theoreticalmodelingoflong-timedrugreleasefrom nitrosalicyl...

Research ArticleTheoretical Modeling of Long-Time Drug Release fromNitrosalicyl-Imine-Chitosan Hydrogels through MultifractalLogistic Type Laws

Anda-Mihaela Craciun 1 Mihaela Luminita Barhalescu 2 Maricel Agop 34

and Lacramioara Ochiuz 5

1Petru Poni Institute of Macromolecular Chemistry GrigoreGhicaVoda Alley Iasi Romania2Constanta Maritime University Department of General Engineering Sciences 104 Mircea cel Batran Street900663 Constanta Romania3Gheorghe Asachi Technical University of Iasi Faculty of Machine Manufacturing and Industrial ManagementDepartment of Physics D Mangeron Bvd No 73 700050 Iasi Romania4Romanian Scientists Academy 54 Splaiul Independentei Blvd Bucharest 050094 Romania5Faculty of Pharmacy Department of Pharmaceutical Technology University of Medicine andPharmacy ldquoGrT Popardquo Iasi Romania

Correspondence should be addressed to Mihaela Luminita Barhalescu mihaelabarhalescucmu-edueu

Received 18 February 2019 Revised 25 April 2019 Accepted 19 May 2019 Published 14 August 2019

Academic Editor Giuseppe Pontrelli

Copyright copy 2019 Anda-Mihaela Craciun et alis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Drug release is a complex phenomenon due to the large number of interdependent side effects that occur simultaneouslyinvolving strong nonlinear dynamics erefore since their theoretical description is difficult in the classical mathematicsmodelling we have built a theoretical model based on logistic type laws validated by the correlations with the experimental datain a special case of drug release from hydrogels e novelty of our approach is the implementation of multifractality in logistictype laws situation in which any chaotic system characterized by a small number of nonlinear interactions gets memory andimplicitly characterization through a large number of nonlinear interactions In other words the complex system polymer-drugmatrix becomes ldquopseudo-intelligentrdquo

1 Introduction

Hydrogels are an important class of materials with a largerange of applications in key domains as biomedicine [1]hygiene [2] environment protection [3] agriculture [4]and so on Among the hydrogels those based on naturaland biopolymers are of crucial significance for bio-applications due to their biocompatibility and bio-degradability mandatory features for their safe in vivo useand for limiting the environmental pollution [5ndash7] In thisregard the most suited polymer for preparation of thehydrogels targeting bioapplication is chitosan a bio-polymer derived from the natural polymer chitin [8] echitosan endows the hydrogels containing it along with the

required biocompatibility and biodegradability with otherremarkable properties as antitumor antimicrobial anti-fungal fat binding and very important with the ability toencapsulate a large range of drugs by strong physical forcesfor a further prolonged delivery Moreover due to its in-trinsic properties the use of chitosan-based hydrogels asdrug matrix led to multifunctional systems with enhancedbiological possibilities which could be further improved bya proper choice of the chitosan crosslinker [9ndash15]

Another important aspect related to the developing ofdrug delivery systems is related to the efficient use of thedrugs limiting thus their side effects on (i) the human bodyand (ii) the environment As an example many antitumordrugs eg 5-fluorouracil which are extensively used in the

HindawiComputational and Mathematical Methods in MedicineVolume 2019 Article ID 4091464 10 pageshttpsdoiorg10115520194091464

treatment of a wide variety of tumor lines [9] showed thattheir systemic administration led to the interference withthe structure and function of DNA of the tumor cells butunfortunately affects also nontarget normal cells More-over part of them is excreted in the environment andrecent studies revealed their toxicity and genotoxicity intozebrafish indicating them as potential negative contami-nants [16] is is the reason why the researchers focustheir attention to the local control release of the antitumordrugs which should limit their side effects on the humanbody and on the environment also

In this context targeting the preparation of hydrogelsfor the local therapy of cancer chitosan has been reactedwith nitrosalicylaldehyde when biocompatible hydrogelswith high cytotoxicity against the HeLa cells were ob-tained [11] Moreover the hydrogels demonstrated fasthydrogelation in physiological pH and moderate swellingwhile maintaining their shape features which recommendthem for local therapy of cancer Taking into consider-ation these findings they were further investigated asmatrix for drug delivery systems when demonstratingprolonged drug release [17] For a better understanding ofthe drug release mechanism we proposed to model thedrug release kinetics both by means of empiric laws and bymeans of logistic type laws

e novelty of our approach is the implementation ofmultifractality in logistic type laws for the description of thedrug release processes In such a situation the complexsystem polymer-drug matrix gets memory becomingldquopseudo-intelligentrdquo

2 Experimental

21 Materials Chitosan of low-molecular weight nitro-salicylaldehyde of 98 purity diclofenac sodium salt (DCF)of purity higher of 99 and phosphate buffer solution werepurchased from Aldrich and used as received

22 Preparation ofDrugDelivery Systems e drug deliverysystems which are the subject of the present paper wereobtained according to a procedure already reported [17]Shortly they were prepared by in situ hydrogelation ofchitosan with nitrosalicylaldehyde (NSA) in the presenceof diclofenac sodium salt (DCF) based on the iminationreaction along with the self-ordering of the newly formedimine units [11] In such a way the new imine unitsformed by the reaction of amine units of chitosan withnitrosalicylaldehyde segregated into ordered clusterswhich played the role of crosslinking nodes Varying themolar ratio of the aminealdehyde functional units aseries of four drug delivery systems A2DndashA5D wereprepared (Figure 1) It should be mentioned here that theamount of drug has been kept constant while the amountof chitosan and nitrosalicylaldehyde was varied in order toobtain drug delivery systems with different crosslinkingdensities which should influence the drug releasemechanism

23 In Vitro Investigation of the Drug Release In order tomonitor the drug delivery profile of the studied formu-lations equal amounts of the synthetized A2DndashA5Dsystems were prepared as pills by pressing with a hy-draulic press (2 Nm2) e pills were introduced intosealed vials containing 10ml of PBS at 37degC At pre-determined moments 2 mL of supernatant was with-drawn and replaced with 2mL of PBS e supernatantsamples were collected and analyzed by quantitativeabsorption spectroscopy by measuring the absorbance ofthe DCF drug e values of the absorbance obtained atdifferent investigation times were fitted on the DCFcalibration curve [9 17] e cumulative release of the5FU was estimated from the BeerndashLambert law All theexperiments were realized in triplicate e absorbancespectroscopy was run on a Horiba spectrophotometer Aschematic representation of the in vitro drug releaseinvestigation is given in Figure 2

