# The Biomechanical Analysis of Tibial Implants Using Meshless Methods: Stress and Bone Tissue Remodeling Analysis

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Formulation

#### 2.1. Meshless Methods Formulation

#### 2.1.1. Influence Domains

#### 2.1.2. Shape Functions

#### 2.2. Weak Form and Discrete System of Equations

#### 2.3. Bone Tissue Remodeling Algorithm

^{3}, and porosity p is calculated from ${V}_{holes}/{V}_{sample}$, where ${V}_{holes}$ is the total volume of holes.

^{3}to differentiate between trabecular and cortical bone.

## 3. Tibia and Implant Numerical Models

## 4. Results and Discussion

#### 4.1. Structural Analysis of the Proximal Tibia

#### 4.2. Structural Analysis of the Implant and Influence of Implant Length

#### 4.3. Bone Remodeling Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Fernandes, D.A.; Poeta, L.S.; Martins, C.A.d.Q.; Lima, F.d.; Rosa Neto, F. Balance and quality of life after total knee arthroplasty. Rev. Bras. Ortop. (Engl. Ed.)
**2018**, 53, 747–753. [Google Scholar] [CrossRef] - Yueh, S.; Noori, M.; Mahadev, S.; Noori, N.B. Finite Element Analysis of Total Knee Arthroplasty. Am. J. Biomed. Sci. Res.
**2021**, 14, 6–15. [Google Scholar] [CrossRef] - Filip, A.C.; Cuculici, S.A.; Cristea, S.; Filip, V.; Negrea, A.D.; Mihai, S.; Pantu, C.M. Tibial Stem Extension versus Standard Configuration in Total Knee Arthroplasty: A Biomechanical Assessment According to Bone Properties. Medicina
**2022**, 58, 634. [Google Scholar] [CrossRef] [PubMed] - Peters, C.L.; Erickson, J.; Kloepper, R.G.; Mohr, R.A. Revision total knee arthroplasty with modular components inserted with metaphyseal cement and stems without cement. J. Arthroplast.
**2005**, 20, 302–308. [Google Scholar] [CrossRef] - Cintra, F.F.; Yepéz, A.K.; Rasga, M.G.S.; Abagge, M.; Alencar, P.G.C. Tibial Component in Revision of Total Knee Arthroplasty: Comparison Between Cemented and Hybrid Fixation. Rev. Bras. Ortop. (Engl. Ed.)
**2011**, 46, 585–590. [Google Scholar] [CrossRef] - Lonner, J.H.; Klotz, M.; Levitz, C.; Lotke, P.A. Changes in bone density after cemented total knee arthroplasty: Influence of stem design. J. Arthroplast.
**2001**, 16, 107–111. [Google Scholar] [CrossRef] [PubMed] - Shannon, B.D.; Klassen, J.F.; Rand, J.A.; Berry, D.J.; Trousdale, R.T. Revision total knee arthroplasty with cemented components and uncemented intramedullary stems. J. Arthroplast.
**2003**, 18, 27–32. [Google Scholar] [CrossRef] [PubMed] - Kang, S.G.; Park, C.H.; Song, S.J. Stem fixation in revision total knee arthroplasty: Indications, stem dimensions, and fixation methods. Knee Surg. Relat. Res.
**2018**, 30, 187–192. [Google Scholar] [CrossRef] - Crawford, D.A.; Berend, K.R.; Morris, M.J.; Adams, J.B.; Lombardi, A.V. Results of a Modular Revision System in Total Knee Arthroplasty. J. Arthroplast.
**2017**, 32, 2792–2798. [Google Scholar] [CrossRef] - Bottner, F.; Laskin, R.; Windsor, R.E.; Haas, S.B. Hybrid component fixation in revision total knee arthroplasty. Clin. Orthop. Relat. Res.
**2006**, 446, 127–131. [Google Scholar] [CrossRef] - Ni, G.X.; Lu, W.W.; Chiu, K.Y.; Fong, D.Y. Cemented or uncemented femoral component in primary total hip replacement? A review from a clinical and radiological perspective. J. Orthop. Surg. (Hong Kong)
**2005**, 13, 96–105. [Google Scholar] [CrossRef] [PubMed] - Kadir, M.R.A. Computational Biomechanics of the Hip Joint; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Dorr, L.D.; Wan, Z.; Gruen, T. Functional results in total hip replacement in patients 65 years and older. Clin. Orthop. Relat. Res.
