Tabletability and compactibility of α-lactose monohydrate powders of different particle size. II: predicted relationships

Abstract This study aimed to evaluate a material sparing method to predict tabletability and compactibility relationships. Seven α-lactose monohydrate powders with varying particle size were used as test materials. The compressibility of the powders was determined experimentally, whereas tabletability and compactibility profiles were derived both experimentally and predicted. In the prediction method, two experimental compression parameters (Kawakita and Heckel plastic stiffness) and a single tensile strength reference value were used, all necessary data obtained from a single compression experiment. For both predicted and experimental relationships, compaction and tableting parameters (performance indicators) were calculated. The correction for viscoelastic recovery was successful in generating compressibility profiles that corresponded to the series of experimental out-of-die tablet porosities. For both the tabletability and compactibility, the experimental and predicted profiles showed a high degree of similarity. Good correlations were obtained between the predicted and experimental compaction and tableting parameters. It is concluded that the hybrid prediction method is a material sparing method, which can give good approximations of tabletability and compactibility relationships. The prediction method has the potential to be included as a part of a protocol for the characterisation of the tableting performance of particulate solids. GRAPHICAL ABSTRACT


Introduction
The fabrication of solid products by shaping powder into a solid specimen by confined compression is a common manufacturing method of diverse types of solid products, including medicines.The success of the compaction process requires that the powder can cohere into a tablet of sufficient mechanical strength.In a pharmaceutical context, mechanical strength is measured typically as the force required to split or break the tablet under unconfined loading condition.In tableting research, cylindrical compacts are often used, and the fracture force is often transformed into a tablet tensile strength, a procedure also referred to as the Brazilian test (Li and Wong 2013).In order to assess the ability of a powder to form a tablet comprehensively, the evolution of tablet tensile strength in relationship to another variable is typically used.Such relationships can be categorised into two types: tablet strength evolution as a function of a process variable and tablet strength evolution as a function of tablet micro-structure.Regarding the former, different process variables have been used, examples of which are applied force or pressure, degree of compression, work of compression and compression time.Among these, the tensile strength vs. applied pressure relationship is the dominating type used today.Regarding the latter, the dominating type of relationship is tensile strength vs. tablet relative density or porosity.
The relationship between tablet strength and compaction pressure and the relationship between tablet strength and tablet porosity have been referred to in the literature as tabletability and compactibility, respectively (Tye et al. 2005).These terms have also been adopted in a general chapter on tablet compression characterisation in the US pharmacopoeia (USP <1062> 2017).A number of mathematical expressions describing such relationships are reported in the literature, and a compilation of commonly used expressions has been presented elsewhere (Sonnergaard 2006).In striving to develop such expressions, important aspects include the need to derive compaction performance indicators and the need to predict how variations in tablet porosity or tableting pressure affect tablet tensile strength.
For compactibility, the Ryshkewitch-Duckworth compaction equation (Duckworth 1953;Ryshkewitch 1953) seems to be the dominating expression used in the pharmaceutical literature, by which two compaction parameters can be calculated.For tabletability, currently, there is no model or expression that has received the same recognition, but several approaches are presented in the literature to theoretically calculate the tensile strength of agglomerates or compacts.Some of these approaches are based on a bond summation concept, e.g.Alderborn (2003) and Wunsch et al. (2021), a concept originally developed by Rumpf (1962), rather than on a fracture mechanics model (Mullier et al. 1987).The expression used in this study is based on the utilisation of a bond summation concept, and the bond forces holding particles together are assumed to be proportional to the inter-particulate contact area (Eriksson and Alderborn 1995).The applied compression pressure is used in the calculation of this contact area, hence making it possible to predict a tabletability profile.To do so, the original expression was developed first by applying restriction criteria for the compaction pressure and second, by using a single reference tensile strength of a tablet formed at a single compaction pressure (Persson and Alderborn 2018).Thus, a semi-empirical hybrid model was derived (Persson and Alderborn 2018) with a predictive capacity, enabling a practical application of the model beyond pure data fitting.In order to reduce the impact of variability, the compression experiment needs to be repeated a few times.
It is stated elsewhere (Vreeman and Sun 2022) that this type of three region tabletability profile, comprising three regions with a linear increase in strength between the two pressure thresholds, is a simplification of reality.This is a correct remark, but earlier experiences (Persson and Alderborn 2018) have shown that the simplified model may approximately fit experimental data for a series of different materials.Sonnergaard (2006Sonnergaard ( , 2022) also pointed out that a linear relationship satisfactorily approximates the region of a tabletability relationship, which, from an applied point of view, is the most relevant pressure range, i.e. a marked deviation from a linear profile occurs typically at pressures above the tablet manufacturing pressure range.
The construction of a predicted tabletability profile requires a value of the Heckel parameter derived from a porosity-pressure relationship.Since in situ porosity-pressure data are thus available, it should be possible to couple this dataset to the predicted set of tablet strength values and therefore also calculate a predicted compactibility profile.We now wish to explore the possibility of using the proposed three-stage compaction model to construct both predicted tabletability and compactibility profiles.To that end, predicted relationships were derived and compared with experimental tabletability and compactibility relationships reported previously (Persson et al. 2022) for a series of crystalline lactose powders.The powders had a median diameter ranging from 203 mm down to about 3.5 mm and showed significant differences in tablet forming ability (Persson et al. 2022).The hybrid approach used to calculate the predicted profiles (compressibility, compactibility, and tabletability), including required parameters, is schematically described in Figure 1.The long-term ambition of this research is to establish a detailed protocol that can be used both in academic and industrial settings as a means to characterise particles to be used in solid medicinal products.

