Anomalous Release Kinetics of Prodigiosin from Poly-N-Isopropyl-Acrylamid based Hydrogels for The Treatment of Triple Negative Breast Cancer

This paper presents the anomalous release kinetics of a cancer drug (prodigiosin) frompoly-n-isopropyl-acrylamide (P(NIPA))-based gels. The release exponents, n, which correspond to the drug release mechanisms, were found to be between 0.41 and 1.40. This is within a range that include Fickian case I (n = 0.45) and non-Fickian diffusion (case II) (n > 0.45) for cylindrical drug-loaded structures. The results, however, suggest that the release exponents, n, correspond mostly to anomalous case II and super case II transport mechanics with sigmoidal characteristics. The drug release kinetics of the P(NIPA)-based hydrogels are well described by bi-dose functions. The observed drug release behavour is related to the porosity of the hydrogels, which can be controlled by cross-linking and copolymerization with acrylamide, which also improves the hydrophilicity of the gels. The paper also presents the effects of cancer drug release on cell survival (%), as well as the cell metabolic activities of treated cells and non-treated cells. The implications of the results are discussed for the development of implantable thermosensitive gels for the controlled release of drugs for localized cancer treatment.


Results
Morphology and De-swelling Kinetics. The microstructures of the gels (Fig. 1a-d) present the effects of copolymer ratio on the morphology of P(NIPA)-based copolymers. The homopolymer matrix (Fig. 1a) indicates less porosity with revealing collapsed walls. The result in Fig. 1c clearly shows the structures of the polymers with micro-cracks and micro-pores. The pore diameters of the homopolymer ranged from 1.67-39.20 μm with an average pore diameter of 9.62 μm (Fig. 1a). This hyddrogel structure also revealed interconnecting cracks with crack lengths ranging from 17.77-84.52 μm (average crack length ∼48.70 μm). In the case of P(NIPA)-based co-polymer matrices (Fig. 1b-d), we observed a predominance of micro-pores at multiple locations with increasing amount of acrylamide. P(NIPA)-co-AM (95:5 wt.%) (Fig. 1c) exhibited pore diameters between 0.1 and 36.07 μm (with mean diameter of ∼7.77 μm, whereas P(NIPA)-co-AM (90:10 wt.%) (Fig. 1c) had collapsed cracks between 0.01-121.01 μm (with an average pore diameter of 61.87 μm). The entire structure of P(NIPA)-based co-polymer matrices (85:15 mol%) was highly porous, characterized by micro-voids and micro-pores with pore diameters between 0.05 and 82.07 μm (with an average pore diameter of 25.59 μm) (Fig. 1d).
Furthermore, distribution of pore diameters from the SEM micrographs are presented (Fig. 2). The result shows the dominance and range of pore sizes within the micrographs for the different experimental samples considered. Although large pore diameters promote diffusion, the pore volume and area equally play key roles in the diffusion process of cancer drugs. The result, therefore, complements the SEM micrographs (Fig. 1). In all cases, 0-10.0 μm pore diameter range dominated within the micrographs presented. To some extent, the distribution     www.nature.com/scientificreports www.nature.com/scientificreports/ order of Drug Release. The order of drug release kinetics was determined by plotting the cumulative concentration of drug against t t or 1 2 ( Fig. 4a-d). In the case of the P(NIPA) homopolymer, linear plots in Fig. 4a and c showed that the homopolymer exhibited zeroth order kinetics with R 2 = 0.999. However, linear plots of cumulative concentration versus t 1 2 ( Fig. 4b and c) show that the homopolymer satisfied the Higuchi model (R 2 = 0.999). Hence, the drug release occurs at a constant rate that is independent of drug concentration (zeroth-order model). However, the Higuchi model 16 assumes that the swelling and dissolution of drugs in the polymer matrix are negligible, while the drug diffusivity is constant. This implies that drug diffusion occurs in one dimension 17 . Mechanisms and order of Drug Release. The drug release exponents, n, and the diffusion coefficients, D, were obtained from plots of ln(M t /M ∞ ) versus In t, shown in Fig. 5a-c for P(NIPA)-based hydrogels that were loaded with PG. The results of drug release exponents, n, diffusion coefficients are presented in Tables 2 and 3, respectively. Note that, for a cylindrical geometry, an n value of 0.45 corresponds to a Fickian diffusion mechanism, i.e. diffusion-controlled transport. Anomalous transport (non-Fickian transport mechanism) is characterized by n-values that are greater than 0.45 but less than 0.89 (i.e. 0.45 < n < 0.89). Also, an n value of 0.89 corresponds to a case-II transport mechanism. A case-II anomalous diffusion process isassociated with controlled swelling. Moreover, n values greater than 0.89 are associated with a super-case-II transport mechanism.
