# Mathematical Development of a Novel Discrete Hip Deformation Algorithm for the In Silico Elasto-Hydrodynamic Lubrication Modelling of Total Hip Replacements

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## Abstract

**:**

## 1. Introduction

- When the joint is subjected to intense relative motion and light loading, the hydrodynamic lubrication occurs, so the coupled surfaces are totally separated by the lubricating fluid along the whole contact area;
- The boundary lubrication is established when the high load and the slow relative motion do not allow to obtain the surfaces’ separation, so the lubrication effect is governed by chemical reactions at the contact interface; and the slow relative motion don’t allow to obtain the surfaces’ separation, so the lubrication effect is governed by chemical reactions at the contact interface;
- In the intermediate condition, the load is supported both by asperities in contact and by the lubricating fluid pressure, so it is defined as mixed lubrication.

## 2. Materials and Methods

#### 2.1. Finite Element Model

- The matrix ${\mathit{J}}_{n}$ which relates the nodal pressure vector ${\mathit{p}}_{n}$ to the surface pressure field vector $\mathit{p}$ in (9);$$\begin{array}{c}{\mathit{p}}_{n}={\mathit{J}}_{n}\mathit{p}\end{array}$$
- The matrix ${\mathit{J}}_{\delta}$ which relates the surface deformation field vector $\mathit{\delta}$ to the nodal normal displacement vector ${\mathit{\delta}}_{n}$ in (10); and$$\begin{array}{c}\mathit{\delta}={\mathit{J}}_{\delta}{\mathit{\delta}}_{n}\end{array}$$
- The matrices ${\mathit{J}}_{l}$ and ${\mathit{J}}_{v}$ which collect the free part $\left(l\right)$ or the constrained one $\left(v\right)$ of a vector quantity $\mathit{x}$, useful also to rewrite it in its ordered form ${\mathit{x}}_{o}$ through the matrix ${\mathit{J}}_{o}$ in (11).$$\begin{array}{c}\{\begin{array}{c}{\mathit{x}}_{l}={\mathit{J}}_{l}\mathit{x}\\ {\mathit{x}}_{v}={\mathit{J}}_{v}\mathit{x}\end{array}\to \hspace{1em}\hspace{1em}{\mathit{x}}_{o}=\left[\begin{array}{c}{\mathit{x}}_{l}\\ {\mathit{x}}_{v}\end{array}\right]=\left[\begin{array}{c}{\mathit{J}}_{l}\\ {\mathit{J}}_{v}\end{array}\right]\mathit{x}={\mathit{J}}_{o}\mathit{x}\end{array}$$

#### 2.2. Acetabular Cup and Femoral Head Meshes

- Regarding the acetabular cup, the nodes on the outer surface are fixed because of the cup fixation with respect to the pelvic bone, namely when ${\rho}_{c}=R+H$, while the nodes subjected to the surface pressure are located on the inner surface, namely when ${\rho}_{c}=R$; and
- Regarding the femoral head, the nodes with the $y$-coordinate lower than a constant value depending on a chosen angle ${\varphi}_{0}$ are fixed because of the head fixation with respect to the femoral stem, namely when ${y}_{h}<-r\mathrm{cos}{\varphi}_{0}$, while the nodes subjected to the surface pressure are located on the outer surface, namely when ${\rho}_{h}=r$.

#### 2.3. Acetabular Cup and Femoral Head Coupling through Cubic Interpolation

- Both for the acetabular cup and for the femoral head, the reference frame used in the finite element discretization was rotated by the inclination angle ${\alpha}_{in}$ and the anteversion angle ${\beta}_{av}$ with respect to the anatomical reference frame (Antero/Posterior AP, Proximo/Distal PD and Medio/Lateral ML) through a rotation matrix ${\mathit{R}}_{g}$ defined in (22) [3]; and$$\begin{array}{c}{\mathit{R}}_{g}={\mathit{R}}_{z}\left(\frac{\pi}{2}-{\beta}_{av}\right){\mathit{R}}_{x}\left(-{\alpha}_{in}\right)\end{array}$$
- The head anatomical reference frame is rotated with respect to the cup one by the Flexion/Extension angle ${\theta}_{FE}$, the Adduction/Abduction angle ${\theta}_{AA}$ and the Internal/External Rotation angle ${\theta}_{IER}$ through the rotation matrix ${\mathit{R}}_{hip}$ defined in (23) [3].$$\begin{array}{c}{\mathit{R}}_{hip}={\mathit{R}}_{z}\left({\theta}_{FE}\right){\mathit{R}}_{x}\left({\theta}_{AA}\right){\mathit{R}}_{y}\left({\theta}_{IER}\right)\end{array}$$

## 3. Results and Discussion

#### 3.1. Validation

#### 3.2. Implementation of the Deformation Model in an EHL Lubrication Algorithm

## 4. Conclusions

- The implementation of a routine which elaborates the contact pressure in domain zones characterized by nodes overlapping, in order to consider the model functionality within a mixed elasto-hydrodynamic lubrication algorithm;
- The adaptation of the finite element model to consider viscoelastic materials, adding deformation contributions which depend on the time history of the pressure field through a viscosity matrix;
- The usage of the model for other types of joint replacements such as knee, ankle, shoulders, etc.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$I,J,K,H$ | Linear tetrahedron nodes |