Materials
Seven a-lactose monohydrate powders of different particle sizes were used: Pharmatose Microfine and Lactohale V R LH300 (kindly donated from DFE Pharma, Borculo, The Netherlands) and Pharmatose V R 200M (obtained from DMV, Veghel, The Netherlands).The following abbreviations are, henceforth, used for the powders: CL80, CL130, LH201, LH230, LCMF, LH300, and CL200.Particle size, compression properties, and tabletability and compactibility relationships have previously (Persson et al. 2022) been reported for all these powders.

Powder compression and calculation of in-die compression parameters
A powder mass of 320-400 mg of each lactose powder was uniaxially compressed at 300 MPa (n ¼ 5) in a materials tester (Zwick Z100, Zwick/Roell GmBH & Co., Ulm, Germany) mounted with 11.3 mm circular flat-faced punches.During compression, the upper punch moved at a rate of 10 mm/min.Before each compression, punches and die were lubricated by brushing a 1% magnesium stearate suspension in ethanol over the metal surfaces, followed by air drying.The force-displacement curve was adjusted for elastic system deformation as described previously (Nordstr€ om et al. 2008).
The in-die porosity vs. pressure data were fitted to the Heckel equation (Heckel 1961) in order to calculate the in-die (P y, inÀdie Heckel plastic stiffness), as previously described (Mahmoodi et al. 2013).The engineering strain vs. pressure data were used to calculate the Kawakita b À1 parameter, as previously reported (Persson et al. 2022).The P y, inÀdie and Kawakita b À1 parameter (Table 1) were taken from Persson et al. (2022).

Tablet compaction
The tablets (n ¼ 5) generated at 300 MPa during the compression experiment in the materials tester described above were collected directly after compression and subjected to a diametrical compression test (PharmaTest, PTB511E, Hainburg, Germany).From the fracture force and tablet dimensions, the tensile strength of the tablet was calculated as given elsewhere (Li and Wong 2013).The tensile strength determined was then used as the reference tablet strength (r t, ref ) in the construction of the predicted relationships (see below).It should be noted that the reference tablet strength (r t, ref ) was determined for each individual lactose powder used in the study.
The compaction and strength determination of tablets used for the experimental tabletability and compactibility relationships, by which the predicted relationships were compared, have been reported previously (Persson et al. 2022).For these compaction experiments, a single punch press (Korsch EK0, Hamburg, Germany) was used.It should be noted that the compression time in the single punch press is considerably lower than in the materials tester used to determine the r t, ref , which was used to construct the predicted profiles.This may affect the agreement of tensile strength between tablets compacted in the different equipment at the same compaction pressure.

