---
title: "Derivation of the BRAID Model"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Derivation of the BRAID Model}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r setup, include=FALSE}
library(braidrm)
```

## Background: The Hill Equation

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:

* The model should simplify to traditional Hill dose-response behaviors when
one of the drugs was absent
* The model should, like Loewe additive surfaces, satisfy the condition that the
same agent combined with itelf (perhaps at different dilutions) always be
represented by a non-interacting "additive" surface
* The model should allow the maximal effects of the two drugs to differ
* The model should inlclude at least one parameter governing the nature and
magnitude of interaction

## First Steps: The Assumptions

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.

## Generalizing

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 $n = \sqrt{n_a n_b}$.

## The Final Model

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

$$
\mathbb{D}_A = \frac{F_A\left(\frac{D_A}{{ID}_{M,A}}\right)^{n_a}}
{1+(1-F_A)\left(\frac{D_A}{{ID}_{M,A}}\right)^{n_a}}
$$
$$
\mathbb{D}_B = \frac{F_B\left(\frac{D_B}{{ID}_{M,B}}\right)^{n_b}}
{1+(1-F_B)\left(\frac{D_B}{{ID}_{M,B}}\right)^{n_b}}
$$
$$
F_A = \frac{E_{f,A}-E_0}{E_f-E_0}
$$
$$
F_B = \frac{E_{f,B}-E_0}{E_f-E_0}
$$
$$
n = \sqrt{n_a n_b}
$$
