Chemical Engineering Science 65 (2010) 5921–5928

Contents lists available at ScienceDirect

Chemical Engineering Science

0009-25

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/ces

An improved approach for determining permeability and diffusivity relevantto controlled release

Yiwei Chen a, Yufeng Zhang b, Xianshe Feng a,n

a Department of Chemical Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1b School of Materials Science and Engineering, Tianjin Polytechnic University, Tianjin, 300160, China

a r t i c l e i n f o

Article history:

Received 17 June 2010

Received in revised form

12 August 2010

Accepted 19 August 2010Available online 27 August 2010

Keywords:

Time lag

Mass balance

Permeability

Diffusivity

Controlled release

Drug delivery

09/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ces.2010.08.028

esponding author. Tel.: +1 519 888 4567x365

ail address: xfeng@uwaterloo.ca (X. Feng).

a b s t r a c t

A new approach combining time lag and mass balance for determining the kinetic parameters in

controlled release was proposed in order to minimize measurement errors due to actual experimental

conditions deviating from idealities under which the time lag and mass balance analyses were developed.

Basically, the permeation data were pre-screened for analysis with time lag and mass balance methods

separately, and criteria for data screening were proposed. The upper time limit of short-time permeation

data suitable for analysis with the time lag method was determined based on the extent of concentration

variations across the membrane, and the onset time point of the long-time permeation data for analysis

with the mass balance method was determined based on the time lag of the system. The permeation of

ciprofloxacin hydrochloride through chitosan membranes was chosen as a model system to demonstrate

the validity of this approach for evaluation of permeability and diffusivity.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Controlled drug release systems provide a sustained deliveryrate of drugs while maintaining their concentrations withintherapeutic limits to overcome problems associated with fluctua-tion of drug concentrations in conventional dosage forms(Hoffman, 2008; Langer and Peppas, 1981; Uhrich et al., 1971).Among them, diffusion-controlled reservoir systems have attractedsignificant interest among pharmaceutical and allied industries dueto great ease with which they can be designed to achieve near-zero-order release kinetics (Langer and Peppas, 1981). In such asystem, the drug is surrounded by a polymeric membrane, and thedrug diffuses through the membrane to the receptor side. Sincediffusion is the rate-limiting step, determining the permeabilityand diffusivity of drug molecules in the membrane is therefore animportant subject in evaluating the drug release performance.Ideally, the drug release systems for kinetic studies should bedesigned in such a way that the intrinsic kinetic parameters can bedetermined accurately. The same also applies to such other releasesystems as delivery of pesticides and fertilizers through syntheticmembranes and transdermal drug delivery. However, a carefulreview of the literature work reveals that this is not always thecase, and a common problem is that the test conditions deviatefrom restricting assumptions under which a certain model can be

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used for evaluation of experimental data. There appears to be anoversight of this issue with the use of both the time lag techniqueand mass balance method, which are the two approaches mostwidely used to determine the kinetic parameters of the release viapermeation process.

Three stages can be distinguished as time proceeds duringthe course of drug diffusion through a membrane: (1) an initialstage of transient permeation, followed by (2) pseudo-steady-state permeation when the concentration difference across themembrane is practically constant, and then (3) unsteady statepermeation when the drug concentration at the receptor sidebecomes significantly high. Eventually the permeation ceases tooccur when the transmembrane concentration difference ap-proaches zero. This is illustrated in Fig. 1, which shows theaccumulative amount of drug released at different times. Thetransient permeation results from the fact that it takes time toestablish a concentration profile in the membrane, and there isthus a ‘‘time-lag’’ (y) in the permeation. After elapse of thetransient permeation, the pseudo-steady-state release can bedescribed by (Crank and Park, 1968)

Q ¼PAC0

lt�

l2

6D

� �ð1Þ

where Q is the total amount of permeant passing through themembrane (thickness l and area A) in time t, C0 the initial drugconcentration at source side, P and D are the permeability anddiffusivity coefficients, respectively. Thus, plotting Q versus t will

θ ~3θ0

Am

ount

Per

mea

ted

Q

Transientstate

QuasiSteadystate

TowardsEquilibrium

Time t

Fig. 1. Quantity of permeant received in the receptor side of the membrane during

the course of permeation to illustrate the time lag due to transient permeation.

