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International Journal of Pharmaceutics 239 (2002) 13–22

Adsorption of an ionizable drug onto microspheres:experimental and modeling studies

Vincent Boudy a,*, Nicolas Voute b, Dominique Pradeau c,Jean Claude Chaumeil a

a Ser�ice Recherche et De�eloppement, Pharmacie Centrale des Hopitaux de Paris, 7 rue du fer a moulin, 75005 Paris, Franceb BioSepra s.a, 48 a�enue des Genottes, 95800 Cergy St Christophe, France

c Laboratoire Central d’Analyses, Pharmacie Centrale des Hopitaux de Paris, 7 rue du fer a moulin, 75005 Paris, France

Received 21 May 2001; received in revised form 9 January 2002; accepted 23 January 2002

Abstract

The purpose of this work was to study the in vitro equilibria and the adsorption kinetics of an ionizable drug,indomethacin, onto commercially available cationic polymeric microspheres: DEAE Trisacryl LS and QA TrisacrylLS. Isotherms were fitted to theoretical equations allowing accurate predictions of drug loading at different saltconcentrations. Isotherm measurements were quickly obtained by simple column breakthrough experiments. Thenature of the ion exchange group of the microspheres was observed to be preponderant for adsorption, as the tertiaryamine derivative exhibited 53% more capacity than its quaternary amine counterpart. The maximum equilibriumuptake capacity in a 5 mM Tris–HCl buffer at pH 7.4 is 303 mmol/ml of particle volume, for DEAE microspheres.Transport properties of indomethacin into the tertiary amine microspheres were obtained in agitated contactor.Microbeads loading was completed in a 1–6 min range and was found to be controlled by pore diffusion mechanism.Equilibrium uptake data was fitted to the Langmuir and the mass action law models. Adsorption kinetics were fittedto a pore diffusion model. Good correlation was obtained between the theoretical models and the experimental data.The methodology outlined in this work provided a simple approach of estimating adsorption behavior of drugs ontoion-exchange macroporous microspheres. Although significant indomethacin loading was obtained onto the DEAEmicrospheres, the rapid rate of diffusion is not compatible with sustained release properties sought for this type ofmicrospheres. © 2002 Elsevier Science B.V. All rights reserved.

Keywords: Microspheres; Cation exchange resins; Diffusion; Adsorption

www.elsevier.com/locate/ijpharm

1. Introduction

Permanent embolization has been used withsuccess for the treatment of angioma, aneuvrismalvarix and arteriovenous fistula, as an alternativeto surgical therapy (Djiandjian et al., 1973; Lan-man et al., 1988; Biondi et al., 1990). Emboliza-

* Corresponding author. Tel.: +33-1-46-691-576; fax: +33-1-46-691-535.

E-mail address: [email protected] (V.Boudy).

0378-5173/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved.

PII: S0 378 -5173 (02 )00033 -9

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V. Boudy et al. / International Journal of Pharmaceutics 239 (2002) 13–2214

tion can be achieved by the mechanical occlusionof the vascular lumen with particles of controlledshape and dimensions. However, recanalizationsof previously occluded arteriovenous malforma-tions have been reported (Hall et al., 1989). Ac-cording to other sources, inflammation caused bythe occluding materials has been identified as amajor element of revascularization (Vinters et al.,1986; Thomashefski et al., 1988; Niechajev andClodius, 1990). Therefore, embolization particlesproviding anti-inflammatory activity should be apromising strategy. This objective can be achievedby coupling drug release activity to the occludingproperty of the particle (Altman et al., 1992). Apossible process to obtain such material is toreversibly adsorb ionizable drug with anti-inflam-matory properties onto microspheres with ion-ex-change properties that then deliver the drug witha controlled rate and target a vascular location.Ionizable drug adsorption and release onto ion-exchange matrixes have been previously studied(Farag and Nairin, 1988; Jones et al., 1989; Mo-hamed, 1996). However, little information isavailable on the mechanism of drug adsorptionand drug transport within ion exchange micro-spheres. Moreover, there are no works that allowone to predict the quantity of bound drug onmicrospheres depending on the concentration ofinitial solution of drug and few works too thatcompare drug affinity for different polymers. Inthis study, equilibria and mass transport of in-domethacin in polymer-based candidate emboliza-tion particles were investigated. The aim was todevelop a model that could allow quantitativepredictions and further insight into the transportmechanism. In particular, factors affecting invitro drug loading such as counter-ion and initialdrug concentrations were investigated togetherwith intraparticle mass transport mechanisms. Inthis study, we used as model of embolizing parti-cles, anion exchange resins (DEAE and QA) fortheir specific characteristics for embolization ther-apy (Beaujeux et al., 1991, 1996; Laurent et al.,1996) and as the absorbed drug, sodium in-domethacin. This drug was selected both for thenature of its ionizable charge and for its potentialeffective action to prevent post-embolizationrevascularization (Altman et al., 1992).

