Advertisement

Pharmaceutical Research

, 36:156 | Cite as

Benefits of Fractal Approaches in Solid Dosage Form Development

  • Renata Abreu-Villela
  • Martin Kuentz
  • Isidoro CaraballoEmail author
Expert Review

Abstract

Pharmaceutical formulations are complex systems consisting of active pharmaceutical ingredient(s) and a number of excipients selected to provide the intended performance of the product. The understanding of materials’ properties and technological processes is a requirement for building quality into pharmaceutical products. Such understanding is gained mostly from empirical correlations of material and process factors with quality attributes of the final product. However, it seems also important to gain knowledge based on mechanistic considerations. Promising is here to study morphological and/or topological characteristics of particles and their aggregates. These geometric aspects must be taken into account to better understand how product attributes emerge from raw materials, which includes, for example, mechanical tablet properties, disintegration or dissolution behavior. Regulatory agencies worldwide are promoting the use of physical models in pharmaceutics to design quality into a final product. This review deals with pharmaceutical applications of theoretical models, focusing on percolation theory, fractal, and multifractal geometry. The use of these so-called fractal approaches improves the understanding of different aspects in the development of solid dosage forms, for example by identifying critical drug and excipient concentrations, as well as to study effects of heterogeneity on dosage form performance. The aim is to link micro- and macrostructure to the emerging quality attributes of the pharmaceutical solid dosage forms as a strategy to enhance mechanistic understanding and to advance pharmaceutical development and manufacturing processes.

KEY WORDS

fractal geometry fractal-like kinetics geometric distribution percolation theory solid dosage forms 

Abbreviation

1D

One dimensional

2D

Two dimensional

3D

Three dimensional

αi

Density in the box

ε

Size of box

ξ

Correlation length

ρ, ρc

Relative tablet density and its critical value

Σs

Sum of all clusters

Σi

Sum of all intensity

χ

Size of finite clusters

ΦT

Mechanical tablet property

τ

Exponent limit

a

Constant

a’

Order of the Mittag-Leffler function

b

Constant

Df

Fractal dimension

D0

Capacity dimension

D1

Information dimension

D2

Correlation dimension

DoE

Design of experiment

Dq

Moments of distribution

Ea

Mittag-Leffler function of order a’

EDS

Energy dispersive X-ray spectroscopy

FDA

Food and drug administration

f(α)

Hausdorff fractal dimension

f (α(q))

local fractal dimension at resolution q

gc

Correlation function

h

Exponent in the time domain

i

Index

ICH

International conference on harmonization

k

Constant incorporating geometrical and structural characteristics

m

Critical exponent of a given mechanical tablet property

Mt / M∞

Drug release fraction

N

Number of boxes

n

Release exponent

ns

Cluster size distribution

p

Probability

Pi

Probability of object pixel

pc

Percolation threshold

P

Infinity cluster

q

Generalized fractal dimension

QbD

Quality by design

r

Distance between two points in the same cluster

s

Mass of cluster

t

Time

ws

Site occupied for a cluster with finite mass

Notes

Acknowledgments and Disclosures

Financial support to RAV from Brazilian Federal Agency for Support and Evaluation of Graduate Education within the Ministry of Education of Brazil (CAPES) grant number 99999.001486 is greatly acknowledged. The Spanish Ministry of Economy and Competitiveness and FEDER Funds (European Union) for the support of the project MAT2016-77345-C3-3-P is also gratefully acknowledged.

