Journal of Pharmaceutical and Bioanalytical Science

Kuldeep K Namdev et al.; J. Pharm. Bioanal. Sci., Volume 3, Issue 2, Apr-Jun 2014, 21-28
Journal of Pharmaceutical and
Bioanalytical Science
ISSN: 2278-828X
Review Article
An Outline of Dose - Response Relationship: Emax Modeling and Its
Application in Bioequivalence Study
Rajneesh Singh, Kuldeep K Namdev*, Manoj Singh, Dr. Deepak Chilkoti and Shireen Rao
Fortis Clinical Research Ltd., Haryana, India
History
Abstract
Received : 04 Feb 2014
Accepted : 02 May 2014
Published online: 19 May 2014
The aim of this review article is to discuss the Emax pharmacodynamic model. An essential
component of drug development is to understand the dose-response relationship of a pharmaceutical
compound under identical condition and find out an optimal dose to achieve targeted drug level for
maximal desired pharmacological response. The current article represents the mathematical
considerations of Emax Model which epitomize the dose-response relationship. The complete fate of a
drug can be expressed in terms of Pharmacokinetics (PK) – “what the body does to the drug” and
Pharmacodynamics (PD) – “what the drug does to the body”. Pharmacodynamic study is not only
apprehensive about safety summary of the drug toxicity but also illustrate the dose-efficacy
relationship. The current review describing the application of Emax model in evaluation of
bioequivalence study (study designs & statistical methodology). The application of Emax model in
bioequivalence study is illustrated with an example of orlistat bio-study. The systemic absorption of
orlistat is minimal; therefore pharmacokinetic study of orlistat study drug is not well-organized to
compare the efficacy parameters (Cmax and AUC). The percent of fecal fat excretion (FFE) expressed
as a ratio of the amount of fat excretion over a 24-hour period at steady-state relative to the amount
of daily ingested fat was taken as PD end-point. The data was statistically analyzed using the DoseScale Method incorporating the Emax model by considering the USFDA draft guidance on orlistat.
Key Words
PD ‘Pharmacodynamics’, PK
‘Pharmacokinetic’, BA
‘Bioavailability’ and BE
‘Bioequivalence’
*Corresponding Author
Kuldeep K Namdev
Pharmacokinetic Scientist
Fortis Clinical Research Ltd.;
Haryana, India
INTRODUCTION
Pharmacodynamics is the study of the relationship between the concentration of a drug and the response obtained in
a patient. For human pharmaceuticals therapies, both pharmacokinetic and pharamcodynamic studies play important
role for drug development. Pharmacokinetic is a studies of time course of absorption, distribution, metabolism and
elimination of a drug in a living body while pharmacodynamic is a study of reaction of body to a given drug of
interest. Pharmacokinetics (PK) – “what the body does to the drug” Pharmacodynamics (PD) – “what the drug does
to the body”. PD study is not only concerned with establishing a safety profile of the drug toxicity but also focus on
the dose-efficacy relationship.
This paper present to examine the dose-response relationship, here Dose is drug input to the body or a biological
system and Response- A direct measure of the pharmacologic effect of the drug, based on a variety of clinically
relevant endpoints/biomarkers or clinical effects related to either efficacy or safety. There are few others objective
of PD studies as characterize the “Therapeutic window” and how it varies across subjects, develop dosing regimens
targeting achieved concentrations leading to therapeutic response and decide on dose(s) to carry forward to future
studies. There is a drug receptor located within the target organ or tissue.
When a drug molecule "finds" the receptor, it forms a complex that causes the pharmacologic response to occur.
The drug and receptor are in dynamic equilibrium with the drug–receptor complex. Determining an adequate dose
level for a drug and, more broadly, characterizing its dose response relationship, are key objectives in the clinical
development of any medicinal drug. If the dose is set too high, safety and tolerability problems are likely to result,
while selecting too low a dose makes it difficult to establish adequate efficacy in the confirmatory phase, possibly
leading to a failed program. Hence, dose finding studies are of critical importance in drug development and need to
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Kuldeep K Namdev et al.; J. Pharm. Bioanal. Sci., Volume 3, Issue 2, Apr-Jun 2014, 21-28
ISSN: 2278-828X
be planned carefully. In this paper we focus on practical considerations for establishing efficient study designs to
estimate target doses of interest. We consider optimal designs for both the estimation of the minimum effective dose
(MED) and the dose achieving 100p% of the maximum treatment effect (EDp). These designs are compared with
D-optimal designs for a given dose response model. Extensions to robust designs counting for model uncertainty are
also discussed. [1]
Fig 1: Flow of Dose – Response Relationship: Pharmacodynamic Study
DOSE VS RESPONSE: MATHEMATICAL PHENOMENON [3, 4, 5, 6]
For drug pharmacological studies, many models can be used to represent dose response relationship. The Emax model
is used to represent monotone, concave dose-response shapes. To distinguish it from the more general sigmoid Emax
model it is sometimes also called hyperbolic Emax model. The sigmoid Emax model is an extension of the
(hyperbolic) Emax model by introducing an additional parameter that determines the steepness of the curve at the
ED50 value. The sigmoid Emax model describes monotonic, sigmoid dose-response relationships. The exponential
model is intended to capture a possible sub-linear or a convex dose-response relationship. The linear in log-dose
model is intended to capture concave shapes. The parameter off is not estimated in the code but set to a prespecified value. The logistic model is intended to capture general monotone, sigmoid dose-response relationships.
