# FEM-Based Compression Fracture Risk Assessment in Osteoporotic Lumbar Vertebra L1

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods and Materials

#### 2.1. Problem Formulation

#### 2.2. Structure and Geometrical Properties of the Model

#### 2.3. Mechanical Properties

#### 2.4. The Parameters of Osteoporosis Impact

#### 2.5. Boundary Conditions and Mesh

#### 2.6. Risk Evaluation

## 3. Numerical Results and Discussion

#### 3.1. Static Analysis

#### 3.2. Dynamic Analysis

#### 3.3. Estimation of the Fracture Risk

#### 3.3.1. Approximation of the Maximum von Mises Stress

#### 3.3.2. Stochastic Models and for the Risk Modelling

#### 3.3.3. Fracture Risk Modelling Data, Results, and Discussion

#### 3.4. Influence of Vertebra Cortical Shell Buckling

#### 3.5. Discussion on Model Limitations

#### 3.6. Validation of Mechanical Model

- The printing of the vertebrae model;
- The printing of the cylindrical PLA sample for mechanical properties of PLA to define;
- The compression test of the printed PLA sample;
- The compression test of the printed PLA vertebrae;
- The determination of the mechanical properties of PLA by verifying the obtained load-displacement curve of the cylindrical sample;
- The implementation of the defined stress-strain curve of PLA onto the numerical calculation;
- The comparison of the results of the experimental and numerical studies.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

