Numerical Investigation of the Residual Stress Distribution of Flat-Faced and Convexly Curved Tablets Using the Finite Element Method

Abstract

The stress distribution of tablets after compression was simulated using a finite element method, where the powder was defined by the Drucker–Prager cap model. The effect of tablet shape, identified by the surface curvature, on the residual stress distribution was investigated. In flat-faced tablets, weak positive shear stress remained from the top and bottom die walls toward the center of the tablet. In the case of the convexly curved tablet, strong positive shear stress remained on the upper side and in the intermediate part between the die wall and the center of the tablet. In the case of x-axial stress, negative values were observed for all tablets, suggesting that the x-axial force always acts from the die wall toward the center of the tablet. In the flat tablet, negative x-axial stress remained from the upper edge to the center bottom. The x-axial stress distribution differed between the flat and convexly curved tablets. Weak stress remained in the y-axial direction of the flat tablet, whereas an upward force remained at the center of the convexly curved tablet. By employing multiple linear regression analysis, the mechanical properties of the tablets were predicted accurately as functions of their residual stress distribution. However, the multiple linear regression prediction of the dissolution parameters of acetaminophen, used here as a model drug, was limited, suggesting that the dissolution of active ingredients is not a simple process; further investigation is needed to enable accurate predictions of dissolution parameters.

Tablets are the most common and popular dosage form of drug administration. Tablets are generally manufactured by compressing a mixture of dry powders or granules using metal dies and punches. Various stresses remain on the surface and in the interior of tablets after the compression. The effect of the residual stress distributions of tablets on tablet hardenss^1–8) and tableting failures such as capping and lamination^9,10) have been numerically investigated. The finite element method (FEM), in which the powder is modeled using the Drucker–Prager cap (DPC) model,^9–14) can be applied to modeling the deformation of pharmaceutical powders^11,15) and thus to simulate the residual stress distribution of tablets. Powders are modeled as continuum media in the FEM and the compaction process is identified by boundary-value analysis. The DPC model is often applied to the analysis of the stress distribution and relative-density changes of the tablets during the tableting process. In examples of the use of the DPC model, Han et al.¹²⁾ reported that the density distributions of the tablets were affected by the punch geometry while Wu et al.^9,10) described how tablet failure is more likely to be associated with a band of intensive shear stress generated during the decompression of tablets. Additionally, the density distribution patterns obtained using the DPC model are similar to experimental results.^15–19) However, a few problems remain to be solved. Powders such as microcrystalline cellulose (MCC) and lactose (LAC) have been used to prepare the model formulation in many studies,^6,9–11,15) despite actual pharmaceutical products including many different types of excipients. We previously demonstrated that the residual stress distribution of flat-faced tablets is affected by the formulation factors, such as the quantities of MCC, LAC and corn starch, and is closely related to tablet characteristics, such as the tensile strength and disintegration time.²⁰⁾ Furthermore, we clarified the stochastic relationships between the tableting process variables, the DPC parameters, and the residual stress distribution of tablets by employing a Bayesian network (BN) analysis.²⁰⁾ The BN model showed the probabilistic relationship between process variables, simulation parameters, tablet characteristics, and residual stress distributions. Results demonstrate that the FEM is a useful tool that can be used to help improving our understanding of residual stress and to optimize process variables, which are considered factors not only affecting tablet characteristics but also of the risk of tableting failure.²⁰⁾ Although BN analysis is a superior method of clarifying the overall structure underlying the tableting process, it cannot provide a quantitative correlation between the residual stress distribution and the critical pharmaceutical characteristics.

