A mathematical model for predicting controlled release of bioactive agents from composite fiber structures

A Mathematical Model for Predicting Controlled Releaseof Bioactive Agents from Composite Fiber Structures

Meital Zilberman, Moran SoferDepartment of Biomedical Engineering, Faculty of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel

Received 25 May 2006; revised 18 June 2006; accepted 12 July 2006Published online 27 October 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/jbm.a.30985

Abstract: A mathematical model for predicting bioactiveagent release profiles from core/shell fiber structures wasdeveloped and studied. These new composite fibers, whichcombine good mechanical properties with desired proteinrelease profiles, are designed for use in tissue regenerationand other biomedical applications. These fibers are com-posed of an inner dense polymeric core surrounded by aporous bioresorbable shell, which encapsulates the bioac-tive agent molecules. The model is based on Fick’s secondlaw of diffusion, and on two major assumptions: (a) first-order degradation kinetics of the porous shell, and (b) anonconstant diffusion coefficient for the bioactive agent,which increases with time because of degradation of thehost polymer. Three factors are evaluated and included in

this model: a porosity factor, a tortuosity factor, and a poly-mer concentration factor. Our study indicates that themodel correlates well with in vitro release results, exhibitinga mean error of less than 2.2% for most studied cases. Inthis study, the model was used for predicting proteinrelease profiles from fibers with shells of various initial mo-lecular weights and for predicting the release of proteinswith various molecular weights. This new model exhibits apotential for simulating fibrous systems for a wide varietyof biomedical applications. � 2006 Wiley Periodicals, Inc. JBiomed Mater Res 80A: 679–686, 2007

Key words: protein-eluting fibers; bioresorbable polymers;porous scaffold; medical device; horseradish peroxidase

INTRODUCTION

Tissue regeneration involves the preparation of poly-meric structures that serve as degradable scaffolds forbioactive agents or cells as well as the study of theirstructure and properties.1 However, the key problem ofhow to incorporate bioactive agent molecules into thindelicate structures that construct devices and scaffoldsremains unresolved, since they must be incorporatedinto dense polymeric structures without adverselyaffecting either the scaffold’s properties or the agent’sactivity. Two types of drug-loaded fibers have beenreported previously.2–7 These are monolithic fibers inwhich the drug is dispersed throughout the fiber, andhollow reservoir fibers in which the drug is stored inthe fiber’s internal cavity. However, most such systemssuffer from poor mechanical properties (due to drugincorporation) and/or require destructively high melt-processing temperatures. Most drugs and all proteinscannot withstand high temperatures or endure manyorganic solvents.

In one of our recent studies, we presented a new con-cept of core/shell fiber structures, which successfullymeet these challenges.8 These composite fibers combinea dense poly(l-lactic acid) (PLLA) core fiber and adrug/protein-loaded porous shell structure, i.e., thedrug or protein is located in a separate compartment (a‘‘shell’’) around a melt spun ‘‘core’’ fiber. A schematicrepresentation of the composite fiber structure is pre-sented in Figure 1. This results in good mechanicalproperties as well as in the desired drug release profile.Unlike conventional bulky multifiber structures thatcarry bioactive agent molecules in their pores, weincorporate these molecules directly into the shell (butnot the core) of our composite fibers, rather than in theinterstices between them. Better control over drugrelease can thus be achieved. In our initial studies, theshell was loaded with the model enzyme, horseradishperoxidase (HRP), which is very sensitive to both sol-vents and elevated temperatures and whose activity isa sensitive monitor of damage during processing. Ourresults indicated that HRP preserved 94–100% of its ac-tivity during the fiber preparation process.

In addition to tissue regeneration applications, ournew fibers are ideal for forming thin, delicate, bio-medically important structures for various applica-tions, such as fiber-based endovascular stents thatmechanically support blood vessels while deliveringdrugs for preventing restenosis directly to the blood

Correspondence to: M. Zilberman; e-mail: [email protected] grant sponsor: RAMOT (Horowitz) Foundation,

Tel-Aviv University

' 2006 Wiley Periodicals, Inc.

vessel wall, or bioresorbable wound and burn dress-ings loaded with antimicrobial agents.

