A novel mathematical model quantifying drug release from lipid implants

Journal of Controlled Release 128 (2008) 233–240

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Journal of Controlled Release j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j c o n r e l

A novel mathematical model quantifying drug release from lipid implants F. Siepmann a, S. Herrmann b, G. Winter b, J. Siepmann a,⁎ a b

College of Pharmacy, JE 2491, University of Lille, 3 Rue du Professeur Laguesse, 59006 Lille, France Department of Pharmacy, Ludwig-Maximilians-University Munich, Butenandtstr.5, 81377 Munich, Germany

A R T I C L E

I N F O

Article history: Received 9 January 2008 Accepted 11 March 2008 Available online 18 March 2008 Keywords: Mathematical modeling Drug release mechanism Implant Protein Diffusion

A B S T R A C T A novel mathematical theory is presented allowing for a quantitative description of the various mass transport processes involved in the control of drug (in particular protein) release from lipid implants. Importantly, the model takes into account the simultaneous diffusion of multiple compounds, including the drug and water-soluble excipients, such as release modifiers (e.g., PEG) and drug stabilizers (e.g., HP-β-CD). Also dynamic changes of the implant structure resulting from drug and excipient leaching into the release medium are considered, resulting in a significant time- and position-dependent mobility of the diffusing species within the systems. Furthermore, the limited solubility of the drug and/or excipients under the given conditions in water-filled channels within the implants can be considered. This includes for instance the limited solubility of IFN-α in the presence of dissolved PEG. Importantly, good agreement between the novel theory and experimentally determined protein, PEG and HP-β-CD release kinetics from tristearin-based implants was obtained. In this particular case it could be shown that the precipitation effect of PEG on IFN-α in water-filled pores plays a crucial role for the overall control of protein release. Neglecting this phenomenon and assuming constant apparent diffusion coefficients, significant deviations between theory and experiment are observed. Importantly, the novel mathematical theory also allows for a quantitative prediction of the effects of different formulation and processing parameters on the resulting drug release kinetics. For instance the importance of the initial PEG content of the systems for the resulting IFN-α release kinetics could be successfully predicted. Interestingly, independent experiments confirmed the theoretical predictions and, thus, proved the validity and suitability of the mathematical theory. © 2008 Elsevier B.V. All rights reserved.

1. Introduction The adequate delivery of protein-based drugs to their sites of action in the human body can present a major obstacle during the development of innovative pharmaco therapies. Even if the drug is known to be efficient and able to help the patient, it must first reach its target site to become active. As oral administration of protein-based drugs is not (yet) feasible, frequent parenteral administrations are nowadays generally required (e.g., as in the case of diabetes treatment with insulin). But frequent injections are inconvenient for the patient, often resulting in non-optimal compliance. To overcome these restrictions, two general strategies can be followed: (i) the drug can be administered using parenteral controlled drug delivery systems, or (ii) alternative administration routes (e.g., pulmonary or nasal) may be used. Recently, an inhalable insulin formulation has been marketed under the trade name Exubera [1]. But the development of such advanced drug delivery systems is complex, and major hurdles need to be overcome/addressed. This includes for instance the assurance that reliable and reproducible drug doses are able to reach the deep lungs, the inter- and intra-subject variability, the importance of respiratory ⁎ Corresponding author. Tel.: +33 3 20964708; fax: +33 3 20964942. E-mail address: [email protected] (J. Siepmann). 0168-3659/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jconrel.2008.03.009

diseases/smoking habits as well as considerable development costs. Recently, Pfizer announced that Exubera will be withdrawn from the market in January 2008, because too few patients use this product. When opting for parenteral controlled drug delivery systems for the administration of protein-based drugs, the injection/implantation frequency can be significantly reduced and the drug be effectively protected against physical and chemical threats in the human body. In addition, optimized release patterns can lead to improved therapeutic effects. Unfortunately, the “standard” polymer used to prepare parenteral controlled drug delivery systems poly(lactic-co-glycolic acid) (PLGA) (serving as a biodegradable matrix former) [2,3] is generally not the optimal choice if proteins are to be incorporated. This is because PLGA (being a polyester) is degraded into smaller chain acids (and alcohols) upon contact with water, which can lead to a significant drop in the micro-environmental pH [4,5]. Protein-based drugs are likely to loose their biological activity under these circumstances [6,7]. Importantly, lipids offer an interesting alternative as matrix formers in parenteral controlled drug delivery systems [8–11], avoiding the creation of acidic micro-climates. Of course, acid-stable drugs can as well be incorporated in such lipid matrices and effectively be protected against degradation in the human body. Different types of lipids have been proposed in the past for the preparation of implants as advanced drug delivery systems and

