The bivariate response to additive interacting doses (or BRAID) model
is a parametric response surface model of the effect of combined doses
of two agents. A full specification of the BRAID equation can be found
in vignette("derivation"), but briefly, the BRAID model is
specified by 9 parameters: a baseline minimal effect observed when no
agent is present, three dose response parameters for each agent, an
overall maximal effect parameter, and an interaction paramter \(\kappa\) indicating the presence of a
synergistic or antagonistic interaction.
In the functions of the braidrm package, BRAID surfaces
are specified by a numeric parameter vector. For a full, length-9
parameter vector, the values are the following
IDMA: The dose of median effect of the first agentIDMB: The dose of median effect of the second
agentna: The Hill slope (or sigmoidicity) of the first
agentnb: The Hill slope (or sigmoidicity) of the second
agentkappa: The BRAID interaction parameter, which is
between -2 and 0 for antagonistic surfaces, 0 for BRAID additive
surfaces, and greater than 0 for synergistic surfacesE0: The minimal effect observed when neither agent is
presentEfA: The maximal effect of the first agent,
theoretically observed at infinite concentrationEfB: The maximal effect of the second agent,
theoretically observed at infinite concentrationEf: The overall maximal effect of the combinationThe order of these parameters is meant to mirror the order of
single-agent dose-response parameters in the basicdrm
package: “potency” parameters, “shape” parameters (including
interaction), minimal effect, maximal effects.
In many cases, the response surface being modeled or studied will
only need a subset of these parameters. Nearly all combination analyses
assume (implicitly or explicitly) that the overall maximal effect of a
combination (the parameter \(E_f\))
will be equal to the “larger” of the two individual maximal effects. (We
place “larger” in quotes here is it refers to the effect value that is
further from the minimal effect, but not necssarily the numerically
greater value.) In some cases, it is even assumed that all maximal
effects are simply equal to each other. For simplicity, many
braidrm functions support shortened BRAID parameter vectors
that carry these assumptions. If a BRAID parameter vector with
eight values is passed to a braidrm function, it
is assumed that the ninth parameter Ef has been left out,
and the it should be equal to the “larger” of the two individual maximal
effects. If a BRAID paramter vector with seven values is
passed, it is assumed that parameters EfA and
EfB have been leftout, and that they are both equal to the
seventh value (assumed to be Ef).
Creating a BRAID parameter vector is as simple as creating a numeric vector:
This vector specifies, in order, that:
Because this vector is length seven, parameters EfA and
EfB are implied and assumed to be equal to Ef.
We could specify exactly the same surface with a full-length parameter
vector:
Evaluating a BRAID surface requires the concentration or concentrations of the first drug, the concentration or concentrations of the second, and the BRAID parameter vector. For example:
The first two outputs demonstrate that, as expected, when either drug is present at 1 micromolar, we observe an effect of 50, halfway between the minimal effect and the maximal effect. The third output is noticeably higher, as when both drugs are present at 1 micromolar, the effect of the individual doses is compounded. We can however produce the same effect as 1 micromolar of either drug alone by halving their doses:
This is because the parameter vector represents an additive combination of two drugs with identical pharmacological properties. (Note that this does not work for all response surfaces, or even all BRAID additive surfaces. It is only in the case of identical Hill slopes and maximal effects that BRAID additivity aligns perfectly with the more classical Loewe additivity.) (Loewe and Muischnek 1926)
We can also see that using the full, length-9 parameter vector produces exactly the same results:
Note what happens, however, when we introduce an interaction to the response surface (in this case, as \(\kappa\) is positive, a synergistic interaction):
While the effect of the individual drugs is unchanged, their combined effect is increased. This is the classic pharmacological definition of synergy: an effect in combination which is larger than what would be expected. What precisely is “expected” for any given pair of drugs is one of the primary debates of combination analysis; BRAID additivity is only one answer, albeit the one around which we have built the BRAID system.
