A study on the effect of biconvex compact shape factors on prediction of dilution potential from tensile strength–compaction force data of coprocessed diluents

The purpose of this research was to compare the effect of shape factors of biconvex compacts on prediction of dilution potential from polynomial regression models of area ratio-mass fraction data of novel α-lactose monohydrate-starch orodispersible diluent (α-LSOD) and StarLac®. Ibuprofen-diluent blends were compacted between 4.98 and 29.89 kN. Given the biconvex round compact dimensions (t = axial thickness, w = central cylinder thickness, and d = diameter), tensile strength was computed as σbiconvex=ΦFπdt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma_{{{\text{biconvex}}}} = \Phi \frac{F}{{\pi {\text{d}}t}}$$\end{document}, where F is the breaking force. The shape factor (Φ) was defined as t2d-1,0.14td+0.36wd-1,10/2.84td-0.126tw+3.15wd+0.01\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ \frac{t}{2d} \right]^{ - 1} ,\left[ {\frac{0.14t}{d} + \frac{0.36w}{d}} \right]^{ - 1} ,\left[ {10/\left[ {\left( {\frac{2.84t}{d}} \right) - \left( {\frac{0.126t}{w}} \right) + \left( {\frac{3.15w}{d}} \right) + 0.01} \right]} \right]$$\end{document} for Podczeck (ΦPz), Shang (ΦSh), and Pitt (ΦPt) models, respectively. Area under the curve (AUC) was obtained by second-order polynomial fitting of the tensile strength–compaction force curve followed by integration of the quadratic regression equation between the two compaction force limits. The AUC of each powder system was normalised with the AUC of ibuprofen-free powder to obtain area ratio (AR). The AR was plotted against mass fraction of ibuprofen, and the curvilinear data was fitted by third-order polynomial fitting. Dilution potential was obtained from the regression equation by back extrapolation to zero area ratio. Tukey’s multiple comparison test indicated statistically significant differences between the three shape factors (p < 0.001). The shape factors ΦSh and ΦPt were higher than ΦPz by a factor of 2.4 and 1.2, respectively. The quadratic regression model sufficiently explained the observed relationship between tensile strength and compaction force as indicated by the coefficient of determination whose values ranged between 0.9344 and 0.99843 and 0.93725 and 0.99812 for α-LSOD and Starlac®, respectively. Dilution potential was predicted as AR=B3x3+B2x2+B1x+c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AR = B_{3} x^{3} + B_{2} x^{2} + B_{1} x + c$$\end{document}, where x and its coefficients are ibuprofen mass fraction and regression constants, respectively. The predicted dilution potential of the three tensile strength models were within a close range of ≈110.93±1.02\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx 110.93 \pm 1.02$$\end{document} and ≈116.02±0.62\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx 116.02 \pm 0.62$$\end{document} for both α-LSOD and StarLac®. This study suggests the adoption of the simplest Podczeck model given a biconvex round punch tooling and highlights the suitability of polynomial regression to predict dilution capacity from nonlinear area ratio-mass fraction data.


