OPTIMAL DESIGN AND POPULATION MODELING Prepared By: Pawan Dhamala

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PHARMACEUTICS
OPTIMAL DESIGN AND POPULATION MODELING
R R COLLEGE OF PHARMACY
SUBMITTED BY
Pawan Dhamala
2ND SEM M.PHARM
SUBMITTED TO
MR K MAHALINGAN
ASST PROF, PHARMACEUTICS

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CONTENTS
1. Optimal Design
 Introduction
 Advantages
 Types
2. Population modeling
 Introduction
 Components of population models
3. References

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OPTIMAL DESIGN
Introduction
• There are many situations in which the requirements of a standard experimental design do not fit the
research requirements of the problem.
• There such situations occur when the problem requires unusual resource restrictions, when there are
constraints on the design region, and when a nonstandard model is expected to be required to
adequately explain the response.
• In the design of experiments, optimal designs (or optimum designs[) are a class of experimental
designs that are optimal with respect to some statistical criterion.
• The creation of this field of statistics has been credited to Danish statistician Kirstine Smith.
• In the design of experiments for estimating statistical models, optimal designs allow parameters to
be estimated without bias and with minimum variance.

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• A non-optimal design requires a greater number of experimental
runs to estimate the parameters with the same precision as an optimal design.
• In practical terms, optimal experiments can reduce the costs of experimentation.
• The optimality of a 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.
• Specifying an appropriate model and specifying a suitable criterion function both require
understanding of statistical theory and practical knowledge with designing experiments.

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Advantages
Optimal designs offer three advantages over sub-optimal experimental designs:
• Optimal designs reduce the costs of experimentation by allowing statistical models to be estimated
with fewer experimental runs.
• Optimal designs can accommodate multiple types of factors, such as process, mixture, and discrete
factors.
• Designs can be optimized when the design-space is constrained, for example, when the
mathematical process-space contains factor-settings that are practically infeasible (e.g. due to safety
concerns).

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Types of Optimal design
There are different types of Optimal designs. They are:
1. A Optimality (Average or Trace)
2. C Optimality
3. D Optimality ( Determinant)
4. E Optimality (Eigenvalue)
5. S Optimality
6. T Optimality
7. G Optimality
8. I Optimality (Integrated)
9. V Optimality (Variance).

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 A-Optimality ( Average or Trace)
One criterion is A-optimality, which seeks to minimize the trace of the inverse of the information
matrix. This criterion results in minimizing the average variance of the estimates of the regression
coefficients.
 C-optimality
This criterion minimizes the variance of a best linear unbiased estimator of a predetermined linear
combination of model parameters.
 D-optimality (determinant)
A popular criterion is D-optimality, which seeks to minimize |(X'X)−1|, or equivalently maximize
the determinant of the information matrix X'X of the design. This criterion results in maximizing
the differential Shannon information content of the parameter estimates.
 E-optimality (eigenvalue)
Another design is E-optimality, which maximizes the minimum eigenvalue of the information matrix.

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 S-optimality
This criterion maximizes a quantity measuring the mutual column orthogonality of X and
the determinant of the information matrix.
 T-optimality
This criterion maximizes the discrepancy between two proposed models at the design locations.
Other optimality-criteria are concerned with the variance of predictions.
 G-optimality
A popular criterion is G-optimality, which seeks to minimize the maximum entry in the diagonal of
the hat matrix X(X'X)−1X'. This has the effect of minimizing the maximum variance of the predicted
values.
 I-optimality (integrated)
A second criterion on prediction variance is I-optimality, which seeks to minimize the average
prediction variance over the design space.
 V-optimality (variance)
A third criterion on prediction variance is V-optimality, which seeks to minimize the average
prediction variance over a set of specific points.

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POPULATION MODELING
Introduction
• Modeling is an important tool in drug development; population modeling is a complex process
requiring robust underlying procedures for ensuring clean data, appropriate computing platforms,
adequate resources, and effective communication.
• Although requiring an investment in resources, it can save time and money by providing a
platform for integrating all information gathered on new therapeutic agents.
• Modeling have emerged as important tool for integrating data, knowledge, and mechanisms to
aid in arriving at rational decisions regarding drug use and development.

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• In Figure, presents a brief outline of some areas in which modeling and simulation are commonly
employed during drug development.
• Appropriate models can provide a framework for predicting the time course of exposure and
response for different dose regimens.
• Central to this evolution has been the widespread adoption of population modeling methods that
provide a framework for quantitating and explaining variability in drug exposure and response.

