pharmaceutics

Article A Mathematical Approach to Consider Solid Compressibility in the Compression of Pharmaceutical Powders

Isabell Wünsch 1,2,* , Jan Henrik Finke 1,2, Edgar John 3, Michael Juhnke 3 and Arno Kwade 1,2 1 Institute for Particle Technology, Technische Universität Braunschweig, Volkmaroder Straße 5, Braunschweig 38104, Germany; jfi[email protected] (J.H.F.); [email protected] (A.K.) 2 Center of Pharmaceutical Engineering (PVZ), Technische Universität Braunschweig, Braunschweig 38106, Germany 3 Novartis Pharma AG, Basel 4002, Switzerland; [email protected] (E.J.); [email protected] (M.J.) * Correspondence: [email protected]; Tel.: +49-531-391-65548



Received: 12 February 2019; Accepted: 9 March 2019; Published: 15 March 2019


Abstract: In-die compression analysis is an effective method for the characterization of powder compressibility. However, physically unreasonable apparent solid fractions above one or apparent in-die porosities below zero are often calculated for higher compression stresses. One important reason for this is the neglect of solid compressibility and hence the assumption of a constant solid density. In this work, the solid compressibility of four pharmaceutical powders with different deformation behaviour is characterized using mercury porosimetry. The derived bulk moduli are applied for the calculation of in-die porosities. The change of in-die porosity due to the consideration of solid compressibility is for instance up to 4% for microcrystalline cellulose at a compression stress of 400 MPa and thus cannot be neglected for the calculation of in-die porosities. However, solid compressibility and further uncertainties from, for example the measured solid density and from the displacement sensors, are difficult or only partially accessible. Therefore, a mathematic term for the calculation of physically reasonable in-die porosities is introduced. This term can be used for the extension of common mathematical models, such as the models of Heckel and of Cooper & Eaton. Additionally, an extended in-die compression function is introduced to precisely describe the entire range of in-die porosity curves and to enable the successful differentiation and quantification of the compression behaviour of the investigated pharmaceutical powders.

Keywords: powder compression; tableting; compression equations; solid compressibility; in-die compression analysis

1. Introduction Powder compaction is an important production process in diverse industries, such as food, ceramic and pharmaceutical industry. The prediction of structural and mechanical properties of tablets based on raw material properties and process parameters is still difficult or rather impossible, although the powder compaction process was the object of numerous scientific investigations. The main reason for the incomplete predictability is certainly the still limited process understanding due to the complexity of the powder compaction process. This complexity can be attributed to various influencing parameters [1–5], such as the deformation behaviour, the particle size and shape and the compression stress on the one hand and to different acting micro-processes [6–10], such as particle rearrangement, elastic and plastic deformation of single particles and particle fragmentation on

Pharmaceutics 2019, 11, 121; doi:10.3390/pharmaceutics11030121 www.mdpi.com/journal/pharmaceutics Pharmaceutics 2019, 11, 121 2 of 19 the other hand. These micro-processes do not take place sequentially but occur simultaneously. The contribution of the single mechanisms depends on material properties and applied process parameters. Until today, the individual quantitative characterization of each mechanism is challenging. The deformation behaviour also affects the number of generated, inter-particulate bonds as well as the bonding forces and thus the resulting structural and mechanical properties of the tablet [11,12]. The missing predictability causes the need for a systematic and comprehensive characterization of the compression and compaction behaviour of raw materials as prerequisite for a rationally based formulation and process development. This study is focused on the characterization of powder compressibility. The compressibility is defined as the relationship between solid fraction/porosity and compression stress [8]. Several mathematical models were developed for the description of the compression curves and the derivation of specific compression parameters. An overview is given by Kawakita and Lüdde [13] and Celik [14], for example. The main requirement for a suitable process function is the ability to describe the compression curves of related materials precisely and robustly. The simplest functions are two-parametric equations, such as the common models of Heckel [15] and of Kawakita and Lüdde [13]. These functions only depict a limited part of the compression curve. In contrast, four-parametric models enable the description of the entire compression curve. A common four-parametric process function is the model of Cooper & Eaton [16]. The most common method for the determination of compression curves is the out-of-die analysis and the resulting compressibility is named out-of-die compressibility. In this case, tablets are produced by applying different compression stresses and related tablet solid fraction/porosity is determined after storage of the tablets at defined conditions for a period of time. However, this method is only applicable with limitations when tablet defects, like lamination or capping, occur. Instrumented tablet presses, equipped with force and displacement sensors, enable the determination of the compressibility during compression (in-die compressibility). In this case, solid fraction/porosity is calculated based on the measured force-displacement curve, taking the tablet mass and its solid density into account. The main advantage of in-die analysis is that, in the best case, the performance of only one compression process is sufficient for the comprehensive characterization of powder compressibility. Furthermore, in-die compressibility can be determined even if tablet defects occur. However, in-die compressibility is typically shifted to higher solid fractions/lower porosities compared to out-of-die compressibility, which lead to different specific compression parameters [17–20]. The main reason for the observed differences is the inclusion of elastic deformation in the case of in-die analysis, while out-of-die analysis does not take elastic deformation into account as compressed powders relax during and after unloading. Additionally, physically unreasonable apparent solid fractions above one/apparent porosities below zero are observed for in-die analysis above certain compression stress levels [21,22]. For this reason, in-die data presented in literature are often limited to lower compression stresses that may affect derived specific compression parameters [3,19,23]. One reason for the occurrence of physically unreasonable values is the assumption of a constant solid density [19,21]. The molecular lattice inside the individual particles deforms elastically during compression as well and this leads to a decrease of specific solid volume upon compression [21,24–26]. Consequently, solid density increases with rising compression stress, especially for organic substances. This phenomenon is referred to as solid compressibility. As one example, Boldyreva showed for paracetamol a decrease in specific solid volume of approximately 3% at hydrostatic compression of 400 MPa [24]. Although different researchers mention the influence of solid compressibility as an important influencing factor on in-die analysis, this phenomenon is scarcely considered. Sun and Grant introduced a term for the correction of in-die porosity in dependence of the Young’s modulus of the material [17]. They showed that the deviation between the yield strength derived by in-die analysis and out-of-die analysis increases with decreasing Young’s modulus. Sun and Grant did not differentiate between elastic deformation of the bulk and the single particles and solid compressibility. Additionally, Sun and Grant did not consider the stress-dependency of solid density. The common neglect of the stress-dependency of solid Pharmaceutics 2019, 11, 121 3 of 19 density might be explained by the inaccessibility of this phenomenon during powder compression. The volume change of single crystals can be measured using crystallographic methods, however, such measurements need to be performed by X-ray diffraction in sufficiently resilient dies. Another method for the determination of solid compressibility is mercury porosimetry provided that the single particles are completely enclosed by mercury and that all pores are completely filled with mercury [27,28]. Further reasons for the occurrence of physically unreasonable solid fractions/porosities are uncertainties of the compressed mass, of the measured solid density and from the force and displacement sensors [19,29–31]. Another influencing parameter could be the die diameter, which is assumed to be constant. However, the expansion of the die was not considered yet. These challenges are the reason for the still more frequent use of out-of-die compressibility analysis, although in-die analysis is material and time-saving. Katz et al. introduced a new empirical approach to use the advantages of both methods [18,20]. They calculated out-of-die compressibility based on in-die compressibility under consideration of elastic and viscoelastic recovery. The performance of only two compression experiments, one at relatively low compression stress and one with maximum compression stress, are sufficient for this approach. This method is consequently not applicable if tablet defects occur. However, the characterization of the compression behaviour of materials which tend to form defective tablets is important for formulation and process development. It can be concluded that a method for the calculation of physically reasonable in-die porosities/solid fractions is needed for establishing in-die analysis for formulation and process development. In this study, the influence of solid compressibility and of die expansion on in-die compression analysis is investigated for pharmaceutical materials with considerably different deformation behaviour. Additionally, a mathematical term for the pragmatic consideration of solid compressibility during in-die compression analysis and hence, for the calculation of physically reasonable in-die porosities is introduced. Its applicability combined with the common compression models of Heckel and of Cooper & Eaton, as well as an extended in-die compression function are discussed.

2. Theory Various mathematical models for the description of powder compressibility exist in literature. Common models in industrial pharmacy are the models of Heckel [15] and of Cooper & Eaton [16], which are used in this study.