**1997**, 336, 143–151. [Google Scholar] [CrossRef] [PubMed] - Kozelskaya, A.I.; Rutkowski, S.; Frueh, J.; Gogolev, A.S.; Chistyakov, S.G.; Gnedenkov, S.V.; Sinebryukhov, S.L.; Frueh, A.; Egorkin, V.S.; Choynzonov, E.L.; et al. Surface Modification of Additively Fabricated Titanium-Based Implants by Means of Bioactive Micro-Arc Oxidation Coatings for Bone Replacement. J. Funct. Biomater.
**2022**, 13, 285. [Google Scholar] [CrossRef] [PubMed] - Completo, A.; Simões, J.A.; Fonseca, F.; Oliveira, M. The influence of different tibial stem designs in load sharing and stability at the cement-bone interface in revision TKA. Knee
**2008**, 15, 227–232. [Google Scholar] [CrossRef] [PubMed] - Quevedo González, F.J.; Sculco, P.K.; Kahlenberg, C.A.; Mayman, D.J.; Lipman, J.D.; Wright, T.M.; Vigdorchik, J.M. Undersizing the Tibial Baseplate in Cementless Total Knee Arthroplasty has Only a Small Impact on Bone-Implant Interaction: A Finite Element Biomechanical Study. J. Arthroplast.
**2023**, 38, 757–762. [Google Scholar] [CrossRef] [PubMed] - Quevedo Gonzalez, F.J.; Lipman, J.D.; Sculco, P.K.; Sculco, T.P.; De Martino, I.; Wright, T.M. An Anterior Spike Decreases Bone-Implant Micromotion in Cementless Tibial Baseplates for Total Knee Arthroplasty: A Biomechanical Study. J. Arthroplast.
**2023**, 4–8. [Google Scholar] [CrossRef] [PubMed] - Liu, Y.; Chen, B.; Wang, C.; Chen, H.; Zhang, A.; Yin, W.; Wu, N.; Han, Q.; Wang, J. Design of Porous Metal Block Augmentation to Treat Tibial Bone Defects in Total Knee Arthroplasty Based on Topology Optimization. Front. Bioeng. Biotechnol.
**2021**, 9, 765438. [Google Scholar] [CrossRef] [PubMed] - Liu, Y.; Zhang, A.; Wang, C.; Yin, W.; Wu, N.; Chen, H.; Chen, B.; Han, Q.; Wang, J. Biomechanical comparison between metal block and cement-screw techniques for the treatment of tibial bone defects in total knee arthroplasty based on finite element analysis. Comput. Biol. Med.
**2020**, 125, 104006. [Google Scholar] [CrossRef] - Bhandarkar, S.; Dhatrak, P. Optimization of a knee implant with different biomaterials using finite element analysis. Mater. Today Proc.
**2022**, 59, 459–467. [Google Scholar] [CrossRef] - Apostolopoulos, V.; Tomáš, T.; Boháč, P.; Marcián, P.; Mahdal, M.; Valoušek, T.; Janíček, P.; Nachtnebl, L. Biomechanical analysis of all-polyethylene total knee arthroplasty on periprosthetic tibia using the finite element method. Comput. Methods Programs Biomed.
**2022**, 220, 106834. [Google Scholar] [CrossRef] - Mondal, S.; Ghosh, R. The role of the depth of resection of the distal tibia on biomechanical performance of the tibial component for TAR: A finite element analysis with three implant designs. Med. Eng. Phys.
**2023**, 119, 104034. [Google Scholar] [CrossRef] - Mondal, S.; Ghosh, R. Biomechanical analysis of three popular tibial designs for TAR with different implant-bone interfacial conditions and bone qualities: A finite element study. Med. Eng. Phys.
**2022**, 104, 103812. [Google Scholar] [CrossRef] - ISO 22926:2023; Implants for Surgery Specification and Verification of Synthetic Anatomical Bone Models for Testing. ISO: Geneva, Switzerland, 2023.
- ISO/CD 5092:2023; Additive Manufacturing for Medical. ISO: Geneva, Switzerland, 2023.
- Belinha, J. Meshless Methods in Biomechanics—Bone Tissue Remodelling Analysis; Springer: Porto, Portugal, 2014. [Google Scholar]
- Belinha, J. Meshless Methods in Biomechanics: Bone Tissue Remodelling Analysis; Lecture Notes in Computational Vision and Biomechanics; Springer International Publishing: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- Hardy, R.L. Theory and applications of the multiquadric-biharmonic method 20 years of discovery 1968–1988. Comput. Math. Appl.
