Transcript The Three Main Issues in the Drug Development:

Workshop Topic
BIOAVAILABILITY AND BIOEQUIVALENCE TRIALS:
STATISTICS & PHARMACOKINETIC PRINCIPLES
Presented at:
International Congress on Medical and Care Compunetics
Venue and Date
Netherlands Congress Centre, The Hague
June 2 – 4, 2004
Workshop Presenters
Lakshminarayan Rajaram1 & Sandip K. Roy2
1Department
of Epidemiology and Biostatistics
University of South Florida,Tampa, Florida,
2Exploratory
Clinical Development
PK – Novartis Pharmaceutical Corporations,
East Hanover, New Jersey
The Three Main Issues in the Drug Development:

Safety

Efficacy

Manufacturing
Safety:
 Safety of the drug is tested extensively in animals prior to
use in humans

Safety in humans is tested in short-term studies prior to
conducting a long-term clinical trials

Pharmaceutical companies continue to look for side effects
as part of safety even after the approval for marketing the
drug
As part of efficacy:

It should be established that a new chemical entity has a
desirable biological effect in non-human screening tests

It should be proven in clinical testing for the medical
purpose intended in typical patient groups
Manufacturing should be carried out under stringent
quality control so that each individual dose of medication
has the highest standards of safety and efficacy required.
Phases of Product Development

Pre-Clinical Testing

Phase I of a Clinical Trial

Phase II of a Clinical Trial

Phase III of a Clinical Trial

Phase IV of a Clinical Trial
Pre-Clinical Testing:

To show biological activity of the compound against the
targeted disease and evaluate its safety

Company conducts the laboratory and animal studies to
show this

This phase lasts anywhere from 2 to 3 years

At the end of this testing, company files an Investigational
New Drug (IND) with the Food and Drug Administration
(FDA) for approval to begin to test the drug in humans
Pre-Clinical Testing (Cont’d):
The IND contains the following:

Results of previous experiments

How, where, and by whom the new studies will be
conducted

Chemical structure of the compound

How it is thought to work in the body

Any toxic effects found in the animal studies

How the compound is manufactured
Phase I of a Clinical Trial:

Conducted in about 20 to 100 normal, healthy
volunteers

Studies drug’s safety profile, including the
safe dosage range

Studies pharmacokinetics of the drug

Lasts from several months to a year
Remark: Many drugs are abandoned because Phase I trials
reveal safety problems, poor tolerability, unsuitable PK
profiles, no indication of efficacy (Lack of POC), etc.
Phase II of a Clinical Trial:

Randomized trials lasting several months to two years

Involves several hundred patients

One group receives experimental drug and the other
group receives a standard treatment or placebo

Evaluates the effectiveness of the drug and looks for
side effects
Phase III of a Clinical Trial:
 Randomized, and double-blind or single-blind trials


Involves several hundred to several thousand patients and
typically lasts for several years
Provides a more thorough understanding of the drug’s
 effectiveness
 benefits
 range of possible adverse reactions
New Drug Application (NDA)

Companies conduct the analysis of the data collected from
all three phases.

If the data successfully demonstrate the safety and
effectiveness, an NDA is filed with the FDA for the
approval of the drug

NDAs contain all of the scientific information gathered by
the company, and typically run 150,000 pages or more
Phase IV of a Clinical Trial:

Phase IV studies are done after the drug has been marketed

Phase IV studies continue testing the drug to collect
information about
 its effect in various populations
 any side effect associated with long-term use
7/16/2015
13
Remarks about the Drug Development Process:
 Discovering and developing safe and effective new drugs is
a long, arduous, and an expensive process

Only 5 in 5000 compounds that enter pre-clinical testing
make it to human testing, and one of those 5 tested is
approved

Takes anywhere from 8 to 12 years for an experimental drug
to go from the laboratory to the medicine cabinet

Average cost to bring one drug to market is  $827 millions
(Tufts’s University Report)
Bioavailability

Defined as [13] the rate and extent to which the active
ingredient or active moiety is absorbed from a drug product
and becomes available at the site of action
For drugs that are not intended to be absorbed into the
bloodstream, bioavailability may be assessed by
measurements intended to reflect the rate and extent to
which the active ingredient or active moiety becomes
available at the site of action
Factors that influence the bioavailability of a dosage form
are, but not limited to:
Physical characteristics of the dosage form itself and includes
such items as tablet compression force, particle size, and
dissolution rate
Choice and spacing of the sampling times as they affect the
precision of the estimate of the pharmacokinetic (PK)
parameters
Amount of physical activity permitted during a study
Pharmacokinetics (PK)

Mathematical quantification of the amounts of drug in the
body over time

Study of the processes of bodily absorption, distribution,
metabolism, and excretion of compounds and medicines
PK parameters required in the FDA regulations for an invivo bioavailability study are:
AUC0 - t:
Area under the plasma or blood
concentration-time curve from 0 to t
AUC0 - :
Area under the plasma or blood
concentration-time curve from 0 to 
Cmax:
Maximum concentration
Tmax:
Time to achieve Cmax.
T1/2:
Elimination half-life
Ke:
Rate constant
Assessment of Bioavailability
Bioavailability is generally assessed by the determination of
the following parameters:
AUC0-:
is proportional to the total amount of drug
reaching the systemic circulation, and thus
characterizes the extent of absorption
Cmax:
is a function of both the rate and extent of
absorption. Increases with an increase in dose as
well as increase in absorption rate
Tmax:
reflects the rate of drug absorption, and
decreases as the absorption rate increases
Bioequivalence
Bioequivalence is defined as [13] the absence of a significant
difference in the rate and extent to which the active
ingredient or active moiety in pharmaceutical equivalents
becomes available at the site of drug action when
administered at the same molar dose under similar
conditions in an appropriate designed study
Purpose of BE trials:
Identify pharmaceutical equivalents that are intended to be
used interchangeably for the same therapeutic effect
Methods of Assessing Bioavailability (BA) and
Bioequivalence (BE)
As noted in §320.24 (Types of evidence to establish BA or
BE) [13], there are many in-vivo and in-vitro methods. Brief
discussion of the following methods will be discussed:

Pharmacokinetic Studies

Pharmacodynamic Studies

Comparative Clinical Studies

In Vitro Studies
Pharmacokinetic Studies:
 The definitions of BA and BE, expressed in terms of rate
and extent of absorption of the active ingredient at the site
of action, emphasize the use of pharmacokinetic measures
in blood, plasma, and/or serum to indicate release of the
drug substance from the drug product into the systemic
circulation

If the measurements of the drug or its metabolites in blood,
plasma, or serum cannot be accomplished, measurements of
urinary excretion can be used to document BE
Pharmacokinetic Studies (Cont’d):

A typical study is conducted as a cross-over study

Clearance, volume of distribution, and absorption are
assumed to have less inter-occassion variability compared to
the variability arising from formulation performance.
Therefore, difference between two products because of
formulation factors can be determined
Pharmacodynamic Studies:

Not recommended for orally administered drug products
when the drug is absorbed into the systemic circulation and
a pharmacokinetic approach can be used to assess systemic
exposure and establish BE

In situations where pharmacokinetic approach is not
possible, a suitably validated pharmacodynamic methods
can be used to demonstrate BE
Comparative Clinical Studies:
These studies to determine BE for orally administered drug
products can be appropriate when neither pharmacokinetic
nor pharmacodynamic approach is feasible.
In Vitro Studies:
Under certain circumstances, product quality BA and BE can
be documented using in-vitro approaches (21 CFR
320.24(b) and 21 CFR 320.22(d) (3))
Statistical Approaches to Determining Bioequivalence
There are 3 approaches [12]:

Average Bioequivalence (ABE)

Individual Bioequivalence (IBE)

Population Bioequivalence (PBE)
Average Bioequivalence (ABE)
 Analysis of PK measures (such as AUC and Cmax) is based
on the two one-sided tests to determine whether the average
values for the PK measures (after the administration of T
and R products) are comparable

Involves the calculation of a 90% confidence interval (CI)
for the ratio of the averages of the measures for T and R
products

To establish BE, the CI should fall within a BE limit,
usually 80 – 125% for the ratio of the product averages
ABE (Cont’d):
 Focuses only on the comparison of population averages of a
BE measure of interest

Does not assess
 within-in-subject variability
 subject-by-formulation interaction
ABE (Cont’d):
 A conventional non-replicated design, such as the standard
two-formulation, two-sequence, two-period crossover
design is:
Period
1
2
Sequence 1
R
T
2
T
R
Note: A replicated 2x3, or 2x4 crossover designs can also be
used for ABE, but are not necessary when an average or
population approach is used
Individual Bioequivalence (IBE)

Although not required by the FDA, IBE studies provide
more data and more assurance that generic and brand drug
products are interchangeable

IBE allows the clinicians to assess:
within-subject variability
which examines whether or not the product behaves
the same pharmacokinetically when it is administered
on two occasions in the same subject)
IBE (Cont’d):
subject-by-formulation interaction
which examines:
1.
one product behaves the same
pharmacokinetically in all subjects
2.
the generic and brand name products
behave differently in individuals
The absence of this interaction provides evidence for
interchangeability between brand and generic products
The presence assumes that the excipients and other nonmedical ingredients in a product may influence the
absorption of the active ingredient
IBE (Cont’d):
 FDA recommendation requires the estimation of withinsubject variance of R and T formulations, and subject-byformulation interaction variance component

Standard 2x2 design is not appropriate. Replicated crossover
designs such as:
 2-sequence, 4-period, TRTR and RTRT
 2-sequence, 3-period, TRT and RTR
are recommended by the FDA
Population Bioequivalence (PBE)

Assesses the total variability of the measure in the
population

Approach is based on the comparison of an expected
squared distance between T and R ( E(T-R)2 = (µT-µR)2 +
TT2 +TR2 ) to the expected squared distance between two
administrations of R ( E(R – R*)2 = 2 TR2 )

Uses 2x2, 2x3, and 2x4 crossover designs (more on PBE in
2001 FDA Guidance for Industry: Statistical Approaches
to Establishing BE)

Bioequivalence Trials (BET)
Conducted to ensure that two products do not differ in
safety, efficacy, and compare bioavailabilities when
administered at the same dose

Conducted under the protocols of an ANDA (Abbreviated
New Drug Application)

The AUC is the primary measure of the extent of
absorption or the amount of drug absorbed in the body
which is most often used to assess BE between two products

µT and µR refer to test mean and reference mean
respectively.

To Claim BE:


Truth Table for
Bioequivalence Decisions
the ratio of averages,
that is, µT/µR, must be
within (80%, 125%)
with 90% assurance
Construct 90% CI for
µT/µR and compare it
with (80%, 125%). If
this CI is within
(80%, 125%), then
BE is concluded
TRUTH
Bioequivalent
D
E
C
I
S
I
O
N
Not bioequivalent
Bioequivalent
Right decision
Everyone
gains
Wrong decision
Consumer
loses
Not bioequivalent
Wrong decision
Sponsor
loses
Right decision
Consumer
gains
Assumptions and Model

The primary assumptions of the model are normality
assumptions

In many cases, the distribution of AUC is skewed, and log
transformation of the raw data is performed to remove the
skewness.

Analysis using the log-transformed model provides an exact
90% CI for µT/µR under normality assumptions

FDA is in favor of log-transformed model if transformation
is required

Inter- and Intra-Subject Variabilities
Individual subjects differ widely in their responses to the
drug. Knowledge of the inter- and intra-subject variabilities
will provide valuable information in the assessment of BE.