Viscoelastic correction of porosity-pressure profile
From the compression data, the engineering strain and porosity of the powder during compression (in-die) were calculated.The in-die porosity was subsequently corrected for elastic and viscoelastic recovery based on the method by Katz et al. (2013) in order to predict an out-of-die porosity, hereinafter referred to as the in-die viscoelastic corrected porosity.This method is first based on the correction of the solid fraction for elastic recovery ðSF elast ) according to the following equation: where SF c and SF d represent the solid fraction during compression and decompression, respectively.SF dð0Þ is the solid fraction of a fully unloaded tablet prior to ejection from the die.Second, SF elast was corrected for viscoelastic recovery (SF visc ) according to following equation: where SF elast max is the solid fraction corrected for elastic recovery at maximum applied pressure, and SF outÀofÀdie is the solid fraction of the ejected tablet.The in-die viscoelastic corrected porosity was calculated as 1ÀSF visc : The subtraction used to calculate the SF elast (Equation ( 1)) requires that SF c and SF d are values sampled at identical pressures, i.e. the same compression and decompression pressures are used to obtain matching pairs of solid fraction.In practice, the experimentally recorded compression and decompression pressures vary slightly between each compression experiment.Moreover, since the sampling occurred at a frequency of approximately 1 per 2.5 ms, the number of sampling points differed between the compression and decompression phases.To satisfy the requirement of matching pairs of solid fractions, interpolated solid fractions were therefore calculated as follows: first, a series of pressures, denoted P int , in the range 0.3-300 MPa with a step change of 0.2 MPa was defined.Second, for each P int , the two nearest experimental pressures to P int , one above and one below, were identified for both the experimental compression and the decompression curves, giving a series of very narrow pressure ranges for each curve.Finally, the solid fraction at pressure P int was calculated by interpolation, using linear regression in each of the series of pressure ranges.
Thereafter, by using the visco-elastic corrected porosity-pressure profiles, the visco-elastic corrected Heckel plastic stiffness (P y, corr ) was calcuated with the same procedure as described in Section 2.2.1.

Prediction of tabletability
The expression used in this study for predicting a tabletability profile is based on a model for the evolution of tablet tensile strength with compaction pressure, assuming that particle permanent plastic deformation controls the inter-particulate contact area and tablet tensile strength (Eriksson and Alderborn 1995).The expression gives a linear increase in tablet strength between a lower and an upper compaction pressure threshold (Alderborn 2003).Experimental tableting profiles typically deviate from this model, but experiences indicate that the model captures an experimental profile in a satisfactory way in a practically relevant pressure range (Persson and Alderborn 2018;Persson et al. 2022).Hence, although conceptually, it is a simplification of real tabletability plots, it is considered a satisfactory approximation of such profiles over a wide range of compaction pressures.
The tablet strength expression, the method by which the necessary compaction parameters are derived and the subsequent construction of the predicted tabletability profile have been presented previously (Persson and Alderborn 2018).The method is based on a few compression experiments (described in Section 2.2.1) at a single compression pressure and the determination of the tensile strength of the tablet formed at this compression pressure (denominated subsequently as the reference compression pressure (P ref ) and the reference tablet tensile strength (r t, ref ), described in Section 2.2.2).From the powder compression experiments in the materials tester, the Kawakita b À1 and the Heckel plastic stiffness P y were derived.These compression parameters together with a parameter proportionality factor (a) and r t, ref were subsequently used to construct the theoretical tabletability relationship.
The method involves the following steps: first, the lower critical pressure (P c1 ), which is the lowest pressure for the formation of a coherent tablet (i.e.below this pressure, the tablet tensile strength is zero), is set equal to the Kawakita parameter b À1 , i.e.: Second, the upper critical pressure (P c2 ), which is the pressure at which the maximum tablet tensile strength (r t, max ) is reached and there is no further increase in the tablet tensile strength, is calculated as the sum of the lower critical pressure and the range during which the tablet tensile strength increases linearly with pressure.This range is estimated as the product between the Heckel plastic stiffness and the parameter proportionality factor, giving the following expression: The product of the a parameter and the P y is a compression parameter C (Alderborn 2003) defined as the width of the compaction pressure region during which the tablet tensile strength increases linearly with pressure, i.e. between P c1 and P c2 : Third, the corresponding r t, max at the upper critical compression pressure P c2 is calculated as: Finally, the tablet tensile strength (r t ) at each compaction pressure (P) in the linear region of the relationship can be predicted as, which combined with Equation ( 5) reduces to,

