Permeability coefficient, steady state equation

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. 

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.)... 

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. 

Skin-snake-model percutaneous absorption Relationships between the in vitro permeability of basic compounds through shed-snake skin as a suitable model membrane for human stratum corneum and their physio-chemical properties were investigated. Compounds with low pKa values were selected to compare the permeabilities of the nonionized forms of the compounds. Steady-state penetration was achieved immediately without a lag time for all compounds. Flux rate and permeability coefficient were calculated from the steady-state penetration data and relationships between these parameters and the physico-chemical properties were investigated. The results showed that permeability may be controlled by the lipophilicity and the molecular size of the compounds. Equations were developed to predict the permeability from the MWs and the partition coefficients of basic compounds. 

Transport Properties. Important transmembrane transport parameters of the fibers are Lp, the hydraulic conductivity Pm, the diffusive permeability for a given solute o, the solute reflection coefficient and R, the solute rejection. These coefficients appear in the following equations, which are assumed to be valid at the steady state at each position Z along the fiber wall ... 

In this equation, A is the area of application, D is the apparent diffusion coefficient and h is the diffusional path length (often taken as the thickness of the membrane). The permeability coefficient (A ) is the steady-state flux per unit area divided by the concentration of drug applied in solution and may be calculated from ... 

Table 10 contains some selected permeability data including diffusion and solubility coefficients for flavors in polymers used in food packaging. Generally, vinylidene chloride copolymers and glassy polymers such as polyamides and EVOH are good barriers to flavor and aroma permeation whereas the polyolefins are poor barriers. Comparison to Table 5 shows that the laige molecule diffusion coefficients are 1000 or more times lower than the small molecule coefficients. The solubility coefficients are as much as one million times higher. Equation 7 shows how to estimate the time to reach steady-state permeation / if the diffusion coefficient and thickness of a film are known. 

Carbon Dioxide Transport. Measuring the permeation of carbon dioxide occurs far less often than measuring the permeation of oxygen or water. A variety of methods are used however, the simplest method uses the Permatran-C instrument (Modem Controls, Inc.). In this method, air is circulated past a test film in a loop that includes an infrared detector. Carbon dioxide is applied to the other side of the film. All the carbon dioxide that permeates through the film is captured in the loop. As the experiment progresses, the carbon dioxide concentration increases. First, there is a transient period before the steady-state rate is achieved. The steady-state rate is achieved when the concentration of carbon dioxide increases at a constant rate. This rate is used to calculate the permeability. Figure 18 shows how the diffusion coefficient can be determined in this type of experiment. The time lag is substituted into equation 21. The solubility coefficient can be calculated with equation 2. 

Permeation Properties. The data shown in Figure 2 are the toluene permeation rates of the fluorinated and untreated containers g. toluene/container per day are plotted vs. the time of toluene exposure on a logarithmic scale. These cumulative permeation rates were calculated based on the cumulative weight loss over the time of toluene exposure, as opposed to the differential permeation rates based on the differential weight loss over each time interval. The room temperature permeation rates for the in-situ fluorinated containers were less than 0.01 g./day and, hence, have been rounded up to 0.01 g./day for illustrative purposes. In Figure 2, the,flat portion of the curves for the untreated containers yielded the steady state permeation rates. From these values, the permeability coefficients (P) for the untreated containers were calculated using Equation 1. 

It may be noted in Figure 2 that the in-situ fluorinated containers exposed to toluene at 50°C appear to be just approaching steady state after 1000 hours of solvent exposure. Thus, the cumulative permeation rate of Figure 2 will result in a slight underestimation ( 10%) of the steady state permeability coefficient. This underestimation was corrected for in Table 1 by using the differential weight loss rate in Equation 1. 

In Equation 10.11, C (t) is the concentration of the compound in the SC lipid phase at time t, 5 is its solubility in these lipids (taken to be the product of water solubility and octanol-water partition coefficient K a)> id b is a positive exponent with a value of about 2.7 (Kasting and Saiyasombati, 2001). The value of b was estimated based on an analysis of the Flynn skin permeability database (Johnson et al., 1997), which represents steady-state permeabilities obtained with hydrated human skin in vitro. It is possible that a somewhat higher value of b may apply for volatile disposition on air-dried skin if it is more size selective than hydrated skin. Such a refinement has not been attempted here. 

With this equation the diffusion coefficient can be obtained from the time lag if the membrane thickness is known. This way to calculate the diffusion coefficient will be further referred to as the Tangent method and the procedure is schematically displayed in Figure 4.8, top section. The permeability follows from the steady state pressure increase rate and can be defined as ... 

With Equation (8) it is relatively simple to estimate the permeability coefficient using steady state analyses. As in the case of the hydraulic permeability, the solute permeability can be conveniently expressed as a resistance i.e., l/P. 

The electrode was inserted vertically into distilled water. Distilled water was saturated with nitrogen gas to displace oxygen and other dissolved gases and then saturated with oxygen gas. The electrode was operated at -0.7v, and the reduction current was measured by use of a Hokuto Denko HM-101 anmeter. From the steady state current ioo of the permeation curve, the permeability coefficient P rcm (S.T.P.)-cm/cm -sec-cmHg] can be calculated by the following equation. 

An apparent permeability coefficient, P, was determined from the slope of downstream pressure vs. time curves at steady state conditions. An apparent diffusion coefficient, D, was calculated from the lag-time, 6, using the following equation. 

The problem of non-steady state flow-which is characteristic of absorbency in polymers-is analogous to the non-steady state diffusion of solutes in a porous material [2]. Therefore, much of this chapter focuses on the solution of the equations for non-steady state diffusion In a porous medium. The most challenging aspect of the general problem of transport in porous polymers is relating the microscopic characteristics of the pore space (porosity, tortuosity, connectivity) to the macroscopic property of interest (permeability or diffusion coefficient). This chapter describes some of the methods that can be used to relate microstructure to transport. While most of the models presented are based on the general problem of diffusion in porous polymers, they can be adapted to explore the mathematically equivalent problem of absorbency in polymers. 

The permeability, of a material is calculated when equal amounts of the penetrating species diffuse and leave the film (steady state operation). The diffusion coefficient is given by D, while is estimated using the following equation ... 

One step in diffusion throngh a polymer membrane is the dissolution of the molecn-lar species in the membrane material. This dissolntion is a time-dependent process, and, if slower than the diffusive motion, may limit the overall rate of diffusion. Consequently, the diffusion properties of polymers are often characterized in terms of a permeability coefficient (denoted by P /), where for the case of steady-state diffnsion through a polymer membrane, Fick s first law (Equation 5.2), is modified as... 

A polymeric system is, thus, characterized by three transport coefficients, which are the permeability, the solubility, and the diffusion coefficients. The permeability coefficient, P, indicates the rate at which a permeant traverses polymer film. The solubility coefficient, S, is a measure of the amount of permeant sorbed by the polymer when equilibrated with a given pressure of gas or vapour at a particular temperature. Finally, the diffusion coefficient, D, indicates how fast a penetrant is transported through the polymer system. For steady state permeation of simple gases into a homogeneous film, the permeability coefficient, P, can be written as the product of diffusion coefficient D and solubility S (Equation 11.2) ... 

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