Inter-subject variability:
This can be removed from the comparison of BA
between two formulations by using a crossover design

Intra-subject variability:
This variability in crossover designs is confounded with
the variabilities such as lot-to-lot, product-to-product,
subject by product variabilities. These are difficult to
assess in a crossover design.
Study Designs
The Parallel Design
R
A
N
D
O
M
I
Z
A
T
I
O
N
Group 1
Test
Group 1
Reference

In this design, each subject receives one and only one formulation

Variability in, for example, AUC consists of inter- and intra-subjects
variabilities. Assessment of BE is usually made based on the intrasubject variability.

Parallel design is not able to identify and separate these two variations
since each subject receives the same drug throughout the study.
Therefore, the parallel design is not an appropriate design for BA/BE
studies
Parallel Design (Cont’d):
Parallel designs may be an alternative to crossover if
 inter-subject variability is relatively small compared to
the intra-subject variability

drug is potentially toxic and/or has a very long
elimination half-life

population of interest consists of very ill patients

cost of increasing the number of subjects is much less
than that of adding an additional treatment
Crossover Design:
 Consider a Standard two-formulation, two-sequence, twoperiod crossover design.
R
A
N
D
O
M
I
Z
A
T
I
O
N
Period 1
Sequence 1
R
Sequence 2
T
Period II
W
A
S
H
O
U
T
T
R
Advantages of the Crossover Design
 The crossover design for BA/BE studies has the following
advantages:
 Each subject serves as his/her own control and allows a
within subject comparison between formulations

Removes the inter-subject variability from the
comparison between formulations

Provides the best unbiased estimates for the differences
(or ratios) between formulations with proper
randomization of subjects to the sequence
Carry-Over Effect in a Crossover Design

Effect of a drug might persist after the end of the dosing
period. This effect is called carry-over effect.

Dosing periods should be separated by a washout period of
usually 5 times the half-life of the active drug ingredient.

The direct drug effect is the effect that a drug has during the
period in which it is administered
Statistical Considerations

For BA/BE, it is assumed that there are:
(i) no period effect
(ii) no carryover effect
However, in many cases, they may still be present.

The statistical model that will be used to describe the standard 2x2
crossover design is:
Yijk = µ + Sik + Pj + F(j, k) + C(j-1, k) + eijk, where
Yijk =
µ=
Sik =
Pj =
F(j, k) =
response of the ith subject in the kth sequence at the jth period
overall mean
random effect of the ith subject in the kth sequence
(i goes from 1, 2, 3, ……,nk; k = 1, 2)
fixed effect of the jth period
direct drug effect in the kth sequence which is administered at the jth
period.
C(j-1, k)= fixed first order carry-over effect in the kth sequence which is
administered at the (j – 1)th period
eijk = within-subject random error in observing Yijk
Statistical Considerations (Cont’d)
Assumptions:



{Sik} are independently and identically distributed with
2
mean 0 and variance s
{eijk} are independently distributed with mean 0 and
2
variance t (where t = 1, 2, that is, number of formulations
to be compared)
{Sik} and {eijk} are assumed mutually independent
Inter- and intra-subject variabilities

Estimate of s is used to explain inter-subject variability,
2
while the estimates of t are used to explain intra-subject
variability for the tth formulation
2
Statistical Considerations (Cont’d)
Analysis of Variance:
 Studies the variability in the observed data by partioning the
total sum of squares (SST) of the observations into
components of fixed effects and random error

For example, in a 2x2 crossover study, the partition of
SSTotal into various components is as follows:
carryover effect
SSTotal
period effect
direct drug effect
error
Partition of Total Sum of Squares
SStotal
=
SSwithin
(2(n1+n2) –1)
(n1 + n2)
+
SSbetween
(n1 + n2) -1
carryover
(sequence)
Inter-subject
error
(subject (sequence))
Direct
Period
drug
(2 -1 = 1)
(df = 2 -1 = 1)
Intra-subject
residual (n1 + n2 - 2)
(model residual)
Sample Size and Randomization

A single dose bioequivalence study in a 2x2 crossover
design is conducted in 24 to 36 healthy, normal volunteers

Each subject is randomly assigned to either sequence 1 (RT)
or sequence 2 (TR).

Subjects within sequence RT (TR) receive R (T) during the
first dosing period and T (R) during the second dosing
period
SAS Program for Randomization
proc format;
value groupf 0 = 'RT' 1 = 'TR'; run;
data randomization;
input subj_id @@;
formulations = ranuni(12345);
datalines;
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
27 28 29 30 31 32 33 34 35 36
;
proc rank data=randomization groups = 2 out = new;
var formulations;
run;
data finalcode; set new;
proc print data=finalcode;
title 'Randomization code for a standard 2 x 2 crossover design with 36 subjects';
title2 'RT and TR are referred to as sequence 1 and sequence 2 respectively';
var subj_id formulations;
format formulations groupf.;
run;
Result of the above SAS Program
Obs subj_id formulations
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
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36
1
2
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RT
TR
TR
RT
RT
TR
RT
TR
TR
TR
TR
RT
RT
TR
TR
TR
TR
RT
RT
RT
RT
TR
RT
TR
TR
RT
RT
RT
TR
RT
TR
RT
TR
TR
RT
RT
Database Containing Time and Concentration Values
A single-click of the following button opens up an
Access database containing values of Time and
Concentration for individual subjects
Time_Concentration_Database
Mathematics of PK Computations
Computation of AUC
 Linear interpolation using the trapezoidal rule is one
of the many methods that exist to compute the AUC
[11].

