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https://doi.org/10.5599/admet.1568

Thickness of the aqueous boundary layer in stirred microtitre plate permeability assays (PAMPA and Caco-2), based on the Levich equationǂ

Alex Avdeef


Puni tekst: engleski pdf 827 Kb

str. 249-252

preuzimanja: 93

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Ključne riječi

Hrčak ID:

289909

URI

https://hrcak.srce.hr/289909

Datum izdavanja:

6.12.2022.

Posjeta: 243 *



In permeability assays using microtitre plates, either based on cellular models (e.g., Caco-2, MDCK) or PAMPA (parallel artificial membrane permeability assay) [1-3], the thickness of the aqueous boundary layer (ABL) has been approximated by:

(1)
ADMET-10-1568-e001.jpg

where Daq is the diffusivity, f is the stirring frequency (RPM), and K and α are fitted constants [4-9].

Based on testosterone Caco-2 measurements, Karlsson and Artursson [5] reported α = 1 and implied K = 0.57 x 10-6 cm/s. Adson et al. [6] reported K = 4.1 x 10-6 cm/s and α = 0.8, also for testosterone. Orbital shakers were used to agitate the microtitre plates during the assay, as is the common practice in cellular assays. Both groups noted that the K parameter is a function of aqueous diffusivity, kinematic viscosity, and geometrical factors. Adson et al. [6] pondered on the α factor being greater than the theoretically expected value of 0.5 and reasoned that the asymmetric hydrodynamic conditions of the Transwell plates may have led to the elevated values.

In a PAMPA study of 53 ionizable molecules, Avdeef et al. [9] determined the ABL permeability, PABL, using the pKaflux method at four different stirring speeds (49, 118, 186, 622 RPM). Efficient individual-well magnetic stirring (using the Gut-Box device) was used in their study. Since PABL = Daq / hABL, the constants K and α can be determined for each molecule by linear regression based on log PABL = log K + α log f. The pKaflux method uniquely made such an analysis possible. The least-squares refined parameters (based on several molecules) were reported as K = 23.1 x 10-6 cm/s and α = 0.709. The implicit assumption in the analysis was that for a given rate of stirring, there is a unique ABL thickness for all molecules.

In the above three studies, different values of α were reported, all greater than the theoretical value of 0.5 expected from the solution to the convective diffusion model partial differential equation, based on the rotating disk geometry, according to Levich [10]. In the theoretical model, the thickness of the ABL may be calculated from:

(2)
ADMET-10-1568-e002.jpg

where ν is the kinematic viscosity (cm2/s). If the Levich equation were applicable to microtitre plate permeability assay geometries, thenEq. (2) suggests that K = 0.201 ν- 1/6 Daq2/3, provided α were 0.5 inEq. (1). Hence, each molecule in the permeability assay would be expected to have its own hABL value, depending on its diffusivity. According to Pohl et al. [11], such "theoretical predictions ...[are]... widely ignored." Moreover, using ion-selective microelectrodes, Pohl and coworkers unequivocally showed that hABL varied with ionic substances at a given level of stirring.

In this Communication, it is hypothesized that the theoretical α = 0.5 was obscured in prior Caco-2 and PAMPA microtitre plate permeability studies [5-9], either because (i) K was evaluated without explicit consideration of the Daq term from the Levich equation, and/or (ii) the data were not of sufficient sensitivity to reveal the theoretical values. We proceeded to test the hypothesis with PAMPA data by re-arranging the Levich equation into a parametric form. Combining PABL = Daq / hABL withEq. (2) and converting into the logarithmic form:

(3)
ADMET-10-1568-e003.jpg

with the theoretical constants a = log (0.201 w-1/6) = -0.356 (25 °C) and b = 0.5. We appliedEq. (3) to the (PABL, f) data of Avdeef et al. [9], augmented with additional measurements at 21 and 313 RPM (Table 1), and found a = -0.731 and b = 0.505 (r2 = 0.93, SD = 0.09, F = 50, n = 6). The plot of the data used in the re-analysis is shown inFigure 1. The slope factor, 0.505, is so close to the theoretical value that we propose to simply use the theoretical value henceforth. Substituting the new parameters intoEq. (3) and converting the resulting equation to the form ofEq. (2) results in:

(4)
ADMET-10-1568-e004.jpg

FromEqs. (2) and(4), hABLPAMPA/hABLLevich = 2.4. The geometry of the rotating disk apparatus allows the convective flow to reach closer to the rotating surface (thus diminishing the thickness of the pure diffusion layer) compared to the geometry of magnetically stirred PAMPA wells. The stirring of Caco-2 plates by orbital shakers produces even a greater ratio, hABLCaco-2/hABLLevich, indicating less "efficient" stirring [9].Table 2 shows sample calculations usingEqs. (2) and(4) for three drugs, widely ranging in size.

Pohl et al. [11] suggested that if a single reference compound is used to calibrate the geometrical factor, then calculations of subsequent hABL should be according to the diffusivity dependence in the Levich equation:

(5)
ADMET-10-1568-e005.jpg

Eq. (5) was experimentally verified with several combinations of ions and buffers by Pohl et al. [11], using pH and other ion-selective microelectrodes to directly measure the change in concentrations in the aqueous boundary layer adjacent to black lipid membranes.

In conclusion, the stirring frequency exponent of -1/2 in the theoretical Levich expression appears to apply to PAMPA assays, where efficient individual-well magnetic stirring (> 20 RPM) is used. The same may be true for Caco-2 assays, although additional measurements at varied stirring speeds would make this a more confident assertion. If a single molecule is used as a stirring calibrant, then it seems reasonable to use the scaling suggested byEq. (5) with microtitre plate data. AsTable 2 suggests, the error in calculating hABL based on unscaled hABLref can be as high as 30 %. Hence, it is prudent to incorporateEq. (5) in the calibration procedure. This is especially important to bear in mind for nonionizable molecules since the pKaflux method cannot be directly applied to them. This is of practical importance in PAMPA and perhaps cellular assays as well.

Notes

[1] ǂ This contribution is dedicated to the memory of Dr. Konstantin Tsinman. The work here is based on discussions with Konstantin before his tragic passing in 2020 from Covid-19. During the discussions, he kindly shared the data reported here at 21 and 313 RPM.

References

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Avdeef A.; Nielsen P.E.; Tsinman O.. PAMPA - a drug absorption in vitro model. 11. Matching the in vivo unstirred water layer thickness by individual-well stirring in microtitre plates. Eur. J. Pharm. Sci. 22 (2004) 365-374. https://doi.org/10.1016/j.ejps.2004.04.009. https://doi.org/10.1016/j.ejps.2004.04.009

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Floating objects

Figure 1. The averaged log PABL - 2/3logDaq vs. log f (RPM) plot of ionizable molecules, with PAMPA measurements done at six different stirring speeds. The data are from Avdeef et al.[9], augmented with previously unpublished measurements at 21 and 313 RPM. The values in parentheses refer to the RPM values.
ADMET-10-1568-g001.jpg
Table 1. Aqueous boundary layer permeability data
f (RPM)log PABL - ⅔ log Daq aSDn b
210.037 c0.11915
490.004 d0.1545
1180.294 d0.1046
1860.435 d0.34051
3130.495 c0.26849
6220.737 d0.22222

[i] a PABL is aqueous boundary layer permeability determined by the pKaflux method [2,8,9].

[ii] b Number of measurements averaged.

[iii] c This work.

[iv] d Values averaged from ref [9].

Table 2. Aqueous boundary layer (ABL) thickness at 300 RPM
COMPOUNDt (℃)ν (cm2/s) aDaq (cm2/s) bhABLLevich (μm) chABLPAMPA (μm) d
vincristine250.008933.49E-062033
370.006974.67E-062135
testosterone250.008935.58E-062339
370.006977.47E-062541
benzoic acid250.008938.21E-062644
370.006971.10E-052847

[i] a Values of kinematic viscosity, V, were taken from Riddick and Bunger [12].

[ii] b Diffusivity, Daq, calculated by the procedure described elsewhere [8].

[iii] c Values of the ABL thicknesses, hABLLevich, were calculated byEq. (2).

[iv] d hABLPAMPA calculated byEq. (4).


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