Kinetic Analysis of Drug Release From Nanoparticles

-PURPOSE. Comparative drug release kinetics from nanoparticles was carried out using conventional and our novel models with the aim of finding a general model applicable to multi mechanistic release. Theoretical justification for the two best general models was also provided for the first time. METHODS. Ten conventional models and three models developed in our laboratory were applied to release data of 32 drugs from 106 nanoparticle formulations collected from literature. The accuracy of the models was assessed employing mean percent error (E) of each data set, overall mean percent error (OE) and number of Es less than 10 percent. RESULTS. Among the models the novel reciprocal powered time (RPT), Weibull (W) and log-probability (LP) ones produced OE values of 6.47, 6.39 and 6.77, respectively. The OEs of other models were higher than 10%. Also the number of errors less than 10% for the models was 84.9, 80.2 and 78.3 percents of total number of data sets. CONCLUSIONS. Considering the accuracy criteria the reciprocal powered time model could be suggested as a general model for analysis of multi mechanistic drug release from nanoparticles. Also W and LP models were the closest to the suggested model RPT


INTRODUCTION
Nanotechnology has been received immense attentions in various branches of science including medical and pharmaceutical sciences (1). Numerous research articles in the field of pharmaceutical nanotechnology have been appeared in literature citation. Our focus, however, is on some recently published review articles that deal with pharmaceutical nanotechnology (2)(3)(4)(5)(6)(7). The drug release from pharmaceutical nanoparticles is a major determinant in its biological effect, thus evaluation of drug release kinetic is of paramount importance in the field. The use of kinetic models is often helpful in elucidating release mechanisms which in turn can be of use in the control of drug release. Another advantage of the kinetics is to represent several release data with one or two parameters. To the best of our knowledge a comprehensive kinetic representation of drug release from nanoparticles for the purpose of finding a general model has not been reported in the literature. Parameters obtained from such a general model can be employed for comparing different delivery systems of a drug as well as correlating with bioavailability data. It is obvious that the comparison and correlation cannot be made with parameters obtained from data fitting to various kinetic models. In the present work release data of 32 drugs from 106 nanoparticle formulations gathered from various research articles was analyzed using 10 well known (8,9) as well as 3 models developed in our laboratory (10,11). Also, a theoretical justification of the best models is provided from a general rate expression describing both the Noyes-Whitney dissolution law as well as Fick's first law of diffusion. _________________________________________

Kinetic models
The kinetic models were those which have been used in interpretation of drug release data (8)(9)(10)(11). The equations of the kinetic models are presented in Table 1.

Data
The release data for kinetic analysis was collected from various research articles the list of which is seen in Table 2. The name of drugs, type of nanoparticles, number of data in each set and the method of release measurement are included in the Table. a Parameters of models 1-10 were obtained by linear regression and those of 11-13 were calculated via nonlinear regression. F denotes fraction of drug released up to time t. k 0 , k f , k H , p, k P , k 1/3 ,k 1/2 , k 2/3 , t d , β, Z 0 , Z 0 ', q, q', n, k 1-n , k n-1 , m and b are parameters of the models. Z and Z' are probits of fraction of drug released at any time. Z 0 and Z 0 ' are the values of Z and Z' when t = 0 and t= 1, respectively. The relationships between Z and Z' with F are given by: Where Z = (t -t 50% )/σ and Z' = (logt -logt 50% )/σ'. σ and σ' are relevant standard deviations (8,64).

Methods
The coordinates of each point in the release profile of drug nanoparticles in the original paper were measured carefully and the data was employed in the subsequent kinetic analysis. The accuracy and prediction ability of the models were compared by calculation of absolute percent error (E) for each set as well as overall mean percent error (OE) for all sets are given by equations 1 and 2: (1) (2) F cal,i and F obs,i denote calculated fraction and observed fraction of drug released at the i th sample, respectively. The value of N is the number of data in each set. The number of single figure OEs associated with the models expressed as percentage of total number of sets was also employed to select suitable models.

RESULTS
The release data of all 106 formulations was fitted to 13 models mentioned in Table 1. The overall error, number of errors expressed as percentage of total number of sets, NEs, less than 5, 10 and 12% for the models are seen in the same table. The values of E, squared correlation coefficient (R 2 ) and the parameters of reciprocal powered time model for the nanoparticle formulations are given in Table  2.

DISCUSSION
Kinetic study of drug release is often useful in obtaining one or two physically meaningful parameters which are employed for comparative purposes and relating the release parameter with important parameters such as bioavailability. Furthermore a kinetic parameter can be used to study the influence of formulation factors on the drug release for optimization as well as control of release. The models having both single figure overall error (OE) and number of single figure OEs higher than 75% of total number of sets were considered as suitable general models. According to Table 1  The advantage of the model over other models is the fact that from its parameters, b and m, the most meaningful release parameter release half life (t 50% ) i.e., time required for 50% of drug to be released, can be calculated from t 50% = m 1/b . This relationship is readily obtained by substitution of 0.5 for F, t 50% for t into the model (equation 13 in Table 1) and subsequent solution for t 50% . The release half life is conversely related to release rate. The parameters of RPT model together with t 50% for 106 formulations examined are seen in Table 2.