Y. Chen et al. / Chemical Engineering Science 65 (2010) 5921–59285922

yield a straight line with an intercept on the time-axis equal to

y¼l2

6Dð2Þ

where y is the so-called time lag. Consequently, the permeabilityand diffusivity coefficients can be determined from the slope andintercept of Q–t graph, respectively. The partition coefficient (K)can further be calculated from K¼P/D. This is the basis of the timelag method. It should be clarified that the time lag is the interceptof Q versus t plot on the time axis obtained by extrapolating thepseudo-steady-state permeation data. The above equations aresubject to several assumptions: (i) the membrane is initially freeof the drug, (ii) the drug concentration at donor side (CD) isconstant (i.e., CD¼C0), (iii) the drug concentration at receptor side(CR) is zero or negligible in comparison to the donor concentra-tion. Also assumed is concentration-independent partition anddiffusion coefficients in the membrane, and this assumption canbe relaxed by the use of apparent or mean quantities (Koros andChern, 1987). However, the assumptions of infinite reservoir andinfinite sink conditions at the two sides of the membrane aremore restricting, and a violation of these assumptions may lead tosignificant errors. Thus the experimental design is particularlyimportant because in practice the reservoir concentration even-tually depletes whereas the sink concentration gradually in-creases. In general, this issue can be addressed by proper design ofexperiments so as to minimize the concentration changes duringthe course of permeation (see, for example, Nakatsuka andAndrady, 1992; Guilherme et al., 2002; Sanli and Asman, 2004;Watanabe et al., 2009; Mittal et al., 2008), although there havebeen rare cases of significant variations in the reservoir concen-tration while still applying the above equations to evaluatein vitro permeation of anticholinergics in epidermal membrane(Bosman et al., 1998), presumably due to skin membranes beingviable for only for a rather short period of time (Shah, 1996).

Another point that should be stressed, but sometimes over-looked is that the time lag is determined by extrapolating thepseudo-steady-state permeation data, while a pseudo-steadystate is reached after approximately three times of the time lag(Crank, 1975). That is, the permeation experiments should beconducted for a sufficiently long period of time. Otherwise, if thepseudo-steady-state permeation is not reached when a seeminglyconstant flux is observed because of the low diffusivity, the time

lag technique will result in an underestimate in the permeabilityand time lag, and an overestimate in the diffusivity. This couldbe the case for the measurements of transdermal permeationof anticholinergics where each permeation experiment wasterminated in only 25 h while the time lag was reported to be10–16 h (Bosman et al., 1998). In general, the diffusion coefficienthas an order of magnitude of 10�13–10�8 cm2/s, and as such thetime lag of diffusion can be from a few minutes to days(Nakatsuka and Andrady, 1992; Watanabe et al., 2009; Shahet al., 1994). It should be further clarified that the time lag isneither the time that the permeant takes to cross the membraneand begin to release nor the time required to reach a steady state,as misperceived (see, for example, Fang et al., 1998; Saikumar andJaccodine, 1993; Yi, 1995; Wagner et al., 2001; Bader et al., 2009).As mentioned earlier, it is a time-like quantity extrapolated from(pseudo)steady-state permeation data.

For finite source/receptor conditions, an alternative method forthe determination of permeability is based on variations in theconcentrations at the donor and receptor sides of the membrane.From mass balance for the permeant at both sides of themembrane

VDðC0�CDÞ ¼ VRCR ð3Þ

where VD is the volume of donor source. The permeation ratethrough the membrane is given by

VRdCR

dt¼

AP

lðCD�CRÞ ð4Þ

where VR is the volume of the receptor side. Integrating Eq. (4)from t¼0 to t¼t and rearranging gives

�ln 1� 1þVR

VD

� �CR

C0

� �¼

PA

l

1

VDþ

1

VR

� �t ð5Þ

This equation accounts for different volumes of the source andthe receptor. It can also be expressed as

�lnm0�VtCR

m0

� �¼

PA

l

1

VDþ

1

VR

� �t ð6Þ

where m0(¼VDC0) is the total amount of drug in the system andVt(¼VD+VR) is the total source and receiving volume. WhenVR¼VD¼V, Eq. (5) reduces to