FEM | Finite element method |

FE | Finite element |

IVD | intervetebral disk |

BV/TV | Bone volume vs total volume |

QCT | Quantitative computational tomography |

BMD | Bone mineral density |

DXA | Dual-energy X-ray absorptiometry |

## References

- Agrawal, A.; Kalia, R. Osteoporosis: Current Review. J. Orthop. Traumatol. Rehabil.
**2014**, 7, 101. [Google Scholar] [CrossRef] - Lin, J.T.; Lane, J.M. Osteoporosis: A review. Clin. Orthop. Relat. Res.
**2004**, 425, 126–134. [Google Scholar] [CrossRef] - Cooper, C.; Cole, Z.A.; Holroyd, C.R.; Earl, S.C.; Harvey, N.C.; Dennison, E.M.; Melton, L.J.; Cummings, S.R.; Kanis, J.A. The IOF CSA Working Group on Fracture Epidemiology. Secular trends in the incidence of hip and other osteoporotic fractures. Osteoporos. Int.
**2011**, 22, 1277–1288. [Google Scholar] [CrossRef] - Cummings, S.R.; Melton, L.J., III. Epidemiology and outcomes of osteoporotic fractures. Lancet
**2002**, 359, 1761–1767. [Google Scholar] [CrossRef] - Doblaré, M.; García, J.M.; Gómez, M.J. Modelling bone tissue fracture and healing: A review. Eng. Fract. Mech.
**2004**, 71, 1809–1840. [Google Scholar] [CrossRef] - odygowski, T.; Kakol, W.; Wierszycki, M.; Ogurkowska, B.M. Three-dimensional nonlinear finite element model of the human lumbar spine segment. Acta Bioeng. Biomech.
**2005**, 7, 17–28. [Google Scholar] - Su, J.; Cao, L.; Li, Z.; Yu, B.; Zhang, C.; Li, M. Three-dimensional finite element analysis of lumbar vertebra loaded by static stress and its biomechanical significance. Chin. J. Traumatol.
**2009**, 12, 153–156. [Google Scholar] - Jones, A.C.; Wilcox, R.K. Finite element analysis of the spine: Towards a framework of verification, validation and sensitivity analysis. Med. Eng. Phys.
**2008**, 30, 1287–1304. [Google Scholar] [CrossRef] - Crawford, R.P.; Cann, C.E.; Keaveny, T.M. Finite element models predict in vitro vertebral body compressive strength better than quantitative computed tomography. Bone
**2003**, 33, 744–750. [Google Scholar] [CrossRef] - Maquer, G.; Schwiedrzik, J.; Huber, G.; Morlock, M.M.; Zysset, P.K. Compressive strength of elderly vertebrae is reduced by disc degeneration and additional flexion. J. Mech. Behav. Biomed. Mater.
**2015**, 42, 54–66. [Google Scholar] [CrossRef][Green Version] - Provatidis, C.; Vossou, C.; Koukoulis, I.; Balanika, A.; Baltas, C.; Lyritis, G. A pilot finite element study of an osteoporotic L1-vertebra compared to one with normal T-score. Comput. Methods Biomech. Biomed. Eng.
**2010**, 13, 185–195. [Google Scholar] [CrossRef] - McDonald, K.; Little, J.; Pearcy, M.; Adam, C. Development of a multi-scale finite element model of the osteoporotic lumbar vertebral body for the investigation of apparent level vertebra mechanics and micro-level trabecular mechanics. Med. Eng. Phys.
**2010**, 32, 653–661. [Google Scholar] [CrossRef][Green Version] - Garo, A.; Arnoux, P.J.; Wagnac, E.; Aubin, C.E. Calibration of the mechanical properties in a finite element model of a lumbar vertebra under dynamic compression up to failure. Med. Biol. Eng. Comput.
**2011**, 49, 1371–1379. [Google Scholar] [CrossRef] - Kim, Y.H.; Wu, M.; Kim, K. Stress Analysis of Osteoporotic Lumbar Vertebra Using Finite Element Model with Microscaled Beam-Shell Trabecular-Cortical Structure. J. Appl. Math.
**2013**, 2013, 285165. [Google Scholar] [CrossRef] - Wierszycki, M.; Szajek, K.; Łodygowski, T.; Nowak, M. A two-scale approach for trabecular bone microstructure modeling based on computational homogenization procedure. Comput. Mech.
**2014**, 54, 287–298. [Google Scholar] [CrossRef][Green Version] - Wolfram, U.; Gross, T.; Pahr, D.H.; Schwiedrzik, J.; Wilke, H.J.; Zysset, P.K. Fabric-based Tsai-Wu yield criteria for vertebral trabecular bone in stress and strain space. J. Mech. Behav. Biomed. Mater.
**2012**, 15, 218–228. [Google Scholar] [CrossRef] - El-Rich, M.; Arnoux, P.J.; Wagnac, E.; Brunet, C.; Aubin, C.E. Finite element investigation of the loading rate effect on the spinal load-sharing changes under impact conditions. J. Biomech.
**2009**, 42, 1252–1262. [Google Scholar] [CrossRef] - Pothuaud, L.; Carceller, P.; Hans, D. Correlations between grey-level variations in 2D projection images (TBS) and 3D microarchitecture: Applications in the study of human trabecular bone microarchitecture. Bone
**2008**, 42, 775–778. [Google Scholar] [CrossRef] - Hans, D.; Barthe, N.; Boutroy, S.; Pothuaud, L.; Winzenrieth, R.; Krieg, M.-A. Correlations Between Trabecular Bone Score, Measured Using Anteroposterior Dual-Energy X-Ray Absorptiometry Acquisition, and 3-Dimensional Parameters of Bone Microarchitecture: An Experimental Study on Human Cadaver Vertebrae. J. Clin. Densitom.
**2011**, 14, 302–312. [Google Scholar] [CrossRef] - Kanis, J.A.; Oden, V.; Johnell, O.; Johansson, H.; De Laet, C.; Brown, J.; Burckhardt, P.; Cooper, C.; Christiansen, C.; Cummings, S.; et al. The use of clinical risk factors enhances the performance of BMD in the prediction of hip and osteoporotic fractures in men and women. Osteoporos. Int.
**2007**, 18, 1033–1046. [Google Scholar] [CrossRef] - Marshall, D.; Johnell, O.; Wedel, H. Meta-analysis of how well measures of bone mineral density predict occurrence of osteoporotic fractures. BMJ
**1996**, 312, 1254–1259. [Google Scholar] [CrossRef][Green Version] - Taylor, B.C.; Schreiner, P.J.; Stone, K.L.; Fink, H.A.; Cummings, S.R.; Nevitt, M.C.; Bowman, P.J.; Ensrud, K.E. Long-term prediction of incident hip fracture risk in elderly white women: Study of osteoporotic fractures. J. Am. Geriatr. Soc.
