1. Introduction
To improve the effectiveness and safety of drug use and issues such as bioavailability or fluctuations in the level of drug concentration in the blood, it is important to control the drug release rate [
1,
2]. Therefore, it is necessary to provide a mechanism to release the drug from a drug delivery system at a controlled and optimal rate. Controlled drug delivery systems can cause desirable effects such as stable drug release, reducing the number of required doses, or better adaptation of the drug to the patient’s condition [
3]. One of these systems, that is of great interest, is the hydrogel-based drug delivery system [
4,
5,
6]. Hydrogels are composed of 3D hydrophilic polymer networks that are able to absorb a large amount of water and medicinal substances. They are effective and popular carriers in drug delivery systems due to the ability to slowly release drugs through a diffusion mechanism [
7,
8], their physical similarity to body tissue [
9], low protein absorption in the body, and the ability to be used for both hydrophilic and hydrophobic drugs [
10]. The mechanism of these systems is such that, first, water penetrates into the volume of the matrix so that the hydrogel contains a large amount of water and its volume increases greatly. This volume increase can be in response to a biological stimulus. This increase in volume causes the polymer chains to open and allows the drug to slowly diffuse into the dissolution medium [
11,
12].
When a conventional or controlled drug delivery system is used for drug release, due to the importance of the release rate, the factors affecting the drug release rate from the used system should be investigated. This issue has received more attention in recent years due to the emphasis on the use of the concept of quality by design (QbD) by the U.S. Food and Drug Administration (FDA). With the QbD initiative, routine quality control tests, such as dissolution testing (DT) performed after the production process, can be reduced or even stopped if the influencing parameters are controlled [
13]. This goal can be achieved by conducting experiments or simulating the drug release process from a drug delivery system.
Modeling the behavior of hydrogel-based systems, such as hydrogel tablets, is currently a topic of great interest because, as with all phenomena, mathematical models are a necessary step to achieve a deep understanding of the process. This work leads to targeted design and reduction of costs due to frequent tests [
14]. One of the key factors in choosing the type of modeling approach is paying attention to the type of responses that are needed. For example, the approach can only provide an analysis of the mass transfer process or focus on the phenomenon’s mechanical behavior. In a general classification, the approaches to modeling the drug release from hydrogel-based systems can be divided into the following three categories: 1—the drug release fitting, 2—statistical and neural network models, 3—models with a mechanical approach [
14].
In drug release fitting models, which are the most widely used modeling approach due to their simplicity, the percentage of drug released from the matrix to the total mass of the initial drug is considered as a function of time. In such modeling, modeling of the amount of drug release over time from one or more drug delivery systems using equations such as Higuchi, zero-order, Korsmeyer–Peppas, or other famous equations is usually attempted [
15,
16,
17].
In statistical and neural network models, the prediction of some important responses of the release profile based on a series of controllable parameters during the production of drug delivery system is attempted. These responses usually include cumulative percentage of released drug, matrix swelling rate, etc. Controllable parameters during the production of drug delivery systems usually include formulation, geometry, tablet surface hardness, etc. For example, in the research carried out by Barmpalexis et al., using neural network techniques, linear regression, and higher-order regressions, they were able to accurately predict responses such as mean dissolution time, maximum solvent absorption, and several other responses based on the combination ratio of four substances in drug production [
18]. Studies such as Yekpe et al.’s study with emphasis on the importance of tablet surface hardness and tablet coating weight on the drug release rate [
13] and Frankiewicz and Sznitowska’s study with emphasis on the importance of production temperature and coating spray pressure on the tablet on the drug release rate [
19] are studies with this type of modeling approach.
