# A Proposal for a Novel Formulation Based on the Hyperbolic Cattaneo’s Equation to Describe the Mechano-Transduction Process Occurring in Bone Remodeling

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## Abstract

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## 1. Introduction

## 2. Modeling the Functional Adaptation of Bone

## 3. Poro-Mechanical Model

- The displacement $\mathit{u}=\mathit{x}-\mathit{X}$;
- The Lagrangian porosity $\varphi $.

- the Infinitesimal Lagrangian strain tensor, $\mathit{E}$, in terms of its components,$${E}_{ij}(\mathit{X},t)=\frac{1}{2}\left({u}_{i,j}+{u}_{j,i}\right)$$
- The change of the Lagrangian porosity$$\zeta (\mathit{X},t)=\varphi (\mathit{X},t)-{\varphi}^{*}(\mathit{X},t)$$

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

RMS | root mean square |

PDE | partial differential equation |

c.p.t. | cycles per ${t}_{ref}$ |

FRF | frequency response function |

FFT | fast Fourier transform |

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**Figure 1.**The rate of bone remodeling $a\left(S\right)$, and the specific surface function $H\left(\varphi \right)$.

**Figure 3.**Stimulus generated from the mechanical load of Figure 2, with Cattaneo’s correction.

**Figure 4.**Frequency response function (FRF) of the sample linking the mechanical excitation and the stimulus.

**Figure 5.**Stimulus generated from the mechanical load of Figure 2 without Cattaneo’s correction.

**Figure 6.**Displacement, ${u}_{1}$, in the direction of the mechanical load of Figure 2, with Cattaneo’s correction at the probe point $(0.5L,0.25L)$.

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**MDPI and ACS Style**

Scerrato, D.; Giorgio, I.; Bersani, A.M.; Andreucci, D.
A Proposal for a Novel Formulation Based on the Hyperbolic Cattaneo’s Equation to Describe the Mechano-Transduction Process Occurring in Bone Remodeling. *Symmetry* **2022**, *14*, 2436.
https://doi.org/10.3390/sym14112436

**AMA Style**

Scerrato D, Giorgio I, Bersani AM, Andreucci D.
A Proposal for a Novel Formulation Based on the Hyperbolic Cattaneo’s Equation to Describe the Mechano-Transduction Process Occurring in Bone Remodeling. *Symmetry*. 2022; 14(11):2436.
https://doi.org/10.3390/sym14112436

**Chicago/Turabian Style**

Scerrato, Daria, Ivan Giorgio, Alberto Maria Bersani, and Daniele Andreucci.
2022. "A Proposal for a Novel Formulation Based on the Hyperbolic Cattaneo’s Equation to Describe the Mechano-Transduction Process Occurring in Bone Remodeling" *Symmetry* 14, no. 11: 2436.
https://doi.org/10.3390/sym14112436