Let C0, C1, C2, ……, Ck be the plasma or blood
concentrations obtained at time points t0 (time 0), t1,
t2, …., tk, respectively. The AUC from t = 0 to t = tk
is given by
C C 
AUC(0  t )   
t  t 
2 

k
k
i 1
i 1
i
i
i 1

AUC0-: The AUC from 0 to  can be estimated as
AUC (0  )  AUC (0  t k ) 
Ck

Where
Ck = concentration at the last measured sample
 = slope of the terminal portion of the log
concentration-time curve multiplied by (-2.303),
also referred to as an elimination rate.
 = -2.303*1 where 1is the slope obtained from
fitting a linear regression to the logarithmic
of the concentration values during the
elimination phase.

Elimination half-life is

Cmax:
Tmax:
0.693

Maximum concentration
Time to achieve Cmax.
Following few slides have a SAS program that
 imports time and concentration data[11] from an Access
database
 converts them to a SAS dataset
 computes all of the above mentioned pharmacokinetic
parameters
 draws graphs of Concentration Vs. Time profile for each
subject
libname hague_04 'c:\hague_04\';
options nodate nonumber;
************************************************;
* A macro to convert an Access Table, time_conc_data
* an Access database called, database_forPKdata.mdb,
* to a SAS data set, called, PKDATA.SAS7BDAT;
%macro convert(table, sasfile);
proc import out = &sasfile
datatable = "&table"
dbms = access2000 replace;
database = "c:\hague_04\database_forpkdata.mdb";
dbpwd = "pkdata";
run;
proc print data=&sasfile;
title "Tabulation of the data from the SAS dataset, &sasfile";
run;
%mend convert;
****************************************;
* The following statement executes the macro;
%convert (time_conc_data, pkdata);
****************************************;
****************************************;
Following segment computes Cmax and Tmax ****;
data one;
set pkdata;
cmax = max(of c1-c13);
array c{13} c1-c13;
array t{13} t1-t13;
do i = 1 to 13;
do j = 1 to 13;
if c{i} = cmax then tmax = t{i};
end;
end;
drop i j;
****************************************;
*********************************************************************;
** The following segment computes AREA UNDER the CURVE (AUC) between **;
** a set of consecutive time points
**;
data two;
set one;
array c{13} c1-c13;
array t{13} t1-t13;
do i =1 to 13;
if i=1 then area = 0;
else area = 0.5 * ( c{i} + c{i-1} ) * ( t{i} - t{i-1} );
output;
end;
drop i;
*************************************************************;
** Following segment arranges AUC values computed above in the form *;
** of a row so as to make it ready for summing up the AUC values
*;
** to get AUC from 0 to 32
*;
data three;
set two (keep = subj_id area);
proc transpose data=three out=four prefix = auc;
by subj_id;
run;
data five;
set four;
array a{*} auc1-auc13;
do i = 1 to 13;
auc0_32 = sum (of a{*});
end;
drop i;
*****************************************************************;
* The following segment creates a data set, SIX with subj_id, CMAX, TMAX *;
* C13 (concentration at the last time point, AUC0_32 for each subject
*;
data six;
set one (keep = subj_id t1-t13 c1-c13 cmax tmax c13);
set five(keep = auc0_32);
**************************************************************************************;
*************************************************************;
** The following segment computes Ck (concentration at the last time
*;
** point. Sets all time points that are LESS THAN or EQUAL to TMAX *;
** value to a SAS missing value period (.). Reason for this is we do
*;
** need time points that are greater than TMAX value and their
*;
** corresponding CONCENTRATION values to fit linear regression in *;
** order to find BETA that will be used in computing ke, the
*;
** elimination rate constant
*;
data seven;
set six;
ck = c13;
array t{13} t1-t13;
array c{13} c1-c13;
do j = 1 to 13;
do i = 1 to 13;
if (t{i} le tmax) then t{i} = .;
end;
end;
drop i j;
run;
*************************************************************;
data eight;
set seven (keep = subj_id t1-t13);
run;
data nine;
set seven (keep = subj_id c1-c13);
run;
*****************************************************************;
** The following segment TRANSPOSES the data set, EIGHT by subj_id to get*;
**columns of data, one for subj_id and another for TIME
*;
proc transpose data=eight out=ten prefix=time;
by subj_id;
run;
*****************************************************************;
*******************************************************;
** The following segment TRANSPOSES the data set, NINE,
*;
** by subj_id to get two columns of data, one for subj_id and
*;
** another for CONC
*;
proc transpose data=nine out=eleven prefix=conc;
by subj_id;
run;
***********************************************************;
** The following segment creates a data set, TWELVE which contains *;
** 3 columns of data. one for subj_id, second one for TIME, and the
*;
** third one for CONC
*;
data twelve;
set ten (keep = subj_id time1);
