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Modeling of Drug (Albumin) Release from Thermosensitive Chitosan Hydrogels Roman Zarzycki, Grzegorz Rogacki, Zofia Modrzejewska,* and Katarzyna Nawrotek Department of Environmental Engineering Systems, Technical University of yodz, ul. Wolczanska 213, 90-924 yodz, Poland ABSTRACT: A theoretical model of drug release in hydrogels was designed taking into account desorption processes on the solid phase surface and diffusion in pores. The model describes kinetics with two constant parameters independent of concentration and time: DAB, the diffusion coefficient of component A in the liquid, and k, the transfer coefficient describing the rate of desorption of component A. The model was verified using experimental data concerning the release of bovine albumin from thermosensitive chitosan gels. The process of albumin release from hydrogels produced from chitosan glutamate can be described by parameters DAB = 2.0 109 m2/s and k = 1.02 106 m/s. Calculations based on the model prove that it properly correlates experimental data.
1. INTRODUCTION Processes of controlled drug release constitute the main direction of researches dedicated to technologies of new drug forms. The researches are carried out mainly to acquire new carriers which enable release of drugs at a proper rate and on lesions only. The new drug forms are developed to achieve better accessibility and effectiveness of therapeutic agents at reduced side effects. Special attention is paid to carriers responsive to external stimuli which can be applied in a so-called targeted therapy. Most often such systems are formed which show volumetric phase transition in response to temperature and pH changes. Hydrogels are a promising carrier material due to such properties as biocompatibility, hydrophility, that is, the presence of water in the structure, elasticity, and possibility changes in physicochemical properties induced by external environment (pH, temperature). Drugs can be incorporated into hydrogel matrices in two ways: (1) postloading and (2) in situ loading. In the postloading method a hydrogel matrix is formed and then the drug is absorbed into this matrix. For an inert hydrogel system diffusion is the major force for drug uptake. Drug release will be determined by diffusion and/or gel swelling. For hydrogel containing drug-binding ligands the release will be determined by the drugpolymer interaction and drug diffusion. In the in situ loading, polymer precursor solution is mixed with drugs or drugpolymer conjugates. Hydrogel network formulation and drug encapsulation are accomplished simultaneously. The drug release will be determined by diffusion, hydrogel swelling, reversible drugpolymer interactions, or degradation of labile covalent bonds.14 Cross-linking and swelling abilities make it possible to modulate release of the drug incorporated in the matrix. This method of drug administration offers a possibility of releasing therapeutic agents such as peptides or proteins. In the case of thermosensitive hydrogels which show volumetric phase transition at physiological temperature, (sol in gel or gel in sol) drugs can be administered parenterally, that is, as injections to lesions difficult to access, or they can be used as a “targeted drug carrier” in which drug molecules are released in the organ or tissue characterized by local hyperthermia.59 r 2011 American Chemical Society
On the basis of numerous experimental investigations and theoretical studies, it was found that drug release rate depended mostly on the following processes: (i) drug diffusion inside a carrier, (ii) swelling of a carrier matrix, (iii) chemical reactions and related changes in the carrier structure. These issues will not be considered in this study because they have been discussed extensively in the literature, including our earlier paper.10 Excellent reviews are given in the papers of refs 1124. In view of our own experiments, it seems that in certain cases one more variable can have a strong effect on both drug absorption and release from a hydrogel pellet. This includes sorption processes which take place on the hydrogel solid surface—the issue discussed in the present paper. A model of the release proposed in this paper is limited to these hydrogels which do not change their shape. The model was verified using experimental data concerning the release of bovine albumin from thermosensitive chitosan gels.