Most commercially available pharmaceutical tablets have gently convex surfaces that make swallowing easy in oral drug administration. It is likely that the tablet shape affects the stress distribution remaining in the tablets,^3,10) and thereby critical pharmaceutical characteristics. The present study investigates the effect of the tablet shape identified by the surface curvature on the residual stress distribution of tablets that were prepared according to pragmatic formulation and process variables. In addition, we analyzed the data obtained using multiple linear regression (MLR) models. Recently, nonlinear approximation techniques, such as the use of an artificial neural network,²¹⁾ genetic programming,²²⁾ and thin-plate spline interpolation,^23,24) have been employed to ensure the highly reliable prediction of pharmaceutical characteristics. However, the mathematical theory of these nonlinear techniques is rather complex and it is often difficult to analyze the mechanistic means of individual causal factors. Compared with these techniques, MLR analyses give us the linear combination of causal factors and can better clarify the mechanistic effects of the residual stress distribution on pharmaceutical characteristics, such as the hardness, disintegration time, and drug dissolution profile, while considering differences in tablet shape.

Experimental

Materials

Acetaminophen was purchased from Mallinckrodt Japan Co., Ltd. (Tokyo, Japan). Lactose (LAC, Tablettose^® 80) was purchased from Meggle Japan Co., Ltd. (Tokyo, Japan). Cornstarch (CS) was purchased from Nippon Starch Chemical Co., Ltd. (Osaka, Japan). Microcrystalline cellulose (MCC; Ceolus^® PH-101) was purchased from Asahi Kasei Chemicals Co., Ltd. (Tokyo, Japan). Low-substituted hydroxypropyl cellulose (L-HPC) was obtained from Shin-Etsu Chemical (Tokyo, Japan). Magnesium–stearate (Mg–St) was purchased from Wako Pure Chemical Industries, Ltd. (Osaka, Japan). Other reagents used were of analytical grade.

Preparation of the Model Tablets

All ingredients listed in Table 1 were dried at 75°C for 24 h and sieved through a 60-mesh screen. The sieved ingredients were accurately weighed according to the designated formulations and all ingredients, except Mg–St, were blended in a polyethylene bag for 2 min. The appropriate amount of Mg–St as lubricant (0.5–2.0%) was then added to the mixture, which was mixed for 1 min. A 200 mg quantity of the mixed powder was compressed with a force of 4–8 kN into a flat-faced or convexly curved tablet (8 mm in diameter) using a tableting machine (AUTOTAB-500; Ichihashi-Seiki Company, Ltd., Kyoto, Japan). Hard chrome plating punches with a flat-face and curvature radii of 16 and 12 mm were used to prepare the flat-faced tablet (FLAT) and convexly curved tablets (R16, R12), respectively.

Table 1. Formulation of Acetaminophen Tablets

Excipient	Quantity/tablet (mg)	Ratio (%)
Acetaminophen	50.0	25.0
Lactose (LAC)	31.0	15.0
Corn starch (CS)	25.0	13.0
Microcrystalline cellulose (MCC)	85.0–88.0	42.5–44.0
Low-substituted hydroxypropyl cellulose (L-HPC)	5.0	2.5
Magnesium–stearate (Mg–St)a)	1.0–4.0	0.5–2.0
Total	200.0	100.0

a) Contents of Mg–St were adjusted by the amounts of MCC.

Determination of Tablet Characteristics

The tensile strength (TS), the disintegration time (DT), percentage of acetaminophen dissolved at 3 min (D_3 min), and 50% dissolution time (t_50%) were determined as critical pharmaceutical characteristics. The breaking force of tablets was determined using a tablet hardness tester (PC-30; Okada Seiko, Tokyo, Japan). TS values of flat-faced and convexly curved tablets were calculated using Eqs. 1 and 2,²⁵⁾ respectively:   

(1)

  

(2)

where F is the breaking force, D is the tablet diameter and t is the total thickness of the tablet and W is the thickness of the band part. The disintegration test was performed according to the JP16 disintegration test for tablets using a disintegration tester (NT-20H; Toyama Sangyo Co., Ltd., Osaka, Japan) and water (as a test medium) at 37±2°C. DT was defined as the time required for the complete disappearance of a tablet and its particles from the tester net. DT was measured for three tablets of each formulation. The dissolution test was performed according to JP16 dissolution test No. 2 (the paddle method) at 50 rpm (NTR-6100 A; Toyama Sangyo Co., Ltd., Osaka, Japan). The dissolution medium used was 900 mL of distilled water at 37±0.5°C. The samples were collected and filtered after 1, 3, 5, 10, 15 and 30 min. The concentration of acetaminophen was measured spectrophotometrically at 244 nm using a V-650 spectrophotometer (Japan Spectroscopic Co., Ltd., Tokyo, Japan). The dissolution rates of three tablets of each formulation were measured.