Of course, the most common usage of the BRAID model is fitting it to
experimental data to summarize and quantify that data. The main
workhorse function for this task is braidrm, which produces
a fit object of class braidrm. We can see it in action with
the example data-set additiveExample:
head(additiveExample)
#> concA concB truth measure
#> 1 0 0.000 0.000000000 0.09014655
#> 2 0 0.125 0.001949318 -0.07605997
#> 3 0 0.250 0.015384615 -0.07342708
#> 4 0 0.500 0.111111111 0.18117181
#> 5 0 1.000 0.500000000 0.55059399
#> 6 0 2.000 0.888888889 0.90448671additiveFit <- braidrm(measure ~ concA + concB, additiveExample)
summary(additiveFit)
#> Call:
#> braidrm.formula(formula = measure ~ concA + concB, data = additiveExample)
#>
#> Lo Est Hi
#> IDMA 0.6820 0.8036 0.9442
#> IDMB 0.8082 0.9536 1.1083
#> na 2.0211 2.4200 3.1179
#> nb 2.0755 2.6070 3.4491
#> kappa -0.4527 -0.2071 0.0827
#> E0 -0.0609 0.0029 0.0591
#> EfA 0.9531 0.9948 1.0287
#> EfB 0.9503 1.0020 1.0334
#> Ef NA 1.0020 NAThe data frame additiveExample is a synthetically
generated set of response surface measurements, and contains four
columns: concA, containing the dose of drug A for each
measurement; concB, containing the dose of drug B for each
measurement; truth containing the ground truth response
generated by an additive response surface with effect ranging from 0 to
1; and measure, a noisy measurement sampled from a normal
distribution centered on the ground truth with a standard deviation of
0.07. By default, braidrm() fits an eight-paramter BRAID
surface (one which assumes the overall maximal effect is equal to the
larger of the two individual maximal effects) to the given data with a
moderate prior on the interaction parameter \(\kappa\). (See
vignette("modelsAndConstraints") for more details on
specifying the BRAID model to be fit, and
vignette("bayesianKappa") for a more in-depth explanation
of the Bayesian stabilization of \(\kappa\)). By default
braidrm() also estimates bootstrapped confidence intervals
on all fit parameters, as can be seen the printed summary; note in
particular that 0 lies within the confidence interval on \(\kappa\), indicating no statistically
significant deviation from BRAID additivity. (Estimating confidence
intervals can take several seconds; so if you are running many fits and
do not need the confidence intervals, you can forgo them by setting the
parameter getCIs to FALSE.)
We get a very different result, however, when we fit the example
data-set synergisticExample, which has the same format as
additiveExample but was generated with a synergistic
parameter vector with a \(\kappa\) of
2:
synergisticFit <- braidrm(measure ~ concA + concB, synergisticExample)
summary(synergisticFit)
#> Call:
#> braidrm.formula(formula = measure ~ concA + concB, data = synergisticExample)
#>
#> Lo Est Hi
#> IDMA 0.9087 1.0398 1.1765
#> IDMB 0.8897 1.0259 1.2158
#> na 2.3985 2.9116 3.7821
#> nb 2.0480 2.4990 3.1903
#> kappa 1.6173 2.1258 2.5755
#> E0 -0.0818 -0.0300 0.0187
#> EfA 0.9552 1.0080 1.0274
#> EfB 0.9144 0.9848 1.0224
#> Ef NA 1.0080 NAThough most of the parameter estimates are very similar (indeed the generating parameter vectors are identical except for \(\kappa\)), the confidence interval on \(\kappa\) lies well above zero, centered quite close the true value of 2.
Another useful fitting function is findBestBraid(),
which uses the Bayesian or Akaike information criterion to select among
several candidate models (again, see
vignette("modelsAndConstraints") for more details) Akaike (1974). This allows the user to stabilize
underdetermined values to reasonable defaults and reduces the frequency
of wildly implausible over-fitting. We have run the function on
antagonisticExample, which, as the name implies, contains a
synthetically generated set of response surface measurements taken on an
antagonistic response surface with a true \(\kappa\) of \(-1\). The usage of
findBestBraid() is very similar to that of
braidrm():
bestFit <- findBestBraid(measure ~ concA + concB, antagonisticExample,
defaults = c(0,1))
summary(bestFit)
#> Call:
#> findBestBraid.formula(formula = measure ~ concA + concB, data = antagonisticExample,
#> defaults = c(0, 1))
#>
#> Lo Est Hi
#> IDMA 0.9651 1.0210 1.1417
#> IDMB 0.9590 1.0267 1.1645
#> na 2.4189 2.8288 3.3656
#> nb 2.3304 2.6675 3.0672
#> kappa -1.1236 -1.0175 -0.8834
#> E0 NA 0.0000 NA
#> EfA NA 1.0000 NA
#> EfB NA 1.0000 NA
#> Ef NA 1.0000 NAThe defaults parameter here indicates that, absent any
other evidence, we expect the minimal effect to 0 and the maximal
effects to be 1, so simpler models that assume these fixed values and
explain the data equally well should be preferred. This time the
confidence interval on \(\kappa\) lies
well below zero and comfortably encloses the true value of \(-1\). No confidence intervals on either
minimal or maximal effects are included, indicating that the most
parsimonious model was one which fixed E0 at 0 and the
three maximal effects at the default 1.