References

  1. 1.
    Sacks LV, Shamsuddin HHB, Yasinskaya YIB, Bouri KC, Lanthier MLD, Sherman REA. Scientific and regulatory reasons for delay and denial of FDA approval of initial applications for new drugs, 2000-2012. J Am Med Assoc. 2014;311:378–84.CrossRefGoogle Scholar
  2. 2.
    Laske S, Paudel A, Scheibelhofer O, and the author team Sacher S, Hoermann T, Khinast J, et al. A review of PAT strategies in secondary solid oral dosage manufacturing of small molecules. J Pharm Sci 2017;106:667–712.Google Scholar
  3. 3.
    Advances T, Characterization M, Ferreira AP, Gamble JF, Leane MM, Park H, et al. Enhanced understanding of pharmaceutical materials through advanced characterisation and analysis. AAPS PharmSciTech. 2018;19:3462–80.CrossRefGoogle Scholar
  4. 4.
    Mishra V, Thakur S, Patil A, Shukla A. Quality by design (QbD) approaches in current pharmaceutical set-up. Expert Opin Drug Deliv. 2018;15:737–58.PubMedCrossRefPubMedCentralGoogle Scholar
  5. 5.
    Sun CC. Microstructure of tablet—pharmaceutical significance, assessment, and engineering. Pharm Res. 2017;34:918–28.PubMedCrossRefPubMedCentralGoogle Scholar
  6. 6.
    Calvo NL, Maggio RM, Kaufman TS. Characterization of pharmaceutically relevant materials at the solid state employing chemometrics methods. J Pharm Biomed Anal. 2018;147:538–64.PubMedCrossRefPubMedCentralGoogle Scholar
  7. 7.
    Gray VA. Power of the dissolution test in distinguishing a change in dosage form critical quality attributes. AAPS PharmSciTech. 2018;19:3328–32.PubMedPubMedCentralCrossRefGoogle Scholar
  8. 8.
    Helešicová T, Pekárek T, Matějka P. The influence of different acquisition settings and the focus adjustment on Raman spectral maps of pharmaceutical tablets. J Drug Deliv Sci Technol. 2018;47:386–94.CrossRefGoogle Scholar
  9. 9.
    Kann B, Windbergs M. Chemical imaging of drug delivery systems with structured surfaces–a combined analytical approach of confocal Raman microscopy and optical Profilometry. AAPS J. 2013;15:505–10.PubMedPubMedCentralCrossRefGoogle Scholar
  10. 10.
    Abreu-Villela R, Schönenberger M, Caraballo I, Kuentz M. Early stages of drug crystallization from amorphous solid dispersion via fractal analysis based on chemical imaging. Eur J Pharm Biopharm. 2018;133:122–30.PubMedCrossRefPubMedCentralGoogle Scholar
  11. 11.
    Gamble JF, Tobyn M, Hamey R. Application of image-based particle size and shape characterization systems in the development of small molecule pharmaceuticals. J Pharm Sci. 2015;104:1563–74.PubMedCrossRefPubMedCentralGoogle Scholar
  12. 12.
    Balant LP, Gex-fabry M. Modelling during drug development. Eur J Pharm Biopharm. 2000;50:13–26.PubMedCrossRefPubMedCentralGoogle Scholar
  13. 13.
    Markl D, Zeitler JA. A review of disintegration mechanisms and measurement techniques. Pharm Res. Pharmaceutical Research. 2017;34:890–917.PubMedPubMedCentralGoogle Scholar
  14. 14.
    Hunt A, Ewing R, Ghanbarian B, Selker J, Horton R, Sahimi M. Percolation theory for flow in porous media. Lecture Notes in Physics 880. Third Edit. Heidelberg/ Germany; 2013.Google Scholar
  15. 15.
    Aguilar-de-Leyva Á, Campiñez MD, Casas M, Caraballo I. Design space and critical points in solid dosage forms. J Drug Deliv Sci Technol. 2017;42:134–43.CrossRefGoogle Scholar
  16. 16.
    Valsami G. Macheras P Determination of fractal reaction dimension in dissolution studies Eur J Pharm Sci. 1995;3:163–9.Google Scholar
  17. 17.
    Patwardhan K, Asgarzadeh F, Dassinger T, Albers J, Repka MA. A quality by design approach to understand formulation and process variability in pharmaceutical melt extrusion processes. J Pharm Pharmacol. 2015;67:673–84.PubMedCrossRefPubMedCentralGoogle Scholar