2. Experimental section

2.1. Materials

Microspheres, DEAE Trisacryl LS (designed asDEAE microspheres) and QA Trisacryl LS (de-signed as QA microspheres) were obtained fromBioSepra s.a. (Cergy Saint Christophe, France).These resins were originally designed as chro-matographic media for biomacromolecule purifi-cation. These materials are hydrophilicmacroporous ion-exchangers prepared as spheri-cal semi-rigid microbeads of 80–160 �m diameterwith a mean diameter of 120 �m. They are ob-tained by free radical polymerization in a w/oemulsion system of cationic acrylic derivativeswith a hydrophilic acrylic monomer and arecrosslinked with a hydroxylated acrylic bifunc-tional monomer. The ionizable groups are strongquaternary amines for the QA resin and a 70/30mixture of weak tertiary amines and strong qua-ternary amines for the DEAE resin. The intra-particle porosity of the ion-exchangers is 69% ofthe bead volume. The interparticle porosity of apacked bed of microspheres is 35% of the bedvolume. By strict control of the polymerizationconditions, a macroporosity is obtained with anexclusion molecular weight limit of the micro-sphere pore of approximately 107 Da.

2.2. Solute

Sodium indomethacin (MW=433, pKa=4.5)was supplied by Merck Sharp & Dohme-Chibret(Rahway, USA). Solutions of concentrations0.325–5.20 g/l were obtained by dissolving appro-priate amounts in 5 mM Tris–HCl pH 7.4 buffer.Indomethacin was fully soluble in the range ofconcentration investigated.

2.3. Column experiments

Equilibrium isotherm was measured usingcolumn assays. The column adsorption experi-ments were carried out with a 6.6 mm i.d., 100mm length glass column, a peristaltic pump and aUV detector equipped with a flow through cell.The output of the UV detector was recorded with

V. Boudy et al. / International Journal of Pharmaceutics 239 (2002) 13–22 15

a strip chart recorder. The temperature was main-tained at 25 °C. Column effluent volumes wereaccurately measured by collecting the column out-let into a burette. After column packing, the resinwas equilibrated with 20 column volumes of 5mM Tris–HCl pH 7.4 buffer. Then, an in-domethacin solution with a defined concentrationwas loaded onto the column at a superficial veloc-ity of 210 cm/h. Solute concentration at thecolumn outlet was continuously detected by UVadsorbance at 410 nm. This wavelength was se-lected as it allows linear detection of the in-domethacin concentration in the studied range.The loading was stopped when the column outletconcentration of indomethacin was equal to thestock solution concentration. The equilibrium ca-pacity of the column microspheres was deter-mined from the experimental columnbreakthrough curve.

At equilibrium, the amount of solute held intothe column is equal to the surface area above thebreakthrough curve and can be determined withthe following equation:

Q=C0Vb−� Vb

0

C(�) d� (1)

where Q is solute held in the column; C0, soluteconcentration in the feed; C is solute concentra-tion in the column effluent; Vb is volume forwhich C=C0.

With the assumption of a sigmoid shape for thebreakthrough profile, Eq. (1) can be simplified asfollows:

Q=C0V50% (2)

where V50% is volume at which C=0.5×C0.The solute concentration in the microsphere

phase (C� ) in equilibrium with the feed stock con-centration C0 is then found with Eq. (3).

C� =Q−Q0−Qp

Vc(1−�0)(3)

where Q0 is the amount of solute held in thesystem void volume (including tubing volume anddead zone in the column headers and UV mea-surement cell); Qp is the amount of solute held inthe total porosity of the column; Vc is the columnvolume; and �0 is the interparticle porosity.