Simulation of the Residual Stress Distributions of Tablets Using the FEM

Half of the cross-sectional axisymmetric part (right side) was analyzed using the two-dimensional FEM (Fig. 1). The die wall and upper punches were assumed as rigid bodies. The interaction between the powder, die wall, and upper punch was modeled, and the friction of the contacts was set to 0.1, in accordance with a previous paper.²⁰⁾ On the symmetry axis, nodes were restrained to move horizontally, and the nodes at the bottom boundaries were allowed to move only vertically. The upper punch moves vertically with compression. The normal x-axial and y-axial stresses and the shear stress distributions were simulated using the FEM, where the powder was defined by a DPC model. The FEM simulation was conducted under the condition that the pressure from the upper punch was released, but the tablet was restricted by the die wall (Fig. 1). The normal and the shear stresses were previously defined as follows: The x-axial stress (σ_x) acts horizontally, increasing from the left to the right. The y-axial stress (σ_y) is vertically and increases from the bottom. The shear stress is given as a tensor quantity composed of two elements, τ_xy and τ_yx, and the τ_xy value was used as an index of the shear stress, because their magnitude is exactly same. The DPC model is described below.

Fig. 1. Flow Chart of FEM Analysis

An axisymmetric two-dimensional model (right half) was used. The powder was modeled using the DPC model.

DPC Model

The plastic deformation, elastic deformation, and internal friction of powders were represented using the DPC model. The constitutive equations attributed to the DPC model were fully described in a previous paper.²⁰⁾ Briefly, the DPC model parameters that have to be measured from the experiments are mainly Young’s modulus (E), the Poisson ratio (ν), the internal friction angle (α^y), and three plastic deformation parameters; i.e., a maximum possible plastic volumetric strain (W₁^c) and two cap hardening exponential terms (D₁^c and D₂^c). A direct shear tester (NS-V100; Nanoseeds Corp., Gifu, Japan) was used to measure the failure envelope and to estimate the α^y value. Before measurement, all ingredients were dried at 75°C for 12 h. The powder (2.0 g) was added to the shear cell. Shear stresses were then measured when the powder bed was compressed at 10, 40, and 60 N. The observation data were plotted on a plane of the axial shear and stress. The failure envelope was estimated from the linear approximation of the axial shear versus stress plots. The failure envelope was measured for three powders of each formulation. Elastic moduli (E and ν) were estimated using an instrumented hydraulic press (TK-TB20KN; Tokushu Keisoku Co., Ltd., Yokohama, Japan). The axial upper/lower punch forces and displacements, and the radial die-wall pressure were measured during compaction. Compression and decompression speeds were set at 1 mm/s and 350 mg of sample powder was used. The relative packing density at 2 MPa was measured according to the volume of the gap between the upper and lower punches to calculate the initial density (D_i) of the powder bed. The DPC parameters described above were obtained as average for three trials. Plastic deformation parameters (W₁^c, D₁^c and D₂^c) were statistically estimated using the volume change as an indicator. Other parameters were set as arbitrary values.

Computer Programs

The FEM analysis of the tableting process, in which the powders were modeled by the DPC model, was performed using ANSYS^® version 14.5 (ANSYS Inc., Canonsburg, PA, U.S.A.). The MLR analysis was performed using JMP^® version 8 (SAS Institute Inc., Cary, NC, U.S.A.).