  18. 18.
    Yu LX, Amidon G, Khan MA, Hoag SW, Polli J, Raju GK, et al. Understanding pharmaceutical quality by design. AAPS J. 2014;16:771–83.PubMedPubMedCentralCrossRefGoogle Scholar
  19. 19.
    Brereton RG, Jansen J, Lopes J, Marini F, Pomerantsev A, Rodionova O, et al. Chemometrics in analytical chemistry—part I: history, experimental design and data analysis tools. Anal bioanal Chem. Anal Bioanal Chem. 2017;409:5891–9.PubMedCrossRefPubMedCentralGoogle Scholar
  20. 20.
    Willecke N, Szepes A, Wunderlich M, Remon JP, Vervaet C, De Beer T. A novel approach to support formulation design on twin screw wet granulation technology: understanding the impact of overarching excipient properties on drug product quality attributes. Int J Pharm. 2018;545:128–43.PubMedCrossRefPubMedCentralGoogle Scholar
  21. 21.
    Watson TJN, Nosal R, Lepore J, Montgomery F. Misunderstanding design space : a robust drug product control strategy is the key to quality assurance. J Pharm Innov. 2018;13:10–2.CrossRefGoogle Scholar
  22. 22.
    Stamatis SD, Kirsch LE. Using manufacturing design space concepts for stability risk assessment—gabapentin NIPTE/FDA case study. AAPS PharmSciTech. 2018;19:2801–7.PubMedCrossRefPubMedCentralGoogle Scholar
  23. 23.
    Dai S, Xu B, Shi G, Liu J, Zhang Z, Shi X, et al. SeDeM expert system for directly compressed tablet formulation: a review and new perspectives. Powder Technol. 2019;342:517–27.CrossRefGoogle Scholar
  24. 24.
    Fuertes I, Caraballo I, Miranda A, Millán M. Study of critical points of drugs with different solubilities in hydrophilic matrices. Int J Pharm. 2010;383:138–46.PubMedCrossRefPubMedCentralGoogle Scholar
  25. 25.
    Janssen HK, Stenull O. Linear polymers in disordered media: the shortest, the longest, and the mean self-avoiding walk on percolation clusters. Phys Rev E Stat Nonlinear Soft Matter Phys. 2012;85:011123.CrossRefGoogle Scholar
  26. 26.
    Saberi AA. Recent advances in percolation theory and its applications. Phys Rep. 2015;578:1–32.CrossRefGoogle Scholar
  27. 27.
    Saremi S, Sejnowski TJ. Correlated percolation, fractal structures, and scale-invariant distribution of clusters in natural images. IEEE Trans Pattern Anal Mach Intell. 2016;38:1016–20.PubMedCrossRefPubMedCentralGoogle Scholar
  28. 28.
    Liu J, Regenauer-Lieb K. Application of percolation theory to microtomography of structured media: percolation threshold, critical exponents, and upscaling. Phys Rev E Stat Nonlinear Soft Matter Phys. 2011;83:016106.CrossRefGoogle Scholar
  29. 29.
    Ding J, Asta M, Ritchie RO. On the question of fractal packing structure in metallic glasses. Proc Natl Acad Sci. 2017;114:8458–63.PubMedCrossRefPubMedCentralGoogle Scholar
  30. 30.
    Finner SP, Kotsev MI, Miller MA, Van Der Schoot P. Continuum percolation of polydisperse rods in quadrupole fields: theory and simulations. J Chem Phys. 2018;148:034903.PubMedCrossRefPubMedCentralGoogle Scholar
  31. 31.
    Coniglio A. Cluster structure near the percolation threshold. J Phys A Math Gen. 1982;15:3829–44.CrossRefGoogle Scholar
  32. 32.
    Malthe-Sorenssen A. Percolation and disordered systems-a numerical approach. Norway: Oslo; 2015.Google Scholar
  33. 33.
    Hunt A. Relevance of percolation theory to power-law behavior of dynamic processes including transport in disordered media. Complexity. 2009;45435:13–27.CrossRefGoogle Scholar
  34. 34.
    Leuenberger H, Leu R, Bonny JD. Application of percolation theory and fractal geometry to tablet compaction. Drug Dev Ind Pharm. 1992;18:723–66.CrossRefGoogle Scholar
  35. 35.
    Grassi M, Grassi G. Application of mathematical modeling in sustained release delivery systems. Expert Opin Drug Deliv. 2014;11:1299–321.PubMedCrossRefPubMedCentralGoogle Scholar
