Skip to main content
SpringerLink
Log in

A reduced-order model of the spine to study pediatric scoliosis

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

Abstract

The S-shaped curvature of the spine has been hypothesized as the underlying mechanical cause of adolescent idiopathic scoliosis. In earlier work, we proposed a reduced-order model in which the spine was viewed as an S-shaped elastic rod under torsion and bending. Here, we simulate the deformation of S-shaped rods of a wide range of curvatures and inflection points under a fixed mechanical loading. Our analysis determines three distinct axial projection patterns of these S-shaped rods: two loop (in opposite directions) patterns and one Lemniscate pattern. We further identify the curve characteristics associated with each deformation pattern, showing that for rods deforming in a Loop1 shape the position of the inflection point is the highest and the curvature of the rod is smaller compared to the other two types. For rods deforming in the Loop2 shape, the position of the inflection point is the lowest (closer to the fixed base) and the curvatures are higher than the other two types. These patterns matched the common clinically observed scoliotic curves—Lenke 1 and Lenke 5. Our S-shaped elastic rod model generates deformations that are similar to those of a pediatric spine with the same sagittal curvature characteristics and it can differentiate between the clinically observed deformation patterns.

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
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • Antman S (2006) Nonlinear problems of elasticity. Applied Mathematical Sciences. Springer, New York

    Google Scholar 

  • Audoly B, Pomeau Y (2010) From hair curls to the non-linear response of shells. Elasticity and geometry. OUP Oxford, Oxford

    MATH  Google Scholar 

  • Belytschko T, Andriacchi T, Schultz A, Galante J (1973) Analog studies of forces in the human spine: computational techniques. J Biomech 6(4):361–371

    Article  Google Scholar 

  • Brink RC, Schlösser TP, van Stralen M, Vincken KL, Kruyt MC, Hui SC, Viergever MA, Chu WC, Cheng JC, Castelein RM (2018) Anterior–posterior length discrepancy of the spinal column in adolescent idiopathic scoliosis—a 3d ct study. Spine J 18(12):2259–2265

    Article  Google Scholar 

  • Burwell RG (2003) Aetiology of idiopathic scoliosis: current concepts. Pediatr Rehab 6(3–4):137–170

    Article  Google Scholar 

  • Castelein RM, Veraart B (1992) Idiopathic scoliosis: prognostic value of the profile. Eur Spine J 1(3):167–169

    Article  Google Scholar 

  • Castelein RM, van Dieen JH, Smit TH (2005) The role of dorsal shear forces in the pathogenesis of adolescent idiopathic scoliosis—a hypothesis. Med Hypotheses 65(3):501–508

    Article  Google Scholar 

  • Cheng JC, Castelein RM, Chu WC, Danielsson AJ, Dobbs MB, Grivas TB, Gurnett CA, Luk KD, Moreau A, Newton PO, Stokes IA, Weinstein SL, Burwell RG (2015) Adolescent idiopathic scoliosis. Nat Rev Disease Primers 1(1):15030

    Article  Google Scholar 

  • Chu WC, Lam WM, Ng BK, Tze-ping L, Lee K-M, Guo X, Cheng JC, Burwell RG, Dangerfield PH, Jaspan T (2008) Relative shortening and functional tethering of spinal cord in adolescent scoliosis—result of asynchronous neuro-osseous growth, summary of an electronic focus group debate of the IBSE. Scoliosis 3(1):8

    Article  Google Scholar 

  • Crisco JJ, Panjabi MM (1992) Euler stability of the human ligamentous lumbar spine. Part I: theory. Clin Biomech (Bristol, Avon) 7(1):19–26

    Article  Google Scholar 

  • Crisco JJ, Panjabi MM, Yamamoto I, Oxland TR (1992) Euler stability of the human ligamentous lumbar spine. Part II: experiment. Clin Biomech (Bristol, Avon) 7(1):27–32

