# Mathematics-aided quantitative analysis of diffusion characteristics of pHEMA sponge hydrogels.

код для вставкиСкачатьASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING Asia-Pac. J. Chem. Eng. 2007; 2: 609–617 Published online 10 August 2007 in Wiley InterScience (www.interscience.wiley.com) DOI:10.1002/apj.062 Research Article Mathematics-aided quantitative analysis of diffusion characteristics of pHEMA sponge hydrogels X. Lou,1 * S. Wang2 and S. Y. Tan1 1 2 Chemical Engineering Department & Nanochemistry Research Institute, Curtin University of Technology, Bentley WA 6102, Australia School of Mathematics and Statistics, University of Western Australia, Crawley WA 6009, Australia Received 4 April 2007; Revised 6 June 2007; Accepted 24 June 2007 ABSTRACT: This study reports the current progress in quantitative analysis of the release characteristics of pHEMA spongy hydrogels using prednisolone 21-hemisuccinate sodium salt as a model drug. Extraction of effective diffusion coefficients of the drug from various pHEMA matrices was made using a novel mathematical model that handles both boundary layer and initial burst effects. Drug loading level and entrapment efficiency were also determined. The computed diffusion coefficients and the drug loading capacity in relation to the device porous structure and drug concentration of the loading solution, as well as the size of device are discussed. Mathematical modelling proves to be a powerful tool not only for establishing and interpreting structure and performance relationships but also for handling experimental ambiguity. 2007 Curtin University of Technology and John Wiley & Sons, Ltd. KEYWORDS: pHEMA; diffusion coefficient; mathematical modelling; drug delivery system INTRODUCTION Quantitative analysis of drug release characteristics and their relation to the properties of a drug carrier are critical to the design and development of an effective controlled drug delivery system which aims to optimise a drug delivery process compatible with the absorption rate of the human body and requires a device to supply and release therapeutic agents to the desired location with a precise therapeutic dose for a prolonged period. The ultimate effect and of great significance to a controlled delivery system include the optimal drug response, minimum side effects, prolonged drug efficacy, and therefore less drug wastage and follow-up care, lower risk and better patient’s compliance. Controlled delivery devices are generally composed of polymers as matrices, which are capable of releasing the drugs continuously over time periods ranging from hours to days or months. In a diffusion-controlled polymeric system, the release of drugs is controlled by the drug diffusing through a complex porous path in the polymer matrix, which is driven by the concentration gradient between the carrier and the drug release medium or body fluid. The effective diffusion coefficient of a drug in a delivery system, critical to its functionality and performance, is largely determined by *Correspondence to: X. Lou, Chemical Engineering Department & Nanochemistry Research Institute, Curtin University of Technology, Bentley WA 6102, Australia. E-mail: X.Lou@curtin.edu.au 2007 Curtin University of Technology and John Wiley & Sons, Ltd. the structural parameters of the system and the interactions between the drugs and the transport matrix. When these properties are unknown or difficult to quantify, extraction of the effective diffusion coefficient of the system becomes a major concern. Various techniques have been developed for the identification of effective diffusivities of matrices with limited pore sizes. Little has been done on such materials as phase-separation spongy pHEMA, which possess a range of pore sizes from several to a hundred micrometers. Phase-separation spongy pHEMA hydrogels are often produced in the presence of a large quantity of water (reportedly above 40–45%) (Peppas and Lustig, 1986a; Chirila et al ., 1993; Lou et al ., 1999; 2000). In comparison with their homogenous and transparent analogue, which has been extensively investigated and well known for their applications as in contact lens and intraocular lens, the spongy pHEMA shows an opaque and softer form because of the pores that are large in both size and number. These materials encourage cell and tissue integration and have demonstrated excellent performance in ophthalmic implants (Chirila et al ., 1998; Crawford et al ., 