  36. 36.
    Mitra S, Saha D, Sensharma A. Percolation in a distorted square lattice. Phys Rev E American Physical Society. 2019;012117:1–7.Google Scholar
  37. 37.
    Caraballo I, Fernández-Arévalo M, Holgado MA, Rabasco AM. Percolation theory: application to the study of the release behaviour from inert matrix systems. Int J Pharm. 1993;96:175–81.CrossRefGoogle Scholar
  38. 38.
    Timonin PN. Clusters ’ size-degree distribution for bond percolation. Physica A. 2018;492:2292–300.CrossRefGoogle Scholar
  39. 39.
    Essam JW. Percolation theory. Reports Progess Phys. 1980;43:833–912.CrossRefGoogle Scholar
  40. 40.
    Caraballo I. Factors affecting drug release from hydroxypropyl methylcellulose matrix systems in the light of classical and percolation theories. Expert Opin Drug Deliv. 2010;7:1291–301.PubMedCrossRefPubMedCentralGoogle Scholar
  41. 41.
    Casas M, Aguilar-de-leyva Á, Caraballo I. Towards a rational basis for selection of excipients: excipient efficiency for controlled release. Int J Pharm. 2015;494:288–95.PubMedCrossRefPubMedCentralGoogle Scholar
  42. 42.
    Ramírez N, Melgoza LM, Kuentz M, Sandoval H, Caraballo I. Comparison of different mathematical models for the tensile strength-relative density profiles of binary tablets. Eur J Pharm Sci. 2004;22:19–23.PubMedCrossRefPubMedCentralGoogle Scholar
  43. 43.
    Krausbauer E, Puchkov M, Betz G, Leuenberger H. Rational estimation of the optimum amount of non-fibrous Disintegrant applying percolation theory for binary fast disintegrating formulation. J Pharm Sci. 2007;95:2145–57.Google Scholar
  44. 44.
    Luginbühl R, Leuenberger H. Use of percolation theory to interpret water uptake, disintegration time and intrinsic dissolution rate of tablets consisting of binary mixtures. Pharm Acta Helv. 1994;69:127–34.CrossRefGoogle Scholar
  45. 45.
    Kuentz M, Leuenberger H. Modified Young’s Modulus of microcrystalline cellulose tablets and the directed continuum percolation model. Pharm Dev Technol. 1998;3:13–9.PubMedCrossRefPubMedCentralGoogle Scholar
  46. 46.
    Kuentz M, Leuenberger H. Pressure susceptibility of polymer tablets as a critical property : a modified Heckel equation. J Pharm Sci. 1999;88:174–9.PubMedCrossRefPubMedCentralGoogle Scholar
  47. 47.
    Kuentz M, Leuenberger H, Kolb M. Fracture in disordered media and tensile strength of microcrystalline cellulose tablets at low relative densities. Int J Pharm. 1999;182:243–55.PubMedCrossRefPubMedCentralGoogle Scholar
  48. 48.
    Kolesnikova A, Zakinyan A, Dikansky Y. Microstructure formation and macroscopic dynamics of ferrofluid emulsion in rotating magnetic field. EPJ Web Conf. 2018;185:09004.CrossRefGoogle Scholar
  49. 49.
    Draief M. Epidemic processes on complex networks: the effect of topology on the spread of epidemics. Phys A Stat Mech its Appl. 2006;363:120–31.CrossRefGoogle Scholar
  50. 50.
    Bao L, Ma J, Long W, He P, Zhang TA, Nguyen AV. Fractal analysis in particle dissolution: a review. Rev Chem Eng. 2014;30:261–87.CrossRefGoogle Scholar
  51. 51.
    Pippa N, Dokoumetzidis A, Demetzos C, Macheras P. On the ubiquitous presence of fractals and fractal concepts in pharmaceutical sciences: a review. Int J Pharm. 2013;456:340–52.PubMedCrossRefPubMedCentralGoogle Scholar
  52. 52.
    Salat H, Murcio R, Arcaute E. Multifractal methodology. Physica A. 2017;473:467–87.CrossRefGoogle Scholar
  53. 53.
    Ruschin-Rimini N, Ben-Gal I, Maimon O. Fractal geometry statistical process control for non-linear pattern-based processes. Institute Ind Eng. 2013;45:355–73.Google Scholar
  54. 54.
