STATISTICAL ANALYSIS OF PHARMACOKINETIC DATA---
BIOEQUIVALENCE STUDY 
  By   Qingmin Guo
    
   
   
   
   
   
   
   
   A THESIS
 Submitted to 
Michigan State University 
in partial fulfillment of the requirements 
for the degree of 
BiostatisticsŠMaster of Science
   2015
   ABSTRACT   STATISTICAL ANALYSIS OF PHARMACOKINETIC DATA--
BIOEQUIVALENCE STUDY
  By  Qingmin Guo
   Bioequivalence (BE) studies are widely carried 
out in the pharmaceutical 
industry. The assessment 
of BE adopted by the Food and Drugs Adminis
tration (FDA) is a moment-based criterion 
evaluating log-transformed pharm
acokinetic responses such as 
Area Under the Curve (AUC), 
Maximum Concentration (
max
C), which are usually estimated from drug plasma time profiles. 
Average BE (ABE) is based solely on the co
mparison of population averages but not on the 
variances, while Population BE (P
BE) and individual BE (IBE) approaches include comparisons of 
both averages and variances. The objective of this thesis is to review the standard approaches to 
statistical analyses of pharmacokinetic data. It also covers estimation of AUC, 
max
Cand other 
pharmacokinetic (PK) pa
rameters as introduced in a Non-Compartmental Analysis (NCA) 
approach and Compartmental Models
 Analysis approach. Widely c
ited data sets from the published 
literature are used to illustrate these two approaches. They show the benefits of parameter 

estimation and subsequent statistical inference with an appropriate compartmental model, even 

though the model fitting coul
d be a little complicated. 
iii ACKNOWLEDGEMENTS
    
  I would never have been able to finish my th
esis without the guidance of my committee members 
Dr. Joseph Gardiner, Dr. Zhehui Luo and Dr. Do
rothy Pathak, help from 
friends, and support from 
my family. 
  iv TABLE OF 
CONTENTS
    
  LIST OF TABLES . . . . . . . . . . . . . .
 . . . . . . . . . . . . . . . . . . . . . . . . . . . vi LIST OF FIGURES . . . . . . . . . . . . .
 . . . . . . . . . . . . . . . . . . . . . . . . . . . vii CHAPTER 1 INTRODUCTION . . . . .
 . . . . . . . . . . . . . . . . . . . . . . . . . . .   1 1.1 Pharmacokinetics (PK) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
1 1.2 Area Under the Curve (AUC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
1 1.3 Peak Concentration (
max
C). . . . . 
. . . . . . . . . . . . . . . . . . . . . . . . . . 
2 1.4 Bioequivalence (BE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
2 1.5 Crossover Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
3  1.5.1 Treatment Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
4  1.5.2 Period Effect . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 
4  1.5.3 Carryover Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
4 1.6 Statistical Model . . . . . . . . . . . 
.. . . . . . . . . . . . . . . . . . . . . . . . . . 
5 1.7 Average Bioequivalence (ABE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
7 1.8 Population Bioequivalence (PBE) . . . . . . . . . . . . . . . . . . . . . . . . . . . 
9 1.9 Individual Bioequivalence (IBE) . . . . . . . . . . . . . . . . . . . . . . . . . . . .  10 
 CHAPTER 2 BIOEQUIVALENCE . .
 . . . . . . . . . . . . . . . . . . . . . . . . . . . .   12 2.1 ABE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
 12 2.2 Sample Size Calc
ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  18 
 2.2.1 Formula for sample size. . . .  . . . . . . . . . . . . . . . . . . . . . . . . . .  18 
 CHAPTER 3 AUC ESTIMATION . . . .
 . . . . . . . . . . . . . . . . . . . . . . . . . . .   20 3.1 Definition of PK Parameters. . . . . . 
.. . . . . . . . . . . . . . . . . . . . . . . . .  20 
 3.1.1 Elimination Rate Constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . .  20 
 3.1.2 Absorption Rate Constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  21 
 3.1.3. Half-life . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .  21 
 3.1.4. Bioavailability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  21 
 3.1.5. Volume of Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  21 
 3.1.6. Clearance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  22 
3.2 Non-Compartmental Analysis (NCA) . . . . . . . . . . . . . . . . . . . . . . . . . . 22 
3.3 One Compartment Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  24 
 3.3.1 AUC estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  26 
 3.3.2 Tmax
 and Cmax
 . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .  26 
 3.3.3 T
1/2 
(half life) estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  27 
 3.3.4 Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  28 
 3.3.5. Model for C(t) . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . .  28 
CHAPTER 4 DISCUSSION . . . . . . . .
 . . . . . . . . . . . . . . . . . . . . . . . . . . .   33 v BIBLIOGRAPHY . . . . . . . . . . . . . .
 . . . . . . . . . . . . . . . . . . . . . . . . . . .  37  vi 
 LIST OF 
TABLES     
   Table 1 
The standard 
22 crossover design . . . . . . . . . . . . . . . . . . . . . . 
. . . . . 3 
 Table 2 
Expected Cell Means for Model (1)
  . . . . . . . . . . . . . . . . . . . . .  . . . . . 
6  Table 3 
Expected Means and Effects for Model (1)
  . . . . . . . . . . . . . . . . . . . . . . 
6  Table 4 
Bioequivalence Types and Evaluation Criteria
 . . . . . . . . . . . . . . . . . . . 
9  Table 5 
Expected Cell Means for 
log(AUC)  . . . . . . . . . . . . . . . . . . . . .  . . . . . 
 13  Table 6 
Expected Means and Effects for 
log(AUC
)  . . . . . . . . . . . . . . . . . . . . . . 
 13  Table 7 
TTEST output for log(AUC)  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
 14  Table 8 
GLIMMIX output for 
log
(AUC)  . . . . . . . . . . . . . . . . . . . . .  . . . . . 
 15  Table 9 
Expected Cell Means for 
log (Cmax
)  . . . . . . . . . . . . . . . . . . . . .  . . . . . 
 15  Table 10 
Expected Means and Effects for 
log (Cmax
)  . . . . . . . . . . . . . . . . . . . . . . 
 15  Table 11 
TTEST output for log (Cmax
) . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . 
 17  Table 12 
GLIMMIX output for 
log
 (Cmax
)  . . . . . . . . . . . . . . . . . . . . .  . . . . . 
 17  Table 13 
Comparison of AUC by NCA and compartment model based estimates . .. . . . . 
 30  Table 14 
Comparison of Cmax
 by NCA and compartment model based estimates . . . . . . . 
 31  Table 15 
Comparison of Tmax
 by NCA and compartment model based estimates . . . . . . . 
 31 vii 
LIST OF 
FIGURES
     
 Figure 1  Schema of plasma concentration-vs.-time curve
  . . . . . . . . 
 . . . . . . . .   2 
Figure 2  Profiles over 
Treatment A and B for 
log(AUC) in two periods
  . . . . . . . . 13 
Figure 3  Treatment A vs. B Agreement of 
log(AUC) in two periods
  . . . . 
 . . . . . . . . 14 
Figure 4  Profiles over 
Treatment A and B for 
log(Cmax
) in two
 periods  . . . . . . . . 16 
Figure 5  Treatment A vs. B Agreement of 
log(Cmax
) in two periods  . . . . 
 . . . . . . . . 16 
Figure 6  One compartmental 
model with i.v. administration
  . . . . . . . . .
. . . . . . . . 24 
Figure 7  One compartmental model with extravascular administration
  . . . . . . . . 25 
Figure 8  Individual concentration profiles by NCA
  . . . . . . . . . . . . 
 . . . . . . . . 30 
1 CHAPTER 1 INTRODUCTION 
 
 1.1   Pharmacokinetics (PK) 
PK is the study of the movement of drugs in 
the body, involving the processes of absorption, 
distribution, metabolism, and excretion (ADME), incl
uding the rate and extent of each of these 
processes. The science of PK concerns on how th
e body converts an active drug molecule into 
metabolites, the time course of drugs in the body, and what the body does to the drug. PK data 

are often derived from blood (serum or plasma) sa
mples in small to medium-size datasets of 
individuals over time.  

