Suppose we have a set of drugs (or other treatments
which we misclassify as ``drugs'' for the purpose of our analysis.)
These drugs are supposed to be treatments for some disease, * e.g. *,
HIV infection. Our goal is to assess whether, and to what extent,
treatment with a particular combination of these
drugs is more or less effective than some other such combination
treatment. Such combination treatments include the case
of treatment with a single drug, so we can also compare the efficacy
of two distinct single drugs with our method.

For each treatment with a combination of drugs
$D$_{1},..., D_{ k } ,
the data we have is a set of $k+1$ dimensional points of the form
$(d$_{ 1 },...,d_{ k },y) ,
where $d$_{ j } is
the dose of drug $D$_{ j } (measured
in any desirable units) and $y$
is the * response * observed in the subject
represented by the point at hand.

The observed response is always non-negative, and is generally a
value in the interval $[0,1]$.
Such a value $y$ denotes the relative
amount or level of disease present * after * the drug treatment.
Thus a small response value indicates a better response than does
a larger response value. Crudely, a response value $y$ should be
measured such that $100(1-y)$ is the percent improvement observed.
For example, let $w$_{ 0 } denote
a subject's white blood cell count
before treatment and let $w$_{ 1 }
denote that subject's white blood cell
count after treatment; then the response for that subject might be
$1-(w$_{ 1 }-w_{ 0 })/w_{ 0 }
which is 1 minus the precentage improvement observed.
The value 1 or greater denotes no suppression of disease, and values
near 0 indicate substantial suppression of disease.
Another example is that where we measure a quantity that is directly
related to the level of disease present, such as the concentration of PSA
present, normalized to lie in the interval $[0,1]$,
although here, as in most
cases, a percentage response is more appropriate, since the amount
of disease present prior to treatment can vary greatly.

For a single drug, a so-called logistic dose-response curve
often provides an adequate description of the effect of treatment
with that drug. * We assume that such a model for single drugs
is appropriate here. * (Although, see the remarks below on a
non-parametric approach.) The general logistic dose-response function
for a single drug is:

In our case, we assume that $a=1$ (* i.e. *,
we have no suppression
of disease at 0 dose) and that $h=0$ (* i.e. *,
we approach complete
suppression of disease as the dose becomes sufficiently large.)
Thus, $y(d)=1/(1+(d/c)b)$. This
dose-response curve generally ``decays''
sigmoidally from the value 1 to the value 0
as the dose $d$ increases.
If $b>1$, we have faster decay,
if $0\; <\; b\; <\; 1$, we have slower decay, if
$b=0$, $y(d)$ is the constant
value .5, and if $b<0$, our dose-response
curve * inceases * instead of decaying.

We may construct a single drug dose-response model for given
dose-response data $(d$_{ 1 },v_{ 1 }),...,
(d_{ m },v_{ m }) by estimating
the parameters $c$ and $b$ that
fit the model function $y$ to
our data via weighted least-squares minimization. An example showing the
use of the MLAB mathematical and statistical modeling system[1]
to read-in the dose-response data,
define the single-drug logistic dose-response model function,
provide initial guesses for
the parameters $c$ and $b$, fit
our model to the data to obtain the
least-squares estimates of $c$ and $b$,
and finally, to graph the results,
is given below. (Often it is necessary to impose constraints on the
parameters $c$ and $b$ in
order to get a successful fit.
This can be done in MLAB, but it is not
needed in the example given here.)

* dv=read(d1data,100,2) * fct y(d) = 1/(1+(d/c)^b) * c = 3; b = 1 * fit (b,c), y to dv final parameter values value error dependency parameter 4.166513485 0.3632194177 0.05378619542 B 2.886642188 0.06692784189 0.05378619542 C 6 iterations CONVERGED best weighted sum of squares = 2.664797e-02 weighted root mean square error = 4.921934e-02 weighted deviation fraction = 5.824418e-02 R squared = 9.833524e-01 * draw points(y,0:6!100) * draw dv lt none pt circle ptsize .02 * left title "response" font 18 * bottom title "drug dose" font 18 * top title "fit of y(d) = 1/(1+(d/c)'.5ub'.5d)" font 7 * title "c = "+c at (.5,.8) * title "b = "+b at (.5,.75) * view