2002; Hicks et al ., 2006). They have also shown potential in such applications as drug delivery systems and tissue scaffolds (Lou et al ., 2004). In some degree, the drug diffusion mechanisms in the pHEMA sponge matrix become more complicated owing to the contributions from both solute diffusion and convection (Peppas and Lustig, 1986b; 610 X. LOU, S. WANG AND S. Y. TAN Asia-Pacific Journal of Chemical Engineering Lou et al ., 2004). Accurate determination and establishment of drug transportation parameters and their correlations with the porous structure of such hydrogels become important for the optimal design of an applicable device to deliver target drugs in a controlled fashion. Investigation on the transportation phenomena and diffusion coefficient of various molecules in porous pHEMA matrices would provide valuable information for their extended applications in the biomedical field. In this work, we use a newly developed mathematical model for the extraction of effective diffusion coefficients of a model drug, prednisolone 21-hemissucinate sodium salt, in a series of porous pHEMA spongy matrices. Boundary-layer effect and the initial burst phenomenon in a diffusion process were treated mathematically. Devices made of pHEMA sponges containing various pore sizes were fabricated into cylindrical geometry. Drug release profiles of each device were monitored and their effective diffusion coefficients computed. The effect of the porous structure on the estimated effective diffusion coefficient, the drug loading capacity and the drug entrapment efficiency are discussed. MATERIALS AND METHODS Materials HEMA (Bimax, ophthalmic grade) was used as received. The cross-linking agent 1,5-hexadiene-3,4diol (DVG) with a purity of 97% was supplied by Sigma-Aldrich. An aqueous solution of 10 wt% ammonium persulphate (APS) (Ajax Chemicals) was used together with N , N , N , N -tetramethylethylene diamine (TEMED) (Aldrich Chemical Co.) as initiators. Prednisolone 21-hemissucinate sodium salt powder (Sigma Chemical Co., Belgium) was used as a model drug. Deionised water was used for all experiments in the study. Preparation of porous pHEMA discs Nine sets of pHEMA discs were prepared according to the formulas and sizes given in Table 1. The amount of each chemical was adjusted proportionally depending on the total volume of monomer mixture required. To cast the polymer discs, HEMA and water were well mixed in a beaker followed by addition of the cross-linking agent (DVG) and the initiators (APS and TEMED). The solution was then distributed in a 24-well cell culture plate. Volume of the monomer mixture in each well was determined by the height of discs required. Polymerisation was carried out at room temperature for 3 h and then at 50 ◦ C for 24 h. Following the polymerisation, the samples were removed from their moulds and immersed in deionised water for 4 weeks with daily water exchange to remove residual monomers and oligomers. The purified discs were kept in deionised water for further work. Measurements on pHEMA discs Several measurements were carried out on each set of pHEMA discs to determine their equilibrium water content (EWC), water and polymer volume fractions (1 and 2 , respectively) and polymer density of fully hydrated pHEMA (ρp,w ), as well as that of dried pHEMA (ρp,d ). These include Ww,a : the weight of a fully hydrated disc in air, Wd,a : the weight of the same disc in air after dehydration, Ww,w : the weight of a fully hydrated disc in water, and Wd,w : the weight of the same disc in water after dehydration. EWC, as a weight percentage, is given by Eqn (1): EWC = Ww,a − Wd,a × 100% Ww,a (1) Table 1. Formulation for pHEMA discs. Sample ID A B B1 B2 B3 C D D1 E Diametera (mm), height (mm) 14, 14, 14, 14, 7, 14, 14, 14, 14, 7 7 14 3 7 7 7 3 7 HEMA (g) Water (g) DVG (µl) APS (10%) (µl) TEMED (µl) 5 4 4 4 4 3 2 2 1.5 5 6 6 6 6 7 8 8 8.5 50 40 40 40 40 30 20 20 15 100 80 80 80 80 60 40 40 30 100 80 80 80 80 60 40 40 30 a Data were taken on the basis of the moulds used for device preparation and are only indicative of the size variation. 