Numerical Simulation on Pharmaceutical Powder Compaction Lianghao Han1,a, James Elliott1,b, Serena Best1,b and Ruth Cameron1,c A.C. Bentham2, A. Mills2, G.E. Amidon2 and B.C. Hancock2 1
Department of Materials Science and Metallurgy, University of Cambridge, CB2 3QZ,UK 2
a
Pfizer Global R&D, Sandwich, CT13 9NJ, UK
[email protected], b
[email protected], c
[email protected] , d
[email protected]
Keywords: powder compaction, constitutive model, finite element method, granular material
Abstract: In this paper, we present a modified density-dependent Drucker-Prager Cap (DPC) model with a nonlinear elasticity law developed to describe the compaction behavior of pharmaceutical powders. The model is implemented in ABAQUS with a user subroutine. Using microcrystalline cellulose (MCC) Avicel PH101 as an example, the modified DPC model is calibrated and used for finite element simulations of uniaxial single-ended compaction in a cylindrical die. To validate the proposed model, finite element simulation results of powder compaction are compared with experimental results. It was found that finite element analyses gave a good prediction of both the loading-unloading curves during powder compaction and the compaction force required for making a tablet with a specified density. Further, the failure mechanisms of chipping, lamination and capping during tabletting are investigated by analysing the stress and density distributions of powders during the three different phases of the tabletting processes, i.e. compression, decompression and ejection. The results indicate that the model has excellent potential to describe the compaction process for generic pharmaceutical powders. Introduction As a widely used dosage form in drug delivery, tablets have many advantages over other dosage forms, such as low cost, long term storage stability, good tolerance to temperature and humidity and ease of use by the patient. However, some common defects such as sticking, picking, chipping, capping and lamination, can occur during tabletting by uniaxial die compaction. Although several simple theories have been advanced to explain the causes of failure [1, 2], a more detailed stress analysis and simulation of the tabletting process are essential in order to quantitatively predict the occurrence of tablet failure and how it may manifest. Moreover, the tabletting process simulation can also help to understand the influence of tooling, lubrication and compaction kinematics (e.g. compaction speed and compaction sequence), and provide the guidance for improving the formulation and the tooling optimisation design. The phenomenological models, such as critical state models like the Cam-Clay and Drucker-Prager Cap (DPC) models, which were originally developed for geological materials in soil mechanics, have turned out to be well suited for modelling the powder compaction, especially in powder metallurgy [3-7]. Recently, Drucker-Prager Cap models have been used for the compaction analyses of pharmaceutical powders [8,9]. However, the unsuitable material parameter identification of DPC models may result in the unrealistic simulation of the decompression phase [8]. Furthermore, the linear elasticity model [9] is not suitable to describe the observed nonlinear unloading behaviour of pharmaceutical powders and to understand the elastic recovery of powders during unloading and after ejection. This is particularly important, since the elastic recovery may produce catastrophic flaws and initiate the cracks within compacts, causing the compaction failures such as chipping, lamination and capping [10, 11]. In this paper, a modified density-dependent DPC model with a nonlinear elasticity law was developed and implemented in ABAQUS by using user subroutines [12]. The nonlinear elasticity law as a function of the relative density and stress was used to describe the nonlinear unloading behaviour during powder compaction. Consequently, a new experimental calibration procedure was developed for the modified DPC model based on the uniaxial compaction tests using an
instrumented die. Using these parameters, finite element simulations were performed on the compaction process of microcrystalline cellulose (MCC) Avicel PH101 powders: compression, decompression, and ejection. The density and stress distributions during tabletting were investigated and used to analyse three typical failure mechanisms: chipping, capping and lamination. Constitutive Model Since the material parameters of pharmaceutical powders are density-dependent [9, 13], a modified density-dependent DPC model was used to describe the compaction behavior of pharmaceutical powders in the present work. Figure 1(a) shows one-quarter of the full 3D yield surfaces of a density-dependent DPC model in the principal stress space, where ρ is the relative density of the compact, and the symmetry axis of the yield surfaces is σ1 = σ 2 = σ 3 . Figure 1(b) shows a 2D schematics in p-q space, where p and q are the equivalent hydrostatic pressure stress and the Mises equivalent stress, respectively, and expressed in terms of the principal stresses as p = (σ 1 + σ 2 + σ 3 ) / 3 and q = 0.5 ⎛⎜ (σ − σ )2 + (σ − σ )2 + (σ − σ )2 ⎞⎟ . For a uniaxial cylindrical die ⎝ 1
2
2
3
1
3
⎠
compaction test, the hydrostatic pressure stress and Mises equivalent stress are respectively p = (σ z + 2σ r ) / 3 and q = σ z − σ r , where σ z is the axial stress and σ r is the expressed as: radial stress.