**1990**, 19, 163–208. [Google Scholar] [CrossRef] - Raggatt, L.J.; Partridge, N.C. Cellular and molecular mechanisms of bone remodeling. J. Biol. Chem.
**2010**, 285, 25103–25108. [Google Scholar] [CrossRef] [PubMed] - Sabet, F.A.; Najafi, A.R.; Hamed, E.; Jasiuk, I. Modelling of bone fracture and strength at different length scales: A review. Interface Focus
**2016**, 6, 20–30. [Google Scholar] [CrossRef] [PubMed] - Carter, D.R.; Hayes, W.C. The compressive behavior of bone as a two-phase porous structure. J. Bone Jt. Surgery. Am. Vol.
**1977**, 59, 954–962. [Google Scholar] [CrossRef] - Goldstein, S.A. The mechanical properties of trabecular bone: Dependence on anatomic location and function. J. Biomech.
**1987**, 20, 1055–1061. [Google Scholar] [CrossRef] - Rice, J.C.; Cowin, S.C.; Bowman, J.A. On the dependence of the elasticity and strength of cancellous bone on apparent density. J. Biomech.
**1988**, 21, 155–168. [Google Scholar] [CrossRef] - Martin, R.B. Determinants of the mechanical properties of bones. J. Biomech.
**1991**, 24, 79–88. [Google Scholar] [CrossRef] - Pauwels, F. Gesammelte Abhandlungen zur Funktionellen Anatomie des Bewegungsapparates; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Cowin, S.C.; Hegedus, D.H. Bone remodeling I: Theory of adaptive elasticity. J. Elast.
**1976**, 6, 313–326. [Google Scholar] [CrossRef] - Cowin, S.C.; Sadegh, A.M.; Luo, G.M. An Evolutionary Wolff’s Law for Trabecular Architecture. J. Biomech. Eng.
**1992**, 114, 129–136. [Google Scholar] [CrossRef] [PubMed] - Rodrigues, H.; Jacobs, C.; Guedes, J.M.; Bendsøe, M.P. Global and local material optimization models applied to anisotropic bone adaptation. In Proceedings of the IUTAM Symposium on Synthesis in Bio Solid Mechanics, Copenhagen, Denmark, 24–27 May 1998; Springer: Berlin/Heidelberg, Germany, 1999; pp. 209–220. [Google Scholar]
- Carter, D.R.; Fyhrie, D.P.; Whalen, R.T. Trabecular bone density and loading history: Regulation of connective tissue biology by mechanical energy. J. Biomech.
**1987**, 20, 785–794. [Google Scholar] [CrossRef] [PubMed] - Whalen, R.T.; Carter, D.R.; Steele, C.R. Influence of physical activity on the regulation of bone density. J. Biomech.
**1988**, 21, 825–837. [Google Scholar] [CrossRef] - Carter, D.R.; Orr, T.E.; Fyhrie, D.P. Relationships between loading history and femoral cancellous bone architecture. J. Biomech.
**1989**, 22, 231–244. [Google Scholar] [CrossRef] [PubMed] - Belinha, J.; Natal Jorge, R.M.; Dinis, L.M. Bone tissue remodelling analysis considering a radial point interpolator meshless method. Eng. Anal. Bound. Elem.
**2012**, 36, 1660–1670. [Google Scholar] [CrossRef] - Belinha, J.; Jorge, R.M.N.; Dinis, L.M.J.S. A meshless microscale bone tissue trabecular remodelling analysis considering a new anisotropic bone tissue material law. Comput. Methods Biomech. Biomed. Eng.
**2013**, 16, 1170–1184. [Google Scholar] [CrossRef] - Taddei, F.; Pani, M.; Zovatto, L.; Tonti, E.; Viceconti, M. A new meshless approach for subject-specific strain prediction in long bones: Evaluation of accuracy. Clin. Biomech.
**2008**, 23, 1192–1199. [Google Scholar] [CrossRef] [PubMed] - Liu, B.; Wang, H.; Zhang, N.; Zhang, M.; Cheng, C.K. Femoral Stems With Porous Lattice Structures: A Review. Front. Bioeng. Biotechnol.
**2021**, 9, 772539. [Google Scholar] [CrossRef] - Marques, M.; Belinha, J.; Dinis, L.M.J.S.; Natal Jorge, R. A brain impact stress analysis using advanced discretization meshless techniques. Proc. Inst. Mech. Eng. Part H J. Eng. Med.
**2018**, 232, 257–270. [Google Scholar] [CrossRef]