2. EXPERIMENTAL SECTION Materials and Method. Thermosensitive gels were produced from low-viscous chitosan (Sigma; molecular mass, 329 kD; deacetylation degree DD, 83%) with 2 wt % glutamic acid as a solvent. Gels were prepared by adding 2 g of β-sodium glycerophosphate dissolved in 2 mL of water to 18 g solutions of chitosan glutamate of polymer concentration 2 wt %. Such polymer concentration in the stock solution ensures appropriate mechanical strength and at the same time elasticity of structure after gelation.25 A standard substance to release was Bovine Serum Albumin (BSA). The albumin was supplied to the structure in the amount 0.25, 0.63, and 1.25 g per 1 g of chitosan. The ratio of BSA to polymer was determined in such a way that the albumin could be bound with the polymer and occur freely in hydrogel pores. The protein was added both prior to neutralization and after the neutralization with β Received: November 24, 2010 Accepted: March 29, 2011 Revised: March 18, 2011 Published: March 29, 2011 5866
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Industrial & Engineering Chemistry Research glycerophosphate. The samples had the form of cylinders 30 mm in diameter and 20 mm high. The system could be transformed from the solution into gel at physiological temperature (37 C). To specify the lowest critical solution temperature (LCST), rheological curves representing a change of viscosity with temperature growth were determined. Rheological properties of the prepared solutions were identified in the coneplate system of a Bohlin CVO 120 rotary densitometer. A CP 2/40 measuring system (cone of the angle 2 and diameter 40 mm) was used. The albumin was released to redistilled water to avoid the impact of additional substances such as phosphates in the phosphate buffer on the release kinetics. To eliminate the effect of electrostatic action of protonated amino groups present in a chitosan molecule, we used water of pH ≈ 6 which corresponded to the value of the isoelectric point of glucosaminechitosan pI = 6.3. The release process was carried out in an Erweka apparatus which meets the requirements of Pharmacopeia. In the present experiments a blade system was used, the process of release was carried out up to the volume 900 mL, at the temperature 37 C and mixing rates 50 and 100 rpm. Structures of the gel samples after lyophilization were observed by a FEI Quanta 200F scanning electron microscope.
3. THEORETICAL: A MODEL OF THE RELEASE PROCESS A system composed of a bath with drug concentration CAf is assumed in the model. (Initial concentration CAf0 can take on different values, including zero.) In the bath there are n identical hydrogel pellets with strictly defined shapes. To describe the processes which occur in the system, it is necessary to formulate mass balances of a drug (component A) for (i) fluid contained in the pellet’s pores, (ii) solid phase of the pellet, and (iii) fluid in the bath (the fluid outside the sample). Additionally, three relations describing dislocation kinetics of component A should be given: (1) for the processes which proceed in the fluid inside the pellet’s pores, (2) for sorption process, (3) for the process that takes place in the bath. Moreover, it is necessary to know sorption equilibrium. The following assumptions are made to derive the model equations: the pellet shape and structure do not change during the process the pellet is porous and porosity is constant, independent of the position inside the pellet at the beginning the drug is uniformly distributed in the whole volume diffusion in the pellet pores is described by Fick’s law on the liquidsolid interface there is thermodynamic equilibrium which is described by one of sorption isotherms no chemical and biochemical reactions occur in the system Additionally, it is assumed that the drug, after being released from the structure, is accumulated in the bath and there is no external diffusion resistance (the bath is perfectly mixed). This assumption corresponds to the conditions of laboratory experiments. In future works this assumption will be changed. To formulate balance equations for pellets, differential volume of the pellet dV will be considered (Figure 1). Pellet porosity is denoted by ε and specific surface related to mass unit of the solid body by a [m2/kg S]. Skeleton density of the solid body is Fm [kg S/m3]. S denotes the mass of solid phase of hydrogel.
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Figure 1. (a) Pellet differential volume dV; (b) determination of flux directions.
