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


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.


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



One dimensional


Two dimensional


Three dimensional


Density in the box


Size of box


Correlation length

ρ, ρc

Relative tablet density and its critical value


Sum of all clusters


Sum of all intensity


Size of finite clusters


Mechanical tablet property


Exponent limit




Order of the Mittag-Leffler function




Fractal dimension


Capacity dimension


Information dimension


Correlation dimension


Design of experiment


Moments of distribution


Mittag-Leffler function of order a’


Energy dispersive X-ray spectroscopy


Food and drug administration


Hausdorff fractal dimension

f (α(q))

local fractal dimension at resolution q


Correlation function


Exponent in the time domain




International conference on harmonization


Constant incorporating geometrical and structural characteristics


Critical exponent of a given mechanical tablet property

Mt / M∞

Drug release fraction


Number of boxes


Release exponent


Cluster size distribution




Probability of object pixel


Percolation threshold


Infinity cluster


Generalized fractal dimension


Quality by design


Distance between two points in the same cluster


Mass of cluster




Site occupied for a cluster with finite mass


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.


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© 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

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