2.1. Model of Heckel The model of Heckel [15] was originally developed for the compaction of metal powders with a mainly ductile deformation behaviour. Heckel assumed that the porosity decrease during powder compaction follows a first order kinetic:

1 ln = k·σ + A (1) ε ax where ε is the porosity, σax is the axial compression stress and k and A are constants. Heckel found that the constant k correlates with the reciprocal value of yield strength σ0 for metals and is thus a measure of the plasticity of a material [32]. Hersey and Rees introduced the term mean yield pressure Py for this correlation [33]: 1 1 k = = (2) 3σ0 Py

2.2. Model of Cooper & Eaton Cooper and Eaton describe the compression process of ceramic powders by two steps: the filling of inter-particulate pores in the size range similar to the particle size and the filling of smaller pores [16]. They hypothesize particle rearrangement as the main deformation mechanism responsible for the first Pharmaceutics 2019, 11, 121 4 of 19 step and plastic deformation and particle fragmentation as the main deformation mechanisms of the second step. Cooper and Eaton consider the volume change in dependence on compression stress and used two exponential terms for the description of both mechanisms:     ∗ V0 − V k1 k2 V = = a1 exp − + a2 exp − (3) Pharmaceutics 2019, 11, x FOR PEER VREVIEW0 − V∞ σax σax 4 of 19 where V* is the relative volume change, V0 is the initial volume, V∞ is the powder volume at infinite ∗ 𝑉 −𝑉 𝑘 𝑘 𝑉 = =𝑎 𝑒𝑥𝑝 − +𝑎 𝑒𝑥𝑝 − (3) compression stress and a1, a2, k1 and𝑉k−𝑉2 are constants,𝜎 with the boundary𝜎 condition of the sum of a1 and a2 equalling one. a1 and a2 are the fractions of theoretically possible compression and describe where V* is the relative volume change, V0 is the initial volume, V∞ is the powder volume at infinite the maximum possible volume reduction attributed to each respective mechanism. k and k are compression stress and a1, a2, k1 and k2 are constants, with the boundary condition of the sum1 of a12 characteristicand a2 equalling compression one. a1 and values, a2 are which the fractions indicate of thetheoretically stress range, possible in whichcompression the particular and describe mechanism the dominatesmaximum the volumepossible change. volume Inreduction this study, attrib theuted initial to specificeach respective volume Vmechanism.0 is calculated k1 and using k2 theare data of thecharacteristic first measurement compression point values, of the which force-displacement indicate the stress curve. range, The in which specific the solid particular volume mechanism determined by heliumdominates pycnometry the volume is usedchange. as In specific this study, volume the initial at infinite specific compression volume V0 is stresscalculatedV∞ .using the data of the first measurement point of the force-displacement curve. The specific solid volume determined 3. Materialsby helium pycnometry is used as specific volume at infinite compression stress V∞.

® Microcrystalline3. Materials cellulose (MCC, Vivapur 102, JRS Pharma, Rosenberg, Germany), anhydrous lactose (Lac, Lactose Anhydrous NF DT, Sheffield Bio Science, Norwich, NY, USA), anhydrous Microcrystalline cellulose (MCC, Vivapur® 102, JRS Pharma, Rosenberg, Germany), anhydrous dicalcium phosphate (DCPA, DI-CAFOS® A150, Chemische Fabrik Budenheim, Budenheim, Germany) lactose (Lac, Lactose Anhydrous NF DT, Sheffield Bio Science, Norwich, NY, USA), anhydrous and paracetamoldicalcium phosphate (Para, Novartis (DCPA, PharmaDI-CAFOS AG,® A150, Basel, Chemische Switzerland) Fabrik were Budenheim, selected as pharmaceuticalBudenheim, materialsGermany) while and magnesium paracetamol stearate (Para, (Faci, Novartis Carasco, Pharma Italy) AG, was Basel, used Switzerland) as lubricant. were selected as Thepharmaceutical characteristic materials particle while sizes magnesium of MCC, stearate Lac and (Faci, DCPA Carasco, are Italy) in a comparablewas used as lubricant. range but differ in width,The while characteristic Para is clearly particle finer sizes (Table of MCC,1). TheLac and particle DCPA shape are in ofa comparable all model materialsrange but differ is irregular in (Figurewidth,1). MCCwhile mainlyPara is clearly consists finer of (Table elongated 1). The primary particle particlesshape of all and model of approximately materials is irregular spherical agglomerates(Figure 1). or MCC aggregates. mainly Theconsists primary of elongated particles primary of Para particles are approximately and of approximately rectangular. spherical In contrast, Lac andagglomerates DCPA consist or aggregates. of irregularly The primary shaped particles aggregates of Para and are agglomerates.approximately rectangular. In contrast, Lac and DCPA consist of irregularly shaped aggregates and agglomerates. Table 1. Characteristic particle sizes and solid density. Table 1. Characteristic particle sizes and solid density. 3 x µ x µ x µ 3 MaterialMaterial10 [ x10m] [µm] x5050 [[µm]m] x90 [µm]90 [ m] ρs [g/cmρ]s [g/cm ] MCCMCC 36 36 125 125 261 261 1.5537 ± 1.5537 0.0080± 0.0080 ParaPara 4 4 25 25 72 72 1.2863 ± 1.2863 0.0001± 0.0001 LacLac 12 12 147 147 342 342 1.5524 ± 1.5524 0.0077± 0.0077 DCPADCPA 77 77 190 190 302 302 2.7718 ± 2.7718 0.0097± 0.0097

FigureFigure 1. Particle 1. Particle images images of theof the pharmaceutical pharmaceutical powderspowders MCC, MCC, Lac, Lac, DCPA DCPA and and Para Para determined determined by by scanningscanning electron electron microscopy. microscopy.

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4. Methods

4.1. Powder Characterization Solid density of the powders was determined using the helium pycnometer ULTRAPYC 1200 (Quantachrome Instruments, Boynton Beach, FL, USA). The powders were dried under vacuum for 24 h before the measurement. Double measurements with 10 measurement points each were performed and the mean values (Table1) were used for the calculation of apparent in-die porosities. Additionally, the true density of paracetamol was determined using the X-ray powder diffraction (Bruker D8 Advance, Billerica, MA, USA). Three measurements were performed at room temperature and Rietveld refinement was applied. Furthermore, particle size analysis was performed using the laser light diffraction instrument HELOS (Sympatec, Clausthal-Zellerfeld, Germany) in combination with the dry dispersion unit RODOS for the excipients and the wet dispersion unit CUVETTE for Para.

4.2. Determination of the Bulk Modulus The solid compressibility of the powders was characterized using the mercury porosimeter PoreMaster® GT60 (Quantachrome Instruments, Boynton Beach, FL, USA). The penetrometer with a volume of 0.5 cm3 was filled with an appropriate amount of powder. The low pressure operation (up to 344 kPa) was used for the filling of the penetrometer with mercury and is not considered for the determination of solid compressibility because of the probably incomplete enclosure of the single particles by mercury. The maximum applied pressure of the high pressure operation was 414 MPa. Four samples per material were characterized and high pressure operation was repeated five times per sample to ensure elastic deformation as the reason for volume changes. Before the analysis, several blank measurements were performed for the consideration of the volume change due to the elastic deformation of the penetrometer and of mercury. The elastic compressibility of an isotropic material under hydrostatic pressure can be described by the bulk modulus K: dp K = −V (4) 0 dV where V0 is the volume at ambient pressure and p is the hydrostatic pressure and dp can be replaced by p−p0 and dV by V−V0. p0 is the atmospheric pressure and can be neglected for powder compression. This results in V (p) p S = 1 − (5) VS,0 K where VS is the specific solid volume dependent on the hydrostatic pressure and VS,0 is the specific solid volume at atmospheric pressure. The bulk modulus is hence the reciprocal of the slope of the measured volume change during mercury intrusion and was determined by linear regression using the software Excel 2010with the solver add-in (Microsoft, Redmond, WA, USA). It has to be noted that not the whole pressure range is considered because of non-linearity at lower pressure (frequently < 150 MPa), which indicates the presence of still unfilled pores and the still insufficient enclosure of the single particles by mercury. The bulk modulus was determined for all performed measurements and the average value was calculated per material.

4.3. Tableting Compaction experiments were performed using the compaction simulator Styl’One Evolution (MEDEL’PHARM, Beynost, France), which is equipped with force and displacement sensors. The die and the punches were manually pre-lubricated with magnesium stearate. 450 mg powder of each MCC, Lac and Para as well as 900 mg of DCPA were compressed (n = 6) with a maximum compression stress of 400 MPa using 11.28 mm flat-faced, round punches. The compression profile of a camshaft was simulated with a maximum compression velocity of approximately 20 mm/s. The weight of the tablets was measured after compaction and in-die analysis was performed considering Pharmaceutics 2019, 11, 121 6 of 19 machine deformation. Additionally, experiments using an instrumented die (11.28 mm Euro B) (MEDEL’PHARM, Beynost, France) were performed for the determination of the radial die wall stress with a maximum compression stress of 300 MPa. The stress ratio, which is defined as the ratio between radial and axial stress, is calculated based on the maximal axial and radial stresses. The tablets were stored under constant ambient conditions for at least 24 h. Afterwards, the weight of the tablets (n = 20) was measured and geometrical dimensions were determined using the tablet tester MultiTest 50 FT (Dr. Schleuniger, Aesch, Switzerland). The data were used for the calculation of out-of-die tablet porosity.

4.4. Application of the Compression Equations The fitting of the models to experimental data was performed using regression by minimization the sum of squared errors. The software Excel 2010 (Microsoft, Redmond, WA, USA) was used for the fitting of the models of Heckel and of Cooper & Eaton. The fitting of the extended in-die compression function was performed using MATLAB (Version: R2015b, MathWorks, Natick, MA, USA). Additionally, out-of-die Heckel analysis was performed by application of the model of Heckel to the out-of-die porosities between 200 MPa and 400 MPa.