Now let us consider a two-drug combination treatment. The dose-response
model in this situation is a function of two arguments,
$y(d$_{ 1 },d_{ 2 }) ,
that predicts the response to the combination dose
$(d$_{ 1 },d_{ 2 }) .
Let us suppose that drug $D$_{ 1 }
and drug $D$_{ 2 } have no interaction.
Moreover, we postulate that
$y(d$_{ 1 },0)=1/
(1+(d_{ 1 }/c_{ 1 })^b_{ 1 }) and
$y(0,d$_{ 2 })=1/(1+(d_{ 2 }/c_{ 2 })
^b_{ 2 }) . (The '^' symbol denotes the exponentiation
operation; we must use it due to the inadequacies of HTML.)
We assume that the parameters
$c$_{ 1 } , $b$_{ 1 } ,
$c$_{ 2 } , and $b$_{ 2 }
are known due to fitting the
single-drug logistic dose-response model to given
single-drug dose-response data for drug
$D$_{ 1 } and separately for
drug $D$_{ 2 } . In analogy to the terminology used with
multivariate distribution functions, we may call the single-drug
logistic functions $y(d$_{ 1 },0)
and $y(0,d$_{ 2 }) the * marginal *
dose-response functions for the two-argument dose-response function
$y$.

Let $h$_{ 1 } be
the value such that our non-interaction two-drug
dose-response model satisfies $y(\; h$_{ 1 },0)=r , and
let $h$_{ 2 } be the
value such that $y(0,\; h$_{ 2 })=r .
Now based on our no-interaction hypothesis, we can follow Bunow and
Weinstein [2], and geometrically
define $y(d$_{ 1 },d_{ 2 }) by postulating
that $y(d$_{ 1 },d_{ 2 })=r
for $(d$_{ 1 },d_{ 2 }) on
the line-segment connecting
$(\; h$_{ 1 },0)
and $(0,\; h$_{ 2 }) . Note it would not be
appropriate to define
$y(d$_{ 1 },d_{ 2 })=
y(d_{ 1 },0)+y(0,d_{ 2 }) since
drug $D$_{ 1 } and
drug $D$_{ 2 } * compete *
to suppress the disease, even
if they act independently; that is, when
drug $D$_{ 1 } has acted to suppress
some of the disease, there is less ``disease''
present for drug $D$_{ 2 }
to apply to, and conversely.
The line-segment connecting
$(\; h$_{ 1 },0) and $(0,\; h$_{ 2 })
is $L$_{ r } :=
(d_{ 1 },d_{ 2 }) | (d_{ 1 },d_{ 2 })=
a ( h _{ 1 },0)+(1- a )(0, h _{ 2 }),
0 __<__ a __<__ 1, and we specify that
$y(d$_{ 1 },d_{ 2 })=r for
$(d$_{ 1 },d_{ 2 }) belongs to the set
$L$_{ r } .
This is appropriate because,
when drug $D$_{ 2 }
is the same as drug $D$_{ 1 } ,
the predicted response to a
combination dose $(d$_{ 1 },d_{ 2 })
* must * be the same as the
predicted response to a dose of size
$d$_{ 1 }+d_{ 2 }
of drug $D$_{ 1 } (or $D$_{ 2 } )
alone!
Any drug-interaction model which fails to satisfy this
condition cannot be a statistically-correct model.

Now note that, if $z=1/(1+(d/c)b)$, then $1=((1\; /\; z)-1)-1/b(d/c)$.

Recall that $y(\; h$_{ 1 },0)=r=
1/(1+( h _{ 1 }/c_{ 1 })^[b_{ 1 }]) .
Thus $1=((1\; /\; r)\; -1)^[-1/b$_{ 1 }]
( h _{ 1 }/c_{ 1 }) , so
$a\; =((1\; /\; r)\; -1)^[-1/b$_{ 1 }]
( a h _{ 1 }/c_{ 1 }) .
Also $y(0,\; h$_{ 2 })=r=
1/(1+( h _{ 2 }/c_{ 2 })^[b_{ 2 }]) , so
$1=((1\; /\; r)\; -1)^[-1/b$_{ 2 }]
( h _{ 2 }/c_{ 2 }) , and thus
$1-\; a\; =((1\; /\; r)\; -1)^[-1/b$_{ 2 }]
((1- a ) h _{ 2 }/c_{ 2 }) .