**2004**, 52, 1479–1486. [Google Scholar] [CrossRef] - Kanis, J.A.; Oden, A.; Johansson, H.; Borgstrom, F.; Strom, O.; McCloskey, E. FRAX and its applications to clinical practice. Bone
**2009**, 44, 734–743. [Google Scholar] [CrossRef] - Kanis, J.A.; Johnell, O.; Oden, A.; Johansson, H.; McCloskey, E. FRAX and the assessment of fracture probability in men and women from the UK. Osteoporos. Int.
**2008**, 19, 385–397. [Google Scholar] [CrossRef] - Timothy, H.K.; Brandeau, J.F. Mathematical modeling of the stress strain-strain rate behavior of bone using the Ramberg-Osgood equation. J. Biomech.
**1983**, 16, 445–450. [Google Scholar] - Nazarian, A.; von Stechow, D.; Zurakowski, D.; Muller, R.; Snyder, B.D. Bone Volume Fraction Explains the Variation in Strength and Stiffness of Cancellous Bone Affected by Metastatic Cancer and Osteoporosis. Calcif. Tissue Int.
**2008**, 83, 368–379. [Google Scholar] [CrossRef] - Abaqus FEA, SIMULIA Web Site. Dassault Systèmes, Retrieved 2017. Available online: https://www.3ds.com/ (accessed on 25 July 2019).
- Linthorne, N.P. Analysis of standing vertical jumps using a force platform. J. Sports Sci. Med.
**2010**, 9, 282–287. [Google Scholar] [CrossRef] - Dodson, B.; Noland, D. Reliability Engineering Handbook; CRC Press LLC Main Office: Boca Raton, FL, USA, 1999; p. 592. [Google Scholar]
- Melton, L.J., III; Achenbach, S.J.; Atkinson, E.J.; Therneau, T.M.; Amin, S. Long-term mortality following fractures at different skeletai sites: A population-based cohort study. Osteoporos. Int.
**2013**, 24, 1689–1696. [Google Scholar] [CrossRef] - R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2017; Available online: https://www.R-project.org/ (accessed on 25 July 2019).
- Pinheiro, J.; Bates, D.; DebRoy, S.; Sarkar, D.; R Core Team (2017). Nlme: Linear and Nonlinear Mixed Effects Models. R Package Version 3.1-131. 2017. Available online: https://CRAN.R-project.org/package=nlme (accessed on 25 July 2019).
- Fields, A.J.; Eswaran, S.K.; Jekir, M.G.; Keaveny, T.M. Role of trabecular microarchitecture in whole-vertebral body biomechanical behavior. J. Bone Miner. Res.
**2009**, 24, 1523–1530. [Google Scholar] [CrossRef] - Roux, J.P.; Wegrzyn, J.; Arlot, M.E.; Guyen, O.; Delmas, P.D.; Chapurlat, R.; Bouxsein, M.L. Contribution of trabecular and cortical components to biomechanical behavior of human vertebrae: An ex vivo study. J. Bone Miner. Res.
**2010**, 25, 356–361. [Google Scholar] [CrossRef] - Jaumard, N.V.; Bauman, J.A.; Weisshaar, C.L.; Guarino, B.B.; Welch, W.C.; Winkelstein, B.A. Contact pressure in the facet joint during sagittal bending of the cadaveric cervical spine. J. Biomech. Eng.
**2011**, 133, 071004. [Google Scholar] [CrossRef] - Venables, W.N.; Ripley, B.D. Modern Applied Statistics with S, 4th ed.; Springer: New York, NY, USA, 2002; ISBN 0-387-95457-0. [Google Scholar]
- Mann, K.A.; Miller, M.A. Fluid-structure interactions in micro-interlocked regions of the cement-bone interface. Comput. Methods Biomech. Biomed. Eng.
**2014**, 17, 1809–1820. [Google Scholar] [CrossRef] - Souzanchi, M.F.; Palacio-Mancheno, P.; Borisov, Y.A.; Cardoso, L.; Cowin, S.C. Microarchitecture and bone quality in the human calcaneus: Local variations of fabric anisotropy. J. Bone Min. Res.
**2012**, 27, 2562–2572. [Google Scholar] [CrossRef][Green Version] - Polikeit, A.; Nolte, L.P.; Ferguson, S.J. Simulated influence of osteoporosis and disc degeneration on the load transfer in a lumbar functional spinal unit. J. Biomech.
**2004**, 37, 1061–1069. [Google Scholar] [CrossRef] - Helgason, B.; Perilli, E.; Schileo, E.; Taddei, F.; Brynjólfsson, S.S.; Viceconti, M. Mathematical relationships between bone density and mechanical properties: A literature review. Clin. Biomech.
**2008**, 23, 135–146. [Google Scholar] [CrossRef] - Genant, H.K.; Wu, C.Y.; van Kuijk, C.; Nevitt, M.C. Vertebral fracture assessment using a semiquantitative technique. J. Bone Min. Res.
**1993**, 8, 1137–1148. [Google Scholar] [CrossRef] - Lakes, R.S.; Katz, J.L.; Sternstein, S.S. Viscoelastic properties of wet cortical bone: Part I, torsional and biaxial studies. J. Biomech.
**1979**, 12, 657–678. [Google Scholar] [CrossRef] - Lakes, R.S.; Katz, J.L. Viscoelastic properties of wet cortical bone: Part II, relaxation mechanisms. J. Biomech.
**1979**, 12, 679–687. [Google Scholar] [CrossRef] - Lakes, R.S.; Katz, J.L. Viscoelastic properties of wet cortical bone: Part III, A non-linear constitutive equation. J. Biomech.
**1979**, 12, 689–698. [Google Scholar] [CrossRef] - Burczinski, T. Multiscale Modelling of Osseous Tissues. J. Theor. Appl. Mech.
**2010**, 48, 855–870. [Google Scholar] - Wolf, J. Das Gesetz der Transformation der Knochen; Hirschwald: Berlin, Germany, 1892. [Google Scholar]
- Goldenblar, I.I.; Kopnov, A. Strength of Glass Reinforced Plastics in the Complex Stress State. Polym. Mech.
**1966**, 1, 54–60. [Google Scholar] [CrossRef] - von Mises, R. Mechanik der festen Körper im plastisch deformablen Zustand Göttin. Nachr. Math. Phys.
**1913**, 1, 582–592. [Google Scholar] - Hill, R. The Mathematical Theory of Plasticity; Oxford, U.P.: Oxford, UK, 1950. [Google Scholar]
- Tsai, S.W. Strength Theories of Filamentary Structures. In Fundamental Aspects of Fibre Reinforced Plastic Composites; Schwartz, R.T., Schwartz, H.S., Eds.; Interscience: New York, NY, USA, Chapter 1; 1968. [Google Scholar]
- Korenczuk, C.E.; Votava, L.E.; Dhume, R.Y.; Kizilski, S.B.; Brown, G.E.; Narain, R.; Barocas, V.H. Isotropic Failure Criteria Are Not Appropriate for Anisotropic Fibrous Biological Tissues. J. Biomech. Eng.
**2017**, 139, 071008. [Google Scholar] [CrossRef] - Wilcox, B.; Mobbs, R.J.; Wu, A.M.; Phan, K. Systematic review of 3D printing in spinal surgery: The current state of play. J. Spine Surg.
**2017**, 3, 433–443. [Google Scholar] [CrossRef]

