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Ockham's Razor Applied on Pharmaceutical Powder Compaction Models

Published:November 30, 2020DOI:https://doi.org/10.1016/j.xphs.2020.11.026

      Abstract

      This investigation contains a critical survey of a collection of five well-known and often used standard models in pharmaceutical technology. The basic idea is to use the recognised Ockham's razor as a tool in search for simpler methods or explanations for these models. The study includes the indirect tensile strength of tablets, Pitt's equation for tensile strength of biconvex tablets, Adams' model for strength of agglomerates, Weibull's distribution for variability of strength measurements and Heckel's equation for compressibility. In all these cases simpler and equally valid solutions and explanations are presented and subsequently preferred rather than the original.

      Keywords

      Introduction

      William of Ockham (1287–1347) was an English friar and medieval philosopher. He is known as the creator of the principle of Ockham's razor: Pluralitas non-est ponenda sine necessitate, “Plurality should not be posited without necessity” Another formulation is: “When two competing theories make exactly the same predictions, the simpler one is the better.” The term razor refers to the choice, where the most complex with most assumptions is shaved away.
      It is important to notice that the two hypotheses or models must have equal validity. The planets motion around the sun may be described by a circular movement as proposed by Copernicus or the elliptical orbits later established by Kepler. Copernicus's model is simpler, but Kepler's model fits the astronomical observations of Tycho Brahe, and accordingly is the right one.
      Another example is the ideal gas law (PV = nRT) which is almost beautiful in its simplicity, however it is only valid for noble gases at infinitely low concentrations. The more complex van der Waals equation accounts for deviations caused by intermolecular forces and molecular size.
      Ockham's razor has frequently been presented as a categorising tool in other scientific arears, such as chemistry,
      • Hoffman R.
      • Minkin V.I.
      • Carpenter B.K.
      Ockham's Razor and chemistry.
      medicine,
      • Spitzer A.L.
      • Barcia A.M.
      • Schell M.T.
      • et al.
      Applying Ockham's razor to pancreatitis prognostication: a four-variable predictive model.
      and statistics.
      • Jefferys W.H.
      • Berger J.O.
      Ockham's razor and Bayesian analysis.
      In pharmaceutical and other material sciences there is a natural and reasonable interest in connecting estimated parameters from a mathematical model to physical characteristics of materials. In particular coefficients estimated in compressibility models are subject to interpretations. The commitment for physical explanations is so pronounced, that even deviations from a model, that does not fit the data is explicated.
      • Paronen P.
      • Ilkka J.
      Porosity–pressure functions.
      ,
      • Shapiro I.
      Compaction of powders X. Development of a general compaction equation.
      It is often observed that the linking between estimated parameters and physical characteristics is based on correlation rather than functional relationship. Furthermore, the associations are often disturbed by lack of precision in terminology.
      It is here appropriate to quote a senior expert witness in a patent case, explaining to an obviously confused legal judge: “Wrong use of definitions seems to be a pharmaceutical discipline in itself”.
      Experience through many years as reviewer confirms this statement. Crushing force of tablets is called hardness, pycnometric density - well described in pharmacopoeias - is often referred to as true density. The established definition made in 1982 by Leuenberger
      • Leuenberger H.
      The compressibility and compactibility of powder systems.
      : compactibility was changed in 1998 to tabletability by Joiris et al.
      • Joiris E.
      • Martino P.D.
      • Berneron C.
      • Guyot-Hermann A.M.
      • Guyot J.C.
      Compression behavior of orthorhombic paracetamol.
      The SI standard for force N (Newton) is frequently replaced by Kg or Kp (Kilopond). The term reproducibility described in ICH Guidelines,
      ICH Harmonised Tripartite Guideline
      Validation of analytical procedures: text and methodology Q2(R1), (CPMP/ICH/381/95).
      is mixed up with repeatability.
      The purpose of this paper is to apply Ockham's razor on the following five different compactibility
      • Leuenberger H.
      The compressibility and compactibility of powder systems.
      or compressibility models often used in pharmaceutical scientific literature. In all cases simpler models compared to the established will be presented.
      • Indirect tensile strength of tablets
      • Pitt's equation for tensile strength of biconvex tablets
      • Adams' model for strength of agglomerates.
      • Weibull versus normal distribution and Weibull modulus as measure for brittleness.
      • Heckel's model for compressibility.
      It is not the intention to go into details with advanced models for crack propagation in brittle materials or stress analysis of individual compacts, but more to focus on aspects related to quality in pharmaceutical development and production.