Mathematical models for controlled release of bio-active agents from bioresorbable matrices are basedon diffusion aspects, on the structural characteristicsof the matrix polymer and its degradation and swel-ling, and on the micro-environmental pH changesinside polymer matrix pores that are due to the deg-radation products. Prediction of the drug release pro-file using a model is obviously very useful in thedevice’s design phase, since the model enables fastevaluation and tuning of the various parameters forachieving an optimal release profile, while reducinglaboratory tasks to a minimum.9,10

Early drug delivery models described either systemsbased on nondegradable matrices or surface-erodingsystems.11 Several more recent models described bulk-eroding systems, in which the drug is physically im-mobilized. For example, Siepman and Gopferich12

quantified drug release from slab-shaped PLLA andpoly(dl-lactic acid-co-glycolic acid) (PDLGA) matrices.Their model was based on Higuchi’s classical pseudo-steady-state equation for oversaturated, planar, nonde-grading polymeric films, where the permeability of thedrug within the polymer matrix was assumed to in-crease with time, due to cleavage of the polymer’sbonds. Another possibility for simulating the effect oferosion on the diffusion process is to use a diffusioncoefficient, which increases with time. Various theoriestherefore related the drug diffusion coefficient inside adegradable polymer directly to its molecular weight,since short chains offer less restriction to drug diffu-sion than do long chains. This was one of the mainassumptions in the models of Charlier et al.13 for pre-dicting the release rate of Mifepristone from 50/50PDLGA bulk-eroding films of various molecularweights, and Faisant et al.14 who examined therelease rate of 5-fluorouracil from 50/50 PDLGAmicrospheres. Each of them used an additional as-sumption regarding the polymer’s first-order degra-dation kinetics, in order to calculate an appropriatetime-dependent diffusion coefficient, which they usedin Fick’s laws of diffusion. Zhang et al.15 addressedthree mechanisms for drug release in a microsphericmatrix, namely dissolution of the drug from the poly-

mer matrix, diffusion of the dissolved drug, and ero-sion of the matrix.

It should be noted that most drug delivery systemsare either in the form of films or microspheres, and soare the above-described models. Sagiv et al.,16 how-ever, developed a specific model for predicting pro-tein release from monolithic PLLA fibers. Since PLLAdegrades relatively slowly, they assumed that poly-mer erosion is negligible during the release time; thusallowing them to use a constant diffusion coefficientso as to simplify the model. This model is thereforeobviously not applicable to our core/shell fiber struc-tures. However, our promising results raised the needfor a suitable model that could facilitate and shortenthe design process of the core/shell fiber structures.This need was the driving force for the currentresearch, in which a mathematical model was devel-oped for predicting protein release profiles from ourcomposite fiber structures. The hypotheses of ourstudy are: (1) A model based on Fick’s laws will beable to provide good prediction of protein releaseprofile from our new core/shell fiber structures, and(2) the protein release profile from the fibers isaffected by the molecular weights of the protein andthe host polymer, and also by the emulsion’s formu-lation parameters.

CASE DEFINITION AND MODELASSUMPTIONS

A mathematical model for predicting drug/proteinrelease from our novel core/shell fiber structures wasdeveloped in this study, using Matlab 6.1. The releaseprofiles that were predicted using this model werecompared with that of the experimental results forcertain fiber types.

The experimental system

About 8-cm long fibers, composed of an inner PLLAcore fiber, were coated with a porous 75/25 PDLGAshell. The porous shell was fabricated using the‘‘freeze-drying’’ technique, where an organic solution(containing PDLGA and chloroform as solvent) andan aqueous solution (containing distilled water andHRP) were mixed together in a test tube. Each PLLAfiber was then dipped in the resulting emulsion andimmediately frozen in liquid nitrogen. The water andthe solvent were removed by sublimation, leaving theHRP molecules mechanically trapped in the porousPDLGA shell surrounding the PLLA fibers.8 The HRPload in the porous shell of all fibers was 5% (w/w; rel-ative to the polymer weight). Various shell types wereprepared and emulsions of organic:aqueous (o:a) phaseratios of 8:1 and 16:1 were obtained. Polymer contents

Figure 1. A schematic representation of a core/shell fiberstructure. [Color figure can be viewed in the online issue,which is available at www.interscience.wiley.com.]