Prediction of compactibility
Since the compaction parameters of the tabletability relationship are derived from a powder compression experiment, the change in tablet porosity with compaction pressure has been recorded.This set of tablet porosity-compaction pressure data can also be used to predict, in addition to the tabletability profile, a compactibility profile by coupling the measured tablet porosity data to the predicted tablet strength data.The procedure used for the calculation of tablet porosities and for the pairwise connection of the two series of data was the following: the calculation of out-of-die tablet porosities, using the correction of elastic and viscoelastic recovery in Equations ( 1) and ( 2), gave in-die viscoelastic corrected porosities at a series of compression pressures P int : To couple the in-die viscoelastic corrected porosity pair-wise to the predicted tensile strength for tablets at the same pressures, the r t, pred was calculated by Equation ( 7) at each interpolated compression pressure P int : Thereafter, predicted compactibility relationships were plotted.

Experimental tabletability parameters
Experimental tabletability relationships have been reported earlier for CL200 (Persson and Alderborn 2018) and for CL80, CL130, LH201, LH230, LCMF, and LH300 (Persson et al. 2022) as well as some tabletability parameters (Table 2) derived from these relationships.In this work, the lower and upper critical compression pressures, P c1 and P c2 , were additionally calculated from these relationships.The procedure used for calculating these two parameters was the following: P c1 was calculated as the x-intercept using linear regression (R 2 >0.967) for the four lowest applied compaction pressures for all powders, except for CL80.For CL80, the two lowest compaction pressures were disregarded due to a large deviation from linearity; thus, the next four applied compaction pressures were used instead.P c2 was determined using the intersection of the linear regression line (R 2 >0.971) for pressures < 700 MPa and the maximum tablet tensile strength (r t, max ), with the latter being taken from Persson et al. (2022).The compression parameter C, i.e. the width of the linear region, was calculated from the experimental tabletability relationships reported previously (Persson and Alderborn 2018) as the reciprocal of the slope (R 2 >0.971) of the relationship between the relative tensile strength ( rt rt, max ) and the effective pressure (P À P c1 ) for pressures < 700 MPa.Two experimental parameter proportionality factors were calculated as the ratio of C and P y, inÀdie and the ratio of C and P y, corr and denoted a inÀdie and a corr , respectively.

Predicted tabletability parameters
The predicted tabletability parameters were calculated as described above.In addition, the slopes (k c ) of the predicted tabletability profiles were calculated using the predicted strength values at three compression pressures, i.e.P c1, pred , P ref , and P c2, pred : P c2, pred were calculated (Equation (4)) using P y, corr and a ¼1.8 as a generic value for all powders.
2.5.Calculation of compactibility parameters 2.5.1.Experimental and predicted compactibility parameters Two compactibility parameters were derived from the compactibility relationships and the same procedure was applied for both experimental and predicted relationships.The parameters were derived by fitting the tablet porosity-tablet tensile strength data to the Ryshkewitch-Duckworth compaction equation (Duckworth 1953;Ryshkewitch 1953): and calculating the slope of the relationship (k R ) and the intercept (i.e. the compact tensile strength at zero porosity, r 0 ) as compactibility parameters.
The compaction data used to calculate the experimental parameters for all powders have been reported previously (Persson et al. 2022).In this study, the experimental compactibility parameters were re-calculated and differ thus slightly from the earlier reported parameters.The reason is that, in order to compare experimental and predicted parameters, the same compaction pressure range needs to be used.This range was dictated by the compression pressure used in the powder compression experiments (P ref ¼ 300 MPa).Both experimental (R 2 >0.968) and predicted parameters were obtained by linear regression of the Ryshkewitch-Duckworth compaction equation within a porosity (e) range corresponding to compression pressures of 30-300 MPa.