But then, when $(d$_{ 1 },d_{ 2 }) belongs
to the set $L$_{ r } , $d$_{ 1 }=
a h _{ 1 } and
$d$_{ 2 }=(1- a ) h _{ 2 } for
some value $a$ such that $0$__<__ a __<__ 1]
, and we have specified that
$y(d$_{ 1 },d_{ 2 })=r . Next, note that

This Bunow-Weinstein two-argument dose-response function for
non-interacting drugs is
a logically-correct generalization of a single-drug
logistic dose-response function. This implicit function
can be defined in MLAB as shown below. Note the care that is
taken to avoid division by zero and raising zero to a negative power.

function y1(d1)=1/(1+(d1/c1)^b1) function y2(d2)=1/(1+(d2/c2)^b2) function y(d1,d2)=if d1=0 then y2(d2) else if d2=0 then y1(d1) else if y1(d1)<.00001 and y2(d2)<.00001 then 0 else if y1(d1)>.99990 and y2(d2)>.99990 then 1 else root(z,1e-11,1-1e-11,(d1/c1)*(1/z-1)^(-1/b1)+(d2/c2)*(1/z-1)^(-1/b2) -1)

The MLAB commands used to produce a
picture of the ``response-surface'' defined by our model
function $y$ for two
non-interacting drugs with identical marginal curves
defined by $c$_{ 1 }=c_{ 2 }=4.1665
and $b$_{ 1 }=b_{ 2 }=2.8866 are given below.
We assume the commands defining the functions
$y$_{ 1 } , $y$_{ 2 } ,
and $y$
have been executed.

/* define c1=c2=2.8866 and b1=b2=4.1665 */ c1=2.8866; c2=c1; b1=4.1665; b2=b1 /* draw a 3D perspective view of the non-interaction drug response surface for two drugs having the same marginal single drug dose-response curves. */ s = cross(0:6!20,0:6!20) s col 3 = y on s draw s linetype net cmd3d("axes") view

The produced picture is shown below.

We may continue the above MLAB dialog to produce
a contour map corresponding to the function $y$
with $c$_{ 1 }=c_{ 2 }=4.1665
and $b$_{ 1 }=b_{ 2 }=2.8866
corresponding to the surface shown above.
The contour map shows some level lines where
$y(d$_{ 1 },d_{ 2 })
is constant.

delete w3 /* draw a contour map corresponding to the surface above. */ draw contour(s) linetype vmarker left title "drug-1 dose" bottom title "drug-2 dose" top title "c1=c2="+c+" b1=b2="+b view

We may further continue the above MLAB dialog to produce
a contour map corresponding to the function $y$
in the case where $c$_{ 1 }=2.08325 ,
$c$_{ 2 }=4.1665 and
$b$_{ 1 }=b_{ 2 }=2.8866 .

delete w /* draw a contour map of the non-interaction combination drug response surface where c1 is reduced to half of c2. */ c1 = c1*.5 s = cross(0:6!20,0:6!20) s col 3 = y on s draw contour(s) linetype vmarker left title "drug 1 dose" bottom title "drug 2 dose" top title "c1="+c1+" c2="+c2+" b1=b2="+b1 view

Finally, we may further continue the above MLAB dialog to produce
a contour map corresponding to the function $y$
in the case where $c$_{ 1 }=c_{ 2 }=4.1665 ,
$b$_{ 1 }=1.4433
and $b$_{ 2 }=2.8866 .

delete w /* draw a contour map of the non-interaction combination drug response surface where b1 is reduced to half of b2. */ c1 = c2; b1 = b1*.5 s col 3 = y on s col 1:2 draw contour(s) linetype vmarker left title "drug 1 dose" bottom title "drug 2 dose" top title "c1=c2="+c+" b1="+b1+" b2="+b2 view

Note that the Bunow-Weinstein non-interacting two-drug dose-response function can be immediately generalized to the case of $k$ non-interacting drugs. In general we have:

Now we may consider drug combination treatments where the drugs
involved may interact synergistically or antagonistically. To
analyze such treatments, we need to have estimated the parameters
$c$_{ 1 } , $b$_{ 1 } ,
$...$, $c$_{ k } ,
$b$_{ k } appearing in the
marginal single-drug dose-response functions for the $k$ drugs
under consideration, so that the Bunow-Weinstein non-interaction
model is specified.

