log p

Alternative Measures of Lipophilicity:From Octanol–Water Partitioning to IAM Retention

COSTAS GIAGINIS, ANNA TSANTILI-KAKOULIDOU

Department of Pharmaceutical Chemistry, School of Pharmacy, University of Athens, Panepistimiopolis, Zografou,Athens 157 71, Greece

Received 13 June 2007; revised 8 October 2007; accepted 8 October 2007

Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jps.21244

Correspondeþ30-210-727453E-mail: tsantili@

Journal of Pharm

� 2008 Wiley-Liss

2984 JOURN

ABSTRACT: This review describes lipophilicity parameters currently used in drugdesign and QSAR studies. After a short historical overview, the complex nature oflipophilicity as the outcome of polar/nonpolar inter- and intramolecular interactionsis analysed and considered as the background for the discussion of the differentlipophilicity descriptors. The first part focuses on octanol–water partitioning ofneutral and ionisable compounds, evaluates the efficiency of predictions and providesa short description of the experimental methods for the determination of distributioncoefficients. A next part is dedicated to reversed-phase chromatographic techniques,HPLC and TLC in lipophilicity assessment. The two methods are evaluated fortheir efficiency to simulate octanol–water and the progress achieved in the refinementof suitable chromatographic conditions, in particular in the field of HPLC, is outlined.Liposomes as direct models of biological membranes are examined and phospolipo-philicity is compared to the traditional lipophilicity concept. Difficulties associatedwith liposome–water partitioning are discussed. The last part focuses on Immobilis-ed Artificial Membrane (IAM) chromatography as an alternative which combinesmembrane simulation with rapid measurements. IAM chromatographic retention iscompared to octanol–water and liposome–water partitioning as well as to reversed-phase retention and its potential to predict biopartitioning and biological activitiesis discussed. � 2008 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci

97:2984–3004, 2008

Keywords: lipophilicity; octanol–water
system; reversed phase chromatographicretention; extrapolated retention factors; liposome–water partitioning; IAM chromato-graphy
INTRODUCTION

Lipophilicity, expressed by the logarithm of octanol–water partition coefficient log P or distributioncoefficient log D, if ionised molecular species

nce to: Anna Tsantili-Kakoulidou (Telephone:0; Fax: þ30-210-7274747;pharm.uoa.gr)

aceutical Sciences, Vol. 97, 2984–3004 (2008)

, Inc and the American Pharmacists Association.

AL OF PHARMACEUTICAL SCIENCES, VOL. 97, NO. 8, AUG

are present, constitutes a physicochemical propertyof paramount importance in medicinal chemistry. Itplays an essential role in absorption, distribution,metabolism and elimination (ADME) character-istics of drugs while affecting also their pharma-codynamic and toxicological profile.1–4 Historically,the importance of lipid solubility in biologicalactivity was demonstrated already at the dawnof 20th century, when Meyer5 and Overton6

independently published their work on narcosis.They were the first who used oil–water partition

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ALTERNATIVE MEASURES OF LIPOPHILICITY 2985

coefficients in their studies, stimulating theinterest for further investigation, such as the workof Gaudette and Brodie7 who found a parallelismbetween heptane–water partition coefficients ofcertain drugs and their penetration rate throughthe blood brain barrier. However, it was Hansch andFujita8 and Leo et al.9 who organised the systematicresearch on lipophilicity, introducing the octanol–water system as a reference to construct anempirical scale for this property which could beused in the newly developing at that time QSARanalysis, known also as Hansch analysis. Sincethen, octanol–water partition coefficient in itslogarithmic form, log P, became a characteristicconstant for a chemical substance. Log P valueshave been compiled in commercially availabledatabases.10–12 Nowadays, Pomona College Med-Chem Database contains more than 53000 log Pvalues, among them 11000 are characterised ashigh quality belonging to the ‘star’ list.12 More to thepoint, Hansch and his coworkers created the basisfor the calculation of lipophilicity, recognizing thepartially additive character of log P. Up to now agreat number of QSAR models have been published,the majority of which include lipophilicity asdescribed by octanol–water partition coefficient,while optimum log P values (log Po) for the penetra-tion through certain biological barriers have beenestablished.13–16 The dominant role of octanol–water partition coefficients in the pharmacokineticprocesses, as well as in ligand–macromolecule inter-actions triggered further research on the develop-ment of rapid methods for their experimentalassessment, among them reversed-phase chromato-graphic techniques, as well as of suitable calculationmethods, so that screening of compound libraries isnow possible. On the other hand, guidelinesreferring to druglike characteristics include upperlimits for log P values,17,18 since there is strongevidence that high lipophilicity is associated withundesired drug features, like extensive and un-predictable metabolism, high plasma protein bind-ing or accumulation to tissues.19 The recognitionof the imperative role of lipophilicity in drugdesign led to the organisation of three successfulinternational Symposia focused especially on thisproperty.20–22

Despite its successful application in drug design,the octanol–water system has received a lot ofcriticism throughout these years, as an isotropicmedium with only a superficial similarity tobiomembranes. Partitioning into liposomes andImmobilised Artificial Membrane (IAM) chromato-graphy are gaining increasing interest, especially

DOI 10.1002/jps JOUR

the latter which combines simulation of cellmembranes with rapid measurements.23–25

Lipophilicity, however, is not merely a usefulparameter in drug design. Being the outcome ofmore than one component, it encodes a highcontent of structural information which can beused to understand better the behaviour of asolute in the biological environment.26 In thisaspect, the differences between the variouslipophilicity descriptors may concern intermole-cular recognition forces, developed between themolecule and the two phases, or intramolecularinteractions, which may differ in their manifesta-tion between the different biphasic systems. In thepresent review, the complex nature of lipophilicitywill serve as the background for the comparison ofalternative measures of lipophilicity. Consideringoctanol–water as the reference system for lipo-philicity, weaknesses associated with the effi-ciency of predictions are outlined and similarities/dissimilarities with reversed-phase chromato-graphic indices are discussed. Liposome partition-ing and IAM chromatography are treated in ananalogous manner and their merit to correlatebiological data is evaluated.