3 Theoretical Considerations

e release kinetics of the studied formulations was firstinvestigated by fitting the resulted release data on theequations of 5 different models zero order first orderHiguchi Korsmeyer-Peppas and HixsonndashCrowell [18] eirequations are presented below

(i) Zero-order model Mt K0t where Mt is theamount of DCF released in the time t and K0 is thezero-order release constant

(ii) First-order model log Mt Kt2303 where Mt isthe amount of DCF released in the time t and K0 isthe zero-order release constant

(iii) Higuchi model Mt KHt(12) where Mt is theamount of DCF released in the time t and KH is theHiguchi dissolution constant

0 20 40 60

A2D

A3D

A4D

A5D

DCF (mg)NSA (mg)Chitosan (mg)

Figure 1 Graphical representation of the reagents weight used forhydrogel preparation

2 Computational and Mathematical Methods in Medicine

(iv) KorsmeyerndashPeppas model (MtMinfin) Ktn where(MtMinfin) is the fraction of DCF released at the timet K is the release rate constant and n is the releaseexponent

(v) HixsonndashCrowell model W130 minusW13

t K middot t whereW0 is the initial amount of DCF Wt the remainingamount of DCF at time t and K a constant

Due to the empirical character of these equations therelease kinetics was further investigated through nonlineardynamics in the form of logistic type laws

e polymeric matrix loaded with drug is bothstructurally and functionally a complex system hereafterreferred to as polymer-drug matrix In this case thecomplexity refers both to the collective behaviour of thepolymer-drug matrix (dictated by the extremely highnumber of nonlinear interactions between its structuralunits) and to the constraints that the polymeric-drugmatrix system supports in relation to the ldquoenvironmentrdquogenerated by various biological structures From such aperspective the polymer-drug matrix evolves away fromthe state of equilibrium (at the ldquoedge of chaosrdquo betweendeterminism and randomness) in a critical state builtfrom an ldquoarcheologyrdquohistory of ldquounpredictable and un-expected eventsrdquo through feedback cycles self-organizedstructures etc thus the archeologyhistory is groundedas the main feature of the polymer-drug matrix

e absence of archeologyhistory is specific to chaoticsystems Here the behaviour of any such system is dic-tated by the relatively small number of nonlinear in-teractions between its structural units and it has thefundamental property the ldquochaotic orderrdquo us thechaotic systems can be assimilated with a subset ofcomplex systems without history In such context thefollowing question arises is it possible like a chaoticpolymer-drug matrix without history to mimic thecomplex polymer-drug matrix a system with history eanswer is affirmative only to the extent that the chaoticsystem (chaotic polymer-drug matrix) is ldquoassignedrdquo a typeof archeologyhistory In this regard let us first notice thatthe polymer-drug matrix as a chaotic system can ldquosup-portrdquo a release mechanism based on the nondimensionallogistic type law [19]

dM

dt RM 1minus

M

K1113874 1113875 (1)

with

Rgt 0 Kgt 0 M(0) Minfin (2)

where M is the amount of drug released at time t Minfin is theamount of drug released after an infinitely long time that canbe approximated with the amount of drug initially loadedinto the polymer matrix and R and K are the structureconstants specific to the release mechanism Indeed theseldquosituationsrdquo are verified directly taking into account that thesolution of equation (2) in the following form

M(t) MinfinK

Minfin + exp(minusRt) KminusMinfin( 1113857 (3)

can describe for different values of R K and M complexrelease dynamics for various temporal sequences (seeFigures 3ndash6)

A possible procedure for assigning archeologyhistory topolymer-drug matrix is through the multifractality of therelease curves (continuous curves with varying degrees ofnondifferentiability hereinafter referred to as multifractalrelease curves) Consequently [20 21] (i) any physicalvariable describing the release dynamics becomes dependentboth on spatial and time coordinates and on the scaleresolutions of multifractal type mathematically such asituation is possible considering that any physical variableacts as the limit of a family of functions the functions beingnondifferentiable for null scale resolutions of multifractaltype and differentiable for non-null scale resolutions ofmultifractal type (ii) the ldquolawsrdquo describing the release dy-namics become invariant with respect to spatial and tem-poral coordinates transformations as well as withtransformations of scale resolutions of multifractal type (iii)the constrained release dynamics on continuous and dif-ferentiable release curves in an Euclidean space aresubstituted with release dynamics free of any constraints oncontinuous release curves with different degrees of non-differentiability (release curves of multifractal type) in amultifractal space moreover the release curves of multi-fractal type works also as geodesics of the multifractal space(iv) between two points of a multifractal space there are aninfinity of geodesics and thus of multifractal release curves[20 21]

In such a context the indiscernibility of the releasecurves is a result of multifractalisation by multi-stocasticizationmdashsimultaneously any monofractalizationfrom the singularity spectrum of the release dynamics being

PBS

Drug delivery system

10080604020

0

05 1 2 3 4 8 24 48 72 96 120

144

168

192

216

A2DTime (hours)

Dru

g re

leas

e (

)

t1 t2 t3 t4 t5 t6

Abso

rban

ce (a

u)

Wavelength (nm)

275

Fitting on thecalibration curve

Abso

rban

ce (a

u)

Concentrationof the diclofenanc (mgmL)

Figure 2 In vitro drug release investigation

Computational and Mathematical Methods in Medicine 3

compatible with a stochastic process However the dis-cernibility of the release curves involves a selection process(obviously at scale resolution of finite multifractal type) ofmultifractal space geodesics based on a measurement pro-cess from the multitude of multifractals manifested at agiven moment in the singularity spectrum of the releasedynamics only one confers globality through the type of thestochastic process

In such a context was analyzed the dynamics of releasefrom different types of drug delivery systems such asmicroparticles [22] nanoparticles [23] and carriers forthe transfer of genetic material [24] proving thus itsuniversal character [25] Its universal character has beenenhanced by the application in the study of variouschemical and physical systems such as plasma [26ndash29]and nanofluids [30] but also various other dynamics[31ndash43]

In these conditions the release dynamics of the polymer-drug matrix considered now as complex system can bedescribed by the generalized nondimensional logistic typelaw

dM

dt RM 1minus

M

K1113874 1113875

F(α)

1113890 1113891 (4)

with

M M[t μ F(α)]

R R[μ F(α)]

K K[μ F(α)]

M[0 μ F(α)] Minfin[μ F(α)]

