Review of theoretical passive drug absorption models: Historical background, recent developments and limitations

PHARMACEUTICA ACTA HELVETIAE

ELSEVIER Pharmaceutics Acta Helvetiae 71 (1996) 309-327

Review

Review of theoretical passive drug absorption models: Historical background, recent developments and limitations

Gian Camenisch a,b, Gerd Folkers b, Han van de Waterbeemd aT *

a F. Hofjkznn-La Roche Ltd., Pharma Research-New Technologies, CH-4070 Basel, Switz,erland b ETH Ziirich, Deparrment of Pharmacy, Winterthurerstrasse 190, CH-8057 Ziirich, Switzerland

Received 3 April 1996; accepted 30 April 1996

Abstract

In the drug discovery process the optimization of a promising lead to an orally bioavailable drug remains a difficult task. Recent progress in the understanding of the role of physicochemical properties in membrane permeability relevant to important processes such as drug absorption and blood-brain barrier crossing, brings rational drug delivery more within reach. In the last thirty years a number of theoretical transport and absorption models have been developed to describe mathematically how a drug is being passively transported from its site of administration to its site of action and how a compound passes a membrane. The goal of such models is to rationalize the physical significance of the observed non-linear structure-permeability relationships. The models are based on various views on the composition of the biological membranes and on the underlying diffusion and distribution mechanisms. Often simplifications reducing the mathematical complexity are made. We review here a selection of the most important models and discuss modem views on the role of lipophilicity and various pathways through membranes.

Keywords: Distribution; Drug transport; Drug absorption; Absorption models; Distribution models; Membrane permeation; Lipophilicity; Pore diffusion

1. Introduction

In order to exert a therapeutic effect a drug should have high affinity and selectivity for its intended biological target ( = receptor), and also reach a sufficient concentration at this site. A receptor is defined as a protein or a protein complex in or on a cell that specifically recognizes and binds to a compound acting as molecular messenger. In a broader sense, the term receptor is often used as a synonym for any specific drug binding site, as opposed to non-specific interactions, e.g. to plasma proteins (van de Waterbeemd et al., 1996a). Factors which influence the possibility of the drug to reach this receptor have to be considered. This process is affected by the liberation of the drug from its formulation environment, its subsequent transport from the site of application to the directly adja-

* Corresponding author. Fax +41 (61) 6881075.

cent compartment (= absorption), its transport to deeper compartments (= distribution), its biotransformation and its elimination from the body (Hildebrand, 1994). These steps together determine the percentage of drug available in the circulation for further distribution to the macro- molecular receptor site ( = bioavailability). The extent to liberate from the formulation the incorporated drug in a sufficient amount is herein of a particular importance (= pharmaceutical availability). Thus, the whole process determining the bioavailability of a drug can be divided into two phases - first, into a pharmaceutical phase where drug liberation takes place, and second, into a so-called pharmacokinetic phase which describes the course of the drug in the body. As a result of these two processes a biological response (= biological activity) leading to a

therapeutic effect may be obtained. Passive diffusion is the most significant transport mech-

anism for the majority of drugs, with the physicochemical properties of both the drug and the permeation barriers

0031-6865/96/$15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved PII SO03 I -6865(96)0003 1-3

310 G. Camenisch et al. /Pharmaceutics Acta Helvetiae 71 (1996) 309-327

(membranes) being the major rate determinants for transport (Higuchi et al., 1981; Martin, 1981; Seydel and

Schaper, 1982). For the membrane its lipophilic character (Higuchi et al., 1981) the presence of water pores (Higuchi et al., 1981) the membrane surface area available for

absorption (Levine, 1970) and the presence of stagnant

diffusion layers (Hayton, 1980) are all considered to be important factors influencing membrane transport. The characteristics of a drug influencing transport rates and permeability are size, lipophilicity, hydrogen bonding ca- pability and the extent of ionization. Various mathematical models have been developed to describe the relation between drug structure and membrane permeation.

The objective of this paper is to review the relationships between permeation processes and physicochemical properties of drug compounds in order to identify differences, similarities, advantages and disadvantages between a number of key absorption models currently in use. Particularly, we discuss modern views on lipophilicity and its components molecular size and hydrogen bonding potential.

2. Membrane transport

2.1. Biological membranes

According to the so-called fluid-mosaic-model biomembranes are composed of a lipid-double-layer (Pfeifer et al., 1995). The double layer results from the orientation of the amphiprotic lipids (phospholipids, glycolipids, cholesterol) in the aqueous medium. Because of the high flexibility of the membrane lipids they are able to perform transversal/lateral movements within the membrane. Thus, biological membranes behave as a kind of two-dimen- sional liquid (= fluid). In the membrane different proteins are embedded performing different functions. Some proteins form selective ion-channels (Na+, K+, Ca*‘, Cl-). By the interaction of membrane proteins at the contact surfaces between single cells so-called tight junctions are

formed. In most membranes, with the exception of endothelial cells in the blood-brain barrier, these tight junctions contain fenestrae, which can be regarded as pores filled with water (Tanaka et al., 1995). The dimensions of these pores have been estimated to be in the range of 3-10 A (Lennemls, 1995). However, the radii are not fixed. Depending on the observed biomembrane and the presence of mechanistic control nutrients (e.g., Ca’+, glucose and amino acids) the pore size may vary (Artursson, 1991; Noach, 1994). Also the number of the tight junctions depends on the biomembrane type. For the small intestine, it amounts about 0.01% of the whole surface (Artursson, 1991). Thus, the surface area of the lipoid part of a

biological membrane is much greater than the aqueous part defined by the tight junctions. It has been suggested that the whole surface of the biomembrane carries fixed an- ionic charges (Rubas et al., 1994) while others believe that only the tight junctions contain fixed negative charges

(Conradi et al., 1996).

2.2. Drug transfer processes through a membrane

The fluid-mosaic-model provides also an interesting

structural model for the understanding of different relevant drug transport processes across a membrane (Fig. 1) (Pfeifer et al., 1995). Compounds are able to move based on their thermic energy in the direction of a concentration gradient (= passive diffusion). For charged molecules

(ions) the electrochemical potential difference across a biological membrane may be an additional driving force

for passive diffusion (Mutschler, 1995). Cations migrate to the negatively charged membrane side (basolateral) and anions to the positively charged membrane side (apical). A basic requirement for passive diffusion is the foregoing dissolution of a compound (Pflegel and Pfeifer, 1980). Passive diffusion through a biomembrane may occur

through its lipoid structures ( = transcellular pathway), and through its water-filled pores (= intercellular, paracellular, pore or tight junction pathway). For transcellular passive diffusion a drug has to overcome two aqueous-lipid interfaces. Drug transport through an interface between two different phases is called penetration. By contrast, drug transport through a whole membrane is called permeation.

transcellular passive diffusion

paracellular passive diffusion

faciliated diffusion

active transport

t\

filtration

endocytosis

Fig. 1. Schematic representation of the different drug membrane transport

processes. Modified after Forth et al., 1987.

G. Camenisch et al. / Pharmaceutics Acta Heluetiae 71 (1996) 309-327 311

apical membrane

$ basolateral membrane

Fig. 2. Potential mechanisms for transcellular diffusion of a solute.

Modified after Burton et al., 1993.