**Figure 3.**Schematic representation of the essential and natural boundary conditions applied to (

**a**) the three-dimensional model and (

**b**) the two-dimensional implant model.

**Figure 4.**Implant model. (

**a**) 1—tibial plate, 2—cement, 3—tibia; (

**b**) simplified assembled model; (

**c**) simplified model representation showing the implant and surrounding bone.

**Figure 5.**Von Mises stress results: (

**a**) selected points in the model (A and B are two representative points of load application and, therefore, are used to evaluate the displacement and points 1 to 9 are the points selected to evaluate stress since each point is further from the fixed ending); (

**b**) von Mises stress at each point calculated with the FEM and the RPIM.

**Figure 6.**Displacement field for each material, where the maximum displacements ${\left|u\right|}_{max}$ are 0.050, 0.016, and 0.010 for model 1, model 2, and model 3, respectively.

**Figure 9.**(

**a**) Evaluation of osteointegration using the FEM along the iterations; (

**b**) indication of the main resorption (indicated by numbers 1 to 3) and growth areas after implant insertion.

Implant Length | Element Type | Nodes | Elements |
---|---|---|---|

0 | 4-node tetrahedral | 2083 | 9522 |

12 mm | 4-node tetrahedral | 5273 | 1171 |

30 mm | 4-node tetrahedral | 6908 | 1487 |

40 mm | 3-node plane stress | 1901 | 3630 |

40 mm | 4-node tetrahedral | 6299 | 1360 |

Young’s Modulus [GPa] | Poisson’s Ratio | |
---|---|---|

Implant—Ti-6Al-4V | 110 | 0.34 |

Low-stiffness bone | 5 | 0.33 |

Healthy cortical bone | 17 | 0.33 |

High-stiffness bone | 25 | 0.33 |

Method | Young’s Modulus | Point | u_{x} [mm] | u_{y} [mm] | u_{z} [mm] | |u| [mm] |
---|---|---|---|---|---|---|

FEM | 5 GPa | A | −0.0281 | −0.0158 | −0.0451 | 0.0555 |

B | −0.0241 | −0.0193 | −0.0035 | 0.031 | ||

17 GPa | A | −0.0082 | −0.0046 | −0.0132 | 0.0163 | |

B | −0.007 | −0.0056 | −0.001 | 0.0091 | ||

25 GPa | A | −0.0056 | −0.0031 | −0.009 | 0.0111 | |

B | −0.0048 | −0.0038 | −0.0007 | 0.0062 | ||

RPIM | 5 GPa | A | −0.0286 | −0.0161 | −0.0471 | 0.0575 |

B | −0.0246 | −0.0203 | −0.004 | 0.0322 | ||

17 GPa | A | −0.0084 | −0.0047 | −0.0138 | 0.0169 | |

B | −0.0072 | −0.0059 | −0.0011 | 0.0094 | ||

25 GPa | A | −0.0057 | −0.0032 | −0.0094 | 0.0115 | |

B | −0.0049 | −0.004 | −0.0008 | 0.0064 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pais, A.; Moreira, C.; Belinha, J.
The Biomechanical Analysis of Tibial Implants Using Meshless Methods: Stress and Bone Tissue Remodeling Analysis. *Designs* **2024**, *8*, 28.
https://doi.org/10.3390/designs8020028

**AMA Style**

Pais A, Moreira C, Belinha J.
The Biomechanical Analysis of Tibial Implants Using Meshless Methods: Stress and Bone Tissue Remodeling Analysis. *Designs*. 2024; 8(2):28.
https://doi.org/10.3390/designs8020028

**Chicago/Turabian Style**

Pais, Ana, Catarina Moreira, and Jorge Belinha.
2024. "The Biomechanical Analysis of Tibial Implants Using Meshless Methods: Stress and Bone Tissue Remodeling Analysis" *Designs* 8, no. 2: 28.
https://doi.org/10.3390/designs8020028