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 rise from it. It has also been shown that fractional differential equations can describe certain the experimental dataset more accurately. Fractional kinetics was introduced in the pharmaceutical literature in [1] and since then a number of applications of fractional pharmacokinetics have appeared such as drug modelling (diclofenac [5], valporic acid [6]), development of non-linear in vitro-in vivo correlations [7] etc. In most of these cases fractional models have been found superior to their equivalent ordinary PK models, using fewer parameters/compartments. However, wider application has been hindered by the lack of appropriate software.
Objectives:
- Develop algorithms and routines to numerically solve nonlinear fractional differential equations
- Implement these routines in NONMEM
- Use these routines to simulate and estimate data to verify the validity of the algorithms and of the implementation
Methods: To begin with, we developed the main algorithm for first order nonlinear fractional differential equations, which was a generalization of the classical Adams–Bashforth–Moulton algorithm for first order equations. The method uses a predictor-corrector logic, where a predicted value is computed via the explicit PI rectangular rule and then a corrected value using the implicit PI trapezoidal rule according to Diethelm [4] and Garappa [3]. The same logic was used to develop an algorithm containing both fractional and integer order derivatives, for multi-compartment modelling. These algorithms where modified to optimize running time while maintaining accuracy and also to include multiple dose cases. The algorithms where implemented in NONMEM via custom Fortran subroutines called in the control file. We ran simulations for two models: the simple anomalous diffusion model presented in [1] and given by the following FDE:
(oD_t^a)C=-kC
where the (oD_t^a) operator is the Caputo fractional derivative of order a computed for the time interval (0,t), and a same model for the case of multiple doses. For the simple diffusion model, the typical value for the rate constant was set to k=0.58 while the order of the FDE was set to a=0.84. The distribution volume had a typical value of V=10. All parameters, but the order a, had lognormal interindividual variability. The OMEGAS where 0.063 for the k constant and 0.31 for the volume. The sigma was set to 0.1. The model with the multiple dose subroutine, was ran for one extra dose of y=(initial dose)/2 and a time tdose=6. The simulated data were then used for the estimation of the model parameters to check the performance of the methodology.
Results: In the simple diffusion model, the estimates for the model parameters where as follows: k=0.619 with an estimated error of 0.0281 and RSE=4.2 %, a=0.836 with an estimated error of 0.00551 and RSE=0.63% and finally the distribution volume was estimated at V=9.06 with an estimated error of 0.86 and RSE=9.9%. For the multidose case the results where similar with k=0.622 and RSE=11.1%, V=9.08 with RSE=8.1% and a=0.829 with RSE=3%. The provided subroutine and control file can be used as a template to implement any model with fractional order differential equations. Advantages include that there is no need to change anything in the datafile 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 stepsize etc used in all differential equational solvers. Furthermore multiple doses are implemented which unlike the ODEs for FDEs it is a non-straightforward issue with mathematical implications due to the memory effects of the latter.
Conclusion: We provide a general purpose fractional differential equation solver written in Fortran that can be called from NONMEM and can estimate parameters for nonlinear mixed effects models defined by fractional differential equations. We thus fill the gap that hindered the use of fractional models in pharmacometrics. However this is a first version of this effort and more work is need to improve the user interface in terms of NONMEM integration.
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] Diethelm K, Ford NJ, Freed AD (2002) A predictor-correctorapproach for the numerical solution of fractional differential equations. Nonlinear Dyn 29(1):3–22. doi:10.1023/A:1016592219341
[5] 25. Popovic ́ JK, Atanackovic ́ MT, Pilipovic ́ AS, Rapaic ́ MR, Pili-povic ́ S, Atanackovic ́ TM (2010) A new approach to the compartmental analysis in pharmacokinetics: fractional time evolution of diclofenac. J Pharmacokinet Pharmacodyn37(2):119–134
[6] Popovic ́ JK, Dolic ́anin D, Rapaic ́ MR, Popovic ́ SL, Pilipovic ́ S,Atanackovic ́ TM (2011) A nonlinear two compartmental fractional derivative model. Eur J Drug Metabol Pharmacokinet 36(4):189–196
[7] Kytariolos J, Dokoumetzidis A, Macheras P (2010) Power lawIVIVC: an application of fractional kinetics for drug release and absorption. Eur J Pharm Sci 41(2):299–304
Reference: PAGE () Abstr 9573 [www.page-meeting.org/?abstract=9573]
Poster: Methodology - New Modelling Approaches