For transcellular diffusion in principle two potential mechanisms exist (Fig. 2) (Burton et al., 1993). First, the solute distributes (penetrates) into the membrane and diffuses within the membrane to the basolateral side (Fig. 2a). This lateral diffusion is thought to be only possible in the exoplasmic/inner leaflet of the membrane, because diffusing within the exoplasmic/outer leaflet is precluded by the structure of tight junctions which are anchored in the outer

leaflet and therefore block a lateral diffusion in this leaflet (Noach, 1994; Conradi et al., 1996). However, this diffusion modus requires an initial passage of the solute to the inner leaflet and a subsequent return to the outer leaflet. Secondly, solutes may diffuse across the apical cell membrane and enter the cytoplasm before exiting the cell across the basolateral membrane (Fig. 2b). Till now there is no evidence of preference for one or the other way. All passive diffusion processes may be superimposed by an osmotic water flow, able to drag along the molecules (= solvent drag) (Pfeifer et al., 1995; Lennemls, 1995). A special form of solvent drag is the filtration on the aqueous pores as function of a hydrostatic water flow. Another kind of passive diffusion is given by the so-called jiicili-

tuted diffusion. This is a form of carrier transport in the direction of the concentration gradient and is therefore not requiring additional energy. Carrier transport is performed by different proteins embedded in biomembranes. Impor- tant systems include the H+/organic cation antiport system, the organic anion transport system, and the dipeptide transporter (Naasani et al., 1995). By contrast, P-glyco-

proteins are proposed to play a role in active efflux transport mechanisms (Hunter et al., 1993; Schinkel et al., 1995). Active transport denotes carrier transport processes consuming energy since transport often takes place against a concentration gradient. Very large molecules are transported by invagination of the membrane and subsequent

vesicularisation and devesicularisation (endocytosis or

trunscytosis).

2.3. Mathematical description of passive drug transport

2.3.1. Passive diffusion in solution

The rate of diffusion of a solute is expressed by Fick’s first law (Flynn et al., 1974):

dm _ = -DA; dt

where dm is the amount of solute (mg) diffusing in the time dt (s) across an area A (cm’) under the influence of a concentration gradient dC/dh (mg mol-’ cm-‘). D is known as the diffusion coefficient (cm2 s-l). It is not strictly constant but varies with concentration at constant temperature. The negative sign in Eq. (1) is necessary because diffusion occurs in the opposite direction to that of increasing concentration.

The diffusion coefficient of spherical particles that are of colloidal dimensions (i.e., considerably larger than the solvent molecules) is given by Eq. (2), which is known as the Stokes-Einstein equation.

RT EC_

6vrN,

where r is the radius of the spherical particle (cm), q is the viscosity of the liquid medium (Pa s), R is the gas constant, T is the thermodynamic temperature (K), and N,

is Avogadro’s constant. Eq. (2) does not apply to non- spherical particles because the frictional force that opposes their movement will vary with the orientation of the

particles.

2.3.2. Passive dijjkion through a membrane

The diffusion flow through a simple membrane not having any pores and separating two equal liquid compartments (same buffer media), under steady-state conditions (i.e., the amount of drug in the membrane is small compared to the amount of drug in the environment) can be

Fig. 3. Passive transport in a two-compartment model divided by a simple membrane according to Fick’s first law (Flynn et al., 1974).

312 G. Camenisch et al./ Pharmaceutics Acta Hebetiae 71 (1996) 309-327

described by applying Fick’s first law (Fig. 3) (Flynn et al., 1974; Hilgers et al., 1990):

dC.4 DP A p=,,(C,-C*)=P$g(C,-C*)

dt A A

= k”bS( c, - CA) (3a) In this equation dC,/dt is the increase of drug concen-

tration in the acceptor chamber during a time considered (mg s-’ ml -I), D is the diffusion coefficient through the membrane (cm* s- ’ >, A the membrane surface (cm2), V,

the solvent volume in the acceptor chamber (ml), C, the initial solute concentration in the acceptor chamber (mg/ml), C, the initial concentration of solute in the donor chamber (mg/ml), h the thickness of the membrane

(cm) and P the partition coefficient between the membrane and its environment. For a given compound in a given system, a coefficient Peyk can be defined. P$ is called the permeability coefficient (cm/s). The product of PC:: and A / VA is the so-called observed absorption rate constant kobs (l/s). The reciprocal of the permeability coefficient is called permeation resistance Robs (s/cm). In a similar way the decrease of drug concentration in the donor chamber can be described (Hilgers et al., 1990).

From Eq. (3a) the following useful relationships can be derived:

pzz _ np _ kobsVA h A (3b)

DAP A kobs = -=p obs _

VA h erm V,

dC,V, dQ ~ = dt = Pe;;A(C, - CA)

dt

dC*V, -=J=P;:(c,-CA)

dtA

(34

where dQ/dt (mg s-l> and J (mg s-’ cm-2) indicate the influx of substance into the acceptor chamber during a time considered. These equations interrelate the various descriptors PlE, kobs, Robs, dQ/dt, J, and CA that are commonly used in the literature to describe absorption processes.

2.4. Role of lipophilicity

As described by Eq. (3a), the permeability through a simple membrane depends on the partition coefficient P between this membrane (memb) and its environment (envir). The partition coefficient, usually expressed in its

.o

.z

2 & d

Lipophilicity

Lipophilicity

Lipophilicity

d)

L

Lipophilicity

Fig. 4. Often observed lipophilicity-permeability relationships. (a) Linear

dependence. (b) Bilinear dependence reaching a plateau value at high

lipophilicity values ( = hyperbolic dependence). (c) Sigmoid dependence.

Cd) Parabolic dependence ( - - ) and bilinear dependence having a local

maximum (--_).

logarithmic form (log P), constitutes a measure of lipophilicity and is defined according to (Fig. 8a):

C p+-EE (4)

L envir

Plotting permeability (Pzz, kobs, etc.) through any biological membrane (small intestine, colon, blood-brain barrier, skin, Caco-2, etc.) versus lipophilicity, linear (Fig. 4a) (Kakemi et al., 1967a), hyperbolic (Fig. 4b) (Leahy et al., 1989; Wagner and Sedman, 1973), sigmoid (Fig. 4c) (Ho et al., 1977, 1983), parabolic (Hansch, 1969, 1971) or bilinear (Fig. 4d) (Kubinyi, 1976) dependencies have been observed, depending on the data set used, mostly involving rather homologous compounds. However, when data for unrelated compound sets are used, such regular relationships may easily be obscured.

2.4.1. Determination of partition coeficients The experimental determination of partition coefficients

between biological membranes and their natural environments is a very difficult task. Therefore, model systems simulating biological membranes have been proposed. Since the pioneering work of Fujita et al., the system 1-octanol-water was established as a good system simulating the above mentioned distribution (Fujita et al., 1964). In particular the water saturated 1-octanolic phase is highly structured (Smith et al., 1975). However, other solvent

G. Camenisch et al./ Pharmaceutics Acta Heluetiae 71 (1996) 309-327 313

systems deserve attention as well. As I-octanol is an amphiprotic solvent, having hydrogen-bond donor and acceptor properties, other systems were proposed yielding

information complementary to I-octanol data (van de Wa- terbeemd and Testa, 1987). Chloroform was proposed as hydrogen-bond donor solvent, PGDP ( = propylene glycol dipelargonate), different esters (e.g., n-butyl acetate) and ethers (e.g., di-n-butyl ether) as hydrogen-bond acceptors, and alkanes (e.g. n-heptane, cyclohexane) as inert solvents. The classification of the organic solvents given above implies that they will interact differently with a given solute, resulting in different partition values. Differences between log P values from different solvent systems therefore contain information on hydrogen bonding properties

of the partitioning solute (see Section 2.4.3). These solvent systems are believed to simulate different kinds of biological membranes (e.g., skin, blood-brain barrier, intestinal membrane, etc.). Biological membranes are structured as bilayers and membrane phospholipids have charged head groups (e.g., zwitterionic). As the above-mentioned solvent systems do not reflect these features, other systems for partition coefficient determination, using agglomerates (liposomes) of different membrane lipids (cholesterol, phospholipids, etc.) or different synthetic tensides, were proposed (Gobas et al., 1988). It was shown, that only for very lipophilic and high molecular size compounds a loss of linear correlation between such ‘lipid’ systems and the standard 1-octanol system is observed (Gobas et al., 1988). A further practical advantage of water-saturated 1-octanol is that it is a reasonably good solvent for many organic compounds, while other organic solvents have a more limited range due to solubility problems. A further reason to refer to 1 -octanol-water partition coefficients is the fact that large compilations of experimental log P values are available, various theoretical approaches to estimate log P values are based on this solvent system, and that theories referring to drug absorption as presently reviewed are all based on this system. Various aspects of its use have recently been compiled (Pliska et al., 1996).

