Evaluation of the intact nephron hypothesis using optimal design methodology

Introduction: Renal dose adjustment generally assumes a linear relationship between renal drug clearance (CLR) and glomerular filtration rate (GFR). The theory underpinning this practice is the intact nephron hypothesis (INH) [1]. Studies designed to test INH do not generally consider optimisation of design factors like sampling time, values of GFR, sample size, etc. [2]. The renal drug studies based on the European Medicines Agency (EMA) guidelines were sufficiently powered to detect non-INH, while both the EMA and the United States Food and Drug Administration (FDA) performed poorly in terms of precision of the parameters [3]. The rationale of this work was to explore whether optimisation of these designs may potentially improve parameter precision for both linear and non-linear models.

Aim: To construct an optimal study design that serves the dual purpose of being robust for parameter estimation and for discrimination between models for linear (INH) and non-linear (non-INH) renal drug handling for drugs that are predominantly renally secreted.

Methods: The population analysis framework was a standard two-stage method common for phase 1 designs. The first stage was assumed to be performed according to standard non-compartmental analysis methods and is not considered further. This study was concerned with the second stage, where the relationship between the dependent variable CLR and the independent variable GFR is reviewed. The design space was given by the range of GFR across the study population. The models for CLR were:

  1. M1: a linear model based on the INH scenario

CL= θ1· GFR

  1. M2: a nonlinear model based on the non-INH scenario

CL= θ1· GFRθ2

where, GFR is measured as the clearance of an exogenous probe, θ1 is the linear coefficient parameter and θ2 is the exponent parameter. In this study, variability between individuals and uncertainty in terms of the standard error of the parameters are indistinguishable and an additive residual error model was assumed to accommodate both sources.

The hypercube ln D (HClnD) optimality criterion, which provides a robust design to account for uncertainty in the parameter space, and the Ds optimality criteria, which provides a discriminatory design between nested models, were combined to form a robust compound optimality criterion. Purpose-built code in MATLAB was used for constructing the compound optimal design criterion. The optimal design was evaluated using simulation (MATLAB) and estimation (NONEMEM). The performance of the optimal designs were evaluated by calculating study power to discriminate between the two models, and the relative standard error (RSE) and bias.

Results: The optimal support values of GFR, based on the compound optimal criterion, were 12, 16, and 96 mL/min. The support values of 12 and 16 mL/min are noted to be close and within the residual error of their measurement and are essentially, therefore, replicate support points. The optimal design achieved 90% power with a total of 8 subjects (2 subjects at each of three support points and an additional two subjects allocated to the support points). With the standard sample size of 24 subjects (8 subjects in each of the three support values), RSE% of θ1 was 13% and θ2 was 20%. All designs with power > 80% provided unbiased estimates of both the θ1 and θ2 parameters.

Conclusions: A standard study size of 24 subjects was adequate to precisely estimate the linear coefficient and exponent used to define a non-linear (non-INH) relationship between GFR and CLR. The optimal designs considered here were more efficient, requiring the inclusion of subjects from only three renal function groups, compared to the regulatory recommendations of five renal function groups, while being appropriately powered and providing precise parameter estimates. Therefore, the optimal design proposed here can provide a more cost-effective and efficient design alternative to the existing phase 1 renal drug studies.


  1. Bricker NS, Morrin PA, Kime SW, Jr. The pathologic physiology of chronic Bright’s disease. An exposition of the “intact nephron hypothesis”. Am J Med. 1960;28:77-98.
  2. Pradhan S, Duffull SB, Walker RJ, Wright DFB. The intact nephron hypothesis as a model for renal drug handling. Eur J Clin Pharmacol. 2018. doi:10.1007/s00228-018-2572-8.
  3. Pradhan S, Wright DF, Duffull SB. Evaluation of designs for renal drug studies based on the European Medicines Agency and Food and Drug Administration guidelines for drugs that are predominantly secreted. Br. J. Clin. Pharmacol. 2020.