Edited by: Emilio Isaac Alarcon, University of Ottawa, Canada
Reviewed by: Jianmeng Wang, First Affiliated Hospital of Jilin University, China; Silviya Petrova Zustiak, Saint Louis University, United States
This article was submitted to Biomaterials, a section of the journal Frontiers in Bioengineering and Biotechnology
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The success of medical therapy depends on the correct amount and the appropriate delivery of the required drugs for treatment. By using biodegradable polymers a drug delivery over a time span of weeks or even months is made possible. This opens up a variety of strategies for better medication. The drug is embedded in a biodegradable polymer (the “carrier”) and injected in a particular position of the human body. As a consequence of the interplay between the diffusion process and the degrading polymer the drug is released in a controlled manner. In this work we study the controlled release of medication experimentally by measuring the delivered amount of drug within a cylindrical shell over a long time interval into the body fluid. Moreover, a simple continuum model of the Fickean type is initially proposed and solved in closed-form. It is used for simulating some of the observed release processes for this type of carrier and takes the geometry of the drug container explicitly into account. By comparing the measurement data and the model predictions diffusion coefficients are obtained. It turns out that within this simple model the coefficients change over time. This contradicts the idea that diffusion coefficients are constants independent of the considered geometry. The model is therefore extended by taking an additional absorption term into account leading to a concentration dependent diffusion coefficient. This could now be used for further predictions of drug release in carriers of different shape. For a better understanding of the complex diffusion and degradation phenomena the underlying physics is discussed in detail and even more sophisticated models involving different degradation and mass transport phenomena are proposed for future work and study.
Over the years different drug carriers have been developed and tested for drug delivery and targeting applications. In terms of materials, polymers are the ones mostly used, perhaps due to their simple forming properties in combination with easily tunable properties. In drug release the release phenomenon varies in complexity depending on the design and types of materials involved. For polymeric materials the mechanisms of drug release are normally directly linked to drug diffusion, dissolution, and degradation of the carrier matrix. However, other factors, such as interactions of the material and the drug, can also influence the release kinetics. In addition to physicochemical and morphological properties, the drug location within the matrix, and the drug solubility are key parameters governing the release kinetics and, therefore, the efficiency and efficacy of the treatment. It has been suggested that degradable materials could provide a steady and tunable release kinetics for different therapeutic applications. Furthermore, it was postulated that the use of combinatory materials for the design of drug release systems has the potential for improving drug bioavailability together with predictable release kinetics. Many efforts have been directed toward the development of biodegradable composite materials for drug delivery and targeted controlled release in terms of reproducible and predictable release kinetics in order to meet the therapeutic demands (Ginebra et al.,
This study focuses on the release of gentamicin and clodronate disodium bisphosphonate embedded or not in hydroxyapatite within in a polylactic acid matrix. Similar systems have already been used and are until now in the focus of medical interest (e.g., for the case of gentamicin in Schmidt et al.,
However, a considerable effort has also been made for modeling the degradation-drug release behavior in these and other drug carrying systems. One of the objectives is to enable and to accompany a fast and rational design of soluble drug release devices (see e.g., Lee,
If the symmetry of the considered carrier proves to be high then it becomes possible to reduce the simulation problem to the solution of a transient partial differential equation with one spatial dimension (e.g., Siepmann and Siepmann,
This paper is a first preliminary attempt to create this awareness. Initially experimental findings will be presented and then correlated with an essentially analytical diffusion model, which explicitly accounts for the underlying drug carrier geometry and, hence, becomes more than just a curve fit. In fact the predicted diffusion coefficients can be considered as geometry independent and useful when assessing the release times from other drug carriers of different geometry.
In this section the investigated drugs, their containment in hydrolyzable, polymer-based matrices (the carriers), the involved dissolution into a body-like fluid, and the corresponding measurement of the drug concentration in that fluid as a function of time will be described. It will be shown that various stages must be distinguished and a (verbal) explanation for their occurrence will be given.
