Skip to main content
SpringerLink
Log in

Beam theory for rapid strain estimation in the mouse tibia compression model

  • Original Paper
  • Published:
Biomechanics and Modeling in Mechanobiology Aims and scope Submit manuscript

This article has been updated

Abstract

The mouse tibia compression model is a leading model for studying bone’s mechanoadaptive response to load. In studying this mechanoadaptive response, (FE) modelling is often used to determine the stress/strain within the tibia. The development of such models can be challenging and computationally expensive. An alternate approach is to use continuum mechanics based analytical theories, such as beam theory (BT). However, applying BT to the mouse tibia requires the fibula be neglected, introducing error in the stress/strain distribution. While several studies have applied BT to the mouse tibia, no study has explored the accuracy of this approach. To address these questions, this work investigates the use of BT in determining stress/strain within the mouse tibia. By comparing BT against FE modelling, it was found that BT can accurately predict tibial stress/strain if correction factors are applied to account for the effect of the fibula. The 25, 37, 50 and 75% cross sections are studied. Focusing on the 37% cross section, without correction, BT can have errors of approximately 21.6%. With correction, this is reduced to 6.6%. Such correction factors are presented. The developed BT model is applicable in the diaphysis and distal metaphysis, where the assumptions of BT are valid. This work verifies BT for determining localised strains in a mouse tibia compression model. This is anticipated to provide efficiency dividends, allowing for high throughput modelling of the mouse tibia, advancing study of bone’s mechanoadaptive response.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Change history

  • 21 January 2022

    The Electronic Supplementary Materials were missed in the original publication. They have been included now

References

  • Adebayo OO, Ko FC, Goldring SR, Goldring MB, Wright TM, van der Meulen MC (2017) Kinematics of meniscal- and acl-transected mouse knees during controlled tibial compressive loading captured using roentgen stereophotogrammetry. J Orthopaedic Res 35(2):353–360. https://doi.org/10.1002/jor.23285

    Article  Google Scholar 

  • Albiol L, Cilla M, Pflanz D, Kramer I, Kneissel M, Duda GN, Willie BM, Checa S (2018) Sost deficiency leads to reduced mechanical strains at the tibia midshaft in strain-matched in vivo loading experiments in mice. J R Soc Interface 15(141):20180012. https://doi.org/10.1098/rsif.2018.0012

    Article  Google Scholar 

  • Bab I, Hajbi-Yonissi C, Gabet Y, Müller R (2007) Tibio-fibular complex and knee joint, Springer US, Boston, MA, pp 171–181. https://doi.org/10.1007/978-0-387-39258-5\_16

  • Berman AG, Clauser CA, Wunderlin C, Hammond MA, Wallace JM (2015) Structural and mechanical improvements to bone are strain dependent with axial compression of the tibia in female c57bl/6 mice. PLOS ONE 10(6):1–16. https://doi.org/10.1371/journal.pone.0130504

    Article  Google Scholar 

  • Brassey CA, Margetts L, Kitchener AC, Withers PJ, Manning PL, Sellers WI (2013) Finite element modelling versus classic beam theory: comparing methods for stress estimation in a morphologically diverse sample of vertebrate long bones. J R Soc Interface 10(79):20120823. https://doi.org/10.1098/rsif.2012.0823

    Article  Google Scholar 

  • Carriero A, Pereira A, Wilson A, Castagno S, Javaheri B, Pitsillides A, Marenzana M, Shefelbine S (2018) Spatial relationship between bone formation and mechanical stimulus within cortical bone: combining 3d fluorochrome mapping and poroelastic finite element modelling. Bone Rep 8:72–80. https://doi.org/10.1016/j.bonr.2018.02.003

    Article  Google Scholar 

  • Charles JP, Cappellari O, Spence AJ, Hutchinson JR, Wells DJ (2016) Musculoskeletal geometry, muscle architecture and functional specialisations of the mouse hindlimb. PLOS ONE 11(4):1–21. https://doi.org/10.1371/journal.pone.0147669