set eleven (keep = subj_id conc1);
rename time1 = time;
rename conc1 = conc;
run;
*******************************************************;
** The following segment does regression analysis on the data
*;
** values in the elimination phase
*;
data reg;
set twelve;
logc = log10(conc);
run;
proc reg data=reg noprint outest = est; * OUTEST = EST outputs
parameter estimates into a SAS data set called EST;
model logc = time;
by subj_id;
run;
***************************************************************;
data regout;
set est (keep = time); * time1 is the BETA estimate from the
regression using model logc = time1;
rename time = beta1;
* Renaming time1 to BETA1;
run;
**************************************************************;
********************************************************;
** The following segment creates a PERMANENT SAS dataset,
*;
** called, FORANALYSIS, with all the necessary variables, and
*;
** then computes Ke, T_HALF, and AUC(0 - infinity)
*;
data hague_04.foranalysis;
set seven (keep = subj_id cmax tmax ck auc0_32);
set regout (keep = beta1);
ke = (-2.303)*beta1; ** BETA1 is the parameter estimate using the
model logc = time1 where logc = log10(conc);
** ke = elimination rate;
t_half = 0.693/ke;
** plasma elimination half-life;
auc0_inf= auc0_32 + (ck/ke);
run;
proc print data=hague_04.foranalysis;
title 'Tabulation of subj_id, CK, CMAX TMAX, BETA1, KE, T_HALF,';
title2 'AUC0_32, and AUC0_INF for all subjects';
var subj_id ck cmax tmax beta1 ke t_half auc0_32 auc0_inf;
run;
***********************************************************;
** The following segment creates plots of CONCENTRATION
*:
** versus TIME for each subject
*;
symbol1 color=red interpol=join value=dot height=1;
***********************************************************;
data plot1; set pk (keep = subj_id t1-t13);
data plot2; set pk (keep = subj_id c1-c13);
proc transpose data=plot1 out= plot1a prefix = time; by subj_id; run;
proc transpose data=plot2 out= plot2a prefix = conc; by subj_id; run;
data for_plot; set plot1a; set plot2a;
rename time1=time; rename conc1 = concentration;
**********************************************************;
proc gplot data=for_plot;
title "Plot of CONCENTRATION vs. TIME";
plot concentration*time / haxis=0 to 35 by 5 vaxis=0 to 5 by 1
hminor=2 vminor=1 cvref=blue caxis=blue ctext=red;
by subj_id;
run; quit;
PK_Parameters_Computation.SAS
sequence
seq_no
period
treat
auc0_32
sequence
seq_no
period
treat
auc0_32
RT
1
1
1
74.675
TR
2
1
2
74.825
RT
1
2
2
73.675
TR
2
2
1
37.350
RT
1
1
1
96.400
TR
2
1
2
86.875
RT
1
2
2
93.250
TR
2
2
1
51.925
RT
1
1
1
101.950
TR
2
1
2
81.675
RT
1
2
2
102.125
TR
2
2
1
72.175
RT
1
1
1
79.050
TR
2
1
2
92.700
RT
1
2
2
69.450
TR
2
2
1
77.500
RT
1
1
1
79.050
TR
2
1
2
50.450
RT
1
2
2
69.025
TR
2
2
1
71.875
RT
1
1
1
85.950
TR
2
1
2
66.125
RT
1
2
2
68.700
TR
2
2
1
94.025
RT
1
1
1
69.725
TR
2
1
2
122.450
RT
1
2
2
59.425
TR
2
2
1
124.975
RT
1
1
1
86.275
TR
2
1
2
99.075
RT
1
2
2
76.125
TR
2
2
1
85.225
TR
2
1
2
86.350
RT
1
1
1
112.675
TR
2
2
1
95.925
RT
1
2
2
114.875
TR
2
1
2
49.925
RT
1
1
1
99.525
TR
2
2
1
67.100
RT
1
2
2
116.250
TR
2
1
2
42.700
RT
1
1
1
89.425
TR
2
2
1
59.425
RT
1
2
2
64.175
TR
2
1
2
91.725
RT
1
1
1
55.175
TR
2
2
1
114.050
RT
1
2
2
74.575
SAS Code for GLM Model (Data in previous slide are from Chow and Liu)
***********************************************;
libname hague_04 'c:\hague_04\';
options nodate nonumber;
proc format;
value seq_nof
1 = 'RT'
2 = 'TR';
value periodf
1 = 'Period I’ 2 = 'Period II';
value treatf
1 = 'B_Reference‘ 2 = 'A_Test';
***********************************************;
PROC IMPORT OUT = fromxls_sas
DATAFILE= "c:\hague_04\beprg.xls"
DBMS= EXCEL2000 REPLACE;
GETNAMES=YES;
RUN;
**********************************************;
** The data are for AUC(0 - 32);
data one;
set fromxls_sas;
format seq_no seq_nof. period periodf. treat treatf.;
*********************************************;
proc glm data=one; ** This segment is to print GLM output;
class treat period seq_no subj_id;
model auc0_32 = seq_no subj_id(seq_no) period treat / ss3;
test h = seq_no e = subj_id(seq_no) / htype=3 etype=3;
SAS Program for ANOVA
lsmeans treat period / stderr pdiff;
lsmeans seq_no / stderr pdiff e = subj_id(seq_no);
estimate 'A_Test - B_Reference' treat 1 -1;
run;
Analysis of Variance Table
Dependent Variable: AUC0_32 (Area Under the Curve from t = 0 to t = 32)
Source of
Variation
DF
Sum of
Squares
Mean Square
F Value
P-value
1
276.00021
276.000
0.37
0.5468
16211.4887
736.886
4.41
0.0005
1
62.79188
62.79188
0.38
0.5463
Period
(period)
1
35.96672
35.96672
0.22
0.6474
Residuals
(model error)
22
3679.42953
167.24680
47
20265.67703
Inter-subjects
Carry-over
(sequence)
Residuals
22
(subject (sequence))
Intra-subjects
Direct drug
(treat)
Corrected Total
Remark: The partition of sums of squares follow the description given on slide#
44
Interpretation of ANOVA output