The anomalous transport class was further investigatedand was found to be dominated by a sigmoidal function 6 with a goodness of fit, R 2 = 0.970. Among the classes of sigmoidal functions investigated, the bi-dose response function gave higher R 2 values than the dose-response function 18 and the sigmoidal Richards's second function 19 . The bi-dose response function obtained for the sigmoidal equation is given by:  www.nature.com/scientificreports www.nature.com/scientificreports/ plots. The parameters of the bi-dose response function are summarized in Table 4 (37 °C) and Table 5 (at different temperatures, 43 and 45 °C).
It is of interest to note that the values of n obtained for P(NIPA)-co-AM (95:5 wt.%) and P(NIPA)-co-AM (85:15 wt.%) (at 37 °C) were 0.41 and 0.42, respectively. Although these n values are close to those of Fickian diffusion, they are more clearly less than 0.45. Furthermore, the anomalous n values also exhibited a sigmoidal class of diffusion. The fitted bi-dose sigmoidal plots (S-shape) are shown (Fig. 6a). Sigmoidal anomalous transport was also observed for P(NIPA)-based hydrogels copolymerized with 5 wt.% AM(P(NIPA)-co-AM (95:5 wt.%)) during release at 45 °C. These had an n value of 0.67 (Table 2).
In vitro Effect of Drug Release. In vitro effect of drug release on cells was described using cells viability study where the drug loaded hydrogels were disinfected and placed in cells seeded well plates. Trypan blue stain was use with the aid of a hemocytometer to estimate the percentage cell survival (survival %) after 72 h period (Fig. 7). The results for MDA MB 231 breast cancer cell line showed an earlier response to drug. This shows that cell survival (%) was more affected in the cancer cell line at the early hours of drug release. The effect of PG release on the metabolic acitivity of normal human cell line (MCF 10A) and breast cancer cell line (MDA MB 231) are presented by the percentage alamarblue reduction as shown in Fig. 8.

Discussion
The SEM images of the P(NIPA)-based hydrogels showed irregular and heterogeneous mesoporous characteristics structures of the hydrogels (Fig. 1a-d). Prior work has shown that the increased porosity (with increasing AM content) is also associated with increased cross-linking [5][6][7] . In any case, the porosity of the P(NIPA)-based gels provides pathways for drug diffusion through the P(NIPA)-based structures. This suggests that the presence of these pores affect the resulting swelling ratios. The gel swellings observed prior to de-swelling represents the affinity and exchange of enthalpy between the liquid phase and the solid phase of the gel matrix. A continuous decrease with similar de-swelling rates was observed in Fig. 3a. In contrast, the initial fast de-swelling rates decreased with www.nature.com/scientificreports www.nature.com/scientificreports/ increasing time at 43 °C (Fig. 3b). However, initially, fast de-swelling rates at 45 °C were followed by a plateau and subsequent fast de-swelling (Fig. 3c). In any case, the amount of de-swelling increased with increasing porosity ( Fig. 3a-d) and temperature (Fig. 3a-c).   www.nature.com/scientificreports www.nature.com/scientificreports/ The slopes and shapes of the bi-dose response curves (S-shape) can provide some insights into the potential effects of specific dose levels of toxicity. The gradients (h 1 > h 2 ) (Tables 4 and 5) from the curves (Fig. 6a) correspond to changes in the amount of drug release. A transient increase in the gradient means an increase in drug concentration and a higher risk of a toxic response and vice versa. It is important to note here that the plotted curves without any initial steepness (A 1 < A 2 ) showed a minimal increase in drug concentration until the concentrations increased progressively with time ( Fig. 6b). Figure 6a indicates a higher threshold dose for P(NIPA)-co-AM (95:5 wt.%), compared to that of P(NIPA)-co-AM (90:10) at 37 °C. Since the higher threshold dose, implies initial effective release, the flow through the P (NIPA) -co-AM (95:5 wt. %) began with a higher dose. However, the slopes increased with increasing concentration. This implies a greater dose response. Hence, P(NIPA)-co-AM (90:10) has a higher dose release than P(NIPA)-co-AM (95:5 wt.%). Similarly, P(NIPA)-co-AM (90:10 wt.%) hydrogel at 43 °C reported a higher threshold dose than P(NIPA)-co-AM (95:5 wt.%) (at 43 °C) (Fig. 6b). In general, an increase in drug release was observed with increasing temperature.
