Abstract
This study proposes a numerical approach for simulating bone remodeling in lumbar interbody fusion (LIF). It employs a topology optimization method to drive the remodeling process and uses a pixel function to describe the structural topology and bone density distribution. Unlike traditional approaches based on strain energy density or compliance, this study adopts von Mises stress to guide the remodeling of LIF. A novel pixel interpolation scheme associated with stress criteria is applied to the physical properties of the bone, directly addressing the stress shielding effect caused by the implanted cage, which significantly influences the bone remodeling outcome in LIF. Additionally, a boundary inverse approach is utilized to reconstruct a simplified analysis model. To reduce computational cost while maintaining high structural resolution and accuracy, the scaled boundary finite element method (SBFEM) is introduced. The proposed numerical approach successfully generates results that closely resemble human lumbar interbody fusion.
References
1.
Mobbs
, R. J.
,
Phan
, K.
,
Malham
, G.
,
Seex
, K.
, and
Rao
, P. J.
, 2015
, “Lumbar Interbody Fusion: Techniques, Indications and Comparison of Interbody Fusion Options Including PLIF, TLIF, MI-TLIF, OLIF/ATP, LLIF and ALIF
,” J. Spine Surg.
, 3
(1
), pp. 2
–18
.10.3978/j.issn.2414-469X.2015.10.052.
Kotani
, Y.
,
Abumi
, K.
,
Ito
, M.
,
Sudo
, H.
,
Abe
, Y.
, and
Minami
, A.
, 2012
, “Mid-Term Clinical Results of Minimally Invasive Decompression and Posterolateral Fusion With Percutaneous Pedicle Screws Versus Conventional Approach for Degenerative Spondylolisthesis With Spinal Stenosis
,” Eur. Spine J.
, 21
(6
), pp. 1171
–1177
.10.1007/s00586-011-2114-x3.
Gagliardi
, M. J.
,
Guiroy
, A. J.
,
Camino-Willhuber
, G.
,
Joaquim
, A. F.
,
Carazzo
, C. A.
,
Yasuda
, E.
,
Cabrera
, J. P.
, and
Morales Ciancio
, A. R.
, 2022
, “Is Indirect Decompression and Fusion More Effective Than Direct Decompression and Fusion for Treating Degenerative Lumbar Spinal Stenosis With Instability? A Systematic Review and meta-Analysis
,” Global Spine J.
, 13
(2
), pp. 1
–13
.10.1177/219256822210983624.
Kaiser
, M. G.
,
Eck
, J. C.
,
Groff
, M. W.
,
Ghogawala
, Z.
,
Watters
, W. C.
,
Dailey
, A. T.
,
Resnick
, D. K.
, et al., 2014
, “Guideline Update for the Performance of Fusion Procedures for Degenerative Disease of the Lumbar Spine. Part 17: Bone Growth Stimulators as an Adjunct for Lumbar Fusion
,” J. Neurosurg.,
21
(1
), pp. 133–
139
.10.3171/2014.4.SPINE143265.
Prendergast
, P. J.
, 1997
, “Finite Element Models in Tissue Mechanics and Orthopaedic Implant Design
,” Clin. Biomech.
, 12
(6
), pp. 343
–366
.10.1016/S0268-0033(97)00018-16.
Garcı́a
, J. M.
,
Doblaré
, M.
, and
Cegoñino
, J.
, 2002
, “Bone Remodelling Simulation: A Tool for Implant Design
,” Comput. Mater. Sci.
25
(1–2
), pp. 100
–114
.10.1016/S0927-0256(02)00254-97.
Erdemir
, A.
,
Guess
, T. M.
,
Halloran
, J.
,
Tadepalli
, S. C.
, and
Morrison
, T. M.
, 2012
, “Considerations for Reporting Finite Element Analysis Studies in Biomechanics
,” J. Biomech.
, 45
(4
), pp. 625
–633
.10.1016/j.jbiomech.2011.11.0388.
Taylor
, M.
, and
Prendergast
, P. J.