P-value of 0.5468 for carry-over implies insignificant evidence for the
presence of unequal carry over effect, suggesting the use of data from
both periods to make inference on the direct drug effect.

P-value of 0.5463 for direct drug implies no significant direct drug
effect was found. Hypothesis of the equality of two formulations does
not imply the bioequivalence of two formulations.

P-value of 0.6474 for period implies the failure to reject the null
hypothesis of no period effect
Remarks:
 The 2x2 crossover design provides estimates and tests for the period
effect, direct drug effect, and the carryover effect.

It does not provide any inference on the subject-by-formulation, the
formulation-by-period, and the sequence-by-period interactions. For
these, a higher order crossover designs are needed.
Statistical Methods for Average Bioavailability
A few of the statistical methods that will be discussed are:

Confidence Interval Approach

Interval Hypothesis Testing

Bayesian Method

Non-parametric Method
Confidence Interval Approach
1. The Classical (Shortest) Confidence Interval

Construct the CI for (µT/µR), and compare that CI with
(80%, 120%) limits

If the constructed CI falls within the limits, then the two
formulations are considered bioequivalent
2. Westlake Symmetric Confidence Interval


As the equivalence limits are in the symmetric form such as
-20% to 20%, Westlake suggested that the CI be adjusted to
be symmetric about 0 for the difference (or about unity for the
ratio
The test formulation is concluded to be bioequivalent to the
reference formulation if |difference mean| < 0.20µR
The following SAS code will produce both types of CIs discussed above:
***********************************************************;
The following segment creates a temporary SAS data file from FOR_BE.SAS7BDAT,
and then computes a few new variables, ALPHA, PVALUE, TVALUE, DIFF_MEAN
and RATIO_PCT;
data one;
set for_BE;
limit =90;
*LIMIT is degree of confidence, that is 95, 90, etc.;
low_tol = 0.80;
*LOW_TOL = lower limit of the tolerance range;
high_tol = 1.20;
*HIGH_TOL = upper limit of the tolerance range, that is 1.20;
alpha = (100 - limit)/100;
pvalue = 1 - (alpha/2);
tvalue = tinv (pvalue, df);
diff_mean = test_mean - ref_mean;
ratio_pct = 100*(diff_mean/ref_mean) + 100;
format ratio_pct 6.2;
********************************************************************;
*
Following segment computes the CLASSICAL (SHORTEST)
*
CONFIDENCE INTERVAL.
*
If the CI is wholly contained within in the TOLERANCE RANGE of
*
(0.80, 1.20), then it implies BIOEQUIVALENCE;
data conventional_CL;
set one;
upp_limit = diff_mean + tvalue*rmse*sqrt(2/n);
upp_perct = 100*upp_limit/ref_mean + 100;
low_limit = diff_mean - tvalue*rmse*sqrt(2/n);
low_perct = 100*low_limit/ref_mean + 100;
low_tol = 100*low_tol;
high_tol = 100*high_tol;
if (low_tol < low_perct) and (upp_perct < high_tol) then bioequivalence = "Yes";
else bioequivalence = "No";
format upp_perct low_perct 6.2;
**********************************************************;
**************************************************************;
** The following segment computes Westlake CI;
data westlake;
set temp;
sum = -(diff_mean/rmse)*sqrt(2*n);
inc = 2;
if sum > 0 then k1 = -sum;
else if sum le 0 then k1 = sum;
k2 = sum - k1;
again: i = 0; j = 0;
low_tail = probt(k1, df);
*PROBT is the cumulative;
high_tail = 1 - probt(k2, df); * t distribution function;
prob_area = low_tail + high_tail;
if low_tail > alpha then do;
k1 = 2*k1; k2 = sum - k1; goto again;
end;
if high_tail > alpha then do;
k1 = k1 + sum; k2 = sum - k1; goto again; end;
**************************************************************;
do while (inc > 0.0001);
k1 = k1 + inc; k2 = k2 -inc; low_tail = probt(k1, df); high_tail = 1 - probt(k2, df);
prob_area = low_tail + high_tail;
if prob_area > alpha then do; k1 = k1 - inc; k2 = k2 + inc;
inc = inc/5; j +1;
end;
i + 1;
end;
****************************************************************;
lower_limit = diff_mean + k1*rmse*sqrt(2/n);
lower_pct = 100*lower_limit/ref_mean + 100;
upper_limit = diff_mean + k2*rmse*sqrt(2/n);
upper_pct = 100*upper_limit/ref_mean + 100;
low_tol = 100*low_tol;
high_tol = 100*high_tol;
if (low_tol < lower_pct) and (upper_pct < high_tol) then bioequivalence = "Yes";
else bioequivalence = "No";
format lower_limit upper_limit 7.2 lower_pct upper_pct 6.2;
SAS Program to get 2
types of CIs, and
hence, to prove BE
Conventional Confidence Interval
If the CI is wholly contained within the tolerance range, (0.80, 1.20),
then, it implies BIOEQUIVALENCE between the test and reference means
Results are in original units and in percentage from the reference mean
pk_parameter rmse
auc0_32
ref_mean test_mean
12.9324 82.5594 80.2719
low_limit
diff_mean
upp_limit
-8.69805
-2.2875
4.12305
low_tol high_tol
low_perct
ratio_pct
upp_ perct
bioequivalence
80
89.46
97.23
104.99
Yes
120
95% Westlake Confidence Intervals
pk_parameter
rmse
auc0_32
12.9324 82.5594
upper_limit
8.84
lower_pct
89.29
ref_mean
ratio_pct
97.23
test_mean
lower_ limit
diff_mean
80.2719
-8.84
-2.2875
upper_pct
bioequivalence
110.71
Yes
Interval Hypothesis Testing
Schuirmann’s Interval Hypotheses for BE
 Let L and U be the clinically meaningful lower and upper limit.
Often they are chosen to be 20% of the reference mean (µR)

The interval hypotheses for bioequivalence are:
H0: µT - µR <= L
or
µT - µR ≥ U
vs.
Ha: L < µT - µR < U

The bioequivalence is shown by rejecting the null hypothesis of
bioINequivalence.