In the case of the absence of a chemical/biochemical reaction each mass balance equation consists of the three terms: inflow outflow ¼ accumulation
ð1Þ
Particular terms of the mass balance (component A) for fluid in the pellet’s pores described by eq 1 have the following forms. Inflow: component A flows to the control volume by means of diffusion at the rate NA [kg A/(m2 s)] and by desorption Ndes A [kg A/(m2 s)]. It should be kept in mind that diffusion and desorption surfaces are different. Outflow: The outflow from the control volume takes place only by diffusion and is NA þ dNA. Accumulation: Since the outflow is not equal to the inflow, there is accumulation (in this case negative accumulation) which will cause a concentration change in time dCA/dt, so the variation of component A in time will be dV 3 ε 3 (dCA/dt). Substituting particular terms to eq 1 we obtain dV 3 ε 3
dCA dV ¼ dNA ε NAdes 3 dV ð1 εÞ 3 a 3 Fm dx dt
ð2Þ
and after reduction dCA dNA NAdes 3 a 3 Fm ð1 εÞ ¼ ε dt dx
ð3Þ
The balance of component A for solid body contains only two terms: the outflow and (negative) accumulation. After substitution to eq 1, both streams give ð1 εÞ 3 Fm 3 dV
dq ¼ NAdes ð1 εÞFm 3 a 3 dV dt
ð4Þ
and after reduction it becomes dq ¼ NAdes 3 a dt
ð5Þ
where q is mass of component A adsorbed by unit solid mass, kg A/kg S. The balance of component A in the bath contains two terms. This mass of component A which diffuses through the external surface of the pellet will cause an increase in the concentration of component A in the bath: Vf
dCAf ¼ NA jr ¼ R 3 Az dt
ð6Þ
where Vf is the bath volume (m3), CAf is the concentration of component A in the bath (kg A/m3), r is the direction of A transport inside the pellet, R is the pellet edge, and Az is the external surface of all pellets (m2). 5867
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Relevant equations of diffusion and desorption kinetics have the form dCA NA ¼ DAB dx
For the concentration of component A in the liquid,
ð7Þ
¼
kðCA
CA Þ
bqmax 3 CA 1 þ b 3 CA
CA
¼
¼ gnalðqÞ q bðqmax qÞ
ð11Þ
ς
and
Γ¼
Ψ ¼ k3a
DAB 3 t R2
Vf V
ð23, 24Þ
ð25Þ
ð1 εÞ R 2 F ε DAB 3 m
ð1 εÞ R 2 K ¼ k3a ε D
ð26Þ ð27Þ ð28Þ
AB
The set of eqs 1318 can be simplified to the form: dQ ¼ ΩðY Y Þ dτ ! dY z DY D2 Y ¼ þ ΨðY Y Þ dτ ξ Dξ Dξ2 dYf DY ¼ 3Γ Dξ dτ
ð14Þ
Y ¼
ð15Þ τ ¼ 0, ð16Þ ð17Þ ð18Þ
where CA0, q0, and CAf0 are the initial concentrations of component A in the liquid, solid, and bath, respectively. The third condition (18) indicates process symmetry in the pellet. Equations 1315 and relevant boundary conditions can be reduced to the following dimensionless forms.
τ
where: V is the total solid phase volume. R2 ε ðCAo CAf o Þ Ω ¼ k3a qo qfo DAB 1 ε
with the following boundary and initial conditions: t ¼ 0 : CA ¼ CA0 , q ¼ q0 , CAf ¼ CAf 0 r ¼ R CA ¼ CA0 DCA r ¼0 Dr
ð22Þ
Further on, to obtain a dimensionless form of eqs 1318, the following simplexes were defined: volume simplex
ð13Þ dq ¼ kaðCA CA Þ dt dCAf DCA n 3 4 3 π 3 R 2 ¼ DAB dt Dr r¼R Vf
r , R
ð12Þ
where qmax is the maximum sorption ability, (kg A/kg S) and b is the constant (m3/kg A). Equations 3, 5, 6, 7, 8 and one of eqs 9 or 11 describe in detail the release of component A into the bath. A detailed form of the equations is obtained assuming a determined pellet shape. So, for the release of component A from n spherical pellets, each with radius R, and upon substitution of kinetic eqs 7 and 8 to balance eqs 3, 5 and 6, we have ! dCA 2 DCA D2 CA ð1 εÞ Fm ¼ DAB þ 2 þ ðCA CA Þk 3 a r Dr ε dt Dr
ð20, 21Þ
where qf0 is the apparent concentration: qf0 = lang(CAF0 * ). For transport path and process time:
ð10Þ
or the form of a reverse function (gnal lang1) CA
CA CAf 0 CAf CAf0 Yf ¼ CA0 CAf 0 CA0 CAf 0
For the solid concentration in the hydrogel: q qf 0 Q ¼ q0 qf 0
ð9Þ
When the classical Langmuir isotherm is used in the description, it has the form q¼
Y ¼
ð8Þ
where k is the transfer coefficient describing the rate of desorption of component A (m/s), CA* is the concentration of component A in liquid near the surface being in adsorption equilibrium with the solid (kg A/m3), and DAB is the effective diffusion coefficient of component A in the liquid, m2/s. The phase equilibrium equation is written in the form q ¼ langðCA Þ