4.5. Estimation of Radial Die Expansion The elastic expansion of the die in radial direction during powder compression is estimated under assumption of rotationally symmetric and height-independent stress conditions inside the die wall based on Hook’s law [34]: σ ν·σ ε = rad − tan (6) rad E E where εrad is the elastic strain in radial direction, σrad is the stress in radial direction, σtan is the elastic stress in tangential direction, E is the Young’s modulus and ν the Poisson’s ratio. The radial stress was measured using an instrumented die, while the tangential stress for a cylinder with a thick wall (ra/ri ≥ 1.2 according to DIN 2413) can be estimated as follows [35]:

r2·p  r2  σ = i 1 + a (7) tan 2 2 2 ra − ri r where ri is the inner radius, ra is the outer radius and p is the inner pressure acting on the die wall. The estimation of die expansion was performed for Euro D compression tools with an inner radius of 5.64 mm and an outer radius of 19.05 mm using a Young’s modulus of 210 GPa and a Poisson’s ratio of 0.28, which are typical values of stainless steel. The influence of the die holder and the groove are neglected.

5. Results and Discussion

5.1. Compression Behavior of the Model Materials The in-die compression curves of the four materials differ considerably (Figure2). The slope of the curve of Para is the highest at the beginning of the compression process (below approximately 50 MPa) indicating the lowest resistance against compression. In contrast, DCPA shows the overall highest apparent in-die porosity and the lowest porosity decrease. Hence, the effective compression of DCPA is the lowest. MCC and Lac show behaviour in between that of the aforementioned. Furthermore, it has to be noted that physically not reasonable negative apparent in-die porosities are reached for Para and MCC at compression stresses above 200 MPa. The reasons for the occurrence of negative in-die porosities are discussed in the further sections. Pharmaceutics 2019, 11, 121 7 of 19 Pharmaceutics 2019, 11, x FOR PEER REVIEW 7 of 19

Figure 2. Apparent in-die compressibility ofof MCC,MCC, Lac,Lac, DCPADCPA andand Para.Para. 5.2. Reasons for Apparent in-Die Porosities Below Zero 5.2. Reasons for Apparent in-Die Porosities Below Zero A fundamental parameter for the calculation of porosities is the solid density, which is commonly A fundamental parameter for the calculation of porosities is the solid density, which is determined by helium pycnometry. Helium pycnometry is applicable regardless of the degree of commonly determined by helium pycnometry. Helium pycnometry is applicable regardless of the crystallinity in contrast to X-ray diffraction. However, variations of the solid density determined by degree of crystallinity in contrast to X-ray diffraction. However, variations of the solid density helium pycnometry are often reported due to the influence of material properties and measurement determined by helium pycnometry are often reported due to the influence of material properties and conditions [31,36,37]. The solid density of Para determined by helium pycnometry is slightly lower measurement conditions [31,36,37]. The solid density of Para determined by helium pycnometry is (1.2863 ± 0.001 g/cm3) compared to the true density determined by X-ray diffraction (1.2936 ± 0.0027 slightly lower (1.2863 ± 0.001 g/cm3) compared to the true density determined by X-ray diffraction g/cm3). The slightly lower density determined by helium pycnometry might be explained by the (1.2936 ± 0.0027 g/cm3). The slightly lower density determined by helium pycnometry might be consideration of closed intra-particulate pores and flaws, while these are not taken into account explained by the consideration of closed intra-particulate pores and flaws, while these are not taken by evaluation of X-ray diffraction. The applied high stress during powder compression causes into account by evaluation of X-ray diffraction. The applied high stress during powder compression particle deformation and fragmentation, which may lead to the disappearance of closed pores and causes particle deformation and fragmentation, which may lead to the disappearance of closed pores flaws. Therefore, the usage of the true density determined by X-ray diffraction seems to be useful and flaws. Therefore, the usage of the true density determined by X-ray diffraction seems to be useful for powder compression. Since often the solid density determined by helium pycnometry is used, for powder compression. Since often the solid density determined by helium pycnometry is used, it it is necessary to consider the influence of the differences between solid density and true density is necessary to consider the influence of the differences between solid density and true density on the on the calculation of in-die porosities. The lower solid density determined by helium pycnometry calculation of in-die porosities. The lower solid density determined by helium pycnometry causes a causes a slight shift of the apparent in-die porosity εapp,He curve of Para to slightly lower porosities slight shift of the apparent in-die porosity εapp,He curve of Para to slightly lower porosities compared compared to the curve calculated using the true density determined by X-ray diffraction (εapp,XRD) to the curve calculated using the true density determined by X-ray diffraction (εapp,XRD) (Figure 3). (Figure3). However, in both cases physically not reasonable negative apparent in-die porosities are However, in both cases physically not reasonable negative apparent in-die porosities are reached at reached at high compression stresses. The apparent in-die porosities are calculated based on the high compression stresses. The apparent in-die porosities are calculated based on the assumption of assumption of a constant solid density during powder compression, although solid compressibility a constant solid density during powder compression, although solid compressibility causes the causes the increase of solid density with rising compression stress [19,21], especially for organic increase of solid density with rising compression stress [19,21], especially for organic materials. materials. Therefore, the bulk modulus measured by mercury porosimetry (Table2) is used for Therefore, the bulk modulus measured by mercury porosimetry (Table 2) is used for the calculation the calculation of a stress-dependent specific solid volume according to Equation (5) based on the of a stress-dependent specific solid volume according to Equation (5) based on the specific solid specific solid volume determined by helium pycnometry. The resulting in-die compression curve εc,Hg volume determined by helium pycnometry. The resulting in-die compression curve εc,Hg is clearly is clearly shifted to the positive porosity range, especially at compression stresses above 200 MPa, shifted to the positive porosity range, especially at compression stresses above 200 MPa, as Figure 3 as Figure3 shows exemplarily for Para. The same trend is found for MCC and Lac (Table 2). The bulk shows exemplarily for Para. The same trend is found for MCC and Lac (Table 2). The bulk modulus modulus of the inorganic DCPA could not be determined by mercury porosimetry because of reaching of the inorganic DCPA could not be determined by mercury porosimetry because of reaching the the detection limit. The bulk modulus of DCPA is hence to be expected clearly higher compared detection limit. The bulk modulus of DCPA is hence to be expected clearly higher compared to the to the three organic materials. The bulk modulus of DCPA could be alternatively calculated using three organic materials. The bulk modulus of DCPA could be alternatively calculated using the the Young’s modulus and Poisson’s ratio. Beam bending of compactates or double compaction with Young’s modulus and Poisson’s ratio. Beam bending of compactates or double compaction with a a fully instrumented compaction simulator can be used for the determination of Young’s modulus and fully instrumented compaction simulator can be used for the determination of Young’s modulus and Poisson’s ratio of compactates with defined porosities [38–40]. Extrapolation to zero porosity can then Poisson’s ratio of compactates with defined porosities [38–40]. Extrapolation to zero porosity can then be applied for the derivation of properties of the solid, which are needed for the calculation of the bulk be applied for the derivation of properties of the solid, which are needed for the calculation of the modulus. Additionally, high-pressure diffraction studies of single particles could be performed [24]. bulk modulus. Additionally, high-pressure diffraction studies of single particles could be performed [24].

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FigureFigure 3. ApparentApparent in-die in-die compressibility compressibility of of Para Para calculated calculated using using the the solid solid density density determined determined by by heliumhelium pycnometry εapp,He andand by by X-ray X-ray diffraction diffraction εεapp,XRDapp,XRD andand correction correction of of the the apparent apparent in-die in-die compressibilitycompressibility εεc,Hgc,Hg withwith the bulkbulk modulusmodulus determined determined by by mercury mercury porosimetry. porosimetry. The The filled filled areas areas mark markthe 95% the confidence 95% confidence interval in regardingterval regarding density densityεapp,He εandapp,Heε app,XRDand εapp,XRDor rather or rather the bulk the modulusbulk modulusεc,Hg. εc,Hg. Table 2. Bulk modulus determined by mercury porosimetry and minimal in-die porosities.