THE NATURE OF LIPOPHILICITY

The dual nature of lipophilicity as the outcome ofnonpolar and polar interactions is well establish-ed.26–30 Nonpolar interactions are equated withhydrophobicity and can be expressed by stericterms like molecular volume, polarizability, molarrefractivity. They contribute positively to lipophi-licity. More complex are the polar interactionswhich include ion–dipole, dipole–dipole or hy-drogen bond interactions and can be expressedby electronic constants, dipole moments or hy-drogen bond parameters. A global polarity para-meter, designated as L, has been proposed byTesta.27 Intramolecular interactions, promoted bycertain structural characteristics, like substitu-ents in aromatic or heterocyclic systems, presenceof conjugated systems in the molecules, internalhydrogen bonds, also affect lipophilicity consider-ably influencing the intermolecular forces.31–33

In addition, conformational changes inducedby polar or nonpolar environment may rendermolecules more hydrophilic or more lipophilicthan expected, depending on the nature of thegroups buried within the molecule or exposed tothe solvent.29,34 Ionisation dramatically alters alltypes of intramolecular interactions involving the

NAL OF PHARMACEUTICAL SCIENCES, VOL. 97, NO. 8, AUGUST 2008

2986 GIAGINIS AND TSANTILI-KAKOULIDOU

charged centre, limiting the validity of additivityrules established for neutral compounds.35,36

The outcome of polar/nonpolar inter- andintramolecular interactions can be summarisedin Eq. (1):

Lipophilicity ¼ hydrophobicity � polarity (1)

The above-mentioned issues refer to any biphasicsystem and are solvent (phase) dependent. Thedifferences between the systems can be demon-strated by the so-called solvatochromic analysis.37–39

Partition coefficients are factorised into steric andpolar terms which express different types of inter-molecular interactions as described in Eq. (2):

log P ¼ vV=100 þ aA þ bB þ sS þ eE þ c (2)

where V represent Mc Gowan’s characteristicmolar volume, A and B (formerly a and b) expressoverall solute’s hydrogen-bonding acidity andbasicity respectively, S (formerly p�) is a combin-ed dipolarity/polarizability term reflecting thesolute’s capacity to elicit orientation and induc-tion forces and E (formerly R) is excess molarrefractivity. v, a, b, s, e, c are constants whichcharacterise the system and are derived bymultiple linear regression analysis.40,41

The terms in Eq. (2) are significant when thegiven type of interaction elicited between thesolute and the two phases is not of equal energy.The coefficients of the terms reflect then themagnitude of this difference.

According to this type of analysis octanol–water partition coefficients differ from alkane(hexadecane)–water partition coefficients mainlyin the acidity term A, which is not significant inthe first case but has a large negative contributionin the latter. This is shown in Eqs. (3) and (4)derived for a large number of data measured in then-octanol–water system and in the alkane–watersystem respectively:39

log P ¼ 3:81V þ 0:034A � 3:460B � 1:054S

þ 0:562E þ 0:088

n ¼ 614; r ¼ 0:997 s ¼ 0:116

(3)

where n refers to the number of data derived fromthe n-octanol–water system.

log P ¼ 4:28V � 3:587A � 4:869B � 1:617S

þ 0:667E þ 0:087

n ¼ 370; r ¼ 0:998; s ¼ 0:124

(4)

JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 97, NO. 8, AUGUST 2008

where n refers to the number of data derived fromalkane–water system.

The contribution of the hydrogen bond basicityterm B in octanol–water partitioning (Eq. 3) hasbeen widely recognised. Some authors comparedsuccessfully the basicities of solutes estimated byequations of type 2 to those experimentallydetermined in CCl4.42,43

Partitioning into n-Octanol

n-Octanol–water is the widely accepted referencesystem for the determination of lipophilicity.n-Octanol offers certain advantages which ad-vocate for its suitability to simulate biologicalmembranes. Initially, these advantages wereattributed to its structure, namely the presenceof a hydrophobic chain with a polar head, as wellas to its moderate water saturation, which allowsrelatively easier and more reliable experimentaldetermination.9 The hydroxyl group in then-octanol molecule has a hydrogen bond capabilityboth as donor and acceptor. However, membranesand receptors are different as far as theirhydrogen bond properties are concerned. Thus,they may contain amphiprotic groups, and in suchcases n-octanol is the most suitable lipophilicphase, or they may contain mainly hydrogenbond donors or hydrogen bond acceptors. In thefirst case chloroform–water may provide a bettersimulation, while in the second case dibutyletheror propyleneglycoldiperlagonate have been sug-gested as lipophilic phases. Finally, it is possiblethat no hydrogen bond sites are available, so analkane–water system or dichloroethane–watersystem would be preferable.39,44–48

The degree of water saturation of organicsolvents may affect the accuracy of the partition-ing experiments, since very high or very low watersaturation values are associated with practicaldifficulties. For n-octanol, the degree of saturationis 0.25 mol % and, as already commented, itpermits more reliable measurements. More thanthat, the water present in wet octanol seems toplay an essential role in the interaction withsolutes rendering its structure more complex thananticipated for an isotropic system.49,50 Accordingto the results of NMR and near IR as well as ofX-ray diffraction analysis, the water is notuniformly dispersed in octanol, but it formsclusters of four molecules. The water moleculesare surrounded by about 16 octanol molecules,which orientate their hydroxyl groups towards the

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water molecules constructing a network of hydro-gen bonds. On the other hand, the hydrocarbonchains create a water free region similar with thatin the interior of lipid bilayer with a dielectricconstant e¼ 8.51–53

Partitioning into an organic solvent–watersystem is generally entropy driven, althoughoctanol has an exothermic heat of transfer(negative enthalpy), due to H-bond stabilisationof the transferred solute. Thermodynamic data aredifferent regarding cyclohexane–water partition-ing for which a positive enthalpy has been found.54

Figure 1. Log D/pH profile of (A) mefenamic acid(B) alprenolol, in presence of 0.01 M (&) and 0.15 M(&) KCl. Data calculated according to Prolog D (Pallas v3.3.2.4).