Model calibration
In a previous paper (Persson et al. 2022), compactibility and tabletability of a series of lactose monohydrate powders of varying particle size were experimentally studied, and the physical significance of two compression parameters for each of the two relationships was discussed.The predicted tabletability and compactibility profiles presented in this paper are derived for the same powders and are based on compression parameters obtained from a single compression experiment, i.e. the Kawakita b À1 parameter which is transformed into P c1 , and the Heckel plastic stiffness P y , which is transformed into the compression parameter C (the width of the linear region of the tableatability plot).Since the transformation (calibration) factors for P c1 and C are a priori unknown, they need to be calibrated using the experimental tabletability parameters.These experimental parameters were calculated from the tabletability profiles (dotted curves in Figure 2) by assuming that the profiles obeyed the three-stage model (Alderborn 2003), as shown in Table 2.
Regarding P c1 , the overall trend obtained was a decreased P c1 with a decreased original particle diameter.Thus, a decrease in particle size increased the ability of the powder to form a tablet of measurable strength.The same trend was also obtained for the effect of particle size on the Kawakita b À1 parameter.Thus, a correlation was obtained between P c1 and b À1 , albeit with a slope larger than unity for these powders, which indicates that the b À1 should be multiplied with a calibration factor of about 3 to represent P c1 : However, it was earlier concluded that for a set of other powders (Persson and Alderborn 2018), b À1 without further correction gave a reasonable approximation of P c1 : Another calibration factor than one for b À1 was not considered necessary; however, the issue deserves to be further studied for more materials.
For C, the transformation of P y into C is done by a factor denoted as the parameter proportionality factor (a), which hence represents the model calibration factor for particle plasticity.This factor could conceptually be understood as a constraint factor, i.e. a factor that transforms one indication of particle plastic stiffness into another.In this transformation, either an in-die or an out-ofdie indication of P y can potentially be used.Hence, both in-die and out-of-die indications of P y were derived.
The in-die porosity-pressure relationships displayed a typical non-linear reduction in powder bed porosity (e) with increased pressure (solid curves in Figure 3).Compared to the porosity of the ejected tablets (dotted curves in Figure 3), the in-die tablet porosities were generally lower than the ejected tablet porosities.In order to compensate for the elastic recovery during ejection, the in-die data were corrected for viscoelastic recovery, according to Katz et al. (2013).The corrected in-die profiles (dashed curves in Figure 3 and referred to as in-die corrected porosity-pressure profiles), agreed well with the ejected tablet porosity profiles with the exception of LH230 (Figure 3(e)), for which the ejected tablet porosities were generally higher than the calculated in-die corrected tablet porosities.The obtained difference between the indie and in-die corrected porosity-pressure profiles subsequently influenced the calculated plastic stiffness P y (Table 1).The P y, inÀdie increased consistently with reduced particle size.In contrast, the P y, corr did not show a similar consistent trend, and there was a tendency of a reduced P y, corr with reduced particle size, i.e. the opposite trend to P y, inÀdie : Moreover, the P y, corr was generally considerably higher than the P y, inÀdie , i.e. the P y, corr was a factor of 1.6-2.7 larger than the P y, inÀdie , but the difference between them tended to decrease with decreased particle size.
If a compactibility profile is to be predicted, it is preferable to use approximations of ejected tablet porosities rather than in-die tablet porosities.The difference between in-die and ejected tablet porosities evidently affected the plastic stiffness P y , which subsequently will affect the value of the parameter proportionality factor a: Lower values of a were obtained using P y, corr in the calculations (Table 2).Moreover, the variation in a corr was lower than for a inÀdie , and no particle diameter-dependent difference was obtained between the powders for a corr : Thus, the particle size effect on a inÀdie could be explained predominantly by variations in particle elastic deformation during compression, a difference that hence was eliminated for the a corr : In practical applications of the tabletability model, such as in formulation development, it is ideal to use a single generic value of the parameter proportionality factor a, which reasonably represents a wide range of powders with varying compression and compaction properties.Moreover, it is preferable to use an a-value based on the in-die corrected data (a corr ), and it is evident that the variation in a corr for the lactose powders used in this study was lower than for a inÀdie : In the following discussion, the same a corr for all powders is used in the prediction method.The value chosen for a corr was 1.8, which is the median value of the cumulative distribution of the in-die corrected parameter proportionality factor (Table 2) and approximately the mean value.