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various preparation techniques have been described [8,12,13]. Importantly, the in vitro and in vivo efficiency of numerous devices has been successfully demonstrated [14–17], and it can be expected that this type of advanced drug delivery system will steadily gain in importance in the future. However, yet little is known on the underlying mass transport mechanisms from this type of devices [18–20]. Different mass transport phenomena are likely to be involved, such as the diffusion of water, drug and water-soluble excipients, the dissolution of water-soluble compounds, the creation of water-filled pores within the lipid implants, as well as time- and position-dependent changes in the matrix structure (in particular porosity) upon exposure to the release medium. So far, the relative importance of these processes is unknown and no mathematical model is available allowing for a better understanding of the underlying drug release mechanisms. Only simplified mathematical theories have been reported up to date, often neglecting the simultaneous diffusion of several species and dynamic changes in the matrix structure of the systems. Generally, timeand position-independent diffusion coefficients are assumed, neglecting changes in the implant porosity due to drug and excipient leaching. For instance, a simple mathematical theory based on Fick's second law was recently fitted to sets of experimentally determined interferon α-2a (IFN-α) release kinetics from tristearin-based, polyethylene glycol (PEG)containing implants [20]. The model considers only drug diffusion with constant diffusivities. Importantly, significant deviations between theory and experiment were observed, indicating that not all of the involved mass transport phenomena were adequately taken into account. The major aim of this study was to develop a novel mathematical theory being more comprehensive than the existing ones and able to quantitatively describe drug release from lipid implants. In particular, multi-component mass transport, dynamic changes in the implant structure as well as potentially limited drug solubility were to be considered. The major two benefits of this model are intended to be: (i) the possibility to better understand how drug (in particular protein) release can be controlled from lipid implants, and (ii) the possibility to quantitatively predict the effects of different formulation and processing parameters on the resulting drug release kinetics (and, thus, to allow for a facilitated optimization of this type of advanced drug delivery systems). 2. Materials Rh-interferon α-2a (IFN-α, Mw = 19.237 Da, stock solution: 1.695 mg/mL in 25 mM acetate buffer pH 5.0, containing 120 mM sodium chloride) was provided as a gift by Roche Diagnostics (Penzberg, Germany). To increase protein stability, IFN-α was colyophilised with hydroxypropyl-β-cyclodextrin (HP-β-CD; Merck, Darmstadt, Germany) after pH adjustment to 4.2 with acetic acid. Tristearin (Dynasan 118; Sasol, Witten, Germany), polyethylene glycol 6000 (PEG; Clariant, Gendorf, Germany), and all other materials (VWR, Darmstadt, Germany) were of high purity grade. 3. Experimental methods 3.1. Implant preparation IFN-α-loaded, tristearin-based implants were prepared by compression as previously described in detail [9,19]. Briefly, the IFN-α/HPβ-CD lyophilisate was blended with different amounts of PEG and tristearin, filled into the cylindrical matrix of a compaction tool (diameter: 5 mm), and compressed using a hydraulic press (Maassen, Eningen, Germany). 3.2. Porosity measurements The porosity of the implants (before and upon exposure to the release medium) was determined using a helium pycnometer

(accuPyc 1330, Micromeritics, Moenchengladbach, Germany). Prior to the measurements, the implants were vacuum dried (VO 200 vacuum chamber, Memmert, Schwabach, Germany) to constant weight. Based on the obtained true implant volume (Vtrue) and geometric implant volume (Vgeometric) [determined by measuring the cylinder height and diameter with a digital calliper rule (Digimatic CD-15CD, Mitutoyo, Oberndorf, Germany)], the porosity (ɛ, in %) could be calculated as follows: e¼

