Permeability coefficient, steady state

The criterion Dt/i = 0.5 to reach steady-state permeation is also useful when measuring the permeability coefficient steady-state data wiU be reached after an experiment time t = fl2D. Other values of as a function of D and thickness i are presented in Fig. 11.21. 

Under conditions of nonlimiting interfacial kinetics the normalized steady-state current is governed primarily by the value of K y, which is the relative permeability of the solute in phase 2 compared to phase 1, rather than the actual value of or y. In contrast, the current time characteristics are found to be highly dependent on the individual K. and y values. Figure 16 illustrates the chronoamperometric behavior for K = 10, log(L) = —0.8 and for a fixed value of Kf.y = 2. It can be seen clearly from this plot that whereas the current-time behavior is sensitive to the value of Kg and y, in all cases the curves tend to be the same steady-state current in the long-time limit. This difference between the steady-state and chronoamperometric characteristics could, in principle, be exploited in determining the concentration and diffusion coefficient of a solute in a phase that is not in direct contact with the UME probe. 

Thus, the rate of change for the cumulative mass of diffusant passing through a membrane per unit area, or the flux of diffusant, j, may be evaluated from the steady-state portion of the permeation profile of a drug, as shown in Eq. (3). If the donor concentration and the steady-state flux of diffusant are known, the permeability coefficient may be determined. 

This equation teaches us that the total stead-state flux (total rate of permeation across a membrane in the steady state of permeation), dM/dt, is proportional to the involved area (A) and the concentration differential expressed across the membrane, AC. In an experiment, flux is the experimentally measured parameter while A and AC are fixed in value when setting up an experiment. The value of the permeability coefficient, Ptotai, is what is calculated upon completion of an experiment using Eq. (8). The permeability coefficient, besides having the specific attributes ascribed to it, is... 

In PAMPA measurements each well is usually a one-point-in-time (single-timepoint) sample. By contrast, in the conventional multitimepoint Caco-2 assay, the acceptor solution is frequently replaced with fresh buffer solution so that the solution in contact with the membrane contains no more than a few percent of the total sample concentration at any time. This condition can be called a physically maintained sink. Under pseudo-steady state (when a practically linear solute concentration gradient is established in the membrane phase see Chapter 2), lipophilic molecules will distribute into the cell monolayer in accordance with the effective membrane-buffer partition coefficient, even when the acceptor solution contains nearly zero sample concentration (due to the physical sink). If the physical sink is maintained indefinitely, then eventually, all of the sample will be depleted from both the donor and membrane compartments, as the flux approaches zero (Chapter 2). In conventional Caco-2 data analysis, a very simple equation [Eq. (7.10) or (7.11)] is used to calculate the permeability coefficient. But when combinatorial (i.e., lipophilic) compounds are screened, this equation is often invalid, since a considerable portion of the molecules partitions into the membrane phase during the multitimepoint measurements. 

Given the low permeability of the antioxidant across MDCK cell monolayers and its large membrane partition coefficient, efflux kinetic studies using drug-loaded cell monolayers cultured on plastic dishes could yield useful information when coupled with the following biophysical model. The steady-state flux of drug from the cell monolayer is equal to the appearance rate in the receiver solution ... 

The choice of flow rates in perfusion experiments is an important consideration as it may affect hydrodynamics [35], ABL thickness [30], intestinal radius [34], intestinal surface area [45], and time to reach steady-state conditions [32], all of which can impact on Peff estimates. The intestinal radius has implications for the estimation of the permeability coefficient. The most widely used estimate for the rat intestinal radius is 0.18 cm [34], These authors found that there was a small change in intra-luminal pressure with an increase in flow... 

It may be concluded from these various arguments that the random walk concept will not in itself give rise to an optimum partition coefficient for biological effect, except in the early stages of the build up to the steady state. A maximal internal availance at an optimal value of P only arises when uptake competes with chemical decay or an excretion process with a different permeability relationship. The explanation of an optimum P value is more likely to involve effects on the form of the pulse of toxicant reaching the site of action, which result in the effect of the chemical transferred, rather than the availance, being greatest at a particular polarity. 