α α DF( 1113857

(5)

where M is the amount of released drug of multifractaltype at time t Minfin is the amount of drug released ofmultifractal type after an infinite time equal to theamount of drug initially loaded into the polymer matrix R

and K are the structure constants specific to the releasemechanism of multifractal type μ is the scale resolution ofmultifractal type and F F(α) is the singularity spectrumof singularity index α an index functionally dependent ofthe fractal dimension DF of the release curve Let us notethat the singularity spectrum F(α) will allow the identi-fication of some ldquouniversality classesrdquo in the field ofldquodynamical systems with releaserdquo

In order to solve equation (4) let us substituteV[t μ F(α)] 1M[t μ F(α)] with V[0 μ F(α)] V0[μ F(α)] We will have the following equation

dV

dt minus

1M2

dM

dt minus

RM

M2 1minusM

K1113874 1113875

F(α)

1113890 1113891 (6)

dV

dt minus

R

M+

RMF(α)minus1

KF(α) minusRV +

RV1minusF(α)

KF(α) (7)

Next let us introduce a new substitution given by Z

VF(α) with Z0 VF(α)0 us a multifractal differential

equation of Bernoulli type is obtained in the following form

RMinfin

k2

k

M (t)

t

Figure 3 Release curves of the same krangMinfin and various R

described by the solution of the logistic type equation (1)

k

Minfin

t

M (t)

R

Figure 4 Release curves of the same klangMinfin and various R


k1

Minfin

t

M (t)

k2

Figure 6 Release curves of the same R and various k with klangMinfindescribed by the solution of the logistic type equation (1)

k2

Minfint

M (t)

k1

Figure 5 Release curves of the same R and various k with krangMinfindescribed by the solution of the logistic type equation (1)


dZ

dt F(α)V

F(α)minus1dV

dt

F(α)VF(α)minus1 minusRV +

RV1minusF(α)

KF(α)1113890 1113891

minusF(α)RVF(α)

+ F(α)R

KF(α)

minusF(α)RZ + F(α)R

KF(α)

(8)

is equation admits the following solution

Z[t F(α)] Z0 exp[minusF(α)Rt]

+1

KF(α)1minus exp[minusF(α)Rt]1113864 1113865

(9)

e above solution has in the variable V the followingexpression

V[t μ F(α)] 1113890VF(α)0 exp[minusF(α)Rt]

+1


1F(α)

(10)

While in the variable M it becomes

M[t μ F(α)] 11138901

MF(α)infin

exp[minusF(α)Rt]

+1


minus(1F(α))

MinfinK1113864MF(α)infin

+ KF(α) minusM

F(α)infin1113960 1113961exp[minusF(α)Rt]1113865

minus(1F(α))

(11)

e stationary points can be calculated as above eyare

M1 0

M2 K(12)

e derivative of function F[M R K F(α)] RM[1minus(MK)F(α)] is

Fprime(M) R 1minus[1 + F(α)]M

K1113874 1113875

F(α)

1113896 1113897 (13)

As Fprime(M1) Rrang0 M1 0 is an unstable stationarypoint In turn the stationary point M2 K is asymptoticallystable because Fprime(M2) minusF(α)Rlang0 Concluding the re-alized mass will tend to the carrying capacity over time

e inflexion point of the solution is given by means ofcondition

Fprime(0) 0 (14)

which implies the following expression

M K

[F(α) + 1]1F(α) (15)

4 Results

Applying the procedure of in situ hydrogelation describedin Section 2 a series of 4 drug delivery systems based onchitosan nitrosalicylaldehyde and diclofenac sodium saltnoted A2DndashA5D (Figure 1) were prepared in view ofmodelling their drug release characteristics As the mor-phology of the drug delivery systems drastically influencesthe release mechanism the A2DndashA5D samples were firstlysubjected to scanning electron microscopy measurementsAs can be seen in Figure 7 the samples were porouswithout visible DCF crystals pointing for its sub-micrometric encapsulation is has been attributed to thestrong intermolecular forces which aroused between thepolycationic chitosan and negatively charged DCF anion[9 17 44 45]

e monitoring of the DCF release from the under studysystems is represented in Figure 8 As can be seen the releasewas significantly affected by the encapsulation pathway FromA2D and A5D systems in which the DCF drug was confinedas micrometric crystals the release occurred faster whilefrom A3D and A4D systems in which the drug was finedispersed the release produced slower e encapsulationpathway played a key role in the total amount of the drugreleased from the matrix too As can be seen after 8 hoursthe A2D and A5D released almost entire amount of the drugwhile the A3D and A4D only 70 is can be explained bythe stronger intermolecular forces developed in the case of thefine dispersion of the drug into the hydrogel matrix forceswhich surpassed the interfacial ones developed in the case ofdispersion of the drug as microcrystals [17]

e release data were fitted on the equations of fivedifferent standard (empiricalsemiempirical) models [18]zero order first order Higuchi KorsmeyerndashPeppas andHixsonndashCrowell

As can be seen in Figure 9 all the obtained release dataproved a good fitting on the HixsonndashCrowell model in-dicating that the release mechanism is mainly controlled bythe dissolution velocity and less by the diffusion through thematrix On the other hand in the case of the samples with ahigher crosslinking density a good fitting has been obtainedalso for the Higuchi and KorsmeyerndashPeppas models in-dicating that the drug release is also strongly influenced bythe diffusion process through the hydrogel network

Taking into account our theoretical model presented inSection 3 we present next in Figure 10(a) the three di-mensional qualitative dependence of the amount of drugreleased M as a function of time t and the global fractaldimension dictated through the singularity spectrum F(α)In such conjecture a time linear dependence is observed Bya convenient choice of the normalization parameters thecorrelations with the experimental data are illustrated inFigure 10(b)


(a) (b) (c) (d)

Figure 7 SEM microimages of the studied drug delivery systems (a) A2D (b) A3D (c) A4D (d) A5D

A4DA3D

A2DA5D

1 2 3 4 8 24 48 72 96 120 144 168 192 21605Time (hours)

0

20

40

60

80

100

Dru

g re

leas

e (

)

Figure 8 Monitoring of the DCF release from the A2DndashA5D formulations

A2A3

A4A5

1 2 3 40

Time (hours)

0

10

20

30

40

50

Cum

ulat

ive d

rug

rele

ase (

)

(a)

A2DA3D

A4DA5D

2 40

Time (hours)

04

08

12

16

Log

cum

ulat

ive

dru

g re

leas

e

(b)

A2A3

A4A5

14 2107

SQRT (time)