Since the determination of partition coefficients is very time demanding, alternative lipophilicity descriptors, based on various chromatographic methods (thin-layer chromatography TLC, reversed-phase high-performance liquid chromatography RP-HPLC, centrifugal partition chromatography CPC, capillary electrophoresis CE) have been developed. Particularly RP-HPLC methods using different stationary phases (alkyl-bonded silica, end-capped silica, phospholipid-coated silica, polymer-coated silica, etc.> and using different solvents (methanol, acetonitrile, micellar solvents, etc.) are widely used (van de Waterbeemd et al., 1996b). The lipophilicity index measured by these RP- HPLC methods is called the capacity factor k (or some-

times written as k’), mostly expressed in its logarithmic form logk.

2.4.2. Partition coejj?cients and their dependence on chem-

ical structure

Numerous investigations concerning structural contributions influencing distribution have been reported (Kubinyi, 1979a; van de Waterbeemd and Testa, 1987; Leo, 1993). These investigations have led to the development of various approaches to estimate 1-octanol-water log P values from molecular structure (Leo, 1993, 1996). The merit of such approach is that log P values of structural proposals can be evaluated prior to synthesis. Partition coefficients encode two major structural contributions, namely a cavity or volume related term reflecting the energy needed to create a cavity in the solvent and an exoergic interactive term which results from solute-solvent interactions. The

volume parameter V can account for the cavity term, and the polarity parameter was designated A. Thus, the partition coefficient can be described by (Testa and Seiler, 1981; El Tayar et al., 1992):

logP=aV-A (5)

For a simple solvent-water system (e.g., l-octanol- water), the A term includes dipole-dipole interactions and, especially, H-bonding capacity of the solute.

2.4.3. Correlation between partition coefJicients from dif-

ferent solvent systems

Collander showed that log P values from different solvent pairs are linearly correlated (Collander, 1950, 1951). Thus, log P values from any solvent system ( = solv) can be converted into P values of their 1-octanol (= act)

system by:

log PSolv = blog POct + d or PSOlv = d P& (6a)

As seen in Eq. (5), this equation is only valid, if the organic solvents have similar physical properties, particularly similar hydrogen-bonding capacity (Leo et al., 197 1). These restrictions have prompted Seiler to define a parameter XI, describing the hydrogen-bonding capacity of a given solute. ZIH is calculated as the difference between the I-octanol-water partition coefficient and the partition coefficient of a cyclohexane-water system (Seiler, 1974). An almost identical approach was proposed by Fujita et al. using the indicator HB to account for hydrogen-bonding capacity of the solutes (Fujita et al., 1977). Thus, in extension of Eq. (6a), the following extended Collander

314 G. Camenisch et al. / Phamaceutica Acta Heloetiae 71 (1996) 309-327

equations were defined (van de Waterbeemd and Testa,

1987):

log Pcyclohex = log P,,,t + CZ, + b (6b)

and

log P,,i,,gas = alog P,,, + HB + b (64

More recently this concept has been applied to studies of membrane permeation. The difference between log P values from the 1-octanol-water system and an alkane- water system (Eq. (6d)) appears to correlate to membrane permeation, including transport through the blood-brain barrier (Young et al., 1988; van de Waterbeemd and Kansy, 1992).

Alog P = log P,,, - log Palkane (6d)

Since the 1-octanol-water system in most cases was shown to be an acceptable simulation system for biological membranes, it can be assumed that both processes permeation and partitioning involve similarities in hydrogen

bonding. Thus, as a first approximation Eq. (6a) can be used for the correlation between biological membrane- water partition coefficients and l-octanol-water log P val-

ues.

2.4.4. Rate constants of drug partitioning For a partitioning process between an aqueous phase,

with solute concentration Caq, and an organic phase, with solute concentration Cars, the partition coefficient can be defined as:

C obs

‘_F=k’ kobs (7)

d4 2

where kpbs is the rate constant of drug partitioning from the aqueous phase to the organic phase and kzbs is the corresponding reverse constant.

The rate constants for drug partitioning can be determined experimentally in three-compartment systems, made up from two separated aqueous phases and an organic phase which is in contact with both aqueous phases (Fig.

Fig. 5. Three-compartment model for the determination of rate constants

for drug partitioning (Lippold and Schneider, 1975a,b; Kubinyi, 1976). aql: aqueous application compartment. org: lipid compartment. aq2:

aqueous receiver compartment. kpbs: rate constant for transport of drug

molecules from aqueous to lipid phase. kzhs: rate constant for transport of drug molecules from lipid phase to aqueous phase.

obs logk

b log P

Fig. 6. Bilinear relationship between the forward rate constant of the drug

partitioning kPhs and the backward rate constant of the drug partitioning

kqhs, respectively, and the partition coefficient P according to Eq. (9).

Based on Eq. (lo), Jc~” for lipophilic compounds approaches a plateau

value k,,, and kyhs for hydrophillic compounds approaches a plateau

value korg (compare Fig. 7).

5) (Rosano et al., 1961; Doluisio and Swintosky, 1964; Kubinyi, 1979a), or in a two-compartment water-organic solvent system (van de Waterbeemd et al., 1980, 1981).

Using experimental data of Lippold and Schneider, Kubinyi derived for the relationship between kp”’ and kib’ (Lippold and Schneider, 1975a,b; Kubinyi, 1976):

k, obs =

- pkpb” + c (8)

where p and c are constants. If either and kibS or kp”’ in this equation are substituted by Eq. (7), the following bilinear relationships can be derived (Kubinyi, 1979a):

log k pb”=logP-log( pp+ 1) +c (9a)

logkzb”= -log( /3P+ 1) +c (9b)

As long as /3P is smaller than 1, the kpbs values are linearly dependent on the partition coefficient P. As P increases, BP + 1 becomes larger than 1 and kpbs approaches a constant value (Fig. 6). The physicochemical meaning of the statistically obtained constants in Eqs. (9a) and (9b) was unclear. Van de Waterbeemd developed an interfacial drug transfer model rationalizing these bilinear results (Fig. 7) (van de Waterbeemd et al., 1980, 1981). According to this model the transfer of a compound from an aqueous to an organic phase can be described by a three-step mechanism. The transport towards the solvent interface is a diffusion process (rate constant k;), which is followed by an energy step when the drug is transferred from water to organic solvent (rate constant k,). The last step is a diffusion-controlled step away from the interface (rate constant k;‘). For the transport between equal volumes of water and organic solvent (Vaq = VO,> and for reversible diffusion steps, it can be assumed that k’, = ki

G. Camenisch et al. /Pharmaceutics Acta Heluetiae 71 (1996) 309-327 315

= k,, and ky = k’; = korg. The essential assumption in this model is that k, E k, B k,, E korg (i.e., equilibrium at the aqueous-organic interface expressed by the partition coefficient P is established instantaneously). This assumption implies that there is no accumulation of transferring solute at the interface and so the transport process can be described by steady-state kinetics as follows (van de Water- beemd et al., 1980, 1981):

logk pb”=logP-log ( LOa)

+ log korg

where (compare Eq. (3)),

naq A k,, = -

haq Kq (lla)

D’org A korg = -

horg Kq (lib)

Thus, based on Eq. (2>, k, and korg can be considered as constant at least for a series of closely related com-

pounds.

