Comparison of null models for combination drug therapy reveals Hand model as biochemically most plausible

Null models for the effect of combination therapies are widely used to evaluate synergy and antagonism of drugs. Due to the relevance of null models, their suitability is continuously discussed. Here, we contribute to the discussion by investigating the properties of five null models. Our study includes the model proposed by David J. Hand, which we refer to as Hand model. The Hand model has been introduced almost 20 years ago but hardly was used and studied. We show that the Hand model generalizes the principle of dose equivalence compared to the Loewe model and resolves the ambiguity of the Tallarida model. This provides a solution to the persisting conflict about the compatibility of two essential model properties: the sham combination principle and the principle of dose equivalence. By embedding several null models into a common framework, we shed light in their biochemical validity and provide indications that the Hand model is biochemically most plausible. We illustrate the practical implications and differences between null models by examining differences of null models on published data.

Dividing by dc yields If we let tend dc −→ 0, also x 1 → x 0 , the difference quotient approaches and we get the above ODE.

Representations of the ODE
For completeness, and as they are referred to in the Supplementary Information, we mention here again the alternative representations of the ODE. This is an exact copy of the main article. Using the inverse derivative formula, we express the differential equation (1) as 1 or in integral representation In terms of sensitivities it is 2 Effect-sensitivity curves

Derivation of the effect-sensitivity formula for Hill curves
The dose-effect behavior is often modeled by Hill curves. For Hill curves of the form EC 50,A a n A 1 + EC 50,A a n A , the derivative is Thus, for the sensitivity we find

Conversion of dose-effect and effect-sensitivity curves
Starting with a strictly increasing, piecewise differentiable dose-effect function f A , its inverse exists and its sensitivity is . For a given positive piecewise continuous effect-sensitivity function s A , the inverse f −1 is obtained by dy.
Since by this calculation f −1 A is a strictly increasing function, it is injective and its inverse f A exists. Alternatively, f A is obtained as the unique strictly increasing solution of the autonomous ODE d da x(a) = s A (x(a)), x(0) = E min,A . In order for the above calculation to be valid, we must assume that if lim x→E min,A s A (x) = 0, the integral is finite. For the class of s A as given by the above formula for Hill curves, this assumption is satisfied, because 1 s A behaves like (x − E min,A ) −γ near E min,A with γ = 1 − 1 n A ∈ (0, 1) and thus integrates to a finite value. Note, that in the limit case the solution x(a) ofẋ = s A (x) approaches E min,A only in the limit a → −∞, not in a finite amount of dose.
Numerically the integral (6) or equivalently the ODEẋ = s A (x), x(0) = E min,A must be treated with care whenever s A (E min,A ) = 0 because it allows the unfavored constant solution x ≡ E min,A as well as solutions that are initially constant and exit E min,A at an arbitrary dose value.
In the numerical implementation we solved this problem by setting the ODE's initial value to x 0 = ε = 1e-7. Details can be found in Sections 6 and 7. Let α be the constant potency ratio, i.e., for any effect level x, f −1 A (x) = αf −1 B (x), it holds for the derivatives: Then by (2) it follows for the combined curve f AB,λ of child agent C λ .
Consequently f AB,λ , f A and f B are pairwise in constant potency relation. Let x be a given effect level. Now, we prove that the dose pair (a, b) lies on the straight line connecting 3.2 Proof of the sham combination principle for the Hand, the Loewe and the Tallarida model , then A and B have constant potency ratio α = 1. In this case the Loewe, the Hand and the Tallarida models

Disproof of the sham combination principle for the Bliss and the HSA model
The sham combination principle for the Bliss model is addressed in (Foucquier and Guedj, 2015). The HSA model if λ ∈ (0, 1) and f A is strictly increasing.
3.4 Proof of the commutativity for the Hand, the Loewe, the Bliss and the HSA model • The Hand model is commutative in A and B. f AB,λ = f BA,1−λ because both satisfy the same ODE. Hence switching the roles of A and B along with their weights does not alter the combined dose-effect curve.
• The formulas for effects E Bliss and E HSA are symmetric in A and B.
• The Loewe isobole equation is symmetric in A and B. The isoboles determine the effect surface uniquely.

Disproof of the commutativity for the Tallarida model
The proof for the asymmetry in the Tallarida model is given in (Lorenzo and Sánchez-Marin, 2006).