Ionisation Effect in Partitioning

Partition coefficient refers to the neutral monomerspecies. If ionic species coexist with the neutralspecies, the distribution coefficient is considered asthe ratio of the sum of the concentration of allspecies in octanol to the corresponding concentra-tion sum in water. If partition of ions in octanol isneglected, log D can be converted to log P by meansof a correction term, based on the Henderson–Hasselbach equation.55 Eqs. (5) and (6) are used formonoprotic acids and bases, respectively, denotingthat distribution coefficients depend on pH assum-ing that only the neutral form partitions into thelipid phase.56

Log D ¼ log P � logð1 þ 10pH�pKaÞ (5)

Log D ¼ log P � logð1 þ 10pKa�pHÞ (6)

Theoretically, the slope of the linear part ofEqs. (5) and (6), plotting log D versus pH, is j1j,although deviations may be observed.36 In case ofextended ionisation, the nature, as well as theconcentration of the counter ions define the lowestlevel asymptote corresponding to the partitioncoefficient of the ionised species. Figure 1 showsan example of lipophilicity profiles, log D versuspH, for an acid (mefenamic acid) and a base(alprenolol) in presence of 0.01 M and 0.15 M KCl(data calculated according to Prolog D, imple-mented in Pallas v. 3.3.2.4, see also LipophilicityPrediction Section). As a rule of thumb, thedifference between partition coefficients of theneutral and ionised form is considered to be �3 logunits for bases and �4 log units for acids at 0.15 MKCl.56,57 This general rule may not be valid if theionised species produce charge delocalisation,contain hydroxyls adjacent to carboxylic groupsor as a result of steric hindrance.57,58 Specialattention deserves the lipophilic behaviour of


ampholytes. Ordinary ampholytes (pKaacid>pKabasic) exhibit an acid and base lipophilicity/pH profile described by Eq. (7).

log P ¼ log D

þ logð1 þ 10pKa1�pH þ 10pH�pKa2Þ (7)

where pKa1 is the basic pKa and pKa2 is theacidic pKa.

In the case, however, of close overlapping pKa’sor if pKaacid<pKabasic, zwitterionic species areformed around the isoelectric point. The presenceof a positive and a negative charge alters theintramolecular interactions and often contra-dictory results are reported in literature concern-ing their lipophilicity.59–61 Partial neutralisationoccurring via through-bond and through-spaceinteractions leads to distribution coefficienthigher than that corresponding to the cationicor anionic species, but lower than the log P ofthe neutral form. Charge compensation results ina bell shaped lipophilicity profile with constant



lipophilicity in a pH range defined by the two pKa

values. Thus, zwitterionic ampholytes are con-sidered to behave like ‘lipophilicity buffers’. Insome cases structural factors do not allow chargecompensation and a U-shaped log D/pH profilemay be obtained.59,61

Lipophilicity Prediction

Lipophilicity prediction is essential in early drugdiscovery process and this necessity was recog-nised from the beginning of the QSAR era. Theintroduction of hydrophobic substituent constant,p, in the 1960s was followed by the fragmentalsystems of Rekker and Mannhold,62 Leo63 andHansch and Fujita,8 based on the additive andconstitutive character of log P. The latter isencoded in C log P, which is considered by manyauthors as the benchmark software. Calculation isperformed by summation of fragmental contribu-tions, while certain corrections are needed toaccount for deviations from additivity. The nextstep was the development of atomic contributionsystems, which are based on atom types, takinginto consideration their neighbouring structuralenvironment. The advantage of atomic contribu-tion systems is that most of them do not requirecorrection terms.64–67 Recently, nonlinear sys-tems based on the construction of artificial neuralnetworks (ANN) have been proposed. Electroto-pological state indices are used as input data, thusincorporating electronic effects in the calculationprocedure.68,69 Atom types have also been used inthe construction of ANN for log P prediction.70

Despite the large arsenal of calculation systems,lipophilicity prediction is not an easy task. Oftenfor complex molecules different systems yielddifferent estimates, which may considerablydeviate from the true log P value.29,33 The sourcesfor prediction errors are associated with thecomplexity in the inter- and intramolecularinteractions involved in partitioning process aswell as with conformational effects, as alreadydiscussed. To this point it should be noted that themolecular lipophilicity potential (MLP) approach,developed by Testa et al. aimed to confrontthose problems. MLP is a useful tool in molecularmodelling, however it has not found generalapplication in lipophilicity screening of large datasets.71 Moreover, the systems described so farpredict the lipophilicity of the neutral species.Limited choices are available for the predictionof distribution coefficients and the results are


susceptible to larger errors, since ionisation mayalter the additivity rules as previously comment-ed. Ionisation correction terms are based onHenderson–Hasselbach equation, while the ionicstrength should also be considered. Prolog Dimplemented in Pallas software (CompuDrugInternational, Inc., Sedona, AZ) calculates log Dvalues at any pH and any K/Na]þ/[C]l� concentra-tion,72 while ADME Boxes (PharmaAlgorithms,Torontto, Canada) and ACD Labs consider astandard ionic strength of 0.15 M KCl/NaCl.

A classification of the different methods forcalculation of lipophilicity along with the relevantsoftware programs can be found elsewhere.73

Measurement of Partition/Distribution Coefficients

Different experimental protocols for log P/log Ddetermination have been suggested in literature.The classical shake-flask method for direct parti-tioning experiments is tedious and time consum-ing, not suitable for degradable compounds, lessamenable to automation, while it presents limita-tions concerning the log P or log D range that canbe reliable measured. Detailed reviews havedescribed the associated pitfalls and difficulties,as well as how to choose the best conditionsfor reliable measurements.74–76 For ionisablecompounds potentiometric dual phase pH metrictitration is possible.77–79 Partition coefficients arecalculated according to the difference in the pKa

values, in presence and absence of n-octanol (or anyother immiscible organic solvent), as described byEqs. (8a) and (8b) for acids and bases respectively:

P ¼ ð10poKa�pKa � 1ÞVH2O=Vorg

for monoprotic acids(8a)

P ¼ ð10�ðpoKa�pKaÞ � 1ÞVH2O=Vorg

for monoprotic bases(8b)

where poKa is the apparent pKa in presence of theorganic solvent.

Centrifugal chromatography offers another pos-sibility for direct log P or log D measurements.80

Attempts for the development of high throughputmeasurements scaled down to 96-well microtitreplate techniques have also been reported.81

Summarizing The Nature of Lipophilicity Sec-tion n-octanol–water partition or distributioncoefficients remain through the decades a usefultool in the design of new drugs while they providevaluable information for the understanding of

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solutes in the biological environment. Calculationsystems permit a rapid lipophilicity screening oflarge drug libraries. However due to the interplayof different forces involved in n-octanol–waterpartitioning, calculations may be less reliable forcomplex structures and in particular for chargedspecies. Thus, experimental determination at alater drug development stage remains necessary.Although there is considerable progress to faci-litate partitioning experiments, many researchersdirect their efforts in the exploitation of user’sfriendlier chromatographic techniques, which willbe discussed in the following section.