�ln 1�2CR

C0

� �¼

2PA

Vlt ð7Þ

which is widely used in the literature using permeation setuphaving equal volumes of source and receptor compartments(Wisniewski and Kim, 1980a,1980b; Gilbert et al., 1988; Lee et al.,1988; Jin and Song, 2006). Plotting the logarithmic term againsttime will produce a straight line, and the permeability can becalculated from its slope. The mass balance approach, whichaccounts for variations in the source and receptor concentrations,appears to be more general and impose less restriction on theexperimental conditions than the time-lag method. Even for near-zero-order releases, the mass balance method gives reportedly apermeability coefficient close to the true value of the membrane(Tavelin et al., 2002). However, this is not without reservation.The underlying assumptions for the mass balance approach arethat (i) the amount of permeant within membrane is negligible,and (ii) the permeant is received in the receptor instantly as soonas it leaves the donor source. That is, the time required for theconcentration profile to establish in the membrane, which wouldcause transient permeation, is neglected. Consequently, thepermeability is essentially calculated using the CR data at time t

and 0, although the data at early stage of permeation, which areaffected by ‘‘transient’’ permeation significantly, should beexcluded. Otherwise, the permeability will be underestimated,

θ ~3θ0

Transient stateTowards

Equilibrium

Slope

Underestimated slope

-ln

[(m

0-V

tCR)/

m0]

Time t

Fig. 2. Illustration of underestimation in permeability calculated on the basis of

the conventional mass balance method.

Time lag �Permeation

dataPermeabilitycoefficient P

Mass balance method

Diffusioncoefficient D

Time-lag method

Fig. 3. A new approach combining time lag and mass balance methods: using the

time-lag technique to determine diffusivity from the short-time permeation data

and using modified mass balance equation to determine permeability from the

long-time permeation data. The terminating point for the short-time permeation

data is determined based on tolerance in concentration variations, and the onset

point for the long-time permeation data is determined based on the time lag

determined from the short-time permeation data.

Y. Chen et al. / Chemical Engineering Science 65 (2010) 5921–5928 5923

as illustrated in Fig. 2, which depicts how the slope of�ln½ðm0�VtCRÞ=m0� versus t curve changes with time. Thisproblem is particularly important in view of the low diffusioncoefficients generally encountered. Unfortunately, little cautionhas been exercised in determining permeability coefficients basedon the mass balance approach. For instance, experiments forpermeation of progestin through a series of hydrogel membraneswere all carried out within 162 h, in spite of a significantly longperiod of transient permeation as estimated from the time lag (upto 380 h) based on the diffusivity reported (Zentner et al., 1978).Unless the diffusivity is fast enough that the transient part can beneglected, the accuracy in the permeability so determined iscompromised. Unlike the time lag technique, the diffusivitycannot be determined using the mass balance approach.

In view of the above, we propose a new approach thatcombines the time-lag and mass balance methods for determiningboth permeability and diffusivity, taking advantage of thecomplementary characteristics of the two methods. For thepurpose of demonstration, the permeation of ciprofloxacinhydrochloride through chitosan membranes was used as a modelmembrane-enclosed reservoir system for drug delivery. Cipro-floxacin hydrochloride is an antimicrobial agent widely used fortopical treatment of infections, and chitosan, a polysaccharidewith many amino and hydroxyl groups, has attracted significantattention for pharmaceutical applications because of its excellentbiocompatibility, biodegradability, antimicrobial activity andenhanced wound healing properties (Wang et al., 2007).

2. The new approach

The idea of the new approach is to evaluate the short-timepermeation data with the time lag technique and the long-timepermeation data with the mass balance method to account forconcentration variations, as illustrated in Fig. 3. It involves twosteps: (i) determine the upper limit of time beyond which theconcentration variation is no longer acceptable for the time lagmethod, which requires infinite source and sink conditions, and(ii) based the time lag obtained, determine the onset point atwhich the impact of transient permeation at the early stage is nolonger important so that the mass balance can be applied. Thus, amodified equation based on mass balance needs to be derived forevaluation of the permeability coefficient. The permeabilityobtained from the time-lag method based on short timepermeation and that obtained from mass balance based on

long-time permeation can thus be compared. The procedure ofthis approach is described in the following sections.

2.1. Step 1: evaluation of time-lag and diffusivity

A plot of Q versus t is constructed, and the long time permeationdata will be excluded in order to apply Eqs. (1) and (2). Theconcentration difference across the membrane is the driving forcefor permeation, and it changes from (C0�0) at beginning to (CD–CR)at time t. Let us assign a relative change in the transmembraneconcentration that is no greater than Z to relax the assumptions ofconstant source and zero sink concentrations

1�CD�CR

CD�0

� �rZ ð8Þ

Combining with Eq. (3) gives

CRrZC1 ð9Þ

where CN (¼m0/Vt) is the equilibrium concentration the systemtends to reach when the permeation ceases. For a given value of Zassigned, the upper limit of CR is defined by Eq. (9), and thisdetermines where to truncate the Q versus t data for use in thetime-lag evaluation. A series of Z values (for example, 2%, 5%, 10%)may be used, and if the time lag or permeability can be obtainedconsistently within the experimental error in the measurements,the applicability of the time-lag approach is justified.

