\beta \in [-\frac_2}}}, -\infty ]\). Then, the feasible condition is

$$ \beta > -\frac_2}}} . $$

When \(\gamma _1 = \gamma _2\), we have \(-C__1}/c_ = -C__2}/c_\) using Eq. 5.\(\square \)

Corollary 1

(Singularity lines of the Greco model) The Greco model becomes singular along the vertical and horizontal lines

$$ c_1 = -\frac_1}}, \qquad c_2 = -\frac_2}}. $$

For \(\beta >0\), these singularity lines lie in the negative region and thus outside the domain (\(c_1,c_2>0\)). For \(\beta <0\), the lines intersect the positive region.

The above results clarify the feasibility domain of the Greco model. The global condition \(\beta \ge 0\) ensures that all the combinations \((c_1,c_2)>0\) yield well-defined and biologically interpretable responses, corresponding to non-antagonistic interactions. When \(\beta < 0\), feasibility is restricted to certain regions of the dose–effect surface, as captured by the isobole-level conditions. In this case, singularity lines appear within the positive quadrant, reflecting parameter choices for which the Greco formulation yields biologically implausible or undefined responses. These insights indicate that while the Greco model can represent synergistic and antagonistic interactions, its practical applicability in clinical or pharmacological settings requires respecting specific parameter constraints to ensure biological interpretability across the relevant domain.

Feasibility of the Minto model Theorem 3

(Global feasibility) Consider the Minto model with \(I_1(c_1) > 0\), \(I_2(c_2) > 0\), and \(\theta \in [0,1]\). Singularities occur if and only if \(\beta \ge 4\). In that case, the singularity points exist at

$$ \theta _ = \frac}}, $$

which both lie strictly within [0, 1].

Proof

The singularity condition is given by \(U_(\theta )=0\), i.e.,

$$ \beta \theta ^2 - \beta \theta + 1 = 0. $$

The discriminant is \(\Delta = \beta ^2 - 4\beta = \beta (\beta -4)\).

If \(0< \beta < 4\), then \(\Delta < 0\), so no real roots exist.

If \(\beta = 4\), then \(\Delta = 0\), yielding one double root at \(\theta =\tfrac\).

If \(\beta > 4\), then \(\Delta > 0\), producing two distinct real roots

$$ \theta _ = \frac}}, $$

and both satisfy \(\theta _ \in [0,1]\).

If \(\beta < 0\), then \(\Delta > 0\) but the corresponding roots satisfy \(\theta _<0\) and \(\theta _>1\), hence they lie outside the the biologically interpretable domain.

Therefore, singularities arise if and only if \(\beta \ge 4\).\(\square \)

Corollary 2

(Singularity lines in the \((I_1,I_2)\) plane) Under the assumptions of Theorem 3 with \(\beta \ge 4\), the singular set in the \((I_1,I_2)\) plane consists of singular line(s) through the origin:

$$ I_2 = m_\pm \, I_1,\qquad m_\pm =\frac =\frac\,}}\,}}, $$

where \(\displaystyle \theta _\pm =\frac}}\).

Proof

Using Eq. 10,

$$ \theta =\frac=\theta _\pm \in (0,1). $$

Solving for \(I_2\) gives

Writing \(\theta _ = \frac}}\) yields \(m_\pm =\frac}}}}\). For \(\beta =4\), \(m=1\). Then, this reduces to the single line \(I_2=I_1\).\(\square \)

Corollary 3

(Mapping of singularities to the \((c_1,c_2)\) plane) For \(I_1=\tfrac_1}}\) and \(I_2=\tfrac_2}}\), the singular sets map to straight lines through the origin in the input level plane:

$$ c_2 \;=\; \kappa _\pm \, c_1,\qquad \kappa _\pm \;=\; \frac_2}}_1}}\, m_\pm \;=\; \frac_2}}_1}}\, \frac\,}}\,}} \, . $$

Proof

By substituting \(I_1 = \tfrac_1}}\) and \(I_2 = \tfrac_2}}\) into Corollary 2, the feasibility condition can be expressed in terms of the input variables \((c_1,c_2)\). For the specific case of \(\beta =4\), this condition simplifies to the single line

$$ c_2 = \frac_2}}_1}}\,c_1, $$

which characterizes the corresponding singularity in the input space.\(\square \)

Theorem 4

(Isobole-level feasibility) For any fixed effect level \(y \in (0,1)\), the corresponding isobole

$$ \frac(\theta )} = I_y, \qquad I_y>0, $$

is well-defined if and only if \(U_(\theta ) > 0\) for all admissible \(\theta \in (0,1)\).

Hence, isobole-level feasibility coincides with the global feasibility condition given in Theorem 3.

The feasibility analysis of the Minto model indicates that its validity depends critically on the value of the interaction parameter \(\beta \). When \(\beta < 4\), the model remains globally feasible with no singularities, ensuring well-defined drug–drug response surfaces. For \(\beta \ge 4\), however, singularities arise along specific lines in both the normalized \((I_1,I_2)\) space and the input \((c_1,c_2)\) plane, leading to regions where the model loses biological interpretability. Hence, while the Minto model can capture a broad range of drug interactions, its reliable application requires carefully avoiding parameter regimes that induce singular behaviour.