Cell counting with the hemocytometer confirmed cell survival (%) of 33% after 72 h for MDA MB 231. Overall, the results indicated an exponential decrease in cell survival (%) as a function of incubation time. On the other hand, the survival (%) of the treated cells was significantly lower than the control (non-treated) cells (p < 0.05). However, the effect of drug concentration on cell survival is outside the scope of the current paper.
The metabolic activities of non-treated cells increased throughout the 72 h duration of the study for both types of cells. Conversely, the treated cells seeded on the drug-loaded gels exhibited much reduced metabolic activities compared to the non-treated cells, as indicated by the % alamar blue reduction values (Fig. 8a,b). Within the drug-loaded gel groups, there was no clear significant difference due to the polymer or copolymer ratio. However, the results clearly indicated an exponential decrease in % alamar blue reduction value as a function of incubation time (Fig. 8a) and fairly constant values in Fig. 8b. This indirect measurement of cell viability, and by extension, cell growth kinetics kinetics demonstrated that, drug release into the culture medium hindered cell metabolism and growth over time.
From the statistical test that were carried out, there was a mean differences in the metabolic activities, of the non-treated and treated cells with p-values less than 0.05. At the 0.05 level, the difference of the population means was significantly different from the test difference (0). For the MCF 10 A cells, p-values from multiple comparison therefore, suggest statistical significance between the metabolic activities of non-treated cells compared to cells treated with the PG drug. Metabolic activities showed marginal statistical difference between samples treated with drug released from P(NIPA)-co-AM (95:5 wt%) and P(NIPA)-co-AM (90:10 wt%). However, there were no   www.nature.com/scientificreports www.nature.com/scientificreports/ statistical differences when the results of the homopolymer was compared with P(NIPA)-co-AM (95:5 wt%) and P(NIPA)-co-AM (90:10 wt%) p-values were greater than 0.05.
The results obtained from the comparison of untreated MDA MB 231 cells with treated group suggest a statistical significance between the metabolic activities. However, there was no statistical differences based on samples treated with drug released from the different polymer ratios. Moreover, there were no significant differences between the metabolic activities of the normal and cancer cells after treatment over 72 h period.
The implications of the above results are very significant for the controlled delivery of prodigiosin cancer drug 6,7 to tumor sites 20 . First, they show clearly that there is a need to use the knowledge of the release kinetics to guide the controlled delivery of prodigiosin to be within the therapeutic windows. There is also a need to consider  www.nature.com/scientificreports www.nature.com/scientificreports/ the possible anomalies and super-case transport mechanisms, and the phenomena that could lead to local cytotoxicity during drug release from implantable P(NIPA)-based devices for localized cancer treatment.
Further work is clearly needed to develop implantable drug delivery systems that can deliver cancer drugs, such as prodigiosin, to regions adjacent to tumor sites, following surgery to remove solid tumors from regions such as the breast. Such scenarios could benefit from an implantable device that contains P(NIPA)-based hydrogels that are encapsulated in poly-di-methyl-siloxane (PDMS) packages with micro-channels that deliver the drugs from the P(NIPA) hydrogels to the surrounding tissue 3,6,7 . There is also the potential to explore the combined effects of enhanced cancer drug release and hyperthermia during drug release at 43 and 45 °C 10 . These are clearly some of the opportunities and challenges for future work.

