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DRUG STUDIES: USING MATHS TO IDENTIFY OPTIMUM DOSAGE
Only a handful of potential pharmaceutical ingredients pass clinical tests and get approved for the market. By
deploying better statistical procedures in dose-finding studies, the number of successful substances could be
increased.
Potential new drugs undergo rigorous testing prior to approval. Most of them do not pass; a mere 0.01 to 0.02
per cent are greenlighted for market release. However, some candidates are unjustly rejected, suspects the
mathematician Prof Dr Holger Dette, because pharmaceutical companies do not use the optimal dose when carrying
out final tests on human subjects. Together with PhD student Kirsten Schorning and the bio-statistics department at
“Novartis”, he has developed a novel mathematical method which helps to set up more efficient dose-finding
studies.
Prior to being approved, drugs undergo three clinical test phases. In phase 1, the substance is tested on humans
for the first time. The aim is to establish how well it is tolerated, how it is distributed in the body and how it is
processed and excreted. In phase 2, the aim is to research its effects in tests with several hundred patients and to
determine the optimal dose. Finally, in the third phase, the optimal doses that were determined in phase 2 are
applied in tests with several thousand patients over a long period of time. But what happens if, rather than being
optimal, the doses used in phase 3 are either too strong or too weak? If the first is the case, the substance will
probably not pass the tests, because the side effects are too severe. If the dosage is too weak, on the other hand, it
may not have any effect whatsoever. The aim of phase 2 is always to find the minimum effective dose, i.e. the
potency that leads to the required effect: which, for example, lowers the blood pressure by a certain factor without
causing any too severe side effects. But how can the optimal dose be found? To date, participants in phase 2 have
been subdivided into several groups of the same size. The applicable dose range, e.g. 0 to 150 milligram, has likewise
been evenly distributed (or evenly on a logarithmised scale), so that a specific dose would be applied to each group.
The first group would, for example, receive 0 milligram active ingredient, i.e. a placebo; the second group 30
milligram, the third 60 milligram, the fourth 90 milligram etc.
According to statistician Holger Dette, this approach is only suboptimal. Deploying more mathematics when
planning the study would help determine the optimum dose much more accurately. But how? First, one has to
understand the way dose and effect correlate in the mathematical model. Pharmacokinetic research has shown that
there are essentially only a few different functions for describing all dose-response relationships of active
ingredients. The different function types can be determined on the basis of chemical reaction equations using
differential equation theory, a traditional branch of mathematics. An example for such a model is the EMAX model:
f(x)=a+bx/(c+x)
The function f(x) assigns a certain value to each dose value x; a, b and c are ingredient-specific parameters. If one
knew the dose-response function and the parameters a, b and c for the new drug, one could easily read the
minimum effective dose from the graph or calculate it using the formula. The problem: when developing a new drug,
researchers know neither the model nor the parameters that describe the dose-response relationship.
Consequently, pharmaceutical enterprises have to draw up good models for describing the dose-response
relationship following the tests in phase 2, in order to approach the optimal dose as closely as possible. Assigning six
dose increments to all participants equally is not the best solution, however, says the team from the Institute of
Statistics in Bochum. Using a simplified example, Holger Dette explains why it does matter how many subjects are
tested with regard to the respective dose increment: let us assume for simplicity’s sake that the effect of our drug is
linearly dependent from the dose, i.e. we can describe the dependency with a straight line. We collect a number of
measurement values and have a line of best fit pass through them, in order to determine the underlying model. If we
collect all measurement data at only one x-value, i.e. one dose increment, the slope of the line of best fit can be
entirely arbitrary, and we cannot make any statements regarding the minimum effective dose. If we measure at two
x-values, the situation improves. But if we place the dose increments too closely together, the line of best fit can still
conceivably assume many different slopes. It gets better once the chosen doses differ as strongly as possible from
each other.