Theoretical justifications for the best models: Reciprocal powered time model
The rate of drug release from any solid or semi solid delivery system is usually controlled by dissolution and/or diffusion. Regardless of mechanisms involved in the release, its rate under sink conditions can be expressed by a single general equation as follows: (3) w is amount of drug released up to time t and dw/dt is the rate of release. D, S, C s and h are drug molecule diffusion coefficient, effective surface area of drug with release medium, drug solubility in the medium and the length of diffusion path. This equation represents both the Noyes-Whitney law of dissolution (44) applied for dissolution rate limited release as well as the Fick's first law of diffusion used for diffusion rate limited release processes (45,46). In the dissolution process h, the thickness of stagnant aqueous layer around drug particle, is constant but S and sometimes D are varied during the release process (47). On the other hand in a diffusion rate limited release in addition to D, h and sometimes S are variable during the release process. For a complex system such as nanoparticles equation 3 does not seem to include all other factors influencing the release rate among which penetration rate of liquid into the system; hydration, swelling, relaxation, erosion and dissolution of polymer can be mentioned. The extents of liquid penetration and the polymer contributed properties are directly proportional to t 1/2 (48) and powered of t (49)(50)(51)(52), respectively. Thus, these effects are collectively represented as a time dependent variable, X, and equation 3 becomes: Therefore, regardless of release mechanism in order to obtain a general working formula for both dissolution and diffusion rate limited release processes it is assumed that the term DSX/h is variable during the release. The integration of equation 4 between time 0 and t as well as 0 and ∞ yields: Where w ∞ is the total amount of drug released in the medium. The amount of drug remaining to be released, M, is given by: Fraction of drug released, F, at any time is calculated by: Since, the value of is increasing with time whereas the opposite is true for , it is reasonable to assume that the respective integrals are expressed by the power terms in the forms of α t a and α' t -a' . Where α, a, α' and a' are constants. Substitution of the power equations into the far-right side of equation 8 results in: (9) In which b = a + a' and m = α'/α. Reciprocating the far right side of equation 9 and subsequent rearrangements would yield the reciprocal powered time model. Equation 9 is similar to previously published model applied for a simpler solid dispersion systems (11). In the derivation leading to that model only the effective surface area of the drug was assumed to be variable and hence the release mechanism was dissolution rate limited solely. The present equation, however, is extended the application of reciprocal powered time model to both dissolution and diffusion rate limited processes and thus is considered to be a general release equation.

The Weibull model
This is an empirical model which has been widely applied to release data of both rapid and extended release drug delivery systems (53)(54)(55)(56)(57)(58)(59)(60)(61)(62)(63). If it is assumed that some of the factors influencing the overall drug release e.g. effective surface area (S) are solely mass dependent, mass of unreleased drug in the medium (8), while the others are considered to be time dependent, thus equation 4 can be written as: (10) Where kM = S and k't p = X. According to previously published papers (47) as well as (45,46), inclusion of theses in to equation 10 and converting the amounts in to fractions in the subsequent equation yields the following relationship: (11) In which F = w/w ∞ , (1-F) = M/ w ∞ , w ∞ is total amount of drug ultimately released and q includes n, n' and p. The value of k b is related to C s and other constants. Rearrangement of equation 11 gives equation 12: The integration of equation 12 between times 0 and t yields: The derivation here is different from the one provided previously (47) in which constant drug effective surface area and non sink conditions were assumed.

CONCLUSION
Evaluation of 13 kinetic models for drug release from nanoparticles indicated that three models i.e. the novel reciprocal powered time, Weibull and Wagner's log-probability models were superior to the rest of models in that the errors associated with those were single figures and the number of single figure errors was higher than other models. Among the three mentioned models, the reciprocal powered time model was more suitable than the other two ones considering the accuracy criteria used for comparison. In addition to the mentioned superiority of reciprocal powered time model, from its parameters a fundamental parameter i.e., release half life can be calculated which is a most widely used criterion for comparison of drug release rate. Taking into account complex processes involved in drug release from nanosystems, theoretical justifications for the reciprocal powered time as well as Weibull models were provided. Based on the theoretical justifications, the reciprocal powered time and Weibull models seem to be applicable for any drug release mechanisms involving dissolution, diffusion and mixed dissolution -diffusion rate limited processes.  H., and Nah, J. W. Effect of solvent on the preparation of surfactant-free poly(DL-lactideco-glycolide) nanoparticles and norfloxacin