, 2015
, “Four Decades of Finite Element Analysis of Orthopaedic Devices: Where Are we Now and What Are the Opportunities?
,” J Biomech.
, 48
(5
), pp. 767
–778
.10.1016/j.jbiomech.2014.12.0199.
Levadnyi
, I.
,
Awrejcewicz
, J.
,
Gubaua
, J. E.
, and
Pereira
, J. T.
, 2017
, “Numerical Evaluation of Bone Remodelling and Adaptation Considering Different Hip Prosthesis Designs
,” Clin. Biomech.
, 50
, pp. 122
–129
.10.1016/j.clinbiomech.2017.10.01510.
Ambard
, D.
, and
Swider
, P.
, 2006
, “A Predictive Mechano-Biological Model of the Bone-Implant Healing
,” Eur. J. Mech.-A/Solids
, 25
(6
), pp. 927
–937
.10.1016/j.euromechsol.2006.02.00611.
Dicati
, G. W. O.
,
Gubaua
, J. E.
, and
Pereira
, J. T.
, 2022
, “Optimum Parameters for Each Subject in Bone Remodeling Models: A New Methodology Using Surrogate and Clinical Data
,” Eur. J. Mech.-A/Solids
, 91
, p. 104409
.10.1016/j.euromechsol.2021.10440912.
Christen
, P.
,
Ito
, K.
,
Ellouz
, R.
,
Boutroy
, S.
,
Sornay-Rendu
, E.
,
Chapurlat
, R. D.
, and
Van Rietbergen
, B.
, 2014
, “Bone Remodelling in Humans is Load-Driven but Not Lazy
,” Nat Commun.
, 5
(1
), p. 4855
.10.1038/ncomms585513.
Frost
, H. M.
, 1987
, “Bone “Mass” and the “Mechanostat”: A Proposal
,” Anat. Rec.
, 219
(1
), pp. 1
–9
.10.1002/ar.109219010414.
Milne
, T. J.
,
Ichim
, I.
,
Patel
, B.
,
McNaughton
, A.
, and
Meikle
, M. C.
, 2009
, “Induction of Osteopenia During Experimental Tooth Movement in the Rat: Alveolar Bone Remodelling and the Mechanostat Theory
,” Eur. J. Orthod.
, 31
(3
), pp. 221
–231
.10.1093/ejo/cjp03215.
Hollister
, S. J.
, and
Kikuchi
, N.
, 1994
, “Homogenization Theory and Digital Imaging: A Basis for Studying the Mechanics and Design Principles of Bone Tissue
,” Biotechnol. Bioeng.
, 43
(7
), pp. 586
–596
.10.1002/bit.26043070816.
Bagge
, M.
, 2000
, “A Model of Bone Adaptation as an Optimization Process
,” J. Biomech.
, 33
(11
), pp. 1349
–1357
.10.1016/S0021-9290(00)00124-X17.
Kowalczyk
, P.
, 2010
, “Simulation of Orthotropic Microstructure Remodelling of Cancellous Bone
,” J. Biomech.
, 43
(3
), pp. 563
–569
.10.1016/j.jbiomech.2009.09.04518.
Boyle
, C.
, and
Kim
, I. Y.
, 2011
, “Three-Dimensional Micro-Level Computational Study of Wolff's Law Via Trabecular Bone Remodeling in the Human Proximal Femur Using Design Space Topology Optimization
,” J. Biomech.
, 44
(5
), pp. 935
–942
.10.1016/j.jbiomech.2010.11.02919.
Donaldson
, F. E.
,
Pankaj
, P.
,
Cooper
, D. M. L.
,
Thomas
, C. D. L.
,
Clement
, J. G.
, and
Simpson
, A. H. R. W.
, 2011
, “Relating Age and Micro-Architecture With Apparent-Level Elastic Constants: A Micro-Finite Element Study of Female Cortical Bone From the Anterior Femoral Midshaft
,” Proc. Inst. Mech. Eng., Part H
, 225
(6
), pp. 585
–596
.10.1177/204130331039567520.