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pac. J. Chem. Eng. 2007; 2: 609–617 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DIFFUSION CHARACTERISTICS OF pHEMA SPONGE HYDROGELS Water and polymer volume fraction (1 and 2 ) are given by Eqn (2), where Vtotal represents the volume of a fully hydrated disc and Vp the volume of polymer, i.e. the same disc after dehydration. Both Vtotal and Vp are calculated on the basis of Archimedes’ buoyancy principle where the density of water, ρw , was taken as 1.0 mg/cm3 (Eqns (3) and (4)): Vp , and φ1 = 1 − φ2 Vtotal Wd ,a − Wd ,w Vp = ρw Ww ,a − Ww ,w Vtotal = ρw φ2 = (2) 30mL glass vial Teflon Disc 15mL Drug Loaded Device (b) r2 r1 (3) (4) Polymer density of a fully hydrated specimen, ρp,w , and that of the same sample when it is dehydrated, i.e. the dry polymer ρp,d , are calculated using Eqns (5) and (6). Wd,a Vtotal Wd ,a = Vp (a) ρp,w = (5) ρp,d (6) Average values of five measurements for each sample were used. Details of the measurements can be found in our previous work (Lou et al ., 1997; 2000). Drug loading Three drug solutions containing 0.5, 1.0 and 1.5 wt% of model drugs were prepared. After the total removal of water in a freeze drier (Dynavac), pHEMA discs were placed in 10 ml stock solution for up to 10 days to facilitate drug loading. The dimensions and weight of the device were recorded after loading of the drugs. Diffusion experiment The diffusion experiment was carried out in a 30-ml glass vial at room temperature. The device pre-loaded with the drug was placed in the glass vial after being gently blotted with a tissue to remove the excessive drug solution, and its height was measured. Deionised water (15 ml) was then added carefully and the vial gently agitated by an orbital shaker (Chiltern Scientific) at a speed of 45 rpm. Five hundred microlitres of solution was withdrawn from the vial at 15, 30, 45, 60 and 90 min, 2, 3, 4, 6 and 9 h, 1, 2 and 3 days to monitor the concentration change. Teflon disks were used on the top of the device to serve as an impermeable barrier (Fig. 1(a)). The absorbance at 247 nm of the sample 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Figure 1. (a) Diffusion experiment set-up. (b) Top view of a cylindrical device of radius r1 in a container of radius r2 . This figure is available in colour online at www.apjChemEng.com. collected from each vial was determined with a UV spectrometer (GBC Scientific Equipment, Australia). The concentration was calculated on the basis of a calibration curve that was obtained from a series of drug solutions with known concentrations. The amount of drug diffused into the solution at various time points was computed to generate the experimental time-release series data (release profiles). Determination of drug loading level and entrapment efficiency Devices that completed the diffusion experiments were taken out from the diffusion glass vial and further extracted with fresh water. The extracted drugs were quantified using the UV/vis spectroscopy and summed with the quantity of drugs already released into the deionised water during the diffusion experiment. This provided the total amount of drug loaded in a specific device. The drug loading level and entrapment efficiency of each device were then determined by the following equations. Drug loading level = Amount of drugs loaded (mg)/ Dry weight of the device (g) Entrapment efficiency = Drugs loaded (mg)/ Drugs used for loading (mg) Extraction of the effective diffusion coefficient of each device The basic model and its analytical solution On the basis of the diffusion experimental set-up, the top view of a device of cylindrical geometry placed in a diffusion container of similar geometry is depicted Asia-Pac. J. Chem. Eng. 2007; 2: 609–617 DOI: 10.1002/apj 611 612 X. LOU, S. WANG AND S. Y. TAN Asia-Pacific Journal of Chemical Engineering in Fig. 1(b) in which r1 is the device radius and r2 the container radius. As the drug release experiments were carried out on an orbital shaker at a relatively low resolution, it is reasonable to assume that release environment is well mixed along the rotational direction and the diffusion process is radial only (more details can be found in Wang and Lou, 2007). That is to say, for a device that is initially loaded with amount of M 0 of drug, the concentration of drug in liquid in the container is a function of r, and the diffusion process is governed by the following diffusion equation in polar coordinates, ∂ 2 C (r, t) 1 ∂C (r, t) ∂C (r, t) =D , + ∂t r ∂r ∂r 2 × 0 < r < r2 , t > 0 (7) ∂C (r2 , t) = 0, t > 0 ∂r 0 M /Vd , 0 < r < r1 C (r, 0) = 0, r1 < r < r2 (8) (9) where D, the effective diffusion coefficient, is independent of r and C (r, t) is the unknown concentration of the drug at r at time t, and Vd denotes the volume of the device. Using the technique of