680 ZILBERMAN AND SOFER

Journal of Biomedical Materials Research Part A DOI 10.1002/jbm.a

of 13, 15, and 19% (w/v in the organic phase) wereused for each o:a value. The types of fibers examinedare presented in Table I.

Assumptions used while deriving the model’sequations

i. The bulk-eroding PDLGA porous structure ex-hibits first-order degradation kinetics, i.e. thecleavage of its chains occurs at a constant rate.17–19

ii. The protein’s diffusion coefficient within the po-rous structure is only time-dependent and is as-sumed to increase with time proportional to thepolymer’s normalized molecular weight loss (Mwl).

iii. The internal PLLA core exhibits minimal deg-radation, if any, within the considered proteinrelease time-frame. This assumption is basedon one of our previous studies.20

iv. There is no protein flux towards the internalPLLA core.

v. The external environment provides perfect sinkconditions for the released proteins. When con-ducting the experiments the aqueous releasemedium was replaced at each sampling point.

vi. The proteins within the porous PDLGA struc-ture are released only when they find a contin-uous path to the surface, and their release oc-curs solely by diffusing through water. Thisassumption is based on other reports of drugdelivery systems based on porous matrices.10

MODEL EQUATIONS

In this model, the protein concentration within thefiber obviously changes with time. Fick’s second lawof diffusion in cylindrical coordinates (r, y, z)12 wastherefore used, as follows:

@C

@t¼ 1

r

@

@rrD

@C

@r

� �þ @

@yD

r

@C

@y

� �þ @

@zrD

@C

@z

� ��

ð1Þwhere C is the HRP concentration and D is its diffu-sion coefficient within the porous PDLGA structure.

Because of circular symmetry, the HRP concentrationalong y is constant, therefore

@C

@y¼ 0 ð2Þ

Furthermore, in the case of a fiber whose radius issignificantly smaller than its length, the end effectsare negligible, yielding

@C

@z¼ 0 ð3Þ

In conclusion, it may be stated that the diffusion ofproteins from a fiber is limited to the radial axis, i.e.C ¼ f(r, t), thus simplifying equation (1) to

@C

@t¼ 1

r

@

@rrD

@C

@r

� �� ð4Þ

Assuming that the diffusion coefficient is only a func-tion of time simplifies it even further, and the follow-ing equation can be used:

@C

@t¼ 1

r

@

@rrD

@C

@r

� �� ¼ 1

rD@C

@rþ rD

@2C

@r2

� �

¼ 1

rD@C

@rþ @2C

@r2D ð5Þ

Thus, the final diffusion equation is

@C

@t¼ D

1

r

@C

@rþ @2C

@r2

� �ð6Þ

Appropriate initial and boundary conditions shouldbe used in order to solve the diffusion equation (6).Assuming an initial uniform HRP concentration (C0)leads to the following initial condition:

C ¼ C0 at t ¼ 0; r1 < r < r2 ð7Þwhere r1 is the radius of the internal PLLA fiberand r2 is the radius of the entire composite fiber(see Fig. 1). A ‘‘no flux’’ boundary condition was seton the internal wall of the porous structure, whichis also the external wall of the inner PLLA fiber,(r ¼ r1), indicating that the proteins within the po-rous structure are assumed to diffuse only towardthe surface of the composite fiber and not towardits core:

@C

@r¼ 0 at r ¼ r1; t > 0 ð8Þ

The second boundary condition is a ‘‘perfect sink’’ atthe external wall of the composite fiber, i.e.

C ¼ 0 at r ¼ r2; t > 0 ð9ÞUsing the above initial and boundary conditions, thedifferential equation (6) was resolved via the ‘‘pdepe’’Matlab function, and yielded C(r, t), the HRP concen-tration function inside the fiber. Thus, the total massof proteins still trapped within the porous structure

TABLE IFiber Types Used for Obtaining the Experimental Data

FiberType

Emulsion Organic:Aqueous PhaseRatio (O:A)

Polymer Contentin the OrganicPhase (%, w/v) Cp t

1 8:1 15 0.29 3.32 8:1 19 0.58 7.03 16:1 13 0.54 8.04 16:1 15 0.8 11.05 16:1 19 1.4 21.0

MATHEMATICAL MODEL FOR PREDICTING CONTROLLED RELEASE 681


at any given moment could be estimated using thefollowing equation:

MðtÞ ¼Z r2

r1

ACðr; tÞdr ¼Z r2

r1

2prLCðr; tÞdr

¼ 2pLZ r2

r1

rCðr; tÞdr ð10Þ

Since the initial mass of proteins inside the fiber, (M(t¼ 0)),is known, the evaluation of the released mass is given by

MreleasedðtÞ ¼ Mðt ¼ 0Þ � MðtÞ ð11Þ

Estimation of the diffusion coefficient

Previous reports on drug delivery systems based onporous matrices revealed that the drug is released muchmore slowly than that would be expected from the sim-plest consideration of aqueous diffusion.10 The porousstructure apparently slows the progress of the diffusingagents, since they go through a tortuous path on theirway to the matrix surface. The length of this path isunknown, but it is greater than (r – r1) by a tortuosityfactor t, which was initially proposed by Higuchi.17,21,22

Thus, the HRP diffusion coefficient within the po-rous PDLGA structure is assumed to initially have acertain low ‘‘effective’’ value, D0, which is determinedby the microstructure of the matrix, i.e. tortuosity tand porosity e (the volume fraction accessible to thediffusing molecules). As degradation and finally ero-sion progress with time, the diffusion coefficient grad-ually increases, until it reaches HRP’s molecular diffu-sion coefficient, i.e. its diffusion coefficient in water,Dw. The above behavior can therefore be expressedusing the following equations:

D0 ¼ Dwet

ð12Þ

DðtÞ ¼ D0 þ ðDw � D0Þ �MwlðtÞ ð13Þwhere

MwlðtÞ ¼ Mwðt ¼ 0Þ � MwðtÞMwðt ¼ 0Þ

Polson’s semiempirical equation18,23,24 was used forthe protein molecular diffusion coefficient, Dw:

Dw ¼ AT

m M1=3wp

ð14Þ

where Mwp is the protein’s molecular weight, T is theabsolute temperature, m is the fluid viscosity, and A isa constant that varies for each protein.

Tortuosity, t, is an empirical parameter whosevalue was determined by us. The stereology samplingtechnique was used in order to estimate the matrixporosity. This technique enables estimating a sam-

ple’s porosity using SEM (2D) images of samplecross-sections.25 Point-counting estimation was there-fore used, where a grid of points is placed on thefiber’s cross-section image and the estimated porosityis the ratio between the number of points that fall onthe pores themselves and the number of points thatfall on the entire cross-section.

Adding the polymer concentration effect

As mentioned earlier, three polymer concentrations(w/v) were used in the organic phase while fabricatingthe different porous structures: 13, 15, and 19%. Ahigher polymer concentration results in a more viscousorganic phase, thus creating a more stable emulsion.This higher viscosity, along with the obvious higherdensity, are expected to create the following hinderingeffects on the system: (1) slowing the matrix degrada-tion rate due to a lower ‘‘readiness’’ to water penetra-tion; (2) reducing the free volume available for proteindiffusion, leading to a shorter initial burst effect in therelease profile. An additional empirical parameter, Cp,was therefore introduced in order to add the variantpolymer concentration to the model, as follows.

A function describing the decrease in molecularweight with time (degradation profile) for the relevantpolymer (in this case the initial molecular weight ¼100 kDa) had to be used. Data taken from a researchby Wu and Wang19 were used and interpolation wasperformed in order to obtain a good estimation for thedegradation profile of 75/25 PDLGA with an initialmolecular weight of 100 kDa. The estimated degrada-tion profiles of the 100 kDa PDLGA and two otherpolymers, which will be used in this model, are pre-sented in Figure 2.

Figure 2. Normalized degradation profile of 75/25 PDLGAwith various initial polymer’s molecular weights. The molecu-lar weights are indicated. [Color figure can be viewed in theonline issue, which is available at www.interscience.wiley.com.]



The normalized molecular weight curves appa-rently fit a function of the type: Mw(t)¼ exp(�t/B).

Since the polymer concentration was hypothesizedto alter the degradation rate, it was added to themodel as follows:

MwlðtÞ ¼ 1 � MwðtÞ ¼ 1 � exp�t

B

� �

¼ 1 � exp�Cp

Bt

� �ð15Þ

According to theory, the degradation of bulk-erodingpolymers follows first-order kinetics, i.e. plotting thenatural logarithm of the molecular weight vs. thehydrolyzing time yields a nearly linear curve.17–19

The B coefficient was thus found using the ‘‘leastsquares’’ algorithm.