Results and Discussion

Residual Stress Distribution

The results of the measurement of Young’s modulus (E), Poisson ratio (ν), an internal friction angle (α^y), an initial density (D_i), and plastic deformation parameters (W₁^c, D₁^c and D₂^c) of the test powders are summarized in Table 2. For typical examples of the residual stress distributions of tablets after decompression, cross-sectional images of half of the axisymmetric part (right side) are shown in Fig. 2. In the FLAT tablets, weak positive shear stress (τ_xy) remained from the top and bottom die walls toward the center of the tablet. A weak negative shear was seen in opposite directions against positive τ_xy values. Almost no change in the shear stress distribution was observed irrespective of variation in the compression force (4 to 8 kN) and the content (0.5 to 2.0%) of Mg–St as a lubricant. In the R12 tablets, a positive τ_xy value remained in the neighborhood of the upper punch. This may arise from the tableting process, in which only the upper punch moves vertically to compress the powder whereas the lower punch is fixed during the compression. In contrast to the FLAT tablets, the compression force near the edge of the R12 tablets may act in the direction normal to the tablet surface; the upper punch force may then slip off inward rather than vertically to the peripheral area of convexly curved tablets. It has also been reported that the propagation of force from the die wall to the tablet interior was considerably different between the flat and convexly curved tablets.¹⁶⁾ When the compression force increased from 4 to 8 kN at the lower level of lubricant (0.5%), the strong positive τ_xy value remained on the upper side and in the intermediate part between the die wall and the center of the tablet. This stress distribution weakened with increasing lubricant content from 0.5% to 2.0%. In the case of x-axial stress (σ_x), negative values were observed in all areas of all tablets, suggesting that the x-axial force always acts from the die wall toward the center of the tablet. This may be because the simulation was conducted with the die wall as a restriction. In FLAT tablets, the negative σ_x value remained from the upper edge to the center bottom. The degree of negative x-axial stress (from the die wall to the center of the tablet) was enhanced with increasing compression force and increasing content of lubricant from 0.5 to 2.0%. It is likely that the segregation tendency in the powder bed observed at a low compression force (4 kN) and low content of lubricant (0.5%) slightly reduces with increasing compression force and increasing lubricant content. The σ_x distribution in R12 was quite different from that in the FLAT tablet. Strong negative stress remained on the border of the die wall and weakened approaching the center of the tablet. Increasing the compression force and the lubricant content resulted in a much higher σ_x value close to the die wall. Rather weak stress remained in the vertical direction (σ_y) in the FLAT tablet. In contrast, the upward force remained at the center of R12. The σ_y stress remaining at the center of the tablet may cause tableting failure such as sticking, and such failure tends to take place in the convexly curved tablets rather than in the flat-faced tablets.

Table 2. Simulation Parameters for DPC Model

Mg–St (%)	Force (kN)	Elastic moduli		Internal friction angle (°)	Initial density (mg/mm3)	Plastic deformation
		Young’s modulus (GPa)	Poisson ratio			W1c	D1c	D2c
0.5	4	3.01±0.74	0.0766±0.0058	25.4±0.7	0.690±0.023	0.600	0.0168	5.42E-09
	6	5.75±0.22	0.0742±0.0271
	8	6.51±0.03	0.0772±0.0032
1.0	4	2.90±0.14	0.0782±0.0029	24.1±0.3	0.694±0.024	0.587	0.0210	4.18E-09
	6	4.42±0.15	0.0794±0.0025
	8	5.74±0.24	0.0839±0.0037
1.5	4	2.96±0.31	0.0826±0.0000	23.2±0.5	0.692±0.031	0.600	0.0170	5.5E-09
	6	4.43±0.15	0.0831±0.0018
	8	5.44±0.33	0.0908±0.0033
2.0	4	2.70±0.95	0.0846±0.0160	21.2±1.3	0.707±0.015	0.578	0.0171	7.03E-09
	6	4.51±0.27	0.0851±0.0048
	8	5.57±0.32	0.0943±0.0054

Young’s modulus, Poisson ratio, the internal friction angle and the initial density are represented as the mean±S.D. for three determinations. Plastic deformation parameters were statistically estimated from the powder compression data (see ref. 20).