    Jelcic Z, Hauschild K, Ogiermann M, Picker-Freyer KM. Evaluation of tablet formation of different lactoses by 3D modeling and fractal analysis. Drug Dev Ind Pharm. 2007;33:353–72.PubMedCrossRefPubMedCentralGoogle Scholar
  55. 55.
    Yao B, Imani F, Sakpal AS, Reutze EW, Yang H. Multifractal analysis of image profiles for the Characterization and detection of defects in additive manufacturing. J Manuf Sci Eng. 2017;140.Google Scholar
  56. 56.
    Imani F, Yao B, Chen R, Rao P, Yang H. Fractal pattern recognition of image profiles for manufacturing process monitoring and control. Int Manuf Sci Eng Conf. 2018;1.Google Scholar
  57. 57.
    Lopes R, Betrouni N. Fractal and multifractal analysis: a review. Med Image Anal. 2009;13:634–49.PubMedCrossRefPubMedCentralGoogle Scholar
  58. 58.
    Chen Z, Liu Y, Zhou P. A comparative study of fractal dimension calculation methods for rough surface profiles. Chaos, Solitons and Fractals. 2018;112:24–30.CrossRefGoogle Scholar
  59. 59.
    Gould DJ, Vadakkan TJ, Poché RA, Dickinson DME. Multifractal and Lacunarity analysis of microvascular morphology and remodeling. Microcirculation. 2011;18:136–51.PubMedPubMedCentralCrossRefGoogle Scholar
  60. 60.
    Blavatska V, Janke W. Multifractality of self-avoiding walks on percolation clusters. Phys Rev Lett. 2008;101:1–4.CrossRefGoogle Scholar
  61. 61.
    Adler C, Teleki A, Kuentz M. Multifractal Characterization of pharmaceutical hot-melt Extrudates. Pharm Res Pharmaceutical Research. 2017;34:321–32.PubMedPubMedCentralGoogle Scholar
  62. 62.
    Adler C, Teleki A, Kuentz M. Multifractal and mechanical analysis of amorphous solid dispersions. Int J Pharm. 2017;523:91–101.PubMedCrossRefPubMedCentralGoogle Scholar
  63. 63.
    Lopez-Sanchez P, Schumm S, Pudney PDA, Hazekamp J. Carotene location in processed food samples measured by Cryo in-SEM Raman. Analyst. 2011;136:3694–7.PubMedCrossRefPubMedCentralGoogle Scholar
  64. 64.
    Abreu-Villela R, Adler C, Caraballo I, Kuentz M. Electron microscopy/energy dispersive X-ray spectroscopy of drug distribution in solid dispersions and interpretation by multifractal geometry. J Pharm Biomed Anal. 2018;150:241–7.PubMedCrossRefPubMedCentralGoogle Scholar
  65. 65.
    Angulo JM, Esquivel FJ. Multifractal dimensional dependence assessment based on tsallis mutual information. Entropy. 2015;17:5382–401.CrossRefGoogle Scholar
  66. 66.
    Waliszewski P. The quantitative criteria based on the fractal dimensions, entropy, and lacunarity for the spatial distribution of cancer cell nuclei enable identification of low or high aggressive prostate carcinomas. Front Physiol. 2016;7:1–16.CrossRefGoogle Scholar
  67. 67.
    Risović D, Poljaček SM, Furić K, Gojo M. Inferring fractal dimension of rough/porous surfaces-a comparison of SEM image analysis and electrochemical impedance spectroscopy methods. Appl Surf Sci. 2008;255:3063–70.CrossRefGoogle Scholar
  68. 68.
    Kirchmeyer W, Wyttenbach N, Alsenz J, Kuentz M. Influence of excipients on solvent-mediated hydrate formation of piroxicam studied by dynamic imaging and fractal analysis. Cryst Growth Des. 2015;15:5002–10.CrossRefGoogle Scholar
  69. 69.
    Stenull O, Janssen HK. Multifractal properties of resistor diode percolation. Phys Rev E Stat Nonlinear Soft Matter Phys. 2002;65:036124.CrossRefGoogle Scholar
  70. 70.
    Shah V, Booth DE. A fractal dimension-based method for statistical process control. Int J Oper Res. 2012;14:187–99.CrossRefGoogle Scholar
  71. 71.
    Cho Y, Shin J-H, Costa A, Kim TA, Kunin V, Li J, et al. Engineering the shape and structure of materials by fractal cut. Proc Natl Acad Sci. 2014;111:17390–5.PubMedCrossRefPubMedCentralGoogle Scholar
  72. 72.