In the experiments the dissolution behavior of two different drugs was investigated, namely of GentaMicin (GM) and of clodronate disodium BisPhosphonate (BP). Some information about their chemical properties can be found in
Chemical composition of drugs and drug carrier matrix.
Gentamicin (drug 1) | C21H43N5O7 | 477.6 | Solubility in H2O: 50 mg/ml | |
Clodronate disodium bisphophonate (drug 2) | CH2Cl2Na2O6P2 | 287.85 | Soluble in H2O | |
Polylactic acid (matrix) | (C3H4O2) |
74 250 | Crystallinity ≈35% |
Two different release media mimicking the body fluid were used for dissolution and diffusion, i.e., phosphate buffered saline solution for the GM and a tris-HCl buffer solution for the BP. The reason for this choice was the procedure applied when measuring the concentration of the released drug, which was sensitive to phosphorus (see the description in section 2.2). Both solutions had a pH of 7.4 and were kept at a “body temperature” of 37 ± 0.1°C. The drugs were stored in a nanoporous matrix made of PolyLactic Acid (PLA), either directly or first embedded in HydroxylApatite (HAp). More specifically coralline HAp was used, and the interested reader can find more information on this topic in corresponding publications by the authors (Ben-Nissan,
Choosing a matrix without or with drug embedding and two solutions led to four different experimental scenarios with corresponding concentration measurements, namely GentaMicin contained in PolyLactic Acid (PLA GM), GentaMicin loaded in HydroxylApatite and then contained in PolyLactic Acid (PLA HAp GM), BisPhosphonate contained in PolyLactic Acid (PLA BP), and, finally, BisPhosphonate (BP) loaded in HydroxylApatite (HAp) and then contained in PolyLactic Acid (PLA HAp BP). We proceed to explain the details of the measurements.
Drug loading to hydrothermally converted coralline HAp was conducted in a vacuum controlled rotavapor with the appropriate amount of either GM or BP mixed with HAp particles to give 10%w/w drug loading. The solution casting method was used during the development of the polymer film composites (either just enriched with the drug or with HAp loaded with drugs) where the PLA was first dissolved in chloroform under room temperature. Then it was mixed with drugs or HAp particles under a magnetic stirrer. After that it was sonicated for 10 min and casted on a petri dish. The solvent in the casted samples was allowed to evaporate under vacuum for 48 h. Finally a thin polylactic acid composite film resulted, which was cut into 2 cm pieces, the thickness of which was around 0.2 mm,
Our assumption is that the drug is homogeneously distributed within the containment. This is confirmed by the Scanning Electron Micrograph (SEM) shown in
SEM of the drug containment.
The drug release was conducted in a buffer solution at a volume large enough to provide complete dissolution of the drug loaded in the samples. The concentration of drug in buffer solutions was measured by using a UV-vis spectrophotometer (Agilent Technologies, Australia) for GM. The quantification of released BP was determined by using 31P-NMR (Agilent Technologies, Australia).
The raw data of the current (average) concentrations of the drug,
where
in dimensionless units.
More specifically, the release medium in liquid form of 15 ml was filled into Falcon™ conical tubes of 17 mm diameter and 120 mm length (see
The experimentally determined fraction of cumulative release,
Fractional cumulative release of GM from PLA thin film composite in PBS solution (pH 7.4, 37°C and 100 rpm) for fifteen weeks. Error bars refer to mean standard deviation of triplicate experimental data.
Fractional cumulative release of BP from PLA thin film composite in TrisHCl buffer solution, pH 7.4, at 37°C and 100 rpm for 11 weeks. Error bars refer to mean standard deviation of replicate experimental data.
The different slopes in the plots indicate that it becomes necessary to distinguish different stages of drug release due to different physical phenomena. We will now attempt to give reasons for the observed behavior based on the schematics shown in
Illustration of release stages for GM without and with HAp.