From a PK profile, pharmacokinetic parameters can be estimated such as Area Under the Curve 
(AUC), Maximum Concentration (
max
C), Time to Maximum Concentration (
max
T), Half Life  
(1/2
T), Bioavailability, Clearance, Volume of Dist
ribution, etc. These PK parameters are very 
useful in optimization of the dosage form and dose interval. 
 1.2   Area Under the Curve (AUC) 
AUC 
has units of concentration
×time (e.g., 
/mghrL
×), is a measure of the total systemic 
exposure of a drug integrated over time. AUC is usually estimated from concentration-time data. 
There are two major approaches to estimation of AUC: one is Non-Compartmental Analysis 

(NCA), which calculates the AUC following the tr
apezoidal rule by adding up the area under the 
curve between consecutive time points, the requi
rement of the Food and Drugs Administration 
(FDA)
[1][2] for AUC estimation; the other is compartmental modeling analysis, which will be 
discussed in chapter 3.  
2 1.3   Peak Concentration (
maxC) max
Crefers to the maximum (or peak) serum concen
tration that a drug achieves in a specified 
compartment or test area of the body after the 
drug has been administrated and prior to the 
administration of a second dose, i.e., in brief, 
max
Cis the maximum concentration observed. 
 Figure 1  
Schema of plasma concentration-vs.-time curve. 
 In bioequivalence studies, the key pharmacoki
netic parameters are log-transformed AUC 
and max
C.[3] Figure 1 shows the schema of AUC, 
max
Cand max
T in the plasma concentration -vs.-
time curve after a single oral drug dose, cited from 
Atkinson AJ, et al. 2007
[4] with a little 
modification. max
Tis Time to Peak Concentration, the term
 used in pharmacokinetics to describe 
the time at which the 
max
C is observed. 
 1.4   Bioequivalence (BE) BE studies are widely carried out in the pha
rmaceutical industry. The US Food and Drugs 
Administration -
Code of Federal Regulations (
FDA-
CFR) 
 definition
[5] of BE is that the absence 
3 of a significant difference in the rate and extent to which the active ingredient in pharmaceutical 
equivalents or pharmaceutical alte
rnatives becomes available at the site of drug action when 
administered at the same molar 
dose under similar conditions in an 
appropriately designed study.
 Although it is seen sometimes that measures used
 in BE study are pharmacological or clinical 
end-points, the most sensitive measures used in BE studies
[6] are drug concentrations in the 
blood. From subject-level time and concentration data, subject-level AUC and 
max
Ccan be 
estimated, which form the BE outcome data for statistical analyses. 
If the drugs' plasma 
concentration curves are superimposable, then th
ese drugs are considered as bioequivalent in 
extent and rate of absorption.
 The FDA guidelines for BE studies recommend a minimum of 12 samples collected over time 
following drug administration with an additional sample prior to dosing.
[2] These investigations on BE are best made through randomized clinical
 trials, and BE of tw
o drugs is assessed by 
analysis of logarithmic transformed AUC and 
max
C typically obtained from a crossover design.  
 1.5   Crossover Design
 Crossover Design is probably the most comm
only used statistical design for comparing 
bioequivalence between two formulations of a 
drug. We shall refer to a two-sequence, two-
period, crossover design as the standard 2
2 crossover design, also called AB|BA design.  
Table 1  The standard 22 crossover design 
Crossover Designs for Two Fo
rmulations Period 1 Period 2 
Sequence AB = 1 A B 
Sequence BA = 2 B A 
A standard 2 2×crossover study will generate paired outcomes 
12(,)
YY in two sequences of 
subjects: (a) In sequence AB subj
ects receive drug A in period 1 and are crossed over to drug B 
4 in period 2; (b) In sequence BA subjects receive drug B in period 1 and are crossed over to drug 
A in period 2, as shown in Table 1. The dosing 
periods are separated by a washout period of 
sufficient length for the drug received in the first period to be completely metabolized or 

excreted from the body. Now, let's discuss here
 treatment effect, period effect and carry-over 
effect. 
 1.5.1  Treatment Effect. 
The objective of a cross-over trial is to focus attention on within-
patient treatment differences, the difference betwee
n different measurements
 in the same subject, also called within-subject difference. The difference between these measurements removes any 

component that is related to the differences betw
een the subjects, which is called ‚subject-effect™, 
from the comparison. 

 
1.5.2  Period Effect. 
The within-subject difference could al
so be thought of as a comparison 
between the two treatment periods, which is 
the reason why usually one group of subjects 
received the treatments in the order AB and the 
other group received the treatments in the order 
BA. 

 
1.5.3  Carryover Effect. 
A carryover effect is defined as effect of the treatment from the 
previous time period on the response at the curren
t time period. The presence of carryover is an 
empirical matter.
[7] It depends on the design, the setting, the treatment, and the response. The 
washout periods are usually included in the desi
gn, to allow the active effects of a treatment 
given in one period to be washed out of the body 
before each subject begins the next period of 
treatment. The disadvantage of the 
22×crossover trial is that several important effects, such as 
5 carryover effect and interaction effect are aliased with each other. Therefore, the carryover effect 
cannot be removed just by randomization alone in 
2 2× crossover design. Washout period has to 
be clarified, usually 5 Half-life time (
1/2 
T) is necessary. 
The objective of crossover design is to estimate the treatment effect, the expected difference in 
mean of logarithm transformed AUC and 
max
C of drug A versus drug B obtained from each 
subject. In each sequence, Treatment A and Tr
eatment B produce a difference measure, provided 
that the period effect can be assumed to be cons
tant, the information from both sequences can be 
combined to obtain the estimate of expected difference. Since the difference measure is within 

one subject, the difference removes 
any ‚subject-effect™ from the comparison. 
 1.6   Statistical Model Consider a statistical model without considering carryover effect used by Byron Jones and  
Michael G. Kenward.
[7] Let
ijk
Y=response in 
k-th patient, in 
j-th period for i-th treatment. There 
is implied ‚nesting™ of treatment in period.
 The model is: 
      ijkijkijk
Yµ
=++++
    
(1) where 
µ is effect of an overall mean;
   
iis effect of 
i-th treatment effect, 
i=1, 2, ..., t;  jis effect of the 
j-th period effect, 
k=1, 2, ..., p;   
k  is random effect associated with the 
k-th subject.  6 ijk
is random error associated with the 
k-th subject who received the 
i-th treatment in the 
j-th 
period, k=1, 2, ..., ni. ()2~0,
kN, 2~(0,)
ijk
N, k and ijk
are independent. 
()()222
  ijkkijk
VarYVar

=+=+=
, and 2,')
( ijkijk
CovYY
= for all
jj. In this random subject-effects model, 
2  is the inter-subject variability and 
2 is the within-
subject variability. 
In Table 2, there are only four sample means 
111221
.,.,.,
yyy
and 
22.y. 11yrepresents the mean 
of samples of treatme
nt 1 and period 1, ..., 
21.yrepresents the mean of samples of treatment 2 
and period 1; Table 3 lists the m
ean of treatment and period, peri
od effect and treatment effect 
with regard to this 
22× crossover design.  
Table 2  Expected Cell Means for Model (1) 
Sequence Period 
1 2 1 1111
.yµ=++
 2222
.yµ=++
 2 2121
.yµ=++
 1212
.yµ=++
  Table 3  Expected Means and Effects for Model (1) 
Mean of Treatment A 
1112112
½(..)=½½
yyµ
++++
 Mean of Treatment B 
2122212
½(..)=½½
yyµ
++++
 Mean of Period 1 
1121112
½(..)½½
yyµ
+=+++
 Mean of Period 2 
1222212
½(..)½½
yyµ
+=+++
 Treatment (A-B) Effect
 12 Period (1-2) Effect 
12  7 It is known from empirical studies that 
after logarithmic tr
ansformation, AUC and 
max
C are normally distributed or may be 
assumed to be approximately no
rmally distributed. The core 
modeling component of SAS proc GLIMMIX ca
n be illustrated for both fixed and random 
subject-effects models.
[7]
 Suppose there are variates including
 subject, period, direct treatment 
and  response variables in the da
taset, model fitting and inference 
for fixed subject-effect models 
follow conventional ordinary least squares (O
LS) procedures and for random subject-effect 
models Restricted Maximum Like
lihood (REML) analyzes are used.
[7] The benefit of crossover design is that each subj
ect serves as their own control, and statistical 
efficiencies are gained with re
spect to power and precision. Alt
hough crossover design has great 
advantages, it also brings a pot
ential disadvantage: (1) Carryove
r may be confounded with direct 
treatment effects.  (2) There are at least 2 pe
riods, patients may withdraw from the trial, or 
become "lost to follow-up". 
 Special consideration is needed while doing statistical analysis for the crossover design. The 