Prediction method performance
The prediction method performance is discussed below in the following two aspects: first, a qualitative comparison between experimental and predicted tabletability and compactibility relationships based on their visual similarity and second, the correlations between two parameters derived from the profiles.
3.2.1.Tabletability 3.2.1.1.Visual similarity.A high degree of similarity was generally obtained between the complete experimental and predicted tabletability relationships (Figure 2).The r t, ref (squared open symbols at 300 MPa) was slightly higher for the three coarsest lactose grades than the corresponding tensile strength of tablets compacted at similar compaction pressure in the single punch press.This difference in tensile strength may be due to the considerably lower compression rate in the materials tester compared to the single punch press.Thus, slightly higher slopes were obtained for the predicted relationship compared to the experimental for these powders.
The prediction method gives a linear relationship in the middle region of the tabletability profiles.The experimental profiles deviated from linearity in two ways: a non-linear part directly above the P c1 and a bending when r t, max was approached.The deviations between the predicted and experimental relationships are illustrated by the residuals, i.e. the difference between experimental data points and the predicted tensile strength at the same compaction pressure (Figure 4).The deviations were affected by the original particle diameter of the powders.For the powders of a median particle diameter >10 mm (Figure 2(a-d)), the residuals (Figure 4(a)) were mostly negative; furthermore, their absolute values increased steadily with compaction pressure over the major part of the pressure range due to a difference in slope between the predicted and experimental relationships.There was only a limited bending of the experimental profiles at low applied pressures, but at the highest pressures, the residuals increased due to the bending of the experimental relationship before the tablet strength plateau was reached.However, over the major part of the pressure range, the experimental relationships were nearly linear.The experimental relationships for powders of a particle diameter <10 mm had a more pronounced sigmoidal profile (Figure 2(e-g)), i.e. bended relationships at both the low and high compaction pressures, deviations also shown by the progression of the residuals with compaction pressure (Figure 4(b)).At low pressures, the residuals were generally negative and decreased, but after a pressure of 100-200 MPa, the residuals increased and became positive at the highest pressures.
For most powders, the predicted linear relationship was within the confidence limits of the experimental profiles (red regions in Figure 2) over the major part of the data points.Moreover, the correlation coefficients, calculated by linear regression of the experimental data points in the compaction pressure range between P c1 and P c2 , ranged from 0.959 to 0.998 (Table 3).Thus, a linear evolution of the tabletability relationships represents a satisfactory approximation of the experimental data (Sonnergaard 2022).In summary, based on a visual similarity between the experimental and predicted relationships, it is concluded that an excellent concordance was obtained.
3.2.1.2.Correlation between parameters.Two tableting parameters were calculated as descriptors of the experimental and predicted tabletability relationships (Persson et al. 2022): one endpoint parameter (r t, max ) that represents the plateau region of the tabletability profile and one rate parameter (k c ) that represents the slope of the linear region of the profile.The experimental and predicted parameters were strongly dependent on the original particle size (Table 3); additionally, for both tableting parameters, the prediction method rank ordered the powders, similar to the experiments.
For the r t, max vs. r t, max pred relationship (Figure 5(a)), a slope below one and an intercept above zero were obtained due to a tendency that the predicted values were overestimated for the coarser powders, especially for CL80, and underestimated for the finer powders (Table 3 and Figure 2).However, a nearly linear relationship was obtained between the r t, max and r t, max pred (Figure 5(a)), with reasonable correlation.Hence, the prediction method gave approximate estimates of the maximum tablet tensile strengths.
For k c , a linear regression of a plot of experimental vs. predicted data (Figure 5(b)) showed a slope close to one, an intercept close to zero and a high correlation coefficient.Thus, the prediction method gave very good estimates of the evolution in tablet strength with compaction pressure.
The agreement between experimental and predicted tabletability relationships is affected by a number of factors, including the similarity of the overall shape of the tabletability profile, the used value of the parameter proportionality factor a corr and the reference tablet tensile strength.In view of this complexity, it is concluded that the prediction method gave very good approximations of the tableting rate parameter, i.e. the gradient of the nearly linear part of the profile.Thus, the tableting rate parameter represents an interesting, valuable descriptor of the tableting performance of powders, which can be used in databases for the characterisation of powder performance.Regarding the end-point parameter, i.e. the plateau of the profile, the prediction method  gave reasonable approximations, which represent valuable complementary additions to the rate parameters but with the less prediction capability.
3.2.2.Compactibility 3.2.2.1.Visual similarity.The maximum compression pressure used in the prediction method was 300 MPa, and the tablet porosity obtained at 300 MPa, i.e. about 0.1, represented the lower tablet porosity used in the prediction of compactibility.The predicted compactibility profiles showed the typical non-linear increase in tablet tensile strength with decreased tablet porosity (Figure 6), and the predicted and experimental relationships almost overlapped with each other for most powders in the range of tablet porosities.A larger difference between the predicted and experimental relationships was obtained for the powders having the largest and smallest median particle diameters, i.e.CL80 and LH300.For CL80, the progression of the predicted and experimental profiles differed at the highest tablet porosities and for LH300, the predicted profile was generally shifted towards higher tablet tensile strength.For the residuals between the predicted and experimental compactibility profiles (Figure 7), except for LH300 (squared open symbols in Figure 7(b)), low and almost constant residuals were obtained for tablet porosities above about 0.15.Below this tablet porosity, the residual became larger both in positive and negative directions, i.e. the predictions became worse at low tablet porosities.
In summary, based on a visual similarity assessment, it is concluded that a good agreement was obtained between the experimental and predicted compactibility relationships.