RESULTS AND DISCUSSION

As mentioned earlier, the experimental data usedto validate this model were taken from one of ourrecent studies.8 The chosen core/shell fiber types andtheir semiempirical Cp and t values are presented inTable I. The predicted HRP release profile was com-pared with that of the experimental release profile foreach type of composite fiber structures, and theresults are presented in Figure 3. A very good fit was

Figure 3. HRP release profile from various core/shell fiber structures containing 5% (w/w) HRP. The experimental results(red line) are compared with the predicted results (blue line): (a) o:a ¼ 8:1, 15% (w/v) polymer, (b) o:a ¼ 8:1, 19% (w/v) polymer,(c) o:a ¼ 16:1, 13% (w/v) polymer, (d) o:a ¼ 16:1, 15% (w/v) polymer, and (e) o:a ¼ 16:1, 19% (w/v) polymer. The mean errorfor each type is presented. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]



obtained for all studied fibers, except for the fiberwith an o:a of 8:1 and 15% (w/v) polymer, where thepredicted HRP release during the first week is lowerthan the experimental HRP release (Fig. 2a). Hence,these results support our first hypothesis about goodpredictability of a model based on Fick’s laws.

As explained above, two main emulsion types werecreated by using a constant organic phase volumewith two different aqueous phase volumes: o:a of 8:1and 16:1. The fibers fabricated with a higher o:a (16:1)exhibit a more tortuous diffusion path, leading tohigher values of the tortuosity factor (Table 1). Fur-thermore, the tortuosity factor within both 8:1 and 16:1‘‘groups’’ increases with the increase in polymer con-

tent. Therefore, either increasing the emulsion’s o:aratio (i.e., decreasing the aqueous phase volume) orincreasing the polymer content, results in a decreasein the free space available for diffusion, leading to ahigher tortuosity factor, which in turn leads to a lowerrelease rate of the bioactive agent from the fiber’s shell.These results are in agreement with our second hy-pothesis, saying that the emulsion formulation param-eters affect the release profile.

Since a higher polymer content leads to an emulsionwith a more viscous and dense organic phase, weassumed that the resulting solid porous structure willtend to absorb less water, resulting in slower hydroly-sis and hence degradation, leading to a shorter and

Figure 4. The effect of the polymer’s initial molecular weight on the predicted HRP release profile from various core/shellfiber structures containing 5% w/w HRP. Molecular weight: upper curve (red line) 40 kDa, center curve (blue line) 100 kDa,lower curve (green line) 160 kDa; (a) o:a ¼ 8:1, 15% w/v polymer, (b) o:a ¼ 8:1, 19% w/v polymer, (c) o:a ¼ 16:1, 13% w/v poly-mer, (d) o:a ¼ 16:1, 15% w/v polymer, (e) o:a ¼ 16:1, 19% w/v polymer. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com.]



more moderate burst effect. Following this hypothesis,Cp was introduced into the model in a way that altersthe porous structure’s degradation rate. This is alsosupported by the experimental results, which demon-strate that as the polymer content increases, the matrixdegradation decreases, leading to a smaller initialburst release.

It is interesting to compare the Cp and t values of dif-ferent composite fiber types and elucidate the effect ofprocessing conditions on these parameters (throughmicrostructure). For example, the Cp value of the 16:1sample of fibers fabricated with the same polymer con-tent of 15% (w/v) is 2.8 times higher than that of the 8:1sample, and for 19% (w/v) polymer fibers the Cp valueof the 16:1 sample is 2.4 times higher than that of the8:1 sample. A similar trend was obtained for t: the tvalue of the 16:1 sample of fibers fabricated with a poly-mer content of 15% (w/v) is 3.3 times higher than thatof the 8:1 sample, and for 19% (w/v) polymer fibers thet value of the 16:1 sample is 3.0 times higher than thatof the 8:1 sample (Table 1). This consistent behavior ofboth parameters as a function of the polymer concen-tration can simplify the model. Although it cannot bedemonstrated without additional experiments, it seemsthat certain calibration curves and/or functions may bedeveloped in order to simplify the model.