ALTERNATIVE MEASURES OFLIPOPHILICITY-REVERSED-PHASECHROMATOGRAPHIC INDICES

Reversed-phase chromatographic techniques, inparticular high performance liquid chromato-graphy (RP-HPLC) and thin layer chromatogra-phy (RP-TLC) have proved to simulate octanol–water partitioning and are considered as popularalternatives for lipophilicity assessment. Theyoffer several practical advantages, includingspeed, reproducibility, insensitivity to impuritiesor degradation products, broader dynamic range,online detection and reduced sample handlingand sample sizes.82–87 These advantages haveattracted considerable interest and literature isrich in research articles, which investigate therelationship of chromatographic retention withoctanol–water partitioning and the commonfactors underlying the two processes.84,87–92 Theselection of either technique is associated withthe state of the art concerning their technology.Especially RP-HPLC got wide acceptance inlipophilicity assessment and has officially beenrecommended by the OECD.93 Moreover, allassumptions dealing with the complex nature oflipophilicity as the outcome of intermolecularand intramolecular interactions, involving elec-tronic, steric, or conformational effects, embracechromatographic retention, as well.

The lipophilicity indices measured by RP-HPLCand RP-TLC are derived by the retention time tr

and the migration distance relative to the solventfront, Rf, both converted to the logarithm of theretention factors log k and RM, according toEqs. (9) and (10), respectively:

log k ¼ logtr � to

to

� �(9)


tr being the retention time of an unretained solute.

RM ¼ log1

Rf� 1

� �(10)

Rf being the migration distance of the solute Zf

divided by the distance of the solvent front fromthe starting point Zs.

Rf ¼Zf

Zs(11)

Isocratic retention factors represent a relativescale of lipophilicity and are converted to log P orlog D values via Collander type Eqs. (12) and (13):

log Pðor log DÞ ¼ a log k þ b (12)

log Pðor log DÞ ¼ a RM þ b (13)

Isocratic retention factors, log k and RM, arepreferred by some authors, as they require fewerexperiments.94 However care should be taken touse an adequate training set of solutes in respect tothe solutes for which the log P or log D values haveto be determined, since often isocratic retentionfactors may lead to under- or overestimation oflipophilicity.89,95 In this aspect, extrapolated reten-tion factors, corresponding to pure water as mobilephase and termed as log kw, RMw are consideredas more representative lipophilicity indices, sincetheir values are of the same order of magnitudewith octanol–water log P and log D.82–87 In fact,extrapolated log kw or RMw values are derivedusing the linear part of the log k/w or RM/wrelationships according to Eqs. (14) and (15),respectively:

log k ¼ �S’þ log kw (14)

RM ¼ �S’þ RMw (15)

where w is the concentration of the organicmodifier in the mobile phase.

In many cases, when log kw or RMw values areused in equation of type (12) or (13), thecorresponding regression coefficients a and b tendto approach 1 and 0, respectively. Considerableresearch efforts are directed towards the stan-dardisation of chromatographic conditions, whichguarantee 1:1 correlation between extrapolatedchromatographic indices and log D values. In thisfield, outstanding progress has been achieved inthe case of RP-HPLC. Currently, standardisationof chromatographic conditions for basic andneutral drugs at physiological pH has beenestablished and analogous efforts for acidic andzwitterionic drugs are bestowed.96–98 As regards



the RP-TLC, the results are less spectacular, eventhough it is assumed that both reversed phasetechniques are governed by analogous retention.However, it should be noted that the stationaryphase in RP-HPLC undergoes a uniform, rela-tively well-defined transformation during equili-bration process with the mobile phase, while inthe case of RP-TLC a stationary phase with agradient composition is formed upon equilibrationcreating less reproducible conditions. Moreover,important factors in HPLC retention, like themobile pH and mobile phase additives do notaffect the retention on the TLC plates to the sameextent, thus reducing the flexibility to investigatedifferent chromatographic conditions.99–103 Forthis reason, the next paragraph will focus onstationary/mobile phase alternatives in RP-HPLCand reference to RP-TLC will be assigned whennecessary.

CHROMATOGRAPHICCONDITIONS—STATIONARY PHASES

C-18 silanised silica gel is the preferred materialfor reversed phase chromatographic columns andplates for the assessment of drug lipophilicity.However, the interference of silanophilic interac-tions in the partitioning mechanism in RP-HPLCand RP-TLC has been recognised long ago as aserious drawback.104–107 Silanophilic interactionsare attributed to the remaining free silanol sitesand include hydrogen bonding, as well as electro-static forces, especially in the case of positivelycharged compounds, producing considerableincrease in retention. They also depend on thedegree of ionisation of the silanol groups, beingless pronounced at low pH.108,109 The problem isextensively studied in the case of RP-HPLC andpartially faced by the development of columnswith reduced free or accessible silanol sitesand the use of masking agents as mobile phaseadditives. Analogous progress is not achieved forTLC layers for the reasons mentioned above. InRP-TLC, the poor wetability of the plates presentsan additional problem, more pronounced forfully silanised silica gel material. The spreadof overpressure TLC in the future may offer asolution,110,111 but up to now, application ofoverpressure TLC in the field of log P assessmenthas not been reported.

End-capping of the silanol residues by trimethy-lchlorosilane (TMCS) or hexamethyldisilazane(HMDS) is usually performed during manufacture


process, leading to a higher degree of silanisa-tion.105 Hence, base deactivated silica represents apacking material more suitable (e.g. BDS C-18) forbasic solutes. In addition, recent technology has ledto the development of polar embedded and polarend-capped stationary phases, which are consid-ered to be further deprived from silanophiliceffects.91,96,112,113 They contain polar functionalgroups such as amide, carbamate, ether orsulphonamide, incorporated at the bottom of thealkyl-bonded chains. These functional groupsprovide electrostatical shielding to the surfacesilanol sites. The LC-ABZþ and the Discovery-RP-Amide-C16 stationary phases belong to thepolar embedded type columns, which have beenused for lipophilicity determination. However,these packing materials may exhibit other polarinteractions with analytes, such as the stronginteraction between the polar embedded groupsand the phenolic analytes.113 In respect to the polarend-capped columns, a second reaction is used tobond a short carbon chain (usually C3–C4) witha polar end to the surface silanol sites.114 Afavourable advantage for both types is the factthat a higher degree of orientation for the alkylchains is achieved and thus they can be used withmobile phases containing high amounts of water oreven pure water without the problem of hydro-phobic collapse.112–114 The pH limitation of theabove-mentioned columns lies in the range 2.5–7.5.Thus, for strong bases determination of retentionfactors corresponding to the neutral form cannot beachieved.