Conclusions
I. This paper presents the kinetics of drug release from P(NIPA)-based hydrogels that are copolymerized with acrylamide. The swelling and drug release characteristics are elucidated for temperatures in the regime between normal physiological conditions (∼37 °C) and hyperthermic regimes (43-45 °C).  www.nature.com/scientificreports www.nature.com/scientificreports/ temperature. Drug diffusion was governed by the volume phase transition temperatures. Gels turn to collapse more readily at temperatures above their respective response temperatures. Extensive discussion on the effect of diffusion and the lower critical solution temperature (LCST) on P(NIPA)-based hydrogels, loaded with PG has recently been studied 6 . IV. The slopes of the bi-dose response curves can be helpful in relating local drug concentrations to therapeutic and toxic levels associated with the release of the prodigiosin cancer drug. In general, a transient increase in the gradient of the bi-dose plots means an increase in the local drug concentration. This corresponds to a higher risk of a local toxic response. However, the plotted curves (without any initial steepness) showed only a minimal increase in drug concentration until the concentrations increased significantly with time. V. The in vitro effect of drug release on cell survival (%) and cell metabolic activities in comparison to the reference (non-treated cells) were presented. The drug inhibited the cell growth of a normal human cell line and a cancer cell line.

Materials and experimental Methods
Materials. The prodigiosin (PG) that was used in this study was obtained from the Biotechnology Advanced APS is used as a radical initiator, while 40 μl of TEMED was used as a catalyst via gently swirling the mixture to decompose persulfate ions release, to give free radicals (molecules with an unpaired electrons) (Fig. 9). The unpaired electron then continues to develop into macromolecules, especially when MBA is crosslinked with AM. After the addition and reaction of TEMEDwith the gel solution, the polymerisation process of P(NIPA)-based hydrogels were then terminated within 5-10 s via the interaction of the gel solutions with oxygen radicals in the environment. That is, the gel solutions in test tubes were completely openedand samples were exposed to oxygen to terminate the reactions and solid gels were obtained within 3 mins. Hence, polymerization of P(NIPA)-based hydrogels was carried out by free radicals polymerization as presented in our prior work 7 . The P(NIPA)-based co-polymers were polymerized with the AM monomer to produce P(NIPA)-based co-polymers with 5, 10 and 15 wt.% of acrylamide. These correspond to: P(NIPA)-co-AM (95:5 wt.%); P(NIPA)-co-AM (90:10 wt.%), and P(NIPA)-co-AM (85:15 wt.%), respectively. Summary of the methodology on the gel polymerization is presented (Fig. 9).
The polymerized hydrogels were subject to several washes, soaking and de-swelling in double deionized water at room temperature to remove, through leaching, any unreacted acrylamide or chemicals salts. A sudden corrosion of the stainless steel razor blades that were used to slice the gels was indicative of the presence of unreacted chemicals/chemical residue in the polymerized gels and this was not observed after thoroughly washing the gels.
The gels were then sliced into the required thicknesses/geometries, washed and dried under vacuum with an isotemp vacuum oven (model 280A, Fisher Scientific, Waltham, MA, USA), connected to a high vacuum pump (Edwards, E2M265503) set at −20 in Hg for 2 h at 40 °C to drive out excess water. The water release from the samples was constantly discarded. Subsequently, gels were completely dried at room temperature until constant weights were achieved, prior to soaking them in drug solutions. Though drying had the tendency to collapse the gels, P(NIPA)-based hydrogels have the ability to recover their natural porosity by expanding, stretching and uncoiling their folded chains when soaked in fluids/drug solutions.
De-swelling Kinetics. The drug affinity, hydrophilicity, and swelling of the hydrogels were studied by incubating dried gels (cylindrical structures) in drug solutions (PG). This was carried out in an orbital incubator shaker (Innova 4300, New Brunswick Company Inc. NJ, USA) at a controlled temperature (27 °C). Incubation was controlled until an equilibrium swollen mass ratio was obtained. This occurred within 72 h of incubation. De-swelling experimentswere then carried out in an orbital incubator shaker, operated at physiological conditions (37 °C at 60 rpm). De-swelling experiments were also performed at 43 and 45 °C to study possible effects of hyperthermic temperatures on drug release kinetics. The de-swelling ratios (DSR) of prodigiosin from P(NIPA)-based hydrogels were obtained from the expression: where M t is the mass of a hydrogel at a time, t, and M eq is the equilibrium mass of the swollen hydrogel. The percentage water absorbability (swelling ratio, %) is given by 21 : www.nature.com/scientificreports www.nature.com/scientificreports/ where M d is the dry mass of a hydrogel at time t = 0 and M a is the mass of a hydrogel in water after 72 h. The hydrophobicity was determined at the equilibrium swelling in deionized water (S eq ). This is given by: where M s is the swollen mass and M d is the dry mass of the gel after soaking in 1 ml of distilled water for 72 h. The hydrophilic swelling index (H-index) for the gels was obtained from: where S eq (70 vol.% Is) is the equilibrium swelling mass of a gel in aqueous 70% iso-propanol for 72 h.