      Indirect Tensile Strength of Tablets

      In material sciences like mineralogy, geology and metallurgy, permanent deformation in the form of plasticity or brittle fracture are of vital interest. Fracture strength is tested according to standards (e.g.ASTM) in multiple loading conditions. The strength may be tested by: beam bending, compressive or tensile stress, 3- or 4-point bending and torsion. A standard procedure is to normalize the force needed to rupture with dimensions of the specimen and the testing equipment. The general assumption in these test procedures is that the break will occur perpendicular and not as a partial camping tendency.
      A unique exception from these practical orientated methods is the so-called Brazilian test or indirect tensile strength where 2/π is added as a correction factor in Eq. (1).
      TS=2πFDh
      (1)


      Where TS is the indirect tensile strength in the center of the tablet, F the force needed to rupture and D and h diameter and height of the tablet.

       Critical Issues in the Tensile Strength Concept

      The concept of tensile strength is based on a theoretical geometric calculation and has several assumptions.
      USP advisory panel on physical test methods, new general test chapter “tablet breaking force” proposed by the USP advisory panel on physical test methods.
      • • The tablet is an isotropic body (independent of direction of measurement).
      • • Hooke's law is obeyed (meaning a perfect elastic body).
      • • The modulus of elasticity in compression and in tension is the same.
      • • Ideal point loading occurs.
      None of these assumptions is demonstrated to be valid for porous and heterogeneous tablets with substantial differences in internal porosity.
      Besides these unfulfilled assumptions other problematic issues are published.
      Claesson and Bohloli
      • Claesson J.
      • Bohloli B.
      Brazilian test: stress field and tensile strength of anisotropic rocks using an analytical solution.
      used an analytical solution, and found that the principal tension at the disc centre for anisotropic rocks was far more complicated than expressed by Eq. (1). Significant linear relationships between strength and compaction pressure was found for 11 materials, without any concern of failure mode, use of 2/π or location of initial crack of the tablets.
      • Sonnergaard J.
      Quantification of the compactibility of pharmaceutical powders.
      Hilden et al.
      • Hilden J.
      • Polizzi M.
      • Zettler A.
      Note on the use of diametrical compression to determine tablet tensile strength.
      concluded, by comparing the indirect method with 3-point bending, that tablets are likely to fracture under shear by the diametrical method.
      It was claimed
      • Podczeck F.
      Methods for the practical determination of the mechanical strength of tablets - from empiricism to science.
      that it in principle is impossible to measure a work of failure for tablets as the force and the displacement vector are perpendicular to each other. However, work of failure exists, it is observable and measurable. The work of failure is operative in the quantitative characterisation of brittleness of tablets.
      • Rees J.E.
      • Rue P.J.
      Work required to cause failure of tablets in diametral compression.
      ,
      • Sonnergaard J.
      A new brittleness index for compacted tablets.
      A simple alternative to Eq. (1) was proposed by de Jong
      • de Jong J.A.H.
      Tablet properties as a function of the properties of granules made in a fluidized bed process.
      as the specific crushing strength, SCS:
      SCS=FDh
      (2)


      No assumptions are connected to Eq. (2), except absence of lamination tendencies. It is obvious that the informative value of the two expressions is exactly the same, and the tensile strength expression is the most complex with most and unsupported assumptions. It was claimed
      • Podczeck F.
      Methods for the practical determination of the mechanical strength of tablets - from empiricism to science.
      that the failure mechanism depends on the use of well-defined and appropriate test conditions, including test speed, stiffness and cleanliness of test platen surfaces, sensitivity of load cells and correct positioning of the test specimens. The aim might be to make the well-functioning crushing force method much more complicated than it is. The effect of test speed is only observed with extremely slow and totally unreasonable speed settings.
      • Sonnergaard J.
      • Jensen C.G.
      • Poulsen L.
      • Lokind K.B.
      Comparative investigations of tablet crushing force testers.
      More than 3000 tablets were tested in an inter-laboratory comparison of 16 commercial crushing force testers.
      • Sonnergaard J.
      • Jensen C.G.
      • Poulsen L.
      • Lokind K.B.
      Comparative investigations of tablet crushing force testers.
      It was demonstrated that the diametral crushing test is a stable and uncomplicated method in expressing the mechanical quality of compacts.