Colabella
, L.
,
Cisilino
, A. P.
,
Fachinotti
, V.
, and
Kowalczyk
, P.
, 2019
, “Multiscale Design of Elastic Solids With Biomimetic Cancellous Bone Cellular Microstructures
,” Struct. Multidiscip. Optim.
, 60
(2
), pp. 639
–661
.10.1007/s00158-019-02229-321.
Bachmann
, S.
,
Pahr
, D.
, and
Synek
, A.
, 2023
, “Hip Joint Load Prediction Using Inverse Bone Remodeling With Homogenized FE Models: Comparison to micro-FE and Influence of Material Modeling Strategy
,” Comput. Methods Prog. Biomed.
, 236
, p. 107549
.10.1016/j.cmpb.2023.10754922.
Mathai
, B.
,
Dhara
, S.
, and
Gupta
, S.
, 2022
, “Bone Remodelling in Implanted Proximal Femur Using Topology Optimization and Parameterized Cellular Model
,” J. Mech. Behav. Biomed. Mater.
, 125
, p. 104903
.10.1016/j.jmbbm.2021.10490323.
Dagan
, D.
,
Be'ery
, M.
, and
Gefen
, A.
, 2004
, “Single-Trabecula Building Block for Large-Scale Finite Element Models of Cancellous Bone
,” Med Biol Eng Comput
, 42
(4
), pp. 549
–556
.10.1007/BF0235099824.
Mukherjee
, K.
, and
Gupta
, S.
, 2016
, “Bone Ingrowth Around Porous-Coated Acetabular Implant: A Three-Dimensional Finite Element Study Using Mechanoregulatory Algorithm
,” Biomech. Model. Mechanobiol.
, 15
(2
), pp. 389
–403
.10.1007/s10237-015-0696-725.
Park
, S. W.
,
Choi
, J. H.
, and
Lee
, B. C.
, 2018
, “Multi-Objective Optimization of an Automotive Body Component With Fiber-Reinforced Composites
,” Struct. Multidiscip. Optim.
, 58
(5
), pp. 2203
–2217
.10.1007/s00158-018-2008-126.
Goda
, I.
,
Ganghoffer
, J. F.
,
Czarnecki
, S.
,
Czubacki
, R.
, and
Wawruch
, P.
, 2019
, “Topology Optimization of Bone Using Cubic Material Design and Evolutionary Methods Based on Internal Remodeling
,” Mech. Res. Commun.
, 95
, pp. 52
–60
.10.1016/j.mechrescom.2018.12.00327.
Jang
, I. G.
,
Kim
, I. Y.
, and
Kwak
, B. B.
, 2009
, “Analogy of Strain Energy Density Based Bone-Remodeling Algorithm and Structural Topology Optimization
,” ASME J. Biomech. Eng.
, 131
(1
), p. 011012
.10.1115/1.300520228.
Harrigan
, T. P.
, and
Hamilton
, J. J.
, 1992
, “An Analytical and Numerical Study of the Stability of Bone Remodelling Theories: Dependence on Microstructural Stimulus
,” J. Biomech.
, 25
(5
), pp. 477
–488
.10.1016/0021-9290(92)90088-I29.
Beaupré
, G. S.
,
Orr
, T. E.
, and
Carter
, D. R.
, 1990
, “An Approach for Time‐Dependent Bone Modeling and Remodeling—Theoretical Development
,” J. Orthop. Res.
, 8
(5
), pp. 651
–661
.10.1002/jor.110008050630.
Tanaka
, M.
, and
Adachi
, T.
, 1994
, “Preliminary Study on Mechanical Bone Remodeling Permitting Residual Stress
,” JSME Int. J. Ser. A, Mech. Mater. Eng.
, 37
(1
), pp. 87
–95
.10.1299/jsmea1993.37.1_8731.
Tsubota
, K. I.
,
Adachi
, T.
, and
Tomita
, Y.
, 2002
, “Functional Adaptation of Cancellous Bone in Human Proximal Femur Predicted by Trabecular Surface Remodeling Simulation Toward Uniform Stress State
,” J. Biomech.