When the logarithmic transformation of the data is considered, the
hypotheses become:
H0: µT / µR <= L
or
µT / µR ≥ U
vs.
Ha: L < µT / µR < U
where L = exp(L) and U = exp(U)

This interval hypotheses can be decomposed into two sets of onesided hypotheses as follows:
H01: T - R <= L vs.
and
H02: T - R ≥ U vs.
Ha1: T - R > L
Ha2: T - R < U

The first set of hypotheses is to verify that the BA of the test
formulation is not too low, while the second set is to verify that the
BA is not too high

Relatively low BA refers to the concern of efficacy of the test
formulation while the relatively high BA refers to the concern of
safety

Rejection of H01 and H02 imply that L < µT - µR < U, and hence, T
and R are equivalent
SAS Program for Schuirmann ‘s Interval Hypotheses Method for Assessing BE
data schuirmann;
set one;
low_value = (low_tol - 1)*ref_mean;
high_value = (high_tol - 1)*ref_mean;
t_left = (diff_mean - low_value) / (rmse*sqrt(2/n));
t_right = (-high_value + diff_mean) / (rmse*sqrt(2/n));
if (abs(t_left) ge tvalue) and (abs(t_right) ge tvalue) then bioequivalence = "Yes";
else bioequivalence = "No";
pr_high = 1 – probt (t_right, df);
pr_low = 1 – probt (t_left, df);
total_probability = pr_high + pr_low;
format pr_high pr_low total_probability 7.5;
proc print;
SAS Program for
SCHIRMANN’s test
title 'SCHUIRMANN 2 one-sided method for testing the bioequivalence';
title2 'Probability area (PR_HIGH) <= ALPHA/2 and probablity area (PR_LOW) <=ALPHA/2';
title3 'That is, the total probability <= ALPHA, implies bioequivalence between';
title4 'the test and reference means';
var pk_parameter rmse ref_mean test_mean diff_mean ratio_pct pr_low pr_high
total_probability alpha bioequivalence t_right t_left;
run;
SAS Output for Schirmann’s test
SCHUIRMANN 2 one-sided method for testing the bioequivalence
Prob. area (PR_HIGH) <= ALPHA/2 and prob. area (PR_LOW) <=ALPHA/2
That is, the total probability <= ALPHA, implies bioequivalence between
the test and reference means
pk_parameter rmse
ref_mean test_mean
auc0_32
12.9324 82.5594 80.2719
pr_high
0.00048

total_probability alpha
0.00050
0.1
diff_mean
-2.2875
bioequivalence
Yes
ratio_pct
97.23
TU
-5.03565
pr_low
0.00002
TL
3.81017
Since | TL| and | TU| are both greater than t(0.05, 22) = 1.717, the
null hypotheses H01 and H02 are rejected at the 5% level of
significance, and hence, the formulations are bioequivalent
Anderson and Hauck’s Test for Testing the Interval
Hypotheses given on slide#65

Compute the probability of the null hypothesis. Decide bioequivalence
if this p-level is less than 

The conclusion of rejecting the null hypothesis H0 of
bioINequivalence in favor of bioequivalence is based on a small pvalue.


This test is always more powerful than Schuirmann’s two one-sided
tests
The drawback of this test is that it may conclude bioequivalence even
when the intra-subject variability becomes very large. For a study with
low precision, Andersen and Hauck’s method may be in favor of
bioequivalence regardless of the large intra-subject variability.
SAS Program for Obtaining Andersen and Hauck p-values
**********************************************************************;
* The following segment computes ANDERSON and HAUCK p-values;
* If the PROBABILITY AREA is less than or equal to ALPHA, then it
* implies BIOEQUIVALENCE between the test and reference means;
data anderson_hauck;
set one;
low_value = (low_tol - 1)*ref_mean;
high_value = (high_tol - 1)*ref_mean;
tvaluel = (diff_mean - (high_value + low_value)/2)/ (rmse*sqrt(2/n));
delta = (high_value - low_value) / (2*rmse*sqrt(2/n));
tadj_high = abs(tvaluel) - delta;
tadj_low = -abs(tvaluel) - delta;
pr_high = probt(tadj_high, df);
pr_low = probt(tadj_low, df);
prob_area = pr_high - pr_low;
if prob_area le alpha then bioequivalence = "Yes";
else bioequivalence = "No";
format prob_area 7.5;
SAS Codes for Andersen
and Hauck’s Test
proc print;
title 'Anderson and Hauck p-value method for testing the bioequivalence';
title2 'Probability area of <= ALPHA implies BIOEQUIVALENCE';
title3 'between the test and reference means';
var pk_parameter rmse ref_mean test_mean diff_mean ratio_pct alpha prob_area
bioequivalence;
run;
**********************************************************************;
Results for Andersen and Hauck’s Test
Anderson and Hauck p-value method for testing the bioequivalence
Probability area of <= ALPHA implies BIOEQUIVALENCE
between the test and reference means
pk_parameter
auc0_32
alpha
0.1
rmse
ref_mean
12.9324 82.5594
prob_area
0.00045
test_mean
80.2719
diff_mean
-2.2875
ratio_pct
97.23
bioequivalence
Yes
p-value = 0.00045 < 0.05, and hence, the rejection of the null hypotheses of bioINequivalence
leads to bioequivalence of the two formulations
Bayesian Methods

Statistical methods studied so far for the assessment of BE were
derived based on the sampling distribution of the estimate of the
parameter of interest such as direct drug effect, which is assumed to
be fixed but unknown

Although CIs and hypotheses testing on the unknown direct drug
effect can be done based on the sampling distribution of the estimate,
there is little information about “the probability of the unknown direct drug
effect being within the equivalent limit, (LU)”

To have a certain assurance on this probability, Bayesian approach
assumes that the unknown direct drug effect is a random variable and
follows a prior distribution

In practice, investigators do have some prior knowledge of the profile
of the blood or plasma concentration-time curve. This knowledge can
be used to choose an appropriate prior distribution of the unknown
direct drug effect
Bayesian Methods (Cont’d)

A different prior distribution can lead to a different posterior
distribution which will have an impact on the statistical inference on
the direct drug effect. Hence, a critical issue in Bayesian approach is
how to choose a prior distribution.