The amount of solute held in the system voidvolume and column porosity can be washed outby a cleaning step with pure buffer. The totalporosity (�T) of the column corresponds to thebead interparticle porosity (�0) and the bead intra-particle porosity �p.

�T=�0+ (1−�0)�p (4)

The frontal chromatography technique de-scribed above was also used for the measurementof total ion exchange capacity of the micro-spheres. In this situation, the solute was a 0.1 MHCl solution. The conductivity of the columneffluent was continuously measured and recorded.Prior to the measurement the column was regen-erated with 5 column volume of 1 M NaOH andrinse with distilled water. The total ion capacity ofthe resin was determined with Eq. (3).

2.4. Batch experiments

Measurement of adsorption kinetics was per-formed in a 500 ml agitated contactor. In-domethacin concentration in solution wasmonitored by continuously recirculating a streamthrough a UV spectrophotometer, equipped withan analytical cell and a peristaltic pump. Resi-dence time in the measurement loop was mini-mized. The stream was drawn through a stainlesssteel frit to prevent recirculating beads in theloop. At the start of the experiment, the vesselwas filled with an appropriate volume of a solu-tion (50–200 ml) at a given concentration (0.65–2.60 g/l). The experiment was started when ca. 1ml of microspheres equilibrated with the bufferand superficially dried was quickly introducedinto the vessel. The UV signal was recorded on astrip chart recorder and converted to concentra-tion through a calibration curve. The temperaturewas maintained at 25 °C. The amount of soluteadsorbed by the media at each time was obtainedby the following mass balance equation:

C� (t)=(C0−C(t))

H(5)

where C� (t) is the amount of solute adsorbed perunit volume of particle at t time; H is microspherevolume to liquid phase volume ratio; and C(t) issolute concentration in the liquid phase at t time.


Titration of a 8.5 ml of DEAE microspherespreviously regenerated with 1 N NaOH, exten-sively rinsed with distilled water and resuspendedin 40 ml of 0.5 M KCl was performed with 100mM HCl.

3. Results and discussion

3.1. Equilibrium of adsorption

The adsorption results of indomethacin ontoDEAE and QA microspheres are shown on Figs.1 and 2.

Two approaches were used to model the soluteuptake by the ion-exchange resins: the mass ac-tion model and the Langmuir formalism. Asshown below, both approaches lead to a uniquemathematical expression, in the case of a univa-lent solute, as the indomethacin.

The mass action model (MA) represents thesolute ion-exchange adsorption process as a stoi-chiometric exchange of liquid phase solute andbound counterions (Whitley et al., 1989). Theuptake is given by:

Fig. 2. Equilibrium uptake of indomethacin on QA micro-spheres in 5 mM Tris–HCl buffer pH 7.4 at different counte-rion concentrations: 4.2 mM (closed circles); 50 mM (opencircles); 140 mM (crossed circles). Dotted lines represent pre-dictions of the Langmuir model with parameters in Table 2.Continuous lines represent the mass action model withparameters in Table 3.

�ResinCounterion+Solute � Resin �Solute

+�Counterion

where � is the effective charge of the solute, whichis pH dependent. The counterion is assumed to bemonovalent.

The apparent equilibrium constant of the ionexchange process is defined as:

Keq=�C

C��I

I

��

(6)

where C, I are the solute and counterion concen-trations in the liquid phase, and C, I the concen-trations in the microsphere phase.

Unlike strong cationic resin, the number ofionizable groups on the weakly cationic resin isrelated to the pH and the total number of chargesby the Henderson–Hasselbach relationship:

pH=pKa+ log� �

1−�

�(7)

where � is the degree of ionization of the resin.However, some of charged groups are not

available for exchange with the solute due to asteric hindrance effect as the solute can not diffuseinto the narrower pores.

Fig. 1. Equilibrium uptake of indomethacin on DEAE micro-spheres in 5 mM Tris–HCl buffer pH 7.4 at different counte-rion concentrations: 4.2 mM (closed circles); 50 mM (opencircle); 140 mM (crossed circles). Dotted lines represent predic-tions of the Langmuir model with parameters in Table 1.Continuous lines represent the mass action model withparameters in Table 3.


The number of charged groups on the resinsterically excluded from exchange is given by:

I� =�� (8)

where � is the ionic capacity of the resin (totalnumber of charge/unit volume of bead) and �, thesize exclusion fraction.