Evidently, the experiments with GM easily allow to distinguish several different stages of drug release. In fact one can note four different ones, identifiable by changes in slope or jumps in slope in the release curve. In order to provide some physical justification for this phenomenological observation, we argue as follows:
Stage I (week 1): In the case of the PLA GM measurements, we observe initially a burst. This accelerated release is due to drug particles situated on the outer surface of the matrix migrating into the release medium. The illustration in the first top inset of Stage II (weeks 1–3): PLA GM continues to show a significant release of drug, but the speed of release, i.e., the slope to the Stage III (weeks 3–5): Both PLA GM and PLA HAp GM show a stagnation of drug release. One might think that the process reaches an equilibrium. However, Stage 4 (see below) causes us to develop another hypothesis. This lag phase can be explained by the fact that the PLA matrix keeps degrading; but all of the drug close to the surface has already been released, see the third top and bottom insets in Stage IV (weeks 5–15): In both cases the polymer matrix finally starts to show strong degrees of deterioration, pores are forming through which the GM or the supply from the HAp GM can more easily diffuse, see fourth top and bottom insets in
In case of the BP experiments presented in Stage I (weeks 1 and 2): A strong burst of drug release is visible in the case of PLA BP. The burst is much more pronounced than for the case of PLA HAp BP, since in the latter configuration it is necessary to overcome the additional HAp barrier first and to supply drug to the polymer matrix. It also shows that the drug release for PLA GM and PLA HAp GM is less than for PLA BP and PLA HAp BP. Obviously the diffusion in the two latter cases is easier, maybe due to the smaller size of the BP drug molecules. Stage II (week 2–3): The drug release slows down in both cases, this process can be related to an equilibrium condition. At the same time, the formation of voids due to the degradation of the matrix is on its way. Stage III (week 3–11): In both cases the formation of voids and degradation of the matrix is sluggishly continuing in the case of PLA BP and slightly faster for PLA HAp BP. No strong jumps in the release speed were observed in this and all other stages. This causes us to believe that the corresponding solution fluid might be less aggressive.
In summary we may say that the cases of PLA GM and PLA BP should be describable by standard diffusion equations, whereas in the case of PLA HAp GM and PLA HAp BP a more complex simulation seems to be required. Here the standard diffusion equation should be equipped by a supply term mimicking the provision of drug from the HAp containment to the PLA matrix.
In this paper the emphasis will be on modeling the release processes in context with drug exclusively stored in the PLA matrix through which it diffuses, either slowly as long the polymer stays intact, or more quickly when the polymer gradually deteriorates and gives “more way” to the diffusing drug molecules. More precisely, the results shown in the orange curves of
A quantification of the blue curves in both pictures is more difficult, because the underlying physics should be observed in the modeling. Here the drug has to diffuse first through the HAp to enter the PLA matrix. In other words, the drug is supplied to the matrix and after that it diffuses through the PLA into the solution. The supply will be greater when the HAp starts to deteriorate, but it is still a supply, which will be part of a modified diffusion equation, but not in terms of an adjusted diffusion coefficient of drug diffusing through the polymer matrix. Hence this type of modeling will result in diffusion constants and in parameters characteristic of the supply, i.e., the release of drug into the polymer matrix. For the supply term in the diffusion equation an adequate constitutive equation must be stated. The idea is to base this relation on a micro-model for the drug diffusion through an HAp shell. However, for conciseness of this paper the corresponding quantification is left to future research.