typical method of Null Hypothesis Significance
 Testing (NHST) is designed to assess the 
evidence against the null hypothesis. The null hypoth
esis is rejected if the observed p-value is 
less than the stated significance level 
; if not rejected, the null hypothesis will be retained. 
However, equivalence is not concluded just b
ecause we do not reject null. Here, the Two One-
Sided Tests (TOST) is applied 
to assessing bioequivalence. 
 1.7   Average Bioequivalence (ABE) 
ABE is a conventional method for the BE study, 
which solely compares the population averages 
of a BE measure of interest but not the vari
ances of the measures for the T and R products, 
8 following the FDA 1992 guidance on Statistical
 Procedures for Bioequivalence Studies.
[3][8] TOST has been used to determine whether the ratio of the logarithm transformed averages of the 
measures for the test and reference products were comparable.
[9]   The two null hypotheses of TOST: (1) the mean difference is larger than the upper value of the 
BE limit 
; and (2) the mean difference is below the lower bound of the BE limit -
, versus the alternative hypothesis of the difference falls within the range of the BE limit. 
         
 
where T is Test
, and R is Reference
. BE is established at significance level of 
 if a t-interval of confidence 
()12100%
× is contained in the interval (,)

, which is called Westlake™s Confidence Interval.
[10]
 Therefore, 
to establish BE at significant level of 
0.05
=, a 90% confidence interval should fall within the 
BE limit
(,)

. For PK measures after 
logarithmic tr
ansformation, 
ln 1.25
=,ln 1.25
=,[3] while for PK measures without logarithmic 
transformation, the BE 
limit is a little different.  For the case of no logari
thmic transformation, let
TR/Dµµ= be the ratio of the averages of the 
measures for the Test and Reference products.  
01T02
: /  , : /     :  /  
µµµµµµ<<RUTRLaLTRU
HHH
 4/50.8
L==, and 5/4 1.25
U==on /µµ=TRD that define the region of equivalence.
[3] To establish BE at significant level of 
0.05
=, a 90% confidence interval should fall within the 
BE limit 
(,)
LU. 0102
:,:
TRTR
HHµµµµ

  
:aTR
Hµµ<<9 ABE only compares the population averages of 
a BE measure, assesses no comparison between 
variances of the measure for T and R products, therefore, there are some limitations on ABE.   
FDA (2001)
[3] recommends Population Bioequivalence 
(PBE) and Individual Bioequivalence 
(IBE), which include comparisons of both averages
 and variances of the measure. In Table 4, the 
evaluation criteria are listed for ABE, PBE and IBE. 
Table 4  Bioequivalence Types a
nd Evaluation Criteria 
Bioequivalence/Parameters
 Evaluation  Criteria 
ABE 
Population averages(
TR;µµ) 22TR(  )
Aµµ PBE 
Population averages (
TR;µµ) Total variances (
22;TTTR
) 2222 
TRP
[(  ) ()]/
TTTRTR
µµ
+  or 2222 
TR0P
[(  ) ()]/
TTTRT
µµ
+ IBE 
Population averages (
TR;µµ) Intra-subject variances (
22 ,WTWR
) Subject-by-formula
tion interaction 
variance (
2D) 22222 
TRI
[(  )()]/
DWTWRWR
µµ
++or  22222 
TR0I
[(  )()]/
DWTWRW
µµ
++ ,,API
: specified bounds by the FDA;
20 T, 20W:specified threshold value by the FDA. 
 1.8   Population Bioequivalence (PBE) 
PBE approach uses both the mean and  the variance of log(AUC) and 
max
log(C)
, to assess total 
variability of the measure in the population.
[3][11] For PBE, the parameter of interest is: 
222
2222
20()()TRTTTR
TRPBE
TRTTTR
Tµµµµ+=+  when220TRT
>(Reference-scaled criterion)
 when220TRT
(Constant-scaled criterion)
 10 
In the equation, 
TRµµ is still the mean difference between the test and reference.
2TT is the 
total variance of test, and 
2TR is the total variance of reference.
20 T is the FDA specified 
threshold value, currently the values recommended by the FDA is 
200.04
T=.[11] When the 
total variance of Reference 
2TR is greater than the FDA specified threshold value 
20 T, Reference-scaled criterion is to 
be applied. Otherwise, Constant-s
caled criterion is to be used. 
Currently FDA recommended value for 
P is 1.7448,[11] if 
PBEp
<, and ABE is concluded, 
then PBE is also concluded. 
 1.9   Individual Bioequivalence (IBE)
 IBE approach uses the means and variances 
of T and R, and the subject-by-formulation 
interaction to assess within-subject variability for the T and 
R products, as well as the subject-by-
formulation interaction. 
 when220WRW
>( Reference-scaled criterion) 
 when220WRW
(Constant-scaled criterion) 
 Here, 2WT is within-subject variance for test drug, and 
2WR is within- subject variance for 
reference drug. 
2BT is between-subject variance for test drug, 
2BR is between-subject variance 
for reference drug. 
()22 2 (1)
=+DBTBRBTBR
, assesses the subject-by-
formulation interaction.  is the correlation coefficient be
tween individual average test and 
reference formulation, they both c
ontribute to the IBE determination.
[3][11] 2222
22222
20()()TRDWTWR
WRIBE
TRDWTWR
Wµµ
µµ
++=++11 
20W is specified threshold value, currently the FDA recommended 
200.04
W=.When the 
within variance of reference 
220 WRW
>, Reference-scaled crite
rion is to be applied. Otherwise, Constant-scaled criterion is 
to be used. FDA recommended value for 
I is 2.4948[11], if IBEI
<, and ABE is concluded, then
 IBE is also concluded. 
The two-period two-treatment crossover design is
 the simplest prototype, which is not enough 
for PBE and IBE. When additional periods a
nd/or treatments are considered, the possible 
configurations would increase. Some examples of
 a three-period two-treatment crossover design 
are (i) Sequences ABA and BAB; (ii) Sequences
 AAB, ABA and BAA; (iii) Sequences ABB and 
BAA.[3] Just as in the AB|BA design, we note that the treatment difference can be estimated in 
each period from any of these three designs. The 
benefit of having add
itional periods and/or treatments contributes to detecting if there is car
ryover effect in crossover design. Moreover, the 
IBE can be estimated with high order crossover design in addition to ABE and PBE. 