When the interface equilibrium is reached thetotal number of counterions adsorbed to the sup-port is given by:

It=I� +I� (9)

with It is the total number of counterions ad-sorbed per unit volume of resin and I is thenumber of counterions adsorbed and available forexchange.

A consequence of this model is that unboundsolute can only interact with unhindered ion-ex-change sites.

Electroneutrality on the microsphere phaserequires:

�=I� +I+�C (10)

Combining Eqs. (6) and (10) and Eq. (8) gives:

C=KeqC(�(1−�)−�C)�

I�(11)

This equation defines a single componentisotherm, based on the mass action law.

It should be pointed out that a numericalmethod is required to calculate the solute micro-sphere phase concentration, for a given liquidphase concentration. Competitive multicompo-nent isotherm can be formulated as a simpleextension of the single component equilibrium.Moreover steric hindrance of counterion by ad-sorbed large macromolecules can be included inthe mass action formalism (Brooks and Cramer,1992).

When the solute characteristic charge is one,the MA model may be rearranged to form aLangmuir expression:

C� =�(1−�)

Keq

IC

1+Keq

IC

(12)

where the maximum capacity is defined as C� max=�(1−�) and the association constant

KL=Keq

I,

Eq. (12) can be found directly by assuming asecond-order kinetics binding process.

Solute+Ligand �k des

�kads

SoluteLigand (13)

where Solute is the solute, Ligand the ligand,SoluteLigand the complex and where kads, kdes arethe association and dissociation rate constants,respectively.

The rate equation for the second-order kineticsis given by Eq. (14)

dC�dt

=kads C(C� max− C� )−kdesC� (14)

where C� and C are the solute concentrations in themicrosphere and liquid phase, respectively, andC� max the maximum solute binding capacity.

At equilibrium dC� /dt=0 and thus Eq. (14)yields the Langmuir isotherm:

C� =C� maxKLC1+KLC

(15)

with

KL=kads

kdes

is dependent on the temperature, the pH and thecomposition of the liquid phase.

The data on Figs. 1 and 2 were fitted to theLangmuir model, Eq. (15), by a least squaresprocedure based on the Marquardt–Levenbergmethod (Marquardt, 1963). This approach ismore robust than Scatchard plots, the semi-recip-rocal plot or the double inverse plot that aresusceptible to introduce experimental uncertaintyinto the independent variables (Johnson andFaunt, 1992). Estimated parameters for Eq. (15)as well as standard errors are reported in Table 1.The maximum amount of indomethacin adsorbedby the tertiary amine ion exchanger is 55% higherthan the uptake of the quaternary amine resin.The apparent dissociation constant, defined asKd=1/KL, is dependent on the counter ion con-centration. Within the range of concentration in-


Table 1Langmuir equilibrium parameters for the adsorption of in-domethacin on DEAE microspheres at different counterionconcentrations

C� max (mM)Counterion S.E. KL (mM−1) S.E.(mM)

4.2 3.55303 0.414.230.9 0.60 0.1450 30322.6 0.21 0.03303140

Fig. 4. Plot of the pH titration curves of DEAE microspheres(titration of 8.5 ml of DEAE microspheres by a 100 mM HCl.Microspheres regenerated with 1 N NaOH, rinsed with dis-tilled water and resuspended in 0.5 M KCl).

vestigated the correlation between Kd and I islinear as shown on Fig. 3. As expected for ionexchangers an increase of the counterion concen-tration increases the dissociation constant of theequilibrium. For any counterion concentration,the QA microspheres exhibit a lower affinity forindomethacin compared to the DEAE micro-spheres. At the minimum counterion concentra-tion (4.2 mM) the apparent dissociation constantis lower on the tertiary amine ion exchanger: 282�M compared to 383 �M for the DEAE and QAmicrospheres respectively. The affinity for theresins is weak and will permit significant soluterelease under physiological conditions.

Using Eq. (15), C can be calculated from thequantity of drug desired at the target location,namely C.