2.2. Step 2: evaluation of permeability from long time permeation

data

The impact of transient permeation is considered to havediminished after three times of the time lag. Therefore, the longtime (t43y) permeation data are subjected to mass balanceanalysis. Assume that at time t¼t0(43y) the concentrationsare CR¼a and CD¼b. For time t4t0, the mass balance for thepermeant in the system yields

VDðb�CDÞ ¼ VRðCR�aÞ ð10Þ

Integrating Eq. (4) from t¼t0 to t¼t gives

�ln 1� 1þVR

VD

� �CR�a

b�a

� �� �¼

PA

l

1

VDþ

1

VR

� �ðt�t0Þ ð11Þ

Based on concentration change from t¼0 to t¼t, one has

VDðC0�bÞ ¼ VRða�0Þ ð12Þ

Then Eq. (11) can be rearranged to give

�lnm0�VtCR

m0�Vta

� �¼

PA

l

1

VDþ

1

VR

� �ðt�t0Þ ð13Þ

DonorVDCD

Membrane

Y. Chen et al. / Chemical Engineering Science 65 (2010) 5921–59285924

By selecting an appropriate reference point t0 and plottingF(t)¼�ln½ðm0�VtCRÞ=ðm0�VtaÞ� against t, a straight line will beobtained and permeability coefficient P can be calculated from itsslope.

It may be mentioned that when a permeation experiment iscarried out in a constantly renewed reservoir/receptor system tosimulate certain in vitro release of drugs, the above analysis stillholds. The diffusivity measurement based on time lag from short-time permeation will be more accurate due to constant reservoirconcentration, and the equations for permeability measurementfrom the long time permeation can be modified accordingly.A detailed analysis about such representative systems is subjectedto further investigation in a separate study.

ReceptorVRCR

Fig. 5. Schematic apparatus for permeation experiment.

3. Experimental

3.1. Membrane and ciprofloxacin hydrochloride

Homogeneous chitosan membranes were prepared by thesolution casting technique. First, chitosan flakes (degree ofdeacetylation 99%, molecular weight 100,000) supplied fromKyowa Technos (Chiba, Japan) was dissolved in 2 wt% aqueousacetic acid solutions to form 1 wt% chitosan solution. Then thechitosan solution was filtered to remove undissolved particles andimpurities before a film was cast onto a glass plate and dried in airat room temperature. Finally, the membrane was treated with1 M NaOH solution for 5 h to convert the cationic amine groups(�NH3

+) to the free amine form (�NH2). The membrane wasrinsed thoroughly with deionized water prior to use. The resultingmembrane was a dense and homogeneous film, and thus the drugrelease was by diffusion in the membrane, and not by diffusive orconvective transport in pores as in a porous matrix.

Ciprofloxacin hydrochloride was obtained by reacting cipro-floxacin with excess hydrogen chloride (molar ratio 10:1) at roomtemperature. Both ciprofloxacin and hydrogen chloride solution(2.0 M in diethyl ether) were supplied by Sigma-Aldrich. Thereaction product was a light yellow suspension, which wasfiltered and then washed extensively with anhydrous diethylether to remove excess hydrogen chloride. After drying in air, thepurified ciprofloxacin hydrochloride was an off-white powder.

The molecular structures of chitosan and ciprofloxacin hydro-chloride were shown in Fig. 4.

Fig. 4. Molecular structures of chitosan and ciprofloxacin hydrochloride.

3.2. Permeation experiments

The experimental apparatus is shown schematically in Fig. 5. Itcomprised of a source compartment of 50 ml capacity and areceiving compartment of 2500 ml capacity. The membrane,which had been stored in deionized water, was mountedhorizontally at the bottom of the source compartment, whichwas suspended on top of the receiving compartment. Then thereceiving compartment was filled with 2350 ml of deionizedwater, and the source compartment was lowered to be partiallyimmersed in water. At time zero, the source compartment wasfilled with 30 ml of a drug solution at a predeterminedconcentration (ranging from 0.2 to 0.6 mg/ml) so that permeationstarted to occur. Vigorous agitations were provided mechanicallyin both compartments to eliminate the boundary layer effect, andthis was confirmed by the fact that the measured permeability anddiffusivity remained constant when a more intense agitation wasused. The concentration of ciprofloxacin hydrochloride in thereceptor side during the course of permeation was measured usinga Shimadzu UV–vis Spectrophotometer. The effective area of themembrane for permeation was 7.54 cm2. The experiments werecarried out at room temperature (20 1C), unless specified other-wise, for the purpose of demonstrating the approach presented.

