Estimation of pore size and pore density of biomembranes from permeability measurements of polyethylene glycols using an effusion-like approach

Journal of Controlled Release 49 (1997) 97–104

Estimation of pore size and pore density of biomembranes from permeability measurements of polyethylene glycols using an effusion-like approach a ¨ ¨¨ ¨ Kontturi b , *, Seppo Auriola a , Lasse Murtomaki ¨ b, Kaisa Mari Hamalainen , Kyosti Arto Urtti a a

b

Department of Pharmaceutics, University of Kuopio, P.O.B. 1627, FIN-70211 Kuopio, Finland Laboratory of Physical Chemistry and Electrochemistry, Helsinki University of Technology, Kemistintie 1, FIN-02150 Espoo, Finland Received 6 September 1996; received in revised form 21 January 1997; accepted 27 February 1997

Abstract Biomembranes restrict adsorption and distribution of hydrophilic molecules, like many peptides and oligonucleotides, in the body. The paracellular pathway occupies a very small surface area and is sealed by the junctional complex. In many cases the paracellular pore size (r p ) is comparable with the radius of diffusing molecule (r d ) and so called Renkin correction is used to model the transmembrane diffusion in a quantitative way. In this approach a crucial parameter r d /r p must not be greater than 0.3–0.4; a requirement that is not often fulfilled by experimental data. Consequently, an effusion-like approach was used to estimate pore sizes of the paracellular route and the porosity of the effective barrier of biomembranes. The model membranes were the cornea and conjunctiva of a rabbit eye and the permeating paracellular probes were a mixture of 17 polyethylene glycols of different sizes. It was concluded that the rate determining step for the permeability of the drug through the studied membranes is the probability of finding hydrophilic pores. Another criteria for the effusion-like process was that the effective barrier thickness is small. The effusion-like approach yielded realistic values for the paracellular pore diameter and for the number of the pores in the cornea and conjunctiva. The theory will evidently also be applicable to several other biomembranes, enabling calculation of paracellular pore sizes and porosities; for example nasal, tracheal and intestinal epithelia should fulfill the criteria.  1997 Elsevier Science B.V. Keywords: Effusion; Polyethylene glycol; Biomembranes; Permeability; Pore size

1. Introduction Passive diffusion of compounds (e.g. drugs) across the biomembranes constitutes diffusion through paracellular hydrophilic pathways and transport across lipoidal cell membranes due to lipid-water *Corresponding author. Tel.: 1358 9 4512575; fax: 1358 9 4512580; e-mail: [email protected]

partitioning. When the octanol-water partition coefficient of the compound and its molecular weight are low enough, its permeability is determined by the paracellular route giving information about intercellular junctions [1]. Knowledge of these hydrophilic pores is important when considering the regulation of membrane permeability and the delivery of polar drugs, such as peptides and oligonucleotides, through biological barriers.

0168-3659 / 97 / $17.00  1997 Elsevier Science B.V. All rights reserved. PII S0168-3659( 97 )00078-3

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Tightness of the intercellular space depends on the type of the junctions between cells. The tight junctional routes are among the most important determinants of the membrane permeability and, therefore, they affect drug absorption and distribution. For example, the cell-cell junctions of the endothelia in the sinusoidal veins of the liver and in the vessels of tumors are open, permitting access of large molecules to these sites [2]. In contrast, in several epithelia and the capillary endothelium in the brain, tight junctions between the cells efficiently decrease the diffusion of polar and large molecules [3]. In addition to the width of the tight junctions their number and length are important factors potentially affecting permeation. In most biomembranes, the surface area of the cell membranes is much larger than that offered by tight junctions. Despite its physiological and pharmacokinetic importance in the body, the mass transport occurring in the routes between the cells is not well understood. The rate of permeability through the paracellular route is considered to be due to the diffusion inside the hydrophilic pores. Because in many cases, the efficient pore size (r p ) is comparable with the diameter of the diffusing molecule (r d ), the so called Renkin correction [4] is used to model the process in a quantitative way. However, to explain the obtained experimental results using the Renkin correction, a crucial parameter, r d /r p , must not be greater than 0.3–0.4. The characteristic feature of this correction is that it affects the diffusion coefficient only, keeping the concentration gradient across the biomembrane as a driving force. This approach is certainly in doubt, when the value of the parameter of 0.4 is exceeded because the diffusion process is changed due to the interaction of the pore wall and the transferring molecules [5]. Also, for many biological barriers, an essential fact is the very low porosity of the paracellular routes. This fact will be used in this paper as an argument for the derivation of a tentative theory based on the assumption that the permeability is determined by the probability of finding a pore and not by the hindered transport through a narrow pore. In our preliminary calculations of pore sizes in anterior membranes of the eye, we used diffusion approach and the Renkin correction. This resulted in the values of 0.7–0.8 for the parameter r d /r p . The