  12 
CHAPTER 2 BIOEQUIVALENCE  
   A data set called "pkdata" from STATA documentation
[12] is used here for our illustration. The 
data comprises two concentrations CONCA, C
ONCB in the same n=16 subjects assesses at 13 
time points, including t=0. At baseline the con
centration is zero. Eight patients are randomly 
assigned to sequence 1, which means they take the 
drug A in the period 1, and then take the drug 
B in the period 2; the other 8 patients are assigned
 to sequence 2, take the drug B in the period 1, 
and then take the drug A in the period 2. Assume 
the Drug B is the reference drug (R), and Drug 
A is the test drug (T). We will calculate AUC and 
max
C by Non Compartmental Analysis 
approach. Take the logarithmic
 transformation of AUC and 
max
C as the variables for the ABE 
study. 
 2.1   ABE Consider the parameter AUC. Let 
BRµµ= be the reference group population mean, 
  TAµµ= be the test group population mean. Let 
 DTRµµµ= be the treatment difference, 
ln1.25
= be the lower bound, and 
ln1.25
=(=0.2231) be the upper bounds on 
 DTRµµµ= that define the region of equivalence. ABE involve
s the calculation of a 90% CI for 
 DTRµµµ=, the difference in the means of log-transformed AUC. The ABE will be concluded based on the 
calculated 90% confidence limits falling within 
0.2231  0.2231
TRµµ
. First, the expected 
cell means for 
()logAUC
 in Table 5 are listed following notations used in Table 2.  
13 
  Table 5  Expected Cell Means for 
()logAUC
 Sequence Period 
1 2 1 11.5.0069
y= 22.4.9270
y= 2 21.5.0077
y= 12.4.8740
y=  Similarly, the expected means and effects for 
()logAUC
in Table 6 are listed following notations 
used in Table 3.  
Table 6  Expected Means and Effects for 
()logAUC
 Mean of Treatment T 
1112
½(..)=4.9404
+yy Mean of Treatment R 
2122
½(..)=4.9673
+yy Mean of Period 1 
1121
½(..)5.0073
yy+= Mean of Period 2 
1222
½(..)4.9005
yy+= Treatment (T-R)  Effect -0.0269
 Period (1-2) Effect 0.1067
   Figure 2  
Profiles over treatment A and B for 
()logAUC
 in two periods. 
 The objective of this study with cross-over design is to focus attention on within-subject 
treatment differences. Figure 2 s
hows profiles over treatment for 
crossover designs. The subject-
14 
profiles in Figure 2 are plotted for each sequenc
e the change in each subject™s response over the 
two treatment periods, which 
show no strong treatment effect or period effect.  
 Figure 3  
Treatment A vs. B Agreement of 
()logAUC
 in two periods. The treatment agreement in Figure 3 is plotte
d for the response associated with the second 
treatment against the response associated with 
the first treatment.  The figure indicates the 
strength of the treatment effect is small, and the treatment effect A-B is negative. The spread of 

points within sequence AB being wider indicates the bigger between-subject variability.  
Table 7  TTEST output for 
()logAUC
 Treatment Method Mean Lower 
Bound 
  90% CL Mean   Upper 
Bound 
Assessment Diff (1-2) Pooled -0.0269 -0.2231 < -0.1462 0.0924 < 0.2231 Equivalent 
Diff (1-2) Satterthwaite -0.0269 
-0.2231 < -0.1503 0.0965 < 0.2231 Equivalent 
 For crossover design, TOST option of PROC 
TTEST requests Schuirman
™s TOST equivalence 
test, with the option of specifying the equivale
nce bounds. After log - tran
sformation of PK data, 
given the BE limit is (-0.2231, 0.2231), the assessment
 of BE is finally shown in Table 7. 
Exactly the same calculations can be carried out in PROC GLIMMIX with LSMEANS statement 

(1) compute least squares (LS) means of fixed effects (2) compute the 90% CI for LS-mean 
15 
difference, and (3) see if 90% CI falls in the stated BE limits (-
, ).  The BE limits are (-0.2231, 
0.2231) in ABE evaluation. 
Table 8  GLIMMIX output for 
()logAUC
 Estimates 
Label Estimate Standard 
Error 
DF t Value Pr> |t| Lower Upper 
T-R -0.02690 0.06774 14 -0.40 0.6973 -0.1462 0.09241 
 In Table 8, the PROC GLIMMIX output shows 
that the 90% CI (-0.1462, 0.09241) falls within 
the range (-0.2231, 0.2231), the ABE is concluded for 
()logAUC
 at significance level 
0.05
= Table 9  Expected Cell Means for 
()max
logC
 Sequence Period 
1 2 1 11.1.9763
y= 22.2.5126
y= 2 21.2.0101
y= 12.2.4469
y= Similarly, the expected cell means for 
()max
logC
 in Table 9 are listed following notations used 
in Table 2.  ()max
logC
 of Drug T and Drug R are es
timated by PROC TTEST and PROC 
GLIMMIX procedures. 
Table 10  Expected Means and Effects for 
()max
logC
 Mean of Treatment T 
1112
½(..)=2.2116
+yy Mean of Treatment R 
2122
½(..)=2.2614
+yy Mean of Period 1 
1121
½(..)1.9932
yy+= Mean of Period 2 
1222
½(..)2.4798
yy+= Treatment (T-R) Effect -0.0498
 Period (1-2) Effect -0.4866
  Similarly, the expected means and effects for 
()max
logC
 in Table 10 are listed following 
notations used in Table 3.  16 
Figure 4 shows profiles over treatment for crossove
r designs. The subject-profiles in Figure 4 are 
plotted for each sequence the change in each 
subject™s response over the two treatment periods, 
which show no strong treatment effect but maybe a period effect.  
 Figure 4  
Profiles over treatment A and B for 
()max
logC
in two periods. 
   Figure 5  
Treatment A vs. B Agreement of 
()max
logC
in two periods.  The treatment agreement in Figure 5 is plotte
d for the response associated with the second 
treatment against the response associated with 
the first treatment.  The figure indicates the 
17 
strength of the treatment effect is small, and the treatment effect of A-B is negative. Substantial 
location differences between the two sequences indicate a strong period effect. 
Table 11  TTEST output for 
()max
logC
Treatment Method Mean Lower 
Bound 
  90% CL Mean   Upper 
Bound 
Assessment Diff (1-2) Pooled -0.0498 -0.2231 < -0.1280 0.0285 < 0.2231 Equivalent 
Diff (1-2) Satterthwaite -0.0498 
-0.2231 < -0.1285 0.0290 < 0.2231 Equivalent 
 For crossover design, TOST option of PROC 
TTEST requests Schuirman
™s TOST equivalence 
test, with the option of specifying the equiva
lence bounds. After logarithmic transformation of 
PK data, given the BE limit is (-0.2231, 0.2231), the 
assessment of bioequivalence is finally 
shown in Table 11. 
Table 12  GLIMMIX output for 
()max
logC
 Estimates 
Label Estimate Standard 
Error 
DF t Value Pr> |t| Lower Upper 
T-R -0.04976 0.04379 14 -1.14 0.2749 -0.1269 0.02737 
 Similarly use PROC GLIMMIX with LSMEANS statement: (1) compute LS means of fixed 