separation of variables, the solution to Eqn (7) satisfying Eqn (8) is given by the following series (Wang and Lou, 2007): C (r, t) = ∞ An J0 n=0 αn r r2 e −Dαn t/r2 2 2 (10) where J0 denotes the zero-order Bessel function of the first kind and An is a coefficient to be determined for each n = 0, 1, 2, . . .. The constants α0 , α1 , . . . , αn , . . . are the non-negative roots of the equation, J1 (αn ) = 0, where J1 is the first-order Bessel’s function of the first kind. The coefficient An is determined on the basis of the initial condition in Eqn (9) and is given as follows: A0 = M 0σ 2 and Vd An = 2M 0 σ J1 (σ αn ) for n = 1, 2, . . . . Vd αn J02 (αn ) (11) where σ = r1 /r2 (details can be found in Wang and Lou, 2007). Since α0 = 0 and J0 (α0 ) = 1, Eqn (10) then becomes C (r, t) = ∞ M 0 σ 2 2M 0 σ J1 (σ αn ) + Vd Vd n=1 αn J02 (αn ) αn r 2 2 e −Dαn t/r2 , 0 < r < r2 (12) × J0 r2 Equation (12) defines an analytical solution to Eqns (7)–(9). 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Finding Mt , total mass released at time t For the above model, the total mass released, Mt , at time t can be determined by integrating C (r, t)rdrdφ over the solution region, 0 ≤ φ ≤ 2π and r1 ≤ r ≤ r2 , where C is given by Eqn (12) Mt = M 0σ 2 4M 0 σ 2 Vc (Vs − Vd ) − Vd Vd ∞ 2 J1 (σ αn ) −Dαn2 t/r 2 2 × e α 2 J 2 (αn ) n=1 n 0 (13) where Vs is the volume of the solution in the container. For t → ∞, the total mass release in infinite time is given as follows: M∞ = M 0σ 2 (Vs − Vd ) Vd Dividing Eqn (13) by M∞ yields ∞ Mt 4 J12 (σ αn ) −Dαn2 t/r 2 2 =1− e M∞ 1 − σ 2 n=1 αn2 J02 (αn ) (14) Equation (14) defines the ratio of the mass released from the device into the liquid in the time interval [0, t] and the total mass release from the device in infinite time. The initial burst An initial burst often appears in a release process mainly because of excessive drug load near the surface of a device or/and some free drugs left on the device surface during the loading process. In this case, the initial release rate is substantially greater than that of the rest of the process. It is also possible that the initial release rate is smaller than the normal rate if a device is overly pre-washed with the intension to remove the free drugs on the device surface, which is a common practice in the lab. In either case, the computed effective diffusion coefficient would differ substantially from the actual one should the process be treated as a single phase. To overcome this problem, the diffusion process might be divided into two phases: the initial burst and a normal diffusion with the following assumptions: D= D0 , D1 , 0 < t < tc , t > tc , where D0 and D1 are effective diffusion coefficients for the burst phase and the normal phase respectively, and tc is the threshold time. All of these parameters are yet to be determined. From the section ‘Basic Model and its Analytical Solution’ one may see that when 0 ≤ t ≤ tc , the concentration C (r, t) is given by Eqn (12) with D = D0 , Asia-Pac. J. Chem. Eng. 2007; 2: 609–617 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DIFFUSION CHARACTERISTICS OF pHEMA SPONGE HYDROGELS while C(r, t) for t > tc can be derived as follows using the same argument as that for Eqn (10): An J0 n=0 αn r r2 e −D1 αn t/r2 , for t > tc 2 2 (15) Considering the continuity condition at tc that C (r, tc− ) = C (r, tc+ ), the coefficient An is given below A0 = M 0σ 2 2M 0 σ J1 (σ αn ) and An = Vd Vd αn J02 (αn ) × e −(D0 −D1 )αn /r2 for n ≥ 1 2 2 By substituting these into Eqn (15), the expression of C (r, t) at t > tc is given by the following equation: ∞ M 0 σ 2 2M 0 σ J1 (σ αn ) + Vd Vd n=1 αn J02 (αn ) αn r 2 2 e −(D0 −D1 )αn t/r2 , t > tc × J0 r2 C (r, t) = (16) Using the same argument as that for Eqn (14), the Mt for the entire process is then expressed as ratio M ∞ ∞ J12 (σ αn ) 4 1 − 2 1 − σ n=1 αn2 J02 (αn ) 2 2 Mt ×e −D0 αn t/r2 , = ∞ J12 (σ αn ) M∞ 4 1 − 2 1 − σ n=1 αn2 J02 (αn ) 2 2 ×e −αn (D1 (t−tc )+D0 tc )/r2 , Drug loading level and drug entrapment efficiency All drug loading and release experiments were carried out in duplicate. There were no significant variations observed between the two sets of experimental results (see Fig. 2 the comparison of the loading data). For 0 ≤ t ≤ tc , Comparison Between Duplicate Experiments (In 0.5%, 1.0% and 1.5% drug solutions repectively) t > tc . (17) RESULTS AND DISCUSSION Physical properties of porous pHEMA discs Drug loading level, mg/g C (r, t) = ∞ are summarised