pore size calculations were based on the results of in vitro permeability studies with a paracellular probe, polyethylene glycol [6], through cornea and conjunctiva of the eye using mannitol as the reference compound. In addition, on the basis of the anatomy, it is known that the density of the pores in these membranes is very low per unit area [7]. This means that the permeating molecules are mostly colliding with the cell membrane and only occasionally finding a paracellular pore through which the diffusion can take place. This kind of situation is familiar from the kinetic theory of gases and is called effusion. For the effusion of gases, it is characteristic that no considerable mass flow occurs in the direction of the pore. The same condition prevails also for the mass transfer of the present case as indicated by high value of parameter r d /r p , when applying the Renkin correction. Therefore, it is concluded that the rate determining step of the permeability of the drug through the studied membranes is not the diffusion through the paracellular routes but the probability to find the hydrophilic pores. This approach also means that the efficient pore size (r p ) is clearly larger than the pore size obtained using Renkin correction.

2. Theory Assuming that the number of pores per unit area and the efficient pore size are so small that practically no appreciable mass flow occurs in the direction of the orifice, the transport across the membrane takes place because the drug molecules happen to hit the pore in the wall. Therefore, the rate determining step in the transport is not the hindered diffusion in narrow pores but an effusion like process. Firstly, in many analyzed cases [8,9] the parameter r d /r p reaches very high values (e.g. 0.7–0.8) indicating that the effective diffusion coefficient must be of orders of magnitude lower than the bulk diffusion coefficient [10]. Naturally, these high values also manifest the wrong theoretical approach. Secondly, diffusion inside a pore, the radius of which is nearly the same as that of the diffusing drug molecule, does not obey the basic rules of the random walk process on which the microscopic theory is based on. This conclusion is evident due to the interaction of pore wall with the molecule. And thirdly, in the mem-

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branes considered in this study, the lipid matrix invades such a considerable part of the membrane surface, that only a very small portion is left for the hydrophilic pathways. To model this kind of behaviour, consider the movement of the drug molecules in a volume V. This situation is described in Fig. 1. This volume element represents a pattern of drug molecules being a jump length ( l) away from the membrane surface with occasional holes. According to Einstein-Smoluchowski equation this jump length is

RT D 5 ]]] 6phr s NA

] l 5Œ2Dt

1 l A n 5 ]N ] ] 4 t V

(1)

where D is the diffusion coefficient of the drug molecule and t is the jump time. Adapting Einstein’s view [11] we assume that l is determined by the solvent, resulting in [12]

S D

Vm l5 ] NA

1/3

(2)

where Vm is the partial molar volume of the solvent and NA is the Avogadro number. It should be pointed out here that in some textbooks l is taken to be the diameter of diffusing molecule, instead of the diameter of the solvent. However, this approach can be shown to lead, in our case now, to a wrong result, as will be discussed later on. In addition to these straightforward assumptions, we further assume that Stokes law for the diffusion coefficient is valid.

99

(3)

where h is the viscosity of the liquid and r s the radius of the diffusing drug molecule, and R and T have their usual significance. Eq. (3) assumes the molecule to be spherical, which corresponds our experimental conditions. To derive the flux, we first consider the collision frequency (n ) of the drug molecules with the membrane surface. It can be written as (4)

where N is the number of the drug molecules in the volume V, A is the surface area of the membrane, and l /t describes the average speed of the drug molecules. The coefficient 1 / 4 is due to the fact that, on average, a quarter of the drug molecules at the pattern distance l from the membrane surface, collide with that surface. Taking into account V5 Al (see Fig. 1) and Eq. (1), we obtain 1 N 1 ND n 5] ]5] ] . 4 t 2 l2

(5)

According to Eq. (5), the flux J expressing the flux of collision with the membrane surface, i.e. J5 n /(ANA ) can be written, noticing that the amount of drug n5N /NA and its concentration c5n /V, in the form of 1 nD 1 cVD JA 5 ] ] 5 ] ]] 2 l2 2 l2

(6)

which leads to an expression 1 cD J 5 ] ]. 2 l

(7)

With the aid of Eq. (7) the actual flux Jh presenting the flux of compound through the membrane, can now be derived as JA h 1 cDA h 1 cDe Jh 5 ]] 5 ] ]] 5 ] ]] A 2 lA 2 l

Fig. 1. The movement of drug molecules in a volume V. The drug molecules are a jump length l away from the membrane surface with occasional holes. A is the surface area of the membrane and a h the area of a hole.