Figure 1 Schematics of a density-dependent Drucker-Prager cap model: (a) 3D yield surfaces in principal stress space (1/4 model) (b) 2D model in p-q space The yield surface of the DPC model is pressure-dependent, and includes three segments: a shear failure surface, providing dominantly shearing flow, a “cap,” which intersects the equivalent pressure stress axis, and a transition region between these segments, introduced to provide a smooth surface. The shear failure surface is defined by two parameters: internal friction angle, β , and cohesion, d . To define the yield surfaces of the DPC model, a total of five parameters, β , d , pa , R , pb , are required, all are functions of the relative density. Typically, the experimental observation on pharmaceutical powder compaction using flat-faced punches indicates a nonlinear elastic behavior during unloading, as shown in Fig. 2. Therefore a nonlinear elasticity law was proposed. The bulk and shear moduli may have the forms: K = K( p, ρ ) and G = G(q, ρ ) , which satisfy the condition that the elastic behavior must be path-dependent [14]. Both are no longer constants but functions of the relative density and the stress. For the sake of simplicity and easy numerical implementation, the unloading curve, AA’B was described with two linear segments, AC and CB. Consequently, to identify the material parameters of the modified DPC model, a new calibration procedure was also developed to extract the material parameters, based on the die compaction tests using an instrumented compaction simulator [15]. The modified density-dependent DPC model with the proposed nonlinear elasticity law has been implemented into the finite element package, ABAQUS/Standard, by writing a user-defined-field subroutine USDFLD.
Figure 2 Typical stress-strain relationship during the quasi-static die compaction of lubricated Avicel PH101: (a) axial stress vs. axial strain (b) radial stress transmission Results and Discussions As a demonstration, the compaction properties of microcrystalline cellulose (MCC) Avicel PH101 powders mixed with 1% w/w magnesium stearate, one of the most commonly used excipients in pharmaceutical industry, was measured by using a compaction simulator with an 8 mm diameter instrumented die (Phoenix Calibration & Services Ltd, Bobbington, UK), and used to extract the model parameters for the modified DPC model [15]. All the tests were carried out under quasi-static loading at a 0.1mm/s compaction speed. The die compaction process was simulated using ABAQUS/Standard. A 2D axisymmetrical model was adopted, due to the axisymmetry of the geometry and the loading conditions. The flat-faced upper/lower punches and the cylindrical die were modelled as analytical rigid bodies without any deformation, while the powder was modelled as a deformable body by axisymmetric elements. To compare with experimental results, an 8 mm diameter die and flat-faced upper/lower punches were used in the simulation. Figure 3 shows a finite element model to simulate the manufacture of flat-faced round tablets by uniaxial single-ended compaction. In the analysis, three phases of the quasi-static compaction process were simulated: compression, decompression and ejection: First, MCC Avicel PH101 powders with a 6 mm filling height in the die were compressed to a specified maximum compaction height by an upper punch while the lower punch was fixed; then the upper punch was removed from the die for decompression; and finally the lower punch was moved upward to eject tablets. The wall friction effect was considered by adopting a Coulombic boundary condition on the interfaces of the powder-die wall and powder-punch, and adjusted by changing die-wall friction coefficients.
Figure 3 Finite element mesh: (a) loose powder (b) compacted powder To validate the proposed model, single-ended quasi-static compaction tests were simulated, without considering die-wall friction. The loading-unloading curves of powders compacted to four different densities were predicted. The compaction force required to make a specified density tablet
was also predicted, and compared with the experimental results. Figure 4 shows a good finite element prediction. The compaction density was calculated from the weight and dimensions of powders in the die, while the tablet density was measured after the tablet was ejected from the die.
