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A mechano-biological model of multi-tissue evolution in bone

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Abstract

Successfully simulating tissue evolution in bone is of significant importance in predicting various biological processes such as bone remodeling, fracture healing and osseointegration of implants. Each of these processes involves in different ways the permanent or transient formation of different tissue types, namely bone, cartilage and fibrous tissues. The tissue evolution in specific circumstances such as bone remodeling and fracturing healing is currently able to be modeled. Nevertheless, it remains challenging to predict which tissue types and organization can develop without any a priori assumptions. In particular, the role of mechano-biological coupling in this selective tissue evolution has not been clearly elucidated. In this work, a multi-tissue model has been created which simultaneously describes the evolution of bone, cartilage and fibrous tissues. The coupling of the biological and mechanical factors involved in tissue formation has been modeled by defining two different tissue states: an immature state corresponding to the early stages of tissue growth and representing cell clusters in a weakly neo-formed Extra Cellular Matrix (ECM), and a mature state corresponding to well-formed connective tissues. This has allowed for the cellular processes of migration, proliferation and apoptosis to be described simultaneously with the changing ECM properties through strain driven diffusion, growth, maturation and resorption terms. A series of finite element simulations were carried out on idealized cantilever bending geometries. Starting from a tissue composition replicating a mid-diaphysis section of a long bone, a steady-state tissue formation was reached over a statically loaded period of 10,000 h (60 weeks). The results demonstrated that bone formation occurred in regions which are optimally physiologically strained. In two additional 1000 h bending simulations both cartilaginous and fibrous tissues were shown to form under specific geometrical and loading cases and cartilage was shown to lead to the formation of bone in a beam replicating a fracture healing initial tissue distribution. This finding is encouraging in that it is corroborated by similar experimental observations of cartilage leading bone formation during the fracture healing process. The results of this work demonstrate that a multi-tissue mechano-biological model of tissue evolution has the potential for predictive analysis in the design and implementations of implants, describing fracture healing and bone remodeling processes.

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Abbreviations

\(\varphi _\mathrm{TOT} \) :

Total volume fraction

\(\varphi _{i,\mathrm{TOT}} \) :

Total volume fraction of bone, cartilage and fibrous tissues, where \(i=B,C or F\)

\(\varphi _V \) :

Total volume fraction of free space

\(\varphi _i^I \) :

Volume fraction of immature bone, cartilage and fibrous tissues, where \(i=B,C or F\)

\(\varphi _i^M \) :

Volume fraction of mature bone, cartilage and fibrous tissues, where \(i=B,C or F\)

\(\varepsilon _I \) :

First principal strain

\(\varepsilon _{II} \) :

Second principal strain

\(\varepsilon _Y \) :

Yield strain

\(\varepsilon _{k,N} \) :

Normalized principal strain where \(k=I or II\).

\(f_{i,k} \left( {\varepsilon _{k,N} } \right) \) :

Function relating the normalized principal strain with the rate of change of the activation time \(t_{\mathrm{act},i}\), where \(i=B, C or F\) and \(k=I or II\).

\(a_{i,k}^\varepsilon ,b_{i,k}^\varepsilon \) :

and k\(c_{i,k}^\varepsilon \) Characteristic coefficients which define \(f_{i,k} \left( {\varepsilon _{k,N} } \right) \) where \(i=B, C or F\) and \(k=I or II\).

\(t_{\mathrm{act},i} \) :

Activation time for each tissue, where \(i=B,C or F\)

\(t_\mathrm{act}^\mathrm{Bound} \) :

Gaussian distribution used to limit the growth of the activation time

e.:

Euler‘s number

\(p^\mathrm{Bound}, q^\mathrm{Bound}\) and \(r^\mathrm{Bound}\) :

Coefficients used to define \(t_\mathrm{act}^\mathrm{Bound}\)

t :

time

D :

Diffusion tensor

\(\Delta \) :

Laplacian

I :

Identity matrix

\(\lambda _i \) and \(\varPhi _i \) :

Diffusion rate coefficients, where \(i=B, C or F\)

\(\alpha _i\) :

Immature tissue growth rate, where \(i=B,C or F\)

\(\beta _i\) :

Tissue resorption rate, where \(i=B,C or F\)

\(\gamma _i \) :

Tissue maturation rate, where \(i=B,C or F\)

\(T_i^G\) :

Immature tissue growth function

\(T_i^R\) :

Tissue resorption function

\(T_i^M \) :

Immature to mature tissue maturation function

\(\theta _I\;\hbox {and}\;\theta _{II} \) :

Direction of the principal stresses

\(\otimes \) :

Tensor product

\(T_i\) :

Effective range of \(t_{\mathrm{act},i} \), where \(i=B,C or F\).

\(T_i^\mathrm{Min}\) and \(T_i^\mathrm{Max} \) :

The maximum and minimum values of \(T_i ,\) where \(i=B,C or F\).

\(T_{i,\mathrm{GT}}\) :

Coefficient used to scale \(T_{i}^G\) where \(i=B,C or F\)

\(k_i^R , \quad l_i^R \) and \( m_i^R \) :

Coefficient used to define \(T_{i}^R\) where \(i=B,C or F\)

\( d_i^M ,\) \(e_i^M \) and \(f_i^M \) :

Coefficient used to define \(T_{i}^M\) where \(i=B,C or F\)

\(E_\mathrm{TOT} \) :

Material Young’s modulus

\(E_i^I\) :

Young’s modulus of immature tissues, where \(i=B,C or F\)

\(E_i^M\) :

Young’s modulus of mature tissues, where \(i=B,C or F\)

\(E_V \) :

Young’s modulus of the free space

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Authors and Affiliations

  1. LBM – Institut de Biomécanique Humaine Georges Charpak, ENSAM Paris, Paris, France

    Jamie Frame, Pierre-Yves Rohan & Rachele Allena

  2. Centre des Matériaux – CNRS : UMR7633, MINES ParisTech, PSL Research University, Paris, France

    Laurent Corté

  3. Matière Molle et Chimie – CNRS : UMR 7167, ESPCI Paris, PSL Research University, Paris, France

    Laurent Corté

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Correspondence to Jamie Frame.

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Communicated by Francesco dell’Isola.

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Frame, J., Rohan, PY., Corté, L. et al. A mechano-biological model of multi-tissue evolution in bone. Continuum Mech. Thermodyn. 31, 1–31 (2019). https://doi.org/10.1007/s00161-017-0611-9

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