Correlation between parameters.
In order to calculate compaction parameters from the predicted and experimental tensile strength vs. porosity relationships (the extrapolated tablet strength at zero porosity r 0 and the rate of change of tablet strength with porosity k R , parameters which conceptually correspond to the tabletability parameters r t, max and k c ), the compactibility data were plotted according to the Ryshkewitch-Duckworth expression.This expression presupposes that the natural logarithm of the tablet tensile strength relates linearly to the tablet porosity.One should, however, note that the experimental compactibility data obeyed the Ryshkewitch-Duckworth equation only approximately over a wide range of tablet porosities due to a systematic deviation from linearity (Persson et al. 2022), which is contrary to the commonly used assumption regarding the appropriateness of this approach.Also, the predicted relationships plotted according to the Ryshkewitch-Duckworth equation clearly captured the slightly bended type of profile, which characterised most of the experimental profiles (plots not shown).In the calculation of the compaction parameters r 0 and k R , the same pressure range was used for both the experimental and predicted tabletability profiles, i.e. matching conditions were used.Regarding r 0 , the experimental and predicted parameters were both strongly dependent on original particle size (Table 4).Moreover, the predicted and experimentally derived values showed a similar rank order, albeit with a changed order for the finest powders LH230, LCMF, and LH300 (there were, however, only small differences in r 0 values for these powders).In most cases, the experimental values were higher than the predicted values, with the largest difference for the finest powders, but a good correlation was obtained between the predicted and experimental r 0 (Figure 8(a)).However, the slope of the r 0 vs. r 0, pred relationship was markedly below one due to an over-prediction for CL80 in combination with an under-prediction for the other powders.
For the rate parameter, the prediction (k R, pred ) gave a gradual reduction in the parameter with decreased particle diameter.It has previously been shown (Persson et al. 2022) that the experimental Ryshkewitch-Duckworth plots for these powders tended to be shifted in parallel to each other, and the slope k R showed a limited dependency on the original particle diameter.Thus, for all powders, a poor correlation was obtained between k R and k R, pred (Figure 8(b)).
For the experimental compaction parameters, a consequence of the strong effect of particle size on r 0 and a limited effect on k R is that these two parameters showed a poor correlation with each other (Figure 9(a)).On the contrary, the predicted compaction parameters k R, pred and r 0, pred showed a good correlation (Figure 9(b)) with a negative gradient, i.e. an increased r 0, pred corresponded to a decreased k R, pred : The prediction method generates a considerably larger number of data points than the experimental method, which is a potential benefit of the former to deliver a better representation of the compaction properties of the powders and thus the r 0 vs. k R relationship.The predicted relationships may thus indicate that both k R and r 0 vary with particle size and correlate with each other.
In summary, the predicted compaction parameters gave a similar rank-order as, and correlated well with, the corresponding experimental parameters, supporting the validity of the prediction method for the prediction of powder compactibility.