ð19Þ
and specifically for equilibrium and the bath concentrations,
and NAdes
CA CAf o CAo CAfo
Y ¼
ð29Þ ð30Þ
ð31Þ ξ¼1
gnalðq0 Q Þ CAf 0 CA0 CAf0 Yf ¼ 0
ð32Þ ð33Þ
Y ¼ 1,
Φ ¼ 1,
ξ ¼ 1,
Y ¼ Yf
ð34Þ
ξ ¼ 0,
DY ¼0 Dξ
ð35Þ
The above model was calculated on the basis of an algorithm in Pascal proposed by the authors. To show advantages of the model, examples of simulation are given below. Three dimensionless modules Γ, Ω, and Ψ in the model describe structural, equilibrium, and kinetic aspects of the process. 5868
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Figure 2. The effect of Γ module on the concentration of component A released to the bath.
Figure 4. The effect of sorption ability of the solid skeleton on release process.
Figure 3. The effect of structural parameter Fm(1 ε)/ε on release dynamics.
Figure 5. The influence of desorption rate K on the release process.
The importance of these modules is shown in the diagrams below. All these diagrams illustrate dimensionless bath concentration Yf as a function of process time τ. Figure 2 shows the effect of module Γ at constant values of Ω and Ψ equal to unity. One can see that with an increase of Γ, which denotes growth of the ratio of bath volume to the volume of fluid contained in the pellet, the final concentration of component A in the bath decreases. Figure 3 shows the effect of parameter Ψ on the desorption curves at constant values of Γ = 3 and Ω = 1. According to the definitions of Ω and Ψ modules (eq 27 and 28), the variability of Ψ in relation to constant Ω should be interpreted as the variability of the structural parameters in the term Fm(1 ε)/ε. At Ψ tending to zero (porosity close to unity), component A is located mostly in the pore fluid and the whole process consists of dilution of this component in common liquid volumes of the pellet and bath. Owing to the increase of Ψ which means an increasing role of solid in the process, growing amounts of component A can be desorbed to the bath. As a result, the concentration of A can reach values closer to unity. Figure 4 presents the influence of parameter Ω on release process at constant Ψ. This combination can be interpreted as an analysis of the effect of sorption equilibrium on the process. A simplifying assumption was made that equilibrium was described by a straight line q = ZL 3 CA* . The model predicts that desorption curves are enclosed within limiting lines for Ω f 0 and Ω f ¥. Small values of Ω indicate that the sorption ability of the pellet skeleton increases. During desorption, the supply of component A becomes so high that the process is limited by diffusion in the pore fluid. In an opposite case, when Ω f ¥, the sorption ability is so small that component A is only in the pore fluid and the
process consists in its diffusive dilution like for Ψ f 0 in the previous case (Figure 3). Figure 5 illustrates the effect of kinetic parameter k/DAB (in the dimensional term K, eq 26) on the process. For all other model parameters being constant, one can observe that the curves are contained within the limiting lines for K f 0 and K f ¥. Small K values can be interpreted as small desorption rates. In this case the whole process is controlled by diffusion of component A through liquid in the pores. With an increase of K the desorption rate is also higher, and for K f ¥ the supply of component A from solid surface is so high that the process is controlled by diffusion again. The above discussion shows that the proposed model, particularly in a dimensionless form, allows us to better analyze the release process and to get a deeper insight into this complex process. The simulation calculations presented above show that it is possible to describe the release of component A from hydrogel. It follows from the calculations that irrespective of the assumed parameters, for the initial time the release rate of component A is always very high (burst effect). This is confirmed by experimental results given in the literature.