Fig. 2. Two-Dimensional Maps for Residual Shear (τ_xy), x-Axial (σ_x) and y-Axial (σ_y) Stress Distributions of FLAT and R12 Tablets Using FEM Analysis

MLR Analysis of Pharmaceutical Characteristics

TS and DT were selected as basic mechanical properties of tablets. To compare the flat-faced and convexly curved tablets, the breaking force measured was transformed to the TS value using Eqs. 1 and 2. TS for flat-faced tablets can be obtained using Eq. 1. Pitt et al. proposed Eq. 2 for estimation of the TS of a convexly curved tablet as an empirical equation obtained from a set of data for gypsum discs in a statistical manner.²⁵⁾ Although the use of Eq. 2 may be restricted to a limited range of tablet shapes and materials, we successfully calculated the TS of convexly curved tablets by using Eq. 2. D_3 min and t_50% were selected as parameters exhibiting dissolution profiles of acetaminophen. D_3 min was selected because the largest difference between the tablets was observed at 3 min after starting the dissolution test. The parameter t_50% is widely recognized as a general standard for dissolution profiles. These data are summarized in Table 3. The tablet characteristics (TS, DT, D_3 min, t_50%) were predicted as functions of the residual stress distribution. As shown in Fig. 3, nine points of each residual stress distribution were selected as representative points, and 27 stress values were then selected from the shear, normal x-axial, and y-axial stress distributions. In the MLR analysis, a stepwise backward regression method was applied to the selection of significant factors, and factors with p>0.25 were successively excluded from the analysis. As a judging standard, p>0.25 was used as a default value in JMP^® software. Lowering the p value increases the risk of involving insignificant factors, while significant factors are eliminated by tightening the criterion.

Table 3. Pharmaceutical Characteristics Determined for FLAT, R16 and R12 Tablets

Tablet shape	Mg–St (%)	Force (kN)	TS (MPa)	DT (log(s))	D3 min (%)	t50% (min)
FLAT	0.5	4	0.93±0.08	1.07±0.02	78.3±5.2	1.56±0.55
FLAT	1	4	0.80±0.10	1.12±0.02	78.8±6.9	1.86±0.11
FLAT	1.5	4	0.98±0.15	1.26±0.05	59.2±25.1	3.52±2.71
FLAT	2	4	0.53±0.05	1.32±0.02	24.2±10.2	8.73±3.33
FLAT	0.5	6	1.28±0.22	1.36±0.06	90.6±7.1	1.77±0.09
FLAT	1	6	1.14±0.34	1.09±0.02	78.3±5.2	1.95±0.47
FLAT	1.5	6	1.01±0.01	1.34±0.01	78.8±6.9	1.70±0.07
FLAT	2	6	0.93±0.01	1.43±0.02	59.2±25.1	5.84±2.03
FLAT	0.5	8	1.66±0.27	1.47±0.05	24.2±10.2	1.71±0.07
FLAT	1	8	1.40±0.05	1.17±0.02	90.6±7.1	2.20±0.67
FLAT	1.5	8	1.28±0.14	1.45±0.00	74.3±12.1	1.90±0.26
FLAT	2	8	1.16±0.06	1.39±0.02	89.4±3.4	4.27±1.75
R16	0.5	4	0.68±0.07	0.95±0.00	30.5±15.3	2.54±0.24
R16	1	4	0.65±0.04	1.14±0.02	92.6±5.6	2.14±0.48
R16	1.5	4	0.58±0.06	1.24±0.01	76.9±22.4	5.47±2.24
R16	2	4	0.51±0.02	1.38±0.00	80.3±14.3	7.63±3.94
R16	0.5	6	0.97±0.01	1.14±0.06	45.7±13.2	1.77±0.07
R16	1	6	1.10±0.13	1.23±0.00	57.6±5.9	1.89±0.22
R16	1.5	6	1.08±0.17	1.25±0.03	68.7±14.7	1.81±0.15
R16	2	6	0.77±0.05	1.39±0.01	34.6±15.8	8.23±4.05
R16	0.5	8	1.60±0.05	1.22±0.02	29.3±15.5	1.86±0.07
R16	1	8	1.29±0.06	1.41±0.00	92.6±4.1	1.73±0.42
R16	1.5	8	1.13±0.10	1.42±0.04	74.4±9.9	1.84±0.20
R16	2	8	0.84±0.06	1.53±0.01	77.5±10.5	3.14±1.21
R12	0.5	4	0.79±0.10	0.94±0.06	30.0±16.1	2.42±1.12
R12	1	4	0.60±0.13	1.18±0.03	88.9±3.8	2.04±0.43
R12	1.5	4	0.62±0.07	1.28±0.00	83.7±16.7	3.19±0.88
R12	2	4	0.36±0.06	1.35±0.05	86.1±14.6	5.65±2.36
R12	0.5	6	1.06±0.20	1.09±0.02	56.4±15.3	1.79±0.09
R12	1	6	1.05±0.12	1.29±0.01	84.4±12.7	1.80±0.07
R12	1.5	6	1.24±0.19	1.30±0.02	69.5±12.8	3.07±0.29
R12	2	6	0.68±0.07	1.45±0.02	44.2±17.2	4.65±2.23
R12	0.5	8	1.46±0.06	1.16±0.02	32.6±10.1	1.86±0.07
R12	1	8	1.20±0.04	1.38±0.04	89.4±4.8	1.53±0.21
R12	1.5	8	1.26±0.10	1.42±0.02	90.4±8.9	2.39±0.16
R12	2	8	0.92±0.02	1.54±0.01	48.3±7.3	3.04±0.82