Mandallaz and Mau proposed a Bayesian method for assessing the BE
for the ratio (µT / µR) with the following assumptions:
 no carryover effect
 effects from subjects are fixed
 number of subjects in each sequence is the same
SAS Program for Mandallaz and Mau
***************************************************************************;
* The following segment computes MANDALLAZ & MAU p values;
data mandallaz;
set one;
t_left = (-high_tol*ref_mean + test_mean) / (rmse*sqrt( (1 + high_tol**2)/n));
t_right = (-low_tol*ref_mean + test_mean) / (rmse*sqrt( (1 + low_tol**2)/n));
pr_high = 1 - probt(t_right, df);
pr_low = probt(t_left, df);
prob_area = pr_high + pr_low;
if prob_area le alpha then bioequivalence = "Yes";
else bioequivalence = "No";
format prob_area 7.5;
proc print;
title 'Mandallaz and Mau method for testing BIOEQUIVALENCE';
SAS Program for
Mandallaz and Mau
var pk_parameter rmse ref_mean test_mean diff_mean ratio_pct alpha prob_area bioequivalence;
run;
Results for Mandallaz and Mau Method for Testing
Bioequivalence
pk_parameter
auc0_32
prob_area
0.00026
rmse
12.9324
ref_mean
82.5594
bioequivalence
Yes
test_mean
80.2719
diff_mean
-2.2875
ratio_pct
97.23
alpha
0.1
Non-parametric Methods

Wilcoxon-Mann-Whitney two one-sided test

Distribution-Free Confidence Interval Based on the Hodges-Lehmann
Estimator

Bootstrap Confidence Interval
Analysis of Variance Table for the Log-Transformed Data on Slide#58
Dependent Variable: Log_AUC0_32 (Log of the Area Under the Curve from t = 0 to t = 32)
Source of
Variation
DF
Sum of
Squares
Mean Square
F Value
P-value
1
0.086
0.086
0.75
0.39
2.524
0.115
3.08
0.005
1
0.01
0.01
0.26
0.612
Period
(period)
1
0.009
0.009
0.25
0.625
Residuals
(model error)
22
0.819
0.037
47
3.44
Inter-subjects
Carry-over
(sequence)
Residuals
22
(subject (sequence))
Intra-subjects
Direct drug
(treat)
Corrected Total
Example (Treat = 1 is Reference Formulation and Treat = 2 is TEST formulation)
subject
period
sequence
treat
auci
subject
period
sequence
treat
auci
1
1
1
2
5144.90
14
2
2
2
16598.30
1
2
1
1
7896.05
14
1
2
1
14947.23
2
2
2
2
13461.79
15
1
1
2
10187.12
2
1
2
1
5627.24
15
2
1
1
9434.20
3
2
2
2
9017.21
16
2
2
2
5573.57
16
1
2
1
4712.56
3
1
2
1
12166.00
17
1
1
2
6770.99
4
1
1
2
9609.00
17
2
1
1
8665.00
4
2
1
1
8259.00
18
1
1
2
7013.16
5
1
1
2
5705.00
18
2
1
1
6611.13
5
2
1
1
14263.15
19
2
2
2
4995.75
6
2
2
2
5394.74
19
1
2
1
6894.94
6
1
2
1
2816.55
20
2
2
2
6043.94
7
2
2
2
3178.30
20
1
2
1
8046.22
7
1
2
1
4700.59
21
1
1
2
1641.67
8
1
1
2
12619.47
21
2
1
1
1819.00
8
2
1
1
9849.77
22
1
1
2
7459.77
9
2
2
2
6428.47
22
2
1
1
5619.67
9
1
2
1
8233.23
23
2
2
2
5486.20
23
1
2
1
7855.16
10
1
1
2
5971.00
24
1
1
2
5361.95
10
2
1
1
5496.87
24
2
1
1
6918.62
11
2
2
2
4782.47
25
2
2
2
7035.65
11
1
2
1
5315.52
25
1
2
1
5404.58
12
2
2
2
4481.00
26
1
1
2
6406.18
12
1
2
1
5895.29
26
2
1
1
8177.03
13
1
1
2
5037.00
27
2
2
2
5319.24
13
2
1
1
3448.48
27
1
2
1
5078.90
Following are the two SAS programs that perform
bioequivalence analysis on the raw data and its logtransformed data.
SAS for BE on “Untransformed” Data
SAS for BE on “Transformed” Data
Test Product 1
Test Product 2
Test Product 3
Log-Transformed data
90% CI
95% CI
90% CI
95% CI
90% CI
95% CI
Test (LS
Means)
8.745661
8.745661
4.155917
4.155917
2.391472
2.39147
2
Ref (LS Means)
8.787235
8.787235
4.180363
4.180363
2.504866
2.50486
6
rmse
0.2637
0.2637
0.14929371
0.149294
0.14743703
0.14743
7
sqrt(2/n)
0.2722
0.2722
0.2722
0.2722
0.2722
0.2722
diff_mean
-0.0416
-0.0416
-0.0244
-0.0244
-0.1134
-0.1134
df
25
25
25
25
25
25
Limit
90
95
90
95
90
95
Alpha
0.10
0.05
0.10
0.05
0.10
0.05
tvalue
1.7081
2.0595
1.7081
2.0595
1.7081
2.0595
Upper Limit
108
111
105
106
96
97
Lower Limit
85
83
91
90
83
82
References
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2. Wijnand, H.P. The determination of the absolute bioavailability for drug substances with long elimination half-lives (with
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1995
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M. Dekker
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Equivalence of Average Bioavailability”, J. of Pharmacokinetics and Biopharmaceutics, vol 15, No 6, p657-680
19. W.W. Hauck and S. Anderson, “A New Procedure for Testing Equivalence in Comparative Bioavailability and Other
References (Cont’d)
20. S.F. Francom, C.M. Metzler, Sample Sizes for Bioequivalence Studies