Given the data $(d$_{ i1 },d_{ i2 },...,
d_{ik},v_{ i }) for
$i=1,...,m$, we may compute the
predicted * non-interaction *
response values $r$_{ i }=y(d_{ i1 },
d_{ i2 },..., d_{ik}) . If we have
no interaction, each $r$_{ i }
value should be approximately the same
as the corresponding observed response value $v$_{ i } .
If not, then we can conclude there is evidence for a synergistic
or antagonistic effect, and the size and direction of this
effect can be estimated from the response differences
$v$_{ i }-r_{ i } .
(Note that for the same combination of drugs,
we might have * both * synergistic effects for dose-pairs in
some regions and
antagonistic effects in different dose-pair regions.)

It may be convenient to consider the fractional differences
$p$_{ i }:=1-v_{ i }/r_{ i } ;
$100p$_{ i } is the percent change
from the non-interaction
predicted response $r$_{ i } that is
observed in the actual response $v$_{ i } .
If $v$_{ i }_{ i }>0 .
If $v$_{ i }>r_{ i } , the disease
was suppressed less than the non-interaction
model would predict and $p$_{ i }<0 .
Crudely, we might say that our drug-combination exhibits
a $100(p$_{ 1 }+...+p_{ m })/m
percent mean synergy interaction effect.

It is interesting to plot the $p$_{ i } -values,
the $v$_{ i } -values and/or the
$r$_{ i } -values versus the associated
* equivalent single-drug-dose * of drug
$D$_{ 1 } (or any other of the drugs
involved) as determined by the Bunow-Weinstein non-interaction
model $y$. If
$(d$_{ 1 },...,d_{ k }) is
the vector of dose values
corresponding to the data-value
$v$_{ i } and the predicted non-interaction
value $r$_{ i } , then the equivalent
drug $D$_{ 1 } dose is the value $e$
such that $y(e,0,...,0)=r$_{ i } .
An MLAB function involving the
root-operator can be defined to compute such equivalent dose values,
but $e$ can be directly computed as
$c$_{ 1 } / ((1 / r_{ i })-1)^[-1/b_{ 1 }]
(with due care for $r$_{ i }=0 and for
$r$_{ i }=1 .)
Such plots may be useful in seeing how the synergy effect varies
with dose.

Now one way to decide if the $r$_{ i } -values are,
over all, sufficiently greater than
the $v$_{ i } values to conclude
that we have statistically-significant
synergy is to test the hypothesis that the mean
of the $p$_{ i } -values
is less than or equal to 0 versus the alternate hypothesis that the
mean of the $p$_{ i } -values
is greater than 0. A similar test can
be used to assess whether we have statistically-significant antagonism.

It is probably most appropriate to use a non-parametric test such as
the Wilcoxon 2-sample paired-data test in MLAB
on the $r$_{ i } -values versus the
$v$_{ i } -values. And we can check our
outcome with a paired-data $t$-test on the
$p$_{ i } -values, even
though the normal-distribution assumption
used there may be invalid.

Note that in order to decide
which of two drug combination treatments is better
we need only compare the $p$_{ i } -values
for treatment 1 with the
$p$_{ i } -values for treatment 2.

Let us look at an example using MLAB to perform an analysis
of some two drug combination responses at various doses. We have a file
called ``a3azt.in'' which contains $n$
lines of three numbers, where
the first number is the administered dose of drug
$D$_{ 1 } ,
the second number is the administered dose of drug
$D$_{ 2 } ,and the
third number is the response measured as the amount of disease
present after treatment (in this case, the response is the amount
of virus assayed.) Each line in our data file is thus an ``observation''
from a single ``experiment''.