4. RESULTS Characteristics of Thermosensitive Chitosan Gels. Figure 6 shows the rheological characterization of the tested solutions for chitosan glutamate with and without albumin. Structures of the obtained gels after lyophilization are shown below in Figure 7 and Figure 8. Rheological curves shown in Figure 6 indicate clearly that chitosan gels are gelled at the temperature close to the normal body temperature. After addition of the albumin, the phase transition temperature practically does not change. 5869
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Figure 6. Rheological characterization of chitosan glutamate with and without albumin. Figure 8. Scanning electron micrograph of hydrogel with albumin after lyophilization.
Figure 9. Albumin release profile for a sample containing 0.25 g protein per 1 g chitosan obtained from experimental data and model verification. Figure 7. Scanning electron micrograph of hydrogel without albumin after lyophilization.
and then the mean square error for n measuring points in a given series, rffiffiffiffiffiffiffiffiffiffiffi Ri 2
These gels are characterized by high porosity. After lyophilization pore dimensions are 1030 μm.
5. DISCUSSION: VERIFICATION OF THE MODEL Figures 911 show experimental data which determine release profiles for three different initial amounts of protein in the structure and their description by the model. The profiles were determined on the basis of experimental data from at least three samples taken at the same time. Using the model, initial calculations were made for several combinations of simplexes Ω, Ψ, and K (all of them contain kinetic parameters DAB and k being searched). Each relation Yf = f(τ) obtained in this way is equivalent to calculations for a much bigger number of combinations of the parameters defined by these simplexes (eq 2628). The initial calculations were used to estimate parameters DAB and k. Next, using the model, we searched in the range A ∈ (DAB,k) for a minimum difference between theoretical CAfi and experimental CAfexpi concentrations. For each measuring time ti the instantaneous square error Ri2 ¼ ðCAf i CAf expi Þ2
ð36Þ
R2 ¼
∑i n
100
ð37Þ
were calculated. The release profiles for both rates overlap. The process of albumin release from hydrogels produced from chitosan glutamate for studied concentrations can be described by two constant parameters DAB = 2.0 109 m2/s and k = 1.02 106 m/s. The best known models presented by Peppas and Higuchi well describe the profiles, but the description is limited to a specified concentration. In Higuchi’s model, diffusion coefficients are the function of the concentration of an agent immobilized in the structure and the function of time.26 In Peppas’ model, the constant of the structural system is also the function of an immobilized agent concentration.27 The model proposed in this paper enables the description of kinetics with two constant parameters independent of concentration and time. The internal diffusion coefficient designed in this model is 2 orders of magnitude higher than that presented in the literature (in normal and neoplastic tissues it is 5.8 ( 1.3 1011 and 6.3 ( 1.9 1011 m2/s, respectively28). When albumin is present in the living tissue structure, interactions between them are possible and the release process can slow down significantly, for 5870
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described by Newton eq 8. To use the model, experimental determination of the sorption curve is necessary. Simulation calculations prove that the model predicts the experimentally observed burst effect.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ REFERENCES
Figure 10. Albumin release profile for a sample containing 0.63 g protein per 1 g chitosan obtained from experimental data and model verification.
Figure 11. Albumin release profile for a sample containing 1.25 g protein per 1 g chitosan obtained from experimental data and model verification.
instance due to the forces of electrostatic attraction. In theory, the release could be enhanced if the forces of electrostatic repulsion would dominate. However, it follows from the quoted researches that most probably the process is slow. In our experiments we tried to reduce electrostatic interactions between supplied proteins (albumin) and carrier structure (chitosan) by carrying out the process in redistilled water at pH values close to the isoelectric point of chitosan, that is, when amino groups have no load. In this case, the release process should be determined mainly by albuminwater interactions. This shows that when albumin is released from the hydrogel to water at pH 6, the diffusion is easier due to electrostatic interactions of the albumin in hydrogel pores with water environment into which it is released. The pH value of water is 6 (in the presence of Hþ ion excess) and albumin in the pores (at this pH) has a negative load (isoelectric point 4.64.729), so the electrostatic interactions can occur.
6. CONCLUSIONS A mathematical model to describe the release of active substances from hydrogels is proposed. The idea of the model is to include into the description the sorption processes which proceed on the solid phase surface. The rate of desorption is
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