The relationship between dose and effect of drugs is not linear, however; it is described by more complex
models. The RUB team has found a way to calculate how many patients should be optimally treated with which dose
if the underlying function is not exactly known. For this method, it is sufficient to have a certain idea about the
possible models: following many years of pharmaceutical research, it has emerged that there are only a few
fundamentally different models in existence that can be used for describing the dose-response relationship;
moreover, initial information regarding ingredient-specific parameters would have already been gathered during
phase 1 of the clinical trials.
“Essentially, all that we have to do for our method is solve an extreme value problem,” says Holger Dette. “What
we do is basically the same as curve sketching in school – the only difference is the functions we use. The functions
have an additional important property: they are concave.” In order to identify the maxima of a concave function,
one has to take its first derivative and equate it with zero (info). The functions that the RUB mathematicians handle
are rather more abstract than school-level mathematics, but the principle is similar. They solve an infinitedimensional extreme value problem. They use a matrix to describe a specific design of experiments:
ξ=((d1,d2,…,dn)¦(P1,P2,…,Pn))
The variable n indicates how many different doses are to be tested. d1,d2,…,dn describe the doses to be
calculated, and P1,P2,…,Pn describe the relative percentage of patients to be calculated who will be receiving the
respective dose. In order to ascertain the accuracy of the estimate for the minimum effective dose depending on the
design of experiments, the team uses mathematical statistics methods to determine a function f(ξ) which describes
the accuracy of the estimate depending on the experimental design. If the variables are assigned numeric values –
for example 0 milligram active ingredient for one-eighth of the patients, 80 milligram active ingredient for onefourth of the patients etc. – the result is a value that indicates how exactly the optimum dose could be calculated
using that design of experiments. By maximising this function, Dette and Schorning thus determine the best-possible
experimental set-up. The mathematical challenge is the fact that it is impossible to indicate the upper limit for the
number of different doses. Thus, the extreme value problem is rendered infinite-dimensional.
Using the new method made in Bochum, researchers could produce a potentially more accurate estimate of the
minimum effective dose in phase 2 of the clinical trials than with the methods deployed hitherto. It sounds simple,
but this kind of mathematics has not, as yet, arrived in practical applications – even though Holger Dette has been
developing the new method since 2008. There are two reasons for the long delay. “First, this is due to logistical
restrictions,” he explains. “The method deployed to date requires test participants to take perhaps two, three or
seven pills. With our method, we have calculated that they would have to take 3.78 pills.” Today, Dette and
Schorning have enhanced their method so that logistical restrictions would be taken in consideration. “Still, it takes
some time for something like that to become widely accepted,” says the RUB researcher. “That is because it’s not
easy to convince clinicians to give up a tried-and-tested approach. We say, for example: in order to find the
minimum effective dose, they should test only three different dose increments if the dose-response relationship is
described using the EMAX-model. And they say: Yes, but perhaps we aren’t using the correct model after all.”
Therefore, Dette and Schorning have optimised their method and laid it out so that it is applicable for a large class of
possible models and does not fail if it is based on the wrong dose-response curve. Still, scepticism has to be
overcome first. “This is because if something goes wrong, it naturally causes immense costs,” says Dette. After all, it
is not the clinician who ultimately decides if the drug will be released to the market, but a public agency – in the USA
it is the “Federal Drug Administration”, short FDA. Dette: “If we’re lucky, FDA will one day include our method in
their guidelines.” Then, doctors would be able to apply it without having to worry that their clinical trial will be
rejected by the agency.
Julia Weiler
Info
FINDING THE EXTREME VALUES OF CONCAVE FUNCTIONS
A function f(x) is a calculus that assigns precisely one value to every x-value. Concave functions have a specific
property: if two arbitrary points are selected in the function graph and connected with a straight line, that line is
always below the graph. If an extremum does exist, it is therefore always a maximum. In order to find the maximum
of a concave function, the first step is to determine the first derivative f‘(x), following clearly defined rules.
f(x)=-4x2+4x
f' (x)=-8x+4
The maximum of the concave function is thus the x-value where the first derivative equals zero. The first
derivative indicates the slope of the function.
0=-8x+4
8x=4
x=0,5
The example function has an extremum at x = 0.5. The respective tangent has the slope 0. Because the function is
concave, it is certain that the extremum is a maximum.