Permeability coefficient, calculation

In these experiments, the measured helium flux through the membrane was less than the flux predicted on the basis of the average bulk concentrations. Consequently, the helium permeability coefficients calculated from observed membrane flux and the bulk partial pressures are lower than the pure gas values obtained by the membrane supplier or independently by us. At the same time, observed nitrogen coefficients are higher than predicted. 

Van Zyl et al. reported on the diffusion of ipratropium through porcine bronchial epithelium tissue [74], In principle, ipratropium is administered via the respiratory tract by inhalation to treat pulmonary diseases associated with bronchoconstriction. Therefore, pulmonary absorption by bronchial tissue determines its local efficacy and was thus investigated in a diffusion cell in vitro. Bronchial epithelium was equilibrated in PBS and discs of 4 mm2 were mounted on that diffusion cell separating the donor and receiver compartment. The donor compartment contained the drug dissolved in PBS (1 mg/ml) and the receiving chamber was permanently flushed with a low flow (1.5 ml/h) of PBS thus allowing time-resolved fractionation for subsequent direct analysis by LC-ESI MS/MS in MRM mode. Transition to the product ion at m z 124 was monitored for quantification (Table 9). The transfer of ipratropium was characterized by the flux (about 220 ng/cm2/min) and the permeability coefficient calculated to be 1.6 x 10-8 cm/s. 

When the diffusion process is not ideal, the diffusion coefficient calculated from Equation 6 for a stationary state process is the mean diffusion coefficient corresponding to a particular concentration difference. Methods of determining the true diffusion coefficient pertaining to a particular concentration were reported by Crank (9). The permeability coefficient calculated from Equation 4 is also a mean value corresponding to a particular vapor pressure difference if the diffusion process is not ideal. 

When the porosity properties of the medium ate reasonably uniform, the permeability coefficient calculated using observed fluxes by rearrangement of Eq. (20.5-2) has a well-defined meaning. For media with fine pores, slip flow in which gas molecules experience wall interactions frequemly as well as gas-gas viscous interactions must be considered. In such cases, the observed permeability coefficient calculated from Eq. (20.5-2) depends on the average pressure Ip = (p, -I- P2)/2] and may be represented by Eq. (20.5-3) ... 

Mathai and Singh have estimated the permeability coefficient P, using the formula P = kD where k is the partition coefficient and D is the diffusivity. They have used both parallel and series models to calculate P. The experimental values are always greater than measured values. The poor agreement between the experimental and calculated values is attributed to the polar-polar interaction between the epoxy group and nitrile group. 

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... 

While the model helps us understand the chemical structure dependencies of skin permeability, it isn t all that useful for calculating permeability coefficients because of the many iffy assumptions it contains. A different tack has to be taken to gain a sense of the limits, especially the upper limit, of cutaneous drug delivery. There is no lower limit. Even proteins penetrate intact skin to some extent. Some idea of the upper... 

Equation (15) ostensibly allows one to estimate permeability coefficients in units of cm/s. However, as with any parameter calculated from statistically drawn relationships, such estimates have to be taken with a grain of salt, because the absolute error of estimation for a single compound can be large. 

The rough brush stroke agreement between model and experiment is illustrated by the results shown in Fig. 14, for which the correspondences of theoretical with experimental permeability coefficients for the compounds listed in Table 2, (3-adrenegic blockers studied by Lee et al. [207,208] and Schoenwald and Huang [191], are plotted. The calculated values utilized the physical model with pores [205]. Characteristic of... 

The equations used to calculate permeability coefficients depend on the design of the in vitro assay to measure the transport of molecules across membrane barriers. It is important to take into account factors such as pH conditions (e.g., pH gradients), buffer capacity, acceptor sink conditions (physical or chemical), any precipitate of the solute in the donor well, presence of cosolvent in the donor compartment, geometry of the compartments, stirring speeds, filter thickness, porosity, pore size, and tortuosity. 

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. 

Shah et al. [51] demonstrated the use of a donor-receptor compartment apparatus separated by a cell monolayer to estimate membrane transport parameters. Permeability coefficients, P, were calculated as... 

Other important factors dictated by the solute are solubility and ionization state. If the compound has very limited solubility either intrinsically or at the experimental pH, it is frequently possible to do a quick calculation to determine if the experiment is even possible. That is, if the donor concentration is very dilute, one can estimate the receiver concentration which would be obtained for a given solute permeability coefficient and determine if it is within the limits of detection of the assay. 

The permeability coefficients and molecular radii are known. The effective pore radius, R, is the only unknown and is readily calculated by successive approximation. Consequently, unknown parameters (i.e., porosity, tortuosity, path length, electrical factors) cancel, and the effective pore radius is calculated to be 12.0 1.9 A. Because the Renkin function [see Eq. (35)] is a rapidly decaying polynomial function of molecular radius, the estimation of R is more sensitive to small uncertainties in the calculated molecular radius values than it is to experimental variabilities in the permeability coefficients. The placement of the perme-ants within the molecular sieving function is shown in Figure 9 for the effective... 

Pparaceii = permeability coefficient accounting for molecular size restricted diffusion independent of the charge on the molecule calculated from Eq. (47). 

When there is no cell-supporting collagen matrix to consider, PF can be calculated a priori and put into quantitative perspective with other permeability coefficients. Otherwise, PF needs to be experimentally determined. 