, 35
(12
), pp. 1541
–1551
.10.1016/S0021-9290(02)00173-232.
Boyle
, C.
, and
Kim
, I. Y.
, 2011
, “Comparison of Different Hip Prosthesis Shapes Considering Micro-Level Bone Remodeling and Stress-Shielding Criteria Using Three-Dimensional Design Space Topology Optimization
,” J. Biomech.
, 44
(9
), pp. 1722
–1728
.10.1016/j.jbiomech.2011.03.03833.
Epari
, D. R.
,
Kandziora
, F.
, and
Duda
, G. N.
, 2005
, “Stress Shielding in Box and Cylinder Cervical Interbody Fusion Cage Designs
,” Spine
, 30
(8
), pp. 908
–914
.10.1097/01.brs.0000158971.74152.b634.
Palm
, W. J.
, IV,
Rosenberg
, W. S.
, and
Keaveny
, T. M.
, 2002
, “Load Transfer Mechanisms in Cylindrical Interbody Cage Constructs
,” Spine
, 27
(19
), pp. 2101
–2107
.10.1097/00007632-200210010-0000535.
Kanayama
, M.
,
Cunningham
, B. W.
,
Haggerty
, C. J.
,
Abumi
, K.
,
Kaneda
, K.
, and
McAfee
, P. C.
, 2000
, “In Vitro Biomechanical Investigation of the Stability and Stress-Shielding Effect of Lumbar Interbody Fusion Devices
,” J. Neurosurg.
, 93
(2
), pp. 259
–265
.10.3171/spi.2000.93.2.025936.
Chuah
, H. G.
,
Rahim
, I. A.
, and
Yusof
, M. I.
, 2010
, “Topology Optimisation of Spinal Interbody Cage for Reducing Stress Shielding Effect
,” Comput. Methods Biomech. Biomed. Eng.
, 13
(3
), pp. 319
–326
.10.1080/1025584090320818937.
Frost
, H. M.
, 2000
, “The Utah Paradigm of Skeletal Physiology: An Overview of Its Insights for Bone, Cartilage and Collagenous Tissue Organs
,” J. Bone Miner. Metab.
, 18
(6
), pp. 305
–316
.10.1007/s00774007000138.
Wang
, L.
,
You
, X.
,
Zhang
, L.
,
Zhang
, C.
, and
Zou
, W.
, 2022
, “Mechanical Regulation of Bone Remodeling
,” Bone Res.
, 10
(1
), p. 16
.10.1038/s41413-022-00190-439.
Guo
, L. X.
,
Wang
, Z. W.
,
Zhang
, Y. M.
,
Lee
, K. K.
,
Teo
, E. C.
,
Li
, H.
, and
Wen
, B. C.
, 2009
, “Material Property Sensitivity Analysis on Resonant Frequency Characteristics of the Human Spine
,” J. Appl. Biomech.
, 25
(1
), pp. 64
–72
.10.1123/jab.25.1.6440.
Frost
, H. M.
, 1994
, “Wolff's Law and Bone's Structural Adaptations to Mechanical Usage: An Overview for Clinicians
,” Angle Orthod.
, 64
(3
), pp. 175
–188
.10.1043/0003-3219(1994)064<0175:WLABSA>2.0.CO;241.
Frost
, H. M.
, 2004
, “A 2003 Update of Bone Physiology and Wolff's Law for Clinicians
,” Angle Orthod.
, 74
(1
), pp. 3
–15
.10.1043/0003-3219(2004)074<0003:AUOBPA>2.0.CO;242.
Zhang
, C.
,
Zeng
, C.
,
Wang
, Z.
,
Zeng
, T.
, and
Wang
, Y.
, 2023
, “Optimization of Stress Distribution of Bone-Implant Interface (BII)
,” Biomater. Adv.
, 147
, p. 213342
.10.1016/j.bioadv.2023.21334243.
Öhman-Mägi
, C.
,
Holub
, O.
,
Wu
, D.