Table 2. Bulk modulus determined byMin. mercury Apparent porosimetry in-die porosityand minimalMin. in-die in-die porosities. porosity Material Bulk Modulus K [GPa] εapp,He,min [-] εc,Hg,min [-] Bulk Modulus K Min. Apparent in-die porosity Min. in-die porosity εc,Hg,min Material MCC[GPa] 10.22 ± 1.02 εapp,He,min− [-]0.035 0.006[-] ± − MCC Para 10.22 ± 1.02 14.93 3.67 −0.035 0.023 0.0050.006 Lac 27.07 ± 7.13 0.001 0.016 Para 14.93 ± 3.67 −0.023 0.005 DCPA - 0.256 - Lac 27.07 ± 7.13 0.001 0.016 DCPA - 0.256 - The minimal in-die porosities change by approximately 3% for Para, by 4% for MCC and by 1.7%The for Lac.minimal Accordingly, in-die porosities solid compressibility change by approximately has a significant 3% for effect Para, during by 4% for powder MCC compressionand by 1.7% forand Lac. should Accordingly, be considered solid compressibility for the calculation has ofa significant in-die porosities. effect during The porositypowder compression change of DCPA and shouldis expected be considered to be considerably for the calculation lower than of in-die 1.7%. po However,rosities. The the porosity transferability change ofof theDCPA bulk is expected modulus todetermined be considerably by hydrostatic lower than compression 1.7%. However, to uniaxial the transferability compression mightof the bebulk limited. modulus The determined stresses acting by hydrostaticon the single compression particles during to uniaxial uniaxial compression powder compression might be limited. are unknown The stresses and, additionally, acting on the powder single particlescompression during is anisotropicuniaxial powder and dynamic, compression whereas are unknown material and, bulk additionally, moduli are generally powder compression measured at isdefined anisotropic pressures and anddynamic, thermodynamic whereas material equilibrium. bulk Inmoduli addition, are thegenerally stress directed measured to theat diedefined wall pressuresis complex and and thermodynamic depends on material equilibr andium. process In addition, parameters the stress [41 ,directed42]. Additionally, to the die wall the accuracyis complex of andmercury depends porosimetry on material for theand determination process parameters of the bulk [41,42]. modulus Additionally, and uncertainties the accuracy of the of measured mercury porosimetrysolid density for by heliumthe determination pycnometry of has the to bulk be considered. modulus and uncertainties of the measured solid densityAn by additional helium pycnometry influencing factorhas to couldbe considered. be the elastic expansion of the die, which is dependent on the radialAn additional stress acting influencing on the die factor wall. could The radial be the stress elastic during expansion powder of compressionthe die, which is dependentis dependent on onthe the stress radial ratio stress between acting radial on the and die axial wall. compression The radial stress stress, during which powder is in turn compression determined is bydependent the axial oncompression the stress ratio stress between and the radial used materialand axial in compre combinationssion stress, with itswhich deformation is in turnbehaviour determined (Table by the3). axialThe stresscompression ratio in thestress range and from the used approximately material in 0.5 combination MPa to 400 MPawith isits the deformation highest for behaviour MCC and (TablePara because 3). The stress of the ratio ductile in the and range elastic from behaviour approximately of these 0.5 materials MPa to 400 [43 MPa] leading is the tohighest the largest for MCC die andexpansion Para because (Figure of4 ).the The ductile stress and ratio elastic of Lac behaviour is in the of medium these materials range, while[43] leading the lowest to the stress largest ratio die expansionwas found (Figure for DCPA 4). (TableThe stress3), which ratio of correlates Lac is in well the medium with the findingsrange, while of Abdel-Hamid the lowest stress and ratio Betz [was43]. foundAccordingly, for DCPA the (Table expansion 3), which of the correlates die, calculated well with using the Equations findings (6)of andAbdel-Hamid (7), is lower and for Betz Lac [43]. and Accordingly,DCPA. For the the direct expansion comparison of the of die, the calculated influence ofusing the expansionEquations of(6) the and die (7), diameter is lower with for theLac effect and DCPA.of solid For compressibility, the direct comparison the die expansion of the influence coefficient of the was expansion introduced of the and die defined diameter as the with reciprocal the effect of ofthe solid slope compressibility, of the die expansion the die curves expansion (Figure coeffici4). Theent die was expansion introduced coefficients and defined are considerably as the reciprocal larger ofthan the the slope bulk of modulithe die (Tablesexpansion2 and curves3). Die (Figure expansion 4). The leads die to expansion small changes coefficients of the calculated are considerably in-die larger than the bulk moduli (Tables 2 and 3). Die expansion leads to small changes of the calculated in-die porosities, which are additionally recognizable by the estimated difference of porosity at

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Pharmaceutics 2019, 11, x FOR PEER REVIEW 9 of 19 porosities, which are additionally recognizable by the estimated difference of porosity at maximum compressionmaximum compression stress of 400 stress MPa, of comparing400 MPa, comparing in-die porosity in-die determined porosity determined with die expansionwith die expansion to in-die porosityto in-die determinedporosity determined without diewithout expansion die expansion (Table3). (Table The influence 3). The ofinfluence die expansion of die expansion is therefore is negligibletherefore negligible for the calculation for the calculation of in-die porosities. of in-die porosities.

Table 3. Stress ratio, die expansion coefficientcoefficient andand porosityporosity changechange duedue toto diedie expansionexpansion ((nn == 6).

MaterialMaterial Stress Stress RatioRatio [-][-] Die Die Expansion Expansion Coefficien Coefficientt [GPa] Difference Difference of of Porosity Porosity at at 400 400 MPa MPa [-] [-] MCC 0.47–0.81 207 ± 4 3.7 × 10−6 MCC 0.47–0.81 207 ± 4 3.7 × 10−6 Para 0.50–0.85 196 ± 3 4.2 × 10−6 Para 0.50–0.85 196 ± 3 4.2 × 10−6 −6 LacLac 0.45–0.71 0.45–0.71 237 ± 44 2.9 × 1010− 6 −6 DCPADCPA 0.21–0.43 0.21–0.43 382 ± 55 1.1 × 1010− 6

Figure 4. Expansion of the die diameter in dependence on the applied compression stress for Para, Figure 4. Expansion of the die diameter in dependence on the applied compression stress for Para, MCC, Lac and DCPA. MCC, Lac and DCPA. Other influencing parameters for the calculation of apparent in-die porosity could be massOther variations influencing and the parameters accuracy offor thethe displacementcalculation of sensors, apparent as in-die other porosity studies showedcould be [30 mass,31]. Thevariations experimental and the determinationaccuracy of the of displacement all uncertainties sensors, for theas other calculation studies of showed in-die porosities[30,31]. The is fundamentallyexperimental feasibledetermination but highly of expensive.all uncertainties However, for in-die the compression calculation analysis of in-die is an advantageousporosities is methodfundamentally for the characterizationfeasible but highly of powder expensive. compressibility However, and in-die can be compression applied as well analysis for materials, is an whichadvantageous tend to method form defective for the characterization tablets. For this of powder reason, compressibility a pragmatic mathematical and can be applied approach as well for considerationfor materials, ofwhich the physicaltend to effectform ofdefective solid compressibility tablets. For this and reason, other lessa pragmatic important mathematical mechanisms isapproach introduced. for consideration of the physical effect of solid compressibility and other less important mechanisms is introduced. 5.3. Introduction of the Solid Compressibility Term 5.3. Introduction of the Solid Compressibility Term Equation (5) is valid for the volume change under hydrostatic compression. For uniaxial powder compression,Equation the(5) hydrostaticis valid for the pressure volume can change be approximated under hydrostatic as follows compression. [44]: For uniaxial powder compression, the hydrostatic pressure can be approximated as follows [44]: σax + 2·σrad p = 𝜎 +2∙𝜎 (8) 𝑝= 3 (8) 3 where σax is the stress in axial direction and σrad is the stress in radial direction. The axial and radial where σax is the stress in axial direction and σrad is the stress in radial direction. The axial and radial stresses acting on the single particles during uniaxial powder compression are unknown and cannot be stresses acting on the single particles during uniaxial powder compression are unknown and cannot measured yet due to the highly complex stress state inside the powder. In reality, a stress distribution be measured yet due to the highly complex stress state inside the powder. In reality, a stress exists due to the bulk structure. Usually, instrumented tablet presses enable only the measurement distribution exists due to the bulk structure. Usually, instrumented tablet presses enable only the of the overall axial stress acting on the powder. The overall radial stress can be determined using measurement of the overall axial stress acting on the powder. The overall radial stress can be andetermined instrumented using die. an However,instrumented the usage die. However, of instrumented the usage dies of is notinstrumented common because dies is ofnot difficulties common inbecause design, of operationdifficulties and in design, data evaluation operation [ 45and]. da Theta estimationevaluation of[45]. the The stresses estimation acting of on the the stresses single particlesacting on based the single on the particles measured based data ison not the possible, measured yet, data because is not of thepossible, complexity yet, because of influencing of the complexity of influencing parameters. Therefore, the hydrostatic pressure is approximated by the overall axial compression stress acting on the powder. Furthermore, investigations of the

Pharmaceutics 2019, 11, 121 10 of 19 parameters. Therefore, the hydrostatic pressure is approximated by the overall axial compression stress acting on the powder. Furthermore, investigations of the compressibility of single crystals by hydrostatic compression with high pressures up to several GPa show the decrease of specific solid volume with rising hydrostatic pressure according to the Murnaghan equation [46]. This equation implies a stress dependency of the bulk modulus. The stress range applied for tableting is considerably lower compared to the applied stresses for single crystal analysis. Thus, a constant bulk modulus is assumed within the context of this investigation. The bulk modulus K of Equation (5) is replaced by the elastic compressibility factor C, which mainly comprises the influence of solid compressibility. According to these assumptions Equation (5) is modified as follows:  σ  V (σ ) = V · 1 − ax (9) S ax S,0 C where σax is the overall axial compression stress acting on the powder and C is here referred to as elastic compressibility factor. The in-die porosity under consideration of solid compressibility εc is given by Vs(σax) εc(σax) = 1 − (10) Vb(σax) Inserting Equation (9) in Equation (10) leads to the following mathematical relation for the in-die porosity under consideration of solid compressibility εc

σax
VS,0· 1 − C Vs,0 VS,0·σax VS,0·σax εc(σax) = 1 − = 1 − + = εapp(σax) + (11) Vb(σax) Vb(σax) Vb(σax)·C Vb(σax)·C

C describes mainly the elastic compressibility of the solid under uniaxial compression and is associated with the applied determination method. Therefore, C is not a directly measured material property and cannot be equated with the bulk modulus of single crystals. The solid compressibility term can be included in common compression equations, such as the model of Heckel and enable the determination of specific compression parameters under the consideration of solid compressibility.