**Figure 8.**Realizations of the probability density functions of the random variables ${X}_{\beta}$ (

**a**,

**d**); r.v. ${X}_{\delta}$ (

**b**,

**e**); and r.v. ${Y}_{p,R}$ (

**c**,

**f**). Density function realizations of initial independent variables

**Figure 9.**Realizations of the probability density functions of the random variables ${X}_{p,E}$ and ${Y}_{p,R}$ (

**a**) for Case I and (

**c**) for Case II; and empirical cumulative distribution functions ${\widehat{F}}_{Z,i}\left(z\right)$ of r.v. $Z={Y}_{p,E}-{X}_{p,E}$ and failure risks $P{r}_{f,i}$, $i\in \{1,2,3\}$ for Case I (

**b**) and for Case II (

**d**): ${\widehat{F}}_{Z,1}\left(z\right)$ and $p{r}_{f,1}$ when $M\left({G}_{p,E}\right)={p}_{min}$; ${\widehat{F}}_{Z,2}\left(z\right)$ and $p{r}_{f,2}$ when $M\left({G}_{p,E}\right)=1.5{p}_{min}$; ${\widehat{F}}_{Z,3}\left(z\right)$ and $p{r}_{f,3}$ when $M\left({G}_{p,E}\right)=2.5{p}_{min}$, where correlation coefficient $Cvar\left({G}_{p,E}\right)=0.2$ for all $P{r}_{f,i},{\widehat{F}}_{Z,i}\left(z\right),i\in \{1,2,3\}$.

