Computers in pharmaceutical
research and development
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CONTENTS
• Confidence Interval
• Sensitivity analysis
• Non linearity at the optimum
• Optimal designs
• Population modelling
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CONFIDENCE REGIONS
• Confidence region is a multidimensional generalization of a
confidence interval. It is a set of points in an n-dimensional
space, often represented as an ellipsoid around a point which is
an estimated solution to a problem, although other shapes can
occur.
• Confidence regions can be defined for any probability
distribution. The significance level and the shape of the region
can be chosen, and then the size of the region is determined by
the probability distribution. A natural choice is to use as a
boundary a set of points with constant values.
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Interpretation
• It is tempting to interpret a 95% CI by saying that
‘there is a 95% probability that the population mean
lies within the CI’.
Formally, this is not quite correct because the
population mean (µ) is a fixed unknown number: it is
the CI that will vary between samples. In other
words, if we were to draw several independent,
random samples from the same population and
calculate 95% CI from each of them, then on average
19 of every 20 (95%) such CI would contain the true
population mean, and one of every 20 (5%) would
not.
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• As part of malaria control programme it was planned to
spray all the 10,000 houses in a rural area with insecticide
and it was necessary to estimate the amount that would be
required. Since it was not feasible to measure all 10,000
houses, a random sample of 100 houses was chosen and
the sprayable surface of each of these was measured.
The mean sprayable surface area for these
100 houses was 24.2m² and the standard deviation was
5.9m². It is unlikely that the mean surface area of this
sample of 100 houses(xˉ) exactly equals the mean
surface area of all 10,000 houses (µ). Its precision is
measured by the standard error σ/√ n, estimated by
s / √n = 5.9/ √100=0.6m².
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There is a 95% probability that the sample mean of
24.2m² differs from the population mean by less than
1.96 s.e. = 1.96 × 0.6 = 1.2m². The 95% CI is :
95% CI = x – 1.96 × s.e. to x – 1.96 x s.e.
= 24.2 – 1.2 to 24.2 + 1.2
= 23.0 to 25.4m²
It was decided to use the upper 95% Confidence limit in
budgeting for the amount of insecticide required as it
was preferable to overestimate rather than
underestimate the amount. One litre of insecticide is
sufficient to spray 50m² and so the amount budgeted
for was: 10,000 × 25.4/50
= 5080 litres.
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• The mean in the whole population (µ = 24.2m²) is
shown by a horizontal dashed line. The sample means
vary around the population mean µ, and one of the
twenty 95% CI (indicated by a star) does not contain
µ.
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SENSITIVITY ANALYSIS:
• It is a technique used to determine how different values of an
independent variable will impact a particular dependent variable
under a given set of assumption .
• This technique is used within specific boundaries that will
depend on one or more input variables, such as the effect that
changes in interest rates will have a bond price.
• Sensitivity analysis is a way to predict the outcome of a
decision if the situation turns out to be different compared to the
key prediction.
• Its goal is to achieve the relative effect of the variation in a
given parameter value on twhwew .DmulooMdix.ecolm prediction .
• Let y=u(x=0)+L, be the considered model with y € ∆m and Ѳ € ∆q.
• To study the sensitivity of modelling function “u” with respect to the
parameter “Ѳ” by means of the absolute sensitivity coefficient ,
where partial derivatives is given as , ∂µ (X.Ѳ) or by ,
• ∂ Ѳ
• Means of normalized sensitivity coefficient given as ,
• ∑(X.∅)= Ѳ. ∂μ(X.Ѳ)
• µ(X.Ѳ).∂ Ѳ
• Normalisation method serves to make sensitivities comparable
across the variables and its parameters.
• In this ,by sensitivity analysis the mean proportion of the model
values may change due to a given proportion of parameter changes.
• Absolute and Normalised sensitivity coefficient can be computed
analytically or approximatedww nw.uDumloMeixr.icocmally
Above graph shows time course of absolute sensitivity coefficient
of Gompertz model with respect to the parameters
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Above graph shows same sensitivity coefficients expressed as a %
of their maximum value.
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• At time zero ,only Vo parameter influences model output and as the
time progresses Vo rapidly looses its importance and growth and
also decay parameters ‘a’ and ’b’ prevailed in determining variations
in predicted value.
• The model output is computed for some specified “X“ of interest
and for each generated value Ѳ = (Ѳ F) T Producing a small model
value matrix “Umnx “,where N rows “Ui” are given by (Ui)t =U(X1
{Ѳ i}t).
• The (Sprearman nonparametric or Pearson Parametric ) Monte Carlo
correlation coefficient matrix {MCCC} matrix that is Rmxq is then
computed between the generated values of Ѳ and the obtained value
of µ ,is similarly given as R=rkj , where rkj is the correlation
coefficient between column Uk and Ѳj.
• It indicates that higher the correlation coefficient rkj,the higher the
importance of the variationsw wowf.D tuhloMe ixѲ.cojmin producing variations of µk.
Graph shows the course of Pearson MCCC between V and 3D
model parameters for such Monte-Carto simulations with
n=10,000.
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• New method generated is Latin Hypercube sampling scheme,in
which square grid over parameter space is constructed and the cell
of p-dimensional grid are sampled so as to have exactly one sample
in each possible combination OF 1-DIMENSIONAL parameter
intervals.
