Exercise 6
Dose linearity and dose
proportionality
Objectives of the exercise
• To learn what is dose linearity vs. dose
proportionality
• To document dose proportionality using ANOVA
• To test dose linearity/proportionality by linear
regression
• To test and estimate the degree of dose
proportionality using a power model and a
bioequivalence approach
Linearity: an overview
• In drug development, it is essential to
determine whether the disposition a new
drug are linear or nonlinear
Linearity and stationary
• Basic PK parameters (F%, Cl, Vss,…) are
usually independent of the dose (linearity)
and repeated (continuous) administrations
(stationary)
• Otherwise they are
– Dose dependent (non-linearity PK)
– Time dependent (non-stationary PK)
Why is it important for a drug
company to recognize dose
dependent kinetics?
• Drugs which behave non-linearly are difficult
to use in clinics, specially if the therapeutic
window is narrow (e.g.: phenytoin)
– drug monitoring
• Thus, EU guidelines required to
document linearity
Why is it important for a drug company to
recognize dose dependent kinetics?
• Drug development is often stopped if non-linearity
is observed for the usual therapeutic
concentration range
• Non linearity is often observed in toxicokinetics
! (higher doses are tested)
– sophisticated data analysis
– data interpretation: need to know whether non-linearity
exists or not
Linearity and order of a reaction
• Order 1 : linearity
– In a linear system all processes
(absorption, distribution, … are governed
by a first order reaction
• Order < 1 : Michaelis-Menten
• Order zero : perfusion, implant
Dose proportionality
• For a linear pharmacokinetic system, measures
of exposure, such as maximal concentration
(Cmax) or area under the curve from 0 to infinity
(AUC), are proportional to the dose.
AUC  Dose
• This can be expressed mathematically as:
AUC    Dose
Assessment of dose proportionality
1. Analysis of variance (ANOVA) on PK
response normalized (divided) by dose
2. Linear regression (simple linear model
or model with a quadratic component)
3. Power model
ANOVA for dose proportionality
AUC dose1 AUC dose2 AUC dose3
H0:


....
Dose1
Dose 2
Dose3
If H0 not rejected, no evidence against DP
Linear regression
• The classical approach to test DP is first to fit the
PK dependent variable (AUC, Cmax…) to a
quadratic polynomial of the form:
Y    1 ( Dose )  2 ( Dose )  
2
Where the hypothesis is whether beta2 and alpha equal or not zero.
Dose non-proportionality is declared if either parameter is
significantly different from zero.
• If only beta2 is not significantly different from 0, the
simple linear regression is accepted.
Y     (Dose )  
•where alpha is tested for zero equality.
• If alpha equals zero, then Eq. 1 holds and dose
proportionality is declared.
•If alpha does not equal zero, then dose linearity (which is
distinct from dose proportionality) is declared.
Limits of the classical regression
analysis
• The main drawback of this regression approach
is the lack of a measure that can quantify DP;
also when the quadratic term is significant or
when the intercept is significant but close to
zero, we are unable to estimate the magnitude
of departure from DP.
• This point is addressed with the power model
Power model
Power model and dose
proportionality (DP)
• An empirical relationship between AUC
and dose (or C) is the following power
model:

Y  Exp( )( Dose ) Exp( )
In this model, the exponent (beta), i.e. the slope
is a measure of DP.
Power model and dose
proportionality (DP)
• Taking the LN-transformation leads to a linear equation
and the usual linear regression can then applied to this
situation
log e (Y )     log e (dose)  
•where beta, the slope, measures the proportionality
between Dose and Y.
•If beta=0, it implies that the response is independent from
the dose
•If beta=1, DP can be declared.
