Implementation of non-linear mixed effects models defined by fractional differential equations.

Fractional differential equations (FDEs), i.e. differential equations with derivatives of non-integer order, can describe certain experimental datasets more accurately than classic models and have found application in pharmacokinetics (PKs), but wider applicability has been hindered by the lack of appropriate software. In the present work an extension of NONMEM software is introduced, as a FORTRAN subroutine, that allows the definition of nonlinear mixed effects (NLME) models with FDEs. The new subroutine can handle arbitrary user defined linear and nonlinear models with multiple equations, and multiple doses and can be integrated in NONMEM workflows seamlessly, working well with third party packages. The performance of the subroutine in parameter estimation exercises, with simple linear and nonlinear (Michaelis-Menten) fractional PK models has been evaluated by simulations and an application to a real clinical dataset of diazepam is presented. In the simulation study, model parameters were estimated for each of 100 simulated datasets for the two models. The relative mean bias (RMB) and relative root mean square error (RRMSE) were calculated in order to assess the bias and precision of the methodology. In all cases both RMB and RRMSE were below 20% showing high accuracy and precision for the estimates. For the diazepam application the fractional model that best described the drug kinetics was a one-compartment linear model which had similar performance, according to diagnostic plots and Visual Predictive Check, to a three-compartment classic model, but including four less parameters than the latter. To the best of our knowledge, it is the first attempt to use FDE systems in an NLME framework, so the approach could be of interest to other disciplines apart from PKs.

Isothermal Crystallization Monitoring and Time-Temperature-Transformation of Amorphous GDC-0276: Differential Scanning Calorimetric and Rheological Measurements.

Cold crystallization of amorphous pharmaceuticals is an important aspect in the search to stabilize amorphous or glassy compounds used as amorphous pharmaceutical ingredients (APIs). In the present work, we report results for the isothermal crystallization of the compound GDC-0276 based on differential scanning calorimetric and rheometric measurements. The kinetics of isothermal crystallization from the induction time to the completion of crystallization can be described by the classic Johnson-Mehl-Avrami (JMA) equation. The time-temperature-transformation (TTT) diagrams were constructed for two time points-that of induction and that of completion of crystallization. The results show that the rheological measurement for GDC-0276 has a better overall sensitivity in detection of the early stage nucleation and, consequently, detects the onset of crystallization sooner than does the differential scanning calorimetry. Rheological measurements were also used to obtain the temperature dependence of the viscosity of GDC-0276 and the relevant parameters were used in a modified form of the JMA model to describe the temperature dependence of the crystal induction and completion times, that is, the TTT diagram for the material. In the modification, we assumed that the kinetics followed the viscosity to the 0.75 power as suggested by the recent work of Huang et al. (Huang, C., et al., J. Chem. Phys.2018,149, 054503). The relationship and the possible impact on crystallization kinetics of the break-down of the Stokes-Einstein relation in glass-forming liquids are discussed. From the crystallization kinetics modeling, the solid-liquid interfacial surface tension σSL was obtained for GDC-0276 and was compared with that obtained from the melting point depression measurements of the material confined in nanoporous glasses. The differences between the values from the two methods are discussed.

MESOSCOPIC MODELING OF STOCHASTIC REACTION-DIFFUSION KINETICS IN THE SUBDIFFUSIVE REGIME.

Subdiffusion has been proposed as an explanation of various kinetic phenomena inside living cells. In order to fascilitate large-scale computational studies of subdiffusive chemical processes, we extend a recently suggested mesoscopic model of subdiffusion into an accurate and consistent reaction-subdiffusion computational framework. Two different possible models of chemical reaction are revealed and some basic dynamic properties are derived. In certain cases those mesoscopic models have a direct interpretation at the macroscopic level as fractional partial differential equations in a bounded time interval. Through analysis and numerical experiments we estimate the macroscopic effects of reactions under subdiffusive mixing. The models display properties observed also in experiments: for a short time interval the behavior of the diffusion and the reaction is ordinary, in an intermediate interval the behavior is anomalous, and at long times the behavior is ordinary again.