Proof of the associative property for the Loewe and the Hand model
• The Loewe model: In terms of the combined curve f AB,λ , we write the Loewe isobole equation as Using analogous equations for f AB,µ and f C λ Cµ,ν , we can calculate • The Hand model: The combined agents C λ and C µ satisfy (2) .
For arbitrary ν ∈ [0, 1], the grand child agent satisfies the grand child agent formed of the child agents C λ and C µ can indeed be formed by the parent agents A and B at appropriate ratio νλ + (1 − ν)µ.
3.7 Disproof of the associativity property for the Tallarida, the Bliss and the HSA model • The Tallarida model does not satisfy the associativity property. By the choice λ = 0, µ = 1, the associativity property implies commutativity, which the Tallarida model violates.
• For the Bliss model, the case λ = µ = ν = 1 2 , A = B shows that the associative property is violated because • The HSA model is characterized by isoboles that form a rectangle with the dose axes. If the newly allocated coordinate axes are bent, the isobole will be reshaped to an angle of more than 90 • . The characteristic property is then lost. Moreover, the isobole suggested by applying the HSA model on C λ and C µ encloses the original isobole, resulting in a smaller prediction value. This geometric interpretation of the associative property is displayed in the formal definition as well: From we conclude that

Proof of the isoboles' convexity in the Hand model
The isoboles obtained from the Hand model are convex. They are strictly convex if and only if f A and f B exhibit a varying potency ratio.
We prove this property of the Hand model using (i) the integral representation (3) and (ii) the associativity property: • Fix an effect level x, which is in the target domain of f A and of f B . Then for any λ ∈ (0, 1), it holds with i.e., the pair (a, b) predicted by the Hand model to generate E H (a, b) = x lies below the Loewe straight isobole.
• To show convexity, apply the same argument with two child agents C λ , C µ instead of A, B and corresponding points on the isobole at level x. By the associativity property, we can conclude that for any σ ∈ [λ, µ] the point (σc, (1 − σ)c) on the Hand isobole at effect level x lies below the line segment through P 1 and P 2 , completing the proof of the convexity property.
Now that the plan was established we carry out the steps.
• We proceed by first proving (8). For readability setÃ = f −1 A (x),B = f −1 B (x). Factoring out c, and multiplying by Integrating the rates f −1 AB,λ , f −1 A , f −1 B , we get the following representations Define the second and third integrand as h A (y) and h B (y), respectively. Using the integral representation (3) of the ODE and the identity (z −1 + w −1 ) −1 = zw z+w , which holds for any two positive real numbers: Substituting (10) and (11), (9) is equivalent to proving: In order to establish this integral inequality, we use Cauchy-Schwarz for the scalar product For readability we omit the integral bounds 0, x and the integration variable y when calculating Furthermore, equality holds in (12) if and only if f = γg for some γ = 0. For the above f and g, this is equivalent to saying which means that f A and f B exhibit a constant potency ratio.

Derivation of the Loewe limit isobole
Let A and B be a partial and a full agent, i.e. E max,A < E max,B . Define b * := f −1 B (E max,A ). Then the Loewe model assigns an effect value to all dose pairs (a, b) in the domain D = {(a, b)|b < b * } and E L (a, b) < E max,A .
Proof: Take an arbitrary (a 0 , b 0 ) ∈ D. Define the function ϕ : Then ϕ is a continuous function and ϕ On the other hand, all isoboles at effect levels 0 < x < E max,A lie in the domain D. The isoboles at effect levels 0 < x < E max,A cover the domain D. By continuity of E L the limit isobole at the effect level E max,A is forced to be the boundary of the set D, which is the horizontal line {(a, b)|b = b * }.

Numerical implementation of the Hand model
The analysis of the experimental data from (O'Neil et al., 2016)  The different null models were implemented in MATLAB. The equation which defines the reference effect for the Loewe model, was solved by bisection. If the dose pair (a, b) did not lie HSA (a, b). This was done since the (axis parallel) HSA isobole can be seen as continuous extension of the Loewe isobole (see Section 5).
The combined dose-effect curve f AB of the Hand model was computed by solving the ODE (4): The initial condition was set to ε = 1e-7, a value slightly above the ODE solvers absolute error tolerance. The MATLAB solver ode15s was used with the settings AbsTol = 1e-8, RelTol = 1e-5 and the non-negative option.
Numerically, the ODE (1) must not be solved starting in x(0) = E min,A , since the simulation would result in the unfavored solution which is constant x ≡ E min,A , unless the Hill coefficient is 1. If the ODE is simulated starting in x(0) = E min,A +ε, the error for the dose can be approximated. Assuming without loss of generality that E min,A = 0, we obtain For λ ∈ (.25, .75) we even get a better bound.
where we used the inequality of the geometric and arithmetic mean to estimate the integral.