    Article  Google Scholar 

  • Cheong VS, Campos Marin A, Lacroix D, Dall‘Ara E (2020) A novel algorithm to predict bone changes in the mouse tibia properties under physiological conditions. Biomech Model Mechanobiol 19(3):985–1001. https://doi.org/10.1007/s10237-019-01266-7

    Article  Google Scholar 

  • Cheong VS, Roberts BC, Kadirkamanathan V, Dall‘Ara E (2020) Bone remodelling in the mouse tibia is spatio-temporally modulated by oestrogen deficiency and external mechanical loading: A combined in vivo/in silico study. Acta Biomaterialia 116:302–317. https://doi.org/10.1016/j.actbio.2020.09.011

    Article  Google Scholar 

  • Christiansen B, Guilak F, Lockwood K, Olson S, Pitsillides A, Sandell L, Silva M, [van der Meulen] M, Haudenschild D, (2015) Non-invasive mouse models of post-traumatic osteoarthritis. Osteoarthrit Cartilage 23(10):1627–1638. https://doi.org/10.1016/j.joca.2015.05.009

  • Christiansen BA, Bayly PV, Silva MJ (2008) Constrained Tibial vibration in mice: a method for studying the effects of vibrational loading of bone. J Biomech Eng 130(4):44502. https://doi.org/10.1115/1.2917435

    Article  Google Scholar 

  • Das Neves Borges P, Forte AE, Vincent TL, Dini D, Marenzana M (2014) Rapid, automated imaging of mouse articular cartilage by microCT for early detection of osteoarthritis and finite element modelling of joint mechanics. Osteoarthrit Cartilage 22(10):1419–1428. https://doi.org/10.1016/j.joca.2014.07.014

    Article  Google Scholar 

  • De Souza RL, Matsuura M, Eckstein F, Rawlinson SC, Lanyon LE, Pitsillides AA (2005) Non-invasive axial loading of mouse tibiae increases cortical bone formation and modifies trabecular organization: A new model to study cortical and cancellous compartments in a single loaded element. Bone 37(6):810–818. https://doi.org/10.1016/j.bone.2005.07.022

    Article  Google Scholar 

  • Dodge T, Wanis M, Ayoub R, Zhao L, Watts NB, Bhattacharya A, Akkus O, Robling A, Yokota H (2012) Mechanical loading, damping, and load-driven bone formation in mouse tibiae. Bone 51(4):810–818. https://doi.org/10.1016/j.bone.2012.07.021

    Article  Google Scholar 

  • Fan L, Pei S, Lucas Lu X, Wang L (2016) A multiscale 3D finite element analysis of fluid/solute transport in mechanically loaded bone. Bone Res 4(May), https://doi.org/10.1038/boneres.2016.32

  • Frost HM (2003) Bone‘s mechanostat: a 2003 update. Anatom Record Part A Dis Mol Cellul Evol Biol 275A(2):1081–1101. https://doi.org/10.1002/ar.a.10119

    Article  Google Scholar 

  • Gardegaront M, Allard V, Confavreux C, Bermond F, Mitton D, Follet H (2021) Variabilities in \(\mu\)qct-based fea of a tumoral bone mice model. J Biomech 118:110265. https://doi.org/10.1016/j.jbiomech.2021.110265

    Article  Google Scholar 

  • Giorgi M, Dall‘Ara E (2018) Variability in strain distribution in the mice tibia loading model: a preliminary study using digital volume correlation. Med Eng Phys 62:7–16. https://doi.org/10.1016/j.medengphy.2018.09.001

    Article  Google Scholar 

  • Imhauser C, Mauro C, Choi D, Rosenberg E, Mathew S, Nguyen J, Ma Y, Wickiewicz T (2013) Abnormal tibiofemoral contact stress and its association with altered kinematics after center-center anterior cruciate ligament reconstruction: An in vitro study. Am J Sports Med 41(4):815–825. https://doi.org/10.1177/0363546512475205