    Yin X, Li H, Liu R, Chen J, Ji J, Chen J, et al. Fractal structure determines controlled release kinetics of monolithic osmotic pump tablets. J Pharm Pharmacol. 2013;65:953–9.PubMedCrossRefPubMedCentralGoogle Scholar
  73. 73.
    Esquena J, Solans C, Llorens J. Nitrogen sorption studies of silica particles obtained in emulsion and microemulsion media. J Colloid Interface Sci. 2000;225:291–8.PubMedCrossRefPubMedCentralGoogle Scholar
  74. 74.
    Nidhi K, Indrajeet S, Khushboo M, Gauri K, Sen DJ. Hydrotropy: a promising tool for solubility enhancement: a review. Int J Drug Dev Res. 2011;3:26–33.Google Scholar
  75. 75.
    Kosmidis K. Monte Carlo simulations in drug release. J Pharmacokinet Pharmacodyn Springer US. 2019;46:165–72.CrossRefGoogle Scholar
  76. 76.
    Shireen Z, Babu SB. Lattice animals in diffusion limited binary colloidal system. J Chem Phys. 2017;147:054904.PubMedCrossRefPubMedCentralGoogle Scholar
  77. 77.
    Kolesnikov BP. Influence of fractal substructures of the percolating cluster on transferring processes in macroscopically disordered environments. J Phys Conf Ser. 2017;891:012355.CrossRefGoogle Scholar
  78. 78.
    Wool RP. Twinkling fractal theory of the glass transition. J Polym Sci Part B Polym Phys. 2008;46:2765–78.CrossRefGoogle Scholar
  79. 79.
    Leuenberger H, Lanz M. Pharmaceutical powder technology - from art to science: the challenge of the FDA’s process analytical technology initiative. Adv Powder Technol. 2005;16:3–25.CrossRefGoogle Scholar
  80. 80.
    Gray V, Kelly G, Xia M, Butler C, Thomas S, Mayock S. The science of USP 1 and 2 dissolution : present challenges and future relevance. Pharm Res. 2009;26:1289–302.PubMedCrossRefPubMedCentralGoogle Scholar
  81. 81.
    Quodbach J, Kleinebudde P. A critical review on tablet disintegration. Pharm Dev Technol. 2015;00:1–12.CrossRefGoogle Scholar
  82. 82.
    Farin D, Avnir D. Use of fractal geometry to determine effects of surface morphology on drug dissolution. J Pharm Sci. 1992;7:41–56.Google Scholar
  83. 83.
    Lisa L, Peppas NA, Yin F, Boey C, Venkatraman SS. Modeling of drug release from bulk-degrading polymers. Int J Pharm. 2011;418:28–41.CrossRefGoogle Scholar
  84. 84.
    Kosmidis K, Argyrakis P, Macheras P. Fractal kinetics in drug release from finite fractal matrices. J Chem Phys 1. 2003;119:6373–7.CrossRefGoogle Scholar
  85. 85.
    Niederquell A, Kuentz M. Biorelevant dissolution of poorly soluble weak acids studied by UV imaging reveals ranges of fractal-like kinetics. Int J Pharm. 2014;463:38–49.PubMedCrossRefPubMedCentralGoogle Scholar
  86. 86.
    Kosmidis K, Macheras P. On the dilemma of fractal or fractional kinetics in drug release studies : a comparison between Weibull and Mittag-Leffler functions. Int J Pharm. 2018;543:269–73.PubMedCrossRefPubMedCentralGoogle Scholar
  87. 87.
    Papadopoulou V, Kosmidis K, Vlachou M, Macheras P. On the use of the Weibull function for the discernment of drug release mechanisms. Int J Pharm. 2006;309:44–50.PubMedCrossRefPubMedCentralGoogle Scholar
  88. 88.
    Macheras P, Dokoumetzidis A. On the heterogeneity of drug dissolution and release. Pharm Res. 2000;17:108–12.PubMedCrossRefPubMedCentralGoogle Scholar
  89. 89.
    Kopelman R. Fractal reaction kinetics. Science (80- ). 1988;241:1620–6.CrossRefGoogle Scholar
  90. 90.
    Dokoumetzidis A, Macheras P. Fractional kinetics in drug absorption and disposition processes. J Pharmacokinet Pharmacodyn. 2009;36:165–78.PubMedCrossRefPubMedCentralGoogle Scholar
  91. 91.
    Riippi M, Antikainen O, Niskanen T, Yliruusi J. The effect of compression force on surface structure, crushing strength, friability and disintegration time of erythromycin acistrate tablets. Eur J Pharm Biopharm. 1998;46:339–45.PubMedCrossRefPubMedCentralGoogle Scholar