0

10

20

30

40

50

Cum

ulat

ive

dru

g re

leas

e

(c)

Log

(M0M

infin)

A2

A3

A4

A5

ndash16

ndash12

ndash08

ndash04

0402ndash02 06ndash04 0

Log (time)

(d)

Figure 9 Continued


5 Discussion

In the present paper release dynamics are analyzed boththeoretical and experimental for a special polymer-drugmatrix based on standard models (empiricalsemi-empirical) zero order first order Higuchi KorsmeyerndashPeppas and HixsonndashCrowell and on the basis of a newmodel given by a logistic type law

Our approach is based on the following (i) first thelogistic type law (1) is proposed in the description of therelease dynamics In this relation the term RM specifies thedifference between the rate of occurrence and the rate ofdisappearance of the released drug mass while the termRM2K limits the increase in drug mass released by typicalprocesses such as the degradation of the polymeric matrixSuch a release system is a chaotic one its behaviour isimposed by a relatively small number of nonlinear in-teractions between its structural units it does not havearcheologyhistory (or if it contains a ldquointrinsicrdquo one it is

freezedmdashit does not manifest itself ) and it has as thefundamental property the ldquochaotic orderrdquo (ii) then ageneralization of logistic type law (1) is proposed in the formof relation (4) in order to describe the release dynamicsassuming that the release curves are of the multifractal typeSuch a choice has several obvious implications (a) the drugmass released given by (4) becomes dependent both on timecoordinate and on the scale resolutions of multifractal typeMathematically such a situation is possible considering thatthe released drug mass given by (4) acts as the limit of afamily of functions the functions being nondifferentiable fornull scale resolutions of multifractal type and differentiablefor non-null scale resolutions of multifractal type (b) therelease dynamics described by relation (4) is invariant bothin relation to temporal transformations and in relation toscale resolution transformations (c) the constrained releasedynamics onmultifractal type curves in an Euclidian space issubstituted with release dynamics free of any constraints ina multifractal space etc

A2

A3

A4

A5

35

4

45

5

(Unr

elea

sed

fract

ion)

13

32 40 1

Time (hours)

(e)

Figure 9 Linear forms of the zero order (a) first order (b) Higuchi (c) KorsmeyerndashPeppas (d) and HixsonndashCrowell (e) models applied forthe release of DCF from the A2D-A5D formulations

t

F (α)

M (t)

(a)

t

M (t)

A4

A2

A5

A3

(b)

Figure 10 (a) 3D dependence of the amount of drug releasedM as a function of time t and the global fractal dimension (b) theoretical andexperimental correlations


6 Conclusion

Amathematical model that intends to describe the dynamicsof controlled drug release systems should allow (a) to predictthe drug release kinetics and the phenomena involvedavoiding thus repetitive time and money expensive exper-iments (b) to optimize the drug release kinetics (c) toestimate the effect of the polymer matrix design parametersie shape size and composition on the drug release kinetics(d) to predict the global therapeutic efficiency and drugsafety and (e) overall to design a new drug delivery systemdepending on the release kinetics imposed by the therapeuticrequirements

So far successful mathematical models designed todescribe the dynamics of the drug controlled release systemhave been developed based on either an almost intuitiveselection of dominant phenomena relative to the polymermatrix configuration and geometry or just by analysing theexperimental results resulting in a wide variety of empiricalsemiempirical models such as the diffusion model the zero-kinetic model the Higuchi model the KorsemeyerndashPeppasmodel the HixonndashCrowell model the Weibull model theBaker Lonsdale model the Hopfenberg model the Gom-pertz model the sequential layer model the PeppasndashSahlinmodel etc

In our opinion this diversity of models demonstrates theinability of differential and integral mathematical pro-cedures to describe the complexity of controlled drug releasedynamics us the success of the abovementioned modelsshould be understood only sequentiallyprogressively indomains where differentiability and integrity are still validHowever the differential and integral mathematical pro-cedures ldquosufferrdquo when one wants to describe the dynamics ofdrug controlled release systems dynamics involving bothnonlinearity and chaos

e model developed in this paper is precisely intendedto substitute the standard mathematical procedures ie ofdifferential and integral types with nondifferential andnonintegrative mathematical procedures applied in de-scribing the dynamics of controlled drug release systems Insuch a conjecture we first developed the logistic model ofdynamics of drug controlled release systems (where thesystem is only chaotic without memory behaviour dictatedby a relatively small number of nonlinear interactions be-tween its entities and still subject to differential and integralmathematical procedures) en by extension we de-veloped the multifractal logistic model for the dynamics ofcontrolled drug release systems (where the system is com-plex with memory its behaviour being imposed by a largenumber of nonlinear interactions between its entitiessubjected only to nondifferential and nonintegrativemathematical procedures) such a description proves to bereducible to the dynamics of drug controlled release systemsat various scale resolutions

e model was finally validated on the basis of the ex-perimental results on diclofenac sodium salt release fromnitrosalicyl-imine-chitosan hydrogels confirming thus thatcomplex systems ie polymer-drug matrices are pseudo-intelligent systems with memory In such an approach

phenomena as drug diffusion swelling and degradation ofthe polymeric matrix can be described as dynamics ofpseudo-intelligent materials at various scale resolutions

Data Availability

Previously reported experimental data were used to supportthis study and are available at httpsdoiorg101016jjcis201810048

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the European Unionrsquos Horizon2020 Research and Innovation Programme (grant no667387) and Romanian National Authority for ScientificResearch MENmdashUEFISCDI (grant nos PN-III-P1-12-PCCDI2017-0569 and 10PCCDI2018)

References

[1] M Naseri-Nosar and Z M Ziora ldquoWound dressings fromnaturally-occurring polymers a review on homo-polysaccharide-based compositesrdquo Carbohydrate Polymersvol 189 pp 379ndash398 2018

[2] B Cheng B Pei Z Wang and Q Hu ldquoAdvances in chitosan-based superabsorbent hydrogelsrdquo RSC Advances vol 7no 67 pp 42036ndash42046 2017

[3] A Rotaru C Cojocaru I Cretescu M Pinteala et alldquoPerformances of clay aerogel polymer composites for oil spillsorption experimental design and modelingrdquo Separation andPurification Technology vol 133 pp 260ndash275 2014

[4] M-M Timpu S Morariu and L Marin ldquoSalicyl-imine-chitosan hydrogels supramolecular architecturing as acrosslinking method toward multifunctional hydrogelsrdquoCarbohydrate Polymers vol 165 pp 39ndash50 2017