• Main advantage of this scheme is that the required number of
samples does not grow as fast as a regular Monte-Carlo sampling
from the joint distribution of the parameters as the number of the
parameters increases.
• FOR EXAMPLE:-
• The influence of the parameter a on the tumor size is judged from
classic sensitivity seems to increase monotonically to the plateau
thereby reaching above its effect.
• Sensitivity diagram determines the influence of the parameter Vo
which is small in the beginning ,peaks over the range approximately
between 8-12 days.
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• Similarly, a qualitative behaviour of the parameter b under the
sensitivity or MCCC analyses are less
• If its negative then the influence on the volume size increases
according to MCC.
• At different times the courses of the parameters influences on
the VOLUME SIZE between sensitivity and MCCC analysis.
• Also the regression coefficient between model output and
parameter compartment value in small internal volume is
approximately equal to the partial derivatives of one model
output with respect to the components.
• MCC considers the influence of the variation of one parameter
influence on model output in context of the simultaneous
vibration of the other parameters.
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• If rjk, is smaller than 1, then the absolute value and its size depends
on the relative Importance of variation of model output due to
parameter of interest .
• Standard sensitivity analysis only gives indications on the
theoretical single parameter effects ,MCCC would be able to
quantify the effective impact that a parameters variation has real life.
• If arbitrary chosen variability of parameter P1 is small with respect
to variability chosen for parameter P2 , then the effect of P2 will
overshadow the effect of .P1 in MCCC computation.
• In case ,of significant population correlation among the parameter
values ,MCC stimulation should make use of the non –Zero
correlation in [parameters which further would generate variable
parametric samples.
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• USES OF SA:-
• Create a Model.
• Write a set of requirements.
• Design A SYSTEM’
• Make a Decision.
• Risk analysis
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NON – LINEARITY AT THE OPTIMUM
• We have seen how different approximate methods for
constructing confidence regions for the parameters can be
employed, once we believe that a linear approximation is
warranted. The problem now is that of deciding that this is
indeed the case. To this end, it is useful to study the degree of
non linearity of our model in a neighbourhood of the forecast.
• The methods of assessing the maximum degree of intrinsic
nonlinearity that the model exhibits around the optimum
found. If maximum nonlinearity is excessive, for one or more
parameters the confidence regions obtained applying the
results of the classic theory are not to be trusted. In this case,
alternative simulation procedures may be employed to provide
empirical confidence regions.
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OPTIMAL DESIGNS
• Optimal designs are a class of experimental designs that
are optimal with respect to some statistical criterion .
• They allow parameters to be estimated without bias and
minimum variance.
• A non optimal design requires greater number of
experimental runs to estimate the parameters with the
same precision as an optimal design.
• The optimacity of the design depends on the statistical
model and is assessed with respect to a statistical
criterion, which is related to the variance-matrix of the
estimator.
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D – Optimal designs
• D-Optimal designs are one form of design provided by a
computer algorithm.
• These type of computer aided designs are particularly useful
when classical designs do not apply.
• Unlike standard classical designs such as factorial and
fractional factorials , D-optimal design matrices are usually not
orthogonal and effect estimates are co-related.
• These type of designs are always an option regardless of the
type of model the experimenter wishes to fit.
• Eg:first order, first order plus some interactions ,full quadratic
,cubic etc) or the objective specified for the experiment
• Eg: screening response ,surface etc.
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• The reasons for using D-optimal designs instead of standard
classical designs usually fall into 2-categories:
• Standard factorial or fractional factorial designs require too
many runs for the amount of resources or time allowed for the
experiment.
• The design space is constrained(the process space contains
factor setting that are not feasible or are impossible to run.)
Advantages
• Optimal designs reduce cost of experimentation by allowing
the statistical models to be estimated with fewer experimental
runs .
• Optimal designs can accommodate multiple types of factors
such as process,mixture,and discrete factors.
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POPULATION MODELLING
• Population modeling is an important tool in drug
development; population modeling is a complex process
requiring robust underlying procedure for ensuring clean data,
appropriate computing platforms, adequate resources, and
effective communication.
• Although by providing a platform for integrating all
information gathered on new therapeutic agents.
• Population Modelling: is a type of mathematical model that is
applied to the study of population dynamics.
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Objective-
• It is the tool to identify and describe relationship between
subject physiologic characteristics and observed drug response
or exposure.
• Population pharmacokinetics modeling was introduced in 1972
by sheiner.
• It deals with Pk data collected during therapeutic drug
monitoring .
• It was soon expanded to include models linking drug
concentration to response of different subjects.
eg – pharmacodynamic.
• Population parameters was originally estimated either by
fitting combined data from all individuals.
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• Ignoring all individuals difference or by fitting each
individuals data separately and combining the individual
parameters .
• Estimates to generate mean population parameters .
• This methods have inherent problems, which become
worse when deficiencies such as dosing compliance,
Missing samples, and other data errors are present,
resulting in biased parameter estimates.
• BSV, and the covariate effects that quantities and explain
variability in drug exposure.
• This approach also allowed a measure of parameter
precision by generation of SE.
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Population modelling can be done by two – stage model :
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REFERENCE
• Computer Applications In Pharmaceutical Research
and Development by Sean Ekins. A John Wiley &
Sons, INC., Publication.
• www.google.com
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