effects, (2) compute the 90% CI for LS-mean difference, and (3) see if 90% CI falls in the stated 
BE limits (-
, ), here the BE limits are (-0.2231, 0.2231). 
In Table 12, PROC GLIMMIX output shows that
 the 90% CI (-0.1269, 0.02737) falls within the 
range (-0.2231, 0.2231), the ABE is concluded for 
()max
logC
 at significance level 
0.05
=.  Therefore, summarizing the two primary response variables for BE study, 
()logAUC
 and  ()max
logC
, the ABE is concluded for T and R at the significance level 
0.05
=.  18 
2.2   Sample Size Calculation
 Sample size calculation for crossover design in bioavailability and bioequivalence study is an 
essential question, which 
establishes bioequivalence within 
meaningful limits in the case of 
logarithm transformed AUC and
max
C.[13] A minimum number of 12 evaluable subjects should 
be included in any BE study according to FDA guidelines.
[3] Based on Schuirmann™s TOST procedure fo
r interval hypothesis, use the data 
DTRµµµ= as the expected mean differ
ence after logarithmic tran
sformation. If the 100(1-2
)% CI
(),22,22
‹‹‹‹
,DnDn
ttµµ+ of the mean
µDis entirely within the BE limit (-ln1.25, 
ln1.25), then 0H is rejected at significance level 
 and no drug-drug inter
action is concluded; 
otherwise, 0Hfails to be rejected.  
The type-I error 
of the TOST procedure is often set 
as 5%.Total number of subjects should 
provide adequate power for BE demonstration, 
and the adequate power means at least 80% 
power to detect a 20% difference in products™ BE
. In practice the power usually is about 80% - 
90%.  
 2.2.1  Formula for sample size
. Let n = number of subjects required per sequence. 
is the 
significant level, 
 is the type 
error, 
CV= coefficient of variation, 
/100%
eRCVµ=, = the BE limit, 
 is the expected difference compared to expected mean of Reference ,
100TRRµµµ=; ‹is the intra-subject standard deviati
on.  Assuming a normal distribution of 
logarithm transformed PK data (AUC, 
max
C), for 0=, n can be estimated
[13][14] by
 19 
22,22/2,22
‹2[][/]
nnntt
+ The approximate sample size calculation for the TOST tests for 
 > 0, the equation is:  
22,22,22
‹2[][/()]
+nnntt
 The n on the right hand side is unknown (in the degr
ees of freedom). Start with an initial value 
n0, the n1 is calculated. The calculation is iterative until n is almost unchangeable.  
As an illustration, consider the STATA dataset de
scribed at the beginning of this chapter. 
Let 
=0.05, =0.2; For 
()logAUC
, 0.2231
=, ‹=0.1355, use the initial value 
08n=22/22222[(22)(22)]
[1.7611.345]
‹220.13558
0.2231
tntn
n++
 So n per group = 8, and total n = 16.  
The n per group =8 is also estimated from PROC POWER.  The example is for ABE study. The 

number of subjects for PBE or IBE studies can 
be estimated by simulation according to the FDA 
guideline,
[3] which will not be discussed here.
  20 
CHAPTER 3 AUC ESTIMATION 
 
 There are two representative examples of
 one compartmental pha
rmacokinetic models:
[15][16] (A) One compartmental model with i.v. adminis
tration; (B) One compartmental model with 
extravascular administration. 

One compartmental model with i.v. dosing means 
administering a dose of drug over a very short 
time period, there is no absorption rate constant (
ak) considered. A one compartmental model 
with extravascular administration means absorpti
on phase is involved in the whole process. As 
shown in Chapter 1, the max
C and  max
T are simple measures for summarizing the absorption 
process. In one compartme
nt model, assessment of max
T depends on the value of both 
elimination rate constant (
ek) and absorption rate constant (
ak). ()maxlog/
aeaekkTkk=. Let us look into the concepts of 
ek and ak, and how they appear in a pharmacokinetic model. 
 3.1   Definition of PK Parameters  
3.1.1  Elimination Rate Constant 
(ek, units are h-1) describes the rate of decrease in 
concentration per unit time, usua
lly the time unit is hour. It is estimated from the log-linear 
terminal part of the concentration-time curve, 
   
ekslope
=. 3.1.2  Absorption Rate Constant (ak, units are h-1) is the rate of absorption of a drug absorbed 
from its site of application according to assumption of first-order kinetics, which is for a drug 
21 
administered by a route (for exampl
e, oral) other than the intrave
nous. The first-order differential 
equation that governs the drug amount remained
()Xt:()()adXt
kXt
dt=. (0)
DX= is the actual 
dose (mg) that is available to the body for kinetic
s, whereas the oral dose is given in mg/kg.  
Next we define the four
 most useful PK parameters charact
erizing the in vivo disposition of a 
drug.
[15][16] 
 3.1.3  Half-life 
(1/2
T) is given by
()1/2
log2/k
eT=, that is, the time from 
max
T to reach one-
half of the maximum concentration 
max
C.   3.1.4  Bioavailability 
(F, has no unit) is described as the fraction of the extravascular dose of 
the administered drug that reaches the absorption de
pot. If the drug is injected intravenously, it is 
assumed that bioavailability
 F =100%. B
ioavailability generally decreases
 when a 
medication
 is administered via other routes
 (such as orally), such as 
oraliv
ivoral
AUCDose
FAUCDose
×=×, F is often measured by quantifying the "AUC". 
 3.1.5  Volume of Distribution 
(V, the units of volume, e.g., L
) or apparent volume of 
distribution is a pharmacological,
 theoretical volume that the to
tal amount of administered drug 
would have to occupy (if it were uniformly distri
buted), to provide the same concentration as it 
currently is in blood plasma. 
There are two quantities: the concentration 
()Ct in plasma and the amount of drug 
()Xt in tissue. It is assumed that 
()/()
XtCtV
= is constant.   22 
3.1.6  Clearance (CL, the units are volume per time, e.g., L/hr) is called the drug clearance rate, 
can be defined as the volume of plasma which is completely cleared of drug per unit time. 
CL is calculated using the dose administered divi
ded by the subsequent measured AUC,
()(0)
 eOraldoseFX
CLVk
AUCAUC
×===×
, where F is the bioavailability. Unless we have 
information on F, the parameter 
CL/F is only identified, this is called the 
apparent clearance.  With the drug plasma concentration-time profile, AUC can be estimated by NCA, and also by 

compartmental modeling analysis.  

 3.2   Non-Compartmental Analysis (NCA) 
Based on the theory of statistical moments, the mo
ments of a function are used in the analysis of 
pharmacokinetic data.
[17]
 Suppose drug concentration 
()Ct is a real-valued function defined on 
the interval [0, 
); the zero
th moment of ()Ct is 0S: 00()SCtdtAUC
==, fithe area under the 
curve from time zero to infinityfl; and the first moment of 
()Ct is  1S: 10()StCtdtAUMC
==, fithe area under the first moment curvefl, is 
the area under the curve of concentration-time 
versus time curve from time zero to infinity, 
AUMC
 can be used to estimate some other PK 
parameters. 

Non-compartmental Analysis estimate AUC using the trapezoidal rule without making any 

assumption concerning the number of compar
tments. Following the trapezoidal rule, 
concentration-time curve is considered as a series of trapezoids and the AUC estimate is the total 

area of all the trapezoids. 
23 
For non-compartment model, 
(0)
 tAUC is AUC from 0 h to the last quantifiable concentration 
to be calculated; (0)(0)()
ttAUCAUCAUC

=+, represents the total drug exposure over 
time.
(0)
AUC
requires extrapolation of th
e elimination-phase curve beyond the last measurable 
plasma concentration. The ex
trapolation of AUC from 
t to infinity requires several assumptions: 
(1) At low concentrations, drug usually declines
 in mono exponential fashion; (2) The terminal 
elimination rate constant does not change over time or with different concentrations of 

circulating drug; (3) other processes such as ab
sorption and distribution do not play a significant 
role in the terminal phase of 
the pharmacokinetic profile.  These 
assumptions usually are valid in 
almost all PK applications. Therefore, 
(0)
AUC
 can be calculated as
(0)(0)
 C/
tlaste
AUCAUCk

=+, where last
Cis the last observed quantifiable concentration 
and ek is the terminal pha
se rate constant, 
ekslope
=(units: h
-1). When a regression line is 
fitted to terminal phase data points on log-scale, then elimination half-life 
1/2
 log(2)/ 
eTk= can be estimated. With NCA, the observed 
max
C and  max
T are obtained directly from the data 
without interpolation. Non-compartmental analysis allows a simple estimation of AUC. It basically summarizes the 

concentration-time pr
ofile without modeling assumpti
ons. However, non-compartmental 
methods are unable to visualize or
 predict plasma concentration
 time profile for other dosing 
regimens. It assumes the kinetics to be linear and stationary (i.e., time
independent) for simple applications. In more sophisticated analyses of PK data, the one compartment model or multi-
compartment analysis with nonlinear mixed effects models (NLMEM) are increasingly used in 
drug development.
[18] 
24 
We will use the widely cited example of drug theophylline,
[19][20][21] which has serum 
concentrations  measured at 11 time points over 
a 25 hour period in 12 subjects to illustrate the 
NCA method and compare the results with that of a one compartment model.
  3.3   One Compartment Model 
Compartmental modelin
g in pharmacokinetics 
estimate the concentration- time curve using 
kinetic models that depend on the 
rate of drug distribution to 
the different parts of the body.
[15] In a one compartment model, the drug is consid
ered to be distributed instantaneously into 
all 
parts of the body
. The simplest case, 
if i.v. drug
 is received, it 
instantaneously equilibrates within 
the compartment and is eliminated at a constant rate 
ek. For a 
concentration measure 
()Ct, first-
order kinetics is assumed: 
()()edCt
kCt
dt= where 0ek> is the elimination rate constant. We 
get 
()(0)exp()
eCtCkt
= where C(0) is the initial concentr
ation. The elimination rate is h
-1, per hour. (0)(0)/
CXV
= is the initial amount of dose in mg per unit volume in L, where 
V is the 
volume distribution.  
 