3.3. Determination of partition coefficient and degree of swelling

The partition coefficient of ciprofloxacin hydrochloridein chitosan was determined by solute uptake experiments(Nakatsuka and Andrady, 1992; Jin and Song, 2006; Singh andRay, 1990). The membrane pre-swollen in water was placed in aciprofloxacin hydrochloride solution. After equilibrium, the con-centration of the solution (C1) was determined, and then themembrane was taken out from the solution, blotted andimmersed in deionized water of known volume (V) for 24 h. Fromthe concentration of ciprofloxacin hydrochloride released in thewater (C2), the partition coefficient K was calculated by

K ¼C2V

VpðC1�C2Þð14Þ

where Vp refers to the volume of the swollen membrane.

30

40

min

10%

15%

5%

7%

Y. Chen et al. / Chemical Engineering Science 65 (2010) 5921–5928 5925

The degree of swelling (Rsw) of chitosan membrane in waterand the drug solution was determined at 20 1C from mass uptakeusing the following equation:

Rsw ¼Wsw

Wd� 100% ð15Þ

where Wd is the weight of a dry membrane sample and Wsw theweight of the swollen membrane.

10

20

Tim

e la

g,

31Sample number

2 Mean4 5

Fig. 7. Time lag determined from the short-time permeation data at different

levels of tolerance in concentration variations. Experimental conditions are same

as in Fig. 6.

4. Results and discussion

4.1. Time-lag analysis of short-time permeation data

Fig. 6 shows the experimental data for the permeation ofciprofloxacin hydrochloride through a 110 mm thick chitosanmembrane at an initial drug concentration of 0.2 mg/ml. All themembrane thickness data mentioned in this work were thethicknesses of swollen membranes. The transient permeation atan early stage is clearly observed. The terminating point definingthe upper limit of time for applying the time-lag method toanalyze the data at different values of Z (i.e., tolerance of theconcentration variations as a relaxation of the assumptions ofinfinite source and sink conditions for the time lag method) arealso shown in the figure. Obviously, a smaller value of Z means asmaller deviation from ideality and thus better validity inapplying the time-lag method. However, there will be fewer dataavailable for use in determining the time lag by data extrapola-tion, making the parameter estimate less accurate. This can bebetter illustrated by Fig. 7, which shows the values of time lagsobtained from five sets of duplicate permeation experiments withZ being 5%, 7%, 10% and 15%. At Z¼5%, there is a considerablefluctuation in the numerical values of the time lag around themean time lag. The fluctuations are more significant at lower Zvalues because of fewer data points available for the time lagevaluation. The Z values selected should not be too small from atheoretical point of view as well because the corresponding timefor data truncation should be sufficiently large to include pseudo-steady-state permeation data that are used to determine the timelag. When Z increases to 10%, the fluctuation in the time lag isdecreased significantly. The mean values of the time lag appear tobe more or less the same for Z¼5–10%, which justifies theapplicability of the time-lag method under such restrictiveconditions. As discussed before, the Z selected should not be too

0 100 200 300 400 5000

200

400

600

800

1000

CR,1

0-6 m

g/m

l

t, min

η-dependentCR upper limit

η = 15%

10%

7%5%

0.0

0.1

0.2

0.3

Q, m

g/cm

2

Fig. 6. Concentration of ciprofloxacin hydrochloride in the receiving tank as a

function of time. Initial source drug concentration 0.2 mg/ml; membrane

thickness, 111 mm.

big because the validity of the time-lag method will becompromised if the important assumptions under which thetime-lag method is developed are seriously violated. At Z¼15%,for instance, the obtained mean time lag is shown to be clearlydifferent from that obtained with Z¼5–10%, although thefluctuation around its mean did not seem to be necessarily moresignificant. Therefore, the pros and cons with regard to modelvalidity and the accuracy in parameter estimate should beconsidered in selecting an appropriate Z, which is actually ameasure of the level of tolerance in the relaxation of the modelassumptions. Nonetheless, the above results suggest that a Zranging from 5% to 10% appears to be appropriate in selecting theshort-time permeation data for parameter estimate with thetime-lag method for the system studied here. A Z value of 10% wasthus employed in subsequent calculations.

