1. Exercise 6 Dose linearity and dose proportionality

2. 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

3. Linearity: an overview
In drug development, it is essential to determine whether the disposition a new drug are linear or nonlinear

4. 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)

5. Thus, EU guidelines required to document linearity
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

6. 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 !

7. 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

8. Linearity and superposition principle
Principle of superposition
Dose 1
Concentration 1
Dose 2
Concentration 2
Dose 1
Dose 2
Concentration 1 + +
Concentration 2
With a linear system, doubling the dose means doubling the concentration

9. Dose proportionality This can be expressed mathematically as:
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.
This can be expressed mathematically as:

10. Assessment of dose proportionality
Analysis of variance (ANOVA) on PK response normalized (divided) by dose
Linear regression (simple linear model or model with a quadratic component)
Power model

11. ANOVA for dose proportionality
If H0 not rejected, no evidence against DP

12. Where the hypothesis is whether beta2 and alpha equal or not zero.
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:
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.

13. If only beta2 is not significantly different from 0, the simple linear regression is accepted.
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.

14. 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

15. Power model

16. Power model and dose proportionality (DP)
An empirical relationship between AUC and dose (or C) is the following power model:
In this model, the exponent (beta), i.e. the slope is a measure of DP.

17. 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
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.

18. Power model and dose proportionality (DP)
A test for dose proportionality is then to test whether beta=1.
The advantage of this method is that the usual assumptions regarding homoscedasticity often apply and alternative variance models are unnecessary.

19. Power model and DP: A bioequivalence approach

20. 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

21. The test problem for BE

22. 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

23. 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

24. Classical statistical hypothesis: drawback
F% Ref Test
n=1000 n=1000
100
702
652
Statistically different for p  0.05 but actually therapeutically equivalent

25. Classical statistical problem : the drawback
F% Ref Test
n=3 n=3
100 70 30
Not statistically different for p  0.05 but actually not therapeutically equivalent

26. 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

27. 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

28. Bioequivalence: the test problem
The appropriate hypothesis
H01
(Ref -test)
H02
(Ref -test) H0 q1 q2
inequivalent
(Ref -test) H1 q1 q2
equivalent
Observation

29. Bioequivalence: the test problem
The appropriate hypothesis q1 q2
(Ref -test)
H01
H02
5%
5%
two unilateral "t" tests
Can we reject H01?
Can we also reject H02?
YES
YES
Bioequivalent

30. 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.

31. Decision procedures for the power model
DP not accepted
DP not accepted 1 2
the 90 % CI of the slope
80%
+125%
DP accepted

32. 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:
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

33. 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:
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.

34. A working example

35. 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 µg/kg”

36. Linear regression analysis
Unweighted vs. weighted simple linear regression

37. Power model: raw data

38. Power model
Observed Y and Predicted Y for the power (linear log-log ) model with data corresponding to doses ranging from 2 to µg/kg (log-log scale) ; visual inspection of figure 7 gives apparent good fitting.

39. Power model
X vs. weighted (w=1) residual Y for a log-log linear power model with data corresponding to doses ranging from 2 to µg/kg; inspection of figure 8 indicates appropriate scatter of residuals (no bias, homoscedasticity)

40. The univariate CI for the SLOPE (0. 9026-1
The univariate CI for the SLOPE ( ) 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 and the shortest 90% CI of the SLOPE is ; this is the classical shortest interval computed for a bioequivalence problem.

41. a priori confidence interval for BPA dose ratio
the a priori confidence interval for this BPA dose ratio was 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;