$\widehat{n}$ | Tetrahedron face outward-pointing normal |

$A$ | Tetrahedron face area |

$N$ | Shape function matrix |

${p}_{I},{p}_{J},{p}_{K}$ | Nodal pressures |

${p}_{n}$ | Nodal pressure vector |

$\mathsf{\Phi}$ | Nodal force vector |

$q$ | Nodal displacement vector |

$K$ | Stiffness matrix |

${\delta}_{n}$ | Nodal normal displacement vector |

$p$ | Surface pressure field vector |

$\delta $ | Surface deformation field vector |

$C$ | Influence matrix |

$n$ | Boundary conditions vector or dimensionless eccentricity vector |

$x$ | Cartesian coordinates vector |

$\rho ,\theta ,\phi $ | Spherical coordinates |

$R$ | Acetabular cup inner radius |

$H$ | Acetabular cup thickness |

$r$ | Femoral head radius |

${\varphi}_{0}$ | Femoral head fixed nodes angle |

$i,j$ | Surface grid point indices |

${J}_{01}$ | Cubic interpolation matrix from the grid $0$ to the grid $1$ |

$R$ | Rotation matrix |

${\alpha}_{in},{\beta}_{av}$ | Inclination and anteversion angles |

${\theta}_{FE},{\theta}_{AA},{\theta}_{IER}$ | Hip Flexion/Extension, Adduction/Abduction and Internal/External Rotation angles |

$\widehat{r}$ | Radial unit vector |

$E,\nu $ | Young modulus and Poisson ratio |

$\mathrm{RMSE},{R}^{2}$ | Root Mean Squared Error and squared correlation coefficient |

$c$ | Radial clearance |

$h$ | Lubricating fluid gap |

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**Figure 1.**Linear tetrahedron with I, J, K, and H nodes subjected to nodal pressures acting on the face IJK.

**Figure 8.**Acetabular cup surface pressure input in Matlab (surf) and Ansys (black circles) (

**a**); Matlab deformation output (surf) and Ansys deformation output (black circles) related to the acetabular cup (

**b**).

**Figure 9.**Matlab deformation results data against Ansys ones regarding the acetabular cup simulation.

**Figure 10.**Femoral head surface pressure input in Matlab (surf) and Ansys (black circles) (

**a**); Matlab deformation output (surf) and Ansys deformation output (black circles) related to the femoral head (

**b**).

**Figure 11.**Matlab deformation results data against Ansys ones regarding the femoral head simulation.

**Figure 12.**Von Mises stress (

**a**) and strain energy (

**b**) within the acetabular cup simulation in Ansys.

**Figure 13.**Von Mises stress (

**a**) and strain energy (

**b**) within the acetabular cup simulation in Matlab.

Parameter | Value |
---|---|

Femoral head radius r | 14 mm |

Acetabular cup inner radius R | 14.03 mm |

Acetabular cup thickness H | 9 mm |

Femoral head fixed nodes angle ${\varphi}_{0}$ | 30° |

Acetabular cup Young modulus ${E}_{c}$ | 1.1 GPa |

Acetabular cup Poisson ratio ${\nu}_{c}$ | 0.42 |

Femoral head Young modulus ${E}_{h}$ | 245.9 GPa |

Femoral head Poisson ratio ${\nu}_{h}$ | 0.24 |

Pressure gaussian peak ${p}_{0}$ | 10^{7} Pa |

Pressure gaussian θ-translation ${\theta}_{0}$ | 2π/3 rad |

Pressure gaussian φ-translation ${\phi}_{0}$ | π/3 rad |

Pressure gaussian dimensionless width ${\alpha}_{0}$ | 0.2 |

**Table 2.**Input parameters related to the simulation with relative rotations between the acetabular cup and the femoral head.

Parameter | Value |
---|---|

Pressure gaussian peak ${p}_{0}$ | 10^{7} Pa |

Pressure gaussian θ-translation ${\theta}_{0}$ | π/4 rad |

Pressure Gaussian φ-translation ${\phi}_{0}$ | π/4 rad |

Pressure gaussian dimensionless width ${\alpha}_{0}$ | 0.2 |

Anteversion angle ${\beta}_{av}$ | 90° |

Inclination angle ${\alpha}_{in}$ | 45° |

Flexion/Extension angle ${\theta}_{FE}$ | −40° |

Adduction/Abduction angle ${\theta}_{AA}$ | 10° |

Internal/External Rotation angle ${\theta}_{IER}$ | 2° |

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**MDPI and ACS Style**

Ruggiero, A.; Sicilia, A.
Mathematical Development of a Novel Discrete Hip Deformation Algorithm for the In Silico Elasto-Hydrodynamic Lubrication Modelling of Total Hip Replacements. *Lubricants* **2021**, *9*, 41.
https://doi.org/10.3390/lubricants9040041

**AMA Style**

Ruggiero A, Sicilia A.
Mathematical Development of a Novel Discrete Hip Deformation Algorithm for the In Silico Elasto-Hydrodynamic Lubrication Modelling of Total Hip Replacements. *Lubricants*. 2021; 9(4):41.
https://doi.org/10.3390/lubricants9040041

**Chicago/Turabian Style**

Ruggiero, Alessandro, and Alessandro Sicilia.
2021. "Mathematical Development of a Novel Discrete Hip Deformation Algorithm for the In Silico Elasto-Hydrodynamic Lubrication Modelling of Total Hip Replacements" *Lubricants* 9, no. 4: 41.
https://doi.org/10.3390/lubricants9040041