Recently bidentate stationary phases (e.g.Zorbax-extend C-18) that include a propylenebridge, as well as surface modified silica columns(e.g. XTerra C-18), where organic functionalgroups have become a constituent of the silicabackbone, have been developed, allowing the useof mobile phases with pH up to 12.115 However,the applicability of such columns in the lipophi-licity assessment has not been systematicallyinvestigated yet. In a recent publication, 1:1correlation has been reported between log kw

and log P for 40 basic compounds measured atpH 10.5 on a Zorbax-extend C-18 column withoutaddition of any masking agent.116

An alternative choice, the polymer-basedoctadecyl-poly(vinyl alcohol) (ODP) stationaryphase, which is completely devoid of reactivesilanol groups, has also been used for lipophilicitymeasurements.117,118 ODP column presents stabil-ity to acidic and strongly basic conditions (atpH between 2 and 13).119 However, it has been

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reported that the retention mechanism on ODPstationary phase compared to octanol–water par-titioning is controlled by a different balance offorces as deduced by solvatochromic analysis.120

In fact, there was a lower contribution of volumeand hydrogen bond basicity, while the polariz-ability term was found nonsignificant in therelevant equation. Thus, the derived data maynot be so suitable to reproduce the classical logP or log D values. It should be noted that the sameauthors reported identical solvatochromic equa-tions for log P and log kw data derived from aDiscovery RP-Amide column.120

CHROMATOGRAPHICCONDITIONS—MOBILE PHASES

The most widely used organic modifiers forlipophilicity assessment by RP-HPLC or RP-TLC are methanol, acetonitrile or THF. Methanolhas a more water-like structure and is considerednot to disturb the hydrogen-bonding network ofwater. Moreover, during stationary/mobile phaseequilibration in RP-HPLC, methanol moleculesassociate with the stationary phase forming amonolayer, which provides a hydrogen-bondingcapability in better agreement with n-octanol.117

The buffer composition of the aqueous compo-nent in the mobile phase may also play an activerole in retention especially in the case of proto-nated basic compounds, which may form ion pairswith the buffer counter ions. Morpholinepropa-nesulphonic acid (MOPS) is considered as thebuffer of choice for lipophilicity assessment by RP-HPLC.96,97,99 It exhibits a large buffering capacitycoupled to poor ion-pair formation ability dueto its zwitterionic nature and thus it does notinterfere either with solutes or with stationaryphase. The benefit of using MOPS in lipophilicityassessment by RP-TLC has also been reported.However, it should be noted that the migration ofsolutes is not affected considerably by changes inpH and for this reason many authors prefer to usewater instead of buffer in RP-TLC.99,102,103 Thisissue will be further discussed in a followingsection.

The addition of a masking agent is a criticalprerequisite in order to achieve the prevalenceof partition mechanism in RP-HPLC in thecase of neutral and basic dugs, while itseffect is not manifested on RP-TLC.99 Hydro-phobic amines, such as n-decylamine and N,N-dimethyl-octylamine, are considered to be the


most suitable masking agents to suppress silano-philic interactions, especially when combinedwith methanol as organic modifier.96,117 Theireffect on retention is less evident with acetonitrileas organic modifier. Acetonitrile, as a weakhydrogen-bonding solvent, is not capable of sol-vating the stationary phase with sufficient water,thus presumably preventing the positivelycharged amine to be dragged on the column andthus to exert its role as a masking agent.117

Recently, room temperature ionic liquids ofthe imidazolium tetrafluoroborate family, suchas 1-butyl-3-methylimidazolium (BMIM BF4),have been reported as suitable masking agentscombining the silanol masking effect of theimidazolium cation with the chaotropic characterof the BF4� anion.121–123 Nevertheless, the above-mentioned masking agents cannot be used in thecase of anionic compounds, since they would formion pairing with the negatively charged centre.

In the last few years, the addition of smallamounts of octanol in the methanol fraction of themobile phase has proved favourable for lipo-philicity assessment by HPLC.96–98 It is consid-ered to produce an octanol-like character to thechromatographic system, although its role inretention is not fully investigated. Almost 1:1correlation between log D and log kw is obtained inthe case of neutral and basic drugs with mobilephase containing small amounts of n-octanoland n-decylamine using either an ABZ or aBDS column. Based on these conditions, a rapidautomated method to convert log kw values to logP or log D by a general calibration equation hasbeen developed, known as the E log D approach.96

In the case of acidic drugs, the addition ofn-octanol has also been recognised as a criticalfactor to obtain 1:1 correlation between log P andlog kw values, measured at low pH at whichionisation is suppressed.98 At pH 7.4, a 1:1correlation was reported in the case of weak acidsupon addition of n-octanol in the mobile phase,while for fully ionised compounds the affinitywith the stationary phase, although reduced,remained strong and the relevant equationpossessed a large intercept.124 Examples of log Dversus log kw plots are illustrated in Figures 2and 3. Figure 2 concerns neutral and basic drugsin absence and presence of n-octanol using aBDS column, MOPS as buffer and n-decylamineas masking agent (data taken from Ref. 97).Although correlation coefficient is slightly lowerin presence of n-octanol the corresponding equa-tion has a nonsignificant intercept, representing


Figure 2. Log D7.4/log kw relationships for 64 struc-turally diverse basic and neutral drugs in absence(A) and in presence (B) of n-octanol as mobile phaseadditive using a BDS column (data taken from Ref. 97).


1:1 correlation. Figure 3 demonstrates the be-haviour of weak and strong acids on a BDS columnin presence of n-octanol, fitting two differenttrendlines.124

EFFECT OF IONISATION ON RETENTION

To adjust retention factors for ionisation, thesame corrections are used as for the distribution

Figure 3. Log D7.4/log kw7.4 relationships for 22 weakly(^) and 21 strongly (~) ionised acidic drugs in presence ofn-octanol as mobile phase additive using a BDS column(reproduced from Ref. 124 by permission).


coefficients. However, whether the effects ofionisation in the octanol–water partition systemand RP-HPLC are similar remains to be clarified.The organic modifiers are capable of affectingthe pKa of ionised solutes, as well as the acidity ofthe surface silanol groups and the pH of the mobilephase. In general, the pKa of bases decreases andthe pKa of acids increases as the organic modifierconcentration increases. Substantial structure-dependent differences in pKa shifts at a givenorganic solvent composition and pKa variations atdifferent organic solvent composition have beenreported for RP-HPLC.125–127 These effects areminimised in extrapolation procedure, providinga further argument for the use of log kw valuesinstead of isocratic log k in the case of basiccompounds.