Drug Release Under physiological Conditions. P(NIPA)-based hydrogels were loaded with prepared concentrations of prodigiosin (PG). Knowledge from the molecular weight of drug candidate was used to prepare the desired drug concentrations (8 nM/ml) for loading the hydrogels via incubation at room temperature for 72 h to ensure complete saturation. The prepared concentration of drugs was also guided by existing literature 21,22 . Drug release from P(NIPA)-based hydrogels were conducted under in vitro conditions at 37 °C under a mechanical agitation of 60 rpm in a phosphate buffer saline (PBS) solution, pH 7.4. These conditions were considered to simulate physiological conditions. Ultraviolet-visible (UV-Vis) spectrophotometer (Evolution 300, Shimadzu Scientific Instruments Inc., Kyoto, Japan) was used to characterize the absorbance of drug release. Concentrations were obtained from the absorbance's values reported. Drug release from P(NIPA)-based hydrogels usually starts with a rapid release. Due to this, 1 ml of drug solutions were collected at 10 min interval on the first day of drug elution, while daily decanting and replacement with equal volume was done. is the fraction of drug release at time t, k is a geometry constant, and n is the release exponent, which corresponds to the mechanism of drug diffusion.
The constants k and n were obtained from the linear logarithm form of equation (6a). This gives: where the values of k and n were obtained from the intercepts and gradients of the plots of ln(m t /m o ) versus ln t(s). Meanwhile, the diffusion coefficients, D, were obtained from: where δ is the thickness of the dry gel and k is the constant in Equation 6a. A recent study 6 , including the current work, suggest a dominance of anomalous transport mechanisms with 0.45 < n < 0.89, for cylindrical disc specimens 23,25,26 . These correspond to a regime in which non-Fickian diffusion control the drug release. Since such anomalous drug release is less well understood than conventional release, there is a need for further work to improve our basic understanding from P(NIPA)-based gel. In the current study, sigmoidal and two-step type of anomalous behaviours were investigated by plotting The above plots were obtained using OriginPro2017 software package (OriginLab Corporation, Northampton, MA, USA) 27 . The regression coefficient, R 2 , was used as the goodness of fit in the studies of the anomalous behavior. Four kinetic models were used to characterize the release kinetics. This includes zeroth-order, first order, second order, Higuchi model and Korsmeyer-Peppas equation (Table 6). These were used to fit the data obtained from the drug release experiments 28 . Material Characterization. Prior to acquiring morphological characteristics with scanning electron microscopy (SEM), sample preparation involved sputter-coated P(NIPA)-based hydrogels with 4 nm of gold under vacuum using a sputter coater (EMS Quorum, EMS150R ES, Quorum Technologies Ltd., East Sussex, UK). Samples were then mounted on standard SEM sample holder, gently loaded into Phenom ProX desktop SEM (ThermoFisher Scientific, Phenom-World B.V., Eindhoven, The Netharlands). Images were acquired and morphological differences were observed and reported. Pore diameters were determined with ImageJ software package (2015 version) 29 .
Thermal analysis on the lower critical solution temperature (phase transition temperature) withdifferential scanning calorimetric (DSC) analysis was carried out by loading ~10 mg of each sample into an aluminum pan. The mass of the reference sample (empty Al pan) was equally measured. Samples were loaded onto DSC (214 Polyma, Netzsch Instruments, Burlington, MA, USA). Methods for data acquisition and analysis were defined with the instrument control software at a heating rate of 10 °C/min. Data acquisition was done in a nitrogen environment by purging N 2 at a flow rate of 20 ml/min. DSC was done from room temperature (26 °C) to 60 °C to determine the LCST.
In Vitro Effect of Drugs Release on Cell Survival Fraction. Prior to cell culture, P(NIPA)-based hydrogels were disinfected with ethanol. Samples were dipped completely into 70% ethanol untill saturation for 1 h. To ensure a complete removal of the ethanol (which can affect cell viability), the hydrogels were first cleaned ultrasonically for 30 min with autoclaved double deionized water. The samples were also rinsed several times with Dulbecco's phosphate buffered saline (DPBS). The disinfected samples were allowed to dry at room temperature while being exposed to UV light under sterile conditions in a biosafety cabinet before they were subsequently loaded with 8 nmol/ml drug solutions (PG). Samples were also exposed to UV light prior to cell seeding.