       Intra-Batch Variability in Crushing Force and Specific Crushing Strength

      The adjustment with dimensions in SCS or TS is absolutely relevant in model studies of compactibility
      • Leuenberger H.
      The compressibility and compactibility of powder systems.
      and in comparing different formulations in development laboratories. Yet, the importance of this correction in intra-batch characterisation of compact strength is indeed questionable. If the height of a tablet and the crushing force are highly correlated, the variability in SCS would vanish and improve the precision. If, however, the association between crushing force and specific crushing strength is dominant, it follows that the dimension modification may be irrelevant. This issue is presented in Table 1, with the squared correlation coefficients, R2 between crushing force and tablet height in addition to SCS for 10 model compacts (microcrystalline cellulose, lactose and 50:50 mixtures) and 4 commercial tablets. While the relation between tablet height and force is absolutely negligible, the correlation between crushing force and SCS is remarkably significant. The consequence should be, that it is irrelevant and waste of time to normalize the crushing force with dimensions in production quality assurance and in-process control. No relevant or useful information is obtained.
      Table 1The Intra-Batch Correlation Between Crushing Force (F) and Tablet Height (h) and Crushing Force Versus SCS (and TS) for 10 Model Compacts and 4 Commercial Tablets.
      Details in Ref.
      • Sonnergaard J.M.
      Distribution of crushing strength of tablets.
      ,
      • Stanley P.
      • Newton J.M.
      Variability in the strength of powder compacts.
      MaterialsRef.nCrush. Force.CVSCSR2 F/hR2 F/SCS
      N%MPa
      MCC50182052.91.361.000.00040.980
      MCC1001820108.01.272.420.1290.971
      MCC1501820151.42.143.680.00040.972
      MCCLAC50182057.92.591.490.0350.982
      MCCLAC1001820107.31.843.040.0020.981
      MCCLAC1501820160.41.744.830.1540.960
      LAC50182067.64.641.980.0600.992
      LAC1001820114.64.543.570.0010.963
      LAC1501820155.53.704.950.0710.982
      Lactose192812.111.200.450.0020.971
      Atropin1810038.811.142.410.3070.991
      Lucosil182586.85.561.580.0980.997
      Solvezink1825100.111.431.430.0560.961
      Alminox182593.57.011.390.00040.997
      n = number of items, CV = coefficient of variation.
      In Table 1 only the commercial Atropin tablets show a weak but significant relation between height and crushing force. However, the force/strength relationship is much stronger and, as shown later, the dimension correction has no influence on the goodness of fit to the normal distribution.
      In summary, using Ockham's razor, uncritical use of the constant 2/π in Eq. (1) might be questioned.

      Pitt's Equation for Strength of Convex Tablets

      In 1988 Pitt et al.
      • Pitt K.G.
      • Newton J.M.
      • Stanley P.
      Tensile fracture of doubly-convex cylindrical disks under diametral loading.
      published a new equation claimed to describe the indirect tensile strength of double-convex tablets:
      TS=10FπD2(2.84tD0.126tW+3.15WD+0.01)1
      (3)


      Where t is the total thickness and W the height of the cylinder.
      This equation is problematic for several reasons:
      • The equation does not reduce to the standard tensile strength for flat tablets when W = t.
      • The investigation is not made on compacted tablets, but on moulded gypsum discs. This would be expected to give a minor variability; however, the CV-range is 5.5–22.7%. These values are much larger than observed on compacted tablets,
        • Sonnergaard J.
        • Jensen C.G.
        • Poulsen L.
        • Lokind K.B.
        Comparative investigations of tablet crushing force testers.
        ,
        • Schmidt P.C.
        • Römpp H.
        Resistance to crushing of tablets. Do we need a more precise description of the method in the European Pharmacopoeia?.
        where values for model tablets was in the interval 1.8–5.3% and for commercial tablets 5.6–12.8%.
      • The use of ratios of correlated variables is erroneous since the effect will diminish. Interaction between two factors in factorial designed experiments are organized as the product and certainly not by ratios as in Eq. (3). The correlation between factors in Pitt's data are shown in Table 2.
        Table 2Correlation Matrix for the Three Factors in Pitt's Equation.
        t/Dt/WW/D
        t/D1.000
        t/W0.2711.000
        W/D0.565−0.5381.000
      • One single observation out of 28 is for no reason selected as a central constant because the variability was smallest (10.1%). The goodness of fit is then presented as individual deviations from this value.
      • The simplified calculation technique for estimation of the coefficients is most unusual and laborious.
      The standard way to evaluate the quality of a model is to summarize the squared difference between the observed variable and the estimated. In this case the ultimate goal would be, that all combinations of factors provided the same tensile strength.
      Multiple regression techniques and response surface methodology were developed in 1950, and introduced to pharmaceutical sciences in 1976 by Swartz and Flamholz
      • Schwartz J.B.
      • Flamholz J.R.
      • Press R.H.
      Computer optimization of pharmaceutical formulations I: general procedure.
      in this journal. The techniques were used by this author in a Ph.D. project in 1977. In those days it was possible - using punch cards and terminals connected by telephone to a remote main-frame computer - to make calculations termed Stepwise Multiple Regression. Today the same calculations are easily made in Excel on a standard PC.
      Table 3 shows the recalculation by multiple regression of Eq. (3) with the factors proposed by Pitt. The R2 value is improved and the constant is insignificant and excluded. The differences in the estimated coefficients are caused by the correlated factors.
      Table 3Coefficients in Eq. 3 Estimated by the Method Used by Pitt, and by Standard Multiple Regression in Excel.
      FactorPitt's MethodMultiple Regression
      t/D2.842.68
      t/W−0.126−0.109
      W/D3.153.32
      Constant0.01NS
      R20.98980.9908
      NS, Not significant.