(8)

where A h is the effective surface area of the hydrophillic pathways (A h 5ma h ; m is the number of paracellular routes in the area of A and a h is the surface area of an individual orifice) and e is the porosity of the membrane, i.e. e 5 A h /A.

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Taking advantage of Eq. (3), Stokes law, Eq. (8) can be written J RTe 1 1 ]h 5 ]]] ] 5 [slope]]. c 12phNA l r s rs

(9)

This equation predicts that the measured permeability Jh /c is inversely proportional to the radius of the drug molecule, and interestingly, from the slope of this relationship, Jh /c5f(1 /r s ) the porosity e can be evaluated after estimating l from Eq. (2). Furthermore, the extrapolation to zero permeability using this relationship gives an estimation for the critical value of the molecular radius still able to penetrate through the hydrophillic part of the barrier. Thus an obtained critical value serves as an estimate for the pore size of the membrane. An interesting feature in Eq. (9) is that, in terms of molar mass, this equation predicts the permeability to be inversely proportional to the cube root of molecular mass. This can be deduced simply by considering the spherical shape of the molecule. It is noteworthy that the effusion of gases obeys, not the cube root law, but the square root or Graham’s law. This is because the average speed, l /t, of the gas molecules is, according to the kinetic theory of gases, inversely proportional to the square root of the molar mass, while in this case it is proportional to the diffusion coefficient. At this stage, it is reasonable to compare what form Eq. (9) obtains if the jump length is taken to be the radius of the drug molecule, i.e. l 52r s . This simple substitution into Eq. (9) yields a result which predicts the permeability to be inversely proportional to the square of the molecular radius of the drug. Experimental data will verify which of approaches is the correct one. Furthermore, another diagnostic criteria of the present approach was fulfilled. In a diffusion process the extrapolation always gives a value of 1 /r s equal to zero, where as the present approach yielded nonzero values. This kind of diagnostics is a powerful tool when considering the applicability of different approaches.

3. Experimental In vitro paracellular permeability of polyethylene

glycols (PEG) in cornea and in bulbar conjunctiva was studied. The test solution included PEG 200 (Mw /Mn 51.11, 2.0 mg ml 21 ), PEG 400 (Mw /Mn 5 21 1.07, 4.0 mg ml ), PEG 600 (Mw /Mn 51.10, 6.0 mg ml 21 ) and PEG 1000 (Mw /Mn 51.05, 10.0 mg ml 21 ) in glutathione Ringer’s buffer (GBR) [13]. The osmolarity of solutions was between 300–309 mosm and pH was adjusted to 7.65 at 378C with O 2 -CO 2 (95:5) bubbling. The PEGs were obtained from Chemical Pressure Co. (Pittsburgh, PA). New Zealand albino rabbits weighing 3.0–4.5 kg were sacrificed by a marginal vein injection of T-61 vet. (Hoechst, Munich, Germany). The bulbar conjunctivas were obtained from an area under cornea without Tenon’s capsule and corneas were dissected leaving a scleral ring. Six to seven tissues were used for each permeability determination. The tissues were positioned between two rings within 25 min of the death in the perfusion chambers. The exposed surface areas of the cornea and conjunctiva were 1.17 and 0.28 cm 2 , respectively. GBR solution (6.5 ml) was added to the receptor side and immediately thereafter, an equal volume of PEG in GBR was added to the donor side. Constant mixing of the reservoir solution was achieved by bubbling a 0 2 –C0 2 (95:5) mixture which maintained the pH at 7.65. The experiments were conducted at 378C for a period of 4 h. Samples of 1 ml were collected from the receptor chamber at intervals of 30 min and replaced with equal volume of blank GBR buffer. Pure PEG oligomer standards for analytical use were prepared by a modified HPLC method of Escott and Mortimer [14]. The LC instrument used was a Shimadzu LC-6A liquid chromatograph (Kyoto, Japan), equipped with a Kromasil C-8 column, 1 cm325 cm (Eka Nobel, Sweden). The eluent consisted of acetonitrile-water, the acetonitrile content was 15% for purification of PEG 300, 20% for PEG 600, and 22.5% for PEG 1000 (Fluka, Buchs, Switzerland). The PEG mixtures were diluted with water (1:1) and 50 ml was injected to the column, the flow was 3 ml min 21 . The separated PEG oligomer peaks were detected at 195 nm and fractions were collected. The eluent was evaporated in vacuum in tared vials. Molecular weights and purity (.95%) of each fraction was determined by thermospray LCMS. The content of each oligomer (i.e. value C) in