Another approach is the mechanical approach. The main building block of this type of approach is Fick’s law of diffusion. In most of the old models with a mechanical approach, the analytical and sometimes numerical solution of diffusion equations is considered according to the geometry of the problem, and in fact, only the description of the drug component has been discussed [
14]. For example, Arroyo et al., to achieve an optimal composition for the release of indomethacin as an anti-inflammatory cream, solved the diffusion equations in one dimension for a curved shell and presented an analytical solution [
20]. But, the models that have received the most attention from researchers today focus on other aspects of drug release from a drug delivery system. For example, the model also explains the behavior of other components in the system such as water and polymer, describes swelling and erosion phenomena, and can also describe the behavior of drug delivery systems with more complex geometries [
14]. One of the main steps in the introduction and presentation of such models is shown by Siepmann et al.’s research series from 1999 to 2000 [
21,
22,
23], especially the “sequential layer” model [
23]. Although their model could explain inhomogeneous deformations of a drug delivery system, there were still some shortcomings in the model. For example, the deformation was affine despite being inhomogeneous (that is, for example, a cylindrical system always remains cylindrical) or the mass of the polymer in the domain could only be calculated globally and not locally [
24]. In the research carried out by Lamberti et al., an attempt was made to overcome the shortcomings of the previous models, such as the affine deformation [
25]. Although this model had a high ability to describe the behavior of drug release from hydrogel-based tablets, like the model of Siepmann et al., the mass of the polymer could be estimated globally in the entire matrix domain. It causes the mass value of the components in the model to be unrealistic at some points of the domain [
24]. In the research carried out by Caccavo et al. an attempt was made to eliminate the weaknesses of the previous models, which occurred in some of the most important models. Briefly, in the model of Caccavo et al., the equations were considered similar to the model of Lamberti et al., with two main differences: (a) swelling is based on the polymer flux and (b) the mass of the polymer at any point can be estimated locally [
24]. Since the simulations in this research are based on the model of Caccavo et al., only a brief explanation of this model is provided and its details are stated in the next section.
When the phenomenon of drug release is simulated using a powerful mathematical model, it is possible to investigate the effect of different factors on this phenomenon. Usually, a single-point comparison is used to compare two or more drug release profiles (resulting from experimental results or mathematical modeling). For example, the percentages of drug released from several drug carriers are compared at certain time points. The U.S. Food and Drug Administration (FDA) recommends that the use of values that can more comprehensively compare the drug release profiles is preferred [
26]. These values can be used to prove the similarity or difference between two drugs with the same purpose. For example, in the research conducted by Kadry et al., they obtained the Dissolution Efficiency (DE) values for the tablets they printed themselves and the tablets available in the market, and using ANOVA tests, they proved that there is no significant difference in DE values among these drugs. In this way, they concluded the bioequivalence of printed drugs with standard drugs available in the market [
27]. In addition to the DE value, other comparative measures related to release profile have also been used in past studies. For instance, Hossein et al., to reach the optimal formulation for the release of indapamide, used the amount of Mean Dissolution Time (MDT) [
28] and Wei et al. used the values of Area Under the dissolution Curve (AUC) and MDT to evaluate the effectiveness of betaxolol hydrochloride intraocular gel [
29].
Most of the studies conducted in this field are related to the effect of drug formulation [
13,
30,
31,
32] or the effect of appearance properties or characteristics related to drug production, such as tablet coating weight, porosity, dimensions of particles, the pressure of the tablet compression machine, or many other properties [
31,
32,
33]. One of the factors that can affect the release rate is the geometry of the matrix. However, much fewer studies have studied the effect of matrix geometry on drug release rate compared to the factors mentioned above. In a study conducted by Windolf et al., to investigate the effect of different shapes made by 3D printing technology on the drug release rate, they produced tablets with six different geometries. They by obtained MDT values from the release profiles and, by comparing the surface-to-volume ratio for these shapes and establishing a relationship with the MDT value, concluded that the surface-to-volume ratio has an effect on the drug release rate [
34]. In other studies, Karasulu et al. investigated the effect of triangular, hemispherical, and cylindrical geometries on the release profile of theophylline [
35], Triboandas et al. investigated the effect of round and oblong geometries produced by hot-melt extrusion technology on the release profile of itraconazole [
36], and Molavi et al. also investigated the effect of size and flat face, round convex, and oblong convex geometries on the release profile of domperidone [
37].
Therefore, it is stated that although the influence of geometry on the release process has been mentioned in very few articles and in a limited way, it can be further investigated as an effective variable. In most of these studies, the surface-to-volume ratio and, in general, surface-related variables have a significant effect on the drug release rate from tablets. In past studies, the importance of the effect of geometry on the release profile was rarely mentioned and it has never been investigated as an effective factor in determining the responses related to drug release alone. Like the change in the formulation, the ratio of compounds or variables related to tablet production, its geometry, and dimensions have not been looked at as variables capable of predicting the desired responses from the drug release process. Also, based on our knowledge, very few studies have compared common profiles in tablet manufacturing based on release responses. In this research, an attempt is made to fill some of these research gaps.