The quadratic model is intended to capture a possible non-monotonic dose-response relationship.
Fig 2: Dose – Response Relationship: Simple Emax Model
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Kuldeep K Namdev et al.; J. Pharm. Bioanal. Sci., Volume 3, Issue 2, Apr-Jun 2014, 21-28
E=
ISSN: 2278-828X
Emax. Doses
Eo +
ED50 +Dose
Where:
E
: Pharmacodynamic effect
Emax : Difference between maximum achievable response and baseline
ED50 : Doses that produces half – maximal effect, also known as potency i.e. >0 or Dose required to achieve 50%
of the Emax value.
E0
: Baseline (Response in absence of drug)
Dose : Dose of Drug
The Hyperbolic Emax model is used for various clinical pharmacological applications like inhibition, clearance,
protein binding etc. There are few below objective of hyperbolic Emax model that achieve as:
•
To find the expected value of maximum achievable effect.
•
Estimates no effect in the absence of drug.
•
Conform to simple theories, relating dose or concentration to receptor binding and the observed effect.
The ED50 is the dose producing 50% of Emax. There are two other useful doses to remember: ED20 – the
concentration at 20% of Emax. It is ¼ of ED50, ED80 – the concentration at 80% of Emax. It is 4 times the ED50. This
means the hyperbolic Emax model predicts a 16 times change in doses is needed to change the effect from 20% to
80% of Emax. Many drugs seem to have a steeper relationship of concentration and effect so that a smaller change is
required. These steeper relationships can be described by the sigmoid Emax model.
Sigmoid Emax Model
In 1910, the physiologist Hill was investigating the shape of the oxygen – hemoglobin saturation relationship. He
noted it was steeper than the simple binding predictions of the Emax model. By trial and error he found that adding
an exponential parameter to the concentration and ED50 terms in the model the shape of the relationship could be
made steeper. This extra parameter is known as the Hill coefficient. The sigmoid Emax model has wide popularity for
its easy interpretation and mechanistic motivation. [7]
The sigmoid Emax model is shown with four different values for the Hill coefficient. When Hill=1 the curve is the
same as the hyperbolic Emax model, When Hill is greater than 1 the curve is steeper and when it is less than 1 it is
shallower than the hyperbolic Emax model. When Hill=2 it only takes as 4 fold change in doses to go from ED20 to
ED80. When Hill is very large (>10) the dose effect relationship is almost like an on-off switch. The effect turns
‘on’ at a threshold concentration close to the ED50.
Sigmoid Emax Model Function
If the simple Emax model wiil not be effectual for dose-efficacy relationship, then Sigmoid Emax model be effective
tools to justify the relationship between dose and efficacy.
The sigmoid Emax model illustrated as below
E=
Eo +
Emax . Dose n
ED50n + Dosen
There are four parameters in this model as below define:
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Kuldeep K Namdev et al.; J. Pharm. Bioanal. Sci., Volume 3, Issue 2, Apr-Jun 2014, 21-28
ISSN: 2278-828X
Note:
E
Emax
: Pharmacodynamic effect
: Difference between maximum achievable response and baseline
ED50 : Doses that produces half – maximal effect, also known as potency or Dose required to achieve
50% of the Emax value i.e. >0.
E0
: Baseline (Response in absence of drug)
Dose
: Dose of Drug
n
: It is Hill parameters
If n < 1 Dose.-effect relation very flat and if n > 5: all-or-none response and n is shown sensitivity of the doseefficacy relationship.
The hill (n) parameter provides flexibility around the hyperbola. Influence of n the shape of the relation
n = 1: classical Emax or hyperbolic Emax Model
n < 1: upper before ED50 , lower after ED50
n > 1: lower before ED50 , upper after ED50
The dose–response relationship of drugs usually resulted in different concentrations in individuals because of
pharmacokinetic differences in clearance and volume of distribution.