Background
Tablets are versatile solid drug delivery systems obtained by compacting blends of therapeutic principles and excipients into compacts of defined geometry (Armstrong 2007). Depending on the type of punch and die tooling, tablets can be manufactured in compaction presses into various shapes including oval, capsule (oblong), flat round, biconvex round, bullet, barrel, arrow, triangle, modified triangle, square, pentagon, pillow, rhomb, heart, half moon, and so forth (Bauer-Brandl 2007;Pitt and Heasley 2013;Osamura et al. 2017;Yohannes and Abebe 2021). Tablet shape is an important tool for product differentiation and identification (Notenboom et al. 2017). Shaping of tablets to ease swallowing is also clinically meritorious to geriatric, paediatric, and dysphagic patients (Wan et al. 2015;Fields et al. 2015).
The mechanical strength of tablets is a critical quality attribute (CQA) that defines the success of tablet formation and use (Thoorens et al. 2014). Additionally, mechanically stronger tablets are necessary for downstream manufacturing operations such as coating, polishing, and packaging. Throughout its shelf life, tablets are required to be sufficiently strong to withstand transportation and patient handling (Alderborn and Frenning 2017;Yohannes and Abebe 2021). For pharmaceutical manufacturing in-process quality control, tensile strength is used as a measure of mechanical strength of compacts. Tensile strength is defined as the maximum bearable stress prior to fracture and is computed from the tablet dimension and breaking force (Pitt and Heasley 2013;Sabri et al. 2018).
Practically, a tablet is a heterogeneous mixture of several blends of powders in a compressed state. Accordingly, the mechanical strength of tableting diluents (pharmaceutical powders for tablet manufacturing) can be accurately expressed in terms of dilution potential. Dilution potential, or dilution capacity, is a property that measures the ability of a diluent to retain optimal mechanical strength when blended with other powder ingredients (Habib et al. 1996). In designing novel diluents for direct compaction application, dilution potential analysis constitutes a critical preformulation study aimed at validating direct compression propensity (Jivraj et al. 2000;Mirani et al. 2011;Thoorens et al. 2014). More precisely, the dilution propensity quantitatively expresses the maximum amount of active pharmaceutical ingredient that can be incorporated in a given tablet formulation to retain optimal tensile strength (Kuentz and Leuenberger 2000;Haruna et al. 2020).
Experimental determination of dilution potential involves tensile strength measurement as a function of applied compaction forces. The compaction profile of a given powder system in binary blends of a diluent and poorly compactible active pharmaceutical ingredient is analysed (Minchom and Armstrong 1987;Habib et al. 1996). Measurement of tensile strength of tablets with simple shapes such as cylinder (round flat faced) is quite simple and straight forward. In pharmaceutical technology, the Brazilian compression test is commonly used to measure the tensile strength of round flat-faced compacts (Mohammed et al. 2006). Newton and Fell 1970 has defined tensile strength of this type of tablets as: where σ-tensile strength of round flat-faced tablets; F-diametral breaking force; d-compact diameter; t-compact thickness.
Equation 1 is quite applied in early formulation development where simple round flat-faced punch tooling is often utilised to study dilution potential. However, as the punch and die tooling deviate from round flat-faced shapes during technology transfer, the complexity to compute tensile strength increases (Yohannes and Abebe 2021). The true mechanical behaviour of shaped tablets calls for consideration of their divergent dimensions. For biconvex tablets, such as those produced in this experiment ( Fig. 1a), the general expression for computation of tensile strength takes into account the shape factor (Φ) of the tablet as given by Eq. 2 (Razavi et al. 2015).
At least four experimentally determined equations for the estimation of Φ was independently derived in order to describe the relationship between compact dimension and tensile strength (Pitt et al. 2011;Shang et al. 2013;Podczeck et al. 2013;Razavi et al. 2015). The complexity to compute tensile strength of biconvex tablets would be better appreciated considering the shape factors expressed in Eqs. 3, 4, and 5.
Coprocessed diluents are combinations of two or more pharmaceutical excipients specially designed to offer superior compaction performance relative to the individual parent constituents or their physical blends (Gohel and Jogani 2005;Li et al. 2017;Salim et al. 2018Salim et al. , 2021. In addition, orodispersible coprocessed diluents have extended applications in formulation of orally disintegrating tablets for enhancing buccal dispersion and facilitating swallowing in dysphagic patients (Aguilar-Díaz et al. 2012;Chinwala 2020). For these newer-generation excipients, it is critically important to have sound understanding of their mechanical properties. Moreover, given the recent recorded advances in elucidation of microstructural and mechanical properties of composite materials for metallurgical engineering applications (Farag et al. 2016;Reda et al. 2018Reda et al. , 2022Zohdy et al. 2021), continuous and complimentary efforts are needed in critical analysis of mechanical behaviour of novel composite particles engineered for pharmaceutical product manufacturing applications. The purpose of this research was therefore to critically analyse the dilution potential of newly engineered directly compactible α-lactose-starch orodispersible coprocessed diluent (α-LSOD) from tensile strength-compaction force profiles generated using biconvex round-shaped punch tooling. Firstly, the research was aimed at comparing the complexity of computing tensile strength of biconvex compacts according to the three independent equations postulated by Shang et al. 2013;Pitt and Heasley 2013;Podczeck et al. 2013. The second goal was geared towards application of a modified Minchom and Armstrong technique to estimate the dilution capacity from integrated area under the tensile strength-compaction force curves of the α-LSOD diluent relative to a reference coprocessed diluent for direct compression (StarLac ® ). This was followed by third-order polynomial modelling of area ratio-mass fraction curvilinear data and back-extrapolation to zero area ratio using the generated regression equation to predict dilution capacity. The ultimate goal was to arrive at the simplest tensile strength computation model to predict the dilution capacity of the coprocessed diluents given a complicated biconvex punch tooling.