 Figure 6  
One compartmental model with i.v. administration. 
 
  
 
 i.v.
 ek Central 
Compartment 
25 
    Figure 7  
One compartmental model with extravascular administration. 
 For extravascular (such as oral) administration, th
e body receives the drug and is absorbed at 
constant rate ak proportional to the amount of drug 
available for absorption. The drug 
instantaneously equilibrates within the compartment and is eliminated at a constant rate
ek. The first-order differential equations that govern the amounts 
1()Xt:11()()adXt
kXt
dt=, and 2()Xt: 212()()()
aedXt
kXtkXt
dt=.The initial amounts are 1(0)
XD= and 2(0)0
X=; at time 
t, the drug amount in the absorption depot is: 
11()(0)exp()
aXtXkt
=, as shown in Figure 6; and the 
drug amount in the central compartment is 
()12(0)
()exp()exp()
aeaaekXXtktkt
kk=
, as shown in Figure 7. Note that this 
equation makes sense only when 
aekk>. The focus is on the equation 
of central compartment, which can be ex
pressed as the con
centration equation: 
()()exp()exp()
()aeaaekDCtktkt
Vkk
=
. From the equations 
/ekCLV
= and  (0)()
XOraldoseF
=×, an operational expression of drug 
concentration in the central compartment at time 
t is  ()()()exp()exp()
()aeeaaekkoraldoseF
Ctktkt
CLkk
×=
. input 
ak ek Absorption depot, 1()Xt Central  compartment, 
2()Xt 26 
The unknown parameters are 
,ak ,ek CL that must be estimated from observed concentrations
{():0,1,,}
Cttm
=–in individuals over a grid of time points.  
 3.3.1  AUC estimation. 
From the equation of the drug co
ncentration, the AUC is defined as
0()AUCCtdt
= and also denoted by 
(0)
AUC
. From the formula for
()Ct we get
 ()0()exp()exp()
()()11()
.()aeeaaeaeaeea
kkoraldoseF
AUCktktdt
CLkk
kkoraldoseF
oraldoseF
CLkkkkCL
×=
××== Only if the underlying pharmacokinetic model is identified, can the parameters be accurately 
estimated, otherwise this method of 
estimation is not to be recommended.[22]  3.3.2  max
T and max
C estimation.
. Since 
max
T is the time to maximum concentration we 
obtain the maximum value of 
()Ct by solving
()0dCt
dt= and showing that the unique solution is 
indeed the maximum value.
 ()()()exp()exp()
()aeeeaa
aekkoraldoseF
dCt
kktkkt
dtCLkk
×=+. The derivative is a continuous function; at 
t=0 it is positive; as 
t  +, the derivative approaches 
zero. When
aekk>, the solution 
max
T is given by 
()maxlog/
aeaekkTkk=.  To obtain max
C: 27 
()maxmaxmax
()()exp()exp()
()aeeaaekkoraldoseF
CTkTkT
CLkk
×=
. Although we defined 
0()AUCCtdt
=, another quantity of interest is  
max
max()(0,)
0TAUCCtdt
T=. Using the formula 
max
max
01exp()
exp()
TaaakTktdt
k= and repeating the previous calculation 
gives 
()max
(0,)
max0maxmax
max()exp()exp()
()()exp()exp()
11()()exp()exp
()()TaeTea
aeaeea
aeeaea
aee
aee
kkoraldoseF
AUCktktdt
CLkk
kkoraldoseFkTkT
CLkkkkkk
kkoraldoseFkT
oraldoseF
CLCLkkk
×=
×=
××=max
()aakTk The first term on the right hand side is 
(0,)
AUC
 that was calculated previously. 
 3.3.3  1/2
T (half life) estimation
. The half-life 1/2
T is the time from 
max
T to reach one-half of 
the maximum concentration 
max
C. Initially at
 t=
0 the concentration is zero. With the passage of 
time the concentration 
()Ct increases to a peak 
max
C at time 
max
T and then 
()Ct declines to 
zero asymptotically. 
The relationship between elimination rate constant (
ek) and half-life (
1/2
T) is: 
1/2
log2
ekt=. The half-life (
1/2
T) is determined 
by clearance (
CL) and volume of distribution
 (V): 
28 
1/2
log2
VTCL×=. An objective of PK studies is to obtain estimates of parameters from observations of 
concentrations 
{():0,1,,}
Cttm
=– in individuals over a grid of time points. The 
parameterization may allow for some parameters
 to be individual-specific which makes them 
random effects instead of pure constants.  

 3.3.4  Application. Take the widely cited example of drug theophylline,
[19][20][21]
 serum concentrations  measured at 11 time points ove
r a 25 hour period in 12 subjects. First the NCA 
method is used to estimate the PK parameters, comparing the re
sults from the one compartment 
model. As described above, the one compartment model is used to estimate the PK parameters, 
and then estimates of AUC, 
max
C and max
T are derived from the form
ulas. Among the benefits 
of the one compartment model analysis is that the PK parameters are estimated together with 
their standard errors and 95% confidence interv
als. In addition indivi
dual (subject-specific) 
prediction of drug conc
entration can be made.
  3.3.5  Model for
()Ct. Assume normal distribution for ()Ct given 
(,,)
aekkCL
.  The parameters 
(,)
akCL
 are subject-specific, i.
e., random effects, but 
ek is a fixed parameter. All parameters 
are transformed to their logged form: 
()alogk
, ()elogk
 and ()logCL
;and the random effects 
are jointly normal and indepe
ndent of the error term in 
()Ct. Therefore, formally the model is desc
ribed by the equation (for one subject) 
()()()exp()exp()()
()×=
+aeeaaekkoraldoseF
Ctktktt
CLkk
 29 
where 
(i) the error term(s) 
()t are serially independent (within 
subject), normally distributed, mean 
zero and variance
2 . (ii) 
11(),
CLlgb
o=+()22,ablogk
=+()3,elogk
=with 1, 2, 3 as fixed parameters, 
1,b 2b as subject-specific ra
ndom effects Š means 0, covariance matrix 
 (3-parameters 
221212
,,). (iii) 
12(,)
bb independent of the error term. 
Across subjects independence is assumed. Hence 
we can construct a joint likelihood for the 
sample data 
{():0,1}

iCttin
 for the
 n=12 subjects with 11 concentrations assessed at 
the same grid of time points from (0, 25hr). 
Maximum likelihood estimation (MLE) provides es
timates of all model parameters and their 
covariance matrix. There are 7 parameters: 
1, 2, 3, 21, 22, 12, 2.The formulas for 
AUC, 
max
C and max
T for the compartment model now give their estimates. Because 
(,)
akCL
 is individual-specific, we will get individual- specific estimates for the PK parameters. The 
additional big advantage of the compartment model 
is the calculation of standard errors of these 
estimates. 
Table 13 shows the 0inf
AUC
 calculation by NCA approach, and AUC estimates by 
compartment model. It shows the values estimated by these two approaches are very similar 
except subject 1 and subject 10.   