Ionisation correction for RP-TLC RM or RMw

factors is more disputable. Dross et al.100 reportedthat ionisation of basic drugs at pH 7.4 wassuppressed in TLC and RMw values withoutionisation correction correlated better with log P.Analogous results for a series of b-blockersanalogs were attributed to silanophilic interac-tions which compensated the decrease in lipo-philicity of the ionised solutes.101 Malawska102

correlated extrapolated RMw values obtained atpH 7.4 with calculated values of the neutral form,while the same author used water as the aqueouscomponent of the mobile phase for a similar seriesof compounds.103 Investigation of the pH effectfor a set of basic drugs measured at pH 7.4 and11 revealed that, despite differences in migration,a reduced net effect of ionisation may occurred asa result of the formation of gradient conditions onthe TLC plate.99

EFFECT OF CONFORMATION IN RETENTION

The effect of conformation in lipophilicity has beencommented under the relevant section. Analogouseffects, although not necessarily to the sameextent, may be expected in retention. Moreover,topographical factors induced by the anisotropy ofreversed phase stationary phase and the fact thatsolutes approach it gradually with the movementof the mobile phase is an additional reason forconformational effects to be present. In such cases,differences in the partitioning behaviour in theoctanol–water system and the chromatographicsystem may be manifested, thus affecting thequality of the relationships between log D andchromatographic indices and rendering inter-pretations ambiguous.92

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OTHER CHROMATOGRAPHIC DATA ASLIPOPHILICITY—RELEVANT EXPRESSIONS

The slope S of the linear Eqs. (14) or (15) isconsidered to encode significant information onthe lipophilic behaviour of the solute. By someauthors, the slope S is considered to reflect thesolute/solvent interactions during the retentionprocess and is related to the specific hydrophobicsurface area.128 The strong influence of volume inthe slope S, derived by RP-HPLC and RP-TLC,was demonstrated for a series of substitutedcoumarins using multivariate data analysis—partial least squares (PLS).92 Within a series ofsuch compounds, a good relationship betweenthe slope S and log kw or RMw is anticipated(Eqs. 16 and 17), if there are no considerabledifferences in the forces involved in solute/stationary phase interactions (mainly concerninghydrogen-bonding or the extent of silanophilicinteractions).

S ¼ a log kw þ b (16)

S ¼ a RMw þ b (17)

The organic modifier concentration wo, whichproduces an equal molar distribution between thestationary and mobile phase, leading to log k¼ 0,has been proposed as a measure to rank lipophi-licity. The wo indices correspond to the quotient asexpressed in Eqs. (18) and (19):

’o ¼ log kw=S (18)

’o ¼ RMw=S (19)

Based on the wo concept, a fast gradient RP-HPLC method has been proposed by Valko andcoworkers129–132 to determine the chromato-graphic hydrophobicity index (CHI) as a high-throughput alternative to the other lipophilicitymeasures. For this purpose, gradient retentiontimes (tg) are measured and converted to CHIvalues by means of a calibration equation, derivedby a set of standards with well-determined CHI(wo) values (Eq. 20):

CHI ¼ slope x tg þ intercept (20)

The absolute magnitude of the CHI parameterdepends on the values assigned to the set ofstandards. The method has the advantage that,once the calibration equation has been establish-ed, the retention parameter is obtained from asingle fast gradient run, thus saving time andsolvents. The CHI parameter has been reported


to correlate satisfactorily with log P. For a series ofsubstituted coumarins however principal compo-nent analysis (PCA) and partial least squaresanalysis (PLS) showed different informationcontent for wo indices, derived from RP-HPLCand RP-TLC, compared to that of the correspond-ing log kw and RMw values.92

Summarizing, reversed phase chromatographyis a friendly and rapid technique, providingindices which correlate directly with log P or log Dvalues. All assumptions however related to thecomplex nature of lipophilicity should be consi-dered also in chromatographic retention, whilesilanophilic interactions are an additional dif-ficulty to be confronted. RP-HPLC is most popularand current research is directed towards theestablishment of standard conditions, so thatextrapolated log kw values can be converted tolog P or log D by a general calibration equation.The CHI concept on the other hand is valuable forrapid screening of compound libraries, rankingthe compounds according to their lipophilicity, butdoes not provide knowledge on lipophilicity valuesper se.

PARTITIONING INTO LIPOSOMES

Partitioning into liposome–water system offers analternative attracting considerable interest, sincethey are considered to represent direct modelsfor biological membranes. Liposomes constituteanisotropic media, predominantly composed ofamhiphilic molecules, like phospholipids, whichform spherical closed structures of curved bilayersthat entrap part of the surrounding solvent intheir interior. The number of bilayer sheetsvaries from one to several hundreds and smallunilamellar (SUV), large unilamellar (LUV) andlarge multilamellar (LMV) vesicles.133–135 Phos-phatidylcholines are most frequently used toobtain standardised and easily reproducible lipo-some systems.136–138 The physical state of lipo-somes is temperature dependent, being in gelstate at low temperatures and as crystal liquids athigher temperature. Transition temperature islow for liposomes prepared from lipids of naturalsources and relatively high when synthetic lipidshave been used.139

From a thermodynamic point of view, transferinto liposomes has been found to be entropy drivenbelow the gel/liquid transition temperature.54

Above the transition temperature, studies, carriedout using highly sensitive micro-calorimetry,



showed that the process is predominatelyenthalpy driven, in contrast to the well establi-shed ideas of entropy-driven partitioning ofdrugs.140–142

The polar headgroups of phospholipids are ofparticular importance determining the electricalproperties of the membrane surface. Their con-formation can be affected by charged solutes thatpartition into the lipid bilayer. Conformationchanges are expected to be different in presenceof positive or negative charge. Binding of acationic solute to the phosphate anion causes arepulsion of the positively charged quaternarynitrogen towards the water phase, ‘opening’the polar region and permitting the penetrationof the solute into the hydrophobic core.141 Neutralmolecules seem to have no effect on polar head-group conformation.143