Conclusions
The compressibility, tabletability, and compactibility relationships for powder formulations are considered to be critical in order to broadly understand the compaction behaviour of powders (USP <1062> 2017).With the prediction method proposed in this paper, it was possible to derive all three relationships by solely a few in-die powder compressions using a single compression pressure.The compressibility was assessed experimentally, whereas the tabletability and compactibility were predicted based on two experimental compression parameters and a single tensile strength value.
It was shown that the correction for viscoelastic recovery was successful in generating compressibility profiles that corresponded to the experimental out-of-die tablet porosities.The use of the viscoelastic corrections gave similar values to the parameter proportionality factor a despite a great variation in particle size and compression and compaction properties of the powders.Thus, a single value of the factor a could be used in the predictions of tabletability and compactibility.
Despite a systematic difference in the evolution of experimental and predicted tabletability profiles, they showed a high degree of similarity.Moreover, for the compactibility profiles, a good concordance was obtained between the experimental and predicted relationships.Two types of parameters were derived from the profiles: one rate parameter and one end-point parameter, and for the respective type of parameter, a good correlation was obtained between the predicted and experimental parameters.It was thus concluded that the hybrid prediction method can be used to construct both the predicted tabletability and compactibility relationships using only a few experiments that show a high degree of similarity to the corresponding experimental relationships.
In the study, a series of particulate materials of the same chemical and crystalline structure (crystalline a-lactose monohydrate) were used, characterised by a wide range of particle size and compression and compaction properties.Within this range of particle size, a brittle-ductile transition is to be expected (Roberts and Rowe 1987), giving also a variation in particle mechanical properties.It is, thus, believed that the prediction method can be used to characterise tabletability and compactibility of materials that differ in particle size and particle mechanics.The prediction method herein described can, thus, be included as a part of a protocol for the characterisation of the tableting performance of particulate solids (Nordstr€ om et al. 2012).

Figure 1 .
Figure1.Overview of the protocol for the hybrid approach.

Figure 2 .
Figure 2. Comparison between the experimental (dotted curves) and predicted tabletability (solid curves, the squared open symbols show the P c1, pred , r t, ref , and P c2, pred , respectively) relationships for (a) CL80, (b) CL130, (c) CL200, (d) LH201, (e) LH230, (f) LCMF, and (g) LH300.The error bars display the standard deviations, and the red regions show the 99% confidence interval for experimental data points.Note the varying range of the y-axes.Ã Experimental data from Persson et al. (2022).ÃÃ Experimental data from Persson and Alderborn (2018).

Figure 4 .
Figure 4. Tabletability residuals as a function of compaction pressure for powders with an original median particle size (a) >10 mm and (b) <10 mm.

Figure 5 .
Figure 5. Correlation between experimental and predicted tabletability (a) endpoint parameters and (b) rate parameters.The experimental r t, max is obtained from Persson et al. (2022).The error bars denote the standard deviations.The linear regression lines and the linear regression equations are included in the figures.

Figure 7 .
Figure 7. Compactibility residuals as a function of compact porosity for powders with an original median particle size (a) >10 mm and (b) <10 mm.

Figure 8 .
Figure 8. Correlation between experimental and predicted compactibility: (a) endpoint parameters and (b) rate parameters.The linear regression line together with the linear regression equation are included in (a).

Figure 9 .
Figure 9. Correlation between compactibility endpoint and rate parameters for (a) experimental relationships and (b) predicted relationships.

Table 2 .
Descriptors of experimental tabletability relationships.Parameter proportionality factor, calculated using in-die yield pressure.e Parameter proportionality factor, calculated using corrected yield pressure.
b Upper critical pressure.c Compression parameter.d
b Regression coefficient of linear regression line.c Tabletability endpoint parameter.d Predicted tabletability rate parameter.e Predicted tabletability endpoint parameter.f
a Compactibility rate parameter.b Compactibility endpoint parameter.c Regression coefficient of linear regression line.d Predicted compactibility rate parameter.e Predicted compactibility endpoint parameter.