2.4.5. The pH-partition hypothesis The pH-partition hypothesis suggests that partitioning is

not only dependent on the so-called intrinsic partition coefficient of the uncharged compounds (log P), but also on the degree of ionization (Shore et al., 1957; Schanker et al., 1958; Hogben et al., 1959). Therefore pH also affects partitioning. It is important to know what percentage of a drug is non-ionized, as this species has a higher partition coefficient than the ionized fraction (Kubinyi, 1979a). The pH-partition hypothesis even claims that only the non- ionized form of the drug is able to permeate a membrane.

Thus, according to the classical pH-partition theory perme-

Fig. 7. Interfacial transfer of a drug R as a three-step mechanism

a) f-ii-- R

b, f---k R z RH+

[R4rg p=- lR1aq

D= [R&g

Mlaq +IRWaq

[RI P’=

erg +W+l w [Rlaq +M+lq

Fig. 8. Definition of different descriptors describing extraction of a

compound R at an aqueous (aq)-lipid (erg) interface. (a) intrinsic

partition coefficient log P, if only the uncharged compound is present. (b)

Distribution coefficient, D, assuming ionization to influence partitioning

and proposing the charged compound not to be able to distribute into the

organic phase. The definition of D is based on the pH-partition hypothe-

sis. (c) Effective distribution coefficient P’, taking into consideration that

both the uncharged and the charged form of a compound are able to

distribute into the organic phase in principle.

ability is expected to correlate not with the intrinsic partition coefficient but with the so-called distribution coefficient D of the solute, where D is defined as (Fig. 8b) (Leahy et al., 1989):

log D = log P + logfu = log P + log( 1 -f,) (12)

where f, is the fraction unionized and f, the fraction ionized, respectively. The correction terms f, and log(1 - fI) in this equation describe the pH dependence of log D and are different for mono acids, mono bases, diacids, dibases, etc. (Avdeef, 1996). Neglecting partitioning of ionic forms into the organic phase, for the different substance groups different correction terms were derived, such as the following Eqs. (13a) and (13b) for mono acids and mono bases, respectively (van de Waterbeemd and Testa, 1987; Scherrer and Howard, 1977):

acid: log D = log P - log( 1 + 10(pH-PK*)) (‘3a)

base: log D = log P - log( 1 + 10(pK~-pH)) ( 13b)

However, there is no strong evidence that ionized species do not partition into organic environments in particular because during absorption experiments often a so- called pH shift can be seen (Crouthamel et al., 1971). The term pH-shift refers to the observation that the fraction absorbed is often higher than expected from the pH-partition hypothesis. Two types of deviation from the classical pH-absorption curves may be observed. First, there can be

316 G. Camenisck et al. /Pkamaceutica Acta Hebetiae 71 (1996) 309-327

Fig. 9. Classical pH-absorption curves (-_) and observed pH-shifts

(- -) for an acid. (a) Upwards shift of the lower part due to a

substantial permeation of the ionic form of a drug compound. (b) Parallel

shift due to the presence of an unstirred aqueous diffusion layer adjacent

to the membrane.

an upwards shift of the lower part of the curve (Fig. 9a>. This shift is supposed to be due to a substantial permeation of the ionic form of the drug through the membrane (Fig. 10) (Crouthamel et al., 1971; Kakemi et al., 1965; Nogami and Matsuzawa, 1961, 1962). In particular, ion-pair partitioning may be more important than was thought originally (Scherrer, 1984). Thus, for transcellular diffusion it is expected that permeation correlates not with the distribution coefficient D but with the effective distribution coefficient P’ (sometimes written as Papp> of the solute, taking into account a possible extraction of ionized compounds into the organic environment (Fig. 8c) (Kubinyi, 1979a). As a consequence of this additional factor influencing distribution, the correction term in Eq. (13) must be adapted. For mono acids and mono bases, the following empirical equations were proposed, describing the experimental log P’ - pH relationship more appropriate (van de Waterbeemd et al., 1996b):

mono acid: log P’ = log P - (1 - tanh(pK, - pH + 1))

( 14a)

mono base: log P’ = log P - (1 - tanh(pH - pK, + 1))

( 14b)

Second, there can be a parallel shift of the pH-absorption curve from the pH-dissociation curve (to the right for acids and to the left for bases), especially for very lipophilic compounds (Fig. 9b) (Beckett and Moffat, 1968; Kakemi et al., 1969; Schaper, 1982). Biological phenomena which may be responsible for these shifts include the presence of pH differences at the surface of the membrane (= microclimate pH) (Hogben et al., 1959; Blair and Matty, 1974; Blair et al., 1975) and binding of drug to the membrane surface (Kakemi et al., 1967b,1969). Other authors proposed the rate constant for the partitioning step out of the membrane (Wagner and Sedman, 1973) or, as a

I I I I?+ i?+x- /?+ k+

Fig. 10. Different hypothesis for a possible membrane transport of

charged species: (a) active transport using opposite-charged proteins, (b)

as ion-pair more. or less compensating charge, (c) charge compensation by

high lipophilicity and/or charge delocaliaation within the compound

itself, (d) passive diffusion via aqueous pores.

most plausible explanation, the presence of an unstirred aqueous diffusion layer adjacent to the lipid membrane (Suzuki et al., 1970a,b; Ho et al., 1972) to be responsible for these parallel pH-shift phenomena.

2.5. Stagnant aqueous diffusion layers

Hydrodynamics predict the existence of a stagnant liquid layer near a surface. Thus, in series with a membrane such stagnant aqueous boundary layers, often referred to as the unstirred layers, can be assumed (Fig. 11). The bulk liquids are assumed to be homogeneous, which in model studies is achieved by stirring. Similar to the dissolution model proposed by Nernst and Brunner (Nernst, 1904; Brunner, 1904), the aqueous layers are proposed to form important permeation barriers for passive diffusion, especially for lipophilic compounds (compare Fig. 6). Mixing of the bulk phase by convection causes a decrease of the thickness of an aqueous layer and influences therefore permeability. A mathematical description of the hydrody- namic phenomena at a surface, derived for a rotating disk, was given by Levich (Levich, 1962). In vivo the thickness of the stagnant aqueous diffusion layers is influenced by

- - DLaq, DLaq2

Fig. 11. Three-compartment model including stagnant aqueous diffusion layers. aql: aqueous application bulk phase. org: lipid compartment. aq2:

aqueous receiver bulk phase. DL,,,, DL,,,: aqueous diffusion layers.