Engineering of the new directly compactible orodispersible α-lactose-starch orodispersible diluent (α-LSOD)
The coprocessing formula comprises of sodium starch glycolate (SSG) (5%), α-lactose monohydrate (80.75%), and maize starch (14.25%). SSG was cold hydrated in 200-mL distilled water and mixed with partially pregelatinized starch paste in a plastic drum. α-lactose monohydrate was mechanically incorporated into the SSG-starch paste mixture using geometric dilution. The α-LSOD wet mass was screened through 1-mm sieve aperture followed by convective drying at 60-70 °C for 18 h. The dried coprocessed composite was comminuted using ball mill operated on All Purpose Equipment ™ (Orchid Scientific, India) at 200 r.p.m for 10 min. The size reduced powder was fractionated using nest of sieves vibrated for 10 min over Ro-Tap ® test sieve shaker (W.S Tyler Mentor, Ohio, USA). Coarser particle dimensions (> 500 µm) and fines (< 50 µm) were separated from the particle size distribution by mechanical sieving.

Compaction of powders to form biconvex compacts
Homogeneous binary powder blends of α-LSOD at 0%w/w, 20%w/w, 30%w/w, 40%w/w, and 100%w/w ibuprofen mass fractions were achieved using rotating cube blender operated on All Purpose Equipment (AP-01 Plus, Orchid Scientific, India) at 100 r.p.m for 10 min. Biconvex round compacts were formed by compressing 500 ± 5 mg powder samples in manually operated single station tablet press (model NP-RD10, Natoli Engineering Company, Missouri, USA) equipped with 12-mm upper and lower convex (cup)-shaped punch tooling. Each powder sample was compressed at 4. 98, 9.96, 14.95, 19.92, 24.91, and 29.89 kN at a dwell time of 30 s. The punch and die surfaces were sufficiently lubricated with 2% w/v magnesium stearate in acetone. Ejected compacts were stored for 24 h prior to evaluation to allow for sufficient elastic recovery and prevent false tensile strength values. The same procedure was adopted for generation of compaction data using StarLac ® as diluent.

Measurement of tensile strength of biconvex tablets
Tensile strength of the biconvex compact (σ biconvex ) was determined according to the protocol described in the United States Pharmacopoeia (USP 41-NF 36 2018). Digital tablet dimension tester was used for the measurement of the compact dimensions d, w, and t ( Fig. 1a). Diametral breaking force (F) was measured as the load required to trigger diametral fracture determined by stationing a compact between two parallel platens of a digital hardness tester (Model THT-2, Biobase, India) ( Fig. 1b). σ biconvex was calculated according to Eq. 2. The value of Φ was computed according to the three models described independently from previous works as given by Eqs. 3, 4, and 5, respectively (Shang et al. 2013;Pitt and Heasley 2013;Podczeck et al. 2013). ( where Φ Pz , Φ Sh , and Φ Pt denote the compact shape factors computed according to the described models. As shown in Fig. 1a, d is the compact diameter, t is the overall axial thickness, while w is the central cylinder thickness. Measurements were conducted on five randomly selected compacts from each compacted powder blends, and the mean tensile strength value was used for AUC calculations. Shape factors and tensile strength calculations were performed using Microsoft Excel Spreadsheet.