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• All drugs exhibit between-subject variability (BSV) in exposure and response, and many studies
performed during drug development are aimed at identifying and quantifying this variability.
• A sound understanding of the influence of factors such as body weight, age, genotype, renal/hepatic
function, and concomitant medications on drug exposure and response is important for refining
dosage recommendations, thereby improving the safety and efficacy of a drug agent by
appropriately controlling variability in drug exposure.
• Population modeling is a tool to identify and describe relationships between a subject's
physiologic characteristics and observed drug exposure or response.
• Population pharmacokinetics (PK) modeling is not a new concept; it was first introduced in 1972 by
Sheiner et al.
• Although this approach was initially developed to deal with sparse PK data collected during
therapeutic drug monitoring, it was soon expanded to include models linking drug concentration to
response (e.g., pharmacodynamics (PD)).
• Thereafter, modeling has grown to become an important tool in drug development.

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• Population parameters were originally estimated either by fitting the combined data from all the
individuals, ignoring individual differences (the “naive pooled approach”), or by fitting each
individual's data separately and combining individual parameter estimates to generate mean
(population) parameters (the “two-stage approach”).
Components of Population Models
• Population modeling requires accurate information on dosing, measurements, and covariates.
• Population models are comprised of several components: structural models, stochastic models,
and covariate models.
 Structural models are functions that describe the time course of a measured response, and can
be represented as algebraic or differential equations.
 Stochastic models describe the variability or random effects in the observed data, and
 Covariate models describe the influence of factors such as demographics or disease on the
individual time course of the response.

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 Structural models as algebraic equations
• The simplest representation of a PK model is an algebraic equation such as the one representing a
one-compartment model, the drug being administered as a single intravenous bolus dose:
• This model states the relationship between the independent variable, time (t), and the dependent
variable, concentration (C).
• The notation C(t) suggests that C depends on t. Dose, clearance (CL), and distribution volume (V)
are parameters (constants); they do not change with different values of t.
• The dependent and independent variables are chosen merely to extract information from the
equation. In PK, time is often the independent variable.
• However, Equation (1) could be rearranged such that CL is the independent variable and time is a
constant (this may be done for sensitivity analysis for example).

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 Structural models as differential equations
• Some complex pharmacometrics systems cannot be stated as algebraic equations. However, they
can be stated as differential equations. Rewriting Equation (1) as a differential equation:
• A differential equation describes the rate of change of a variable.
• In this example, dC/dt is the notation for the rate of change of concentration with respect to time
(sometimes abbreviated as C′).
• Note that differential equations require specification of the initial value of the dependent variables.
Here, the value of C at time zero (C0) is Dose/V.
• Numerical methods are needed to solve systems of differential equations. Euler's method is a simple
example and can be easily coded.
• Numerically solving Equation (2) requires approximating the value of the variable (C2) after an
increment in time (t2 – t1) based on the previous value (C1) and the implied rate of change (–
CL/V*C1):

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 Stochastic models for random effects
• Population models provide a means of characterizing the extent of between-subject (e.g., the
differences in exposure between one patient and another) and between-occasion variability (e.g., the
differences in the same patient from one dose to the next) that a drug exhibits for a specific dose
regimen in a particular patient population.
• Variability is an important concept in the development of safe and efficacious dosing; if a drug has a
relatively narrow therapeutic window but extensive variability, then the probability of both sub
therapeutic and/or toxic exposure may be higher, making the quantitation of variability an important
objective for population modeling.
 Covariate models for fixed effects
• The identification of covariates that explain variability is an important objective of any population
modeling evaluation.
• During drug development, population modeling develops quantitative relationships between
covariates (such as age) and parameters, accounting for “explainable” BSV by incorporating the
influence of covariates on THETA ( Unknown parameter).

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REFERENCES
1. Rachel T Johnson, Douglas C Montgometry and Bradley Jones. An Expository Paper on Optimal
Design. Quality Engineering 23(3): 287-301.
2. https://en.wikipedia.org/wiki/Optimal_design
3. Mould DR, Upton RN. Basic concepts in population modeling, simulation, and model-based drug
development. CPT Pharmacometrics Syst Pharmacol. 2012 Sep 26;1(9):e6. doi:
10.1038/psp.2012.4. PMID: 23835886; PMCID: PMC3606044.

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THANK YOU