Each datum represents the mean±S.D. for three determinations.

Fig. 3. Sampling Points of Shear (τ_xy), x-Axial (σ_x) and y-Axial (σ_y) Stress Values in FLAT, R16 and R12 Tablets

MLR results for DT, TS, D_3 min and t_50% are summarized in Tables 4–7, respectively. The relationships between predicted and experimental values for these pharmaceutical characteristics are shown in Figs. 4–7. The highest R² and R*² (an R² value adjusted for the number of degrees of freedom) values were observed with the prediction of DT (Table 4). An excellent one-to-one agreement between predicted and experimental values was seen in the scatter plot of DT (Fig. 5). A fairly good approximation was also observed for the prediction of TS (Table 5, Fig. 4). Both the breaking and disintegration of tablets are essentially mechanical properties of the tablets. That is why these pharmaceutical characteristics are dependent on the residual stress distributions. As summarized in Table 3, D_3 min values might be appropriate for exhibiting overall dissolution behaviors because the dissolution of acetaminophen was considerably fast in all tablets. For many model tablets, t_50% values are less than 3 min, and it is rare for t_50% values to be more than 8 min. The MLR prediction of D_3 min was limited compared with the cases of TS and DT (Table 6, Fig. 6). For the FLAT tablets, only the shear stress was a significant factor for prediction, while D_3 min of R12 was predicted mainly as a function of the normal stress. The prediction of D_3 min was reasonable for R12, but poor for R16. The prediction ability of t_50% was poor for all tablets (Table 7, Fig. 7), suggesting that the dissolution of acetaminophen is not a simple process, and further study is needed to be able to predict accurately the dissolution parameters on the basis of mechanistic analysis.

Table 4. MLR Analysis for DT (log(s))

Tablet	FLAT		R16		R12
Factor	Coefficient	95%CI	Coefficient	95%CI	Coefficient	95%CI
Intercept	−4.72	1.53	−0.889	0.162	−1.80	0.27
τxy3	−0.749	0.144			−0.117	0.022
τxy4					−0.369	0.201
τxy5	1.32	0.28
τxy6	−2.28	0.37	−0.396	ns
σx1			−0.0214	ns	0.0468	0.0155
σx2	−0.111	0.072	0.121	0.080
σx3	−0.0201	0.0139
σx4			0.334	0.126
σx5			−0.433	0.125
σx6					−0.0928	0.0155
σx9	0.161	0.030
σy1	0.382	0.073
σy3			−0.0302	0.0208
σy4	0.394	0.042	−0.131	0.044
σy9	−0.200	0.056	0.0100	ns	0.000462	ns
R2	0.963		0.982		0.964
R*2	0.951		0.977		0.958
RMSE	0.0317		0.0237		0.0338