  92. 92.
    Markl D, Strobel A, Schlossnikl R, Bøtker J, Ridgway C, Rantanen J, et al. Characterisation of pore structures of pharmaceutical tablets : a review. Int J Pharm. 2018;538:188–214.PubMedCrossRefPubMedCentralGoogle Scholar
  93. 93.
    Leane M, Pitt K, Reynolds GK, Dawson N, Ziegler I, Szepes A, et al. Manufacturing classification system in the real world: factors influencing manufacturing process choices for filed commercial oral solid dosage formulations, case studies from industry and considerations for continuous processing. Pharm Dev Technol. 2018;23:964–77.PubMedCrossRefPubMedCentralGoogle Scholar
  94. 94.
    Kuentz M, Leuenberger H. A new theoretical approach to tablet strength of a binary mixture consisting of a well and a poorly compactable substance. Eur J Pharm Biopharm. 2000;49:151–9.PubMedCrossRefPubMedCentralGoogle Scholar
  95. 95.
    Nalluri VR, Kuentz M. Flowability characterisation of drug-excipient blends using a novel powder avalanching method. Eur J Pharm Biopharm. 2010;74:388–96.PubMedCrossRefPubMedCentralGoogle Scholar
  96. 96.
    Kaye BH. Characterizing the Flowability of a powder using the concepts of fractal geometry and Chaos theory. Pan Part Syst Characl. 1997;14:53–66.CrossRefGoogle Scholar
  97. 97.
    Hirschberg C, Boetker JP, Rantanen J, Pein-Hackelbusch M. Using 3D printing for rapid prototyping of Characterization tools for investigating powder blend behavior. AAPS PharmSciTech. 2017;19:941–50.PubMedCrossRefPubMedCentralGoogle Scholar
  98. 98.
    Alakoskela JMI, Kinnunen PKJ. Probing phospholipid main phase transition by fluorescence spectroscopy and a surface redox reaction. J Phys Chem B. 2001;105:11294–301.CrossRefGoogle Scholar
  99. 99.
    Demetzos C, Pippa N. Fractal analysis as a complementary approach to predict the stability of drug delivery nano systems in aqueous and biological media: a regulatory proposal or a dream? Int J Pharm. 2014;473:213–8.PubMedCrossRefPubMedCentralGoogle Scholar
  100. 100.
    Demetzos C, Pippa N. Fractal geometry as a new approach for proving nanosimilarity: a reflection note. Int J Pharm. 2015;483:1–5.PubMedCrossRefPubMedCentralGoogle Scholar
  101. 101.
    Pippa N, Pispas S, Demetzos C. The fractal hologram and elucidation of the structure of liposomal carriers in aqueous and biological media. Int J Pharm. 2012;430:65–73.PubMedCrossRefPubMedCentralGoogle Scholar
  102. 102.
    Nematollahi M, Jalali-Arani A, Modarress H. Effect of nanoparticle localization on the rheology, morphology and toughness of nanocomposites based on (poly (lactic acid)/natural rubber/Nanosilica). Polym Int. 2019;68:779–87.CrossRefGoogle Scholar
  103. 103.
    Guichard B, Cassagnau P, Sudre G, Fulchiron R, Ledieu B, Espuche E. Effect of a post-annealing process on microstructure and mechanical properties of high-density polyethylene/silica nanocomposites. J Polym Sci Part B Polym Phys. 2019:1–12.Google Scholar
  104. 104.
    White EV, Fullwood D, Golden KM, Zharov I. Percolation analysis for estimating the maximum size of particles passing through nanosphere membranes. Phys Rev E. 2019;99:022904.PubMedCrossRefPubMedCentralGoogle Scholar
  105. 105.
    Caraballo I, Millan M, Rabasco AM. Relationship between drug percolation threshold and particle size in matrix tablets. Pharm Res. 1996. p. 387–90.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Pharmacy and Pharmaceutical Technology, Facultad de FarmaciaUniversidad de SevillaSevilleSpain
  2. 2.Institute of Pharma TechnologyUniversity of Applied Sciences and Arts Northwestern SwitzerlandMuttenzSwitzerland

Personalised recommendations