3. Results
After the drug release process was simulated for the desired geometries (
Figure 6) and the release profile was obtained for each one, the results were categorized into three parts: 1—results related to each geometry, 2—results related to common features, and 3—comparing geometries with each other. In the first part, the aim is to answer whether, knowing the dimensions of a geometry and its type, it is possible to predict the expected responses of a release profile with appropriate accuracy. For each type of geometry, the drug release process for 20–40 matrices with different dimensions was simulated, and statistical modeling was used to check how accurately the responses of a release profile can be predicted by having the dimensions of a matrix.
The second part relates to whether the common features of the matrices, such as the maximum length, maximum width, or surface-to-volume ratio, have a significant effect on the release profile. This question is also answered by applying statistical modeling, showing that regardless of the type of geometry and simply by having these values in hand, it is possible to make an accurate prediction of the responses resulting from the drug release profile. The reason for presenting this part is that in addition to the fact that the effect of these values on the release profile has been emphasized in past studies [
34,
37,
38], these common features are of special importance in drug design for purposes such as patient compliance [
41]. Thus, this issue is important from the point of view of designing tablets. In the third part, the question whether there is a significant difference between the types of geometries in the responses caused by the release profiles or not is answered. It is noteworthy that all these simulations were carried out by keeping the mass and dose of the drug fixed. This means that if the results of this part are meaningful, during the design of a drug tablet, by only changing the dimensions or the type of geometry and without any change in the dose and the polymer carrier type, according to the drug class and the needs of the patients, the release profile can be adjusted to the target profile as much as possible. It should also be noted that the following parameters are extracted from the release profile: a. 95 rel (the time required to release 95% of the initial mass of the drug), b. 30 min (percentage of drug released after 30 min from the start of the dissolution process), c. area under the dissolution curve (AUC), d. dissolution efficiency (DE), e. mean dissolution time (MDT), and f. variance of dissolution time (VDT).
DE,
MDT, and
VDT are obtained as [
42]:
In these relations, is the amount of drug released until time , is the amount of drug released at the last time of recording the release result, ΔMi means the percentage of drug released between times i and i+1, and is the average of i and i+1.
3.1. Results Related to Each Geometry
Since the tablets’ mass and volume have been kept constant in all the simulations, this causes a strong dependence between the dimensions in all the investigated geometries. If there is a strong dependence between the independent variables in a linear regression model, the results are greatly weakened, and if this dependence exceeds a certain value, the results lose their validity. One of the regression methods that can be used in these cases is partial least squares (PLS) regression [
43]. To estimate the power of the represented models, we use the values of coefficient of determination (R
2) and the mean absolute percentage error (MAPE) simultaneously. In
Table 2, the values of R
2 and MAPE for all the built models that include 48 regression equations, as well as the lower and upper limits of the ranges, are presented. It is revealed that, for example, by having the exact dimensions of the FFBE type geometry, on average, the six examined responses can be predicted with a coefficient of determination of 0.978, while the deviation from the values obtained from the simulations is only about 0.66%. For other geometries, the predictions are approximately as strong as mentioned. It is also inferred that, in general, by having the dimensions of a geometry, with an R
2 of 0.96 on average, the responses related to release can be accurately predicted, while there is about 1.27% deviation from the responses. Among the responses, the lowest average prediction accuracy is for DE, which has an accuracy of about 0.92, while the other responses have coefficients of determination close to 1.
3.2. Results Related to Common Features
During the design of a drug tablet, special attention is paid to features such as maximum length, apparent surface area, or apparent volume because these values can be directly related to patient compliance [
41]. Since these features can be checked for any tablet and with any geometry, it is possible to check all eight chosen geometries simultaneously, this time using common dimensional features. Another issue is the special importance of surface-related variables in the diffusion process. Many studies have emphasized the importance of variables such as surface or surface-to-volume ratio in the release process [
35,
44]. Therefore, in addition to the mentioned features, the effect of variables related to surface area is also investigated in this section, because in some of the previous research, the reason for the difference in drug release profiles between different forms of tablets has been justified by these variables [
37].
According to the past studies and, of course, the introduction of the variable “ratio of maximum height to maximum length”, the following four variables were examined:
In the case of the first variable, the ratio of surface area to volume, it is enough to divide the surface area by volume. Variable number 2 is calculated as shown in
Figure 7. Of course, since all the geometries examined in this research are round tablets, their length and width are the same due to axial symmetry. Variable number 3 is the product of maximum length, width, and height, as shown in
Figure 7. The apparent volume is always greater than the real volume, and only if the geometry is a cube are these two values equal.
