# Synovial Joints. Tribology, Regeneration, Regenerative Rehabilitation and Arthroplasty

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Articular Cartilage as a Basis of Synovial Joints

#### 2.2. Lubrication and Friction in Synovial Joints

#### 2.2.1. Lubrication and Friction Modes

- articular cartilage is a linear porous-permeable two-phase material filled with a linear viscous (Newtonian) fluid;
- synovial fluid is also a Newtonian fluid;
- articular cartilage is a homogeneous layer of thickness H, and the thickness of the synovial fluid film (h) is significantly less than H; h << H;
- the radius of curvature R of the bearing articular surfaces is much larger than H; R >> H;
- the compression of the synovial fluid film is provided by a stepped load in the form of a Heaviside function applied to both bearing articular surfaces.

- articular cartilage material deforms, while the load transfer area increases;
- articular cartilage deformation leads to a decrease in the synovial fluid velocity, thus increasing the time for the formation of the squeezed film;
- synovial fluid in the gap is forced from the central high-pressure region into articular cartilage, and expelled from the tissue at the low-pressure periphery of the load-bearing region;
- tensile hoop stress exists at the cartilage surface despite the compressive squeeze-film loading condition.

#### 2.2.2. Mathematical Models of Squeeze Film Lubrication

#### 2.3. Regeneration and Regenerative Rehabilitation of Articular Cartilage

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$a$ | Tibial length |

$c$ | Clearance |

${c}_{1},\dots ,{c}_{4}$ | Integration constants |

$e$ | Gap between articular surfaces |

$\dot{e}$ | Rate change in the gap between articular surfaces |

${f}_{r}\left(e,\dot{e}\right)$ | Function characterizing the force perceived by articular surfaces |

${\overline{F}}_{r}$ | Dimensionless resistance force to the action of an external load |

$H$ | Articular cartilage layer thickness |

$\widehat{h}$ | Thicknes of poro-elastic articular cartilage layers |

$\tilde{h}\left(t\right)$ | Total layer thickness of synovial fluid |

$h\left(x,t\right)$ | Synovial film thickness |

$\overline{H},\overline{h}$ | Dimensionless parameters $H$ and $h\left(x,t\right)$ |

$K$ | Diffusive drag in articular cartilage |

$L$ | Length of the cylindrical joint model |

$l$ | Synovial fluid constant with the dimension of length: $l=\sqrt{\frac{\eta}{\mu}}$ |

$\overline{L},\overline{l}$ | Dimensionless parameters $L$ and $l$ |

p | Synovial fluid pressure |

$\overline{p}$ | Dimensionless parameter p |

$R$ | Effective radius of curvature of the contact of talus and tibia: $\frac{1}{R}=\frac{1}{{R}_{1}}-\frac{1}{{R}_{2}}$ |

${R}_{1}$ | Radius of talus curvature |

${R}_{2}$ | Radius of tibia curvature |

$t$ | Time |

u, v, w | Velocity field components of fluid media |

$W\left(t\right)$ | Law of change of an external load for unit of length |

$x,y,z$ | Cartesian coordinates of the ankle model |

$\alpha $ | Parameter: $\alpha =\frac{{l}^{2}}{\mathsf{\Phi}}$ |

$\beta $ | Polar angle: $\beta =\pm \frac{a}{R}$ |

${\delta}^{2}$ | Small parameter: ${\delta}^{2}=\frac{{\mu}_{a}}{K{H}^{2}}$ |

$\epsilon $ | Small parameter: $\epsilon =\frac{H}{R}$ |

$\overline{\epsilon}$ | Dimensionless parameter: $\overline{\epsilon}=\frac{e}{c}$ |

$\eta $ | Couple stress synovial fluid constant |

$\theta $ | Polar angle of the ankle model |

$\mu $ | Synovial fluid dynamic viscosity |

${\mu}_{a}$ | Apparent viscosity of the interstitial fluid |

Φ | Permeability of the cartilage matrix |

$\overline{\mathsf{\Phi}}$ | Dimensionless parameter Φ |

## Appendix A

**Table A1.**Numerical values used for the calculations [45].

Parameters | Numerical Values | Units |
---|---|---|

a | $14\times {10}^{-3}$ | [m] |

L | $28\times {10}^{-3}$ | [m] |

c | $7.25\times {10}^{-7}$ | [m] |

R | $3.5\times {10}^{-1}$ | [m] |

R_{1} | $22\times {10}^{-3}$ | [m/s] |

Ф | $2\times {10}^{-14}$ | [m^{2}] |

$\beta $ | $4\times {10}^{-2}$ | [rad] |

$\mu $ | $1\times {10}^{-2}$ | [Pa s] |

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**Figure 1.**Stages of development of ankle osteoarthritis: (

**a**) zero (no changes); (

**b**) the first; (

**c**) the second; (

**d**) third; (

**e**) fourth (the complete obliteration of the joint). Adopted from [8].

**Figure 2.**Stages of development of osteoarthritis of the knee joint according to Kellgren and Lawrence: (

**a**) zero (no changes); (

**b**) the first; (

**c**) the second; (

**d**) third; (

**e**) fourth (severe osteoarthritis). Adopted from [9].

**Figure 3.**Stages of development of osteoarthritis of the hip joint: (

**a**) zero (no changes); (

**b**) the first; (

**c**) the second; (

**d**) third (maximum disappearance of cartilaginous tissue from the joint surfaces). Adopted from [9].

**Figure 4.**Dynamics of publications on topics synovial joints (SJ), articular cartilage (ACar) and synovial fluid from 1950 to 2020 (according to Web of Science data).

**Figure 6.**Coordinate system and location of the main model elements ($Aca{r}_{1,2}$ are articular cartilages). Adopted from [54].

**Figure 7.**Graph of changes the external load in a dimensionless form $n\left(t\right)$ during a normal human walking cycle.

**Figure 8.**Graphs of pressure changes in the squeezed fluid film of the ankle joint at different points in the gait cycle.

**Figure 11.**Change in the dimensionless distributed load ${\overline{F}}_{r}vs.\left(\overline{l},\overline{\epsilon}\right)$ applied to the elements of the ankle joint under kinematic loading.

**Figure 12.**Graphs of the dimensionless pressure $\overline{p}$ distribution on the contact surfaces of the ankle joint elements at constant $\overline{l}$ and variable values of dimensionless displacement $\overline{\epsilon}$.

**Figure 13.**Graphs of the dimensionless pressure $\overline{p}$ distribution on the contact surfaces of the ankle joint elements at constant values of dimensionless displacement $\overline{\epsilon}$ and constant $\overline{l}$.

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**MDPI and ACS Style**

Popov, V.L.; Poliakov, A.M.; Pakhaliuk, V.I.
Synovial Joints. Tribology, Regeneration, Regenerative Rehabilitation and Arthroplasty. *Lubricants* **2021**, *9*, 15.
https://doi.org/10.3390/lubricants9020015

**AMA Style**

Popov VL, Poliakov AM, Pakhaliuk VI.
Synovial Joints. Tribology, Regeneration, Regenerative Rehabilitation and Arthroplasty. *Lubricants*. 2021; 9(2):15.
https://doi.org/10.3390/lubricants9020015

**Chicago/Turabian Style**

Popov, Valentin L., Aleksandr M. Poliakov, and Vladimir I. Pakhaliuk.
2021. "Synovial Joints. Tribology, Regeneration, Regenerative Rehabilitation and Arthroplasty" *Lubricants* 9, no. 2: 15.
https://doi.org/10.3390/lubricants9020015