The data were also used to fit the mass actionmodel, Eq. (12). This model requires the determi-nation of the total ion exchange capacity of themicrospheres. It was measured using frontal chro-matography with a 100 mM HCl titration solu-tion. The total numbers of charged groups basedon the bead volume are 267 and 517 mM, for QAresin and DEAE resin respectively. For the qua-ternary amine ion exchanger, the number of ioniz-able groups is equivalent to the total number ofcharged groups. However, for the tertiary amineresin the number of ionizable groups depended onthe pH, therefore a correction to the total ioniccapacity was made using Eq. (7). Titration curvesfor DEAE microspheres is shown on Fig. 4. Itindicates the presence of two basic groups with apKa value of 6.7 for 70% of the immobilizedcharges and a pKa of 10.7 for the remaining 30%of immobilized charges. With these data, the cal-culated number of charged groups at pH 7.4 is341 mM.

Fig. 3. Correlation between the apparent dissociation constantKd and the counterion concentrations. Uptake of in-domethacin on QA microspheres in 5 mM Tris–HCl bufferpH 7.4 at (open triangles). Uptake of indomethacin on DEAEmicrospheres in 5 mM Tris–HCl buffer pH 7.4 (open circles).


Table 2Langmuir equilibrium parameters for the adsorption of in-domethacin on QA microspheres at different counterion con-centrations

C� max (mM)Counterion S.E. KL (mM−1) S.E.(mM)

2.44.2 2.61196 0.2250 196 25.3 0.47 0.14

10.6 0.16 0.02140 196

tion. A set of adjusting parameters is required foreach counterion concentration with the Langmuirmodel.

3.2. Kinetics of adsorption

Experimental results from indomethacin stirredbatch adsorption kinetics onto DEAE micro-spheres are shown on Fig. 5. The experimentalpoints shown are taken from a continuous chartrecorder trace rather than being individualmeasurements.

Numerous reports indicate that, for ion ex-changers, the kinetics of binding is much fasterthan the rate of diffusion (Ruthven et al., 1984).Therefore, mass transfer in an ion exchanger canbe only affected by two types of resistances inseries: external film mass transfer resistance andintraparticle mass transfer resistance (LeVan etal., 1997). Two types of intraparticle mass transferhave been described: homogeneous particle diffu-sion (transport by diffusion of solute in the ad-sorbed state or solid diffusion) and pore diffusion(transport by diffusion through the liquid con-tained in the pores of the particle). Pore diffusionoccurs when the pore size of a liquid filled poreparticle is large in comparison to the mean freepath of the solute molecule (Li et al., 1995). Thissituation provided an accurate description of thesystem under study for which the ratio of the porediameter to the solute free path was large.

An analytical solution of the mass transportequation for pore diffusion alone was found forthe case of very favorable isotherms assuming afinite fluid volume. The time needed to attain acertain fractional approach to equilibrium is givenby (Theo and Ruthven, 1986):

�pDpC0tr2C� �

=I2−I1 (16)

where

I1=1

6�Rln��3+�3

�3+1��+1

�+�

�3�+

1

�R�3

�tan−1�2�−�

��3

�− tan−1�2−�

��3

��(17)

Best fit parameter estimates for Eq. (12) arereported in Table 3 and the fitted curves areshown in Figs. 1 and 2. Excellent agreement isfound between the experimental data and themodel. Contrary to the Langmuir isotherm, theeffect of the counterion is explicitly accounted forin the mass action model. Therefore, it can beused to predict the effect of the increase in counte-rion concentration on the uptake isotherm.

The difference of indomethacin binding capac-ity of the two ion exchangers can be explained onone hand by the difference in the concentration ofimmobilized charges on each resin, and on theother hand, by a higher steric hindrance factor forthe quaternary amine microspheres compared tothe tertiary amine microspheres. This latter factormay be explained by a higher crosslinking of theQA hydrogel resulting in a lower accessibility ofthe narrower pores.

From the equilibrium results, it can be con-cluded that the DEAE microspheres is the mostsuitable microsphere for the binding of in-domethacin as it exhibits a higher binding capac-ity as well as a lower dissociation constant. TheLangmuir model and the Mass action modelprovide an equally good fit of the data. However,the mass action model is preferable as it includesexplicitly the impact of the counterion concentra-

Table 3Mass action law parameters for the adsorption of in-domethacin on DEAE and QA microspheres

� (mM)Microspheres � Keq

341.00DEAE microspheres 0.136 30.92QA microspheres 24.140.289267.00


Fig. 5. Uptake of indomethacin on DEAE microspheres in 5 mM Tris–HCl buffer pH 7.4 (circle: reduced solution concentration-square: fractional uptake curve). Lines represent the pore diffusion model fitted with parameters in Table 3. (A) Starting batchsolution concentration=2.6 g/l; batch volume=50 ml. (B) Starting batch solution concentration=1.3 g/l; batch volume=100 ml.(C) Starting batch solution concentration=0.65 g/l; batch volume=200 ml. (D) Starting batch solution concentration=0.65 g/l;batch volume=50 ml.