Computation of dilution potential from compaction data
The values of tensile strength were plotted against the range of applied compaction forces for each powder system. The tensile strength-compaction force data was fitted using second-degree polynomial fitting. The quadratic regression equation was integrated between the limits of 4.98 and 29.89 kN to calculate the AUC. The AUC of each of the powder system was normalised with that of 0% ibuprofen (100% coprocessed diluent) to obtain the area ratio. The area ratio was plotted against ibuprofen mass fraction (Minchom and Armstrong 1987;Habib et al. 1996;Kuentz and Leuenberger 2000). The area ratio-mass fraction curvilinear data was fitted by thirdorder polynomial fitting. Both integration and polynomial fittings were conducted using OriginPro, Version 2021b, OriginLab Corporation, Northampton, MA, USA. Using the generated polynomial regression equation, dilution capacity was obtained by back extrapolation to zero area ratio. Shape factor variation at variable compaction forces within each ibuprofen mass fraction was analysed through one-way analysis of variance (ANOVA) followed by Tukey's multiple comparison test using GraphPad Prism version 5.01 for Windows, GraphPad Software, California, USA. Statistical significance was set at alpha level of 5%.

Analysing the effect of Podzeck, Shang, and Pitt shape factors on magnitude of tensile strength
The sample data shown in Table 1 reflects drug-diluent binary mixture compacted at variable compaction forces but with the same punch tooling. The tested validity of Pitts model has shown a w/d ≥ 2.0 which was above the theoretical minimum acceptable value of 1.0. The shape factors decrease with increasing values of d, t, and w. Previous studies found only slight differences between the shape factors, however with Φ Sh being slightly higher (Yohannes and Abebe 2021). Comparing the shape factors for the three models in this study, it was observed that Φ Pt was slightly higher than Φ Pz , but Φ Sh differ significantly to as twice their values. Tukey's multiple comparison test indicated statistically significant differences between the three shape factors (p < 0.05) ( Table 2). The variability can be appreciated when both Φ Sh and Φ Pt were normalized with Φ Pz . The shape factors Φ Sh and Φ Pt were higher than Φ Pz by a factor of 2.4 and 1.2, respectively. Similar trend holds for all other compaction data.
Based on Eq. 2, the tensile strength was directly proportional to the shape factor (Φ). It then follows that the term ( F π d 2 ) was constant in all the three models, hence the shape factor was regarded as the main determinant of variability in the values of the tensile strength measured at each compaction force. Given a biconvex compact with the experimentally determined dimensions (Table 1), it can be deduced that the shape factors were of the order Φ Sh > Φ Pt > Φ Pz . Consequently, for all experimental runs, the magnitude of the calculated tensile strength was of the order Shang > Pitt > Podzeck, respectively. Accordingly, the tensile strength calculated using the model by Shang et al. 2013 was approximately twice that obtained via Heasley 2013 andPodczeck et al. 2013. Similar trend was observed from compaction data of StarLac ® . However, the values of the tensile strength of compacts from StarLac ® -ibuprofen blends were similar indicating equivalent tabletability compared to α-LSOD diluent.
It is noteworthy that although the magnitude of the calculated tensile strength, AUCs, and area ratios varies considerably between Podzeck, Shang, and Pitt models, the predicted dilution capacities were similar as presented subsequently.

Quadratic regression models and Integrated Area Under the Curve
The tensile strength-compaction force profiles and fitted quadratic regression models for each set of powder system are presented in Figs. 2, 3, 4, 5, 6 and 7 for α-LSOD and Starlac ® , respectively. Since the tensile strength varied appreciably with ibuprofen loading and compaction forces, the curves were not superimposable. Lower compaction forces generally resulted in lower tensile strength. To predict the response variable, which is the tensile strength of the biconvex tablets (y = σ biconvex ), the quadratic regression model was described by the general expression: where x variable is the compaction force, while the coefficients B 1 , B 2 , and c are regression constants.
The quadratic regression models sufficiently explained the observed relationship between the tensile strength and compaction forces as indicated by the coefficient of determination (R-squared), whose values ranged between 0.9344 and 0.99843 and 0.93725 and 0.99812 for α-LSOD diluent and Starlac ® , respectively. By integrating between the limits of the compaction forces (4.98 to 29.89 kN), the AUC of each powder system was described by: The observations summarized in Table 3 compared the mechanical properties of the two coprocessed excipients. From the integrated AUC for Podczeck, Shang and Pitt models, it can be inferred that both diluents possess similar deformation physics. The observed trends in the AUCs have explained the following mechanical performance of the diluents.     (i) AUC 100 > AUC 0 AUC 100 and AUC 0 describe the mechanical strength of pure ibuprofen particles and pure coprocessed diluents, respectively. This signifies that pure ibupro-fen is mechanically stronger than the pure coprocessed diluents. (ii) AUC 20 < AUC 0 < AUC 100 .

