Dose Linearity versus Dose Proportionality

In early phase studies of the drug development, dose linearity and dose proportionality are usually tested. It is essential to determine whether the disposition a new drug are linear or nonlinear. Drugs which behave non-linearly are difficult to use in clinics, especially if the therapeutic window is narrow. if non-linearity is observed for the usual therapeutic concentration range, more clinical studies/tests are needed for the drug development program and drug development can even be stopped. EMA guidance “GUIDELINE ON THE INVESTIGATION OF BIOEQUIVALENCE”, “Guideline on the pharmacokinetic and clinical evaluation of modified release dosage forms”, and FDA Guidance “Bioavailability and Bioequivalence Studies Submitted in NDAs or INDs — General Considerations” specifically requires the test of dose linearity or dose proportionality.

The concept of dose linearity and dose proportionality are often confused because they are very closely related. It can be said that the dose proportionality is a special case of dose linearity or a subset of the dose linearity.

To test the dose linearity or dose proportionality, the clinical trials are often designed as:

• Dose escalation study
• Parallel group study with various dose groups
• Cross-over design with various dose groups

In practice, people usually only test for the dose proportionality. To test for dose proportionality, there are generally four approaches:

Analysis of Variance Approach

In this approach, the dose-normalized PK parameters (AUC or Cmax) will be calculated. The dose-normalized values will then be analyzed by ANOVA approach. Dose normalization is simply the PK parameter divided by dose. With AUC as an example, we can construct the hypothesis as the following: 

          H0: AUC(dose1) / Dose 1 = AUC(dose2) / Dose 2 = AUC(dose3) / Dose 3

If null hypothesis H0 is not rejected, there is no evidence against the dose proportionality. The dose proportionality is then declared. 

Linear Regression Approach

In this approach, the linear regression with quadratic polynomial term of dose will be fit. The PK parameters (AUC or Cmax) will be the dependent variable and dose will be the independent variable.

            Y=alpha + beta1*Dose + beta2*Dose^2 + error

Where the hypothesis is whether beta2 and alpha equal to zero. If either beta2 or alpha is significantly different from zero, the dose proportionality will not be declared. In beta2 is not significant different from zero, the above linear regression is simplified as:

           Y=alpha + beta*Dose + error

If alpha is not significantly different from zero, then dose proportionality is declared

If alpha is significantly different from zero, then dose proportionality cannot declared, but the dose linearity can be declared.

Power Model Approach

In this approach, the relationship between PK parameters (AUC or Cmax) and the dose can be described by the following power model.

          Y=exp(alpha) * Dose^beta * exp(error)

This model can be re-written as:

           ln(Y) = alpha + beta*ln(dose) + error

The slope, beta, measures the proportionality between dose and the PK parameters. If beta=0, it implies that the response is independent from dose. If beta=1, the dose proportionality can be declared. The power essentially tests whether or not the beta = 1.

Equivalence (interval) Approach For Power Model Approach

Based on the power model, Brian Smith et al proposed a bioequivalence approach in their paper “Confidence interval criteria for assessment of dose proportionality”. This approach is concisely described in paper by Zhou et al.

 

EXAMPLE:

In a paper by Campos et al, the dose proportionality was evaluated to compare the 120 mg/kg dose versus 60 mg/kg dose. They first normalized the AUC and Cmax to 60 mg/kg dose. The dose normalized values were then used for ANOVA analysis (mixed model approach as described in previous topic since the study design was a crossover design). They concluded the dose proportionality based on the 90% confidence interval of geometric least square mean ratio (0.83-0.88 for AUC and 0.85-0.92 for Cmax) fell within 80-125% equivalence limits.  

REFERENCES:

• Why should you evaluate dose proportionality?
• Statistics and Pharmacokinetics in Clinical Pharmacology Studies
• Using SAS to Assess Dose Proportionality in Dose Escalation Studies
• Exercise 6: Dose Linearity and Dose Proportionality
• Dose Proportionality Vs Linear Pharmacokinetics