       Alternative Models

      From the data published by Pitt a substitute and uncorrelated variable H is easily calculated, as the height of the circle segment H =(t-W)/2. The area of the triangle D∗H is approximately proportional to the area of the circle segment and therefore a part of the breaking zone. An obvious and simple one-parameter model for the specific crushing strength SCS of biconvex tablets would be:
      SCS=FDW+dDH
      (4)


      Which for flat tablets and H = 0 reduces to the standard F/D∗W.
      The parameter d is estimated by multiple regression of Eq. 5
      F=SCS(DW+dDH)
      (5)


      A correlation analysis of all main factors and their interactions shows that a better fit to the data is obtained by replacing D∗H with W∗H and d with w.
      F=SCS(DW+wWH)
      (6)


      Which also reduces to the standard F/D∗W for H = 0.
      An even simpler model for calculating the indirect tensile strength of biconvex tablets was proposed by Podczeck et al.
      • Podczeck F.
      • Drake K.R.
      • Newton J.M.
      Investigations into the tensile failure of doubly-convex cylindrical tablets under diametral loading using finite element methodology.
      The curvature was simply ignored and the standard method Eq. (1) was used with the total thickness as the height of the tablet.
      Table 4 shows the essential results of the 3 alternative models compared with Eq. (3). The simple TS-model has a markedly poorer fit to data compared to the others. With much fewer parameters and only one assumption the model in Eq. (6), should be preferred with reference to Ockham's razor.
      Table 4Summary of Goodness of Fit for the Equations Describing the Variability of Data Published by Pitt et al.
      • Pitt K.G.
      • Newton J.M.
      • Stanley P.
      Tensile fracture of doubly-convex cylindrical disks under diametral loading.
      NameR2SCSCV %TSSource
      Pitt0.9898.558.445.44Eq. 3
      DW + HD0.9879.3817.85.97DW +0.4∗HD
      DW + HW0.9969.2011.45.85DW+3.47∗WH
      Simple TS0.6445.7336.33.64DT∗π/2
      The SCS and TS values are the mean estimates for all data with common CV in per cent.
      Finally, it must be emphasized that the problem with establishing the tensile strength for convex tablets mainly is an academic issue for university people. In industrial in-process control and control of the finished product, the crushing force is a stable and absolutely sufficient quality parameter, as the variability in tablet height is extremely low and uncorrelated with the crushing force, as shown in Table 1.
      Adams’ Equation for Strength of Agglomerates Under Confined Compaction.
      Adams et al.
      • Adams M.J.
      • Mullier M.A.
      • Seville J.P.K.
      Agglomerate strength measurement using a uniaxial confined compression test.
      proposed a mathematical compressibility model, where an estimated parameter τ0 is claimed to depict the mechanical strength of granules in confined compression. The equation was developed, with a few assumptions, based on strength of a single spherical agglomerate:
      lnP=ln(τ0α)+αln(V0V)
      (7)


      P and V are the pressure and volume, V0 is the volume at P = 0 and τ0 and α are constants.
      Eq. (7) in a different form is shown in Eq. 8
      lnP=ln(τ0)ln(α)+αln(V0)αln(V)
      (8)


      In an investigation of 11 different compression equations Adams' model was excluded because the model was “generally over-parameterized”.
      • Machaka R.
      • Chikwanda H.K.
      Analysis of the cold compaction behavior of titanium powders: a comprehensive inter-model comparison study of compaction equations.
      This statement is, in an Ockham context, worthwhile to investigate. Furthermore, it is concerning that the Adams' model may estimate a strength of granules where no granules are present.