I2=1

3Rln��3+�3

�3+1�

(18)

�=�

1−C� (t)C� �

�1/3

(19)

�=�1

R−1

�1/3

(20)

The model assumes that film mass transfer re-sistance is negligible because a high stirring speedis used, the particles are spherical and instanta-neous equilibrium is obtained between the solu-tion in the pore and the surface of the solid.

The transient uptake data in Fig. 5 were fittedto the pore diffusion model, Eq. (16), allowing fordetermination of a pore diffusion coefficient for

each set of conditions. The resulting values ofpore diffusivity are summarized in Table 4. Fig. 5indicates that the uptake was quite rapid withessentially complete saturation achieved within a1–6 min range. Reasonable agreement betweenthe experimental data and the model was obtainedwith a pore diffusion coefficient value in the 2–5.5×10−10 m2/s range. These data are between17 and 47 times smaller than the estimate ofindomethacin free aqueous solution diffusivity(9.4×10−9 m2/s) determined with the equation ofWilke and Chang (Wilke and Chang, 1955). Thehindrance factor of diffusivity in the porous poly-meric materials was within ranges previously re-ported (Guiochon et al., 1994).


Table 4Parameters for stirred tank adsorption of indomethacin onDEAE microspheres

Initial Pore diffusionBatch volume(ml) coefficient (m2/s)concentration

(g/l)

2.60 50 2.1×10−10

1.30 100 4.3×10−10

5.5×10−102000.65500.65 5.3×10−10

�p microspheres porositytotal column porosity�T

� degree of ionization of themicrospheres

� variable defined by Eq. (20)variable defined by Eq. (19)�

total number of charge per unit vol-�ume of bead

C solute concentration in the liquidphase

C0 initial solute concentration in the liq-uid phase

C� max Langmuir equation constantC� � final solute concentration in the

microspheresC� amount of solute adsorbed per unit

volume of microspheresDp pore diffusion coefficient

microsphere volume to liquid phaseHvolume ratio

I counterion concentration in the liq-uid phase

I1 variable defined by Eq. (17)I2 variable defined by Eq. (18)It total number of counterions ad-

sorbed per unit volume of resinI counterion concentration adsorbed

and available for exchange in themicrosphere

I� counterion concentration stericallyexcluded from ion-exchange in themicrosphere

Keq apparent equilibrium constant of theion exchange reaction

KL Langmuir association constantkads, kdes association and dissociation rate

constantsQ amount solute held in the columnQ0 amount of solute held in the system

(void volume)Qp amount of solute held in the total

porosity of the columnr microspheres radius

fraction of solute ultimately boundRto the microspheres

t timecolumn effluent volume at whichV50%

C=0.5×C0.

Based on the value of pore diffusivity, estimatesof kinetics of drug release could be made assumingpore diffusion as the limiting step for desorptioninto a finite or infinite solvent sink. Such estimates(data not shown) indicated that a very rapid drugrelease was obtained (within 1–5 min). Such arelease profile may not be appropriate when sus-tained levels of drug bioactivity is sought. How-ever, it should be pointed out that drug releasemechanisms in vivo in a vascular environment arediffusion limited due the inherent hydrodynamicconditions. The absence of flow of physiologicalliquid will impact the release from microparticlesand result in a slower release compared to the idealin vitro well stirred tank conditions.

The models used in this work provided valuableinsights into the mechanism of drug binding andelution on ion-exchange microspheres. The effectof counterions was explicitly included in themodel, therefore the capacity of the bead could beassessed in various conditions. Such models can beused to characterize or select microspheres de-signed for specific controllable drug release. It canalso be used to design new microspheres by provid-ing information on optimal binding affinity andcapacity.

Appendix A. Nomenclature

size exclusion fraction�

effective charge of the solute�

�0 interparticle porosity of packed bedcolumn


Vb column effluent volume at whichC=C0

Vc packed bed column volume

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