Clinical Oral Investigations

, Volume 20, Issue 6, pp 1271–1278 | Cite as

Multifractal spectrum and lacunarity as measures of complexity of osseointegration

  • Daniel de Souza Santos
  • Leonardo Cavalcanti Bezerra dos Santos
  • Alessandra de Albuquerque Tavares Carvalho
  • Jair Carneiro Leão
  • Claudio Delrieux
  • Tatijana Stosic
  • Borko StosicEmail author
Original Article



The goal of this study is to contribute to a better quantitative description of the early stages of osseointegration, by application of fractal, multifractal, and lacunarity analysis.

Materials and methods

Fractal, multifractal, and lacunarity analysis are performed on scanning electron microscopy (SEM) images of titanium implants that were first subjected to different treatment combinations of i) sand blasting, ii) acid etching, and iii) exposition to calcium phosphate, and were then submersed in a simulated body fluid (SBF) for 30 days. All the three numerical techniques are applied to the implant SEM images before and after SBF immersion, in order to provide a comprehensive set of common quantitative descriptors.


It is found that implants subjected to different physicochemical treatments before submersion in SBF exhibit a rather similar level of complexity, while the great variety of crystal forms after SBF submersion reveals rather different quantitative measures (reflecting complexity), for different treatments. In particular, it is found that acid treatment, in most combinations with the other considered treatments, leads to a higher fractal dimension (more uniform distribution of crystals), lower lacunarity (lesser variation in gap sizes), and narrowing of the multifractal spectrum (smaller fluctuations on different scales).


The current quantitative description has shown the capacity to capture the main features of complex images of implant surfaces, for several different treatments. Such quantitative description should provide a fundamental tool for future large scale systematic studies, considering the large variety of possible implant treatments and their combinations.

Clinical relevance

Quantitative description of early stages of osseointegration on titanium implants with different treatments should help develop a better understanding of this phenomenon, in general, and provide basis for further systematic experimental studies. Clinical practice should benefit from such studies in the long term, by more ready access to implants of higher quality.


Dental implants Osseointegration Fractal dimension Multifractal spectrum Lacunarity 



This work is supported by research grants from CNPq, CAPES and FACEPE (Brazilian research agencies), and MINCYT (Argentinean Ministry of Science, Technology and Productive Innovation).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Daniel de Souza Santos
    • 1
  • Leonardo Cavalcanti Bezerra dos Santos
    • 2
  • Alessandra de Albuquerque Tavares Carvalho
    • 2
  • Jair Carneiro Leão
    • 2
  • Claudio Delrieux
    • 3
  • Tatijana Stosic
    • 4
  • Borko Stosic
    • 4
  1. 1.Departamento de FísicaUniversidade Federal Rural de PernambucoRecifeBrazil
  2. 2.Departamento de Clínica e Odontologia PreventivaUniversidade Federal de PernambucoRecifeBrazil
  3. 3.Departamento de Ingeniería Eléctrica y de ComputadorasUniversidad Nacional del SurBahía BlancaArgentina
  4. 4.Departamento de Estatística e InformáticaUniversidade Federal Rural de PernambucoRecifeBrazil

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