 1

Vtrue Vgeometric

  100

ð1Þ

3.3. Protein precipitation measurements In order to quantify the ability of PEG to precipitate IFN-α, differently concentrated PEG solutions [(2–40% (w/v), in phosphate buffer pH 7.4] were added to a 4.9 mg/mL IFN-α solution. The systems were equilibrated for 2 h at 37 °C (40 rpm, Certomat IS; Braun Biotech, Goettingen, Germany) and potential protein precipitates subsequently separated by centrifugation at 5000 rpm (5 min, 4K15 laboratory centrifuge; Sigma, Osterode, Germany). The IFN-α concentrations in the supernatants were determined by reversed phase HPLC, which was performed using a Jupiter 5u C18 300 Å´ 250 × 4.60 mm column (Phenomenex, Aschaffenburg, Germany). The mobile phase consisted of a 49:51 (v/v) acetonitrile/ultra purified water mixture which was acidified with 0.1% (v/v) trifluoroacetic acid. The flow rate was adjusted to 1 mL/min; UV detection (UV 1000; Thermo Electron Cooperation, Dreieich, Germany) was performed at λ = 215 nm. 3.4. In vitro release studies The protein-loaded implants were placed into TopPac vials (cycloolefin copolymer vials; Schott, Mainz, Germany) filled with 2.0 mL isotonic 0.01 M phosphate buffer pH 7.4 containing 0.05% (w/v) sodium azide. The vials were placed in a horizontal shaker (40 rpm, 37 °C, Certomat IS). At predetermined time points, the release medium was completely exchanged and the amounts of IFN-α, PEG, and HP-βCD in the phosphate buffer were determined as previously described in detail [19]. Briefly, the protein concentration was measured by sizeexclusion chromatography, the PEG concentration was determined using a modification of the ammonium ferrothiocyanate assay described by Nag et al. [21], and the HP-β-CD content was determined using a modification of the method proposed by Basappa et al. [22]. 4. Results and discussion 4.1. Model development Recently obtained experimental results and a simple mathematical model (considering protein diffusion only, with constant diffusion coefficients) revealed that the control of drug release from lipid implants can be complex [19,20]. Consequently, device optimization is not straightforward and might require time- and cost-intensive series of experimental studies. As an example, Fig. 1 shows the fitting of Fick's second law of diffusion (taking into account radial and axial mass transfer in cylinders) from tristearin-based, IFN-α-loaded implants containing 10% PEG. The symbols represent the experimental results, the curve the fitted theory considering time- and positionindependent diffusivity. Clearly, systematic and significant deviations can be observed: Drug release is overestimated at early time points and underestimated at late time points. To overcome these restrictions a novel mathematical theory is proposed, aiming at: (i) a better understanding of the underlying mass transport phenomena, and (ii) the possibility to quantitatively predict the effects of different formulation and processing parameters on the

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Simultaneous diffusion of IFN-α, PEG, and HP-β-CD is described using Fick's second law considering the cylindrical geometry of the implants and taking into account axial and radial mass transfer with time- and position-dependent diffusion coefficients [23]:        Ack 1 A Ac A Dk Ack A Ac rDk k þ rDk k ¼ þ r Ar Ah r Ah Az At Ar Az

ð2Þ

Here, ck and Dk are the concentration and diffusion coefficient of the diffusing species (k = 1: drug; k = 2: PEG; k = 3: HP-β-CD), respectively; t represents time; and r, z and θ denote the radial, axial and angular coordinate, respectively (Fig. 2a). As there is no concentration gradient of any of the three components with respect to θ (Fig. 2b), this equation can be transformed into:     Ack A Ac D Ac A Ac Dk k þ k k þ Dk k ¼ Ar At Ar r Ar Az Az Fig. 1. Experiment (symbols) and simple theory (curve): IFN-α release into phosphate buffer pH 7.4 from tristearin-based implants containing 10% PEG (initial IFN-α/HP-β-CD content: 10%) (adapted from reference 20, with permission).

resulting protein release kinetics. This new model is based on previously presented [19,20] and new experimental results, and takes into account the following series of physical processes occurring during drug release well as the given initial and boundary conditions:

ð3Þ

To minimize computation time, also the symmetry plane at z = 0 (Fig. 2a) is taken into account. Thus, it is sufficient to describe the changes in the IFN-α, PEG, and HP-β-CD concentrations within the 2dimensional rectangle highlighted in Fig. 2b to be able to calculate the mass transport phenomena in the entire cylinder: Upon rotation around the z-axis the upper half of the implant is described, and

• The lipid implants are cylindrical in shape. • IFN-α, PEG, and HP-β-CD are homogeneously distributed throughout the devices before exposure to the release medium (at t = 0). • The lipid implants are slightly porous at t = 0 (as determined using a helium pycnometer). • Upon contact with the release medium, water diffuses into the implants and IFN-α, PEG, and HP-β-CD simultaneously diffuse out of the systems (due to concentration gradients). • The porosity of the lipid matrices steadily increases due to IFN-α, PEG, and HP-β-CD leaching (as confirmed with a helium pycnometer). • The increase in matrix porosity is time- and position-dependent (first surface-near regions become more and more porous, later on also the core of the implants). • The diffusion coefficients of IFN-α, PEG, and HP-β-CD within the implants are directly related to the matrix porosity and are, thus, also position- and time-dependent (the species become more and more mobile with increasing porosity of the implant). • Water penetration into the matrix is much faster than the subsequent IFN-α, PEG, and HP-β-CD diffusion. • Diffusion occurs through water-filled pores; crystalline lipid plates are impermeable for IFN-α, PEG, and HP-β-CD. • The solubility of IFN-α in the water-filled matrix pores strongly depends on the PEG concentration in these channels (the solubility of the protein strongly decreases with increasing PEG concentration, as experimentally determined). At each time point and at each position, the actual PEG concentration and matrix porosity are calculated and used to determine the actual IFN-α solubility. • Non-dissolved IFN-α is not available for diffusion. • Diffusional mass transport occurs in radial as well as axial direction within the cylinders. • Swelling or dissolution of the matrices is negligible within the observation period (visual observation). • Perfect sink conditions are maintained throughout the experiments. In the following, the mathematical equations the theory is based on are given and the principles of the numerical analysis used to solve the obtained set of Partial Differential Equations are briefly described. For further details on the latter aspect, the reader is referred to the literature [23].

Fig. 2. Schematic presentation of a cylindrical lipid implant for mathematical analysis: (a) symmetry plane at z = 0, and (b) rotational symmetry with respect to the angle θ.

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due to the symmetry at the z = 0 plane, the whole cylinder can be considered. The initial condition is based on the fact that all three diffusing components are uniformly distributed throughout the implants before exposure to the release medium (at t = 0). Thus, the IFN-α, PEG, and HP-β-CD concentrations at any position are equal to the respective initial concentrations (c1 initial, c2 initial, and c3 initial, respectively): t¼0

c1 ¼ c1

initial

0 VrVR

0V z VZ

ð4Þ

t¼0

c2 ¼ c2

initial

0 VrVR

0V z VZ

ð5Þ

t¼0

c3 ¼ c3

initial

0 VrVR

0V z VZ

ð6Þ

where R denotes the radius and Z the half-height of the implant. As perfect sink conditions are provided for all diffusing species throughout the experiments, the IFN-α, PEG, and HP-β-CD concentrations at the surface of the implant are considered to be equal to zero upon exposure to the release medium: tN0

c1 ¼ 0

0VrVR

z¼Z

ð7Þ

tN0

c1 ¼ 0

0VzVZ

r¼R

ð8Þ

tN0

c2 ¼ 0

0VrVR

z¼Z

ð9Þ

tN0

c2 ¼ 0

0VzVZ

r¼R

ð10Þ

tN0

c3 ¼ 0

0VrVR

z¼Z

ð11Þ

tN0

c3 ¼ 0

0VzVZ

r¼R

ð12Þ

Due to the symmetries at z = 0 and r = 0 (Fig. 2a), there are no concentration gradients at z = 0 for 0 ≤ r ≤ R and at r = 0 for 0 ≤ z ≤ Z for any of the diffusing components: tN0