G. Camenisch et al./ Pharmaceutics Acta Helvetiae 71 (1996) 309-327 317

mechanic mixing due to agitated flow of the intestinal

fluid or blood flow. Therefore, it is sometimes believed that in vivo the stagnant aqueous diffusion layers are nearly non-existent and have no effect on drug absorption,

2.6. Effect of molecular size

For the diffusion coefficient D in biological membranes and continuous fluid mediums, in accordance with Eq. (2), a dependence on molecular size could be shown (Lieb and Stein, 1971; Walter and Gutknecht, 1986; Xiang and An- derson, 1994). For continuous fluid mediums this effect is relatively small (D a 1 /V”, where V the molecular volume of the solute and II = 0.6) (Walter and Gutknecht, 19861, while in biological membranes a rather strong dependence on molecular size was observed (D a e-Iy ‘“. V”, where n = 0.8 and (Y zz 0.005) (Xiang and Anderson, 1994). The rigid molecular matrix of a membrane is thought to act as a kind of molecular sieve for transcellular diffusion (Bunge and Cleek, 1995). Also for the partition coefficient P a relationship with molecular size has been established (Testa and Seiler, 1981; El Tayar et al., 1992) (see Eq. (5)). B es1 es ‘d b y passive transcellular diffusion

through the membrane lipids, some compounds may go through the paracellular pathway. This is also limited by the molecular size, and is called sieving effect (Leahy et al., 1989). The extent to which paracellular route is available is given by the sieving coefficient 0, which is the ratio of solute concentrations within the connective stream and in the bulk solution. Thus, 0 for a compound not held back by the pores is 1 and for a compound totally retained is 0. Renkin described 0 for spherical solutes diffusing within a cylindrical channel as (Renkin, 1954):

I r \2 @= 1-i I J

.[l -2.104(;)+2.09(~)3-0.95(;)5] (15)

which compares the molecular solute radius r with the pore radius R. Leahy and coworkers derived based on experimental data, an empirical model for a sigmoid relationship between 0 and molecular size according to the following equation:

(16)

where a, b, c and f are parameters to be determined from regression analysis. MW is the molecular weight of a compound often used as a simple accessible molecular size

2000 r

0 ’ 1 /

0 260 520 780 1040 1300

MW Fig. 12. Correlation between the molecular van de Waals volume V [A31

and the molecular weight MW [g/mol] or a heterogeneous data set

(Sietsema, 1989).

descriptor. In Fig. 12 for a heterogeneous data set of 373

compounds (Sietsema, 1989), having molecular weights from 130 up to 1200, a plot of the molecular van der Waals volume V versus the corresponding molecular weight MW is presented, indicating that molecular weight indeed reasonably substitutes molar volume (r = 0.990). It is assumed that only compounds with MW < 200 are able to use the paracellular pore pathway (Lennernls, 1995).

2.7. Effects of charge and other interactive terms on

permeation

The distribution (log D) at an aqueous-lipid interface is a function of molecular size and an interactive term encod- ing for H-bonding, dipole-dipole interactions, ion-ion interactions, etc. (see Eq. (5)). Just like the diffusion coefficient strongly depends on solvent-solute interactions, also for D a relationship with these interactive terms can be expected, To describe this dependence no satisfac- tory theory has been developed so far, making it impossi- ble to provide an exact picture of these diffusion phenomena in liquids and biological membranes (Walter and Gutknecht, 1986). Additionally, the fixed anion charges at the surface of biological membranes are postulated to contribute to an ion-selectivity of permeation (Rubas et al., 1994; Conradi et al., 1996).

318 G. Camenisch et al. /Pharmaceutics Acta Helvetiae 71 (19961309-327

3. Linear drug absorption models

The central paradigm of structure-permeability correlations can be expressed as:

permeability

= f( lipophilicity, molecular size, interactive terms)

As seen above, all these properties are believed to play an important role for drug transport processes. In literature, different modifications of such equations can be found. To rationalize the in some cases observed linear permeability-lipophilicity relationships (see Fig. 4a) some authors postulated simple equations of the type (Kakemi et al., 1967a,b; Houston et al., 1974):

log k “b”=at-blogP (17)

Considering approximatively that the diffusion coefficient D through a membrane is inversely proportional to the square root of the molecular weight (Koizumi et al., 1964a), based on Eq. (3b) permeability can be ‘normal- ized’ for molecular size according to (Rim et al., 1986; Pardridge and Mietus, 1979):

P log Pe;“k = a + blog -

i i m (18)

To account for molecular size dependence of permeation, other authors proposed equations of the type (Potts and Guy, 1992; Lien and Gao, 1995):

log (permeability) = a + blog P + clogMW (19)

From Eq. (5) it is evident, that the parameters log P and MW may be intercorrelated making such equations ques- tionable.

However, generally observed permeability-lipophilicity plots will not be linear (see introduction of Section 2.4) and therefore various non-linear models have been developed (see Section 4). In some approaches the linear approach is obscured by writing log P or log D out in its components. Thus, taking into consideration the dependence of permeability from interactive terms, as hydrogen bonding or dipole-dipole interactions, extended equations of the following type have been considered (Abraham et al., 1995; Potts and Guy, 1995):

log PZ”,” = a+b,rr,H+cCa,H+dCg+eV, (20)

where TF is the solute dipolarity/polarizability, Co,” the solute effective or overall hydrogen-bond acidity, C pp the solute effective or overall hydrogen-bond basicity and V, the McGowan characteristic volume. Again, the danger of interrelationships between the single parameters exists (see Eq. (5)). In addition, all these parameters are rarely available (Potts and Guy, 1995). Published correlations

may be misleading, since sometimes collinearity among descriptors was ignored and mostly the compound sets consist of rather homologue structures (Houston et al., 1974; Potts and Guy, 1992, 1993, 1995; Abraham et al., 1995).

4. Non-linear drug absorption models

Absorption phenomena can also be described by considering the time-course of drug concentration as a function of distribution between alternate subcellular compartments, as the extra- and intracellular aqueous phases and membranes (Balaz, 1994). Thus, the model parameters are given in terms of structure of both the biosystem and the drugs. It can be distinguished between equilibrium models

and non-equilibrium models. In equilibrium absorption models the fate of drugs is described in the time interval starting after distribution (storage, biotransformation) was finished, and terminating before elimination starts. In non- equilibrium models the elimination is included in the description of the drug’s fate. Because of the resulting complexity, the non-equilibrium models are mathematically more tedious.

The influence of temperature on passive membrane transport is mostly not considered and membranes are assumed to behave as coherent lipid layers. Generally, the pharmaceutical availability is neglected completely. The models presented in this review have in common that they are based on data measured in a system using single doses and not using any permeation enhancers (e.g., micelles or cyclodextrins). The pH-partition hypothesis also forms the basis of the vast amount of non-linear absorption models.

4.1. Model&g of membranes without pores

The absorption model devised by Hyde and Lord (Hyde and Lord, 1979) and Kubinyi (Kubinyi, 1979b) is equilibrium based. They considered the distribution of a drug for a simple system with an aqueous compartment (aq), a lipid compartment (erg) and a receptor compartment of interme-

Fig. 13. Equilibrium model according to Kubinyi (Kubinyi, 1979b). aq: aqueous application compartment. org: lipid compartment. ret: receptor

compartment. kpbs: forward and backward rate constants for passage

between the compartments.