 
 30 
 Table 13  Comparison of AUC by NCA and compartment model based estimates
   NCA Compartment model based estimates 
Obs subject 0inf
AUC
 AUC Stderr 
AUC 
Lower 
95% CI 
Upper 
95% CI 
1 1 270.004 144.816 7.00102 129.217 160.416 
2 2 95.050 110.005 5.87935 96.905 123.105 
3 3 107.599 111.415 5.95405 98.148 124.681 
4 4 121.926 118.535 6.25209 104.604 132.465 
5 5 146.878 137.689 6.62215 122.934 152.444 
6 6 87.877 87.023 5.54399 74.670 99.376 
7 7 115.931 106.554 6.28655 92.547 120.561 
8 8 104.732 102.774 5.87002 89.695 115.853 
9 9 96.641 95.818 5.34297 83.914 107.723 
10 10 207.536 154.629 7.40864 138.122 171.137 
11 11 85.472 96.932 5.54048 84.587 109.277 
12 12 126.815 141.073 6.77517 125.977 156.169 
  Figure 8  
Individual concentration profiles by NCA. 
 Figure 8 shows individual concentration profile
s by NCA, AUC of subject 1 and subject10by 
NCA are higher than AUC of most subjects. Higher 
last
Cof these 2 subjects explains the higher 
AUC by NCA, and furthermore,
 explains the discrepancy of
 AUC by NCA and compartment 
model analysis. 
AUC=270 
AUC=207 
31 
Table 14  Comparison of max
C by NCA and compartment model based estimates
   NCA Compartment model based estimates 
Obs subject max
C max
C Stderr 
max
C Lower 
95% CI 
Upper 
95% CI 
1 1 10.50 
10.3420 0.33180 9.60273 11.0813 
2 2 8.33 
8.2205 0.32169 7.50370 8.9372 
3 3 8.20 
8.3918 0.32066 7.67732 9.1063 
4 4 8.60 
8.2855 0.31677 7.57966 8.9913 
5 5 11.40 
9.8981 0.32666 9.17026 10.6259 
6 6 6.44 
6.1226 0.30897 5.43414 6.8110 
7 7 7.09 
6.9453 0.31625 6.24068 7.6500 
8 8 7.56 
7.3470 0.31939 6.63534 8.0586 
9 9 9.03 
7.7379 0.31140 7.04408 8.4318 
10 10 10.21 
9.7244 0.31813 9.01554 10.4332 
11 11 8.00 
7.5903 0.32118 6.87468 8.3060 
12 12 9.75 
9.5540 0.32690 8.82562 10.2824 
 Table 15  Comparison of max
T by NCA and compartment model based estimates 
  NCA Compartment model based estimates 
Obs subject max
T max
T Stderr
max
T Lower 
95% CI 
Upper 
95% CI 
1 1 1.12 2.10624 0.18124 1.70241 2.51008 
2 2 1.92 1.57581 0.16126 1.21650 1.93511 
3 3 1.02 1.48354 0.17806 1.08680 1.88028 
4 4 1.07 2.35712 0.21209 1.88455 2.82969 
5 5 1.00 2.02911 0.15872 1.67545 2.38277 
6 6 1.15 2.28098 0.27775 1.66212 2.89984 
7 7 3.48 3.17439 0.32442 2.45154 3.89723 
8 8 2.02 2.09445 0.22878 1.58470 2.60420 
9 9 0.63 0.66861 0.13729 0.36270 0.97452 
10 10 3.55 3.59319 0.26621 3.00003 4.18634 
11 11 0.98 1.02902 0.15033 0.69407 1.36397 
12 12 3.52 2.72684 0.20907 2.26101 3.19268 
 Similarly, the 
max
C and max
Testimates by NCA approach and compartment model are shown in 
Table 14 and 15. It indicates the values estimate
d by these two approaches are very similar. 
Our analysis shows the benefits of parameter esti
mation and subsequent statistical inference with 
an appropriate compartmental model, even though the model fitting could be a little complicated. 
32 
This pharmacokinetic model is well identified; the parameters can be accurately estimated. If it 
failed AUC estimation is recommended by NCA.
[22]
 Failure to fit a compartment model to a 
given data set could be due to ma
ny factors. As seen the statistical model is highly non-linear, 
introduction of too many random 
effects can be problematic if
 the data set cannot support a 
complex structure. A good approach would start w
ith a ‚fixed™ parameters model to obtain initial 
parameter values for building a more complex model. A few attempts might be needed before a 

stable model can be obtained. 

Extension beyond a one compartment model is 
possible. Two-compartment models view the 
body as a central compartment that receives the drug with transfer from the central compartment 
to a peripheral blood compartment that absorbs th
e drug. Transfer in the opposite direction from 
peripheral to central is also possible. Elimination occurs from the central compartment. 
  33 
CHAPTER 4 DISCUSSION 
 
 The objective of this thesis is to review the st
andard approaches to statistical analyses of 
pharmacokinetic (PK) data. It covers estimation of Area Under the Curve (AUC), Peak 
Concentration (max
C) and other PK parameters and how 
a bioequivalence (BE) study can be 
conducted with crossover design. 
Parameters such as AUC and 
max
C are the key parameters in a 
PK study and used for bioavailability and bioe
quivalence. They are identified as population 
parameters and estimated from observed drug concentration-time profiles.
  
The assessment of AUC adopted by the Food 
and Drugs Administration (FDA) is the Non-
Compartmental Analysis (NCA) approach that es
timates AUC using the trapezoidal rule without 
making any assumption concerning the number of 
compartments. Following the trapezoidal rule, 
concentration-time curve is considered as a series of trapezoids and the AUC estimate is the total 

area of all the trapezoids. The other appr
oach is based on compartmental models.  
 
The one compartment model is used to estimate the PK parameters, such as Absorption Rate 
Constant (ak), Elimination Rate Constant (
ek) and Clearance (
CL). They can be estimated from 
observed concentrations 
{():0,1,,}
Cttm
=– in individuals over a grid of time points, and then 
estimates of AUC, 
max
C and max
T are derived from formulas. They show the benefits of 
parameter estimation and subsequent statistical inference with an appropriate compartmental 
model, even though the model fitting could be 
a little complicated. Among the benefits of the 
one compartment model analysis is that the PK parameters are estimated together with their 
34 
standard errors and 95% confidence intervals.  
In addition individual (subj
ect-specific) prediction 
of drug concentration can be
 made. For Pharmacokinetic/P
harmacodynamic modeling, the 
compartmental pharmacokinetic models are widely 
used, providing continuous description of the 
drug concentration that can serve as 
the input of pharmacodynamic models.
[23]  Fitting of compartmental models can be a complex and lengthy process. As seen the statistical 

model is highly non-linear, introduction of too 
many random effects can be problematic if the 
data set cannot support a comple
x structure. A good approach would start with a ‚fixed™ 
parameters model to obtain initial parameter values for building a more complex model. A few 
attempts might be needed before a stable model can be obtained. If it failed AUC estimation is 
recommended by NCA.
[22] 
 
The widely cited example of drug "theophylline" data
[19][20][21] is used to illustrate these two 
approaches.  Comparison of the PK paramete
rs by NCA and one compartmental model shows 
the parameters estimated from these two methods ar
e very close, the model is identified. Only 
when the underlying pharmacokinetic model parame
ters are identified, can  AUC be accurately 
estimated, otherwise AUC estimation is recommended by the NCA. 