Figure 4 Finite element prediction of compaction behavior of Avicel PH101 powder: (a) loading-unloading curves during compaction (b) maximum compaction force In the practical tabletting process, the die-wall friction effect cannot be neglected. To consider the die-wall friction effect between the interfaces of powder-wall and powder-punch, a wall friction coefficient, µ , was set to 0.0 for the frictionless condition and 0.2 for the friction condition, respectively. Figure 5 illustrates the density distributions of flat-faced tablets after ejection, with or without considering wall friction. It was apparent that the die wall friction gave rise to a non-uniform density distribution. The highest density appeared at the top corner, while the lowest density appeared at the bottom corner. The simulations were consistent with experimental measurements [10, 16].
Figure 5 Density distribution of flat-faced round tablet (a) friction coefficient µ = 0.0 (b) friction coefficient µ = 0.2
Figure 6 Von Mises stress distribution of MCC Avicel PH101 powders compacted by flat-faced punches with a 0.2 wall friction coefficient: (a) maximum compression (b) after decompression (c) emerging from the die (d) during ejection
Figure 6 shows the von Mises stress distribution of powders considering the die-wall friction effect during tabletting. Again, the presence of wall friction gave rise to a non-uniform stress distribution. During the compression and decompression phases, the high stress regions were found at the top corner, while the low stress regions were found at the bottom corner. During the ejection phase (Figs. 6(c) and 6(d)), high stress regions close to the die edge were formed once the powder emerged from the die, due to the radial elastic recovery of part of tablets located outside of the die. The density and stress distributions can be linked with the origins of three typical defects: chipping, capping and lamination (Fig. 7).
Figure 7 Typical failure types of tablets (a) chipping (b) capping (c) lamination
Typically Chipping occurs at the edge of the tablet. As shown in Fig. 5(b), the edge of the tablet was normally the region with a larger density gradient, which could be caused by the wall friction or the punch geometry. The edge of tablets may be broken by the large stress concentration due to the radial elastic recovery (Fig. 6(d)) during ejection, or by the force applied on the tablet edge during the tablet handling after ejection, causing chipping.
Figure 8 A comparison between (a) cracks from X-ray tomographic imaging [8] and (b) von Mises stress distribution (see Fig. 6(c) for details)
Capping generally refers to the top of the tablet or cap becomes detached from the main body, either at the time of compression or after the tablet has left the die. Lamination involves the occurrence of layers in a compact parallel to the tablet face. Since lamination is a precursor to capping, the terms lamination and capping are often used interchangeably [1, 17]. If the wall friction was considered, there exists a local stress concentration at the top corner after decompression (Fig. 6(b)), which may initiate the crack at the top corner, resulting in capping. Once the tablet emerged from the die (Fig. 6(c) and 6(d)), a larger stress concentration was observed due to the radial elastic recovery of the tablet [18]. The stress concentration was affected by the radius of the die edge, namely, the larger the die edge radius, the higher the stress concentration. These local stress concentrations may cause crack initiation, causing capping at the top corner, and a single lamination event or multiple laminations at the other regions. It is not surprising that the stress distribution has a similar pattern to the observed cracking in X-ray tomography experiments, as shown in Fig. 8. As the tablet emerges from the die, the crack may initiate on the top corner with the highest stress (e.g. point A in Fig. 8(b)). The crack propagation direction will depend on both the stress and density distributions. It is more likely that the crack will propagate along the relatively weak powder interface, that is, the low density areas, and the areas with high local stress concentrations (e.g. dashed line AB in Fig. 8(b)).
Conclusion
Although several simple theories can explain the causes of failure during pharmaceutical tabletting, a detailed process simulation and stress analysis are necessary in order to quantitatively predict the tablet quality and the failure mechanism. In this paper, a modified density-dependent Drucker-Prager Cap model with a nonlinear elasticity law was developed and implemented in ABAQUS by using a user subroutine. The proposed model was used to model the tabletting process of MCC Avicel PH101. We simulated not only the compression and decompression phases but also the ejection phase, during which tablet failures took place more frequently. The finite element analyses of the density and stress distributions during tabletting provided a method for the future quantitative assessment of the origins of the tablet failures. Once the relationship between the density and the tablet strengths was built up, it is possible to quantitatively assess the tablet quality and predict the failure probability based on the density and stress distributions from finite element analyses. Acknowledgements
This work has been carried out at the Pfizer Institute for Pharmaceutical Materials Science and the authors would like to acknowledge the funding support of Pfizer Ltd. References
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