DOSES-RESPONSE RELATIONSHIPS AND RELATED MATHEMATICAL CONSIDERATIONS [8.9,
10]
In this section, a mathematical concept is described for evaluation of response to a specific dose (i.e. performed
using continuous dose as X). Here the response vs. doses is nonlinear as mathematical consideration. As below we
illustrate the some of dose-response model.
1.
2.
3.
4.
5.
6.
7.
8.
9.
Linear f (x) = E0 + δx
Linear Log-dose f (x) = E0 + βlog(x + c)
Exponential f (x) = E0 + E1 (ex/ϑ − 1)
Four parameter logistic f (x) = E0 + [Emax −E01+exp[(ED50−x)/φ]]
Five parameter logistic f (x) = E0 + [Emax −E0(1+exp[(ED50−x)/φ])γ]
Hyperbolic Emax f (x) = E0 + [ (Emax * x / ED50+x]
Sigmoidal Emax f (x) = E0 + [ (Emax * xn / ED50 n+xn]
Gompertz f (x) = E0 + [(Emax − E0)e−exp(ϕ(ED50−x))]
Weibull f (x) = E0 + [(Emax − E0)e−exp(b(log(x)−log(ED50)))]
The Non-linear dose response modeling can be performed using continuous dose, X (using PROC NLIN in SAS)
with the following model: Percent change in Y= β0 + (Xγ. β1)/(Xγ + β2 γ ) + e (error), where β0 is the response at
dose = 0, β1 is the optimum % change in Y compared to dose = 0 and β2 is the dose at 50% of β1. A fixed value of γ
can be chosen based on minimum residual sum of squares. Basically, Nonlinear Regression Model is described as
below:
Simple Maximum Effect (Emax) model:
1.
β0 ≡ Mean Response at Dose 0
2.
β1 ≡ Maximal Effect (β0+ β1 = Maximum Mean Response)
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Kuldeep K Namdev et al.; J. Pharm. Bioanal. Sci., Volume 3, Issue 2, Apr-Jun 2014, 21-28
3.
ISSN: 2278-828X
β2 ≡ Dose providing 50% of maximal effect (ED50)
Nonlinear Least Squares Method
This procedure used to estimate the beta value.
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Kuldeep K Namdev et al.; J. Pharm. Bioanal. Sci., Volume 3, Issue 2, Apr-Jun 2014, 21-28
ISSN: 2278-828X
In above mathematical consideration, the relation between doses vs. efficacy response is follow nonlinear trend, and
there is assumption about their respective variable (i.e. efficacy and doses) both variable should be positive and
error term (residual) follow the normal distribution with zero mean and constant variance for true estimate. The
nonlinear regression analysis is used to estimate the slope coefficient (i.e. β). The PROC NLIN in SAS is used to
estimate these functions and find the true relationship between doses vs. response.The objective is to base model
selection on substantive facts and broad understanding with dose-response relationships rather than criteria selected
to ensure convergence of estimators. True dose-response model is typically unknown and choice of a working
model may have a substantial impact on dose selection.
A BIOEQUIVALENCE STUDY WITH PHARMACODYNAMIC PARAMETERS [11, 12, 13]
In this paper, we converse the statistical methodology of bioequivalence study which is based on pharmacodynamic
parameters as discussed in USFDA draft report for orlistat drug. The systemic absorption of orlistat is minimal.
Based on urinary excretion, less than 5% of an oral dose is absorbed. Since orlistat acts locally in the
gastrointestinal tract, systemic absorption is not required for efficacy. Following oral dosing with 360 mg 14Corlistat, plasma radioactivity peaked at approximately 8 hours; plasma concentrations of intact orlistat were near the
limits of detection (<5 ng/mL).
Therefore, the Fecal Fat Excretion FFE will be used as the pharmacological endpoint to evaluate and compare the
efficacy of the orlistat products.
Study Design
The study design of orlistat product need to be conducted as multiple-dose, 3-way crossover consisting of two doses
of reference product and at least one dose of the test product as per USFDA regulation.
Analysis for studies using multiple doses of the study products
Pharmacodynamic bioequivalence study designs using only single doses of the test product are acceptable.
However, multiple doses of both test and reference products may enrich the study data and enhance precision of the
estimated values. The PD study should be conducted as a randomized crossover design with at least 2 doses of the
RLD and 1 dose of the test product. We may include additional doses of the test and reference products to improve
precision of parameters in the Dose-Scale Analysis. The "Dose Scale" assessment of BE incorporates the doseresponse information and provides appropriate estimates of relative BA. Application of the “Dose Scale” method to
PD BE studies is independent of the in vivo PD model/endpoint as well as quantitative aspects of dose-response
relationships.
