# A coupled mechano-biochemical model for bone adaptation

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## Abstract

Bone remodelling is a fundamental biological process that controls bone microrepair, adaptation to environmental loads and calcium regulation among other important processes. It is not surprising that bone remodelling has been subject of intensive both experimental and theoretical research. In particular, many mathematical models have been developed in the last decades focusing in particular aspects of this complicated phenomenon where mechanics, biochemistry and cell processes strongly interact. In this paper, we present a new model that combines most of these essential aspects in bone remodelling with especial focus on the effect of the mechanical environment into the biochemical control of bone adaptation mainly associated to the well known RANKL-RANK-OPG pathway. The predicted results show a good correspondence with experimental and clinical findings. For example, our results indicate that trabecular bone is more severely affected both in disuse and disease than cortical bone what has been observed in osteoporotic bones. In future, the methodology proposed would help to new therapeutic strategies following the evolution of bone tissue distribution in osteoporotic patients.

## Keywords

Mechano-biochemical model Bone remodelling BMU Damage Mineralization RANKL-RANK-OPG pathway## List of symbols

## *Used notation of quantities:*

- \(v_b\)
Bone volume fraction [1]

- \(v_m\)
Mineral volume fraction of bone [1]

- \(v_o\)
Organic volume fraction of bone [1]

- \(v_v\)
Void volume fraction [1]

- \(dam\)
Damage concentration [1]

- \(r_{\alpha }\)
Rate of \(\alpha \)-th interaction \(\left[ \text{ kmol }\cdot \text{ m }^{-3}\cdot \text{ s }^{-1}\right] \)

- \(\mathcal {A}_{\alpha }\)
Affinity of \(\alpha \)-th reaction \(\left[ \text{ J }\cdot \text{ kmol }^{-1}\right] \)

- Open image in new window
The deformation rate tensor Open image in new window and its j-\(th\) invariant; \(j=1\) and \(j=2\) represent rate of volume dilatation and shear rate, respectively [\(s^{-1}\)]

- \({\varvec{\varepsilon }},~\varepsilon ^{(j)}\)
The deformation tensor \(\varepsilon \) and its j-\(th\) invariant [1]

- \(l_{v\alpha },~l_{\alpha \alpha }\)
Phenomenological constants in classical irreversible thermodynamics (CIT)

- \(C_j, [\mathrm{C_j}]\)
j-\(th\) substance and its molar concentration \(\left[ \text{ kmol }\cdot \text{ m }^{-3}\right] \)

- \(\nu _{j\alpha },~\nu '_{j\alpha }\)
Stoichiometric coefficients of substance \(C_j\) entering \(\alpha \)-th reaction or being produced in it (denoted with a prime), respectively [1]

- \(k_{+\alpha },~k_{-\alpha }\)
Forward and backward reaction rate constant of \(\alpha \)-th reaction

- \(\alpha \)
Ash fraction [1]

- \(\nu \)
Poisson ratio [1]

- \(E\)
Young’s modulus \(\left[ \text{ M } \text{ kg }\cdot \text{ m }^{-1}\cdot \text{ s }^{-1}\right] \)

- \(\xi \)
Daily strain history [1]

- \(L\)
Number of loading cases [1]

- \(N_i\)
Number of cycles of i-\(th\) loading case [1]

- \(\bar{\varepsilon }\)
Strain level [1]

- \(\bar{d}\)
Strain rate level [\(s^{-1}\)]

- \(U\)
Strain energy density \(\left[ \text{ kg }\cdot \text{ m }^{-1}\cdot \text{ s }^{-2}\right] \)

- \(f_i\)
Frequency of a considered loading case [\(s^{-1}\)]

- \(T_i\)
Period of a considered loading case [s]

- \(\mathcal {D}_{\alpha }\)
Parameters describing the influence of mechanical stimulus on \(\alpha -th\) interaction [1]

- \(A\)
Parameter used for relating \(\mathcal {D}_{\alpha }\) to \(\xi \) [1]

- \(\mathcal {D}_{\alpha ,ref},~\xi _{ref}\)
Reference values of mechanical stimulus [1]

- \(\varepsilon _{crit}\)
Critical strain value of bone tissue [1]

- \(N_f\)
Fatigue life expectation (number of cycles) [1]