    Article  Google Scholar 

  • Javaheri B, Razi H, Gohin S, Wylie S, Chang YM, Salmon P, Lee PD, Pitsillides AA (2020) Lasting organ-level bone mechanoadaptation is unrelated to local strain. Sci Adv 6(10), https://doi.org/10.1126/sciadv.aax8301

  • Lockwood KA, Chu BT, Anderson MJ, Haudenschild DR, Christiansen BA (2014) Comparison of loading rate-dependent injury modes in a murine model of post-traumatic osteoarthritis. J Orthopaed Res 32(1):79–88. https://doi.org/10.1002/jor.22480

    Article  Google Scholar 

  • Lu Y, Boudiffa M, Dall‘Ara E, Liu Y, Bellantuono I, Viceconti M (2017) Longitudinal effects of parathyroid hormone treatment on morphological, densitometric and mechanical properties of mouse tibia. J Mech Behavi Biomed Mater 75:244–251. https://doi.org/10.1016/j.jmbbm.2017.07.034

    Article  Google Scholar 

  • Meakin LB, Delisser PJ, Galea GL, Lanyon LE, Price JS (2015) Disuse rescues the age-impaired adaptive response to external loading in mice. Osteop Int 26(11):2703–2708. https://doi.org/10.1007/s00198-015-3142-x

    Article  Google Scholar 

  • Miller CJ, Trichilo S, Pickering E, Martelli S, Delisser P, Meakin LB, Pivonka P (2021) Cortical thickness adaptive response to mechanical loading depends on periosteal position and varies linearly with loading magnitude. Front Bioeng Biotechnol 9:504

  • Norman SC, Wagner DW, Beaupre GS, Castillo AB (2015) Comparison of three methods of calculating strain in the mouse ulna in exogenous loading studies. J Biomech 48(1):53–58. https://doi.org/10.1016/j.jbiomech.2014.11.004

    Article  Google Scholar 

  • Oliviero S, Lu Y, Viceconti M, Dall‘Ara E (2017) Effect of integration time on the morphometric, densitometric and mechanical properties of the mouse tibia. J Biomech 65:203–211. https://doi.org/10.1016/j.jbiomech.2017.10.026

    Article  Google Scholar 

  • Oliviero S, Giorgi M, Dall‘Ara E (2018) Validation of finite element models of the mouse tibia using digital volume correlation. J Mech Behav Biomed Mater 86:172–184. https://doi.org/10.1016/j.jmbbm.2018.06.022

    Article  Google Scholar 

  • Oliviero S, Roberts M, Owen R, Reilly GC, Bellantuono I, Dall‘Ara E (2021) Non-invasive prediction of the mouse tibia mechanical properties from microCT images: comparison between different finite element models. Biomech Model Mechanobiol 20(3):941–955. https://doi.org/10.1007/s10237-021-01422-y

    Article  Google Scholar 

  • Patel TK, Brodt MD, Silva MJ (2014) Experimental and finite element analysis of strains induced by axial tibial compression in young-adult and old female c57bl/6 mice. J Biomech 47(2):451–457. https://doi.org/10.1016/j.jbiomech.2013.10.052

    Article  Google Scholar 

  • Pereira AF, Javaheri B, Pitsillides AA, Shefelbine SJ (2015) Predicting cortical bone adaptation to axial loading in the mouse tibia. J R Soc Interface 12(110):20150590. https://doi.org/10.1098/rsif.2015.0590

    Article  Google Scholar 

  • Pickering E, Silva MJ, Delisser P, Brodt MD, Gu Y, Pivonka P (2021) Estimation of load conditions and strain distribution for in vivo murine tibia compression loading using experimentally informed finite element models. J Biomech 115:110410. https://doi.org/10.1016/j.jbiomech.2020.110140