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the polydisperse PEG mixture used for permeation studies was measured by using analytical scale HPLC. The analytical HPLC system was as described above, but the column size was 4.5 mm315 cm. The acetonitrile-water gradient was from 10% to 15% in 7.5 min and 25% in 30 min. The PEG contents in the permeation study samples were measured using thermospray-mass spectrometry (TSP-LC-MC) [15]. The penetrated amounts (in mg) of each PEG oligomers were analyzed and the apparent permeability coefficient Papp (in cm s 21 ), corresponding to Jh /c in Eq. (9), was calculated using Eq. (10) with linear regression analysis. J 1 dQ ]h 5 Papp 5 ]] ] c 60Ac dt

(10)

where A is the surface area of tissue (0.28 or 1.17 cm 2 ), dQ / dt is the permeability rate (mg min 21 ) of the drug across cornea or conjunctiva, c is the initial drug concentration (mg ml 21 ) and 60 is included to convert minutes to seconds.

4. Results The data for the used polydisperse PEG molecules needed in the modelling is presented in Table 1. The intrinsic viscosity ([h ]) and the radius of gyration (r g ) of different molecular weight PEGs have been calculated according to Zeman and Wales [16]. Diffusion coefficients (D` ) of PEGs are based on the article of Chin et al. [17]. The radius of PEG molecules (r s ) was calculated in two different ways: one based on radius of gyration (r s 50.7 r g ) [16,18] and another (r s* ) on Stokes-Einstein equation [19] for spherical solutes. For each PEG species the initial molarity in the donor chamber and a value for Jh /c were obtained as described in experimental section. From this and utilizing Eq. (9), i.e. by plotting Jh /c5f(1 /r s ), we obtain Figs. 2 and 2(b) for cornea and bulbar conjunctiva, respectively. In these figures we also show how the different basis of calculating the molecular radii of PEG molecules affects the curves. From the intercept of the obtained linear slopes, the critical value for the pore size r p , can be estimated, and from the slope itself the porosity (e )

101

Table 1 Characteristic properties of PEG oligomers Mw 21 g mol

[h ] 21 dl g

Molarity 21 mol l

D` ?10 6 2 21 cm s

rg ˚ A

rs ˚ A

r *s ˚ A

238 282 326 370 414 458 502 546 590 634 678 722 766 810 854 898 942

0.033 0.035 0.036 0.038 0.040 0.041 0.042 0.044 0.045 0.047 0.048 0.049 0.051 0.052 0.053 0.054 0.056

0.0033 0.0028 0.0027 0.0033 0.0029 0.0028 0.0022 0.0022 0.0021 0.0017 0.0015 0.0014 0.0014 0.0012 0.0012 0.0012 0.0012

7.3 6.8 6.4 6.1 5.9 5.8 5.7 5.6 5.4 5.35 5.3 5.2 5.18 5.15 5.1 5.0 4.9

6.44 6.95 7.36 7.82 8.26 8.61 8.95 9.35 9.67 10.04 10.34 10.64 10.99 11.27 11.55 11.82 12.15

4.51 4.87 5.15 5.47 5.78 6.03 6.27 6.55 6.77 7.03 7.24 7.45 7.69 8.89 8.09 8.27 8.51

4.50 4.83 S.I3 5.38 5.57 5.66 5.76 5.86 6.08 6.14 6.20 6.3I 6.34 6.38 6.44 6.57 6.70

[h ]50.0212.4?10 24 Mw , 0.73 [16]. r g 53.240?([h ]Mw )1 / 3 [16]. kT r s 50.7 r g [16,18].r s* 5 ]] 6phD`