    Article  Google Scholar 

  • de Reuver S, Brink R, Homans J, Kruyt M, van Stralen M, Schlösser T, Castelein R (2018) The changing position of the center of mass of the thorax during growth in relation to pre-existent vertebral rotation. SPINE 44(10):679–684

    Article  Google Scholar 

  • Duong L, Cheriet F, Labelle H (2006) Three-dimensional classification of spinal deformities using fuzzy clustering. Spine 31(8):923–30

    Article  Google Scholar 

  • Goto M, Kawakami N, Azegami H, Matsuyama Y, Takeuchi K, Sasaoka R (2003) Buckling and bone modeling as factors in the development of idiopathic scoliosis. Spine (Phila Pa 1976) 28(4):364–371

    Google Scholar 

  • Gu S-X, Wang C-F, Zhao Y-C, Zhu X-D, Li M (2009) Abnormal ossification as a cause the progression of adolescent idiopathic scoliosis. Med Hypotheses 72(4):416–417

    Article  Google Scholar 

  • Janssen MMA, Kouwenhoven J-WM, Schlösser TPC, Viergever MA, Bartels LW, Castelein RM, Vincken KL (2011) Analysis of preexistent vertebral rotation in the normal infantile, juvenile, and adolescent spine. Spine 36(7):E486–91

    Article  Google Scholar 

  • Kouwenhoven J-WM, Vincken KL, Bartels LW, Castelein RM (2006) Analysis of preexistent vertebral rotation in the normal spine. Spine 31(13):1467–72

    Article  Google Scholar 

  • Liu Z, Tam EMS, Sun G-Q, Lam T-P, Zhu Z-Z, Sun X, Lee K-M, Ng T-B, Qiu Y, Cheng JCY, Yeung H-Y (2012) Abnormal leptin bioavailability in adolescent idiopathic scoliosis: an important new finding. Spine 37(7):599–604

    Article  Google Scholar 

  • Lucas DB (1970) Mechanics of the spine. Bull Hosp Joint Diseases Orthop Inst 31(2):115–131

    Google Scholar 

  • Lucas DB, Boris B (1961) Stability of the Ligamentous Spine. N.p.: Biomechanics Laboratory, U of California. Print

  • Meakin JR, Hukins DWL, Aspden RM (1996) Euler buckling as a model for the curvature and flexion of the human lumbar spine. Proc R Soc Lond Ser B Biol Sci 263(1375):1383–1387

    Article  Google Scholar 

  • Neelakantan S, Purohit PK, Pasha S (2020) A semi-analytic elastic rod model of pediatric spinal deformity. J Biomech Eng (under review)

  • Nizette M, Goriely A (1999) Towards a classification of Euler–Kirchhoff filaments. J Math Phys 40(6):2830–2866

    Article  MathSciNet  Google Scholar 

  • Pasha S (2019a) 3D deformation patterns of S-shaped elastic rods as a pathogenesis model for spinal deformity in adolescent idiopathic scoliosis. Sci Rep 9(1):16485

    Article  Google Scholar 

  • Pasha S (2019b) 3D spinal and rib cage predictors of brace effectiveness in adolescent idiopathic scoliosis. BMC Musculoskelet Disord 20:384

    Article  Google Scholar 

  • Pasha S, Baldwin K (2019a) Surgical outcome differences between the 3D subtypes of right thoracic adolescent idiopathic scoliosis. Eur Spine J 28:3076–3084

    Article  Google Scholar 

  • Pasha S, Baldwin K (2019b) Preoperative sagittal spinal profile of adolescent idiopathic scoliosis lenke types and non-scoliotic adolescents: A systematic review and meta-analysis. Spine (Phila Pa 1976) 44(2):134–142

    Article  Google Scholar 

  • Pasha S, Aubin CE, Sangole AP, Labelle H, Parent S, Mac-Thiong JM (2014a) Three-dimensional spinopelvic relative alignment in adolescent idiopathic scoliosis. Spine (Phila Pa 1976) 39(7):564–70

    Article  Google Scholar 

  • Pasha S, Aubin C-E, Parent S, Labelle H, Mac-Thiong J-M (2014b) Biomechanical loading of the sacrum in adolescent idiopathic scoliosis. Clin Biomech 29(3):296–303