The MLAB dialog given below shows
how we read our data, normalize the response values
$v$_{ 1 },...,v_{ n }
to lie in
$[0,1]$, extract the data for the marginal dose-response curves
(one data-set where the drug $D$_{ 1 } dose is 0 and
one data-set where the drug $D$_{ 2 } dose is 0),
fit to obtain the estimated
marginal dose-response curve parameters
$c$_{ 1 } , $b$_{ 1 } ,
$c$_{ 2 } , and $b$_{ 2 }
(which then define
our Bunow-Weinstein non-interaction two-drug dose-response
function), compute the predicted non-interaction response
values $r$_{ 1 },..., r_{ n } ,
and then compare the predicted $r$_{ i } -values
with the observed $v$_{ i } -values
to study the nature and amount of
synergism apparent between our two drugs.

/* read-in a3azt.in, normalize it, and extract the data needed to fit the marginals. */ m = read("a3azt.in",1000,3) m col 3 = (m col 3)/maxv(m col 3) m1 = extract(m,2,0) col (1,3); m1 = sort(m1) m2 = extract(m,1,0) col (2,3); m2 = sort(m2) /* define the non-interacting 2-drug combination model and its marginals */ fct y1(d1) = 1/(1+(d1/c1)^b1) fct y2(d2) = 1/(1+(d2/c2)^b2) fct y(d1,d2) = if d1=0 then y2(d2) else if d2=0 then y1(d1) else if y1(d1)<.00001 and y2(d2)<.00001 then 0 else if y1(d1)>.9999 and y2(d2)>.9999 then 1 else root(z,1e-11,1-1e-11,(d1/c1)*(1/z-1)^(-1/b1)+(d2/c2)*(1/z-1)^(-1/b2)-1) /* fit the marginals. the derivative y1'b1 involves the log of d1/c1 which may be 0. Also the derivative y1'c1 involves (d1/c1)^(b1-1) which may become 0^0. y2 has similar problems. To avoid these problems we can use symdsw = 0. */ symdsw = 0 constraints q = {b1 > .00001,b2 > .00001,c1 > .00001,c2 > .00001} c1 = 1; b1 = 1 fit (c1,b1), y1 to m1 with weight ewt(m1), constraints q final parameter values value error dependency parameter 0.039470779 0.0063108802 0.05395208237 C1 1.078644618 0.2041472332 0.05395208237 B1 9 iterations CONVERGED best weighted sum of squares = 1.028776e+01 weighted root mean square error = 8.572280e-01 weighted deviation fraction = 7.954586e-02 R squared = 9.770071e-01 no active constraints c2 = 1; b2 = 2 fit (c2,b2), y2 to m2 with weight ewt(m2), constraints q final parameter values value error dependency parameter 0.204651635 0.0231668295 0.07182693961 C2 2.676935005 0.6159096057 0.07182693961 B2 11 iterations CONVERGED best weighted sum of squares = 4.343704e+01 weighted root mean square error = 1.647669e+00 weighted deviation fraction = 7.585580e-02 R squared = 9.646190e-01 no active constraints /* graph the resulting y1-fit. */ draw points(y1,minv(m1 col 1):maxv(m1 col 1)!100) draw m1 lt none pt circle ptsize .01 left title "response" font 18 bottom title "drug-1 dose" font 18 top title "fit of y1(d1) = 1/(1+(d1/c1)'.5ub1'.5d)" font 7 title "c1 = "+c1 at (.5,.8) title "b1 = "+b1 at (.5,.75) view

delete w /* graph the resulting y2-fit. */ draw points(y2,minv(m2 col 1):maxv(m2 col 1)!100) draw m2 lt none pt circle ptsize .01 left title "response" font 18 bottom title "drug-2 dose" font 18 top title "fit of y2(d2) = 1/(1+(d2/c2)'.5ub2'.5d)" font 7 title "c2 = "+c2 at (.5,.8) title "b2 = "+b2 at (.5,.75) view

delete w /* Now to analyze our drug combination data, we compute our non-interaction response values r, and compare them to the actual data v. If the v's are smaller, we have synergism. Also we will look at the percent changed-response values p.*/ r = y on (m col 1:2) v = m col 3 p = 1-v/'r /* Type out the average percentage-improvement seen in our data compared to the predicted non-interaction response. */ type mean(p) = .593531217 /* Graph the linked-pairs of predicted noninteraction response and observed response for each drug-dose pair used. */ draw v pt square lt none ptsize .01 draw r pt circle lt none ptsize .01 top title "squares:data, circles:non-interaction prediction" left title "amount of disease" bottom title "drug-combination observation number" n = nrows(v) nm = 1:n fct f(x) = if x < 0 then -1 else 1 c = f on (v-r) qv = nm&'v&'c; qr = nm&'r&'c qq = mesh(qv,qr) qv = extract(qq,3,-1) col 1:2 qr = extract(qq,3,1) col 1:2 draw qv lt alternate color green draw qr lt alternate color red view