    Article  Google Scholar 

  • Poulet B, Hamilton RW, Shefelbine S, Pitsillides AA (2011) Characterizing a novel and adjustable noninvasive murine joint loading model. Arthrit Rheumat 63(1):137–147. https://doi.org/10.1002/art.27765

    Article  Google Scholar 

  • Razi H, Birkhold AI, Zehn M, Duda GN, Willie BM, Checa S (2014) A finite element model of in vivo mouse tibial compression loading: influence on boundary conditions. Facta Univ Ser Mech Eng 12(3):195–207

    Google Scholar 

  • Razi H, Birkhold AI, Zaslansky P, Weinkamer R, Duda GN, Willie BM, Checa S (2015) Skeletal maturity leads to a reduction in the strain magnitudes induced within the bone: a murine tibia study. Acta Biomater 13:301–310. https://doi.org/10.1016/j.actbio.2014.11.021

    Article  Google Scholar 

  • Saxon LK, Jackson BF, Sugiyama T, Lanyon LE, Price JS (2011) Analysis of multiple bone responses to graded strains above functional levels, and to disuse, in mice in vivo show that the human lrp5 g171v high bone mass mutation increases the osteogenic response to loading but that lack of lrp5 activity reduces it. Bone 49(2):184–193. https://doi.org/10.1016/j.bone.2011.03.683

    Article  Google Scholar 

  • Silva MJ, Reed KL, Robertson DD, Bragdon C, Harris WH, Maloney WJ (1999) Reduced bone stress as predicted by composite beam theory correlates with cortical bone loss following cemented total hip arthroplasty. J Orthopaedic Res 17(4):525–531. https://doi.org/10.1002/jor.1100170410

    Article  Google Scholar 

  • Silva MJ, Brodt MD, Hucker WJ (2005) Finite element analysis of the mouse tibia: estimating endocortical strain during three-point bending in samp6 osteoporotic mice. Anatom Record Part A Discover Mol Cellular Evol Biol 283A(2):380–390. https://doi.org/10.1002/ar.a.20171

    Article  Google Scholar 

  • Somerville JM, Aspden RM, Armour KE, Armour KJ, Reid DM (2004) Growth of c57bl/6 mice and the material and mechanical properties of cortical bone from the tibia. Calcified Tissue Int 74(5):469–475. https://doi.org/10.1007/s00223-003-0101-x

    Article  Google Scholar 

  • Stadelmann VA, Hocké J, Verhelle J, Forster V, Merlini F, Terrier A, Pioletti DP (2009) 3d strain map of axially loaded mouse tibia: a numerical analysis validated by experimental measurements. Comput Methods Biomech Biomed Eng 12(1):95–100. https://doi.org/10.1080/10255840802178053, pMID: 18651261

  • Sugiyama T, Saxon LK, Zaman G, Moustafa A, Sunters A, Price JS, Lanyon LE (2008) Mechanical loading enhances the anabolic effects of intermittent parathyroid hormone (1–34) on trabecular and cortical bone in mice. Bone 43(2):238–248. https://doi.org/10.1016/j.bone.2008.04.012

    Article  Google Scholar 

  • Sugiyama T, Meakin LB, Browne WJ, Galea GL, Price JS, Lanyon LE (2012) Bone‘s adaptive response to mechanical loading is essentially linear between the low strains associated with disuse and the high strains associated with the lamellar/woven bone transition. J Bone Mineral Res 27(8):1784–1793. https://doi.org/10.1002/jbmr.1599

    Article  Google Scholar 

  • Sztefek P, Vanleene M, Olsson R, Collinson R, Pitsillides AA, Shefelbine S (2010) Using digital image correlation to determine bone surface strains during loading and after adaptation of the mouse tibia. J Biomech 43(4):599–605. https://doi.org/10.1016/j.jbiomech.2009.10.042