Sampling points of residual stresses were shown in Fig. 3. Coefficient: partial regression coefficient, 95%CI: 95% confidence interval, ns: not significant, R*²: adjusted R² for the number of degrees of freedom, RMSE: residual mean squared error.

Table 5. MLR Analysis for TS (MPa)

Tablet	FLAT		R16		R12
Factor	Coefficient	95%CI	Coefficient	95%CI	Coefficient	95%CI
Intercept	−1.14	ns	2.77	0.48	2.89	0.89
τxy1			−1.19	0.86
τxy2	0.858	0.257
τxy4	16.0	5.2
τxy5					0.0986	0.0185
τxy6			−4.74	1.60	0.412	0.180
σx2			−0.156	0.077
σx6			0.643	0.225
σx7	−0.0770	0.0438
σy3			−0.106	0.047
σy4					−0.00741	0.00263
σy8			0.253	0.064
σy9	−0.0298	0.0258
R2	0.818		0.954		0.851
R*2	0.794		0.944		0.837
RMSE	0.144		0.0763		0.136

Sampling points of residual stresses were shown in Fig. 3. Coefficient: partial regression coefficient, 95%CI: 95% confidence interval, ns: not significant, R*²: adjusted R² for the number of degrees of freedom, RMSE: residual mean squared error.

Table 6. MLR Analysis for D_3 min (%)

Tablet	FLAT		R16		R12
Factor	Coefficient	95%CI	Coefficient	95%CI	Coefficient	95%CI
Intercept	−453	160	169	34	298	85.3
τxy1	90.4	ns
τxy3	−29.4	23.7
τxy4	−128	ns
τxy5	50.3	14.8
τxy6	−184	49
τxy7					597	268
τxy9					7.58	4.53
σx1					30.1	8.4
σx2					−19.4	6.7
σx8			3.01	0.96
σy2					27.5	14.8
σy3					5.36	2.92
σy6					−44.7	18.9
σy7					2.53	ns
R2	0.722		0.527		0.831
R*2	0.675		0.513		0.781
RMSE	14.4		17.4		10.9

Sampling points of residual stresses were shown in Fig. 3. Coefficient: partial regression coefficient, 95%CI: 95% confidence interval, ns: not significant, R*²: adjusted R² for the number of degrees of freedom, RMSE: residual mean squared error.

Table 7. MLR Analysis for t_50% (min)

Tablet	FLAT		R16		R12
Factor	Coefficient	95%CI	Coefficient	95%CI	Coefficient	95%CI
Intercept	43.8	13.1	−5.07	4.62	−14.2	7.5
τxy1	−5.50	ns
τxy2	−0.931	0.804
τxy5	−2.80	1.79
τxy6	15.5	4.9			−3.45	1.52
τxy8			−0.242	0.131
R2	0.619		0.279		0.699
R*2	0.570		0.257		0.351
RMSE	1.63		2.38		1.21

Sampling points of residual stresses were shown in Fig. 3. Coefficient: partial regression coefficient, 95%CI: 95% confidence interval, ns: not significant, R*²: adjusted R² for the number of degrees of freedom, RMSE: residual mean squared error.