[5] J Chabbi O Jennah N Katir M Lahcini M Bousmina andA El Kadib ldquoAldehyde-functionalized chitosan-montmoril-lonite films as dynamically-assembled switchable-chemicalrelease bioplasticsrdquo Carbohydrate Polymers vol 183pp 287ndash293 2018

[6] L Marin D Ailincai S Morariu and L Tartau-MititeluldquoDevelopment of biocompatible glycodynameric hydrogelsjoining two natural motifs by dynamic constitutionalchemistryrdquo Carbohydrate Polymers vol 170 pp 60ndash71 2017

[7] C Laroche C Delattre N Mati-Baouche et al ldquoBioactivity ofchitosan and its derivativesrdquo Current Organic Chemistryvol 22 no 7 pp 641ndash667 2018

[8] M C G Pella M K Lima-Tenorio E T Tenorio-NetoM R Guilherme E C Muniz and A F Rubira ldquoChitosan-based hydrogels from preparation to biomedical applica-tionsrdquo Carbohydrate Polymers vol 196 pp 233ndash245 2018

[9] D Ailincai L Tartau Mititelu and L Marin ldquoDrug deliverysystems based on biocompatible imino-chitosan hydrogels forlocal anticancer therapyrdquo Drug Delivery vol 25 no 1pp 1080ndash1090 2018

[10] L Marin D Ailincai M Mares et al ldquoImino-chitosan bio-polymeric films obtaining self-assembling surface and an-timicrobial propertiesrdquo Carbohydrate Polymers vol 117pp 762ndash770 2015


[11] A-M Olaru L Marin S Morariu G Pricope M Pintealaand L Tartau-Mititelu ldquoBiocompatible chitosan basedhydrogels for potential application in local tumour therapyrdquoCarbohydrate Polymers vol 179 pp 59ndash70 2018

[12] D Ailincai L Marin S Morariu et al ldquoDual crosslinkediminoboronate-chitosan hydrogels with strong antifungalactivity against Candida planktonic yeasts and biofilmsrdquoCarbohydrate Polymers vol 152 pp 306ndash316 2016

[13] A Bejan D Ailincai B C Simionescu and L MarinldquoChitosan hydrogelation with a phenothiazine based alde-hyde a synthetic approach toward highly luminescent bio-materialsrdquo Polymer Chemistry vol 9 no 18 pp 2359ndash23692018

[14] M M Iftime and L Marin ldquoChiral betulin-imino-chitosanhydrogels by dynamic covalent sonochemistryrdquo UltrasonicsSonochemistry vol 45 pp 238ndash247 2018

[15] L Marin S Moraru M-C Popescu et al ldquoOut-of-waterconstitutional self-organization of chitosan-cinnamaldehydedynagelsrdquo ChemistrymdashA European Journal vol 20 no 16pp 4814ndash4821 2014

[16] R Kovacs Z Csenki K Bakos et al ldquoAssessment of toxicityand genotoxicity of low doses of 5-fluorouracil in zebrafish(Danio rerio) two-generation studyrdquo Water Research vol 77pp 201ndash212 2015

[17] A M Gajski L Mititelu Tartau M Pinteala and L MarinldquoNitrosalicyl-imine-chitosan hydrogels based drug deliverysystems for long term sustained release in local therapyrdquoJournal of Colloid and Interface Science vol 536 pp 196ndash2072019

[18] A Fifere N Marangoci S Maier A Coroaba D Maftei andM Pinteala ldquoeoretical study on β-cyclodextrin inclusioncomplexes with propiconazole and protonated propicona-zolerdquo Beilstein Journal of Organic Chemistry vol 8pp 2191ndash2201 2012

[19] B Mandelbrot Ae Fractal Geometry of Nature W HFreeman New York NY USA 1983

[20] L Nottale Scale Relativity and Fractal Space-Time A NewApproach to Unifying Relativity and Quantum MechanicsImperial College Press London UK 2011

[21] I Merches and M Agop Differentiability and Fractality inDynamics of Physical Systems World Scientific Singapore2016

[22] D Magop S Bacaita C Peptu M Popa and M Agop ldquoNon-differentiability at mesoscopic scale in drug release processesfrom polymer microparticlesrdquo Materiale Plastice vol 49no 2 pp 101ndash105 2012

[23] A Durdureanu-Angheluta S Bacaita V Radu et alldquoMathematical modelling of the release profile of anthra-quinone-derived drugs encapsulated on magnetite nano-particlesrdquo Revue Roumaine de Chimie vol 58 no 2-3pp 217ndash221 2013

[24] G Cioca M Pinteala E S Bacaita et al ldquoNonlinear behaviorsin gene therapy theoretical and experimental aspectsrdquoMateriale Plastice vol 55 no 3 pp 340ndash343 2018

[25] G Cioca E S Bacaita M Agop and C Lupascu UrsulesculdquoAnisotropy influences on the drug delivery mechanisms bymeans of joint invariant functionsrdquo Computational andMathematical Methods in Medicine vol 2017 Article ID5748273 8 pages 2017

[26] S Gurlui M Agop M Strat G Strat and S Bacaita ldquoEx-perimental and theoretical investigations of anode doublelayerrdquo Japanese Journal of Applied Physics vol 44 no 5pp 3253ndash3259 2005

[27] M Agop D Alexandroaie A Cerepaniuc and S Bacaita ldquoElNaschiersquos ε(infin) space-time and patterns in plasma dischargerdquoChaos Solitons amp Fractals vol 30 no 2 pp 470ndash489 2006

[28] C Ursu O G Pompilian S Gurlui et al ldquoAl2O3 ceramicsunder high-fluence irradiation plasma plume dynamicsthrough space-and time-resolved optical emission spectros-copyrdquo Applied Physics A vol 101 no 1 pp 153ndash159 2010

[29] O Nica D G Dimitriu V P Paun P DMatasaru D Scurtuand M Agop ldquoExperimental and theoretical investigations ofa plasma fireball dynamicsrdquo Physics of Plasma vol 17 no 4article 042305 2010

[30] M Agop V Paun and A Harabagiu ldquoEl Naschiersquos ε(infin)

theory and effects of nanoparticle clustering on the heattransport in nanofluidsrdquo Chaos Solitons amp Fractals vol 37no 5 pp 1269ndash1278 2008

[31] M Colotin G O Pompilian P Nica S Gurlui V Paun andM Agop ldquoFractal transport phenomena through the scalerelativity modelrdquo Acta Physica Polonica A vol 116 no 2pp 157ndash164 2009