It may be pointed out that the receptor volume used in this work(i.e., 2350 ml) was substantially larger than those commonly used inprior studies (in the range of 0.9–53 ml (Nakatsuka and Andrady,1992; Guilherme et al., 2002; Watanabe et al., 2009; Shah et al.,1994). This will help slow down the concentration buildup in thereceptor, thereby allowing a longer period of permeation while stillsuitable for the time-lag analysis. In the measurements of Orange IIpermeability in polyacrylamide hydrogel membranes with anapparatus of equal volumes of the two compartments (50 ml), thetime-lag measurements have been reported to be rather inaccurate(Muniz and Geuskens, 2000). A close look at the receptor concentra-tion revealed that the transmembrane concentration differenceexperienced a 17% reduction during the course of permeation, whichcould be decreased using a larger receptor volume.

4.2. Selection of t0 for long-time permeation data analysis with the

mass balance method

After the time lag had been obtained from the short-timepermeation data, t0 needs to be selected to evaluate the long-timepermeation data based on the modified mass balance method. It isa reference point set to exclude the experimental data in the earlystage of permeation; only the data beyond this time will be usedfor determination of permeability with the mass balance analysis.Fig. 8 shows how the t0 selection on the F(t)–t profile. When t0¼0,that is, the F(t)–t relationship conforms to the original unmodified

Y. Chen et al. / Chemical Engineering Science 65 (2010) 5921–59285926

mass balance Eq. (6), and clearly there is a nonlinear part in theF(t)–t profile during the early period. The nonlinear part, which isdue to the impact of transient permeation at the beginning,gradually diminishes with an increase in the t0 selected. After t0

exceeds about three times of the time lag, the influence oftransient permeation vanishes, and a straight line of F(t) versus t

results with a practically constant slope.This may also be illustrated by Fig. 9, which shows the

permeability coefficient calculated using Eq. (13). As expected,when the t0 values are less than 3y, the permeability coefficientcalculated from the experimental data varies with time, and thereis no consistency in the permeability so obtained. These resultsclearly demonstrate the problems associated with the originalmass balance approach (i.e., t0¼0). In addition, the calculatedpermeability coefficient tends to increase if a larger value of t0 is

Fig. 8. The F(t) versus t curves based on the modified mass balance method for

different values of time t0. Initial source drug concentration 0.3 mg/ml; membrane

thickness, 150 mm.

0 100 200 300 4000

1

2

3

4

5

6

7

t, min

P, 1

0-7cm

2 /s

uniform valuet0 value

0

θ/2

θ

3θ

4θ

5θ

Fig. 9. The membrane permeability calculated using Eq. (13) with different values

of t0. Experimental conditions are same as in Fig. 8.

used. This is not surprising because the impact of transientpermeation in the early stage on subsequent permeation becomesless important as permeation proceeds with time. The permeationrate during transient stage is lower than the permeation rate atsteady state. Thus, the permeability will be underestimated unlessthe influence of transient permeation is eliminated. Therefore,when t0 is large enough (i.e., greater than three times of time lag),a consistent and practically constant value of permeability isobtained. However, using a larger value of t0 means moreexperimental data will be truncated and excluded for use inthe permeability evaluations, and the t0 selected should not beunnecessarily large. In addition, as Eq. (13) suggests, thepermeability can be determined mathematically with only twodata points: the reference point at time t0 and another point attime t. In consideration of experimental errors associated, it isrecommended to determine an averaged permeability from theslope of the F(t)–t plot using t0¼3y.

4.3. Effects of membrane thickness

The applicability of the new approach was further demon-strated with membranes of different thicknesses. The influence ofthe membrane thickness on the release kinetics of ciprofloxacinhydrochloride through chitosan membranes was studied at aninitial drug concentration of 0.2 mg/ml. As shown in Table 1, theswelling ratio of the chitosan membranes in water is shown to beindependent of the membrane thickness, because of the homo-geneousness of the membranes. The overall percentage release ofciprofloxacin hydrochloride after 7 h reached 45% with a mem-brane thickness of 50 mm, and the percentage release of the drugappears to be inversely proportional to the membrane thickness.