Ac1 ¼0 Az

0VrVR

z¼0

ð13Þ

tN0

Ac1 ¼0 Ar

0VzVZ

r¼0

ð14Þ

tN0

Ac2 ¼0 Az

0VrVR

z¼0

ð15Þ

tN0

Ac2 ¼0 Ar

0VzVZ

r¼0

ð16Þ

tN0

Ac3 ¼0 Az

0VrVR

z¼0

ð17Þ

tN0

Ac3 ¼0 Ar

0VzVZ

r¼0

ð18Þ

Furthermore, the limited aqueous solubility of IFN-α in the presence of PEG is taken into account. As it can be seen in Fig. 3, the soluble fraction of the protein significantly decreases in the presence of only a few percentages of PEG (Please note that the soluble fraction only corresponds to the solubility for PEG concentrations ≥4%. This means that the IFN-α solubility is higher than the values indicated below 4% PEG and equal to the IFN-α solubility at ≥4% PEG). Within the water-filled pores of the lipid implants it can be expected that much higher PEG concentrations are attained during drug release. Thus, this precipitation effect is likely to play a crucial role in the present systems. Importantly, non-dissolved IFN-α is not available for diffusion. The increase in IFN-α, PEG, and HP-β-CD mobility within the lipid implants due to drug and excipient leaching is considered based on the following relationship: Dk ðr; z; t Þ ¼

Dk

 eðr; z; t Þ 100

crit

ð19Þ

Fig. 3. Soluble fraction of IFN-α in phosphate buffer pH 7.4 at 37 °C as a function of the PEG content (average +/− SD; n = 3) (please note that only values at ≥4% PEG correspond to the solubility of the protein under these conditions) (adapted from Ref. [20], with permission).

where Dk crit represents a critical diffusion coefficient, being characteristic for the diffusing species, and ɛ denotes the implant porosity in percent. Thus, the consequences of the dynamic changes in the implant structure during drug release on the conditions for the multicomponent diffusion are quantitatively taken into account. [Remark: Please note that diffusion in confined regions can differ from diffusion in bulk fluids. Here, a simplified linear relationship is assumed between the porosity of the system and the diffusivity of the respective species. Thus, the Dk crit values do not necessarily represent the diffusion coefficients in pure release medium.] 4.2. Numerical analysis As the matrix porosity increases upon IFN-α, PEG, and HP-β-CD leaching in a time- and position-dependent manner, there is no analytical solution for the described set of Partial Differential Equations (Eqs. (3–19)). Thus, a numerical technique, based on finite differences is applied. In the following, only a brief description of this method is given, the reader is referred to the literature for any details [23–26]. To be able to calculate the changes in the IFN-α, PEG, and HP-β-CD concentrations within the 2-dimensional rectangle highlighted in Fig. 2b, the implant radius, R, and half-height, Z, are divided into I and J space intervals, Δr and Δz, respectively (Fig. 4a). This generates a grid of (I + 1) × (J + 1) grid points. The time is divided into g time intervals Δt (for most of the simulations I = J = 50 and g = 500,000 were chosen). At t = 0 (before exposure to the release medium), the IFN-α, PEG, and HP-β-CD concentrations are known from the initial conditions (Eqs. (4–6)). Importantly, using Eqs. (3) and (7–19), the concentration profiles of the three diffusing species at a new time step (t = t0 + Δt) can be calculated, when the concentration profiles are known at the previous time step (t = t0): The concentration of a particular species at a certain inner grid point (i × Δr, j × Δz) at the new time step (t = t0 + Δt) is calculated from its concentrations at the same grid point (i × Δr, j × Δz) and the four direct neighbors [(i − 1) × Δr, j × Δz; i × Δr, (j − 1) × Δz; i × Δr, (j + 1) × Δz; (i + 1) × Δr, j × Δz] at the previous time step (t = t0) (Fig. 4b). The respective concentrations at the outer grid points (i = 0 v i = I v j = 0 v j = J) at a new time step (t = t0 + Δt) are calculated using the boundary conditions (Eqs. (7–18)). As all initial concentrations are known, the concentration profiles at t = 0 + Δt, t = 0 + 2Δt, t = 0 + 3Δt, ..., t = 0 + gΔt can be calculated sequentially.