[32] G V Gurlui V-P Paun I Casian-Botez and M Agop ldquoemicroscopic-macroscopic scale transformation through achaos scenario in the fractal space-time theoryrdquo InternationalJournal of Bifurcation and Chaos vol 21 no 2 pp 603ndash6182011

[33] I Gottlieb M Agop G Ciobanu and A Stroe ldquoEl Naschiersquosε(infin) space-time and new results in scale relativity theoriesrdquoChaos Solitons amp Fractals vol 30 no 2 pp 380ndash398 2006

[34] M Stroe P Nica P D Ioannou O Malandraki andI Gavanas-Pahomi ldquoEl Naschiersquos ε(infin) space-time hydro-dynamic model of scale relativity theory and some applica-tionsrdquo Chaos Solitons amp Fractals vol 34 no 5pp 1704ndash1723 2007

[35] C Malandraki A Nicuta and B Constantin ldquoDynamicscontrol of the complex systems via nondifferentiabilityrdquoJournal of Applied Mathematics vol 2013 Article ID 13705612 pages 2013

[36] V Nedeff E Mosnegutu M Panainte et al ldquoDynamics in theboundary layer of a flat particlerdquo Powder Technology vol 221pp 312ndash317 2012

[37] M Agop P Nica and M Gırtu ldquoOn the vacuum status inWeyl-Dirac theoryrdquo General Relativity and Gravitationvol 40 no 1 pp 35ndash55 2008

[38] M Agop and C Murgulet ldquoEl Naschiersquos ε(infin) space-time andscale relativity theory in the topological dimension D 4rdquoChaos Solitons amp Fractals vol 32 no 3 pp 1231ndash1240 2007

[39] I Gottlieb M Agop and M Jarcǎu ldquoEl Naschiersquos cantorianspace-time and general relativity by means of Barbilianrsquosgrouprdquo Chaos Solitons amp Fractals vol 19 no 4 pp 705ndash7302004

[40] M Agop P Ioannou P Nica C Radu A Alexandru andP Vizureanu ldquoFractal characteristics of the solidificationprocessrdquo Materials Transactions vol 45 no 3 pp 972ndash9752004

[41] M Agop V Griga B Ciobanu et al ldquoGravity and cantorianspace-timerdquo Chaos Solitons amp Fractals vol 9 no 7pp 1143ndash1181 1998

[42] C Ciubotariu and M Agop ldquoAbsence of a gravitationalanalog to the Meissner effectrdquo General Relativity and Grav-itation vol 28 no 4 pp 405ndash412 1996

[43] M Agop P E Nica S Gurlui C Focsa V P Paun andM Colotin ldquoImplications of an extended fractal hydrody-namic modelrdquo European Physical Journal D vol 56 no 3pp 405ndash419 2010


[44] E Szajdzinska-Pietek M Pinteala and S Schlick ldquoMoni-toring pH-dependent conformational changes in aqueoussolutions of poly(methacrylic acid)-b-polydimethylsiloxanecopolymer based on fluorescence spectra of pyrene and 13-bis(1-pyrenyl)propanerdquo Polymer vol 45 no 12 pp 4113ndash4120 2004

[45] L Marin M C Popescu A Zabulica H Uji-I and E FronldquoChitosan as matrix for bio-polymer dispersed liquid crystalsystemsrdquo Carbohydrate Polymers vol 95 no 1 pp 16ndash242013


Stem Cells International

Hindawiwwwhindawicom Volume 2018


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of

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Immunology ResearchHindawiwwwhindawicom Volume 2018

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine


Behavioural Neurology

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary andAlternative Medicine

Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom

treatment of a wide variety of tumor lines [9] showed thattheir systemic administration led to the interference withthe structure and function of DNA of the tumor cells butunfortunately affects also nontarget normal cells More-over part of them is excreted in the environment andrecent studies revealed their toxicity and genotoxicity intozebrafish indicating them as potential negative contami-nants [16] is is the reason why the researchers focustheir attention to the local control release of the antitumordrugs which should limit their side effects on the humanbody and on the environment also

In this context targeting the preparation of hydrogelsfor the local therapy of cancer chitosan has been reactedwith nitrosalicylaldehyde when biocompatible hydrogelswith high cytotoxicity against the HeLa cells were ob-tained [11] Moreover the hydrogels demonstrated fasthydrogelation in physiological pH and moderate swellingwhile maintaining their shape features which recommendthem for local therapy of cancer Taking into consider-ation these findings they were further investigated asmatrix for drug delivery systems when demonstratingprolonged drug release [17] For a better understanding ofthe drug release mechanism we proposed to model thedrug release kinetics both by means of empiric laws and bymeans of logistic type laws

e novelty of our approach is the implementation ofmultifractality in logistic type laws for the description of thedrug release processes In such a situation the complexsystem polymer-drug matrix gets memory becomingldquopseudo-intelligentrdquo

2 Experimental

21 Materials Chitosan of low-molecular weight nitro-salicylaldehyde of 98 purity diclofenac sodium salt (DCF)of purity higher of 99 and phosphate buffer solution werepurchased from Aldrich and used as received

22 Preparation ofDrugDelivery Systems e drug deliverysystems which are the subject of the present paper wereobtained according to a procedure already reported [17]Shortly they were prepared by in situ hydrogelation ofchitosan with nitrosalicylaldehyde (NSA) in the presenceof diclofenac sodium salt (DCF) based on the iminationreaction along with the self-ordering of the newly formedimine units [11] In such a way the new imine unitsformed by the reaction of amine units of chitosan withnitrosalicylaldehyde segregated into ordered clusterswhich played the role of crosslinking nodes Varying themolar ratio of the aminealdehyde functional units aseries of four drug delivery systems A2DndashA5D wereprepared (Figure 1) It should be mentioned here that theamount of drug has been kept constant while the amountof chitosan and nitrosalicylaldehyde was varied in order toobtain drug delivery systems with different crosslinkingdensities which should influence the drug releasemechanism

23 In Vitro Investigation of the Drug Release In order tomonitor the drug delivery profile of the studied formu-lations equal amounts of the synthetized A2DndashA5Dsystems were prepared as pills by pressing with a hy-draulic press (2 Nm2) e pills were introduced intosealed vials containing 10ml of PBS at 37degC At pre-determined moments 2 mL of supernatant was with-drawn and replaced with 2mL of PBS e supernatantsamples were collected and analyzed by quantitativeabsorption spectroscopy by measuring the absorbance ofthe DCF drug e values of the absorbance obtained atdifferent investigation times were fitted on the DCFcalibration curve [9 17] e cumulative release of the5FU was estimated from the BeerndashLambert law All theexperiments were realized in triplicate e absorbancespectroscopy was run on a Horiba spectrophotometer Aschematic representation of the in vitro drug releaseinvestigation is given in Figure 2

