An asymptotic description of a basic FcRn-regulated clearance mechanism and its implications for PBPK modelling of large antibodies
Csaba Katai (1), Jeroen Elassaiss-Schaap (1), Shepard J. Smithline (2), Craig J. Thalhauser (2), Sieto Bosgra (2)
(1) PD-value B.V., (2) Genmab B.V.
Introduction:
Physiology-based pharmacokinetic (PBPK) models are important tools for understanding the distribution and clearance of drugs in organisms. Drugs may be cleared from the body in several ways depending on, for example, their molecular weight or how they interact with target receptors. The long half life of large-antibody drugs is usually attributed to the Neonatal Fc Receptors (FcRn) which bind and save antibodies from degradation in the endosomal space of endothelial cells. A mechanistic formulation of the FcRn-regulated endosomal degradation has been proposed and employed successfully by several PBPK models found in the literature [1,2]. However, due to the complexity of their formulation, a deeper understanding of how the parameters in the model affect clearance has remained elusive. A simpler form of the mechanism, consisting of a single plasma space and a single endosomal space, was studied by Patsatzis et al. (2022) [3] – although for unusually high doses – through the lens of a computational singular perturbation analysis. The advantage of their approach is that no input is required from the investigator but interpreting the output of the method may be more difficult. This is in contrast with analytical singular perturbation analyses that require a much higher level of input from the investigator, but the output is more general. Singular perturbation techniques enjoy widespread use in fluid mechanics [4] and physics [5], but are being increasingly applied in the biomedical sciences as well – see for example the analysis of a target mediated drug disposition model by Peletier and Gabrielsson (2012) [6].
Objectives:
- Perform a singular perturbation analysis of the basic FcRn-regulated clearance mechanism outlined by Patsatzis et al. (2022) [3] and thereby obtain a deeper understanding of it.
- Derive accurate approximations for the antibody concentration in plasma and thereby identify the functional dependence of the participating parameters on FcRn-regulated clearance.
Methods:
The equations governing the basic FcRn-regulated endosomal degradation mechanism are first non-dimensionalised. Assumptions on the order of magnitude of the arising dimensionless groups are then made, which are inspired by physiological parameter values reported in the literature. A singular perturbation analysis, involving the method of matched asymptotic expansions, is then performed. The leading-order governing equations at each time scale are identified and solved yielding approximations for antibody concentrations valid over each time scale.
Results:
Analytic approximations for the antibody concentrations are obtained, that are valid over all time scales. The asymptotic functional relationship between the participating parameters and clearance is derived. When clearance, CL, is defined through
Vp dCp/dt = - CL * Cp ,
with Vp and Cp being the plasma volume and concentration of antibodies in the plasma, the expression for the terminal clearance takes the form
CL ~ Vp * Ve/(Vp + Ve) * kdeg/(kon * [FcRn]0) * (CLup/Ve + koff) .
In the above equation Ve is the volume of the endosomal space, the parameters kdeg, kon, koff, and CLup are the rate constants associated with endosomal degradation, antibody-FcRn association and dissociation, and pinocytosis, respectively, while [FcRn]0 is the initial FcRn concentration.
Conclusions:
The analysis shines light on the functional relationship between the model parameters and the basic FcRn-regulated clearance. Curiously, the expression for clearance, valid over the longest characteristic time scale, depends on the ratio between the degradation rate constant and the initial FcRn concentration. This result should caution against estimating the aforementioned parameters in a PBPK setting simultaneously (cf. Shah and Betts, 2012 [1]). For typical physiological parameters and high antibody dose levels the clearance admits a behaviour akin to that for target mediated drug disposition, however, for moderate to low dose levels clearance is "linear". In fact, it is shown that the level of clearance decays to a constant as the concentration of antibodies in the plasma decays.
References:
[1] D. K. Shah and A. M. Betts. Towards a platform PBPK model to characterize the plasma and tissue disposition of monoclonal antibodies in preclinical species and human. J. Pharmacokinet. Pharmacodyn., 39(1):67–86, 2012.
[2] Z. Li and D. K. Shah. Two-pore physiologically based pharmacokinetic model with de novo derived parameters for predicting plasma PK of different size protein therapeutics. J. Pharmacokinet. Pharmacodyn., 46:305–318, 2019.
[3] D. G. Patsatzis, S. Wu, D. K. Shah, and D. A. Goussis. Algorithmic multiscale analysis for the FcRn mediated regulation of antibody PK in human. Sci. Rep., 12(1):1–21, 2022.
[4] M. Van Dyke. Perturbation methods in fluid mechanics. The Parabolic Press, 1975.
[5] C. C. Lin and L. A. Segel. Mathematics applied to deterministic problems in the natural sciences. SIAM, 1988.
[6] L. A. Peletier and J. Gabrielsson. Dynamics of target-mediated drug disposition: characteristic profiles and parameter identification. J. Pharmacokinet. Pharmacodyn., 39(5):429–451, 2012.