Penetration kinetics models

Penetration systems at the air-water interface in which a dissolved amphiphile (surfactant, protein) penetrates into a Langmuir monolayer are interesting models for a better understanding of various complex processes. Most of all, penetration systems can simulate properties of biological membranes typically comprised of lipids mixed with proteins. First penetration experiments have been described by Schulman and Hughes in 1935 [110]. In the 

Studies on the penetration dynamics, i.e., the time dependence of the surface pressure of a penetration system during the adsorption of the soluble surfactant, are rather scarce [112, 113, 114, 115, 116]. 

The penetration kinetics of a component 2 into an insoluble monolayer, can be monitored by measurements of the rate of the surface pressure change An(t). The above discussed integro-differential equation (4.1) derived by Ward and Tordai [3] is again the basis for a theoretical description of penetration processes. As shown in paragraph 2.9, a simple model for the diffusion mechanism of the penetration process can be obtained by using an equation of the following type interrelating the subsurface concentration and the adsorption 1-0, b C3(0,t) 

Here component 2 is the soluble species with the bulk concentration C2(0, t), while component 1 is the monolayer covering a certain part of the interface . b2 and C02 are the adsorption constant and the area per molecule of the penetrating species. A simultaneous numerical solution of Eqs. (4.1) and (4.70) was performed by Fainerman et al. [115] using the collocation method proposed in [59]. The variation of the dynamic surface pressure for mixed monolayers caused by the penetration of the soluble component can be calculated from the equation 

As example of the calculations performed for various monolayer coverages of the insoluble component by Fainerman et al. [115] Fig. 4.11 shows the variation of surface pressure with time An(t) for different initial surface coverage. 

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The situations a) and c) can be described by one and the same model, as for diluted solutions there is no difference in the diffusitivities of the components in dependence of the presence of others. For case b) a particular model is required, as the behaviour of an insoluble monolayer, already existing at the interface, cannot be described by a classical adsorption isotherm. The thermodynamics of such systems was discussed in Section 2.8.2 and a more detailed description on penetration kinetics models will be given in Section 4.3.6. 

T. Ogiso, M. Iwaki, T. Tanino, A. Yono, A. Ito, In vitro Skin Penetration and Degradation of Peptides and Their Analysis Using a Kinetic Model , Biol. Pharm. Bull. 2000, 23, 1346-1351. 

To explain deviations from the ideal gas law, we must look for characteristics of real gas molecules that are ignored in the kinetic model. That model took the view that molecules are noninteracting, infinitesimal points. So, to improve the model, we need to see how interactions play a role and allow for molecules to have a definite size. Actually, these two features are related, because when we say that a molecule has a definite size, we mean that it exerts repulsive forces. When you touch an object, you feel its size and shape because your fingers cannot penetrate into it. That in turn is due to the repulsive forces its atoms exert on the atoms in your fingers. When you dip your finger into a liquid, your molecules repel the molecules of the liquid and push them aside. 

This mechanism is important for compounds that lack sufficient lipid solubility to move rapidly across the membrane by simple diffusion. A membrane-associated protein is usually involved, specificity, competitive inhibition, and the saturation phenomenon and their kinetics are best described by Michaelis-Menton enzyme kinetic models. Membrane penetration by this mechanism is more rapid than simple diffusion and, in the case of active transport, may proceed beyond the point where concentrations are equal on both... 

In order to increase the number of drugs which can be administered transdermally, the barrier function of the skin must be reduced. The kinetic model can be used to assess the role of a penetration enhancer as a function of the physicochemical properties of the drug. In its simplest form a penetration enhancer may be considered to act in one of two ways. Firstly it may increase the permeability of the skin and, secondly, it may additionally modify the partitioning characteristics at the stratum corneum-viable tissue interface. For illustration, two enhancers have been arbitrarily chosen, the first PE1 increases the permeability by a factor of 10, i.e. k- is increased ten fold. The second, PE2, increases k- by a factor of 10 and decreases kg by a similar amount. Thus PE2 additionally reduces the partition coefficient by a factor of 10. The relative effects can be seen by considering two model drug... 

A description of transdermal drug delivery has been produced which is based on the physicochemical properties of the permeant. At this time transdermal delivery is limited to the administration of potent drugs. Higher doses may be accessible if penetration enhancers are incorporated into the formulation. The kinetic model shows what properties these should have and that they are a function of the physico-chemical properties of the drug. Various loss processes, e.g. microbial biotransformation, skin enzyme metabolism can be identified but cannot, as yet, be quantified. 

The kinetic model (4) developed for analysis of the XPS data is based on a system in which the modification of a surface layer of thickness d occurs via both direct and radiative energy transfer processes, while beneath this layer only radiative energy transfer processes are considered to be important. This assumption derives from the fact that the U.V. and vacuum U.V. radiation, emitted from the plasma, is expected to penetrate the... 

Conceptual models of percutaneous absorption which are rigidly adherent to general solutions of Pick s equation are not always applicable to in vivo conditions, primarily because such models may not always be physiologically relevant. Linear kinetic models describing percutaneous absorption in terms of mathematical compartments that have approximate physical or anatomical correlates have been proposed. In these models, the various relevant events, including cutaneous metabolism, considered to be important in the overall process of skin absorption are characterized by first-order rate constants. The rate constants associated with diffusional events in the skin are assumed to be proportional to mass transfer parameters. Constants associated with the systemic distribution and elimination processes are estimated from pharmacokinetic parameters derived from plasma concentration-time profiles obtained following intravenous administration of the penetrant. 