Another important factor, determining theelectrostatic interactions between a solute and amembrane, is the surface charge. The surfacecharge depends on the lipids present in the outerlayer and may be changed by pH. It is measuredby the zeta-potential, which is defined as thesurface charge at a distance of 2 A

´from the

membrane surface.144 Phosphatidylcholine lipo-somes, with a positive and negative group andthereupon a zero net charge in their headgroup,have neutral zeta-potential in a pH range 2–11.The binding of charged solutes to the membranealters the surface charge.145

Phospholipids possess a dipole field that isdetermined by the polar headgroups, the surfacewater molecules and the lipid carbonyls, espe-cially the ester group carbonyls, which linkthe fatty acids to the glycerol backbone. Thedipole field is positive inside the liposome andnegative outside, denoting that the C––O dipole isoriented towards the lipid–water interface.143

The above described issue imply that partition-ing into liposomes, a process known also asphospholipophilicity, involves electrostatic inter-actions as additional intermolecular recognitionforces, not encoded in traditional lipophilicityparameters and could be factorised as expressedby Eq. (21):

Phospholipophilicity

¼ Hydrophobicity � Polarity

þ Ionic bonds (21)

As a result, charged species seem to partitioninto membranes considerably more strongly than


they do into octanol. Thus, in the case of ionisabledrugs, a lower difference is observed betweenthe partitioning of the neutral and the ionisedform.146,147 In this aspect, phospholipophilicitymay be considered to represent a border casebetween partitioning and binding.

Determination of liposome water distributioncoefficients requires that the sample is equili-brated with a suspension of liposomes, followedby a separation procedure and quantitation ofthe sample in the lipid free phase. For theseparation of liposomes from the aqueous compo-nent ultrafiltration, ultrafiltration/centrifugationor equilibrium dialysis are the commonly usedmethods.148–150 Potentiometry, a pH-metric pro-cedure analogous to that applied for solvent/waterdistribution coefficients does not involve a separa-tion step.151,152 H-NMR spectroscopy is anotheralternative to study solute/membrane interac-tions which does not require separation ofphases.153–155 Line-width broadening upon inter-action of solutes with liposomes is a linearfunction of phospholipids concentration, with aslope (called NMR slope), which can used toquantify the degree of this interaction.156

Working with liposomes is associated withdifficulties and considerable care is required,compared to octanol. The preparation of liposomesshould be performed under inert atmosphere andreduced temperatures. They have limited stabilityand liposome/water partitioning studies are moresusceptible to experimental conditions, whileNernst conditions are not always achievable.157

These difficulties limit the use and the wide scaledevelopment of liposomes as an alternative tooctanol in partitioning studies.

IMMOBILISED ARTIFICIALMEMBRANE CHROMATOGRAPHY

The criticism towards octanol as an isotropicmedium with only a superficial similarity tobiomembranes and the difficulties associatedwith the use of liposomes as more representativemodels, have triggered the development ofIAM stationary phases for use in HPLC. IAMchromatography offers a promising alternativeto simulate liposome/water partitioning andcell membrane permeation and has unfoldednew perspectives in the application of HPLCas a tool to mimic specific interactions withphospholipids.158–160

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IAM stationary are prepared from phos-pholipids covalently bonded to a propylamino-silica support material at monolayer densities.Remaining propylamine residues are treatedin a second step to suppress undesired basicfunction on the silica backbone. Moreover, freesilanol groups, although not easily accessible,may interfere in secondary interactions. Themost frequently used IAM column is IAMPC,which contains phosphatidylcholine. In fact, threedifferent types of IAMPC have been introduced inthe market, the single chain IAMPC-DD, thedouble chain IAMPC-MG and IAMPC-DD2, whichdiffer on the way the remaining propylamineresidues are treated. It is reported that doublechain IAM surfaces simulate better naturalphospholipids and thus the resulting chromato-graphic indices correlate better with permeabilitydata.160–162

IAM columns permit the use of aqueousmobile phases without addition of organic modi-fier, leading to directly measured log kw valuesand reducing considerably the time of analysis.The buffer of choice is phosphate buffered saline(PBS) in order to mimic physiological conditions.The pH limitations of the column restrict mea-surement in the pH range 2.5–7.4. Many authorsprefer the use of pH 7.0, which is close tophysiological pH and safer for the column.160,162

In the case of compounds with strong affinityfor the IAM surface, acetonitrile up to 30% ispreferably added and log kw values are obtainedby linear extrapolation. The use of methanol asorganic modifier is avoided, since it affects thestability of the column, leading to methanolysis ofthe phospholipids. Nevertheless, the ageing of thecolumn should be checked from time to time usingstandard compounds.163–165

The structural characteristics of IAM surfaceshave been investigated and compared to those ofliposomes. The synthetic phospholipids linked tosilica-propylamine skeleton are fully saturatedand usually contain acyl chains with 16 carbonatoms (propylamine included), while egg phos-phatidylcholine, the most widely used phospholi-pid for liposome preparation, contains longer acylchains and cis double bonds.133,166 Liposomesconstitute a fluid phase and each vesicle is formedby a phospholipids bilayer, which separatesthe external from the inner aqueous phase. Incontrast, IAM are solid-phase system, where aphospholipid monolayer is covalently linked to thesilica backbone. As a result, the hydrophobic coreis half large in IAM surface than in liposomes and


phospholipids are more ordered and deprivedfrom molecular dynamics. The most importantdifference concerns the density of the polarheadgroups, which in IAM surface is smallerthan in liposomes. However, the polar interfacialregion in IAM surface mimics well that of fluidmembranes. Thus, in both systems the cholinegroups exhibit larger motional fluctuations thanphosphate groups and the distribution of waternear the polar headgroups is the same.167