G. Camenisch et al. /Pharmaceutics Actu Heluetiae 71 (1996) 309-327 319

diate lipophilicity (ret) (Fig. 13). The amount of drug in each compartment is given by the product of concentration Ci and volume v.. The sum of all products is equal to the administered dose S. The distribution of S is given by:

S = CL?4 v”q + Corg Vorg + CR, v,,, (21)

Due to the extremely small volume of the receptor phase (as compared with the volumes of the other phases) the product C,, V,,, always will be small. With P, =

Corg/Caq and P, = Crec/COrg (compare Eq. (7)), C,,, can calculated according to Eq. (22) ( p = Vorg/Vaq>:

CGX = sp2

vT,,( PPI + 1) (22)

If P, is related by an Collander equation to P, (see Eq. (6a)), Eq. (22) can be written in the form of the following bilinear relationship (see Fig. 4d):

logC,, = alog P, - log( pi P, + 1) + const. (23)

For a similar three-compartment system, made up from two separated aqueous phases (aql and aq2) and an organic phase (erg) (see Fig. 5), the concentration Caq2 for any time t can be calculated from Eq. (241, if the forward and backward rate constant of drug partitioning are known (Kubinyi, 1976; van de Waterbeemd et al., 1978).

obs

ca,2 =

k* + kybse(-kphS+2k;bs)f e-kp%

,ybs + 2k;bs 2( kp”’ + 2kib”) - - (24)

2

By a mathematical transformation (Kubinyi, 1979a) and using Eqs. (9a) and (9b) for kp”’ and kzbs, for the dependence of CaqZ on lipophilicity at a fixed time t the following equation can be derived (Kubinyi, 1979a):

log C,,, = log P - 21og( pP + 1) + const. (25)

The dependence of C,,, on lipophilicity and the degree of ionization for equilibrium model systems, made up of an aqueous, a non-aqueous and a receptor compartment, has been treated by Martin (Martin and Hackbarth, 1976; Martin, 1978, 1979). For different models Martin considered the cases where only the un-ionized, where only the ionized, and where both the ionized and the un-ionized species are distributed to the receptor compartment. C,, versus log P curves were calculated for various degrees of ionization. Again in most cases linear relationships are obtained that level off at higher log P values and eventually approach a constant value.

Koizumi et al. applied a relatively simple model to describe the observed permeability-lipophilicity dependencies for absorption processes (Koizumi et al., 1964a,b). The model consists of two aqueous phases (aql and aq2) separated by a lipid membrane (erg) (Fig. 14). It is as-

a% erg

pm

+-I-

aq2

km

Fig. 14. Absorption model according to Koizumi et al. (Koizumi et al.,

1964a,b). aql: first aqueous phase; aq2: second phase; org: lipid mem-

brane; P,,,: partition coefficient between aql and membrane; k,: first-

order rate constant for transport of drug out of the membrane.

sumed that equilibrium partitioning of a solute between the first aqueous phase and the membrane takes place rapidly in comparison with diffusion from the lipid phase into the second aqueous phase. Back diffusion from the second aqueous phase into the membrane is considered negligible (sink conditions). For the disappearance from the first aqueous phase the following equation can be derived:

k kmpm obs _

1 + Pm

In this equation k, is the first-order rate constant for drug transport out of the membrane, and Pm is the partition coefficient between the first aqueous phase and the membrane. To the intrinsic partition coefficient P, P,,, is related by Eq. (27), in which V, and V,,, refer to the volume of the membrane and the first aqueous phase.

Pm= + P i i w

(27)

It was further assumed (Koizumi et al., 1964a) that transfer from the membrane into the second aqueous phase is determined by the diffusion rate through the membrane, in which case k, can be expressed by (compare Eq. (1 lb)):

k, = eAD (28) A is the interfacial area between the lipid and the aqueous phase (assumed to be equal on both sides), D the diffusion coefficient and e a constant. Considering D for small molecules to be inversely proportional to the square root of molecular weight MW and expressing P with help of the 1-octanol-water partition coefficient leads to:

k - obs _ abPoct rn( 1 + UP,,,)

(29)

Eq. (29) for low lipophilicities predicts a linear increase of drug absorption rate with increasing lipophilicity,

320 G. Camenisch et al. / Pharmaceutics Acta Helvetiae 71 (19961309-327

whereas at high lipophilicity values a plateau is reached. According to the pH-partition theory for ionizable drugs, Eq. (26) must be replaced by Eq. (301 (Koizumi et al., 1964a,b):

kobs = k” f” pm 1 If P (30) ’ ‘Ju’mu

where f, denotes the fraction unionized, and k,, and P,,,, are the rate constant for the transport and the partition coefficient of the unionized form, respectively.

Using Eq. (26), Wagner and Sedman (Wagner and Sedman, 1973; Wagner, 19751, have derived equations that relate drug absorption directly to measured lipophilicities as follows:

ohs _ knu p,“,t kl”P,“,t k-d =-

F + P,“,l Q + P,“,t

(31)

JU

where IZ, d and Q are constants for each given system, for drugs in which only the unionized form is absorbed and for a constant pH value. It is assumed, that the diffusion rate constant out of the membrane limits the absorption rate constant, so that kobs + k,, as Pact --f 00. The plot of k Ohs versus Pact is thus a hyperbola with asymptotic value k mu. The existence of an unstirred diffusion layer on the membrane is not questioned, but it is assumed that it represents no resistance to the penetration of the solute.

The model proposed by Stehle and Higuchi assumes stationary aqueous diffusion layers at the interface between the donor and the receptor aqueous phases and the membrane (Fig. 15) (Stehle and Higuchi, 1967, 1972). Apply- ing Fick’s first law (see Eq. (1)) at any location by assuming steady-state conditions (i.e., the concentrations have no time dependency) the membrane transport may be summarized by the following set of equations describing

D 1 2 A

aqueous I : aqueous donor j DLaq, erg DL I receptor phase ; w ; , phase

R j R

Fig. 15. Three-phase model according to Stehle and Higuchi. For a

uncharged solute R the concentration gradient under steady-state condi-

tions is indicated. DL,,,, DL,,,: aqueous diffusion layers. org: lipid compartment. D, 1, 2 and A indicate the different interfaces.

I I ” donor 2 phase

-c-,

Cm p,:, l L p&n ----

4-w P$ II

receptor phase

Fig. 16. Model according to Flynn and Yalkowsky, assuming n different

diffusion barriers in series (Flynn and Yalkowsky, 1972; Flynn et al.,

1974; De Haan, 1985). P&,,: permeability coefficient through every

individual homogeneous barrier in series; Pe$: permeability coefficient

through the composite barrier.

the flux J through each of the three consecutive barriers (compare Eq. (3a) and (3e)).

J _ DDdCD - Cl) Dl - h DI

J,, = ~12Wl - C2)

h 12

J _ WC2 - C*)

2A - h 2A

(=a)

(32c)

From qualitative theoretical considerations Flynn et al. derived the following equation for the apparent permeation through several diffusion barriers in series under steady- state conditions (Fig. 16) (Flynn and Yalkowsky, 1972; Flynn et al., 1974):

In this equation P is the partition coefficient in the ith barrier with respect to the donor phase, i represents every individual homogeneous barrier in series, n denotes the total number of barriers, h is the barrier thickness and R represents the diffusional resistance. For the deviation of Eq. (33) all interfacial resistances are ignored, assuming them to be negligible compared with the diffusive resistances.

Inserting Eq. (32a)-(32c) in Eq. (331, Stehle and Higuchi derived for their model the following hyperbolic relationship (see Fig. 4b):

(34)

When P is small, the second term of the denominator is much larger than the first, so that J increases with P. At further increased values of P, the second term of the denominator becomes smaller, eventually becoming negli-

G. Camenisch et al. / Phanaceutica Acta Heluetiae 71 (1996) 309-327 321

erg DL ’ 4 I

aqueous i donor i phase ! _

: R,ql

J

R 01s

aqueous

b receptor

,: phase

R : a@ ,

Fig. 17. Three-phase absorption model based on the model of Stehle and

Higuchi according to Flynn and Yalkowsky (Flynn and Yalkowsky, 1972;

Yalkowsky and Flynn, 1973). org: lipid membrane. DL,,,, DL,,,: aque-

ous diffusion layers. I?,, , R,,,: resistance of the aqueous diffusion layer

to drug penetration, whereby R,, = R,,, + R,,,. Rorg: resistance of the

membrane to drug penetration. J: flux of the drug.

gible so that the flux is entirely aqueous diffusion layer controlled. Stehle and Higuchi applied their model also to ionizable compounds.