delete w /* Graph the percentage-improvements (positive or negative) seen in each observation. */ draw mesh(nm&'0, nm&'p) lt alternate color yellow draw p pt circle ptsize .01 lt none draw nm&'0 color red top title "fractional change from non-interaction response" bottom title "drug-combination observation number" left title "percent improvement" view

delete w fct ed2(r) = if r=1 then 0 else if r=0 then maxd2 else c2/[(1/r-1)^(-1/b2)] maxd2 = maxv(m col 2) e = ed2 on r draw e&'v color green lt none pt square ptsize .01 /* also show the drug-2 marginal curve */ draw points(y2,minv(m2 col 1):maxv(m2 col 1)!100) top title "solid curve:drug-2 response, squares:data" bottom title "equivalent drug-2 dose" left title "amount of disease" view

/* Now we use the Wilcoxon 2-sample test for paired data to compare our data v with the corresponding predicted non-interaction response values r. */ wil2t(v,r) [Wilcoxon 2 sample signed-rank test: are the medians of the ranks of the observed paired data d1[] and d2[] plausibly equal?] null hypothesis H0: median(d1) = median(d2). The sum of positive ranks W-sample: T+ = 459.000000 The absolute sum of negative ranks W-sample: T- = 1819.000000 The probability P(W > 459.000000) = 0.999989 This means that a value of T+ at least as large as 459.000000 arises about 99.998920 percent of the time, given H0. The probability P(W > 1819.000000) = 0.000011 This means that a value of T- at least as large as 1819.000000 arises about 0.001050 percent of the time, given H0. The probability P[W > 1819.000000 or W < 459.000000] = 0.000022 This means that a rank-sum value at least as extreme as 1819.000000 arises about 0.002160 percent of the time, given H0. : a 6 by 1 matrix TP 1: 67 2: 459 3: 1819 4: .999989202 5: 1.05008711E-5 6: 2.15957906E-5

In the wil2t output vector ** TP** above, if the probability reported
in ** TP[5]** is small then we can reject the null hypothesis of equal
responses, with probability ** TP[5]** of being correct, and we can
conclude that the alternate hypothesis:
$median(v)\mathop{\mathrm{median}}(r)\; math>\; probably\; holds,\; so\; we\; accept\; that\; synergism\; is\; present.\; If$** TP[4]** is small,
then we can conclude that $median(v)>median(r)$
(corresponding to
antagonism), and if ** TP[6]** is small, we can conclude that
$median(v)$ does not equal $median(r)$.

Note that the graphs presented above can be constructed and have interest in the more-general situations of more than two drugs being used in combination.

When we have synergy, we will generally see that the bulk of the observed
$v$_{ i } -values, plotted
with respect to the equivalent dose $e$_{ i }
of one of the drugs, lie * below * the marginal dose-response
curve for that drug. This occurs above, where we compare the marginal
curve for drug $D$_{ 2 } with our
observed response vs. $D$_{ 2 } -equivalent
dose amounts. We may fit a single-drug dose-response model to these
$(e$_{ i },v_{ i }) points;
the difference between this fitted curve and the
single-drug marginal curve is a suitable basis for computing
the degree of synergism or antagonism at that equivalent dose.

Sometimes we may have the situation where a
drug $D$_{ 2 } has no
theraputic effect, but may, nevertheless, modify the effect of
another drug $D$_{ 1 } when used
in combination. In this case,
we should define the Bunow-Weinstein non-interaction model
function so that $y(d$_{ 1 },d_{ 2 })=
y(d_{ 1 },0) (* i.e. *,
$y(d$_{ 1 },d_{ 2 })=
1/(1+(d_{ 1 }/c_{ 1 })^[b_{ 1 }]) .)
If this model does
* not * fit our data, then we have statistically demonstrated
the pure potentiation or inhibition
behavior of drug $D$_{ 2 }
on drug $D$_{ 1 } ,where
drug $D$_{ 2 } , by itself,
has no effect.