    Article  Google Scholar 

  • Pasha S, Hassanzadeh P, Ecker M, Ho V (2019a) A hierarchical classification of adolescent idiopathic scoliosis: identifying the distinguishing features in 3d spinal deformities. PLOS ONE 14(3):1–12

    Article  Google Scholar 

  • Pasha S, Sankar WN, Castelein RM (2019b) The link between the 3d spino-pelvic alignment and vertebral body morphology in adolescent idiopathic scoliosis. Spine Deform 7(1):53–59

    Article  Google Scholar 

  • Pasha S, Smith L, Sankar WN (2019c) Bone remodeling and disc morphology in the distal unfused spine after spinal fusion in adolescent idiopathic scoliosis. Spine Deform 7(5):746–753

    Article  Google Scholar 

  • Pasha S (2020) What causes different coronal curve patterns in idiopathic scoliosis? bioRxiv. https://doi.org/10.1101/2020.01.21.913707.

  • Schlösser TP, Shah SA, Reichard SJ, Rogers K, Vincken KL, Castelein RM (2014) Differences in early sagittal plane alignment between thoracic and lumbar adolescent idiopathic scoliosis. Spine J 14(2):282–290

    Article  Google Scholar 

  • Schlösser TPC, Vincken KL, Rogers K, Castelein RM, Shah SA (2015) Natural sagittal spino-pelvic alignment in boys and girls before, at and after the adolescent growth spurt. Eur Spine J 24:1158–1167

    Article  Google Scholar 

  • Stokes IA, Aronsson DD, Spence H, Iatridis JC (1998) Mechanical modulation of intervertebral disc thickness in growing rat tails. J Spinal Disord 11(3):261–265

    Article  Google Scholar 

  • Weinstein SL, Dolan LA, Cheng JC, Danielsson A, Morcuende JA (2008) Adolescent idiopathic scoliosis. The Lancet 371(9623):1527–1537

    Article  Google Scholar 

Download references

Acknowledgements

Sunder Neelakantan and Prashant K. Purohit acknowledge partial support for this work through an NSF Grant NSF CMMI 1662101. Saba Pasha acknowledges partial support for this work through a research grant from the Scoliosis Research Society and the National Institute of Health (NIH)  R21 AR075971 01A1.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA, 19104, USA

    Sunder Neelakantan & Prashant K. Purohit

  2. Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA, 19104, USA

    Saba Pasha

Authors
  1. Sunder Neelakantan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Prashant K. Purohit
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Saba Pasha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saba Pasha.

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.

Appendix

Appendix

We present the sagittal view of the deformed configuration of the rods presented in Fig. 8 in Fig. 11. We see that the rods bend toward the +ve y- axis, which would be the case given a -ve \(m_x(s)\) moment acting along the length of the rod.

Fig. 11
figure 11

Sagittal view of the deformed configuration of the \({37}^{\circ }\) average curve in Fig. 8

Full size image

We present the equations of the surfaces and the coefficient of determination(\({\varvec{R^2}}\)) used in Fig. 3 in Tables 4 and 5. We present the equations of the loop–Lemniscate classification surfaces in Table 4. We also present the equations of the surfaces generated from the uprightness check in Table 5.

Table 4 Equations of the loop–Lemniscate classification surfaces (Fig. 3 blue surface) for the 3 cases in along with the coefficient of determination
Full size table
Table 5 Equations of the uprightness condition surfaces (Fig. 3 yellow surface) for the 3 cases along with the coefficient of determination. Here, \(x = \kappa _P\), \(y = \kappa _N\) and \(S = f(x,y)\)
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Neelakantan, S., Purohit, P.K. & Pasha, S. A reduced-order model of the spine to study pediatric scoliosis. Biomech Model Mechanobiol 20, 467–480 (2021). https://doi.org/10.1007/s10237-020-01394-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10237-020-01394-5

Keywords

Search

Navigation