According to Ong and Pidgeon, solute partition-ing seems to be the principal retention mechanismin IAM retention, with similar thermodynamics onboth the single-chain and double-chain IAMPCsurfaces. For a set of phenol derivatives, thepartitioning into IAMPC surfaces was found tobe both enthalpy and entropy driven, while forbeta-blockers, it was entropy driven.168 Otherauthors report negative DS values for two acidicdrugs (warfarin and salicylic acid) and positive DSvalues for three basic drugs (lidocaine, propranololand diazepam), whereas negative DH valueswere calculated for all five drugs.169 For neutralcompounds the intermolecular forces resemblethose underlying partitioning in octanol/waterand retention in reversed phase HPLC. Hence,besides the hydrophobic/solvophobic interactions,polar interactions, mainly expressed as H-bondacceptor basicity, are the predominant factorsin IAM retention. According to the results ofsolvatochromic analysis however, the hydrophobicterm seems to have a smaller positive contributionin IAM retention compared to its contribution inoctanol/water partitioning.170 Moreover, it is con-sidered that the IAM surface provides a hydro-phobic environment that resembles a RP-C3 HPLCcolumn, whereas for lipophilicity assessment RP-C-18 stationary phase are used.171 For ionisedspecies, polar interactions are electrostatic innature.168 Therefore, protonated basic compoundsare stronger retained as a result of their interactionwith the phosphate anions of the stationary phase.It is reported that IAM retention of protonated b-blockers was found to be stronger, compared toisolipophilic neutral compounds.172 In a studyconcerning structurally diverse basic and neutralcompounds, the degree of protonation had to beconsidered as an additional parameter in order toobtain a good correlation between log kwIAM andlog D values at pH 7.4.173 Otherwise, a bettercorrelation was obtained with log P values, imply-ing that the decrease in the retention due toionisation was partly compensated by the electro-static interactions. These results are depicted in



Figure 4. In the same study, IAM retention wascompared to reversed phase chromatographicretention. Very characteristically, the strong basemetformin, fully protonated at pH 7.4, eluted withthe dead time in reversed phase HPLC, while itwas retained in IAM chromatography due to theelectrostatic interactions of its positively chargedcentre with the phosphate anions.173 In this aspect,the question may arise whether IAM indices aresuitable to predict passive diffusion of drugs, whichaccording to the well known pH-partition hypo-thesis implies reduced absorption of chargedspecies, or they should rather be used to simulatedrug–membrane interactions. As already com-mented about liposomes, IAM chromatographymay also be considered as a border case betweenpartitioning and binding. Electrostatic forces,however, have been reported to be weaker inIAM chromatography than in liposome partition-ing and the reason may be related to the smallerdensity of the polar headgroups in IAM surfaces.174

The contribution of hydrogen bond seems also to beless important for the affinity to IAM stationaryphase than to liposomes.175

Despite these differences, IAM chromatographicindices have successfully been correlated withliposomes partitioning data; however, such studies

Figure 4. Relationship of (A) log D7.4 (B) log P versuslog kwIAM measured at pH 7.4, for neutral (&) and basic(&) drugs.


include a rather limited number of compounds.169

In the case of basic drugs, silanophilic interactionshave been reported to affect the log kwIAM/pHprofile as compared to the corresponding pH/partition diagram in liposomes.169,175

Less investigated is the effect of conformationin IAM retention. In a study of substitutedcoumarins, such effects, induced by the extendedplanar conformation of some large derivatives,were found to have a similar impact in IAMretention and reversed-phase HPLC and TLC,while they did not affect the n-octanol–waterpartitioning.176 Therefore, they should ratherbe attributed to the particular features of thechromatographic procedure, according to which incontrast to solvent/water partitioning, the solutesapproach the stationary phase with the movementof the mobile phase.

The potential of IAM chromatographic indicesto predict passive transport through variousbiological barriers, as well as to estimate pharma-cokinetic properties and certain pharmacologicalactivities has recently been reviewed by Bar-bato.177 IAM chromatographic indices have beensuccessfully correlated with Caco 2 cells perme-ability and intestinal absorption in rat. In thelatter case, molecular weight should be introducedas an additional parameter to improve thecorrelation.178–180 Other pharmacokinetic para-meters (protein binding, partition to erythrocytes,apparent volume of distribution), as well aspenetration across the blood brain barrier havealso been satisfactorily correlated with IAMindices.159,181,182 Most studies, however, concernsmall data sets of congeneric compounds. Never-theless, the belief that IAM chromatographyshould be considered to be always a better choicefor modelling biological processes is disputed in arecent publication on the similarity between IAMcolumns, conventional HPLC columns, octanol–water partitioning and biopartitioning systems bymeans of solvatochromic analysis.183 To thispoint, it should be noted that the last years othertypes of biomimetic stationary phases, like humanserum albumin (HSA) or a-acid glycoproteincontaining surfaces, are available, creating newperspectives on biochromatography as an alter-native rapid technique to investigate biologicalprocesses.184

SUMMARY AND CONCLUSIONS

Figure 5 provides a rough summary of the mostimportant information discussed in the present

DOI 10.1002/jps

Figure 5. Schematic representation of the basic features of lipophilicity andphospholipophilicity, the related indices and the impact of chromatographic techniquesin lipophilicity/phospholipophilicity assessment.


review. Far for being complete, Figure 5 focuses onthe distinction between lipophilicity and phos-pholipophilicity, as well as on the impact andcommon characteristics of chromatographic tech-niques in the assessment of both properties. Itsummarises the difficulties in direct partitioningexperiments, (which in the case of liposomesare the major obstacle), and the advantages ofreversed phase chromatographic indices with awarning for potential silanophilic interactions.Question marks refer to the parallelism betweenliposome partitioning and IAM retention, denot-ing that further investigations are necessary.

In conclusion, despite criticisms, for more thanforty years n-octanol–water partitioning system isused as a reference system to assess lipophilicity.During these decades, large progress has beenachieved concerning prediction of log P and log D,as well as the understanding of the particularnature of log P as a prerequisite to appreciate thebehaviour of solutes in a lipophilic environment.On the other hand, based on the similarity ofreversed phase retention with n-octanol–waterpartitioning, chromatographic techniques havebeen developed and conditions refined to produce


lipophilicity indices analogous to log P or log D.The turn towards liposomes as more representa-tive models for membranes triggered the devel-opment of IAM stationary phases to overcome thedifficulties associated with liposome partitioningand to guarantee rapid measurements. Althoughthe advantages of IAM chromatographic indices tomimic membrane interactions and/or permea-bility are still under investigation, biochromato-graphy is attracting an increasing interestoffering new perspectives in the research ofbiological processes.

ACKNOWLEDGMENTS

The authors would like to thank CompuDrugInternational, Inc. and Dr. Ferenc Darvas whokindly offered Pallas software v.3.3.2.4 for log Dcalculations.

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