 
BE studies are widely carried out
 in the pharmaceutical industry. 
For small molecule drug 
products, a bioavailability and bioequivalence study are required by FDA for 
approval of generic 
drug products, which contain the exact same activ
e ingredient as the innovator drug. Biosimilars 
are large molecule biological drug products ma
de via living systems. As generic forms of 
35 
biological products instead of the classical generic drugs, biosimilars are only similar to the 
reference product; with no exactly the same activ
e ingredient as the innovator drug. The more 
stringent assessment include safety, purity, and pote
ncy, to show that a follow-on biologic is not 
clinically different from the 
reference biological product.
[24]  Average bioequivalence (ABE) is based solely on
 the comparison of population averages but not 
on the variances, while population bioequivale
nce (PBE) and individua
l bioequivalence (IBE) 
approaches include comparisons of both averages
 and variances. For statistical analyses in a 
bioequivalence study, we used the "pkdata" example with AB|BA design to illustrate how the 
crossover design is applied and ABE is te
sted. SAS procedures PROC TTEST and PROC 
GLIMMIX are applied to the logarithm-transformed AUC and 
max
C to estimate ABE
. Available 
from a public resource even though there are quality 
issues in this data, the "pkdata" example is 
the reasonable example of data with blood c
oncentration time profile
s, from which we can 
estimate AUC and 
max
C, the two key parameters to compare in the BE study.  
 The deficit of the data for BE study includes:
 (1) For a AB/BA design, since there are only 4 
combinations of periods and treatme
nts, the period effect in this particular parameterization is 
aliased with the carryover effect.
[25] Our results show that there is period effect when comparing 
max
log()
Cfor A and B. There is no information available 
if there is carryover effect,  therefore,  
it is not  clear if the period effect is a real period effect. The purpose of this thesis is to illustrate 
how a bioequivalence (BE) study 
can be conducted with crossover 
design, so we claim there is 
no carryover effects. (2) For the AB/BA design, it is not well-suited for comparison of the 
36 
within-unit variance 
2A and 
2B[25] in the statistical model, we have only have one common 
variance
2 for treatment A and treatment B. Theref
ore, total variance of A and B cannot be 
calculated.  The "pkdata" example cannot be used
 for Population BE and individual BE study. 
We will need a replicated crossover design.
[3] 
 In recent years statisticians in the pharmaceuti
cal industry have given a
ttention to developing 
strategies for statistical analyses for 
Biosimilars and 
Biobetters. The FDA recently (April 2015) 
issued guidance on the scientific issues to be considered in demonstrating biosimilarity to a 
reference product.
[26] The FDA™s definition states: 
Biosimilar or biosimilarity 
means that fithe 
biological product is highly similar to the refere
nce product notwithstanding minor differences in 
clinically inactive components,fl and that fithere are no clinically meaningful differences between 

the biological product and the refe
rence product in terms of the safety, purity, and potency of the 
product.fl 

 Therefore, the role that crossover designs have in 
bioequivalence demonstration on a single 
endpoint or outcome measures must now be ex
panded in ways to asse
ss multiple endpoints and 
measures.  It will bring biostatis
ticians and methodologists in pharm
acology together to craft the 
statistical designs and studies for clinical evaluations  that can answer these questions. 
  37 
          BIBLIOGRAPHY 
 
         38 
BIBLIOGRAPHY 
 [1] 
Guidance for Industry--- Exposure-Response Rela
tionships  --- Study Design, Data Analysis, 
and Regulatory Applications. U.S. Department of Health a
nd Human Services Food and Drug 
Administration, Center for Drug Evaluation a
nd Research (CDER), Center for Biologics 
Evaluation and Research (CBER) April 2003. 
[2] 
Guidance for Industry --- Bioavailability and Bioe
quivalence Studies Submitted in NDAs or 
INDs  --- General Considerations. 
U.S. Department of Health 
and Human Services Food and 
Drug Administration Center for Drug Ev
aluation and Research (CDER) March 2014. 
 
[3] 
Guidance for Industry ---Statistical Approac
hes to Establishing Bioequivalence. 
U.S. 
Department of Health and Hu
man Services Food and Drug Ad
ministration Center for Drug 
Evaluation and Research (CDER) January 2001. 

 
[4] Atkinson AJ, Abernethy DR, Daniels CE, et al. 
Principles of clinical pharmacology. Second edition. Academic Press, 2007. 

 
[5] FDA-CFR. Title 21---
Food and Drugs chapter 1--- Food and drug administration 
department of health and human services Subchapter D---Drugs and human use
. Part 320 Bioavailability and Bi
oequivalence, 2014. 
 
[6] Niazi SK. 
Handbook of Bioequivalence Testing. 
Second Edition.CRC Press 2014. 
 
[7] Jones B, Kenward MG. 
Design and Analysis of Cross-Over Trials. Third Edition. Chapman and Hall/CRC, 2014. 

 
[8] 
CENTER FOR DRUG EVALUATION AND RESEARCH --- Guidance for Industry. 
The FDA 
published Good Guidance Practices in February 1997. 

 
[9] Schuirmann DJ
. A comparison of the two one-sided tests procedure and the power approach 
for assessing the equivalence of average bioavailability
. J Pharmacokinet Biopharm. 
1987; 15: 657-80.  [10] 
Westlake WJ
. Symmetrical confidence intervals for bioequivalence trials. 
Biometrics. 
1976; 32: 741-4. 
 
[11] 
McNally RJ
, Iyer H
, Mathew T. Tests for individual and population bioequivalence based 
on generalized p-values. Stat Med. 2003; 22: 31-53.  [12] StataCorp. 
Stata 14 Base Reference M
anual. College Station, TX: Stata Press. Datasets for 
Stata Base Reference Manual, Release 14. 
2015. 
 39 
[13] Chow SC, Jen-pei Liu, JP. 
Design and Analysis of Bioavailability and Bioequivalence 
Studies. Third edition. Chapman and Hall/CRC, 2008. 
 
[14] 
Statistical Approaches To Establishing Bioequivalence
. Appendix 1: statistical issues --- 
Food and Drug Administration. 2002.   
[15] Hedaya MA. 
Basic pharmacokinetics. Second edition
.CRC Press, 2011. 
[16] Benet LZ, Zia-Amirhosseini P. 
Basic principles of pharmacokinetics.
ToxicolPathol.1995; 
23: 115-123. [17] Chow SC, Hsieh TC, Chi E, and Yang J.
A comparison of moment-based and probability-
based criteria for assessment of follow-on biologics. 
Journal of Biopharmaceutical Statistics. 
2010; 20: 31-45. 
 
[18] Wang Z, Kim S, Quinney S, Zhou J, Li L
. Non-compartment model to compartment model 
pharmacokinetics transformation meta-analysis Πa multivariate nonlinear mixed model.BMC 
Systems Biology. 2010; 4: S8. 

 
[19] Pinheiro  JC and Bates DM. 
Approximations to the Loglikelihood Function in the Nonlinear Mixed Effects Model.
 Journal of Computational and 
Graphical Statistics.1995; 4: 12-35  
[20] Wolfinger RD. 
Fitting Nonlinear Mixed Models with the New NLMIXED Procedure
. SUGI 
Proceedings, 1999. 

 
[21] Russek-Cohen E, Ma
rtinez MN, Nevius AB. 
A SAS/IML program for simulating pharmacokinetic data. Computer Methods and Programs in Biomedicine.2005; 78: 39-60. 
 
[22] Buncher  CR. 
Statistics in the Pharmaceutical Industry. 
Second edition. Marcel Dekker Inc, 
2006.  
 
[23] Derendorf H, Meibohm B. 
Modeling of pharmacokinetic/pharmacodynamic (PK/PD) 
relationships: concepts and perspectives. Pharm Res. 1999; 16: 176-85. 
 
[24] Chow SC, Ju C.
 Assessing biosimilarity and interchange
ability of biosimilar products under the Biologics Price Competition and Innovation Act. Generics and Biosimilars Initiative 
Journal.2013; 2: 20-5. 
[25] Vonesh, EF and Chinchilli, VM. 
Linear and Nonlinear Models for the Analysis of Repeated 
Measurements. 
First edition. Marcel Dekker Inc, 1997.
 [26] 
Scientific Considerations in Demonstra
ting Biosimilarity to 
a Reference Product  
Guidance for Industry. U.S. Department of Health a
nd Human Services Food and Drug 
Administration Center for Drug Evaluation a
nd Research (CDER), Center for Biologics 
Evaluation and Research (C
BER).April 2015 Biosimilarity.