       The Adams equation is essentially a two-parameter negative power function:

      P=P1Vα;ln(P)=ln(P1)αln(V)
      (9)


      Where P1 and α are constants. P = P1 when V = 1 and V= P11/a when P = 1. An example of compressibility data fitting the power function is shown in Fig. 1.
      Figure thumbnail gr1
      Fig. 1Example of compressibility data for lactose fitted with a negative power function. Pressure range 5–195 MPa.
      Details in Sonnergaard.
      • Sonnergaard J.
      Impact of particle density and initial volume on mathematical compression models.
      No fundamental change is made by introducing a new constant t or actually rename the product of slope and intercept: t = P1∗α
      P=tαVα;ln(P)=ln(tα)αln(V)
      (10)


      However, when the normalisation parameter V0 is introduced, a compensation is needed. It is notable that V0 from a fitness point of view is superfluous and strictly does not exist, since P = 0 is horizontal asymptote as V → 0. In this context V0 acts as a disturbing and unnecessary factor in a complete fitting model. This is in contrast to the Heckel model
      • Heckel R.W.
      Density-pressure relationships in powder compaction.
      where the density of particles is necessary and the Kawakita model
      • Kawakita K.
      • Lüdde K.-H.
      Some considerations on powder compression equations.
      where an approximation of V0 is required.
      Eq. (10) can now, by inserting ±α ∗lnV0, be written:
      ln(P)=ln(tα)+αln(V0)αln(V0)αln(V)
      (11)


      or
      ln(P)=ln(tα)αln(V0)+αln(V0V)
      (12)


      Linking Adam's Eq. (8) with Eq. (12) leads to:
      τ0=tV0α=P1αV0α
      (13)


      Eq. (13) shows that τ0 is heavily influenced by the normalisation parameter V0. If V0 is substituted with V1, τ0 is the simple product of P1 and α.
      The strong influence of fluctuations in the measured value of V0 was observed and emphasized by Kawakita and Lüdde
      • Heckel R.W.
      Density-pressure relationships in powder compaction.
      and the problem was later confirmed by Nordström et al.
      • Nordström J.
      • Welch K.
      • Frenning G.
      • Alderborn G.
      On the physical interpretation of the Kawakita and Adams parameters derived from confined compression of granular solids.
      The influence of initial volume V0 onτ0 is visualised in Fig. 2, with the data from Fig. 1 as a practical example. The key Kawakita parameter 1/b is included to show the often-noticed relationship between the two parameters.
      Figure thumbnail gr2
      Fig. 2The relationship between τ0 and 1/b versus V0. The fit to the Kawakita model was achieved with R2 values between 0.9977 and 0.9980. For all τ0 values R2 was 0.9979.
      Original data from lactose tablet, Sonnergaard.
      • Sonnergaard J.
      Impact of particle density and initial volume on mathematical compression models.
      In Fig. 2 a change in initial volume of 7.5% results in a 60% change in τ0 and a 28% change in the estimate of 1/b. The product of P1 and α, estimated by least square method in Fig. 1, is exactly the value 119.48 in Fig. 2, confirming the validity of Eq. (13).
      The by Adams et al. observed positive correlation between the strength of sand agglomerates and τ0 might thus be an inverse functional relationship between τ0 and the inserted bulk volume at zero pressure, V0. It would be anticipated that weak agglomerates are more porous and have a large bulk volume. This simple description could thus explain the correlation between τ0 and the agglomerate strength.
      The final Ockham question is now whether τ0 in the Adams equation is an unnecessary partly compensation for the inserted initial volume V0, or is a measure of the strength of agglomerates crushed under the compaction process. The characterisation of the equation as being overparameterized appears to be correct.