In addition to the previous three variables that have been considered in previous studies, the variable of the ratio of maximum height to maximum length has also been investigated. To calculate this parameter, it is enough to divide the height by the length according to
Figure 7, and as a result, a dimensionless variable is obtained. Correlations between variables and responses are specified in
Table A1. It is observed that all the correlations are significant at the significance level of 0.01. Regarding the signs, it is also observed that, for example, as expected, the higher the surface-to-volume ratio, the shorter the release time of the entire drug mass because the sign of the correlation between the two is negative. It can be seen that, on average, the surface-to-volume ratio is the most effective variable, with an average correlation value of 0.89. According to
Table A1, it is concluded that all considered variables have a significant effect on the estimation of responses related to the release. Now, the second question must be answered: Is it possible to accurately predict the drug release profile by having these values, as in the previous part?
In this section, due to the strong correlation between variables, partial least squares regression is used so that we can ignore multicollinearity.
Table 3 provides the details of statistical modeling for all six responses. According to the values of the coefficient of determination and the mean absolute percentage error in
Table 3, the most suitable prediction is related to the response of “30 min”. This variable is presented as a representative of burst release. This high predictive power shows that by having common features, regardless of the type of geometry, the amount of burst release can be predicted, and the type of geometry cannot significantly affect it. But according to other responses, it can be seen that the coefficient of determination is about 0.90 on average, while on average, there is about 13.9% error in the prediction values. These values are significantly less accurate than the previous part of the results, i.e., modeling based on geometry dimensions, because in the previous part, the same values were about 0.96 and 1.5%, respectively, while in many studies, variables related to the surface were introduced as the determining factor in estimating the responses related to the diffusion process. In this part, in addition to this variable, three other variables were also added for modeling, but even so, the models could not provide significant accuracy. The important result is that the surface factor alone cannot accurately predict the release process and the type of geometry significantly affects these responses. Now another question arises as to why, contrary to what was expected, these characteristics, especially the surface factor, cannot be good regressors in the diffusion process. The following results can clarify this issue.
The two series of given information correspond to two matrices of two different geometries (
Table 4). In general, it is expected that the higher the ratio of the surface to the volume of a matrix, the less time it will take for the entire mass of the drug to be transferred to the dissolution medium, because more surface area results in faster release. It can be seen that the surface area (S) and the surface-to-volume ratio (S/V) for matrix #1 are smaller than the same values for matrix #2, but the release of the entire drug mass (95 rel) for matrix #2, contrary to expectation, requires more time. But the amount of burst release (30 min), as expected, is higher for the matrix with more surface area, i.e., matrix #2. Therefore, some responses are in line with our expectations and some are against our expectations. By looking at the variance of dissolution time (VDT) values, it is clear that the drug release process for matrix #2 is a far from linear process, and the percentage of drug released at the time intervals is very different. Therefore, the release rate frequently changes. In order to have a better understanding of the drug release process in these two matrices, we obtain help from
Figure 8a–c. It can be seen in
Figure 8a that initially the amount of drug released from matrix #2 is more than that of matrix #1. Therefore, as expected, the percentage of drug released in the first 30 min is higher for matrix #2, which has a higher surface-to-volume ratio. However, between 500 and 600 min, the two lines cross each other, and from then on, the percentage of drug released from matrix #1 is higher than matrix #2. Now it is better to examine the release rate. In
Figure 8b we see the percentage of the drug released at each moment. It is observed that in the initial moments, the rate of drug release from matrix #2 is higher, but gradually and a little after 200 min, the rate of drug release from matrix #1 surpasses it. Therefore, we can now state that the reason for the longer duration of complete release of the drug from matrix #2 is that after the first few minutes, the release of the drug from matrix #1 gradually speeds up, and the cumulative amount of drug released after about 520 min is more than the amount of drug released from matrix #2. But this increase in the release rate itself is not without reason.
Figure 8c shows the changes in surface-to-volume ratio against time. In this graph, it can be seen that in the initial moments, the ratio of surface to volume in matrix #2 is more than this value in matrix #1, but almost 100 min from the start of dissolution, the two curves cross each other, and gradually it is matrix #1 which has a higher surface-to-volume ratio. It was also pointed out in previous studies that the change in the shape of the matrix over time has a significant effect on the drug release process and can produce unexpected results [
38,
45]. Therefore, these results show from another point of view that the difference in geometry can have a direct and significant effect on the drug release process. This means that it is difficult to predict how the shape of each geometry will change at different moments, and they cannot be identified from the primary features. This section states how our prediction of a drug release profile can be wrong and far from reality without having the type of geometry and only relying on variables such as surface.