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addition, the knowledge of the time- and position-dependent implant porosity allows accounting for the time- and position-dependent increase in the diffusion coefficient of IFN-α, PEG, and HP-β-CD using Eq. (19). Please note that changes in the matrix porosity have not been considered to alter the compounds' concentrations, except for the calculation of the time- and position-dependent protein solubility. For the implementation of the mathematical model the programming language C++ was used (Borland C++ Builder V.6.0). 4.3. Modeling PEG and HP-β-CD release Fig. 5a and b shows the experimentally determined and theoretically calculated PEG and HP-β-CD release kinetics from implants initially containing 10% IFN-α/HP-β-CD and 10% PEG, respectively. Clearly, good agreement was obtained in both cases, indicating that the release of both excipients is primarily controlled by diffusion. Based on these calculations, the critical diffusion coefficients of PEG and HP-β-CD could be determined to be equal to 1.7 × 10− 7 cm2/s and 1.1 × 10− 7 cm2/s, respectively. Please note that due to the time- and position-dependent changes in the matrix structure upon drug and excipient release, the mobility of these species within the implants is not constant. Importantly, also the resulting PEG and HP-β-CD concentrations in the water-filled channels within the systems can

Fig. 4. Schemes illustrating the principles of the applied numerical analysis: (a) definition of the grid dividing the rectangle highlighted in Fig. 2b into small pixels, (b) calculation of the drug, PEG and HP-β-CD concentrations of a particular pixel at a new time point based on the drug, PEG and HP-β-CD concentrations of the same pixel and its four direct neighbors at the previous time point.

Furthermore, the total amounts of IFN-α, PEG, and HP-β-CD within the implants are calculated at each time step (by integrating the respective concentrations with respect to r, z and θ). Based on the spatial protein and excipient concentration profiles at all time points and the experimentally determined implant porosity at t = 0 (before exposure to the release medium), the matrix porosity can be calculated at any time point and any position as follows: It is assumed that the matrix porosity is uniform at t = 0. The increase in porosity at a particular position at a particular time point is calculated based on the knowledge of the IFN-α, PEG, and HP-β-CD loss at this position at this time point. This is a crucial information because it provides the basis for the determination of the protein solubility in a time- and positiondependent manner, taking into account the experimentally determined limited solubility of IFN-α in the presence of PEG (Fig. 3). In

Fig. 5. Theory and experiment: (a) PEG release, and (b) HP-β-CD release into phosphate buffer pH 7.4 from tristearin-based implants initially containing 10% IFN-α/HP-β-CD and 10% PEG [symbols: experimental results (average +/− SD; n = 3); curves: theory].

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be calculated as a function of time and position. Fig. 6a–d shows as an example the resulting excipient concentration profiles within the lipid implants after 1 and 3 days exposure to phosphate buffer pH 7.4, respectively. Clearly, after only 1 day the inner implant cores still contain the initial amounts of PEG and HP-β-CD (Fig. 6a and b). In contrast, steep concentration gradients exist in the surface near regions, being the driving forces for further excipient diffusion. After 3 days exposure, also the inner matrix core becomes more and more depleted from PEG and HP-β-CD (Fig. 6c and d). Most importantly, the PEG concentrations are still significant and in most parts of the implants well above the values causing IFN-α precipitation (Fig. 3) (please note that the concentrations shown in Fig. 6 refer to the entire system, and not only to water-filled pores). Thus, the release of the protein is highly restricted due to the presence of dissolved PEG within the water-filled channels in the lipid implants. Based on the calculated PEG concentrations and on the time- and positiondependent matrix porosity, the resulting IFN-α solubility can be determined at any time point and at any position in the cylinder. Fig. 7a and b shows for example the protein solubility profiles within the implants after 1 and 3 days exposure to phosphate buffer pH 7.4. Very clearly, only in the surface near regions which contain only small amounts of PEG after 1 day, significant amounts of protein can be dissolved and become available for diffusion. In the rest of the implant, IFN-α solubility is very low (Fig. 7a). With time, the PEG-poor regions increase in size, resulting in significantly increasing protein solubility within the implant (Fig. 7b), and allowing for further IFN-α release (Fig. 1, symbols). Thus, this study confirms the role of PEG as a precipitation agent rather than that of a pore former in IFN-α-loaded implants.