      Weibull Model for Strength of Materials

      The Weibull distribution and equation
      • Weibull W.A.
      A statistical distribution function of wide applicability.
      is known to be extremely flexible and utilized in many scientific disciplines. The distribution is used on observations of wind speed
      • Torres J.L.
      • García A.
      • Prieto E.
      • de Francisco A.
      Characterization of wind speed data according to wind direction.
      over mortgage market
      • Lambrecht B.
      • Perraudin W.
      • Satchell S.
      Time to default in the UK mortgage market.
      to microbiology.
      • Hutchinson T.P.
      Graphing the death of Escherichia coli.
      In pharmaceutical tablet technology the Weibull equation is widely used as model for dissolution rate profiles, fitting of force-time profiles in compression studies and depiction of compactibility
      • Leuenberger H.
      The compressibility and compactibility of powder systems.
      curves. Of special interest here, the Weibull distribution is also used to characterize the variability of tablet strength measurements. The inverse Weibull modulus m, immensely related to the coefficient of variability,
      • Stanley P.
      • Newton J.M.
      Variability in the strength of powder compacts.
      is used as characteristic for brittleness of materials.
      Two topics are of special interest in this Ockham connection. The first question is whether tablet strength data are normal or Weibull distributed. The second problem is the choice between a simple calculation of the standard deviation or the complex estimation of the Weibull m parameter.

       Normal or Weibull Distribution

      Numerous investigations in material sciences are dealing with the distribution of repeated strength measurement. The preference of the Weibull distribution is clearly connected to the weakest link concept. The force needed to break a long chain is less than for a short chain, due to the increased possibility of weak links. The Weibull distribution model was introduced to pharmaceutical technology in 1977 by Stanley and Newton.
      • Stanley P.
      • Newton J.M.
      Variability in the strength of powder compacts.
      The weakest link theorem for tablet strength measurement seems contradictory, since it is evident, that the force needed to crush large tablets is greater than for small tablets with the same porosity. This is the simple reason for normalizing the force with tablet dimension (and 2/π!). To overcome this dilemma Stanley and Newton tested whether the tensile strength of several tablet batches was inversely related to the volume of the tablets. It might be mathematically possible to compensate for dimensions in these two opposite ways, but the attempt was unsuccessful or “not entirely adequate”.
      • Stanley P.
      Mechanical strength testing of compacted powders.
      Pitchumani et al.
      • Pitchumani R.
      • Zhupanska O.
      • Meesters G.M.H.
      • Scarlett B.
      Measurement and characterization of particle strength using a new robotic compression tester.
      investigated the crushing strength of sodium benzoate and microcrystalline granules. They found that the relationship between strength ratios and the inverse granule radius “seems to give good agreement with the experimental data”. This observation was unfortunately without accompanying documentation.
      An alternative to the weakest link concept was presented in 1945 by Daniels
      • Daniels H.E.
      The statistical theory of the strength of bundles and threads.
      He showed that the cumulative distribution function of a brittle bundle of fibers converges to the Gaussian (normal) distribution as n → ∞. For a plastic bundle, such convergence is a consequence of the central limit theorem of the theory of probability.
      As the Weibull distribution is extremely flexible, most continuous frequency data will fit the equation. This might be the simple reason for the often-seen conclusion that the normal or Weibull distributions are fitting the data equally well. Fig. 3 shows the normal and Weibull distributions fitted to 100 Atropine tablets. None of the curves matches the data perfect, but the normal distribution is slightly better. Changing the data from crushing force to the normalized strength values (SCS or TS) in Fig. 3b does not alter anything.
      Figure thumbnail gr3
      Fig. 3Cumulative normal and Weibull probability plots of crushing force (a) and crushing strength, SCS (b) for 100 commercial Atropine tablets. Full line normal and dotted line Weibull distribution.
      Data from Sonnergaard.
      • Sonnergaard J.M.
      Distribution of crushing strength of tablets.
      Putting on the Ockham glasses the normal distribution must be preferred as it is simple, well-known and required for data in most parametric statistical test procedures.