5.4. Extension of the Model of Heckel by the Solid Compressibility Term For the application of the solid compressibility term, the Heckel model (Equation (1)) is transformed into its exponential form:

(− σax −A) Py εapp(σax) = e (12)

The limit value of the exponential term is zero and this is the necessary condition for the determination of the elastic compressibility factor C, which is determined by fitting of Equations (11) and (12) to the apparent in-die porosity curve from compaction experiments. Further details of the fitting procedure are summarized in chapter 4.4. The elastic compressibility factor C, determined by the mathematical approach, is comparable with the bulk modulus K characterized by mercury porosimetry for Para and MCC (Table4). The elastic compressibility factor of Lac is somewhat lower compared to the bulk modulus, so that the influence of solid compressibility is slightly overestimated. One possible reason for the deviation is the mathematical method for the determination of C, which is strongly dependent on the applied models. Additionally, the transferability of the bulk modulus determined by hydrostatic compression with a defined pressure to uniaxial powder compression with an unknown and complex stress distribution on the single particles might be limited. Moreover, the back-calculated elastic compressibility factor is also influenced by other mechanisms like die expansion. For DCPA, C is considerably larger than the values for the other three materials which correlates well with the expectation based on mercury porosimetry. The correlation of the elastic compressibility factor with the bulk modulus is generally good and confirms the suitability of the introduced mathematical approach for the consideration of Pharmaceutics 2019, 11, 121 11 of 19 the effect of solid compressibility. The introduced approach provides an inexpensive method for the correction of negative apparent in-die porosities and thus for the convergence of the apparent in-die Pharmaceuticsporosity to 2019 the, real11, x in-dieFOR PEER porosity, REVIEW which is not directly accessible. 11 of 19

Table 4.4. BulkBulk modulus modulus determined determined by mercuryby mercury porosimetry porosimetry and elasticand elastic compressibility compressibility factors derivedfactors derivedby extension by extension of the models of the ofmodels Heckel of and Heckel of Cooper and of & Cooper Eaton with& Eaton the solidwith compressibilitythe solid compressibility term and termby the and extended by the extended in-die compression in-die compression function. function.

Material Bulk Modulus K [GPa] CHeckel [GPa] CCooper &Eaton [GPa] CExtended function [GPa] Material Bulk Modulus K [GPa] CHeckel [GPa] CCooper &Eaton [GPa] CExtended function [GPa] MCC 10.22 ± 1.02 11.51 ± 0.17 11.87 ± 0.12 11.74 ± 0.30 ± ± ± ± MCCPara 10.22 14.931.02 ± 3.67 11.51 15.220.17 ± 0.07 11.87 18.41 ± 0.740.12 15.97 11.74 ± 1.00 0.30 Para 14.93 ± 3.67 15.22 ± 0.07 18.41 ± 0.74 15.97 ± 1.00 Lac 27.07 ± 7.13 14.09 ± 0.03 - 14.80 ± 1.46 Lac 27.07 ± 7.13 14.09 ± 0.03 - 14.80 ± 1.46 DCPA - 150.02 ± 17.24 - 277.06 ± 10.02 DCPA - 150.02 ± 17.24 - 277.06 ± 10.02

The consideration of C for the calculation of in-die porosities leads to a considerable shift of the HeckelThe plot consideration to lower values of C for thePara, calculation MCC and of Lac in-die (Figure porosities 5). The leads linear to region a considerable and by that shift the of applicabilitythe Heckel plot of the to lowerplot in values general, for is Para, extended MCC to and higher Lac (Figurecompression5). The stresses, linear region which and enables by that the applicationthe applicability of the ofHeckel the plot model in general, to a more is suitable extended fit torange higher When compression applying the stresses, correction which the enables whole pressurethe application range can of the be Heckelused for model the evaluation to a more of suitable Para and fit MCC range data. When However, applying the the curves correction are still the bentwhole at pressure maximum range compression can be used stress for the due evaluation to the still of Para very and low MCC porosities data. However, above approximately the curves are 350still MPa. bent atThe maximum real in-die compression porosity at high stress compression due to the stillstresses very is low expected porosities to beabove slightly approximately higher but is not350 experimentally MPa. The real in-die accessible. porosity Additionally, at high compression the slopes stressesof the plots is expected become tosmaller be slightly and this higher results but in is highernot experimentally mean yield pressures accessible. (Table Additionally, 5). However, the slopes the difference of the plots of P becomey and Py,c smaller is very and small this for results DCPA in becausehigher mean of the yield high pressures elastic compre (Tablessibility5). However, factor. the The difference differentiat of Piony and of thePy,c compressionis very small behaviour for DCPA ofbecause the different of the high materials elastic by compressibility Py,c is successful. factor. The The finding differentiation of the highest of the mean compression yield pressure behaviour for DCPA,of the different the lowest materials for MCC by P andy,c is a successful. medium value The finding for Lac of thecorrelates highest well mean with yield the pressure structural for observationDCPA, the lowest (Section for 5.1) MCC and and with a medium literature value [23,47 for,48]. Lac Commonly, correlates well the with yield the pressure structural is used observation for the classification(Section 5.1) and of the with plasticity literature of [materials.23,47,48]. It Commonly, is assumed the that yield low pressure mean yield is used pressures, for the classificationwhich indicate of athe low plasticity resistance of materials. against compression, It is assumed are that an low indicator mean yield for pressures,a high plasticity, which indicate while the a low plasticity resistance is decreasingagainst compression, with increasing are an mean indicator yield pressure for a high [47]. plasticity, According while to the the derived plasticity mean is decreasingyield pressures, with MCCincreasing and Para mean possess yield pressure the highest [47 ].plasticity According followed to the derivedby Lac, while mean DCPA yield pressures, has the lowest MCC plasticity and Para overallpossess (Table the highest 5). plasticity followed by Lac, while DCPA has the lowest plasticity overall (Table5).

(a) (b)

Figure 5. 5. ((aa)) Heckel Heckel plots plots calculated calculated using using the the apparent apparent and and the the corrected corrected in-die in-die porosity; porosity; ( (bb)) corrected corrected Heckel plots and applied model of Heckel for Para, MCC, Lac and DCPA.

Table 5. Parameters derived by Heckel analysis (n = 6).

With Solid Compressibility Term Material Py [MPa] Fit Range R2 [-] Py,T [MPa] Py,c [MPa] Fit range Entire Curve MCC 72.6 ± 1.0 10–50% σmax 89.3 ± 1.2 40–90% σmax 0.9860 543.5 Para 93.5 ± 1.2 6–56% σmax 121.2 ± 2.7 30–80% σmax 0.9739 - Lac 145.6 ± 0.5 25–75% σmax 194.5 ± 0.6 40–90% σmax 0.9935 233.1 DCPA 765.4 ± 1.5 38–88% σmax 771.2 ± 2.5 40–90% σmax 0.9313 1062.6

Pharmaceutics 2019, 11, 121 12 of 19

Table 5. Parameters derived by Heckel analysis (n = 6).