Fig. 10 shows the membrane permeability coefficient calcu-lated from the long-time permeation data by the mass balancemethod (denoted as PM) and the diffusivity and permeabilitycoefficients calculated from the short-time permeation data withthe time-lag method (denoted as DT and PT, respectively). Themembrane thickness did not affect the permeability coefficient inthe experimental range studied. It is interesting to note that thePM values tended to be greater than the PT values. This is notunexpected in consideration of the calculation methods used.Both PM and PT calculations were based on the first law of Fickiandiffusion, which regards the driving force for drug permeation asthe concentration difference across the membrane. As permeationproceeds with time, the drug concentration at the source sidedecreases, while the drug concentration at the receptor sideincreases. The calculation of PM based on the long-time permea-tion with the mass balance method had accounted for concentra-tion variations in the drug at both sides of the membrane.However, the concentration variations were neglected in theevaluation of PT based on the short-time permeation with thetime-lag method, and the initial source drug concentration andzero receptor concentration were used in the calculation. There-fore, there is an underestimation in the permeability measured by

Table 1Degree of membrane swelling in water and percentage drug release after 7 ha.

Membrane thickness

(mm)

RSW in water (%) Percentage drug release after 7 h

50 130 45

73 128 32

100 132 24

125 127 19

150 128 15

a Initial source drug concentration 0.2 mg/ml.

40 80 120 1600

2

4

6

8

10

Membrane thickness, μm

P, 1

0-7 c

m2 /

s

PM

PT

0.0

0.5

1.0

1.5

2.0

2.5

3.0

DT

D, 1

0-8 c

m2 /

s

Fig. 10. The diffusivity (DT) and permeability (PT) determined from the short-time

permeation data and the permeability (PM) determined from the long-time

permeation data. Initial source drug concentration 0.2 mg/ml.

Table 2Degree of membrane swelling in drug solutions percentage drug release after 7 ha.

Initial source drug

concentration (mg/ml)

Degree of swelling in

drug solution

Percentage drug

release after 7 h

0.1 165 20

0.2 157 15

0.3 166 19

0.4 168 19

0.5 166 17

a Membrane thickness 150 mm.

0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

PM

PT

Initial source drug concentration, mg/ml

0

2

4

6

8

10

DT

P, 1

0-7cm

2 /s

D, 1

0-8 c

m2 /

s

Fig. 11. Permeability and diffusivity at different drug source concentrations.

Membrane thicknesses, �150 mm.

Y. Chen et al. / Chemical Engineering Science 65 (2010) 5921–5928 5927

PT, and PM is practically a more realistic representation of themembrane permeability.

Fig. 10 also shows that the DT values are more fluctuated andappear to be a little smaller for thinner membranes. The averagevalue of DT is 1.1�10�8 cm2/s, well within the common range ofdrug diffusivities in various membranes. The permeation of the drugmolecules through the membrane can be considered to occur by athree-step process of sorption–diffusion–desorption. The time-lagmethod assumes a sorption equilibrium of the drug molecules onthe membrane surface, that is, the sorption reaches equilibriuminstantaneously, and the molecular diffusion determines the rate oftransport. However, because of the carboxylic and protonated aminogroups in ciprofloxacin hydrochloride and the hydroxyl and aminogroups in chitosan, there are very strong hydrogen bond andelectrostatic interactions between the drug and the membrane(Wang et al., 2007). Thus if the equilibrium cannot be reachedinstantaneously due to limited rates of drug sorption onto themembrane at the source side and desorption from the membrane atthe receptor side, the observed time lag in permeation will increase,causing an underestimate in the diffusivity coefficient obtained fromthe time-lag calculation because diffusion is considered the only ratecontrolling step. The time required to establish sorption anddesorption equilibria on the membrane surfaces is expected to beless significant on a relative scale for a thicker membrane because ofits longer diffusional paths, resulting in a diffusivity measurementthat is underestimated less significantly. The permeability coeffi-cient, however, will not be affected as both quantities PM and PT arecalculated from the permeation rate data after transient permeationhas elapsed.

As mentioned before, the time-lag method has been used inthe literature to determine diffusivity (D) and permeability (P),from which the partition coefficient can be calculated using P¼DK

(Watanabe et al., 2009; Mittal et al., 2008; Shah et al., 1994).The mass balance method can only be used to determine thepermeability (P), and the partition coefficient (K) can bedetermined separately via sorption experiments in order tocalculate the diffusivity. To our knowledge, there has been nowork on the comparison of the parameters evaluated using thesedifferent approaches, although it is imperative that these para-meters be consistent no matter what measurement techniquesare used. In this work, the partition coefficient (K) of ciprofloxacinhydrochloride in the membrane was measured to be 59.4,irrespective of the membrane thickness as expected. It is

considerably high due to the good affinity between the membraneand the drug molecules. The membrane permeability can thusalso be estimated from K and the diffusivity for comparisons withPT and PM. Based on an average diffusivity of 1.1�10�8 cm2/sfrom the time-lag measurements, the permeability (denoted asPTC) was calculated to be 0.65�10�6 cm2/s, which was close tothe average PM and PT. (0.50�10�6 and 0.32�1016 cm2/s,respectively). This is considered to be very satisfactory in viewof the different experimental techniques used.

