Determination of tensile strength of shaped tablets
Graphical abstract
Introduction
Pharmaceutical tablets are the preferred forms of drug delivery and account for more than 70% of the solid oral dose drug products [3,4]. It is crucial that these tablets reach the patient intact, without any mechanical damage such as breaking, chipping, capping, etc. A common practice to evaluate the mechanical integrity of tablets is based on the tensile strength of the tablets. The tensile strength of a tablet is computed from the dimensions of the tablet and the breaking force during a diametrical compression test. The breaking force is the force at which the tablet breaks. For flat, round tablets, the breaking force is the compression force that is applied in the radial direction of the tablet [5].
However, most commercially available tablets are not flat, round in shape. Even though there is no limit to the types of shapes that can be formed using powder compaction, the most common shapes are doubly-convex round (referred herein as biconvex round or biconvex), capsule shaped, oval, and flat, round (Fig. 1). Particularly, for larger tablets, oval or capsule shapes are preferred to facilitate swallowing and reduce the esophageal transit time [6,7]. The shape of a tablet was also shown to have an effect on the drug release time [8]. Other reasons for having different shapes of tablets include commercial branding and easier identification of tablets by patients.
Flat, round tablets are used to assess the mechanical characteristics of formulations in early drug product development. The assessment includes the compressibility, tabletability, and compactibility of the tablets [9]. Ideally, it is preferable to develop these three curves for all shapes of tablets. However, during transfer of a drug product development technology to the commercial shape of the tablets, there is a lack of understanding on how to compute the tensile strength of shaped tablets.
Irrespective of the shape of a tablet, the diametrical compression test is commonly used to measure the tensile strength. The diametrical compression test as a method to measure the tensile strength of cylindrical shaped (referred herein as flat, round) samples was originally developed by Carneiro in 1943 [10]. This test is also referred to as indirect tensile strength test, splitting test, or Brazilian test [11,12]. The test has been used in various engineering application for several materials including concrete, ceramics, asphalt concrete, rocks, and pharmaceutical tablets. As the list of these materials indicates, the test is appropriate for brittle materials. The tensile strength of materials can also be measured using axial tensile strength test. Jarosz and Parrott [13] tested the tensile strength of tablets using both axial and diametrical tests. Their results show that for most tablets, the axial tensile strength tests gave a lower tensile strength value compared to that of the diametrical compression tests.
In other engineering fields, the mechanical strength of materials is always reported in terms of the maximum tensile stress the materials can support instead of a breaking force (please refer to the following articles for examples [[14], [15], [16], [17], [18], [19]]) The breaking force is commonly used to report mechanical strength of pharmaceutical tablets, but it is not a sufficient parameter to characterize the intrinsic property of the tablets. Reporting the mechanical strength in terms of maximum tensile stress allows for a universal comparison of materials independent of size and shape of tested sample. On the other hand, reporting the mechanical strength in terms of breaking force works only when comparing the mechanical strength for a given shape and size. For example, two flat, round tablets with the same tensile strength may have different breaking force if the thickness of the tablets are not the same even when other parameters (such as diameter, relative density, and compaction pressure) are kept constant. In other words, breaking force is size-dependent but tensile strength is size-independent, at least for the ranges of sizes that are of interest for pharmaceutical tablets.
The biggest hurdle preventing the use of tensile strength as a property to characterize the mechanical strength of shaped tablets is the difficultly of computing the tensile strength from the breaking force and the dimensions of the tablets. The breaking force can be measured directly, therefore, easy to report; whereas it is difficult to get a direct measurement of the tensile stress (and in most cases impossible). In limited circumstances, elastic stresses can be measured through optical techniques like photoelasticity, acoustics, or diffraction contrast tomography, and digital image correlation (DIC) based finite element method (FEM) simulations [20]. The photoelasticity method is possible only for transparent materials such as glass and plastics, and DIC is very complicated and computationally intensive method that only gives values based on a priori assumed models [21,22]. Unfortunately, for most applications, the tensile strength has to be computed, which can be challenging particularly for complicated shapes of tablets. To determine the stress distribution in 3D systems, such as oval and biconvex tablets, numerical solutions or FEM type simulations are usually required.
This paper has two main objectives. The first objective is to compare existing models for tensile strength prediction of convex and capsule shaped tablets. The effect of dimensional parameters, such as tablet diameter, thickness, and cup depth, were investigated for three models which are described in the next section. The second objective is to introduce an FEM model to compute the tensile strength of oval shaped tablets. The FEM model was kept as simple as possible so that the model can be run in short time and minimum effort. However, developing a general model for all types of oval tablets is out of the scope of this paper.
Section snippets
Theory
The distribution of the tensile stress (σxx), compressive stress(σyy), and shear stress (σxy) in flat, round tablets can be computed using Hertz's solution for stress distribution in disks as follows [23]:where x and y are the horizontal and vertical distances from the center of the disk, respectively. P is the applied force along the y-axis, and D is the disk diameter. and In
Comparison of models
The sensitivity of the mathematical models for the tensile strength of biconvex tablets were investigated by varying the dimensions of the tablets. The sensitivity analysis was done for the shape factors by changing W, D, and t independently, which represent changes in thickness of tablet, size of die, and shape of punch, respectively. In addition, the sensitivity of the models was investigated by changing W and t simultaneously. The latter investigation represents changing the shape of a
Tensile strength of biconvex tablets
We compared the models developed by Pitt et al. [39], Shang et al. [1], and Podzeck et al. [4] to find the similarities, differences, and limits of the models. The models were compared based on the shape factor for different sizes and shapes of biconvex tablets. Fig. 4 shows the comparison of for the three models. Overall, the trends and the values of are similar for the three model. is slightly higher than and in all cases that were tested. Fig. 4.a shows the effect
Conclusion
Determining the tensile strength of tablets is essential for understanding the integrity of tablets. Developing a mechanism to determine the tensile strength of shaped tablets is crucial during technology transfer from development to commercial production. Few models are available for biconvex tablets [1,4,39,40] and, to the best of our knowledge, only one model is available for capsule shaped tablets [2]. We found that the difference between the existing models for biconvex tablets is not
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
We thank Dr. Omar Sprockel and Dr. Faranak Nikfar for very helpful discussions and feedback.
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