With Solid Compressibility Term Material Py [MPa] Fit Range R2 [-] Py,T [MPa] P [MPa] Fit Range y,c Entire Curve

MCC 72.6 ± 1.0 10–50% σmax 89.3 ± 1.2 40–90% σmax 0.9860 543.5 Para 93.5 ± 1.2 6–56% σmax 121.2 ± 2.7 30–80% σmax 0.9739 - Lac 145.6 ± 0.5 25–75% σmax 194.5 ± 0.6 40–90% σmax 0.9935 233.1 DCPA 765.4 ± 1.5 38–88% σmax 771.2 ± 2.5 40–90% σmax 0.9313 1062.6

Pharmaceutics 2019, 11, x FOR PEER REVIEW 12 of 19 It can be concluded that the quantification and differentiation of the resistance of the powder againstIt compression can be concluded by Heckel that the analysis quantification is successful. and differentiation However, the of Heckel the resistance model describes of the powder only the linearagainst region compression of the Heckel by Heckel plot analysis and not is the successf entireul. compression However, the curve. Heckel Deviations model describes are visible only inthe the lowlinear and region high compression of the Heckel stress plot and range, not which the entire are alsocompression the reason curve. for the Deviations overall low are goodness-of-fit,visible in the if thelow whole and high compression compression curve stress is considered. range, which The are comprehensive also the reason for differentiation the overall low of the goodness-of-fit, compressibility requiresif the thewhole description compression of the entirecurve in-dieis considered compression. The curves. comprehensive A model, whichdifferentiation is better suitedof thefor thecompressibility description of requires the entire the curve description is the model of the ofenti Cooperre in-die & compression Eaton [16]. curves. A model, which is better suited for the description of the entire curve is the model of Cooper & Eaton [16]. 5.5. Extension of the Model of Cooper & Eaton by the Solid Compressibility Term 5.5. Extension of the Model of Cooper & Eaton by the Solid Compressibility Term The criterion for the correction by the solid compressibility term (Equation (11)) when applying The criterion for the correction by the solid compressibility term (Equation (11)) when applying the model of Cooper & Eaton is to keep the relative volume change in the considered compression the model of Cooper & Eaton is to keep the relative volume change in the considered compression stress range (up to 400 MPa) lower than the limiting value of one. The model of Cooper & Eaton stress range (up to 400 MPa) lower than the limiting value of one. The model of Cooper & Eaton consists of two exponential terms, which are, however, able to describe relative volume changes above consists of two exponential terms, which are, however, able to describe relative volume changes one. Therefore, the fitting of the model including the solid compressibility term was performed using above one. Therefore, the fitting of the model including the solid compressibility term was performed ≤ theusing constraint: the constraint:V* 1. Thus,V* ≤ 1. in Thus, the considered in the considered case, the solidcase, compressibilitythe solid compressibility term is only term applicable is only if theapplicable apparent if relative the apparent volume relative change volume converges change to valuesconverges above to values one, which above isone, only which the case is only for the MCC andcase Para for (FigureMCC and6). Para The (Figure derived 6). elastic The derived compressibility elastic compressibility factors C for factor Paras and C for MCC Para are and comparable MCC are tocomparable the values derivedto the values by the derived model by of Heckelthe model and of are Heckel thus and as well are thus in good as well accordance in good withaccordance the bulk moduluswith theK bulk(Table modulus4). K (Table 4).

1.1 1.1 1.0 1.0

[-] 0.9 *

[-] 0.9 *

0.8 1.05 0.8 [-] 0.7 * 0.7 1.00 0.6 0.6

0.5 0.95 0.5 Para (V* ) 0.4 0.4 c 0.90 MCC (V* ) 0.3 c Rel. volume change V change volume Rel. 0.3

Rel. volume change V change volume Rel. * * * V change volume Rel. Lac (V app) 0.2 V app V c 0.85 0.2 DCPA (V* ) 0.1 Para 100 150 200 250 300 350 400 450 0.1 app MCC Compression stress [MPa] Cooper & Eaton 0.0 0.0 0 50 100 150 200 250 300 350 400 450 0 50 100 150 200 250 300 350 400 450 Compression stress [MPa] Compression stress [MPa] (a) (b)

FigureFigure 6. 6.(a ()a Apparent) Apparent and and corrected corrected relativerelative volumevolume change change for for Para Para and and MCC; MCC; (b ()b Relative) Relative volume volume changechange and and applied applied model model of of Cooper Cooper && EatonEaton for Para, MCC, MCC, Lac Lac and and DCPA. DCPA.

TheThe four-parametric four-parametric functionfunction ofof CooperCooper & Eato Eatonn enables enables the the mathematical mathematical description description of ofthe the wholewhole in-die in-die compression compression curve. curve. The The deviations deviations between between experimental experimental data data and and the the model model areare smallsmall for allfor four all powdersfour powders (Figure (Figure6) and 6) thusand thus the correlation the correlati coefficientson coefficients are are high high (Table (Table6). Obviously,6). Obviously, the the first termfirst mainly term mainly describes describes the volume the volume change change at low at compression low compression stresses stresses (<25 (< MPa), 25 MPa), while while the second the termsecond is almost term zerois almost up to zero 10 MPa up to and 10 primarily MPa and describesprimarily thedescribes volume the change volume at higherchange compression at higher stressescompression (Figure stresses7). (Figure 7).

Table 6. Parameters derived by the fitting of the model of Cooper & Eaton (n = 6).

V*app With Solid Compressibility Term V*c Material a1[-] k1 [MPa] a2 [-] k2 [MPa] a1c[-] k1c [MPa] a2c[-] k2c [MPa] R2 [-] MCC 0.58 ± 0.01 1.46 ± 0.16 0.45 ± 0.01 20.2 ± 1.0 0.57 ± 0.01 1.43 ± 0.20 0.45 ± 0.01 18.7 ± 0.9 0.9998 Para 0.70 ± 0.003 1.40 ± 0.10 0.35 ± 0.003 43.7 ± 2.4 0.69 ± 0.003 1.37 ± 0.10 0.34 ± 0.003 38 ± 2.0 0.9993 Lac 0.52 ± 0.002 0.84 ± 0.00 0.54 ± 0.002 53.6 ± 0.4 0.9963 DCPA 0.45 ± 0.01 0.96 ± 0.10 0.42 ± 0.009 29.2 ± 1.9 0.9926

Pharmaceutics 2019, 11, 121 13 of 19

Table 6. Parameters derived by the fitting of the model of Cooper & Eaton (n = 6).

* * V app With Solid Compressibility Term V c Material 2 a1 [-] k1 [MPa] a2 [-] k2 [MPa] a1c [-] k1c [MPa] a2c [-] k2c [MPa] R [-] MCC 0.58 ± 0.01 1.46 ± 0.16 0.45 ± 0.01 20.2 ± 1.0 0.57 ± 0.01 1.43 ± 0.20 0.45 ± 0.01 18.7 ± 0.9 0.9998 Para 0.70 ± 0.003 1.40 ± 0.10 0.35 ± 0.003 43.7 ± 2.4 0.69 ± 0.003 1.37 ± 0.10 0.34 ± 0.003 38 ± 2.0 0.9993 Lac 0.52 ± 0.002 0.84 ± 0.00 0.54 ± 0.002 53.6 ± 0.4 0.9963 ± ± ± ± PharmaceuticsDCPA 0.45 2019,0.01 11, x FOR 0.96 PEER0.10 REVIEW 0.42 0.009 29.2 1.9 130.9926 of 19

Figure 7. IndividualIndividual terms terms of the model of Cooper & Eaton for Para.

The influenceinfluence ofof thethe solidsolid compressibilitycompressibility term term on on the the derived derived specific specific compression compression parameters parameters is veryis very small small (Table (Table6). 6). Only Onlyk2 kof2 of Para Para clearly clearly decreases. decreases.k k11,, significantsignificant inin thethe lowlow compressioncompression stress range, is the lowest for DCPA followed by Lac, MCC and Para. The resistance against filling filling of large pores thus increases increases in in this this order. order. However, However, the the differences differences between between the the four four materials materials are are small. small. k2 differsk2 differs more more clearly. clearly. In this In thisstress stress range, range, MCC MCC show showss the lowest the lowest resistance resistance against against the filling the fillingof small of poressmall poresby deformation by deformation followed followed by DCPA, by DCPA, Para Paraand Lac. and Lac.The lowest The lowest compression compression resistance resistance was wasexpected expected for MCC for MCC and andthe thehighest highest for forDCPA DCPA base basedd on on the the slopes slopes of of the the compression compression curves. curves. Furthermore, the inapplicability of the solid compressibility term for Lac and DCPA has to be taken into account.account. TheThe differentiation differentiation of of the the materials materials based based on theon resistancethe resistance values values of the modelof the ofmodel Cooper of &Cooper Eaton & is Eaton difficult is difficult as the parameters as the parameters cannot becanno correlatedt be correlated to macroscopically to macroscopically reasonable reasonable rank orders rank ordersof the materials,of the materials, as found as as wellfound by as Sonnergaard well by Sonnergaard [49]. Therefore, [49]. anTherefore, extended an in-die extended compression in-die compressionfunction is developed, function is which developed, is based which on the is modelsbased on of the Heckel models and of of Heckel Cooper and & Eaton.of Cooper & Eaton.