in Table 2. In general, the higher the initial water content, the greater the EWC and water volume fraction. Both polymer volume fraction and wet polymer density display a descending trend with the increase of water content in the initial monomer mixture. However, all dry spongy specimens show a dry polymer density (ρp,d ) in the range of 1.27–1.30. While the observation on dry polymer density demonstrates a full collapse and densification of polymer mass during the dehydrating process at a relative low temperature (50 ◦ C), other results indicate clearly a greater porosity of specimens made in the presence of larger quantity of water which is not unexpected. Quantitative analysis on the porosity of such materials based on micrographic images has been proved difficult, and the results are biased, in particular when the materials are fully hydrated. To examine the porosity effect on the release characteristics of the sponge devices, wet polymer density, which is inversely proportional to the porosity, is used in this paper as a quantitative measure. 350 300 250 200 150 100 50 0 A B C D E A B C D E A B C D E Devices The spongy discs varying in the initial water content in their monomer mixtures show significant differences in their EWC, polymer volume fraction and water volume fraction, as well as in the polymer density, which Exp 1 Exp 2 Figure 2. Drug loading level in various devices. This figure is available in colour online at www.apjChemEng.com. Table 2. Equilibrium water content, volume fraction and density of hydrated pHEMA devices. Sample ID A B C D E HEMA : water ratio EWC (%) ρp,w , Wet polymer (g/cm3 ) ρp,d , Dry polymer (g/cm3 ) Polymer volume fraction (2 ) Water Volume Fraction (1 − 2 ) 50 : 50 40 : 60 30 : 70 20 : 80 15 : 85 51.5 ± 2.1 62.9 ± 0.8 72.4 ± 0.5 76.3 ± 0.5 80.3 ± 2.5 0.547 ± 0.027 0.406 ± 0.009 0.295 ± 0.060 0.250 ± 0.006 0.208 ± 0.027 1.28 ± 0.02 1.27 ± 0.00 1.29 ± 0.00 1.29 ± 0.01 1.32 ± 0.05 0.428 ± 0.021 0.320 ± 0.008 0.228 ± 0.004 0.195 ± 0.003 0.159 ± 0.026 0.572 0.680 0.772 0.805 0.841 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pac. J. Chem. Eng. 2007; 2: 609–617 DOI: 10.1002/apj 613 Asia-Pacific Journal of Chemical Engineering simplicity and convenience, the discussions in this paper are based on one set of experiments only. Conclusions are applicable to all results unless specifically indicated. Figure 3 displays the influence of polymer density and the drug concentration of the loading solutions on the amount of drug loaded into a device. A drastic and near-linear ascending dependence on polymer density is shown in samples C, D and E, while little difference in loading level is seen between A and B. The drug concentration in loading solution has also a more significant effect on samples C, D, E than A and B. It is worthy of mention that sample A and B were respectively prepared in the presence of 40 and 50 wt% of water, which are within (for B) and above (for A) the critical water concentration at which a transparent, and therefore more likely a non-porous pHEMA hydrogel would have been produced. Drug loading and diffusion would be impeded significantly in such a matrix by the non-porous nature, which has been a major limitation for the conventional pHEMA hydrogel being used as a drug carrier for the delivery of large molecules (Oxley et al ., 1993). Since C, D and E were produced in more than 50 wt% of water, they contain large pores ranging from several to hundreds of micrometers (Chirila et al ., 1998; Lou et al ., 2000, 2004). At this water concentration range, the pore size increases further with an increase in initial water concentration, which also leads to a decrease in hydrated polymer density. As the pore voids increase, the drug loading level rises as well, while for A and B, only free volume and very limited pores are available to accommodate the drug molecules (Yasuda et al ., 1969; Yasuda and Lamaze, 1971; Lou et al ., 2004). The slight change of drug loading level between A and B caused by drug concentration might be due to some free drugs left on the device surface. More details on surface free drugs are discussed in a later section. Drug entrapment efficiency, i.e. the ratio of loaded drug and drug used for loading, is important for the application of any useful device. The drug loading onto each device was achieved by a simple diffusion process, which allows the highest loading level and repeated use of the drug solution possible, though the latter is not the intention of this study. Comparison of drug entrapment efficiency in all devices is shown in Fig. 4. The 400 320 240 160 80 0 0.00 0.5 wt% 1.0 wt% 1.5 wt% Sample E Sample D Sample C 0.10 0.20 0.30 Sample B 0.40 Polymwer density, 0.50 Sample A 0.60 g/cm3 Figure 3. Dependence of drug loading level on hydrated polymer density. This figure is available in colour online at www.apjChemEng.com. 