The advantage of building a model for drug/proteindelivery systems is the ability to elucidate the effect ofthe system’s parameters on the release profile of thebioactive agent. In this regard, the effect of the molecu-lar weight of both components, PDLGA and HRP, wasstudied. The effect of the PDLGA molecular weight onthe release rate was examined using the degradationprofiles of polymers with initial molecular weights of40 and 160 kDa, in addition to that of the standard 100kDa used in our experiments. These degradation pro-files were obtained using interpolations based on theexperimental results of Wu and Wang19 and are pre-sented in Figure 2. Our predicted HRP release profilesfor the series of 75/25 PDLGA with three molecularweights are presented in Figure 4 for each of the stud-ied fiber types. The decrease in initial molecular weightalways resulted in an increased HRP release rate. Thisprediction is logical, since a lower initial molecularweight polymer will result in shorter polymer chainsas degradation proceeds, giving rise to an enhanceddrug release rate. It should be mentioned that the pre-dicted release profiles are not accurate, and that theburst release values almost do not change with the ini-tial molecular weight. This occurs mainly because theonly parameter that was changed while making thesepredictions is the matrix degradation profile, leavingthe same tortuosity factor that was calculated for the100 kDa fiber type. However, the tortuosity factor issupposed to increase with an increase in the molecularweight. Our future work will focus on an additionalminimodel that predicts the effect of various parame-

ters on the tortuosity of the polymer and therefore ena-bles more accurate prediction of release profiles.

The effect of HRP’s molecular weight (i.e., size) onits release profile from the various fibers was alsostudied using our model. The results for fiber struc-tures with a shell prepared from an emulsion of 5%(w/w) HRP and 19% (w/v) polymer are presented inFigure 5. The predicted profiles demonstrate that theHRP release rate decreases with the increase in its mo-lecular weight, i.e. higher molecular weight proteinsexhibit a lower diffusion coefficient, which results inlower mobility in water. Since protein release occursby means of diffusion in water, this lower diffusioncoefficient should result in a lower release rate. Theseresults support our second hypothesis, saying that therelease profile is affected by the sizes of the system’scomponents, the bioactive agent, and the host poly-mer. We discovered that the effect of HRP’s size on its

Figure 5. The effect of HRP’s molecular weight on its pre-dicted release profile from core/shell fiber structures contain-ing 5% w/v HRP and 19% w/v polymer. The HRP’s molecu-lar weights are indicated; (a) o:a ¼ 8:1, (b) o:a ¼ 16:1. [Colorfigure can be viewed in the online issue, which is available atwww.interscience.wiley.com.]



release profile is apparently higher than that of thehost polymer’s initial molecular weight.

SUMMARY AND CONCLUSIONS

The aim of this study was to develop a mathematicalmodel for predicting protein release profiles from novelbioresorbable core/shell fiber structures designed to beused in tissue regeneration and other biomedical appli-cations. These novel composite structures are com-posed of a dense polymeric core fiber coated with a po-rous protein-loaded PDLGA shell, and combine desiredmechanical properties with versatile release profiles ofthe active agent. Use of the suggested model affords agood and rapid evaluation of the release profile, ena-bling further economical in vitro/in vivo release studies.

The model is based on Fick’s second law of diffusionand on two major assumptions: (a) first-order degrada-tion kinetics of the porous shell, and (b) a time-depend-ent bioactive agent diffusion coefficient, which increaseswith time because of the degradation of the host poly-mer. The model also uses three empirical parametersthat address the matrix microgeometry: porosity fac-tor, tortuosity factor, and a polymer concentration fac-tor that relates to the polymer concentration in theemulsion from which the matrix was fabricated.

The model correlates well with in vitro release re-sults, exhibiting a mean error of less than 2.2% for moststudied cases. The behavior (values and tendencies) ofthe empirical tortuosity and polymer concentration fac-tors in the model correlated well with theory. Further-more, the model predicts an increased polymer con-centration factor for higher polymer contents, resultingin a smaller burst release and a more moderate releaseprofile.

In this study, the model was used for predictingprotein release profiles from fibers with shells of vari-ous initial molecular weights and for predicting therelease of proteins with various molecular weights(sizes). This new model exhibits a potential for simu-lating fibrous systems for a wide variety of biomedicalapplications.

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A mathematical model for predicting controlled release of bioactive agents from composite fiber structures