According to Yalkowsky and Flynn (Flynn and Yalkowsky, 1972; Yalkowsky and Flynn, 1973) the steady-state flux J through a model membrane as proposed by Stehle and Higuchi, can also be described by

(Fig. 17):

where AC is the concentration difference across the bar-

rier, R, and Rorg are the resistances of the aqueous diffusion layer and the membrane to the solute penetration, and P is the partition coefficient between membrane and luminal aqueous phase. Under physiological conditions sink conditions can be assumed for the aqueous receptor phase. Hence the concentration in this compartment is more or less zero. Thus, AC can be approximated by the concentration of the donor chamber Co.

Using Eq. (35), Flynn and Yalkowsky also introduced the concept of solubility control of absorption since the flux across the lipid membrane depends on the concentration gradient (AC) over the membrane (Flynn and Yalkowsky, 1972; Yalkowsky and Flynn, 1973; Yalkowsky et al., 1974). Therefore, within a homologous series, the decreasing flux at higher lipophilicities was proposed to be due to decreasing aqueous solubilities. When corrected for solubility, all cases could be corrected to the hyperbolic case.

While Yalkowsky and Flynn did not extend their model to a general equation for non-linear regression analyses,

Martin reconsidered their model to obey the following bilinear equation (Kubinyi, 1979a):

log J = alog P - log( dP b + e) + const. (36)

According to Kubinyi this equation can be transformed to Eq. (37) using the mathematical equivalence of log(dPb

+ e) and log i i

d Pb + 1 + loge (Kubinyi, 1979a): e

log J = alog P - log( SPb + 1) + const. (37)

A further simplification is possible if one considers that for 6Pb <( 1 and for 6Pb z+ 1, (6Pb+ 1) can be expressed by ( PP + l>b, where p = al/b (Kubinyi, 1979a):

log J = alog P - blog( pP + 1) + const. (38)

Based on the interfacial drug transfer model (see Fig. 7) van de Waterbeemd and Jansen proposed a model describing membrane diffusion as the sum of two distribution steps (van de Waterbeemd and Jansen, 1981) (Fig. 18). If the stagnant organic layers inside the membrane are thought to be equal at steady-state for the observed forward rate constant k$ the following equation (because of sink conditions the observed backward rate constant k;i” can be neglected) was derived:

log k$ = log kobs

= log P, - log( 2 + p1 P, + p2 P2) + log korg

(39)

k k where p, = F, & = F and P, and P, are the

aql a@ partition coefficients between lipid membrane and aqueous phase 1 or aqueous phase 2, respectively.

De Haan applied Flynn’s Eq. (33) to a two-compartment model similar to the interfacial drug transfer model,

: DL 1 ad lipid IDL I orgz

membrane : %q* ; aqueous

: receptor : phase

Fig. 18. Model according to van de Waterbeemd and Jansen (van de

Waterbeemd and Jansen, 1981) assuming on both sides of each lipid

aqueous interfaces stagnant diffusion layers. DL,, , DL,,,: aqueous

diffusion layers. DL,,, , DL,,,, : organic diffusion layers in the lipid

membrane. kybS, k;bs: rate constants for the forward and reverse transport over the interface 1. kjobs, kibs: rate constants for the forward and reverse

transport over the interface 2. k$, k$“: rate constants for the forward

and reverse drug transport over the entire membrane.

322 G. Camenisch et al. / Pharmaceutics Acta Heloetiae 71 (1996) 309-327

consisting of an aqueous and a lipophilic barrier in series (see Fig. 7). Under steady-state for the observed forward permeability coefficient follows (De Haan, 198.5; De Haan and Jansen, 1985):

(40)

Multiplying both sides by A / Vaq exactly yields Eq. (lOa), indicating that both approaches are identical. Based on these considerations it is evident that:

(41a)

porg = korg Yiq p erm A

(4lb)

Under normal circumstances A and Vaq are constant.

k, and korg can be considered as constant too, at least for a series of closely related compounds.

De Haan applied these considerations to various earlier absorption models (De Haan, 1985). For example the model of Stehle and Higuchi (see Fig. 15) can be rewritten as the following hyperbolic relationship:

(42)

where according to the discussed theory (Y (= korg . Vaq . A-‘) and Pzybs (= (l/P::) + (l/P:%) are constants.

4.2. Modelling of membranes with pores

According to Scheuplein, for independent diffusional pathways that are linked in parallel, the total flux J,,, through the composite is simply the sum of the individual fluxes through the separate routes (Scheuplein, 1966). Thus, for a membrane having an aqueous and a lipoidal pathway in parallel the total amount of substance reaching the acceptor chamber during a defined lapse of time can be described by (Flynn et al., 1974):

dQtO’ -=Pphp,:A’“‘(C,-C,)

dt

= J tOtA’“’ = J OrgAW3 + JhydAW (43) where the superscript hyd denotes the aqueous part of the membrane and the superscript tot the whole membrane including the aqueous and the lipoidal part of the membrane. A’“’ can be calculated as the sum of A“Q and Ahyd.

i DL,, w

aqueous i donor I phase [

: R,,

-1 ZJllJr

: R

hyd ):

I -

Fig. 19. Model according to Ho et al. (Ho et al., 1972) where an aqueous

boundary layer CDL,) is in series with the biomembrane consisting of

parallel lipoidal (erg) and aqueous pore (hyd) pathway. R,,: resistance of

the diffusion layer to drug penetration. Rorg: resistance of the lipoidal

part of membrane to drug penetration. R,,,: resistance of the aqueous

pore part of the membrane to drug penetration.

Thus, based on Eq. (43) the permeability coefficient PC::

of such a barrier can be described as follows (De Haan,

1985):

AW A’@

P tot _ erm - P2 A”! + P.22 - = Pz;yforg + PLi(PfhYd

A mf (44)

where f “g and f hyd are the fractional areas of the parallel hydrophilic and lipophilic diffusional routes, respectively. Pey: is determined by the diffusion rate of the transferring compounds in the channel medium and can therefore for a biological membrane described by (compare Eq. (3b)) (Ho et al., 1977, 1983):

pW= !!!%@ erm

h hyd (45)

where 0 is the sieving coefficient (see Eq. (15) and (16)). Based on Eq. (44) the paracellular route is generally considered as relatively inefficient compared to the transcellular pathway, since the intercellular space in biological membranes is very small compared to the whole surface. Such considerations do not take into account the volumetric fluid flow Jeuid (cm3 s) in the intercellular space (= solvent drag), which determines additionally the passive paracellular drug absorption. Recent data for the intestinal mucosa point out, that this fluid flow for small hydrophilic compounds up to a maximal molecular weight of about 200 affects drug absorption quantitatively (Leahy et al., 1989; Lennernas, 1995; Hill et al., 1994). Thus, Pey: in Eq. (45) should be rewritten as:

(46)

Since forg3 fhyd and Peyz can be considered as constant, at least for a series of closely related compounds in a given

G. Camenisch et al. /Pharmaceutics Acta Heluetiae 71 (1996) 309-327 323

system, the resulting total permeability constant is mainly

affected by the permeability constant through the transcellular organic-lipid route.

