The Hill model of dose-response, also known as the log-logistic function, is one of the most widely used mathematical tools in the analysis of pharmacodynamics. Though there are many, many cases in which the model falls short, its popularity derives from the balance it strikes between simplicity and versatility. Its four parameters - potency, sensitivity, minimal effect, and maximal effect - are easily understood and all carry intuitive pharmacological relevance. So when we set out to develop a similar model for combined action, we chose to make the Hill model its basis, both in form and in the behavior of drugs individually.
We therefore set ourselves the following criteria:
We will suppose that our model will be reprsented by a function \(E(D_A,D_B)\). Our first condition states that the following must be true:
\[ E(D_A,0) = E_A(D_A) = E_0 + \frac{E_{f,A}-E_0}{1+\left(\frac{D_A}{{ID}_{M,A}}\right)^{-n_a}} \] \[ E(0,D_B) = E_B(D_B) = E_0 + \frac{E_{f,B}-E_0}{1+\left(\frac{D_B}{{ID}_{M,B}}\right)^{-n_b}} \]
Our second assumption (additivity for the sham combination of a single compound with itself at a different dilution) can be expressed mathematically. Suppose that “drug A” is actually a single drug diluted by a factor \(R_A\) (so the concentration of the actual drug is \(R_AD_A\)); similarly, “drug B” is the same drug diluted by a factor \(R_B\). Then the does of median effect of our first “drug” is simply \({ID}_{M,A}={ID}_M/R_A\), where \({ID}_M\) is the true dose of median effect; and the dose of median effect for our second “drug” is \({ID}_{M,B}={ID}_M/R_B\). (Obviously the Hill slopes and maximal effects will all equal.) In this particular case then, we have that:
\[ E(D_A,D_B) = E_0 + \frac{E_f-E_0}{1+\left(\frac{R_AD_A+R_BD_B}{{ID}_M}\right)^{-n}} \] This can be rewritten as:
\[ E(D_A,D_B) = E_0 + \frac{E_f-E_0}{1+\left(\frac{R_AD_A}{{ID}_M}+\frac{R_BD_B}{{ID}_M}\right)^{-n}} = E_0 + \frac{E_f-E_0}{1+\left(\frac{D_A}{{ID}_{M,A}}+\frac{D_B}{{ID}_{M,B}}\right)^{-n}} \]
Which is the closed-form expression for a Loewe additive combination of compounds with identical maximal effects and Hill slopes.
The equation above only applies to a very narrow set of circumstances in which the two compounds being combined are basically pharmacologically identical. Indeed, it is a well known result that there is no closed-form expression for a general Loewe additive surface. In order to generate the BRAID model, we must generalize this expression to work with other compound behaviors.
The form of the Loewe additive sham combination can be described as a traditional Hill equation, but with the dose of the single drug replaced with a weighted sum of the two doses present. In a much more general sense, the equation for an additive combination takes this form:
\[ E(D_A,D_B) = E_0 + \frac{E_f-E_0}{1+\left(g_A(D_A)+g_B(D_B)\right)^{-n}} \]
where \(g_A(D_A)\) is a “sensible” function of \(D_A\), and \(g_B(D_B)\) is a “sensible” function \(D_B\). In this case, by “sensible”, we simply mean that it behaves like a rescaled or transformed dose, so \(g_A(D_A)\) is not affected by \(D_B\), \(g_A(0)=0\), and as \(D_A\) increases, so does \(g_A(D_A)\) (that is, \(g_A\) is monotonically increasing). Clearly the equation we derived for the sham combination satisfies these assumption, but hopefully an equation that works for more general combinations can be found as well.
Of course, we want our model to support interaction as well. Traditionally, to introduce interaction to an additive combination, one adds a product term with a coefficient governing the sign and magnitude of the interaction; so \(A+B\) would be come \(A+B+\alpha AB\). This causes very serious problems for combined doses; though we do wish them to behave additively, the doses can never be less than 0. Unfortunately, the expression \(A+B+\alpha AB\) will always become negative for certain non-negative pairs \((A,B)\) if alpha is any value less than zero, meaning this method would not allow us to generate reduced effective doses; in short, the model could not include antagonism. Since antagonism is clearly an observed phenonemon, this is insufficient. Fortunately, the problem can be side stepped by using a square-root interaction: \(A + B + \kappa\sqrt{AB}\). So long as \(A\ge 0\), \(B \ge 0\) and \(\kappa \ge -2\), this quantity will always be non-negative.
So now we have a proposed structure for our genaral combined model:
\[ E(D_A,D_B) = E_0 + \frac{E_f-E_0}{1+\left(g_A(D_A)+g_B(D_B) + \kappa\sqrt{g_A(D_A)g_B(D_B)}\right)^{-n}} \]
Surprisingly, this is enough to almost completely define our model. Setting \(D_A\) or \(D_B\) to zero and using our original assumptions allows us to conclude that
\[ g_A(D_A) = \left( \frac{\left(\frac{E_{f,A}-E_0}{E_f-E_0}\right)\left(\frac{D_A}{{ID}_{M,A}}\right)^{n_a}} {1+\left(1-\frac{E_{f,A}-E_0}{E_f-E_0}\right)\left(\frac{D_A}{{ID}_{M,A}}\right)^{n_a}} \right)^\frac{1}{n} \] \[ g_B(D_B) = \left( \frac{\left(\frac{E_{f,B}-E_0}{E_f-E_0}\right)\left(\frac{D_B}{{ID}_{M,B}}\right)^{n_b}} {1+\left(1-\frac{E_{f,B}-E_0}{E_f-E_0}\right)\left(\frac{D_B}{{ID}_{M,B}}\right)^{n_b}} \right)^\frac{1}{n} \]
In the event that the overall maximal effect parameter \(E_f\) is equal to one or both of the individual maximal effect parameters, the corresponding expressions for \(g_A\) and/or \(g_B\) simplifies considerably; but the BRAID package supports the option of an even higher overall maximal effect, so this parameter \(E_f\) is treated as an additional parameter of the model.
One value that is not treated as a further parameter is the “overall” Hill slope, written here as \(n\). We have found little value in varying this parameter independently, so it is assumed to be an aggregate of the two Hill slopes. In order to be consistent with the sham experiment, it must simplify to be equal to the two individual Hill slopes when they are equal, so an average would seem to be most sensible. However, as Hill slopes are strictly greater than zero, and tend to distribute themselves in a log-normal fashion, a sligthly more reasonable choice is the geometric mean of the two Hill slopes. So the BRAID model assumes that \(n = \sqrt{n_a n_b}\).
Putting all this together, we end up with the full BRAID model in its current form:
\[ E(D_A,D_B) = E_0 + \frac{E_f - E_0}{1+\left(\mathbb{D}_A^{1/n} + \mathbb{D}_B^{1/n} + \kappa\sqrt{\mathbb{D}_A^{1/n}\mathbb{D}_B^{1/n}}\right)^{-n}} \] where
\[ \mathbb{D}_A = \frac{F_A\left(\frac{D_A}{{ID}_{M,A}}\right)^{n_a}} {1+(1-F_A)\left(\frac{D_A}{{ID}_{M,A}}\right)^{n_a}} \] \[ \mathbb{D}_B = \frac{F_B\left(\frac{D_B}{{ID}_{M,B}}\right)^{n_b}} {1+(1-F_B)\left(\frac{D_B}{{ID}_{M,B}}\right)^{n_b}} \] \[ F_A = \frac{E_{f,A}-E_0}{E_f-E_0} \] \[ F_B = \frac{E_{f,B}-E_0}{E_f-E_0} \] \[ n = \sqrt{n_a n_b} \]