It follows that the stirring dependency of the permeability coefficient of the donor and receiver ABLs (i.e., PAm) is AW(v)08 cm/sec. The calculated values of PABL and the effective thicknesses of the ABLs, hABL, are summarized in Table 12. It... 

The initial conditions are CD = CD(0) at t = 0 and CR = 0 at t = 0. Efforts to obtain analytical solutions are tedious and unnecessary. By applying the change in concentrations (or mass) in the donor and receiver solutions with time to the Laplace transforms of Eqs. (140) and (141), the inverse of the simultaneous transformed equations can be numerically calculated with appropriate software for best estimates of a, (3, and y. It is implicit here that P Pap, Pbh and Ke are functions of protein binding. Upon application of the transmonolayer flux model to the PNU-78,517 data in Figure 32, the effective permeability coefficients from the disappearance and appearance kinetics points of view are in good quantitative agreement with the permeability coefficients determined from independent studies involving uptake kinetics by MDCK cell monolayers cultured on a flat dish... 

Instead of using the oral bioavailability of a drug, one can attempt to correlate PM values with permeability coefficients generated from in situ perfused intestinal preparations. Here, one eliminates the complexities of liver metabolism, clearance, and formulation variables. Recently, this type of in vitro-in situ correlation has been conducted using the model peptides (described previously in Section V.B.2). The permeabilities of these model peptides were determined using a perfused rat intestinal preparation which involved cannulation of the mesenteric vein (Kim et al., 1993). With this preparation, it was possible to measure both the disappearance of the peptides from the intestinal perfusate and the appearance of the peptides in the mesenteric vein. Thus, clearance values (CLapp) could be calculated for each peptide. Knowing the effective surface area of the perfused rat ileum, the CLapp values could be converted to permeability coefficients (P). When the permeability coefficients of the model peptides were plotted as a function of the lipophilicity of the peptides, as measured by partition coefficients in octanol-water, a poor correlation (r2 = 0.02) was observed. A better correlation was observed between the permeabilities of these peptides and the number of potential hydrogen bonds the peptide can make with water (r2 = 0.56,... 

Figure 2.8 Effective permeability coefficients (Peff) (xlO-6) of verapamil calculated from appearance in venous blood (P 00 ) and disappearance from lumen (Peff) in the presence and absence of PSC833, midazolam, or ketoconazole in the vascularly perfused intestinal.

The appropriate calculations for determining the flux and permeability coefficient across the buccal mucosa using this approach are detailed in the appendix containing the detailed method used in our laboratory for assessing buccal permeation. 

Either Transwell inserts or side-by-side diffusion chambers can be used for transport studies. Bode et al. have provided an excellent review on this subject [60], Briefly, cells are incubated for 30-60 min with a buffer solution. To initiate the transport study, a transport buffer containing the drug under investigation is added to either the apical or the basal chamber depending on the transport direction of interest. At predetermined time points, the respective receiver chamber is sampled and the withdrawn volume is replaced with the same volume of fresh buffer. The permeability coefficient (Papp) is calculated and the ratio of /apP in the basolateral-to-apical direction versus that in the apical-to-basolateral direction gives the efflux ratio. These sort of transport experiments are well suited to determine if drugs/xenobiotics are substrates of the placental efflux proteins. 

The Caco-2 permeability assay is usually performed in a Transwell device (Figure 18.1). The Transwell contains two compartments a donor and a receiver compartment. The apical donor compartment contains a porous membrane that supports the growth of the Caco-2 monolayer. Caco-2 cells are seeded on the porous membrane. Upon confluency of the cell culture, the compound is added into the donor compartment at a concentration range from one to several hundred micromolar. Samples are collected from the receiver compartment for up to 2 h, then LC-UV or LC-MS methods are used to quantify compound in each sample. The permeability coefficient of the compound is calculated based on the following equation ... 

A further proof of principle was conducted by Cooney et al. who demonstrated the feasibility of the Andersen cascade impactor as a cell compatible deposition device [90], Permeability coefficients of fluorescent isothiocyanate-labeled dextrans after impaction as aerosols on Calu-3 cells were calculated. Deposition did not negatively affect the cell monolayer integrity. 

Substituting hx = 3.6 cm and K ip/w = K - into Eq. 28 Johnson et al. calculated solute lateral diffusion coefficients in stratum corneum bilayers from macroscopic permeability coefficients. Measurements with highly ionized or very hydrophilic compounds were not performed because of the possible transport along a nonlipoidal pathway. Comparison of the computed Aat values with experimentally determined data for fluorescent probes in extracted stratum corneum lipids [47] showed a highly similar curve shape. The diffusion coefficient for the lateral transport showed a bifunctional size dependence with a weaker size dependence for larger, lipophilic compounds (> 350 Da), than... 

PK-Map and PK-Sim (Bayer Technology Services, Wuppertal, Germany), that are based on the models described by Willman et al. [54], In these software packages, the intestinal permeability coefficient can be calculated using a compound s lipophilicity and molecular weight [52,54] and hence, no experimental permeability data is needed. Different to the model described by Willman et al. [54], the commercial prediction tools model the dissolution rate taking the particle size distribution of the solid particles into account (www.pk-sim.com). 

For films, in several series of experiments concerning various thicknesses of various polymers, permeability coefficients have been calculated for a reference thickness of 40 /xm. Units differ for the various gases but are comparable for the different polymers tested with the same gas. The following data (without units) are only given to provide a general idea and cannot be used for designing any parts or goods. 

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