Feasibility of the Finney model Theorem 5

(Global feasibility) For all \(c_1,c_2>0\) and \(\rho >0\), the log-argument

$$ \Phi (c_1,c_2):=c_1+\rho c_2+\beta _4\sqrt $$

is strictly positive for every \((c_1,c_2)\) if and only if

$$ \quad \beta _4>-2. \quad $$

Proof

Let \(x=\sqrt\), \(v=\sqrt>0\). Then \(\Phi (x,v)=x^2+v^2+\beta _4 xv\). By the arithmetic-geometric mean inequality, \(x^2+v^2\ge 2xv\), hence

$$ \Phi (x,v) \ge (2+\beta _4)xv. $$

If \(\beta _4>-2\), the right-hand side is \(\Phi (x,v)>0\) for all \(x,v>0\).

If \(\beta _4=-2\), \(\Phi (x,v)=(x-v)^2\ge 0\) and vanishes along \(x=v\).

If \(\beta _4<-2\), choosing \(x=v\) yields \(\Phi (x,v)=(2+\beta _4)x^2<0\).

\(\square \)

Theorem 6

(Isobole-level feasibility) For any fixed \(y \in (0,1)\), the Finney isobole

$$ c_1 + \rho c_2 + \beta _4 \sqrt = I_y $$

admits a real and nonnegative solution \((c_1,c_2)\) if and only if

$$ \beta _4^2 \;\ge \; 4\big (1 - \eta _y\big ), $$

where \(\eta _y\) is a level-dependent ratio that reduces to \(\eta _y = I_y/A_\) or \(\eta _y = I_y/(\rho A_)\) depending on which variable is fixed. In particular, if \(|\beta _4| \ge 2\), the feasibility condition is always satisfied.

Proof

Set \(x=\sqrt\) and \(v=\sqrt\),

Case A: The isobole equation reduces to a quadratic in v:

$$ v^2+\beta _4 x v+(x^2-I_y)=0 $$

Real (hence nonnegative) solutions exist if and only if the discriminant \(\Delta (x)\) is nonnegative:

$$ \Delta (x)=\beta _4^2 x^2-4(x^2-I_y)=(\beta _4^2-4)x^2+4I_y \ge 0 $$

For \(x^2=c_1\in [0,C_],\)

If \(|\beta _4|\ge 2\), then \(\Delta (x)\ge 4I_y>0\).

If \(|\beta _4|<2\), \(\Delta (x)\) decreases with \(x^2\) and its minimum occurs at \(x^2=C_\), yielding

$$ 4I_y-(4-\beta _4^2)C_\ge 0 \iff \beta _4^\ge 4(1-I_y/C_). $$

Case B: Symmetrically, write the isobole equation in the quadratic in x

$$ x^2+\beta _4 v x+(v^2-I_y)=0 $$

Real (hence nonnegative) solutions exist if and only if the discriminant \(\Delta (v)\) is nonnegative:

$$ \Delta (v)=\beta _4^2 v^2-4(v^2-I_y)=(\beta _4^2-4)v^2+4I_y \ge 0 $$

The same monotonicity argument on \(v^2=\rho c_2\in [0,\rho C_] \) gives

$$ \beta _4^\ge 4(1-I_y/(\rho C_)). $$

Since \(\rho = C_/C_\), these two conditions are equal to each other.\(\square \)

Corollary 4

(Singularity lines) Let \(\Phi (c_1,c_2)\) denote the argument of the logarithm in the Finney surface model. In the positive region \((c_1,c_2>0)\) the singular set \(\\) has the following structure:

If \(\beta _4 = -2\), the singular set reduces to the single line

along which \(\Phi =0\).

If \(\beta _4 < -2\), the singular set consists of two lines

$$ c_2=\frac}\,c_1, \qquad t_\pm =\frac}>0, $$

which bound a sector where \(\Phi <0\) (i.e. the model is undefined in that region).

Proof

From Theorem 6, \(\Phi (x,v) \ge (2+\beta _4)xv\).

If \(\beta _4=-2\), then \(\Phi =(x-v)^2\ge 0\), and \(\Phi =0\) iff \(x=y\), i.e., \(c_2=\rho ^c_1\)

If \(\beta _4<-2\), write \(t=x/v=\sqrt\). Then

$$ \Phi =v^2\,(t^2+\beta _4 t+1). $$

The quadratic has two positive roots \(t_\pm =\frac}>0\), so \(\Phi (x,v)=0\) along \(t=t_\pm \), i.e., along the two lines, and \(\Phi <0\) strictly between them.

\(\square \)

The feasibility analysis of the Finney model highlights that the interaction parameter \(\beta _4\) plays a decisive role in determining whether the log-argument remains strictly positive. For \(\beta _4 > -2\), the model is globally feasible, ensuring biologically interpretable responses across all dose combinations. At the boundary case \(\beta _4=-2\), singularities appear along a single line, while for \(\beta _4<-2\), entire sectors of the input space yield complex-valued responses where the model breaks down. Thus, the Finney formulation accommodates both synergistic and antagonistic interactions, but antagonism beyond the threshold \(\beta _4=-2\) introduces non-feasible regions that limit its practical applicability.