- \(K_i\)
(\(i=t,c\)) Constant of proportionality in fatigue life expectation [1]

- \(\delta _i\)
(\(i=t,c\)) Exponent in fatigue life expectation [1]

- \(dam_i\)
(\(i=t,c\)) Damage in trabecular or cortical bone [1]

- \(C_{c,t_1,t_2},~\gamma _{c,t}\)
Fitted parameters in evolution laws for damage growth [1]

- \(\varepsilon _u\)
Ultimate tensile strain, function of calcium content [1]

- \(v_{b,res}\)
Amount of resorbed bone volume fraction [1]

- \(\rho _{res}\)
Difference in bone tissue density caused by resorption [kg\(\cdot \text{ m }^{-3}\)]

- \(\rho _{max}\)
Maximal amount of bone density (true bone density) [kg\(\cdot \text{ m }^{-3}\)]

- \(J_{Old\_B}\)
Normalised turnover rate in equilibrium [1]

- \(n_{C_j}\)
Normalised concentration of \(j\)-th substance [1]

- \({n_{Old\_B}}_{,max}\)
Maximal possible value of \(n_{Old\_B}\) corresponding to intact bone tissue under the maximal mechanical stimulus \(\xi (\varepsilon _{crit})\) [1]

- \(m_m,~m_o\)
Mineral/organic mass [kg\(\cdot \text{ m }^{-3}\)]

- \(\rho _m,~\rho _o\)
Mineral/organic true densities [kg\(\cdot \text{ m }^{-3}\)]

- \(v_m^{\mathrm{secondary}}\)
Exponential law for secondary mineralisation [1]

- \(v_m^{prim}\)
Mineral volume fraction value at the end of the primary phase [1]

- \(v_j^0\)
Initial value of \(v_j,~j=m,b,o\) [1]

- \(\tau _{BR}\)
Time corresponding to the end of the primary phase [1]

- \(v_m^{max}\)
\(v_m\) corresponding to the maximal mineral content found in bone [1]

- \(\kappa \)
Exponent in secondary mineralisation phase [1]

- \(v_m^{\mathrm{OldB}}\)
Average mineral content of old bone \(Old\_B\) [1]

- \(v_m^{\mathrm{NewB}}\)
Average mineral content of new bone \(New\_B\) [1]

- \(n_{prim}\)
Fraction of bone undergoing primary mineralisation [1]

- \(X^0\)
Initial value of \(X\) (e.g. \(v_b^0,\alpha ^0, [\mathrm{PTH}]_0\))

- \(BT_{ind}\)
Bone tissue index (a measure of total bone mass) [\(\text{ m }^3\)]

- \(BMD_{ind}\)
Bone mineral index (a measure of total mineral mass) [kg]

- \(BMD(i)\)
Bone mineral density at \(i\)-th element [kg\(\cdot \text{ m }^{-3}\)]

- \(V(i)\)
Volume of the \(i\)-th element [\(\text{ m }^3\)]

## Used notation in Appendices:

- \(n_{C_{j,0}}\)
Initial normalised concentration of j-\(th\) substance [1]

- \([\mathrm{C_j}]_{st}\)
Standard concentration of j-\(th\) substance [1]

- \(n_{C_j,st}\)
Normalised equivalent standard concentration of j-\(th\) substance [1]

- \(n_{i_0}^{RRO,C_j}\)
\(C_j \in \{PTH,~estr,~NO\}\), \(i \in \{RKL,~OPG\}\); predicted initial value of \(n_i\) (from a submodel for \(C_j\) influence on RANKL-RANK-OPG pathway) used as input in the RANKL-RANK-OPG model [1]

## Mathematics Subject Classification (2000)

74F25 92C45 92E20 74A15 74B99## Notes

### Acknowledgments

This research was supported by the Instituto Aragones de Ciencias de la Salud through the research project (PIPAMER10/015) and the Spanish Ministry of Science and Technology through the Research Project DPI2011-22413. Further support was received from the CENTEM project, reg. no. CZ.1.05/2.1.00/03.0088, that is partially supported by the ERDF within the OP RDI Programme of the Ministry of Education, Youth and Sports and from the institutional support RVO:61388998.

## Supplementary material

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