    Article  Google Scholar 

  • Tiwari AK, Prasad J (2017) Computer modelling of bone‘s adaptation: the role of normal strain, shear strain and fluid flow. Biomech Model Mechanobiol 16(2):395–410. https://doi.org/10.1007/s10237-016-0824-z

    Article  Google Scholar 

  • Tobias K, Johnston S (2013) Veterinary surgery: small animal, Elsevier Health Sciences, p 999

  • Ugural AC (2011) Advanced mechanics of materials and applied elasticity, 5th edn. Prentice Hall, Upper Saddle River

    Google Scholar 

  • Vøls KK, Kjelgaard-Hansen M, Ley CD, Hansen AK, Petersen M (2020) In vivo fluorescence molecular tomography of induced haemarthrosis in haemophilic mice: link between bleeding characteristics and development of bone pathology. BMC Musculoskeletal Disorders 21(1):1–10. https://doi.org/10.1186/s12891-020-03267-5

    Article  Google Scholar 

  • Vøls KK, Kjelgaard-Hansen M, Ley CD, Hansen AK, Petersen M (2019) Bleed volume of experimental knee haemarthrosis correlates with the subsequent degree of haemophilic arthropathy. Haemophilia 25(2):324–333. https://doi.org/10.1111/hae.13672

    Article  Google Scholar 

  • Wagner DW, Chan S, Castillo AB, Beaupre GS (2013) Geometric mouse variation: implications to the axial ulnar loading protocol and animal specific calibration. J Biomech 46(13):2271–2276. https://doi.org/10.1016/j.jbiomech.2013.06.027

    Article  Google Scholar 

  • Willie BM, Birkhold AI, Razi H, Thiele T, Aido M, Kruck B, Schill A, Checa S, Main RP, Duda GN (2013) Diminished response to in vivo mechanical loading in trabecular and not cortical bone in adulthood of female c57bl/6 mice coincides with a reduction in deformation to load. Bone 55(2):335–346. https://doi.org/10.1016/j.bone.2013.04.023

    Article  Google Scholar 

  • Yang H, Butz KD, Duffy D, Niebur GL, Nauman EA, Main RP (2014) Characterization of cancellous and cortical bone strain in the in vivo mouse tibial loading model using microct-based finite element analysis. Bone 66:131–139. https://doi.org/10.1016/j.bone.2014.05.019

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge Professor Joanna Price, Royal Agricultural University, UK (formerly of University of Bristol, UK), for providing the \(\mu\)CT data used in this study. The authors gratefully acknowledge and thank the facilities and technical support provided by the High Performance Computing service at the Queensland University of Technology. Professor Peter Pivonka gratefully acknowledges support from the Australian Research Council (ARC ITTC, IC190100020). Dr Edmund Pickering gratefully acknowledges support from the QUT Wilson Bequest Fund.

Author information

Authors and Affiliations

  1. School of Mechanical, Medical and Process Engineering, Centre for Biomedical Technologies, Queensland University of Technology (QUT), Brisbane, QLD, Australia

    Edmund Pickering & Peter Pivonka

  2. Centre for Biomedical Technologies , Queensland University of Technology (QUT), QLD, Brisbane , Australia

    Edmund Pickering & Peter Pivonka

  3. Vincent’s Department of Surgery, University of Melbourne, Melbourne, VIC, Australia

    Silvia Trichilo

  4. Veterinary Specialist Services, Brisbane, QLD, Australia

    Peter Delisser

Authors
  1. Edmund Pickering
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Silvia Trichilo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peter Delisser
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Peter Pivonka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Edmund Pickering.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file1 (PDF 3103 KB)

Supplementary file1 (XLSX 62 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pickering, E., Trichilo, S., Delisser, P. et al. Beam theory for rapid strain estimation in the mouse tibia compression model. Biomech Model Mechanobiol 21, 513–525 (2022). https://doi.org/10.1007/s10237-021-01546-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10237-021-01546-1

Keywords

Search

Navigation