4.4. Modeling drug release Knowing the time- and position-dependent changes in the IFN-α solubility and the total (dissolved and non-dissolved) protein concentrations at any time point and at any position in the implants, the resulting protein concentration gradients (considering only dissolved IFN-α, being available for diffusion) can be calculated. Based on this information and the knowledge of the time- and position-dependent implant porosity, the resulting protein release rate can be determined. Fig. 8 shows a comparison of the experimentally measured (filled squares) and theoretically calculated (solid curve) release of IFN-α from tristearin-based implants initially containing 10% IFN-α/HP-β-CD and 10% PEG in phosphate buffer pH 7.4. Clearly, good agreement between the fitted theory and the experimental results was obtained. Importantly, deviations are small and arbitrary. This is in contrast to the previously presented [20] simple mathematical approach illustrated in Fig. 1. In particular, the linear portion of the release profile is much better described with the complex mathematical theory than with the simple model. This can be attributed to: (i) the consideration of limited protein solubility as a function of the PEG concentration within the water-filled pores of the implant, and (ii) the consideration of an increased IFN-α mobility with time due to drug and excipient leaching into the release medium, compensating the monotonic increase in the length of the diffusion pathways (which leads to monotonically decreasing protein release rates in the case of the simple mathematical model). Based on these calculations, the critical diffusion coefficient of IFN-α in these implants could be determined to be equal to 5.9 × 10−8 cm2/s. This value is much lower than that of PEG and HP-β-CD, which can be attributed to the

Fig. 6. Calculated concentration profiles of: (a) PEG after 1 day, (b) HP-β-CD after 1 day, (c) PEG after 3 days, and (d) HP-β-CD after 3 days within tristearin-based implants (initially containing 10% IFN-α/HP-β-CD and 10% PEG).

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significantly higher molecular weight of this protein compared to the two excipients. However, it has to be pointed out that obtaining good agreement between theory and experiment is not yet a real proof for the validity of a mathematical model, when the theory is fitted to the experimental results. A much more reliable indication for the validity of a model is good agreement obtained when comparing theoretical predictions and independent experimental results. Once all system-specific parameters were known, in particular the respective critical diffusion coefficients of IFN-α, PEG, and HP-β-CD, the proposed mathematical theory was used to theoretically predict the protein release kinetics from implants containing initially 20% PEG. The dotted curve in Fig. 8 shows the calculated IFN-α release profile from these systems in phosphate buffer pH 7.4, the open squares the respective independent experimental results. Clearly, good agreement between theory and experiment was obtained, demonstrating the validity of the novel complex mathematical theory for this type of advanced protein delivery systems. The obtained good agreement also indicates that it is highly Fig. 8. Theory and experiment: IFN-α release into phosphate buffer pH 7.4 from tristearin-based implants: Experiment (filled squares) and fitted theory (solid curve) in the case of implants initially containing 10% IFN-α/HP-β-CD and 10% PEG, and theoretical prediction (dotted curve) and independent experimental verification (open squares) in the case of implants initially containing 10% IFN-α/HP-β-CD and 20% PEG (experimental results: average +/− SD; n = 3).

unlikely that the model neglects major mass transport phenomena involved in the control of IFN-α release from the lipid implants. 5. Conclusions The proposed mathematical model considers all relevant mass transport phenomena involved in the control of drug release from lipid implants. Its two major applications are: (i) The elucidation of the underlying drug release mechanisms. For instance, it could be shown that IFN-α precipitation due to the presence of dissolved PEG within water-filled pores plays a significant role for the control of drug release in the investigated systems. (ii) The quantitative prediction of the effects of different formulation and processing parameters (e.g., PEG content, implant height and radius) on the resulting drug release kinetics. Thus, the number of time- and cost-intensive experiments required during product development/optimization can be significantly reduced. Acknowledgements The authors are grateful for the support of this study by the Center of Bavarian-French-Cooperation. References

Fig. 7. Position- and time-dependent solubility of the drug within the lipid implant: (a) solubility profile after 1 day exposure to phosphate buffer pH 7.4, (b) solubility profile after 3 days exposure to phosphate buffer pH 7.4 (please note that the view point is different from that in Fig. 6).

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