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Entrapment efficiency, wt% X. LOU, S. WANG AND S. Y. TAN Drug loading level, mg/g 614 60 50 40 30 20 10 0 A B C Sample ID 0.5 1 D E 1.5 Figure 4. Comparison of drug entrapment efficiency. This figure is available in colour online at www.apjChemEng.com. efficiency of the drug entrapment is generally below 50% for all cases. This is due to the use of excessive volume of loading solutions in the loading experiment. An increase in efficiency is also demonstrated when the device becomes more porous (from A to E) for both 0.5 and 1.0 wt% drug solutions, which is typical for a diffusion-driven loading process. For 1.5 wt% drug solution, a descending trend is shown from device A to E. Some of these devices have turned slightly transparent on their surface after the drug loading, which indicates the occurrence of polymer-drug interaction at such high drug concentration. The interaction first took place in the near-surface region of the discs, which hindered further diffusion of drugs into the matrix. More details will be discussed in a future paper. Computation of effective diffusion coefficients Diffusion processes appear in many areas, including geophysics, engineering, material science and biomedical sciences. In many cases, diffusion coefficients are unknown and need to be identified using experimental or exploratory data. While the diffusion coefficient can be a function of space, time and even concentration of the substance, in practice we normally seek a constant approximation to the diffusion coefficient, i.e. the effective diffusion coefficient that yields a diffusion process matching the observed one optimally in the least-squares sense at the observation time points. Most existing techniques for the extraction of diffusion coefficients are based on either empirical or semi-empirical models for drug delivery mechanisms or on analytical solutions of the diffusion equation (Fu et al ., 1976; Abdekhodaie and Cheng, 1997; Siepmann et al ., 1998; Siepmann and Peppas, 2001; Siepmann and Göpferich, 2001; Grassi and Grassi, 2005). Analytical solutions can be obtained for certain geometries of devices under some assumptions. Some classic results Asia-Pac. J. Chem. Eng. 2007; 2: 609–617 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DIFFUSION CHARACTERISTICS OF pHEMA SPONGE HYDROGELS on analytical solutions for devices of the disc or spherical geometry in ‘well-stirred’ liquids (Crank, 1975) have been widely used in practice. Although these models are popular, they are based on the assumption that the fluid around the device is ‘well stirred’, implying that the concentration is uniform in the liquid. This can never be true in real cases, as the width of a layer can never be zero and only through this layer can the drug diffuse into the external liquid. Another phenomenon in the drug delivery process that has so far not been handled by the existing models is the ‘initial burst’ that can be caused by the experimental error due to the limitation of available techniques in controlling such error. This initial burst phase may severely pollute the true effective diffusion coefficient of a process. Using the mathematical model and the novel technique described in Section on Extraction of the Effective Diffusion Coefficient, the experimental diffusion process was divided into two stages separated by a critical time tc . The effective diffusion coefficient was given as D = D0 when 0 ≤ t ≤ tc and D = D1 when t > tc . D0 and D1 represent, respectively, the diffusion coefficients of the initial burst and a corrected normal diffusion process of the drug. Release profiles of a set of samples are displayed in Fig. 5. The computed diffusion coefficients of each device, D0 and D1 , as well as the least-squares fitting errors and the critical time tc are tabulated in Table 3. In general, D0 is greater than D, indicating a burst effect of excessive drugs on or near the device surface. Four out of thirty cases in this study show a smaller D0 , presumably due to the overly blotted device