Barrow [772] derived a kinetic model for sorption of ions on soils. This model considers two steps adsorption on heterogeneous surface and diffusive penetration. Eight parameters were used to model sorption kinetics at constant temperature and another parameter (activation energy of diffusion) was necessary to model kinetics at variable T. Normal distribution of initial surface potential was used with its mean value and standard deviation as adjustable parameters. This surface potential was assumed to decrease linearly with the amount adsorbed and amount transferred to the interior (diffusion), and the proportionality factors were two other adjustable parameters. The other model parameters were sorption capacity, binding constant and one rate constant of reaction representing the adsorption, and diffusion coefficient of the adsorbate in tire solid. The results used to test the model cover a broad range of T (3-80°C) and reaction times (1-75 days with uptake steadily increasing). The pH was not recorded or controlled. 

Because the radiation in most cases will not penetrate the entire sample, the concentration of the reactant is unlikely to approach zero at infinite time. A plot of remaining concentration vs. time will therefore level off at a value greater than zero. This should be taken into account when selecting the kinetic model for studies of solid-state degradation (Sande, 1996). The solid-state degradation will in some cases appear to consist of a series of consecutive processes with different mechanisms and rates (Carstensen, 1974). Such a stepwise change in reaction rate is most likely caused by an alteration in sample surface and fading of subsequent layers. The concept of reaction order may not be useful for photodecomposition in the solid state (De VUliers et al 1992). 

The studies of adsorption layers at the water/alkane interface give excess to the distribution coefficient of a surfactant, which is a parameter of particular relevance for many applications. Theoretical models and experimental measurements of surfactant adsorption kinetics at and transfer across the water/oil interface will be presented. The chapter will be concluded by investigations on mixed surfactant systems comprising experiments on competitive adsorption of two surfactants as well as penetration processes of a soluble surfactant into the monolayer of a second insoluble compound. In particular these penetration kinetics experiment can be used to visualise separation processes of the components in an interfacial layer. 

With increasing 0[ a decrease of the time necessary to achieve the equilibrium state is observed, and the final value of All increases also significantly. This phenomenon is caused by the decrease of the equilibrium adsoiption value for the soluble surfactant in presence of an insoluble monolayer. There are more recent attempts to describe the penetration kinetics, for example the diffusion model for dissolved homologues and ideal monolayers as developed by Sundaram et al. [113]. 

Fig. 4.11 Model calculations for the penetration kinetics of a soluble surfactant into a monolayer at different surface coverage 01=0 (1), 0.35 (2) and 0.7 (3), D = 5.10-6cmVs, co = 2.28.109 cmVmol, b = 1.69.108 cmVmol according to [ 115]...

An ideal pharmacokinetic model of the percutaneous absorption process should be capable of describing not only the time course of penetration through skin and Into blood (or receptor fluid In a diffusion cell), but also the time course of disappearance from the skin surface and accumulation (reservoir effect) of penetrant within the skin membrane. Neither Pick s Plrst Law of Diffusion nor a simple kinetic model considering skin as a rate limiting membrane only Is satisfactory, since neither can account for an accumulation of penetrant within skin. To resolve this dilemma, we have analyzed the In vitro time course of absorption of radiolabeled benzoic acid (a rapid penetrant) and paraquat (a poor penetrant) through hairless mouse skin using a linear three compartment kinetic model (Figure 5). The three compartments correspond to the skin surface (where the Initial dose Is deposited), the skin Itself (considered as a separate compartment), and the receptor fluid In the diffusion cell. The Initial amount deposited on the skin surface Is symbolized by XIO, and K12 and K23 are first order rate constants. 

The development of a blophyslcally based model of chemical absorption via human skin Is described. The simulation has been used to analyze the In vivo penetration kinetics of a broad range of molecular species. Four first-order rate constants are Identified with the percutaneous absorption process k -penetrant diffusion through the stratum corneum k2 transport across the viable epidermal tissue to the cutaneous microcirculation k - a retardation parameter which delays the passage of penetrant from stratum corneum to viable tissue k - the elimination rate constant of chemical from blood to urine. 

The kinetic model has been used to analyze percutaneous penetration rate data for nine molecules aspirin, benzoic acid, caffeine, chloramphenicol, dlethyltoluamlde, nitrobenzene, salicylic acid ( ) and the methyl and benzyl esters of nicotinic acid ( 9 ). Experimentally, the chemicals ( C-labelled) were applied topically In acetone to the ventral forearm of human volunteers and the urinary excretion of radioactivity was then measured over a five day period ... 

A phenomenological kinetic model of this process has been developed by Chang (47). The catalysts are zeolites of ZSM-5 type which have some unusual properties. The largest hydrocarbons which can penetrate the channel structure of ZSM-5 are in Cio range as, for in-... 

Another S-utilization mechanism has been proposed consisting of a consequence of two phenomena (adsorption and penetration of substrate molecules on specific sites followed by transportation through the cell membrane). In this approach, the following generalized kinetic model equation was derived (Borzani and Hiss, 1979 Hiss and Borzani, 1983) ... 

In these experiments, the amount of herbicide distributed on the plant surface (SRF), beneath the plant cuticle (PEN), and in the air (AIR) was monitored at 7-9 time points over a 24-48-h period. The values of and fcpen, the rate constants for volatilization and penetration of the cuticle, respectively, were obtained from the compound distribution versus time data using a kinetic model and mathematical modeling. The simplest model for foliar absorption of a volatile compound is given in Eq. (8.5) ... 

The model proposed by Menger et al. (Fig. 2) shows two extreme conformations, one in which the hydrocarbon chains are fully extended and another in which they are folded [18, 19], The surface of Menger s micelle is less defined than in the classical model and the surfactants that form the micelle are randomly orientated. The water can penetrate and enter in contact with the hydrophobic part of the surfactants. This model, apart from being more acceptable from an esteric point of view, gives a better explanation than the classical model of a series of experimental results such as viscosity, polarity, or kinetics. 

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