5.6. Extended in-die Compression Function 5.6. Extended in-die Compression Function The extended in-die compression function consists also of two exponential terms (Equation The extended in-die compression function consists also of two exponential terms (Equation (13)) (13)) besides the solid compressibility term (Equation (11)). The counter and denominator of the besides the solid compressibility term (Equation (11)). The counter and denominator of the exponent exponent of the exponential terms are inverted compared with the model of Cooper & Eaton and, of the exponential terms are inverted compared with the model of Cooper & Eaton and, by that, in by that, in accordance to the model of Heckel. The limit value of these terms is zero, so that the solid accordance to the model of Heckel. The limit value of these terms is zero, so that the solid compressibility term is applicable in combination with this function. compressibility term is applicable in combination with this function.  σ   σ  = · −𝜎ax + · −𝜎ax 𝜀,εc,(f it𝜎(σ)ax=𝜀) ε∙𝑒𝑥𝑝−l exp + 𝜀ε∙exp−h exp (13) 𝜎σl 𝜎σh where εll is thethe porosityporosity changechange attributedattributed to the lowlow stressstress processprocess andand σll is the bulk densificationdensification strength whichwhich isis significantsignificant for for the the low low stress stress process, process, while whileεh ε ish theis the porosity porosity change change attributed attributed to the to thehigh high stress stress process process and σandh is σ theh is material the material densification densification strength strength of the of high the stress high process.stress process. The sum The of sumεl and ofε hεl equalsand εh theequals initial the porosity initial porosity at 0 MPa. at 0 MPa. The in-die porosity curves are shifted or rather bent into the positive porosity range compared with thethe apparent apparent in-die in-die porosity porosity curves (Figurecurves 8),(Figure by mathematically 8), by mathematically considering solid considering compressibility. solid Thecompressibility. derived in-die The porosity derived approaches in-die porosity zero for approaches materials withzero lowfor densificationmaterials with strength. low densification The curves strength. The curves of all four materials are bent to lower porosities just before reaching the maximum compression stress. This curvature at maximum compression stress is technically caused but does not represent the behaviour of the powder and therefore can be considered as measurement artefact due to construction, instrumentation and operation of the compaction simulator. Based on the small stress range in which this curvature occurs, it is not significant for the application of the extended in-die compression function. The deviations between in-die porosity and the function are small (Figure 8) and the correlation coefficients (Table 7) are very high for all investigated materials. The derived elastic compressibility factors are in good accordance with the bulk moduli for MCC and Para and are comparable to the values derived by applying the solid compressibility term in

Pharmaceutics 2019, 11, 121 14 of 19 of all four materials are bent to lower porosities just before reaching the maximum compression stress. This curvature at maximum compression stress is technically caused but does not represent the behaviour of the powder and therefore can be considered as measurement artefact due to construction, instrumentation and operation of the compaction simulator. Based on the small stress range in which this curvature occurs, it is not significant for the application of the extended in-die compression function. The deviations between in-die porosity and the function are small (Figure8) and the correlation coefficients (Table7) are very high for all investigated materials. The derived elastic compressibility factors are in good accordance with the bulk moduli for MCC and Para and are Pharmaceutics 2019, 11, x FOR PEER REVIEW 14 of 19 comparable to the values derived by applying the solid compressibility term in combination with the combinationmodel of Heckel with (Table the model4), except of Heckel for DCPA. (Table The 4), elasticexcept compressibility for DCPA. The factorelastic C compressibility of DCPA, derived factor by Cthe of extendedDCPA, derived in-die compressionby the extended function, in-die compress is considerablyion function, higher. is The consider in-dieably compression higher. The curve in-die of compressionDCPA does approximate curve of DCPA neither does theapproximate zero line norneither a plateau the zero value line innor the a plateau applied value compression in the applied stress compressionrange. The in-die stress porositiesrange. The at in-die maximum porosities compression at maximum stress compression are still very stress high (>20%).are still very However, high (>20%).one criterion However, of the one determination criterion of the of determination C is the convergence of C is ofthe the convergence curve to the of zerothe curve line. to This the range zero line.is clearly This range outside is clearly (>1000 outside MPa) the (> 1000 considered MPa) the compression considered stresscompre rangession whichstress range in turn which leads in to turn the leadsmathematically to the mathematically uncertain determination uncertain determination of C. Furthermore, of C. Furthermore, a deviation between a deviation C and between K is visible C and for KLac, is visible which for was Lac, already which discussed was already in Chapter discussed 5.4. in Chapter 5.4.

(a) (b)

FigureFigure 8.8. (a(a)) Corrected Corrected in-die in-die porosity porosity curves curves and appliedand applied extended extended compression compression function; function; (b) Residual (b) Residualplot for the plot fitted for the extended fitted extended compression compre functionssion forfunction Para, MCC,for Para, Lac MCC, and DCPA. Lac and DCPA.

Table 7. Specific compression parameters of the extended in-die compression function (n = 6). Table 7. Specific compression parameters of the extended in-die compression function (n = 6). 2 MaterialMaterial εl [-] εl [-] σl [MPa]σl [MPa] εh [-]ε h [-] σh [MPa]σh [MPa] C C[GPa][GPa] R R[-]2 [-] MCC MCC 0.20 ± 0.20 0.00± 0.00 8.5 ± 8.5 0.6± 0.6 0.51 0.51 ± 0.01± 0.01 79.0 79.0 ± 1.9± 1.9 11.74 11.74 ± ±0.300.30 0.99980.9998 Para Para 0.24 ± 0.24 0.00± 0.00 4.3 ± 4.3 0.1± 0.1 0.28 0.28 ± 0.00± 0.00 99.2 99.2 ± 3.2± 3.2 15.97 15.97 ± ±1.001.00 0.99930.9993 Lac Lac 0.14 ± 0.14 0.01± 0.01 9.4 ± 9.4 0.5± 0.5 0.32 0.32 ± 0.00± 0.00 177.3 177.3 ± 5.6± 5.6 14.80 14.80 ± ±1.461.46 0.99970.9997 DCPA DCPA 0.21 ± 0.21 0.00± 0.00 16.8 16.8± 0.2± 0.2 0.47 0.47 ± 0.00± 0.00 668.6 668.6 ± 3.5± 3.5 277.06 277.06 ± ±10.2010.20 0.99930.9993

TheThe first first exponential term describes the the low compression stress range, while while the the second exponentialexponential term mainly describesdescribes thethe highhigh compressioncompression stress stress range range (Figure (Figure9). 9). A clearA clear distinction distinction of l ofthe the bulk bulk densification densification strength strengthσ lσof of the the four four materials materialsin in thethe lowlow compressioncompression stress range is visible l (Table(Table 77).). σσ l isis most most likely likely mainly mainly dependent dependent on on the the friction friction between between particles particles and and their their Young’s Young’s l modulus.modulus. σσ lrisesrises in in the the order order Para Para < < MCC MCC < < Lac Lac < < DCPA. DCPA. This This correlates correlates well well with with the the experimental experimental findings.findings. The The lowest lowest bulk bulk densification densification strength strength for for Para Para may may be be explained explained by by a facilitated particle l rearrangementrearrangement due due to to the the rectangular rectangular particle particle shape. shape. σσ lofof MCC MCC is is also also low low because because of of a facilitated particle rearrangementrearrangement due due to to the the initially initially loose loose packing packing (initial (initial porosity porosity > 0.7) of > the0.7) elongated of the elongated particles. particles.The particle The rearrangement particle rearrangement of Lac may of Lac be hindered may be hindered due to a closerdue to initiala closer packing initial ofpacking the particles of the particles because of the very broad particle size distribution. The reasons for the highest σl for DCPA might be the hindered particle rearrangement due to a probably already very close packing of the spherical particles and the low elasticity. These are certainly only hypothetical considerations, which have to be proved by further investigations. The material densification strength σh is the lowest for MCC. Para shows a slightly higher resistance against compression followed by Lac. DCPA has the overall highest material densification strength. The material densification strength of Para and MCC can be explained by the mainly elastic and plastic deformation behaviour of these materials. The medium material densification strength of Lac may be traced back to the higher necessary compression stress for the breakage of the agglomerated/aggregated particles and the higher resistance of the primary particles against

Pharmaceutics 2019, 11, 121 15 of 19

because of the very broad particle size distribution. The reasons for the highest σl for DCPA might be the hindered particle rearrangement due to a probably already very close packing of the spherical particles and the low elasticity. These are certainly only hypothetical considerations, which have to be proved by further investigations. The material densification strength σh is the lowest for MCC. Para shows a slightly higher resistance against compression followed by Lac. DCPA has the overall highest material densification Pharmaceuticsstrength. 2019, 11 The, x FOR material PEER densificationREVIEW strength of Para and MCC can be explained by the mainly 15 of 19 elastic and plastic deformation behaviour of these materials. The medium material densification strength of Lac may be traced back to the higher necessary compression stress for the breakage of deformation due to their smaller particle size. The reasons for the considerably higher material the agglomerated/aggregated particles and the higher resistance of the primary particles against densificationdeformation strength due of to DCPA their smaller might particle be the size.high The stiffness reasons and for low the considerably elasticity of higher this material. material Thedensification extended in-die strength compression of DCPA might function be the high is well stiffness-suited and low forelasticity the precise of this description material. of the in-die compressionThe curves extended of all in-die model compression materials function and issu well-suitedccessfully for mathematically the precise description considers of the in-die the physical effect of compressionsolid compressibility. curves of all modelAdditionally, materials andthe successfullyextended mathematicallyin-die compression considers function the physical enables the effect of solid compressibility. Additionally, the extended in-die compression function enables the successful quantification and differentiation of the compression behaviour of the materials successful quantification and differentiation of the compression behaviour of the materials investigated. investigated.

