Nonlinear kinetics processing

Linear (or first-order) kinetics refers to the situation where the rate of some process is proportional to the amount or concentration of drug raised to the power of one (the first power, hence the name first-order kinetics). This is equivalent to stating that the rate is equal to the amount or concentration of drug multiplied by a constant (a linear function, hence linear kinetics). All the PK models described in this chapter have assumed linear elimination (metabolism and excretion) kinetics. All distribution processes have been taken to follow linear kinetics or to be instantaneous (completed quickly). Absorption processes have been taken to be instantaneous (completed quickly), follow linear first-order kinetics, or follow zero-order kinetics. Thus out of these processes, only zero-order absorption represents a nonlinear process that is not completed in too short of a time period to matter. This lone example of nonlinear kinetics in the standard PK models represents a special case since nonlinear absorption is relatively easy to handle mathematically. Inclusion of any other type of nonlinear kinetic process in a PK model makes it impossible to write the... 

Let us now incorporate this nonlinear kinetic process into a non-Markovian transport process described by a CTRW. We write the equations for the densities j (x, t) and p x, t) in the following forms ... 

As this volume attests, a wide range of chemistry occurs at interfacial boundaries. Examples range from biological and medicinal interfacial problems, such as the chemistry of anesthesia, to solar energy conversion and electrode processes in batteries, to industrial-scale separations of metal ores across interfaces, to investigations into self-assembled monolayers and Langmuir-Blodgett films for nanoelectronics and nonlinear optical materials. These problems are based not only on structure and composition of the interface but also on kinetic processes that occur at interfaces. As such, there is considerable motivation to explore chemical dynamics at interfaces. 

Solid-phase organic synthesis (SPOS) exhibits several shortcomings, due to the nature of the heterogeneous reaction conditions. Nonlinear kinetic behavior, slow reactions, solvation problems, and degradation of the polymer support due to the long reaction times are some of the problems typically experienced in SPOS [2], Any technique which is able to address these issues and to speed up the process of solid-... 

Oscillations have been observed in chemical as well as electrochemical systems [Frl, Fi3, Wol]. Such oscillatory phenomena usually originate from a multivariable system with extremely nonlinear kinetic relationships and complicated coupling mechanisms [Fr4], Current oscillations at silicon electrodes under potentio-static conditions in HF were already reported in one of the first electrochemical studies of silicon electrodes [Tul] and ascribed to the presence of a thin anodic silicon oxide film. In contrast to the case of anodic oxidation in HF-free electrolytes where the oscillations become damped after a few periods, the oscillations in aqueous HF can be stable over hours. Several groups have studied this phenomenon since this early work, and a common understanding of its basic origin has emerged, but details of the oscillation process are still controversial. 

This is known as Michaelis-Menten or saturation kinetics. The processes that involve specific interactions between chemicals and proteins such as plasma protein binding, active excretion from the kidney or liver via transporters, and metabolism catalyzed by enzymes can be saturated. This is because there are a specific number of binding sites that can be fully occupied at higher doses. In some cases, cofactors are required, and their concentration may be limiting (see chap. 7 for salicylate, paracetamol toxicity). These all lead to an increase in the free concentration of the chemical. Some drugs, such as phenytoin, exhibit saturation of metabolism and therefore nonlinear kinetics at therapeutic doses. Alcohol metabolism is also saturated at even normal levels of intake. Under these circumstances, the rate of... 

Factors analogous to those affecting gut absorption also can affect drug distribution and excretion. Any transporters or metabolizing enzymes can be taxed to capacity—which clearly would make the kinetic process nonlinear (see Linear versus Nonlinear Pharmacokinetics ). In order to have linear pharmacokinetics, all components (distribution, metabolism, filtration, active secretion, and active reabsorption) must be reasonably approximated by first-order kinetics for the valid design of controlled release delivery systems. 

The structural submodel describes the central tendency of the time course of the antibody concentrations as a function of the estimated typical pharmacokinetic parameters and independent variables such as the dosing regimen and time. As described in Section 3.9.3, mAbs exhibit several parallel elimination pathways. A population structural submodel to mechanistically cover these aspects is depicted schematically in Fig. 3.14. The principal element in this more sophisticated model is the incorporation of a second elimination pathway as a nonlinear process (Michaelis-Menten kinetics) into the structural model with the additional parameters Vmax, the maximum elimination rate, and km, the concentration at which the elimination rate is 50% of the maximum value. The addition of this second nonlinear elimination process from the peripheral compartment to the linear clearance process usually significantly improves the fit of the model to the data. Total clearance is the sum of both clearance parts. The dependence of total clearance on mAb concentrations is illustrated in Fig. 3.15, using population estimates of the linear (CLl) and nonlinear clearance (CLnl) components. At low concentra-... 

Process Transfer Function Models In continuous time, the dynamic behaviour of an ideal continuous flow stirred-tank reactor can be modelled (after linearization of any nonlinear kinetic expressions about a steady-state) by a first order ordinary differential equation of the form... 

In order to avoid the restrictions to complicated adsorptive reactions in the MOC3D, Selim et al. (1990) developed a simulation system based on the multireaction model (MRM) and multireaction transport model (MRTM). The MRM model includes concurrent and concurrent-consecutive retention processes of the nonlinear kinetic type. It accounts for equilibrium (Freundlich) sorption and irreversible reactions. The processes considered are based on linear (first order) and nonlinear kinetic reactions. The MRM model assumes that the solute in the soil environment is present in the soil solution and in several phases representing retention by various soil... 

It is essential that the functionals are positively defined in all of the considered examples to imply stability of the stationary state in the relevant stepwise processes. Strongly nonlinear kinetic schemes need special procedures for analyzing the stability. While doing so, it is easy to demonstrate that the positive definition of functional is indeed the sufficient condition of stability of the chemical process (in the case, naturally, when these functionals exist). 

The stability and other thermodynamic properties of nonlinear kinetic systems far from thermodynamic equibbrium are commonly accepted to analyze by inspecting the system behavior as functions of some external controlling parameters. Chemical affinity of stepwise processes or those related to it values that characterize the remoteness from the equilibrium point of the system are often chosen as such a controlling parameter of general nature. 

The ion-exchange kinetics accompanied by association-dissociation in this model differ, on the one hand, fiom nonlinear sorption processes where the isotherm shape is responsible for the concentration profiles in the bead and for the kinetic rate [63] and, on the other, from conventional ion exchange where the kinetic rate and concentration profiles Cb, Ca are governed by the diffusivity factor D, /Db [16]. In the model both the selectivity (Kb /K b) and the diffusivity (Df/D factors play a role in the IE process. 

If we assume that the SCE can be used to correlate kinetic processes, then a means of testing our earlier conclusions regarding failure of the EtOH-TFE method is provided. According to our proposal, a nonlinear EtOH-TFE plot... 

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