3.3. Comparison of Geometries
In this part, we try to compare the geometries based on their responses from a general point of view. To describe the difference in the average responses in different geometries, we must first measure the significance of these differences. For this, we use one-way ANOVA tests to interpret their results from several perspectives. In
Table A2, the
p-values (sig) indicate that the difference in means for all six investigated responses is significant. But if the average responses have a significant difference only between two groups of the eight investigated geometries, the result of the ANOVA test will show significance. To find out which differences between these groups are significant, we must use post hoc tests. The Tamhane post hoc test is used in this research. To have a better view of the results, the results of this test are completely summarized in
Table 5. The sign “✓” means the difference is significant and the sign “-” means the difference between the averages of two groups is not significant. All the signs on the diagonal line of
Table 5 indicate the lack of significance of the difference because, clearly, the difference in values between two identical groups is always meaningless. Therefore, a total of 28 comparisons have been made for each of the responses (168 comparisons in total). It should be noted that the level of significance in all investigated cases is set at 0.05. Comparing these results, one finds that all responses, except DE, have significant differences in 22 to 24 out of 28 possible situations. It means that only by changing the type of geometry and with the same amount of drug and without the slightest change in the formulation, the time required to release the entire mass of the drug (95 rel), the time the drug remains in the bloodstream (AUC), the ability of the matrix to keep the drug in dosage form (MDT), and the percentage of the burst release of the drug (30 min) change significantly. But the conditions for the DE response are distinct and significant differences were obtained in only 10 cases. Therefore, as stated in the first part of the results, this response is not particularly sensitive to geometry change. It also means that, contrary to the responses mentioned, in most cases, by changing the type of geometry, the bioavailability of the drug does not undergo significant changes. In general, except for DE, the other five responses undergo significant changes by changing the type of geometry of the matrix.
In the case of the 95 rel response, the range of changes was 374 to 2267 min (
Table 2). This means that in some cases, the time required to release the total drug between two geometries is different by about six times. It should be remembered that this difference occurs between two matrices with the exact same formulation and dosage. Thus, once again, these results show that the change in tablet geometry significantly affects the release-related responses. The results show that, on average, EXDC and TO geometries require the maximum and minimum time to release all the initial mass of the drug, respectively. More completely, the order of geometries with the maximum to minimum time required to release all the initial mass of the drug is as follows: 1—EXDC, 2—DC, 3—STC, 4—SHC, 5—FF, 6—FFBE, 7—CR, and 8—TO (
Figure 9a). Regarding the 30 min response, the range of changes was from about 6.5 to 19.7 percent (
Table 2). This amount of difference between the upper and lower limits of the interval is also remarkable considering the how little time is considered after the dissolution start moment, especially considering that, in general, the response related to burst release is of special importance from a biological point of view. The results show that, on average, TO and EXDC geometries have the maximum and minimum percentage of burst release, respectively. More completely, the order of geometries with the maximum to minimum percentage of burst release is as follows: 1—TO, 2—CR, 3—FFBE, 4—SHC, 5—STC, 6—FF, 7—DC, and 8—EXDC (
Figure 9b). The order of the average response for VDT can also be important because the lower the VDT value, the closer the release profile is to the zero-order release curve, and the higher the value, the further away it is from this curve. On average, among the geometries, EXDC and TO have the highest and lowest amount of VDT, respectively. Therefore, the TO geometry has the closest behavior to the zero-order release behavior.
The order of the average of the six investigated responses for the groups (eight geometries) is almost the same as the results that were expected according to the average values of surface-to-volume ratios. But, in some cases, it is different. For example, the average surface-to-volume ratio for SHC and FF geometries is 1.0 1/mm and 0.97 1/mm, respectively. It seems that the higher this value is, the time required to release 95% of the initial drug should be less, while the average time required to release 95% of the initial drug for SHC and FF geometries is 1514 and 1461 min, respectively (
Figure 9a). Similarly, in some cases, expectations about other responses are violated by comparing geometries. Therefore, once again, the results confirm that, despite the significant effect of surface-related variables on the drug release process, the final results do not always match the expectations caused by these variables. Therefore, not only on a case-by-case basis but also with a general look at the types of geometries, the difference in the geometry type produces unexpected results in drug release profile responses.