4.4. Effect of initial donor concentration

The permeation of the drug molecules through water-swollenchitosan membranes was dominated by the degree of swelling ofthe membrane in the aqueous solution. As shown in Table 2, theswelling degree of chitosan membrane in the ciprofloxacinhydrochloride solutions remained constant (ca. 164%) over aconcentration range from 0.1 to 0.5 mg/ml. It was higher than theswelling degree in water because of the strong affinity betweenchitosan and ciprofloxacin hydrochloride due to their stronghydrogen bond and ionic interactions. Seven hours after the startof permeation, the accumulative percentage release of the drugreached �18%, and the relative release rate was not affected bythe initial drug concentration.

The permeability (PM and PT) and diffusivity (DT) coefficients atdifferent initial drug concentrations are shown in Fig. 11. Oneassumption in the parameter evaluation is concentration inde-pendence of the diffusivity, which appears to be satisfactory in theexperimental range studied. However, it needs to be mentionedthat because of the acidity of ciprofloxacin hydrochloride(pHE4.9 at a concentration of 0.6 mg/ml), the physical integrityof the membrane will be compromised at a drug concentration of

Y. Chen et al. / Chemical Engineering Science 65 (2010) 5921–59285928

0.6 mg/ml or higher as the free amine groups in chitosan will begradually attacked, and partial dissolution of the membrane willoccur when the drug concentration is high enough. The drugconcentration was expected to influence the permeability anddiffusivity, but no clear trend was observed within experimentalerror in the concentration range tested. The membrane stabilitycan be improved by crosslinking or blending with other polymersto handle relatively high drug concentrations, which is beyond thescope of this study. Wang et al. (2007) loaded a substantially largeamount of ciprofloxacin hydrochloride in films of chitosan/polyethylene glycol blend (chitosan content 92–98 wt%) forcontrolled release in aqueous media, and no problems in thephysical stability of the films were reported.

It may be pointed out that a simple model system was used inthis work for data analysis and illustration of the approachpresented. In biological membranes (e.g., skin) composing ofmultiple layers with different permeabilities and diffusivities, thepermeability and diffusivity coefficients determined using thisapproach will be apparent effective values. While the apparentpermeability P can be found to be related to the permeabilities ofindividual layers by

l

P¼X

i

liPi

ð16Þ

where li and Pi are thicknesses and permeabilities, respectively, ofindividual layers and l is the total thickness of the membrane(l¼Sli), no simple relationship exists for the diffusivity. None-theless, it is the apparent permeability and diffusivity thatdetermine the overall release kinetics.

5. Conclusions

A new approach combining the time-lag method for analysis ofshort-time permeation data and the mass balance method foranalysis of long-time permeation data was proposed and demon-strated for evaluation of kinetic parameters relevant to controlledrelease of drugs. Basically, the permeation data were pre-screenedfor analysis with time lag and mass balance methods separately.The upper time limit for the short-time permeation data wasdetermined by parameter Z based on concentration variations,and the onset time point for long-time permeation data wasdetermined based on the time lag obtained from the short-timepermeation data. Conventionally, the time lag or mass balancemethod was used alone, and the experiments were rarely at ornear ideal conditions for the two methods to apply, resulting ininaccurate evaluation of diffusivity and permeability. In addition,the initial period of transient permeation was completely ignoredin the traditional mass balance method, which is a significantissue especially for slow diffusing systems. The later has beenaddressed in this work by truncating the permeation data toexclude transient permeation. The present approach, combiningtime lag and mass balance methods, involves truncation, testingand justification of both sets of short- and long-time permeationdata, was shown to give a more reliable and realistic evaluation ofpermeability and diffusivity. For the purpose of demonstration,the permeation of ciprofloxacin hydrochloride through chitosanmembranes was chosen as a model drug release system, and thepermeability and diffusivity were determined for differentmembrane thicknesses and initial drug concentrations.

Acknowledgements

Research support from the Natural Sciences and EngineeringResearch Council (NSERC) of Canada is gratefully acknowledged.The drug permeation experiments were conducted by Heidi Yao.

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