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Kuldeep K Namdev et al.; J. Pharm. Bioanal. Sci., Volume 3, Issue 2, Apr-Jun 2014, 21-28
ISSN: 2278-828X
Primary Pharmacodynamic Parameter
The percent of fecal fat excretion expressed as a ratio of the amount of fat excretion over a 24-hour period at steadystate relative to the amount of daily ingested fat.
Statistical Method: Emax Modeling and Bootstrapping
The pharmacodynamic parameter FE24 (SS) will be statistically analyzed to estimate the relative
bioavailability of test product by fitting simultaneously the within study dose response data of completed
subjects of test and reference products to a Non-Linear Emax Model:
Emax * Dose *Fi
y=
E0 +
-----------------------ED50 + Dose*Fi
Where y = FFE24(ss), i = Treatment indicator (0 = Ref, 1 = Test) with the understanding that F0 = 1, E
=Response, Dose = Administered dose, E0 = Baseline response in the absence of the drug, Emax = Fitted
maximum drug effect, ED50 = Dose required to produce 50% the fitted maximum effect, and F=relative
bioavailability. Analysis will include calculation of estimated values for model parameters Emax, ED50, E0 and
F.
The above analysis will be done using appropriate SAS procedure. Bootstrap samples, of size equal to the size
of dose response data of completed subjects, will be generated by repetitive sampling with replacement and
value of “F” will be estimated by fitting above model to each bootstrap sample. Efron's bias corrected and
accelerated method will be used for estimation of 90% confidence intervals for “F”.
Calculation of Confidence Intervals for F
Determination of BE based on the Dose-Scale method is a two-step procedure. First, using either of the procedures
described above, a within-study dose response relationship is mathematically described by fitting the relevant
version of the Emax model to the mean dose-response data and an estimate for F is obtained. Second, a 90%
confidence interval for F is estimated by a bootstrap procedure and each bootstraps estimation includes the
calculation of F by fitting one of the above models to a "sample dose-response data set", which is generated by
repetitive sampling with replacement.
The Agency has used Efron's method for calculation of 90% confidence interval. The 90% confidence interval for
“F” value must fall within 80-125% to establish bioequivalence. Efron's bias corrected and accelerated (BCA)
method will be used for estimation of 90% confidence intervals for “F”.
SUMMARY AND CONCLUSION
In this reviewed, we exemplify the relationship between dose-concentration-effect of investigational drugs. There
are so many models to explain the relationship between drugs doses vs. efficacy and be fond of linear model: this
model erroneously assumes that the effect can increase with concentrations without limits. Log-linear model:
Impossible to predict the value of E when concentration at zero. Emax model: Broadly used to characterize
pharmacological effects. Sigmoid Emax model: More convenient to fit steep pharmacological responses. Modified
Emax model: Can be applied to pharmacological responses without reaching the maximal effect.
In this present article, we focus on the study design of clinical dose finding studies to produce the information
needed to efficiently and reliably characterize the benefit of a drug over a dose range of interest When study has
minimal systemic absorption or pharmacokinetic study is not possible, in this case phramacodynamic parameters
should be analyze as per drug regulatory recommendation. In this reviewed we have explained the statistical method
for bioequivalence criterion and the dose-response relationship as Emax model and application in evaluation of
bioequivalence.
The dose- response studies is the early publication of the ICH E4 guideline, which is the primary source of
regulatory guidance in this area. The guideline states in its introductory section that the purpose of dose response
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Kuldeep K Namdev et al.; J. Pharm. Bioanal. Sci., Volume 3, Issue 2, Apr-Jun 2014, 21-28
ISSN: 2278-828X
information is “to find the smallest dose with a discernible useful effect or a maximum dose beyond which no
further beneficial effects is seen, but practical study designs do not exist to allow for precise determination of these
doses”. The ICH E4 guideline stresses the importance of obtaining good dose response information by estimating
relevant target doses [13, 14]
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How to cite this article: Kuldeep K Namdev et al, An
Outline of Dose - Response Relationship: Emax Modeling
and Its Application in Bioequivalence Study, J. Pharm.
Bioanal. Sci., Volume 3, Issue 2, Apr-Jun 2014, 21-28
Copyright: © 2014 Kuldeep K Namdev, et al. This is an
open-access article distributed under the terms of the
Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any
medium, provided the original author and source are
credited.
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