Discussion
The concept of powder dilution capacity can best be viewed as the dilution of the tensile strength of a given powder in the presence of another material of inferior, equivalent, or superior mechanical strength. In this research we focused on critical analysis of dilution capacity of novel diluent (α-LSOD) in the presence of ibuprofen powder. Plotting the normalized AUC values (area ratio) as a function of ibuprofen mass fraction formed the basis of prediction of dilution capacity from nonlinear compaction data. The concept described by Minchom and Armstrong 1987 and later modified by Habib et al. 1996 was based on simple linear regression analysis of area ratio-mass fraction data followed by back extrapolation to zero area ratio.
In contrast, this study utilised third-order polynomial regression to predict the drug loading capacity. Polynomial regression is a special form of multiple linear regression that is employed to explain nonlinear relationship between an independent variable and a dependent variable (Ostertagová 2012; Angelini 2019). The polynomial regression model was applied to describe the curvilinear compaction pattern of the studied diluents. From the fitted data (Figs. 8 and 9), the following form of polynomial regression equations was derived for prediction of dilution potential.
where y is area ratio, while all x coefficients and c are regression coefficients, with x being the mass fraction of ibuprofen.
In summary, the dilution capacities of both experimental (α-LSOD) and reference (StarLac ® ) coprocessed diluents have suggested appreciable ibuprofen loading capacities. Ibuprofen and the coprocessed excipients were compactible at all dilution ratios. This suggested that in the presence of ibuprofen, both substances synergistically contributed in interparticle bond formation. The observed mechanical profile of the compacts produced from these diluents could be attributed to multifunctional synergy imparted by coprocessed excipients (Bhatia et al. 2022). Constitutional similarities of α-LSOD and StarLac ® contributed to similarities in their deformation physics and hence similar dilution capacities. Despite being a lactose-based coprocessed excipient, Starlac is less brittle (Özalp et al. 2020). In fact, earlier studies have shown that Starlac ® deforms plastically with minimal elasticity, which implies high mechanical profile (Hauschild and Picker-Freyer 2004).

Conclusions
Quantification of drug loading capacities of pharmaceutical diluents as conceptualized by Minchom and Armstrong was derived from area under the curve of tensile strength-compaction force data. Unlike compaction simulators, non-automated acquisition of such data is tedious and time-consuming. In scenarios where more complex punch shapes other than the more common round flat-faced tooling are used, tensile strength measurement is complicated given the intricacies of accounting for compact diameter (d), central cylinder thickness (w), and overall axial thickness (t). Accordingly, prediction of dilution potential from the tensile strengthcompaction force data would be time-consuming. From simplistic view point, this study suggests the adoption of the simplest Podzeck model compared to the Shang and Pitt models given a biconvex round punch tooling. This study further introduced polynomial regression analysis to model curvilinear data as complimentary to the original technique where simple linear regression was used to model linear association between area ratio-mass fraction data. Using the fitted thirdorder regression model, the average predicted dilution capacities of α-lactose-starch orodispersible coprocessed diluent (α-LSOD) and StarLac ® obtained by extrapolation to zero area ratio were ≈ 110.93 ± 1.02 and ≈ 116.02 ± 0.62 (R 2 > 0.9). However, this study is limited to ibuprofen. Since pharmaceutical powder systems demonstrate broad-ranged anisotropic behaviours, we recommend similar studies using other moderately compressible model drugs and diluents to validate the current postulations.  Mass fraction of Tablettose100-starch-SSG (%) Fig. 8 Third-order polynomial fitting of area ratio-mass fraction data using orodispersible α-lactose-starch diluent was the model diluent