       Coefficient of Variation or Weibull Modulus

      It is generally accepted, that repeated strength measurements of brittle materials will show a greater variation than plastic deforming materials. The Weibull modulus or m-parameter is inversely related to the variability of the data,
      • Sonnergaard J.M.
      Distribution of crushing strength of tablets.
      ,
      • Stanley P.
      • Newton J.M.
      Variability in the strength of powder compacts.
      and is therefore used as an indicator of brittleness.
      It is however, a question how well the brittleness of a given tablet is characterised by the coefficient of variation or the inverse Weibull modulus. Fig. 4 illustrates the relationship between crushing force and normalized standard deviation (CV%) for three model tablets: pure lactose, microcrystalline cellulose and a 50/50% mixture.
      • Sonnergaard J.M.
      Distribution of crushing strength of tablets.
      As expected, the brittle material lactose has the largest variability, while the more plastic mixture and pure MCC may be difficult to categorize. There seems to be no clear relation between crushing force and thereby porosity and variability. It is also interesting to notice the equivalent variability of crushing force and the normalized strength measurements. This confirms the previous conclusion of the lack of importance of the tensile strength concept in this context. No significant difference between the performance of force and strength is observed.
      Figure thumbnail gr4
      Fig. 4The coefficients of variation of 18 batches of model compacts. CV% of crushing force (open symbols) and CV% of SCS or TS (closed symbols). MCC, microcrystalline cellulose; MCCLAC, microcrystalline cellulose and lactose 50/50; LAC, Lactose. N = 20.
      Details in Sonnergaard.
      • Sonnergaard J.M.
      Distribution of crushing strength of tablets.
      Considering the excessive work involved in estimation of the Weibull modulus, compared to the easy calculation of the coefficient of variation, the last method should absolutely be preferred.

      The Heckel Equation. Estimating Yield Pressure or Mean Pressure?

      With 1259 citations in Google Scholar the Heckel compressibility model
      • Heckel R.W.
      Density-pressure relationships in powder compaction.
      is definitely the most used and investigated in pharmaceutical technology.
      ln(11DR)=PK+A
      (14)


      Where DR is the relative density, and K and A are coefficients.
      It was postulated by Heckel, that K or its inverse the yield pressure, YP is a measure of the ability to densify by plastic deformation. This was based on a weak correlation between K and the yield pressure of five plastic deforming metals, excluding the brittle material alumina. It is meaningful to recall that the yield pressure or yield point marks the transition from elastic and reversible deformation to plastic and irreversible. The question is then whether YP is a plastic or an elastic characteristic. An excellent linear relationship was observed between Young's modulus of elasticity and the yield pressure for pharmaceutical relevant materials.
      • Hassanpour A.
      • Ghadiri M.
      Distinct element analysis and experimental evaluation of the Heckel analysis of bulk powder compression.
      ,
      • Sonnergaard J.
      A critical evaluation of the Heckel equation.
      It is documented that the estimate of yield pressure is influenced by variation in the inserted material density.
      • Sonnergaard J.
      A critical evaluation of the Heckel equation.
      A much more fundamental problem is that the calculation of YP is heavily influenced by the maximum pressure on the compact.
      • Hooper D.
      • Clarke F.C.
      • Mitchell J.C.
      • Snowden M.J.
      A modern approach to the Heckel Equation: the effect of compaction pressure on the yield pressure of ibuprofen and its sodium salt.
      ,
      • Patel S.
      • Kaushal A.M.
      • Bansal A.K.
      Mechanistic investigation on pressure dependency of Heckel parameter.
      This fact is in itself a destroying property for a general compression model.
      The Heckel plot is normally separated in three regions: an initial downwards, a linear part and a final upwards curvature. See Fig. 5a.
      Figure thumbnail gr5
      Fig. 5Compression profiles for ibuprofen with the Heckel transformed density (a) and the ordinary density (b) as function of the compaction pressure. The solid line and curve show the fitted values of the Heckel equation.
      Details in Sonnergaard.
      • Sonnergaard J.
      Impact of particle density and initial volume on mathematical compression models.
      Heckel explained the initial deviation from the model as due to particle rearrangement in a non-coherent mass. This description is not valid for pharmaceutical relevant substances. Lactose is often characterised as not fitting the Heckel equation below 50 MPa,
      • Sonnergaard J.
      A critical evaluation of the Heckel equation.
      however a full investigation on lactose tablets compacted at 35.5 MPa was made by Davies and Newton.
      • Stanley P.
      • Newton J.M.
      Variability in the strength of powder compacts.
      Similarly, tablets of ibuprofen were made by low pressures in the non-linear region.
      • Hooper D.
      • Clarke F.C.
      • Mitchell J.C.
      • Snowden M.J.
      A modern approach to the Heckel Equation: the effect of compaction pressure on the yield pressure of ibuprofen and its sodium salt.
      In the linear region it is remarkable that the yield point never is visible as a break or shift on the compressibility curve and that YP often is located outside the linear pressure region.
      • Patel S.
      • Kaushal A.M.
      • Bansal A.K.
      Mechanistic investigation on pressure dependency of Heckel parameter.
      Ibuprofen is a relatively soft API with a low melting point (75-77.5 °C), while a major compressibility at low pressures is expected. This is verified in Fig. 5b with the large initial deformation from 1 to 20 MPa. In this case the linear Heckel part accounts for only 22.8% of the total deformation.
      The final deviation from the straight line at low porosities is sometimes characterised as work or strain hardening,
      • Rowe R.C.
      • Roberts R.J.
      Mechanical properties.
      which is remarkable since the material apparently is softer than predicted by the model. Another proposed description is elastic deformation.
      • Sun C.
      • Grant D.J.
      Influence of elastic deformation of particles on Heckel analysis.
      A third explanation might be, that the deviation only is observable in the transformed presentation and practically non-existent in the original presentation in Fig. 5b.
      Table 5 show the relation between the logarithm to the porosity and the height difference in μm. The decrease in height for every interval is of course the natural exponential base 2.7183. The requirements to distance measurement, calibration and compensation for elastic deformation of machinery will be impossible to achieve with these extreme small distances. The scale is stretched to 13 as presented by Sun and Grant.
      • Sun C.
      • Grant D.J.
      Influence of elastic deformation of particles on Heckel analysis.
      The same authors suggested that the limit for data with porosity less than 0.05 should be interpreted with caution. This recommendation may be relevant in some cases, but for ibuprofen in Fig. 5 it would reduce the linear Heckel fragment from 22.8% to 8.3% of the total deformation.
      Table 5Relationship Between the Heckel Transformed, -ln(Porosity) and the Tablet Height Differences for a 12 mm Diameter Tablet, with 4 mm as Assumed Height at Zero Porosity.
      IntervalPorosityHeight Diff.
      -ln(Poros.)%μm
      3–41.831631.471
      4–50.673811.578
      5–60.24794.259
      6–70.09121.567
      7–80.03350.576
      8–90.01230.212
      9–100.00450.078
      10–110.00170.029
      11–120.00060.011
      12–130.00020.004