Verhallen P.T.H.M., Oomen L.J.P., v.d.Elsen A.J.J.M., Kruger A.J., and Fortuin J.M.H. (1984) The diffusion coefficients of helium, hydrogen, oxygen and nitrogen in water determined from the permeability of a stagnant liquid layer in the quasi-steady state. Chem. Eng. Sci. 39, 1535-1541. 

The integral permeability coefficient P may be determined directly from permeation steady-state flux measurements or indirectly from sorption kinetic measurements 27 521 activity is usually replaced by gas concentration or pressure (unless the gas deviates substantially from ideal behaviour and it is desired to allow for this) and a<>, ax (p0, Pi) are the boundary high and low activities (pressures) respectively in a permeation experiment, or the final (initial) and initial (final) activities (pressures) respectively in an absorption (desorption) experiment. 

In the absense of any dependence of S and DT on penetrant concentration, the sorption and steady-state permeation properties of the membrane are described by effective solubility and permeability or diffusion coefficients given by... 

Permeability Experiments. Three sets of in-vitro diffusion experiments were conducted 1) identical ethanol/saline composition in both diffusion chambers, 2) ethanol/saline in the donor chamber and saline in the receiver, and 3) saline in the donor and ethanol/saline in the receiver chamber. Tritium labeled 3-estradiol was added to the donor side and samples were taken from both compartments at predetermined times and read in a scintillation counter (Beckman Inst., San Ramon, CA). Effective permeability coefficients were then calculated after steady state was reached using the following equation ... 

Experimental Results and Comparisons with the Classical Lipid Barrier Model. Some typical experimental data are presented in Figure 1 for the transport of g-estradiol. In each of the experiments a lag-time of 1.5 to 2.5 hours were followed by linear steady state fluxes. The effective permeability coefficient, Peff> was calculated from such data using Equation 1 under sink conditions (i.e., Cj/K Cr/Kr where, Kj is the partition coefficient between membrane and donor phase and Kr the partition coefficient between membrane and receiver phase.)... 

A new theoretical model will now be described aimed at attempting to provide a possible explanation for the deviations observed in Figure 3. The model assumes that significant porosity prevails in the hairless mouse stratum corneum when ethanol is present. Although it can be assumed, that at low ethanol concentrations (below 50%) ethanol fluidizes lipid bilayers, there is evidence, that ethanol at high concentration (over 50%) may induce significant pore formations in hairless mouse stratum corneum as measured by the substantial increase of tetraethylammonium bromide permeabilities (10). The permeability coefficient P of a solute across a membrane or stratum corneum under steady state conditions may be described by ... 

For mono-disperse pore size distributions a combination of steady state diffusion and flow permeability measurements can be used to characterize the structural parameters which enable consistent values for tortuosity to be defined. These results can be used to predict the dynamic response of a Wicke-Kallenbach cell to short pulses of a tracer gas having a comparatively high diffusivity and enable a reasonable estimate of the effective diffusion coefficient to be obtained. 

One obtains the permeability coefficient with the help of Eq. (9-1) using the slope of the asymptote in Fig. 9-1, which means steady state permeation has been reached ... 

Example 9-6. A constant 0.5 ml/min stream of nitrogen flows through aim long hose having an outer diameter of 3 cm and 5 mm wall thickness. The hose is in contact with air. After steady state permeation is reached the oxygen concentration in the nitrogen is 0.8 ppm (v/v). What is the permeability coefficient P of this hose ... 