3 Theoretical Considerations

e release kinetics of the studied formulations was firstinvestigated by fitting the resulted release data on theequations of 5 different models zero order first orderHiguchi Korsmeyer-Peppas and HixsonndashCrowell [18] eirequations are presented below

(i) Zero-order model Mt K0t where Mt is theamount of DCF released in the time t and K0 is thezero-order release constant

(ii) First-order model log Mt Kt2303 where Mt isthe amount of DCF released in the time t and K0 isthe zero-order release constant

(iii) Higuchi model Mt KHt(12) where Mt is theamount of DCF released in the time t and KH is theHiguchi dissolution constant

0 20 40 60

A2D

A3D

A4D

A5D

DCF (mg)NSA (mg)Chitosan (mg)

Figure 1 Graphical representation of the reagents weight used forhydrogel preparation








dM

dt RM 1minus

M

K1113874 1113875 (1)

with



M(t) MinfinK





PBS


10080604020

0

05 1 2 3 4 8 24 48 72 96 120

144

168

192

216

A2DTime (hours)

Dru

g re

leas

e (

)

t1 t2 t3 t4 t5 t6

Abso

rban

ce (a

u)

Wavelength (nm)

275


Abso

rban

ce (a

u)







dM

dt RM 1minus

M

K1113874 1113875

F(α)

1113890 1113891 (4)

with

M M[t μ F(α)]

R R[μ F(α)]

K K[μ F(α)]


α α DF( 1113857

(5)




dV

dt minus

1M2

dM

dt minus

RM

M2 1minusM

K1113874 1113875

F(α)

1113890 1113891 (6)

dV

dt minus

R

M+

RMF(α)minus1

KF(α) minusRV +

RV1minusF(α)

KF(α) (7)




RMinfin

k2

k

M (t)

t



k

Minfin

t

M (t)

R



k1

Minfin

t

M (t)

k2


k2

Minfint

M (t)

k1



dZ

dt F(α)V

F(α)minus1dV

dt


RV1minusF(α)

KF(α)1113890 1113891

minusF(α)RVF(α)

+ F(α)R

KF(α)


KF(α)

(8)



+1


(9)



+1


1F(α)

(10)


M[t μ F(α)] 11138901

MF(α)infin

exp[minusF(α)Rt]

+1


minus(1F(α))


+ KF(α) minusM


minus(1F(α))

(11)


M1 0

M2 K(12)



K1113874 1113875

F(α)

1113896 1113897 (13)



Fprime(0) 0 (14)


M K

[F(α) + 1]1F(α) (15)

4 Results







(a) (b) (c) (d)


A4DA3D

A2DA5D

1 2 3 4 8 24 48 72 96 120 144 168 192 21605Time (hours)

0

20

40

60

80

100

Dru

g re

leas

e (

)


A2A3

A4A5

1 2 3 40

Time (hours)

0

10

20

30

40

50

Cum

ulat

ive d

rug

rele

ase (

)

(a)

A2DA3D

A4DA5D

2 40

Time (hours)

04

08

12

16

Log

cum

ulat

ive

dru

g re

leas

e

(b)

A2A3

A4A5

14 2107

SQRT (time)

0

10

20

30

40

50

Cum

ulat

ive

dru

g re

leas

e

(c)

Log

(M0M

infin)

A2

A3

A4

A5

ndash16

ndash12

ndash08

ndash04


Log (time)

(d)

Figure 9 Continued


5 Discussion




A2

A3

A4

A5

35

4

45

5

(Unr

elea

sed

fract

ion)

13

32 40 1

Time (hours)

(e)


t

F (α)

M (t)

(a)

t

M (t)

A4

A2

A5

A3

(b)



6 Conclusion







Data Availability




Acknowledgments


References






















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of

























dM

dt RM 1minus

M

K1113874 1113875 (1)

with



M(t) MinfinK





PBS


10080604020

0

05 1 2 3 4 8 24 48 72 96 120

144

168

192

216

A2DTime (hours)

Dru

g re

leas

e (

)

t1 t2 t3 t4 t5 t6

Abso

rban

ce (a

u)

Wavelength (nm)

275


Abso

rban

ce (a

u)







dM

dt RM 1minus

M

K1113874 1113875

F(α)

1113890 1113891 (4)

with

M M[t μ F(α)]

R R[μ F(α)]

K K[μ F(α)]


α α DF( 1113857

(5)




dV

dt minus

1M2

dM

dt minus

RM

M2 1minusM

K1113874 1113875

F(α)

1113890 1113891 (6)

dV

dt minus

R

M+

RMF(α)minus1

KF(α) minusRV +

RV1minusF(α)

KF(α) (7)




RMinfin

k2

k

M (t)

t



k

Minfin

t

M (t)

R



k1

Minfin

t

M (t)

k2


k2

Minfint

M (t)

k1



dZ

dt F(α)V

F(α)minus1dV

dt


RV1minusF(α)

KF(α)1113890 1113891

minusF(α)RVF(α)

+ F(α)R

KF(α)


KF(α)

(8)



+1


(9)



+1


1F(α)

(10)


M[t μ F(α)] 11138901

MF(α)infin

exp[minusF(α)Rt]

+1


minus(1F(α))


+ KF(α) minusM


minus(1F(α))

(11)


M1 0

M2 K(12)



K1113874 1113875

F(α)

1113896 1113897 (13)



Fprime(0) 0 (14)


M K

[F(α) + 1]1F(α) (15)

4 Results







(a) (b) (c) (d)


A4DA3D

A2DA5D

1 2 3 4 8 24 48 72 96 120 144 168 192 21605Time (hours)

0

20

40

60

80

100

Dru

g re

leas

e (

)


A2A3

A4A5

1 2 3 40

Time (hours)

0

10

20

30

40

50

Cum

ulat

ive d

rug

rele

ase (

)

(a)

A2DA3D

A4DA5D

2 40

Time (hours)

04

08

12

16

Log

cum

ulat

ive

dru

g re

leas

e

(b)

A2A3

A4A5

14 2107

SQRT (time)

0

10

20

30

40

50

Cum

ulat

ive

dru

g re

leas

e

(c)

