Influence of study design on choice of the allometric exponent

Background: Identification of appropriate covariate(s) in population PK-PD modelling is a key requirement to account for between subject variability (BSV). Total body weight (WT) is often considered an important covariate for clearance (CL). Models that link CL to WT vary in the literature and typically include linear (exponent = 1) and nonlinear scaling (exponent = 0.67 or 0.75). Fixing the exponent to a biologically plausible value is widely practiced as its estimation is often associated with poor precision. The practice may involve fixing the exponent to a single value and not engaging in further testing or sequential testing with multiple fixed exponent, e.g. 0.67, 0.75, 1 and selecting the best model. It is not known whether standard population PK study designs can support selecting an exponent from a fixed range of choices.

Aim: The aim of this work was to assess the influence of clinical study design on the probability of selecting the true exponent.

Method: A one compartment IV bolus ‘unit’ model with first-order elimination was used to simulate concentration-time data for various sample sizes (N = 10, 20, 50, 100, 200, 500, 1000). Three levels of BSV (low = 20%, moderate = 40%, high = 60%) were included on clearance (CL) and volume of distribution (V). A combined error model was used. Individual CL values were simulated using WT as a covariate with a hypothetically true exponent of 0.75. The covariate distribution model was based on NHANES III (1988-1994) demographic data with individuals selected by non-parametric bootstrap. Subjects aged < 18 years or weighed < 35 kg were excluded. Two covariate models were considered: all-comer (BMI greater or equal to 18 kg/m2) and normal weight (BMI ranged 18-30 kg/m2). Altogether, 42 design scenarios were considered for simulation and 1000 replicate datasets were simulated under each scenario using MATLAB (2016). Estimation of CL and V was conducted using NONMEM 7.2. Each replicate dataset, in the next step, was fitted by fixing the exponent to 0.75 (‘true’), 0.67 (‘false’) and 1 (‘false’) sequentially and inference was based on the objective function value. The power was calculated as the probability of discerning the ‘true’ exponent out of 1000 replicates.

Results: The study power increases with the increasing sample size and decreasing between subject variance. Probability of discerning the exponent of 0.75 from 0.67 was always less than that of discerning 0.75 from 1.0 under a same design.  In the case of moderate BSV (BSV=40%) from the all-comer data set, at least 200 subjects were required to achieve > 80% power if the competing exponent was 1 and 1000 subjects if the competing exponent was 0.67. The number of subjects increased to 500 and > 1000 for the normal weight subject data.

Conclusion: Estimation of the allometric exponent is known to be associated with low precision.  Fixing the allometric exponent to a prior value is therefore a reasonable alternative.  However, study designs with < 200-500 subjects will not be able to distinguish between competing values of the exponent.  It is recommended that the investigator chooses a value that they believe most appropriately corresponds to the biology and not to test this further.