Construction of IMPRES-M, a non-parametric impulse-response modeling method, in the context of varying pharmacokinetic profiles
Jeroen Elassaiss-Schaap, Lorenzo Cifelli and Paul H.C. Eilers
PD-value B.V., Utrecht, the Netherlands
Introduction
While in many pharmacokinetic (PK) and pharmacodynamic (PK-PD) modeling approaches only a a straightforward but continuous description of PK curves is required for further modeling, the process of constructing proper PK models can be laborious. In principle, smoothing methods can be applied in such cases. Several aspects of clinical pharmacology however limit their application , and published methods [1-4] have not led to wide implementation in pharmacometrics. The aspects that complicate the smoothing of PK profiles are, among others, (1) sharp peaks in profiles in combination with elongated tails; (2) variability in sampling; and (3) incomplete capture of longitudinal exposure profiles with samples e.g. taken on a first and last day of repeated dosing, or according to sparse sampling schemes. A versatile method also must be able to (4) handle different and mixed inter-dose intervals and dose levels. Some of these 4 elements have been addressed by published smoothing methods but thus far no single method addresses them all.
Objectives
The objective therefore is to construct a method that can handle a variety of PK curves of longitudinal drug concentration measurements using a non-parametric regression framework.
Methods
We employ P-splines , which offers several advantages over other methods, including conceptual simplicity and computational efficiency [5]. Curves are represented as a sum of B-spline functions, with a penalty applied to differences of their coefficients. We use cubic B-splines and a second-order penalty. The smoother is estimated with weighted least squares, and the degree of smoothness is determined by leave-one-out cross-validation.
We simulated different concentration profiles from multi-compartmental models with proportional errors. These simulations varied in terms of administration type (intravenous or oral), number of compartments, PK parameters, dose administration designs, and sampling density (dense, typical, sparse).
Results
To adapt the P-splines smoother for multiple dose administrations, we define the complete concentration profile as a convolution of impulses and a response function. In the IMPRES-M framework, each dose administration is considered as an impulse with an amplitude proportional to the dose amount, generating a response representing drug concentration . The complete concentration profile is then obtained by summing all these responses, which is accounting for the accumulation of concentration. Moreover, this impulse response framework serves as a powerful tool, enabling to derive and estimate the profile in time intervals without measurements between two doses or in the case of sparse sampling, and additionally in the case of different dose schedule designs.
Additional features were added to the standard non-parametric regression methods to address the other elements mentioned above. The placement of B-splines in time was optimized through a mathematical transformation. Shape constraints were additionally included to ensure further capture of pertinent PK properties, implemented in estimation using the technique of quadratic programming.
The appropriateness of the resulting smoother framework, IMPRES-M, is graphically confirmed for a range of simulated profiles. IMPRESS-M can adequately capture one- or multi-compartmental models upon intra- or extravascular dosing and different absorption profiles with or without accumulation.
Conclusion
We have constructed a new smoother framework for inferring drug concentration profiles under multi-dose administration designs. Our simulation study demonstrated adequate capture of drug concentration profiles across various settings. IMPRES-M successfully addresses shortcomings of existing smoothing procedures in pharmacometric application. The framework (patent pending) allows for efficient capture of pharmacokinetic profiles for simulation and other modeling purposes.
References:
[1] Eilers, P. H. (2005). Unimodal smoothing. Journal of Chemometrics: A Journal of the Chemometrics Society, 19(5‐7), 317-328.
[2] Jullion, A., Lambert, P., Beck, B., & Vandenhende, F. (2009). Pharmacokinetic parameters estimation using adaptive Bayesian P‐splines models. Pharmaceutical Statistics: The Journal of Applied Statistics in the Pharmaceutical Industry, 8(2), 98-112.
[3] Neve, M., De Nicolao, G., & Marchesi, L. (2005, June). Nonparametric identification of population pharmacokinetic models: An MCMC approach. In Proceedings of the 2005, American Control Conference, 2005. (pp. 991-996). IEEE.
[4] Park, K., Verotta, D., Blaschke, T. F., & Sheiner, L. B. (1997). A semiparametric method for describing noisy population pharmacokinetic data. Journal of pharmacokinetics and biopharmaceutics, 25, 615-642.
[5] Eilers, P. H., & Marx, B. D. (2021). Practical smoothing: The joys of P-splines. Cambridge University Press.