A further approach was presented by Higuchi, Ho and coworkers. Based on Eq. (33) and Eq. (44) assuming only a stagnant aqueous layer in front of the outer part of the membrane, a new model of sigmoidal nature (see Fig. 4c) taking into account a parallel diffusion pathway through aqueous pores was derived (Fig. 19) (Ho et al., 1972, 1977). They used the model with the premise that all molecular species (ionized and non-ionized) can pass through the aqueous pores under the influence of osmotic and hydrostatic flow and that the lipid component of the heterogeneous phase is only accessible to the non-ionized species (pH-partition hypothesis). For kobs, the absorption rate constant, Eq. (47) was derived:

A k obs _ _-

VD pez

(47)

In this equation A / V, represents the geometrical surface area/volume ratio of the donor side, PGk, P,“,C and Pe:‘,” are, respectively, the permeation coefficients for the drug in the aqueous diffusion boundary layer, the lipoidal membrane and the pores, and fu represents the non-disso- ciated drug fraction at the membrane surface in the aqueous boundary layer side. According to Higuchi, Ho and coworkers the additional way through the tight junctions is considered as relatively inefficient compared to the transcellular pathway. Therefore, the tailing effect at low lipophilicity values (lower part of the sigmoidal curve of Eq. (47)) is believed to be due to the uptake of these small,

log P$$

Y----= i

log D

Fig. 20. Extended pH-partition model according to Leahy et al. (Leahy et

al., 1989). The curves represent the postulated effects of 1ogD and

molecular weight MW on the permeability (P&) from Eq. (50). The curves are drawn for different molecular weights, varying from 100 to

800 Da.

-4

0 o/

/ . -7

_-- I

-6 -4 -2 0 2 4

log D

Fig. 21. Experimental permeability-lipophilicity relationship for Caco-2

cell permeation (Artursson and Karlsson, 1991). For the main group of

compounds a sigmoidal relationship can be assumed ( - - ). The outliers

below the curve have highest molecular weight in the series (compare

model of Fig. 20), while the two outliers above the curve have a

molecular weight below 200 and thus may use the aqueous pore pathway.

hydrophilic compounds via this inefficient paracellular route. As lipophilicity increases, transcellular permeability increases until a plateau region is reached beyond which permeability becomes independent of lipophilicity. The plateau region corresponds to the situation where membrane transport has become so rapid that convective diffusion through the stagnant layer becomes rate limiting.

Likewise the model of PlCDelfina and Moreno considers paracellular diffusion (Pli-Delfina and Moreno, 198 1). But in contrast to Higuchi and Ho, they did not consider paracellular diffusion as a minor path of absorption. For low and medium molecular weight compounds pore diffusion is seen as the major contributor of absorption and for extremely low molecular weight and highly hydrophilic compounds, the pore route is assumed to be the only means of permeation. Within a homologous series the partition coefficient increases as the molecular weight is increased (see Eq. (5)). Consequently, with increasing partition coefficient, passage through the lipophilic part of the membrane will predominate. Thus, for intermediate- molecular weight (in this case medium partition coefficients) compounds of a series, both mechanisms could be operative, so that the observed absorption rate results from the summation of these parallel pathways:

k obs = kOQ + kbyd (48)

where korg represents the rate constant through the lipophilic membrane, and kbyd accounts for pore diffusion.

Assuming a limiting value k, for kbyd (so that kbyd +

k, if Pact + 0) and a minimum limiting value zero (so

324 G. Camenisch et al. /Pharmaceutics Acta Heluetiae 71 (1996) 309-327

that khyd = 0 when P,,, is sufficiently large) the Wagner- Sedman model (see Eq. (31)) can be expanded according to:

k obs _ km P,“,, k, Q' _-

Q + PA + Q' + P;,

Thus, this tentative equation introduces the concept of a possible contribution of the pore passage to the passive diffusion of low and medium molecular weight compounds, only applicable to a homologous series of compounds. Eq. (49) rationalizes parabolic permeability-lipophilicity relationships (see Fig. 4d).

Taking the molecular size dependence of permeability into consideration the pH-partition model, originally derived for small molecules, was extended by Leahy and coworkers assuming stagnant aqueous layers on both sides of the membrane as proposed by Stehle and Higuchi (Leahy et al., 1989). On theoretical considerations, they derived the molecular size dependence (expressed as molecular weight dependence) for diffusion in the unstirred layers and in the membrane, and for sieving on the aqueous pores. Aqueous diffusivity was seen to be proportional to MW-0.6 and diffusivity within a membrane to be proportional to (MW/32)-’ where s ranges from 3 to 6 for different membrane types. Based on permeability data through rat ileum a non-linear molecular size dependence of sieving effect on aqueous pores was proposed as briefly mentioned in Section 2.6. The extent of the sieving effect was described by the sieving coefficient 0 (see Eq. (15)). They incorporated a mathematical description of these relationships into the model of Stahle and Higuchi, according to the following equation:

MW -’ aMWp0.6 -

( i D ?_

where D is the distribution coefficient (see Eq. (12)), and a, b, c and s are parameters to be determined from regression analysis. This so-called extended pH-partitioning model, describes a separate sigmoid relationship between permeability and lipophilicity for each molecular

weight over the whole range of lipophilicity (Fig. 20).

5. Discussion

The often observed non-linear relationship between passive permeation and lipophilicity can be rationalized in terms of drug ionization and molecular size. Various theo-

retical models of membrane permeation suggest a sigmoidal relationship between permeability and lipophilicity. This general shape is influenced by the presence of aqueous pores and stagnant aqueous layers. However, as we demonstrate in this review, different views on a membrane and differences in mathematical treatment may lead to different interpretations of the most relevant factors. A limitation of these models is that they only describe passive diffusion, which nevertheless is relevant to the majority of drugs. A basic problem for the comparison of different models is given by the considerable number of descriptors for ‘permeability’ and the many unproven as- sumptions and their complex contribution to the results.

Nevertheless, recently obtained experimental permeation data through different biological membranes (skin, blood-brain barrier, intestinal mucosa, Caco-2 cells, etc.) confirm a general sigmoidal permeability-lipophilicity relationship even for heterogeneous data sets (Artursson and Karlsson, 1991; Leahy et al., 1989; Potts and Guy, 1992; Oldendorf, 1974) (Fig. 21). However, the general shape of the curve may be blurred by some compounds diverging from such a regular relationship, showing increased or decreased permeabilities. The appearance of outliers have mostly been explained in terms of active transport (Fens- termacher et al., 19811, solubility (Flynn and Yalkowsky, 1972; Yalkowsky and Flynn, 1973), and more recently the existence of efflux systems (Schinkel et al., 1995). Another suggestion is given by Leahy at al. (Leahy et al., 1989), taking molecular size into account (see Fig. 20). This proposal has rarely focused the attention of other investiga- tors. A possible reason is the lack of good experimental permeation data. It should be realized that variations in molecular size and lipophilicity at the same time leads to other results than a series that is varied only in lipophilicity but all compounds having more or less the same molecular weight.

Hydrogen bonding has been considered as another important factor in membrane permeation. However, it should be realized that hydrogen bonding is a component of lipophilicity. Indeed, analysing the contributions of lipophilicity separately and in combination may provide useful insights in quantitative structure-drug disposition studies leading to a better understanding of the most relevant properties in drug absorption.

Acknowledgements

This paper is part of a Ph.D. study and was supported by Hoffmann-La Roche, Basel.

G. Camenisch et al. / Phannaceutica Acta Heluetiae 71 (1996) 309-327 325

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Documents

Review of theoretical passive drug absorption models: Historical background, recent developments and limitations