Continuum Mechanics and Thermodynamics

, Volume 30, Issue 5, pp 995–1009 | Cite as

New description of gradual substitution of graft by bone tissue including biomechanical and structural effects, nutrients supply and consumption

  • Yanfei LuEmail author
  • Tomasz Lekszycki
Open Access
Original Article


A new description of graft substitution by bone tissue is proposed in this work. The studied domain is considered as a continuum model consisting of a mixture of the bone tissue and the graft material. Densities of both components evolve in time as a result of cellular activity and biodegradation. The proposed model focuses on the interaction between the bone cell activity, mechanical stimuli, nutrients supply and scaffold microstructure. Different combinations of degradation rate and stiffness of the graft material were examined by numerical simulation. It follows from the calculations that the degradation rate of the scaffold should be tuned to the synthesis/resorption rate of the tissue, which are dependent among the others on scaffold porosity changes. Simulation results imply potential criteria to choose proper bone substitute material in consideration of degradation rate, initial porosity and mechanical characteristics.


Bone regeneration Bioresorbable and biodegradable material Microstructure Nutrients supply Numerical simulation 

List of symbols

\(\varvec{\varepsilon }\)


\(\varvec{\sigma }\)



Strain energy

\(\varphi \)



Young’s modulus

\(d_\mathrm{a}\), \(d_\mathrm{s}\)

Normalized sensor/actor cell densities


Mechanical stimulus


Reference stimulus


Nutrients consumption rate

\(z_\mathrm{s}\), \(z_\mathrm{a}\)

Sensor/actor cell activities


Coefficient defining the range of stimulus without cell activity


Degradation rate of bone substitute material

\(s_\mathrm{b}\), \(r_\mathrm{b}\), \(r_\mathrm{m}\)

Synthesis / resorption coefficients



This research was supported by Warsaw University of Technology Dean’s grant for young researchers No. 504/03098/1101 and Polish National Science Centre (NCN) grant Preludium No. 2017/25/N/ST7/02334.


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Authors and Affiliations

  1. 1.Faculty of Engineering ProductionWarsaw University of TechnologyWarsawPoland
  2. 2.Department of Experimental Physiology and PathophysiologyMedical University of WarsawWarsawPoland

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