surface when removal of surface drug solution was intended (Section on Diffusion Experiment). The critical time tc is in the range 900–3600 s. The high end is seemingly larger than expected, which is probably due to the very frequent collection of samples in the first few hours. Figure 5. Time-release series curves based on the new mathematical model and technique. This figure is available in colour online at www.apjChemEng.com. Diffusion coefficient and drug loading level in relation to the drug concentration and device porosity Displayed in Fig. 6(a) and (b) are the diffusion coefficient and drug loading level of each device in relation to the drug concentration in the loading solution and the device porosity. An indicative ascending trend of drug loading level is observed with the increase in both the drug concentration and device porosity (inversely proportional to the polymer density). A similar trend is also demonstrated in the diffusion coefficient, although the dependence of diffusion coefficient on the drug concentration is less noticeable than that of drug loading level. In terms of the dependence on the polymer density, a much smaller slope is apparent in the range Table 3. Computed diffusion coefficient, least-squares fitting error and the critical time. Drug conc. (wt %) 0.50 1.00 1.50 Sample ID D0 (×10−6 cm2 /s) D1 (×10−6 cm2 /s) Squared error (×10−2 ) tc (s) A B C D E A B C D E A B C D E 12.4 10.4 6.97 25.7 24.8 32.0 38.4 18.6 33.5 68.9 51.5 19.5 35.2 101 187 2.18 3.25 1.99 1.28 1.52 4.14 3.43 3.73 7.90 16.1 3.80 3.01 3.80 9.26 49.3 4.71 0.84 3.82 1.85 1.69 1.52 0.93 1.00 0.21 0.88 3.15 1.36 0.39 0.80 0.58 3600 900 1800 900 900 3600 3600 1800 3600 3600 1800 2700 900 2700 1800 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pac. J. Chem. Eng. 2007; 2: 609–617 DOI: 10.1002/apj 615 616 X. LOU, S. WANG AND S. Y. TAN Asia-Pacific Journal of Chemical Engineering Table 4. Diffusion coefficient, D1 × 10−6 cm2 /s, for devices of various sizes. D1 (×10−6 cm2 /s) Sample ID B B1 B2 B3 D D1 a a Diameter, height (mm, mm) 0.5 wt% drug 1.0 wt% drug 14.0, 7.0 14.0, 14.0 14.0, 3.0 7.0, 7.0 14.0, 14.0 14.0, 3.0 2.26 2.76 2.96 – 10.4 12.1 4.07 2.62 2.87 1.61 20.1 18.2 Indicative data based on the moulds used for device preparation. CONCLUSIONS Figure 6. Drug loading level (a) and effective diffusion coefficient (b) in relation to polymer density of a device. 0.35–0.55 for both the diffusion coefficient and drug loading level. Again this is due to the non-porous nature of homogenous pHEMA hydrogels. It is worthy of mention that the change in drug loading level in the porous range, ρp,w = 0.1–0.3, is more significant than that of diffusion coefficient. This is a very useful piece of information in the optimal design of drug delivery systems for which a high loading capacity and a slow release are often desirable. Diffusion coefficient vs size of the device A set of devices of various heights and diameters were prepared and analysed on their diffusion coefficients. Table 4 shows the values of D1 for these devices. Change in device height has little influence on D1 for both the less porous device B and more porous D. There is a small reduction in D1 value shown in the device that has a reduced diameter; however no sufficient data is available to make it conclusive at this stage. 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Materials such as phase-separation spongy pHEMA hydrogels contain much larger pores than their transparent counter materials. Mathematical modelling proves to be a powerful tool not only for the extraction of drug diffusion coefficients from these materials, but also for reducing experimental errors that are not easy to handle in the common practice. The pHEMA hydrogels reported in this study vary significantly from their conventional non-porous analogue in the porous structure, and therefore lead to different performances in applications such as a drug delivery system. The results have shown a great potential for them to be used to deliver large molecules. The geometry parameter of the devices has shown little influence on the diffusion characteristic of these matrices. However, the drug concentration of the loading solution and device porosity affect drug diffusion in various ways. The performance of these matrices can be further improved by altering the porosity of the surface layer of each device, which is under study by our group. 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