Christos Kaikousidis, Aris Dokoumetzidis
Department of Pharmacy, University of Athens
Introduction: Fractional calculus has been used within the last couple of decades, to describe anomalous diffusion (i.e. diffusion not described by Fick’s law) and as a result, the anomalous kinetics that arise from it. It has also been shown that fractional differential equations (FDEs) can describe certain experimental datasets more accurately. Fractional pharmacokinetics was introduced in the pharmaceutical literature in [1],[2] and since then several applications. In most of these cases fractional models have been found superior to their equivalent ordinary PK models, using fewer parameters. However, wider application has been hindered by the lack of appropriate software.
Objectives:
Develop algorithms and routines to solve numerically nonlinear FDEs and implement these routines in NONMEM in the context of nonlinear mixed effects (NLME)
Evaluate the performance of the method by a simulation/estimation study
Test the method on real datasets to show the usefulness of fractional models
Methods: We developed the main algorithm for nonlinear FDEs, which was based on the Gr?nwald-Letnikov scheme [3]. This is a generalization of the linear multistep methods used for solving classical ODE’s [4]. We ran simulations for two models: the simple linear kinetics model presented in [1] (variability assumed only on k and V) and a nonlinear Michaelis – Menten type model with two parameters and a volume distribution V, i.e.:
(_0^C)D(_t^a)C=-(Vmax*C)/(( Km+C)*V)
where the (_0^C)D(_t^a)C is the Caputo fractional derivative of order α of concentration C. Variability was assumed on Vmax, Km and V. The fractional order α was assumed the same for all patients. One hundred data sets were simulated for each model and estimation of the population parameters was carried out for each one of them in order to calculate the bias and precision of the methodology, by the relative mean bias (RMB) and relative root mean square error (RRMSE), respectively. Finally, the method was applied on a real data set of Diazepam from [8] containing data from 23 patients for 72 hours in order to estimate the population model parameters
Results: In the simulation study the RMB and RRMSE values for the linear model were : RMB: k=1.3%, V=-0.36%, α =-0.07%,ω_k=0.39%, ω_V=0.67%,σ=0.18%; RMSE k=17.7%, V=1.51%, α=3.4%,ω_k=77%, ω_V=67%,σ=15.8 %. In the non-linear model: RMB: Vm=-2.57%, Km=-9.44%, V=-0.05%, α=-0.96%, ω_Vmax=0.52%, ω_Km=-4.32%, ω_V=-0.77%, σ=0.63% and the corresponding RRMSEs: Vm=2.04%, Km=22.6%, V=0.7%, α =3.07%, ω_Vmax=23.5%, ω_Km=76.6%, ω_V=32.8% and σ=16.7%. These results indicate that method has good performance with good accuracy and precision for the estimates. Regarding the Diazepam data, the linear model was used to fit the data and the resulting parameters were k=21.2, α=0.52 with ω_k=0.074 while their relative standard errors RSE: k=6.59 %, a=0.92 %, ωk=32.59 %. An important advantage of this model was that using only 2 parameters (k and α), it provided similar results to a classic 3 compartment model with absorption, which uses 6. A VPC also showed that the model was able to reproduce the central trend of the data as well as the variability. Finally, predicted versus observed and residual versus time diagrams verified the goodness of fit.
The provided subroutines and control files can be used as a template to implement any population analysis described by a model of fractional order differential equations. Advantages include that there is no need to change anything in the data file compared to a usual NONMEM run. Arbitrary models can be used with several linear and nonlinear equations of different fractional order, without the need to fine tune anything other than the usual parameters of step size etc. used in all differential equation solvers.
Conclusion: We provide a general purpose fractional differential equation solver written in Fortran that can be called from NONMEM and can estimate population parameters for NLME defined by fractional differential equations. A simulation study is successful in terms of estimating population model parameters. A practical application on Diazepam data gives a good fit verified by various diagnostic plots and a VPC. We thus fill the gap that hindered the use of fractional models in NLME. However, this is a first version of this effort, and more work is needed to improve the user interface in terms of NONMEM integration as well as more applications on real data.
References:
[1] Dokoumetzidis A, Macheras P (2009) Fractional kinetics in drugabsorption and disposition processes. J Pharmacokinet Pharmacodyn 36(2):165–178
[2] Dokoumetzidis A, Magin R, Macheras P (2010b) Fractionalkinetics in multi-compartmental systems. J Pharmacokinet Pharmacodyn 37(5):507–524
[3] Garrappa R (2015) Trapezoidal methods for fractional differ-ential equations: theoretical and computational aspects. MathComput Simul 110:96–112
[4] Lubich, C. On the stability of linear multi-step methods for Volterra convolution equations. IMA J.Numer. Anal. 1983, 3, 439–465.
[5] Greenblatt DJ, Allen MD, Harmatz JS, Shader RI. Diazepam disposition determinants. Clin Pharmacol Ther. 1980 Mar;27(3):301-12. doi: 10.1038/clpt.1980.40. PMID: 7357789.
Reference: PAGE 30 (2022) Abstr 10234 [www.page-meeting.org/?abstract=10234]
Poster: Methodology - Estimation Methods