It is important to note that to use the approach to drug-combination analysis presented here, we must have the marginal single-drug dose-response curves specified by some means. If this is not possible, we can use a direct model-based approach due to Bunow and Weinstein. This approach requires that we fit an explicit model function to our drug combination data, where the model function contains explicit interaction terms. If the estimated interaction parameters are significantly different from their ``no-interaction'' value, then we have evidence for synergy or antagonism (but not both in different regions.)

One such drug-interaction model proposed by Bunow and Weinstein is:

In this model, the term
$a$_{12}d_{ 2 }^[e_{12}] determines an
change in the efficacy of drug
$D$_{ 1 } in the presence of
drug $D$_{ 2 } , which itself has an independent
effect. When $a$_{12} is 0, there is no interaction.
The greater $a$_{12}
and $e$_{12} are, the greater is the
dose-dependent potentiation effect of drug
$D$_{ 2 } on drug $D$_{ 1 } .
When $a$_{12} is negative,
drug $D$_{ 2 } has an inhibitory effect
on drug $D$_{ 1 } . Fitting such
a model requires some care in
the curve-fitting process. Numerical issues such as division by zero
and zero raised to a negative power must be handled.
Generally, appropriate weights
are required, and often good initial parameter guesses and
accompanying constraints are also needed. For example, it
may be appropriate to constrain
$e$_{12} to be non-negative.
MLAB can accept such weights and constraints, and with
some care, can be used to estimate the desired parameters.

It is tempting to suppose that the estimated values of
$C$_{ 1 }
and $B$_{ 1 } determine the
drug $D$_{ 1 } single-drug dose
response curve, and likewise for $C$_{ 2 }
and $B$_{ 2 } , but this
is an unwarranted assumption. What we do know is that if
$a$_{12} is significantly
different than 0 and $e$_{12} is
not a large negative value, then our data prefers to follow
this interaction model, rather than the pure non-interaction
model, and we take this as evidence of interaction. The basic
underlying assumption is still that
the actions of drug $D$_{ 1 }
and drug $D$_{ 2 }
separately are well-described by logistic dose-response curves
as discussed above.

The entire drug-combination analysis method presented here
can be recast in a more-general non-parametric framework.
We can use a * non-parametric estimate * for each of our single-drug
marginal dose-response curves, instead of using the logistic
model.
The Bunow-Weinstein non-interaction model can be defined
with respect to these marginals, and the subsequent analysis
proceeds as before. A suitable non-parametric estimate of a
single-drug dose-response curve can be obtained in MLAB by first using
the function ` MONOT` to * monotonize * the data, and then
using the function ` SMOOTHSPLINE` to obtain an estimating
optimal smoothing spline.

There is a serious practical problem with * any * drug-interaction
analysis method which uses dose-response data as is done here. The
problem is that at high (theraputic-level) doses, for some types of
measurement of the amount of disease present, the
error in the response-values may be so magnified that such measurements
may be predominantly noise, and yet this is likely to be the domain in which
treatment will actually occur. Practically speaking, it is much more
important that there be no serious antagonism present than that our
drug-combination be synergistic; as long as the drugs involved are
not antagonistic and are jointly effective, it is generally
useful to use them in
combination. Of course such possible antagonism can be assayed via
the analysis methods discussed here, subject to the same caveat of high
errors in the response data. You might imagine that if
synergism is seen at moderate
($IC$_{ 50 } -level) doses (where
measurement error may be less problematical), then it will also
be present at higher theraputic doses, but this is not always the
case. The conclusion is that when the drugs involved can be usefully
administered at levels where our observed responses will be predominately
noise because of the nearly complete suppression of disease expected,
drug-combination analysis does not
take the place of clinical
trials using the drugs at theraputic levels in order to assess
the effacacy of the treatment.

**[1]** see www.civilized.com

**[2]** Bunow, Barry J. and Weinstein, John N. ''Combo: A New Approach to
the Analysis of Drug Combinations in Vitro'', Annals N.Y. Acad. of Science
Vol. 616, pp. 490-494, 1990.