As for the permeability measurements, most techniques based on the analysis of transient behavior of a mixed conducting material [iii, iv, vii, viii] make it possible to determine the ambipolar diffusion coefficients (- ambipolar conductivity). The transient methods analyze the kinetics of weight relaxation (gravimetry), composition (e.g. coulometric -> titration), or electrical response (e.g. conductivity -> relaxation or potential step techniques) after a definite change in the - chemical potential of a component or/and an -> electrical potential difference between electrodes. In selected cases, the use of blocking electrodes is possible, with the limitations similar to steady-state methods. See also - relaxation techniques. 

This description is particularly useful because the diffusion (6) coefficient is reasonably well defined in aqueous solutions, it is related to molecular properties in aqueous solutions, and it can be predicted. However, in biologic systems, the observed length X—for a single transport step—will be widely unchanged. Preferably, steady state and sink conditions are studied, which simplifies the Pick law and focuses on permeability as stated above. 

Also in this example, flux J is constant during the steady state and permeability P can be revealed. However, here we have to consider a lag time ti- For a constant flux, we have to wait until the lipid barrier is saturated and shows a constant gradient of the diffusant according to the partition coefficient. The lag time that represents the intercept with the time axis is found by extrapolating the linear part of the curve (Fig. 3). 

As a predictor of the concentration of cisplatin in normal peritoneal tissues, these data indicate a steady-state penetration depth (distance to half the surface layer concentration) of about 0.1 mm (100 tm). If this distance applied to tumor tissue, penetration even to three or four times this depth would make it difficult to effectively dose tumor nodules of 1- to 2-mm diameter. Fortunately, crude data are available from proton-induced X-ray emission studies of cisplatin transport into intraperitoneal rat tumors, indicating that the penetration into tumor is deeper and is in the range of 1-1.5 mm (10). Such distances are obtained from Equation 9.5 or 9.5 only if k is much smaller than in normal peritoneal tissues — that is, theory suggests that low permeability coefficient-surface area products in tumor (e.g., due to a developing microvasculature and a lower capillary density) may be responsible for the deeper tumor penetration. 

A major breakthrough in the study of gas and v or transport in polymer membranes was achieved by Daynes in 1920 He pointed out that steady-state permeability measurements could only lead to the determination of the product EMcd and not their separate values. He showed that, under boundary conditions which were easy to achieve experimentally, D is related to the time retired to achieve steady state permeation throu an initially degassed membrane. The so-called diffusion time lag , 6, is obtained by back-extrapolation to the time axis of the pseudo-steady-state portion of the pressure buildup in a low pressure downstream receiving vdume for a transient permeation experiment. As shown in Eq. (6), the time lag is quantitatively related to the diffusion coefficient and the membrane thickness, , for the simple case where both ko and D are constants. 

In order to evaluate the steady-state water profile in the membrane of a PEFC under given operating conditions, the necessary membrane transport properties required thus include water uptake by the membrane as function of water activity and membrane pretreatment conditions, A(aw) (covered in Section 5.3.1) the diffusion coefficient of water in the membrane as a function of membrane water content, D ) the electroosmotic drag coefficient as a function of membrane water content, (A) and the membrane hydraulic permeability, A hy(i(A). Section 5.3.2 includes a discussion of water transport modes in ionomeric membranes. 

The initial emphasis on evaluation and modeling of losses in the membrane electrolyte was required because this unique component of the PEFC is quite different from the electrolytes employed in other, low-temperature, fuel cell systems. One very important element which determines the performance of the PEFC is the water-content dependence of the protonic conductivity in the ionomeric membrane. The water profile established across and along [106]) the membrane at steady state is thus an important performance-determining element. The water profile in the membrane is determined, in turn, by the eombined effects of several flux elements presented schematically in Fig. 27. Under some conditions (typically, Pcath > Pan), an additional flux component due to hydraulic permeability has to be considered (see Eq. (16)). A mathematical description of water transport in the membrane requires knowledge of the detailed dependencies on water content of (1) the electroosmotic drag coefficient (water transport coupled to proton transport) and (2) the water diffusion coefficient. Experimental evaluation of these parameters is described in detail in Section 5.3.2. 

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