Power model and DP:
A bioequivalence approach
Power model and DP:
issue associated to the classical H0
• If an imprecise study lead to large
confidence intervals around β,
– You cannot reject the classical H0 and you
can conclude to a DP but that is in fact
meaningless
The test problem for BE
Bioequivalence : the test problem
From a regulatory point of view the
producer risk of erroneously rejecting
bioequivalence is of no importance
The primary concern is the protection of
the patient (consumer risk) against the
acceptance of BE if it does not hold true
Bioequivalence : the test problem
Classical test of null hypothesis (I)
H 0 : T - R = 
or T = R
H 1 : T - R  
or T  R
T and R : population mean for test and
reference formulation respectively
Decision on the BE cannot be based on the
classical null hypothesis
Classical statistical hypothesis:
drawback
F%
Ref
n=1000
Test
n=1000
100
702
652
Statistically different for p  0.05 but actually
therapeutically equivalent
Classical statistical problem : the
drawback
F%
100
Ref
n=3
70
30
0
Not statistically different for p  0.05 but
actually not therapeutically equivalent
Test
n=3
Bioequivalence : the test problem
Classical test of null hypothesis
• Can be totally misleading
• Acceptance of B.E. despite clinically relevant
difference between R and T formulation
• Rejection of B.E. despite clinically irrelevant
difference between R and T
Bioequivalence : the test problem
Classical test of null hypothesis
Use of the classical null hypothesis would
encourage poor trials, with few subjects,
under uncontrolled conditions to answer
an irrelevant question
Bioequivalence: the test problem
• The appropriate hypothesis
H01
(Ref -test)
H0
H02
(Ref -test)
q1
inequivalent
q2
equivalent
(Ref -test)
H1
Observation
q2
q1
Bioequivalence: the test problem
• The appropriate hypothesis
q2
q1
(Ref -test)
H01
5%
H02
5%
two unilateral "t" tests
Can we reject H01?
YES
Can we also reject H02?
Bioequivalent
YES
Two unilateral t test and a 90%
• From an operational point of view to
perform 2 unilateral t-tests or to compute
the 90% CI (of the ratio) lead to exactly
the same conclusion.
Decision procedures for the power model
DP not
accepted
1 the 90 % CI of the slope2
80%
DP accepted
DP not
accepted
+125%
Power model: construction of a 90% CI
• If Y(h) and Y(l) denote the value of the dependent
variable, like Cmax, at the highest (h) and lowest (l) dose
tested, respectively, and the drug is dose proportional
then:
Y ( h) h
  Ratio
Y (l ) l
where Ratio is a constant called the maximal dose
ratio.
Dose proportionality is declared if the ratio of
geometric means Y(h)/Y(l) equals Ratio
Construction of a 90% CI
• The a priori acceptable confidence interval (CI)
for the SLOPE (see Smith et al for explanation)
is given by the following relationship:
Ln(0.8)
Ln(1.25)
1
 slope  1 
Ln(dose _ ratio)
Ln(dose _ ratio)
Here 0.8 and 1.25 are the critical a priori values suggested by
regulatory authorities for any bioequivalence problem after a
data log transformation.
A working example
Fist analysis:
an ANOVA to test H0
• Conclusion of the ANOVA: in the present
experiment, there was no evidence
against the null hypothesis of BPA dose
proportionality for BPA doses ranging from
2.3 and 100000 µg/kg”
Linear regression analysis
•
Unweighted vs. weighted simple
linear regression
Power model: raw data
Power model
6
4
2
0
Observed
-2
Predicted
-4
-6
-8
0
2
4
6
8
10
12
Ln_nominal_dose
Observed Y and Predicted Y for the power (linear log-log ) model with
data corresponding to doses ranging from 2 to 100 000µg/kg (log-log
scale) ; visual inspection of figure 7 gives apparent good fitting.
Power model
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
0
2
4
6
8
10
Ln_nominal_dose
X vs. weighted (w=1) residual Y for a log-log linear power model
with data corresponding to doses ranging from 2 to 100 000µg/kg;
inspection of figure 8 indicates appropriate scatter of residuals (no
bias, homoscedasticity)
12
The univariate CI for the SLOPE (0.9026-1.030) as computed by WinNonLin is
a 95% CI computed with the critical ‘t’ value for 20 ddl i.e. t=2.086.
To compute a 90% CI i.e. (1-2*alpha) 100%, the critical “t” for 20 ddl is 1.725
and the shortest 90% CI of the SLOPE is 0.9137-1.019; this is the classical
shortest interval computed for a bioequivalence problem.
a priori confidence interval for BPA
dose ratio
Ln(0.8)
Ln(1.25)
1
 slope  1 
Ln(dose _ ratio)
Ln(dose _ ratio)
• the a priori confidence interval for this BPA dose
ratio was 0.9794-1.0206 it can be concluded that
both the 95 and the 90% CI for the SLOPE were
not totally included in this a priori regulatory CI
and then the BPA dose proportionality cannot be
accepted (proved) for this range of BPA doses;