Some remarks from the viewpoint of continuum physics are now in order. Typically the concentration,
In our simulations we will study the diffusion within a thin film matrix. The film is curled up in form of a cylindrical tube and fully immersed in the solution. For this geometry we will solve a transient diffusion equation of the form
where
with the unknown diffusion coefficient of the drug in the matrix,
where
This is one of the simplest form of an effective diffusion coefficient,
It should be mentioned that including a drain term in the diffusion equation is not the only way to account for the fact that the outflux of drugs stagnates at higher concentration levels in the solution. Indeed, it is possible to put
where
The diffusion flux can be used to obtain the mass outflux of drug,
where d
Hence the total mass of drug released into the solution follows by integration in time from which the drug release can be calculated:
The unknown diffusion constant can now be determined such that the predicted drug release agrees with the actually observed average release fraction
In the next subsections we present solutions to these various initial boundary value problems. We will start with a closed form solution for the concentration field
It was mentioned above that the drug is released from a hollow cylindrical tube, which formed by curling of a square thin film (the “matrix”) of height
In this spirit we argue that the diffusion process from the cylindrical tube into the solution takes place only in radial direction and the transport in axial direction of the tube walls to the solution can be ignored. This seems reasonable, because the cylinder walls are so thin when compared to height of the cylinder axis.
The ordinary Fickean diffusion equation for the concentration field
We will solve this equation only within the wall of a now infinitely long cylindrical tube, i.e., within the region
The initial concentration within the cylinder walls is a constant and given by:
The situation is illustrated in
where
Hence the total output of mass per unit time across the boundaries at
This will lead to a rise of drug concentration in the solution of d
In fact, if we divide this expression by
To find this solution is unfortunately also a numerical task. However, if the boundary conditions are time-independent and given by the value
where
and
They must be found numerically. It turns out that for our case of a very thin wall,
During the derivation of this formula use was made of the relations
Since during the derivation of Equation (22)
Equation (24) forms the basis for our numerical assessment in the next section.
Illustration of the geometry.
where
GM in PLA.
0.0 | 0.399 | 0.517 | 0.607 | 0.623 | 0.639 | 0.792 | 0.805 | 0.814 | 0.823 | 0.844 |
We are now in a position to compute an update for the diffusion coefficient:
For this purpose we have terminated the sum after the first term. This is possible because of the large values of α
The time dependence of diffusion coefficients for GM in PLA.
Now that the diffusion coefficients are known the progression of drug release can be calculated from the following formula:
where
Drug release predicted (blue curve) for GM in PLA in comparison with measurement data (red dots).
After week 5 the last diffusion coefficient was used to predict the drug release if there were no change in the release mechanism. This is indicated by the dashed blue line. Clearly the prediction does not match the actually observed data. Indeed, it has already been noted that after that time the matrix deteriorates strongly. Consequently, a different evaluation strategy for the diffusion is required. In fact, it is still based on Equation (26) with three subtle differences. First,
The resulting diffusion coefficients are indicated by the green dots in
A word of caution is finally in order. Note that there are no physical reasons that the diffusion coefficient decreases during the release process in weeks 1 through 5 and then, after the jump, once more in weeks 5 through 15. The diffusion coefficient should simply remain constant during these two time intervals. Its time-dependent behavior must be attributed to the fact that constant concentrations
Performing a comparison between our diffusion coefficients for GM in PLA is not as straightforward as one wishes it to be, mainly for two reasons. First, the diffusion data in the literature are often obtained by fitting release curves with the point source solution of the Fickean diffusion equation (see e.g., Crank,
In Dunn et al. (
In Table 1 from Zhang et al. (
This conversion formula was applied to the data leading to the values shown in
Diffusion coefficients
2 | - | 4.8 | - | - |
4 | - | 1.7 | - | - |
5 | 1.0 | 1.3 | 11.1 | 73.9 |
7 | - | 1.5 | - | - |
10 | - | 1.0 | 4.1 | 41.1 |
20 | - | - | 1.6 | 31.4 |
BP in PLA.