Fig. 4. Relationship between Predicted and Experimental Values of TS in FLAT, R16 and R12 Tablets

Fig. 5. Relationship between Predicted and Experimental Values of DT in FLAT, R16 and R12 Tablets

Fig. 6. Relationship between Predicted and Experimental Values of D_3 min in FLAT, R16 and R12 Tablets

Fig. 7. Relationship between Predicted and Experimental Values of t_50% in FLAT, R16 and R12 Tablets

Selection of Crucial Stress Factors

To visualize the effect of the residual stress distribution on the pharmaceutical characteristics, extremely significant stress factors (p<0.0001) were marked on the cross-sectional images of the tablets as shown in Fig. 3. Results are shown in Figs. 8–11. For FLAT tablets, the τ_xy2 and τ_xy4 values were both positive and their coefficients were also positive; therefore, these stresses act to enhance tablet hardness. A negative coefficient of τ_xy6 and positive coefficients of σ_x6 and σ_y8 were observed for factors increasing the tablet hardness of R16. As the x-axial force always acts from the die wall to the center, such a force is important in improving the tablet hardness of R16. The normal stress from the lower punch to the upper part acts to enhance the tablet hardness. For the more deeply curved tablets (R12), there were positive coefficients of τ_xy5 and τ_xy6, and a negative coefficient of σ_y4 for significant effects on tablet breaking. In the case of DT (Fig. 9), totally different stress points were selected as important factors affecting DT. There were negative coefficients of τ_xy3 and τ_xy6, and a positive coefficient of τ_xy5, for important factors that reduce DT. The force acting from the lower die wall to the center part (σ_x9) affected the disintegration of FLAT tablets. Furthermore, there were positive coefficients of σ_y1 and σ_y4, and a negative coefficient of σ_y9 for significant factors affecting DT. When the shape of tablets changed from FLAT to R16 and R12, greatly different stress points became important factors affecting DT. On the whole, the shear stresses significantly involve DT in FLAT tablets, while the axial stresses are dominant in R16 and R12 tablets. Although the prediction ability of MLR for D_3 min was limited, the shear stresses (τ_xy5 and τ_xy6) were observed to be significant factors for predicting D_3 min in FLAT tablets, and x-axial stresses were dominant for predicting D_3 min in R16 and R12 tablets (Fig. 10). Only τ_xy6 was recognized as a significant factor in the case of t_50% (Fig. 11).

Fig. 8. Highly Significant Factors (p<0.0001) of Shear (τ_xy), x-Axial (σ_x) and y-Axial (σ_y) Stress Values for MLR Prediction of TS in FLAT, R16 and R12 Tablets

: positive regression coefficient, : negative regression coefficient.

Fig. 9. Highly Significant Factors (p<0.0001) of Shear (τ_xy), x-Axial (σ_x) and y-Axial (σ_y) Stress Values for MLR Prediction of DT in FLAT, R16 and R12 Tablets

: positive regression coefficient, : negative regression coefficient.

Fig. 10. Highly Significant Factors (p<0.0001) of Shear (τ_xy), x-Axial (σ_x) and y-Axial (σ_y) Stress Values for MLR Prediction of D_3 min in FLAT, R16 and R12 Tablets

: positive regression coefficient, : negative regression coefficient.

Fig. 11. Highly Significant Factors (p<0.0001) of Shear (τ_xy), x-Axial (σ_x) and y-Axial (σ_y) Stress Values for MLR Prediction of t_50% in FLAT, R16 and R12 Tablets

: positive regression coefficient, : negative regression coefficient.

Conclusion

Residual stress distributions of FLAT, R16 and R12 tablets were successfully simulated using the FEM, where powders were defined using the DPC model. The shear stress and axial stress distributions were greatly affected by the shape of a tablet. Large shear stress remained in the region of the upper punch in R12 tablets. The x-axial stress distribution was greatly biased in FLAT, but the distribution was considerably different in R16 and R12 tablets. Tablet characteristics, such as TS and DT, were well-predicted as functions of residual stress distributions by employing MLR analysis. In the case of dissolution parameters, the prediction ability was somewhat limited.

Acknowledgments

We thank Mr. Nobuto Okada, Ms. Haruka Nemoto, Ms. Sayaka Nogi, Ms. Yukiko Yokokura, and Ms. Eri Morii for their technical support and many useful discussions. This study was supported by JSPS KAKENHI Grant Number 26460047.

Conflict of Interest

The authors declare no conflict of interest.

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