**Figure 10.**Cross-sectional view of the 3-D finite element model of the vertebra in the sagittal plane (

**a**); bonded and unbonded, with a gap and cortical shells (

**b**,

**c**) respectively.

**Figure 11.**Deformed shape effect at the first bifurcation point A. Figure a–c for the bonded trabecular bone (Figure 10b): (

**a**) when $\delta =0.5$, $t=0.86{t}_{max}$, (

**b**) when $\delta =0.4$ mm and $t=0.85{t}_{max}$, (

**c**) when $\delta =0.2$ mm and $t=0.58{t}_{max}$. Figure d–f for the unbounded trabecular bone (Figure 10c): (

**d**) when $\delta =0.5$ mm and $t=0.32{t}_{max}$, (

**e**) when $\delta =0.4$ mm and $t=0.32{t}_{max}$, (

**f**) when $\delta =0.2$ mm and $t=0.38{t}_{max}$.

**Figure 12.**Load dependency graph. Displacement in horizontal direction at point A (Figure 11).

Property | Lumbar Body | Intervertebra Disc |
---|---|---|

Young modulus, MPa | 8000 | 10 |

Ulimate strength, MPa | 60 | - |

Yield strength, MPa | 40 | - |

Poisson’s ratio | 0.30 | 0.40 |

Model | Thickness, mm | BV/TV | TBS |
---|---|---|---|

Healthy | 0.5 | 0.35 | 1.45 |

Ostseopenea | 0.4 | 0.20 | 1.33 |

Osteoporosis | 0.2 | 0.10 | 1.20 |

Cortical Shell Thickness, mm | BV/TV | TBS |
---|---|---|

0.20 | 0.10 | 1.28 |

0.40 | 0.10 | 1.28 |

0.50 | 0.10 | 1.28 |

0.20 | 0.20 | 1.33 |

0.40 | 0.20 | 1.33 |

0.50 | 0.20 | 1.33 |

0.20 | 0.35 | 1.45 |

0.40 | 0.35 | 1.45 |

0.50 | 0.35 | 1.45 |

**Table 4.**The maximum von Mises stresses of the static analysis calculated by finite element method (FEM) at different $\delta $, $\beta $, and p

Analysis | Thickness, mm, $\mathit{\delta}$ | BV/TV, $\mathit{\beta}$ | External Load, MPa, p | ||||
---|---|---|---|---|---|---|---|

0.15 | 0.30 | 0.45 | 0.60 | 0.75 | |||

Static | 0.5 | 0.10 | 2.9 | 6.1 | 9.3 | 12.3 | 15.2 |

0.20 | 2.0 | 4.1 | 6.1 | 8.2 | 8.5 | ||

0.35 | 1.6 | 3.4 | 5.0 | 6.7 | 8.5 | ||

0.4 | 0.10 | 3.4 | 6.7 | 10.2 | 14.1 | 17.0 | |

0.20 | 2.1 | 4.2 | 6.3 | 8.4 | 10.5 | ||

0.35 | 1.8 | 3.5 | 5.4 | 7.3 | 9.1 | ||

0.2 | 0.10 | 18.9 | 24.1 | 31.3 | 40.8 | 49.4 | |

0.20 | 13.3 | 18.4 | 23.2 | 28.1 | 33.5 | ||

0.35 | 12.1 | 17.4 | 21.0 | 24.5 | 28.0 |

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Maknickas, A.; Alekna, V.; Ardatov, O.; Chabarova, O.; Zabulionis, D.; Tamulaitienė, M.; Kačianauskas, R. FEM-Based Compression Fracture Risk Assessment in Osteoporotic Lumbar Vertebra L1. *Appl. Sci.* **2019**, *9*, 3013.
https://doi.org/10.3390/app9153013

**AMA Style**

Maknickas A, Alekna V, Ardatov O, Chabarova O, Zabulionis D, Tamulaitienė M, Kačianauskas R. FEM-Based Compression Fracture Risk Assessment in Osteoporotic Lumbar Vertebra L1. *Applied Sciences*. 2019; 9(15):3013.
https://doi.org/10.3390/app9153013

**Chicago/Turabian Style**

Maknickas, Algirdas, Vidmantas Alekna, Oleg Ardatov, Olga Chabarova, Darius Zabulionis, Marija Tamulaitienė, and Rimantas Kačianauskas. 2019. "FEM-Based Compression Fracture Risk Assessment in Osteoporotic Lumbar Vertebra L1" *Applied Sciences* 9, no. 15: 3013.
https://doi.org/10.3390/app9153013