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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