Log

(M0M

infin)

A2

A3

A4

A5

ndash16

ndash12

ndash08

ndash04


Log (time)

(d)

Figure 9 Continued


5 Discussion




A2

A3

A4

A5

35

4

45

5

(Unr

elea

sed

fract

ion)

13

32 40 1

Time (hours)

(e)


t

F (α)

M (t)

(a)

t

M (t)

A4

A2

A5

A3

(b)



6 Conclusion







Data Availability




Acknowledgments


References






















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of






















dM

dt RM 1minus

M

K1113874 1113875

F(α)

1113890 1113891 (4)

with

M M[t μ F(α)]

R R[μ F(α)]

K K[μ F(α)]


α α DF( 1113857

(5)




dV

dt minus

1M2

dM

dt minus

RM

M2 1minusM

K1113874 1113875

F(α)

1113890 1113891 (6)

dV

dt minus

R

M+

RMF(α)minus1

KF(α) minusRV +

RV1minusF(α)

KF(α) (7)




RMinfin

k2

k

M (t)

t



k

Minfin

t

M (t)

R



k1

Minfin

t

M (t)

k2


k2

Minfint

M (t)

k1



dZ

dt F(α)V

F(α)minus1dV

dt


RV1minusF(α)

KF(α)1113890 1113891

minusF(α)RVF(α)

+ F(α)R

KF(α)


KF(α)

(8)



+1


(9)



+1


1F(α)

(10)


M[t μ F(α)] 11138901

MF(α)infin

exp[minusF(α)Rt]

+1


minus(1F(α))


+ KF(α) minusM


minus(1F(α))

(11)


M1 0

M2 K(12)



K1113874 1113875

F(α)

1113896 1113897 (13)



Fprime(0) 0 (14)


M K

[F(α) + 1]1F(α) (15)

4 Results







(a) (b) (c) (d)


A4DA3D

A2DA5D

1 2 3 4 8 24 48 72 96 120 144 168 192 21605Time (hours)

0

20

40

60

80

100

Dru

g re

leas

e (

)


A2A3

A4A5

1 2 3 40

Time (hours)

0

10

20

30

40

50

Cum

ulat

ive d

rug

rele

ase (

)

(a)

A2DA3D

A4DA5D

2 40

Time (hours)

04

08

12

16

Log

cum

ulat

ive

dru

g re

leas

e

(b)

A2A3

A4A5

14 2107

SQRT (time)

0

10

20

30

40

50

Cum

ulat

ive

dru

g re

leas

e

(c)

Log

(M0M

infin)

A2

A3

A4

A5

ndash16

ndash12

ndash08

ndash04


Log (time)

(d)

Figure 9 Continued


5 Discussion




A2

A3

A4

A5

35

4

45

5

(Unr

elea

sed

fract

ion)

13

32 40 1

Time (hours)

(e)


t

F (α)

M (t)

(a)

t

M (t)

A4

A2

A5

A3

(b)



6 Conclusion







Data Availability




Acknowledgments


References






















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















dZ

dt F(α)V

F(α)minus1dV

dt


RV1minusF(α)

KF(α)1113890 1113891

minusF(α)RVF(α)

+ F(α)R

KF(α)


KF(α)

(8)



+1


(9)



+1


1F(α)

(10)


M[t μ F(α)] 11138901

MF(α)infin

exp[minusF(α)Rt]

+1


minus(1F(α))


+ KF(α) minusM


minus(1F(α))

(11)


M1 0

M2 K(12)



K1113874 1113875

F(α)

1113896 1113897 (13)



Fprime(0) 0 (14)


M K

[F(α) + 1]1F(α) (15)

4 Results







(a) (b) (c) (d)


A4DA3D

A2DA5D

1 2 3 4 8 24 48 72 96 120 144 168 192 21605Time (hours)

0

20

40

60

80

100

Dru

g re

leas

e (

)


A2A3

A4A5

1 2 3 40

Time (hours)

0

10

20

30

40

50

Cum

ulat

ive d

rug

rele

ase (

)

(a)

A2DA3D

A4DA5D

2 40

Time (hours)

04

08

12

16

Log

cum

ulat

ive

dru

g re

leas

e

(b)

A2A3

A4A5

14 2107

SQRT (time)

0

10

20

30

40

50

Cum

ulat

ive

dru

g re

leas

e

(c)

Log

(M0M

infin)

A2

A3

A4

A5

ndash16

ndash12

ndash08

ndash04


Log (time)

(d)

Figure 9 Continued


5 Discussion




A2

A3

A4

A5

35

4

45

5

(Unr

elea

sed

fract

ion)

13

32 40 1

Time (hours)

(e)


t

F (α)

M (t)

(a)

t

M (t)

A4

A2

A5

A3

(b)



6 Conclusion







Data Availability




Acknowledgments


References






















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















(a) (b) (c) (d)


A4DA3D

A2DA5D

1 2 3 4 8 24 48 72 96 120 144 168 192 21605Time (hours)

0

20

40

60

80

100

Dru

g re

leas

e (

)


A2A3

A4A5

1 2 3 40

Time (hours)

0

10

20

30

40

50

Cum

ulat

ive d

rug

rele

ase (

)

(a)

A2DA3D

A4DA5D

2 40

Time (hours)

04

08

12

16

Log

cum

ulat

ive

dru

g re

leas

e

(b)

A2A3

A4A5

14 2107

SQRT (time)

0

10

20

30

40

50

Cum

ulat

ive

dru

g re

leas

e

(c)

Log

(M0M

infin)

A2

A3

A4

A5

ndash16

ndash12

ndash08

ndash04


Log (time)

(d)

Figure 9 Continued


5 Discussion




A2

A3

A4

A5

35

4

45

5

(Unr

elea

sed

fract

ion)

13

32 40 1

Time (hours)

(e)


t

F (α)

M (t)

(a)

t

M (t)

A4

A2

A5

A3

(b)



6 Conclusion







Data Availability




Acknowledgments


References






















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















5 Discussion




A2

A3

A4

A5

35

4

45

5

(Unr

elea

sed

fract

ion)

13

32 40 1

Time (hours)

(e)


t

F (α)

M (t)

(a)

t

M (t)

A4

A2

A5

A3

(b)



6 Conclusion







Data Availability




Acknowledgments


References






















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















6 Conclusion







Data Availability




Acknowledgments


References






















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of





























































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















theoreticalmodelingoflong-timedrugreleasefrom nitrosalicyl...

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