,
Hall
, R. M.
, and
Persson
, C.
, 2021
, “Density and Mechanical Properties of Vertebral Trabecular bone-A Review
,” JOR Spine
, 4
(4
), p. e1176
.10.1002/jsp2.117644.
Aiyangar
, A. K.
,
Vivanco
, J.
,
Au
, A. G.
,
Anderson
, P. A.
,
Smith
, E. L.
, and
Ploeg
, H. L.
, 2014
, “Dependence of Anisotropy of Human Lumbar Vertebral Trabecular Bone on Quantitative Computed Tomography-Based Apparent Density
,” ASME J. Biomech. Eng.
, 136
(9
), p. 091003
.10.1115/1.402766345.
Song
, C.
, and
Wolf
, J. P.
, 1997
, “The Scaled Boundary Finite-Element method - Alias Consistent Infinitesimal Finite-Element Cell method - For Elastodynamics
,” Comput. Methods Appl. Mech. Eng.
, 147
(3–4
), pp. 329
–355
.10.1016/S0045-7825(97)00021-246.
Song
, C.
, and
Wolf
, J. P.
, 2000
, “Scaled Boundary Finite-Element Method - A Primer: Solution Procedures
,” Comput. Struct.
, 78
(1–3
), pp. 211
–225
.10.1016/S0045-7949(00)00100-047.
Song
, C.
, and
Wolf
, J. P.
, 2002
, “Semi-Analytical Representation of Stress Singularities as Occurring in Cracks in Anisotropic Multi-Materials With the Scaled Boundary Finite-Element Method
,” Comput. Struct.
, 80
(2
), pp. 183
–197
.10.1016/S0045-7949(01)00167-548.
Wang
, C.
,
Han
, J.
,
Li
, Q.
,
Wang
, L.
, and
Fan
, Y.
, 2014
, “Simulation of Bone Remodelling in Orthodontic Treatment
,” Comput. Methods Biomech. Biomed. Eng.
, 17
(9
), pp. 1042
–1050
.10.1080/10255842.2012.73696949.
Fang
, L.
,
Wang
, Z.
,
Chen
, Z.
,
Jian
, F.
,
Li
, S.
, and
He
, H.
, 2020
, “3D Shape Reconstruction of Lumbar Vertebra From Two X-Ray Images and a CT Model
,” IEEE/CAA J. Autom. Sin.
, 7
(4
), pp. 1124
–1133
.10.1109/JAS.2019.191152850.
Holick
, M. F.
, 2004
, “Vitamin D: Importance in the Prevention of Cancers, Type 1 Diabetes, Heart Disease, and Osteoporosis
,” Am. J. Clin. Nutr.
, 79
(3
), pp. 362
–371
.10.1093/ajcn/79.3.36251.
Hartikka
, H.
,
Mäkitie
, O.
,
Männikkö
, M.
,
Doria
, A. S.
,
Daneman
, A.
,
Cole
, W. G.
,
Ala-Kokko
, L.
, and
Sochett
, E. B.
, 2005
, “Heterozygous Mutations in the LDL Receptor-Related Protein 5 (LRP5) Gene Are Associated With Primary Osteoporosis in Children
,” J. Bone Miner. Res.
, 20
(5
), pp. 783
–789
.10.1359/JBMR.05010152.
Park
, S.
,
Park
, J.
,
Kang
, I.
,
Lee
, H.
, and
Noh
, G.
, 2022
, “Effects of Assessing the Bone Remodeling Process in Biomechanical Finite Element Stability Evaluations of Dental Implants
,” Comput. Methods Prog. Biomed.
, 221
, p. 106852
.10.1016/j.cmpb.2022.10685253.
Chun
, B. J.
, and
Jang
, I. G.
, 2021
, “Determination of the Representative Static Loads for Cyclically Repeated Dynamic Loads: A Case Study of Bone Remodeling Simulation With Gait Loads
,” Comput. Methods Prog. Biomed.
, 200
, p. 105924
.10.1016/j.cmpb.2020.105924