can be calculated according to the relation e 512 [slope] phNA l /RT. These results are presented in Table 2. Taking into account the fact that the experiments have been carried out with real samples of biomembranes, the fit with values of the correlation coefficient of 0.96–0.99 to the theory can be regarded as excellent. Here we consider the other interpretation of the results, i.e. choice of the jump length. If the molecular radius of the drug was used as the jump length, and Papp was thus plotted vs. 1 /r s2 , extrapolation to zero permeability would lead to negative values of the abscissa. This is, of course, physically meaningless, and therefore this alternative is ignored. The obtained values for the critical pore size ˚ diameter), c.f. Table 2, are in line with the (15–30 A previous observations. The corneal epithelium has been found to be impermeable to horseradish peroxidase (with molar mass of 40 000 g mol 21 ) indicating that the intercellular spaces of this tissue ˚ [20]. Grass and Robinson [21] are smaller than 30 A have found in a scanning electron microscopic study that the intercellular space in the cornea for the ˚ The diffusion of gold particles is smaller than 50 A. limiting size of the paracellular pathway has been

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˚ (molecular diameter of also estimated to be 12 A ˚ glycerol) [22] or 30 A (inulin) [23]. Furthermore, it should be noticed that a hydration sphere could add considerably to the size of an individual molecule. For PEGs this potential is also high. The molecular radius of PEG based on the radius of gyration (r g ) takes into account the solvent properties, and conse˚ and 31 A ˚ in the cornea quently, the pore sizes 20 A and bulbar conjunctiva, respectively, are more reliable than the pore sizes based on Stokes-Einstein equation. The critical pore sizes were greater than the ones obtained by the Renkin correction. In our case now this method gave between 0.7–0.8 for the ratio of r d /r p resulting in efficient pore sizes between ˚ in diameter both in cornea and conjunctiva. 8–14 A Porosities of the cornea and conjunctiva were low of the order of 10 26 –10 25 (Table 2), being about an order of magnitude higher in the conjunctiva than in the cornea. These values support the idea of permeation limited by the frequency of hitting the pore.

5. Discussion

Fig. 2. The relationship between permeability (in cm s 21 ) and the ˚ in cornea (a) and in conjunctiva radius of the drug molecule (in A) (b). (m) The radius of PEGs (r *s ) has been calculated using Stokes-Einstein equation. (n) The radius of PEGs (r s ) are based on the radius of gyration (r g ). From the slope of the relationship Jh /c5f(1 /r s ) the porosity is calculated according to the theory and from the x-intercept of the regression line, the critical pore size can be calculated. Table 2 Porosities, pore radii and number of pores of the rabbit cornea and conjunctiva Membrane Porosity?10 6 rs r *s ˚ Pore size (A) rs r *s Number of pores?10 26 cm 22 rs r *s

Cornea

0.13 0.21

Conjunctiva (bulbar) 2.45 3.75

10.0 7.3

15.3 8.8

4.26 12.5

33.3 154

As described above the drawback due to the failing of the Renkin correction needs attention. Since the Renkin correction is still based on the diffusion process it cannot explain the paracellular diffusion correctly if the ratio of r d /r p exceeds 0.4. Consequently a new approach is required. Present theory is based on an effusion-like process which takes the rate determining step of the drug transport across a biological barrier to be the probability to find a pore, not the hindered diffusion through a narrow pore. This approach neglects completely the actual transport process inside the membrane. Therefore, it can be used as a model only for the cases where the diffusion length is short enough and the pore density is low. In order to judge when these conditions prevail, knowledge of the morphology of the membrane is needed. Effusion-like approach yielded realistic values for the intercellular space diameter and for the number of pores in the membrane. In the cornea, the first and second flattened apical cell layers are the rate determining barrier in the paracellular permeation of molecules [24]. Conjunctiva contains tight junctions

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only in the very surface cell layer [25]. Consequently, the effective barrier thickness in these membranes is very small and, as stated above, the fraction of the intercellular spaces in these epithelia is very small. Several other biomembranes fulfil these criteria as well. These include blood vessel endothelia with tight junctions such as blood-brain barrier [26], nasal [24,27,28], tracheal [24], alveolar [29], and intestinal epithelia [24]. However, many biomembranes have either sponge like structure with large water channel network (e.g. sclera of the eye, corneal stroma, dermis of the skin, subcutaneous space) or long diffusion pathlength (e.g. stratum corneum). In these cases, the criteria for effusion-like process are not met, and the permeability of the polar permeants is instead described by the diffusion kinetics.

[9]

[10]

[11] [12] [13]

[14]

[15]

6. Acknowledgements This work was supported by the grants from the Academy of Finland and the Finnish Cultural Foundation, the Elli Turunen Fund.

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