       The Statistical Exponential Distribution

      The Heckel equation can be presented as:
      DN=1exp(PYP)
      (15)


      Where DN is the normalized relative density by Holman.
      • Holman L.E.
      The compaction behaviour of particulate materials. An elucidation based on percolation theory.
      In the exponential distribution used in statistics, the cumulative density function is:
      F(x)=1exp(xk)
      (16)


      Where k by definition is the mean value.
      From Eqs. ((15), (16)) it is obvious that the yield pressure estimated by linear regression essentially is the mean value in the exponential distribution. This means that 63.2% of the deformation fitted by the straight line occurs at exactly this pressure. Why the transition from elastic to plastic deformation for all materials always should occur at this mean pressure is really challenging to accept. Three substances (Propranolol HCl., NaCl and MCC) with very different compressibility profiles had exactly the same yield pressure: 77–78 MPa.
      • Sonnergaard J.
      Impact of particle density and initial volume on mathematical compression models.
      However, the extent of the deformation is absolutely unspecified and the relation to compactibility
      • Leuenberger H.
      The compressibility and compactibility of powder systems.
      of the materials is completely absent.
      With all these negative characteristics of the Heckel model, it seems unreasonable that the approval is so massive. One explanation might be that there are so many estimation problems and deviations from that model to explain, that the final goal, which is to find a reliable and useful parameter, disappears. In this way the many papers dealing with estimation problems in the Weibull distribution, with secondary focus on the interpretation of the estimate, might be a worrying example.

      Conclusion

      It is demonstrated for all five models: The Brazilian test, Pitt's equation, Adams' model, Weibull distribution and Heckel's equation, that simpler models or explanations are available.
      • The indirect tensile strength used for other purposes than stress analysis may with Ockham's razor be reduced to a specific crushing strength, where only the dimension of the specimen is involved. Furthermore, the crushing force may be sufficient as measure of mechanical quality in intra-batch characterisation.
      • Pitt's equation is shown to be too complicated compared to a simpler model with equal validity.
      • Adams' model for strength of agglomerates is confirmed to be over-parametrized and the estimate of τ0 is severely confounded with the initial bulk volume V0.
      • The repeated strength measurement of tablets is shown acceptably to follow the normal distribution compared to the Weibull distribution. It is questioned how suitable the coefficient of variation is as indicator for brittleness.
      • It is doubted whether the estimated yield pressure in the Heckel model is a true physical characteristic or simply an estimate of the mean value in the exponential distribution.
      Using Ockham's razor the superfluous elements should be cut away or the model replaced by simpler alternative solutions.

      Disclosures

      This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
      There are no competing interests to declare.

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