Figure 9. Individual terms of the extended in-die compression function for Para. Figure 9. Individual terms of the extended in-die compression function for Para. 5.7. Comparison in- and out-of-die Compressibility 5.7. ComparisonSolid in- compressibility and out-of-die isCompressibility only one part of the overall elastic deformation during powder compression. Another factor is the elastic deformation and relaxation of the framework of the Solid compressibility is only one part of the overall elastic deformation during powder particles, which is not considered by the introduced mathematical approach yet because of considering compression.only the Another compression factor curve is butthe not elastic the decompression deformation curve and so far.relaxation The elastic of deformation the framework of the of the particles,framework which is thusnot the considered main reason forby the the differences introduced between mathematical the in-die compressibilities approach considering yet because of consideringsolid compressibilityonly the compressionεc and the out-of-die curve compressibilitiesbut not the decompressionεT (Figure 10). Therefore, curve theso remainingfar. The elastic deformationdifferences of the provide framework information is onthus the elasticthe main deformation reason of for the frameworkthe differences within thebetween bulk. It canthe in-die be deduced that the elastic relaxation of the bulk framework of Lac is least pronounced while compressibilities considering solid compressibility εc and the out-of-die compressibilities εT the highest values are measured for MCC. It needs to be clarified whether this can be described (Figure 10).by an Therefore, intercorrelation the remaining between solid differences compressibility provide and deformation information mechanisms on the elastic of the material.deformation of the frameworkTherefore, within the consideration the bulk. ofIt thecan decompression be deduced curvethat the is necessary. elastic relaxation The out-of-die of compressibilitythe bulk framework of of Lac is leastPara pronounced cannot be determined while becausethe highest of the occurrence values are of tabletmeasured defects duefor toMCC. capping. It needs The differences to be clarified whetherof this the can compressibility be described curves by alsoan intercorrelation cause differences ofbetween the mean solid yield compressibility pressures derived and by in-die deformation Heckel analysis considering solid compressibility P compared to out-of-die Heckel analysis P mechanisms of the material. Therefore, the considerationy,c of the decompression curve is y,Tnecessary. (Table5). Py,c is lower compared to Py,T because of the consideration of the elastic deformation of the The out-of-die compressibility of Para cannot be determined because of the occurrence of tablet defects due to capping. The differences of the compressibility curves also cause differences of the mean yield pressures derived by in-die Heckel analysis considering solid compressibility Py,c compared to out-of-die Heckel analysis Py,T (Table 5). Py,c is lower compared to Py,T because of the consideration of the elastic deformation of the particle framework. The deformation behaviour of MCC is classified as brittle based on Py,T. This is not reasonable because of the evidently plastic deformation behaviour of MCC [23,48]. The high mean yield pressure Py,T can be traced back to the small slope of the out-of-die compressibility between 200 MPa and 400 MPa. Furthermore, the mean yield pressure is dependent on the considered compression stress range. The compression behaviour of a powder is not only affected by plastic deformation and particle fragmentation but rather by all acting micro-processes including particle rearrangement and elastic deformation. Therefore, the comprehensive characterization of the compression behaviour requires the consideration of the entire compression curves including the effect of all these micro-processes. The introduced in-die approach enables such a comprehensive characterization of powder compressibility even if tablet defects occur. The estimation of the complete elastic recovery of the bulk solely based on in-die compression by a further development of the presented method using the decompression curve is a prospective aim.

Pharmaceutics 2019, 11, 121 16 of 19

particle framework. The deformation behaviour of MCC is classified as brittle based on Py,T. This is not reasonable because of the evidently plastic deformation behaviour of MCC [23,48]. The high mean yield pressure Py,T can be traced back to the small slope of the out-of-die compressibility between 200 MPa and 400 MPa. Furthermore, the mean yield pressure is dependent on the considered compression stress range. The compression behaviour of a powder is not only affected by plastic deformation and particle fragmentation but rather by all acting micro-processes including particle rearrangement and elastic deformation. Therefore, the comprehensive characterization of the compression behaviour requires the consideration of the entire compression curves including the effect of all these micro-processes. The introduced in-die approach enables such a comprehensive characterization of powder compressibility even if tablet defects occur. The estimation of the complete elastic recovery of the bulk solely based on in-die compression by Pharmaceuticsa further 2019, 11 development, x FOR PEER of REVIEW the presented method using the decompression curve is a prospective aim. 16 of 19

Figure 10. Comparison of in-die porosity considering solid compressibility εc with out-of-die c Figure 10.compressibility ComparisonεT for of DCPA, in-die Lac andporosity MCC. considering solid compressibility ε with out-of-die compressibility εT for DCPA, Lac and MCC. 6. Conclusions 6. ConclusionsThe physical effect of solid compressibility, which causes the increase of solid density with rising compression stress, cannot be neglected for the calculation of in-die porosities. The experimental Thedetermination physical effect of the of bulksolid modulus compressibility, is difficult which and expensive. causes the Therefore, increase a mathematical of solid density term forwith rising compressionthe consideration stress, cannot of solid be compressibility neglected for is introducedthe calculation based on of physical in-die considerations.porosities. The This experimental term determinationcan be used of the for bulk the extension modulus of is common difficult mathematical and expensive. models, Therefore, such as the a models mathematical of Heckel andterm for the of Cooper & Eaton providing an inexpensive method for the consideration of the influence of solid consideration of solid compressibility is introduced based on physical considerations. This term can compressibility during uniaxial powder compression. This method can be applied even if tablet be used defectsfor the occur. extension Additionally, of common an extended mathematical in-die compression models, function such as is introducedthe models based of Heckel on the and of Cooper common& Eaton models providing of Heckel an and inexpensive of Cooper & method Eaton. This for function the consideration precisely describes of the entireinfluence in-die of solid compressibilitycompression during curves uniaxial and enables powder the successful compression. differentiation This and method quantification can be of theapplied compression even if tablet defects occur.behaviour Additionally, of the investigated an extended pharmaceutical in-die powders. compression function is introduced based on the The consideration of solid compressibility leads to the approach of the in-die compression curves common models of Heckel and of Cooper & Eaton. This function precisely describes the entire in-die to the out-of-die compression curves. However, differences are still visible because of the elastic compressionrecovery curves of the and bulk, enables which is the not consideredsuccessful yet. differentiation The estimation and of the quantification elastic behaviour of of the the compression bulk behavioursolely of basedthe investigated on in-die compression pharmaceutical by consideration powders. of the decompression curve and thus the further Thedevelopment consideration of the of presented solid compressibility method, is a prospective leads to aim. the Continuing, approach the of systematic the in-die investigation compression of curves to the out-of-diethe influence compression of material parameters curves. (e.g., However, deformation differences behaviour, particleare still size visible and morphology) because of up the to elastic complex mixtures is required to allow the evaluation of the proposed model for realistic formulations. recovery of the bulk, which is not considered yet. The estimation of the elastic behaviour of the bulk solely based on in-die compression by consideration of the decompression curve and thus the further development of the presented method, is a prospective aim. Continuing, the systematic investigation of the influence of material parameters (e.g., deformation behaviour, particle size and morphology) up to complex mixtures is required to allow the evaluation of the proposed model for realistic formulations.

Author Contributions: Conceptualization, all authors, methodology, all authors, formal analysis, I.W., investigation, I.W., data curation, I.W., writing-original draft preparation, I.W., writing-review and editing, all authors, visualization, I.W., supervision, A.K.

Funding: This research received no external funding.

Acknowledgments: We thank Dirk Märtin, Markus Matthis and Arnaud Grandeury (all Novartis Pharma AG) and Tabea Rook, Mehmet Kale and Thorben Höltkemeier (all Technische Universität Braunschweig, Institute for Particle Technology) for experimental support. We acknowledge financial support from the German Research Foundation and the Open Access Funds of the Technische Universität Braunschweig.

Conflicts of Interest: The authors declare no conflict of interest. The co-authors Edgar John and Michael Juhnke are employees of Novartis Pharma AG. Novartis Pharma AG was involved in the design, interpretation and writing of the study, the discussion of data, revision of the manuscript, and the decision to publish the results. Novartis Pharma had no role (effect) on the execution of the study.

References

Pharmaceutics 2019, 11, 121 17 of 19

Author Contributions: Conceptualization, all authors, methodology, all authors, formal analysis, I.W., investigation, I.W., data curation, I.W., writing-original draft preparation, I.W., writing-review and editing, all authors, visualization, I.W., supervision, A.K. Funding: This research received no external funding. Acknowledgments: We thank Dirk Märtin, Markus Matthis and Arnaud Grandeury (all Novartis Pharma AG) and Tabea Rook, Mehmet Kale and Thorben Höltkemeier (all Technische Universität Braunschweig, Institute for Particle Technology) for experimental support. We acknowledge financial support from the German Research Foundation and the Open Access Funds of the Technische Universität Braunschweig. Conflicts of Interest: The authors declare no conflict of interest. The co-authors Edgar John and Michael Juhnke are employees of Novartis Pharma AG. Novartis Pharma AG was involved in the design, interpretation and writing of the study, the discussion of data, revision of the manuscript, and the decision to publish the results. Novartis Pharma had no role (effect) on the execution of the study.

References

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