0.0 | 0.394 | 0.728 | 0.774 | 0.789 | 0.795 | 0.799 | 0.803 | 0.806 | 0.809 | 0.811 |
Diffusion coefficients
A comparison of our diffusion constants for BP in PLA with values from the literature is very difficult, since despite an intensive search such data seems currently not to be available. Billon-Chabaud et al. (
The apparent time-dependence of the diffusion coefficients shown in
Predicted fractional release based on constant diffusion coefficients for GM
We found for GM that
We now turn to the solution of the diffusion equation when using the effective concentration dependent diffusion coefficient according to Temkin, Equation (7). After using the same FE and optimization techniques as before we obtained the results shown in
Predicted fractional release based on effective diffusion coefficients (Temkin model) for GM
Finally Equations (3), (4) were solved (with
Predicted fractional release based on the Robin boundary condition (8) for GM
A first analysis predicted diffusion coefficients for GM of
Marine structures can play a vital role in the treatment of a wide range of human diseases including non-curable one by providing hydrothermally converted coralline HAp for drug loading and release within a polymer-based matrix. In drug delivery, they have demonstrated a potential for a controlled release of clinically active agents. Their combination with degradable polymers in terms of a drug containment facility supplying the drug in a controlled manner to the polymer and then finally to the body fluid has widened their application in orthopedics as implant coatings for the prevention of biofilm.
In this paper experimental results have been presented showing the release behavior without and with such HAp containments. Moreover, the establishment of a quantitative drug release kinetics model can help to speed up the controlled drug release systems manufacturing. Knowing and quantitatively describing the complexity of mechanisms will lead to mastering the release from these devices.
For this reason, a collaborative effort between materials scientists and continuum physicists has been made for the development of a physically sound and closed-form release model, presently only for the case without HAp containment. The experimental results presented in this paper have been careful considered and related to the theoretical aspects in this model incorporating diffusion and degradation of the polymer matrix. Quantitative results for time-dependent diffusion coefficients in a degrading polymer matrix were presented based on this closed-form 1D diffusion model for a thin film curled up to form a thin-walled cylinder. The non-physical nature of the time-dependence was discussed and average values for the diffusion coefficients were compared to former literature data. In addition first numerical investigations, based on FE and FV methods were presented, which confirmed the average values of the time-dependent diffusion coefficients from the analytical model. The presented diffusion coefficients can be considered as geometry independent and they are ready for predicting the release kinetics in other geometries but thin film, for instance, fine structures made from 3D polymer printing. These are in preparation by the authors.
In the future the following remains to be done: First, the presented cylindrical solution model should be evaluated for a continuous input of drug concentrations at the cylinder wall over time from a deteriorating PA matrix as well as HAp containment. More precisely: This will require even more detailed numerical analyses, based on finite element, finite volume, or finite difference methods. The development of such tools is currently underway and will result in time-independent diffusion coefficients covering the time span until massive degradation of the polymer matrix and for times after. It can be expected that the (time-dependent) diffusion constants obtained in this work will serve as good starting values for the iteration procedure involved in the numerical approach. Ideally, a micro-model leading to a constitutive equation for a supply term of drug to the polymer matrix should be developed and then be included in an extended diffusion equation. This relation will have be solved numerically with suitable initial-boundary value data. This would allow to study the effect of a HAp containment supplying drug to the polymer matrix. In a third step this constitutive relation must be extended to account also for the deterioration of the HAp container requiring further numerical analysis. The general idea is to separate the various physical mechanisms and to obtain physically meaningful, geometry independent parameters by linking such models to experimental observations.
All datasets generated for this study are included in the manuscript and/or the supplementary files.
IM and BB-N: experiments; WM and EV: model and theory development; AM: finite volume analysis; BA and WR: finite element analysis.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
IM wants to thank for support through the Deutsche Forschungsgemeinschaft for a TWAS grant (MU 1752/49-1) for his collaborative visit at Berlin University of Technology whilst EV and WM express their thanks to the support by DFG/RFFI grants No. MU 1752/47-1 and 17-51-12055.
1Not to be confused with
2The technical details of that study as well as the FV analysis mentioned further below will be published elsewhere.