Human breast cancer cell line (MDA-MB-231) was used to investigate the effects of the free PG drug and drug released from P(NIPA)-based hydrogels on cell survival (%). The MDA-MB-231 cell line were cultured with Liebovitz's L15 medium supplemented with 10% fetal bovine serum (FBS) and 1% penicillin/streptomycin antibiotic. MCF 10A cell lines were cultured with DMEM/F12 medium that were supplemented with 0.5 μg/ml hyrocortisone, 5% horse serum, 10 μg/ml insulin, 30 ng/ml murine Epidermal Growth Factor, 100 ng/ml cholera toxin, 0.2% amphotericin and 1% Penicillin-Streptomycin. The MCF 10A cell lines were cultured in T75 flask at 37 °C in a humidified incubator 5% CO 2 , whereas the MDA-MB-231 cell lines were in T75 tissue culture flask at 37 °C until 80-90% confluence (log phase of growth) was achieved. The cells were then harvested with 0.25% rypsin-EDTA and 10,000 cells/well were seeded in 24-well plates (n = 4). After 48 h, the cells attained their 70% confluence state (log phase growth). Then, a concentration of 8 nmol/ml of PG drug prepared in the culture medium was added to the wells with 10,000 cells/well (treated cell wells). Cells in wells without drugs were used as control wells with 10,000 cells/well (non-treated cell-wells). The cells were immediately trypsinized and stained at different time just after 6 h, 24  where the total number of cells equals to the number of live plus dead cells counted.
In Vitro effect of Drugs Release on Cell Metabolic Activities. AlamarBlue assay, a fluorometric method for the detection of metabolic activity of cells, was used as an indirect measure of cell viability. MCF 10A and MDA MB 231 cells in their log phase of growth were harvested with trypsin-EDTA and 10,000 cells/ml were seeded on the various PG drug-loaded hydrogels in 24-wells plates (treated groups) and on non-loaded gels (non-treated control) (n = 3 per group). After a 3 h cell attachedment period (time 0), the culture medium was replaced with 1 ml of culture medium containing 10% alamarBlue reagent (Thermo Fisher Scientifi). The plates were incubated in a humidified incubator at 37 °C for 3 h and 100 μl aliquots were transferred into duplicate wells of a black opaque 96-well plate (Thermo Fisher Scientific). The fluorescence intensities were then measured at 544 nm excitation and 590 nm emission using a 1420 Victor3 multilable plate reader (Perkin Elmer, Waltham, MA). All alamarBlue reagents were prepared according to the manufacturer's instructions. The percentage reduction of alamarBlue reagent was determined using equation (9 where FI sample is the experimental fluorescence intensity of samples, FI 10%AB is the fluorescent intensity of the 10% alamarBlue (negative control: oxidized form of alamarBlue), and FI 100%R is the fluorescence intensity of 100% reduced form of alamarBlue (positive control).
statistical Analysis. Statistical analysis was carried out with OriginPro 2017 software package. One-way ANOVA and paired sample t-test were used for multiple comparisons. Statistical significance in the survival (%) of treated and non-treated cancer cells were statistically evaluated based on the differences in population means. The difference in toxicity obtained for free drug and the drug released from the hydrogels was also evaluated. Thus, p < 0.05 was considered to bestatistically significant, unless otherwise stated.

Model Equation Constant
Zeroth-order C t = C o − k o t C t is the amount of drug release (assuming the release occurs rapidly) after a time, t; C o is the initial amount of drug; k o is the Zero-order rate constant.
First-order ln C t = ln C o − k I t C t is the amount of drug release after time, t; C o is the initial concentration of drug; k I is the First-order constant.
Second-order = + k t C t C o II 1 1 C t is the aa mount of drug release after time, t; C o is the initial concentration of drug; k II is the Second-order constant.
Higuichi C t = [D(2C − C s t)] 1/2 C t is the amount of drug release after time, t; C is the initial drug concentration; C s is the drug solubility in the matrix media; D is the diffusivity of drug molecules in the matrix substance.
Korsmeyer-Peppas equation is the fraction of drug release at time t, k is a geometry constant, and n is the release exponent which corresponds to the mechanism of drug diffusion.