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2011, 8(4): 1061-1083. doi: 10.3934/mbe.2011.8.1061

A mathematical model of the compression of a spinal disc

Matthias Ngwa 1, and  Ephraim Agyingi 2,

1. 

Calgary Board of Education, Calgary, AB T2G 2L9, Canada
2. 

School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 14623-5801, United States

Received  January 2010 Revised  April 2011 Published  August 2011

A model is developed of the stress-strain response of an intervertebral disc to axial compression. This is based on a balance of increased intradiscal pressure, resulting from the compression of the disc, and the restraining forces generated by the collagen fibres within the annulus fibrosus. A formula is derived for predicting the loading force on a disc once the nucleus pressure is known. Measured material values of L3 and L4 discs are used to make quantitative predictions. The results compare reasonably well with experimental results.
Keywords: axial compression, fibre displacement., intradiscal pressure, Intervertebral disc.
Mathematics Subject Classification: Primary: 74F10, 74L15, 92C10; Secondary: 92B0.
Citation: Matthias Ngwa, Ephraim Agyingi. A mathematical model of the compression of a spinal disc. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1061-1083. doi: 10.3934/mbe.2011.8.1061
References:
[1]

M. A. Adams and W. C. Hutton, Mechanics of the intervertebral disc, in "The Biology of the Intervertebral Disc" (ed. P. Ghosh), II, CRC Press, Inc., Boca Raton, Florida, (1988), 40-69.

[2]

I. Althoff, P. Brinckmann, W. Frobin, J. Sandover and K. Burton, An improved method of stature measurement for quantitative determination of spinal loading. Application of sitting postures and whole body vibration, Spine, 17 (1992), 682-693.

[3]

G. B. J. Andersson and A. B. Schultz, Effects of fluid injection on mechanical properties of intervertebral discs, J. Biomechanics, 12 (1979), 453-458.

[4]

G. B. J. Andersson and A. Nachemson, Intradiskal pressure, intra-abdominal pressure and myoelectric back muscle activity related to posture and loading, Clin. Orthop., 129 (1977), 156-164.

[5]

B. A. Best, F. Guilak, L. A. Setton, W. Zhu, F. Saed-Nejad, A. Ratcliffe, M. Weidenbaum and V. C. Mow, Compressive mechanical properties of the human annulus fibrosus and their relationship to biochemical composition, Spine, 19 (1994), 212-221.

[6]

M. T. Bayliss, B. Johnstone and J. P. O'brien, Poteoglycan systhesis in the human intervertebral disc: Variation with age, region and pathology, Spine, 13 (1988), 972-981.

[7]

M. T. Bayliss and B. Johnstone, Biochemistry of the intervertebral disc, in "The Lumbar Spine and Back Pain" (ed. M.I.V. Jayson), Longman Group UK Limited, 1992.

[8]

T. B. Belytschko, R. F. Kulak, A. B. Schultz and J. D. Galante, Finite element stress analysis of an intervertebral disc, J. Biomechanics, 7 (1974), 277-285.

[9]

S. Bernick and R. Caillet, Vertebral end-plate changes with ageing of human vertebrae, Spine, 7 (1982), 97-102.

[10]

P. Brinckmann and H. Grootenboer, Change of disc height, radial disc bulge, and intradiscal pressure from discectomy: An in vitro investigation on human lumbar discs, Spine, 16 (1991), 641-646.

[11]

P. Dolan, M. Earley and M. A. Adams, Bending and compressive stresses on the lumbar spine during lifting activities, J. Biomechanics, 27 (1994), 1237-1248.

[12]

Wu Han-Chin and Yao Ren-Feng, Mechanical behaviour of the human annulus fibrosus, J. Biomechanics, 9 (1976), 1-7.

[13]

A. D. Holmes, D. W. L. Hukins and A. J. Freemont, End-plate displacement during compression of lumbar vertebra-disc-vertebra segments and the mechanism of failure, Spine, 18 (1993), 128-135.

[14]

D. W. L. Hukins, Disc structure and function, in "The Biology of the Intervertebral Disc" (ed. P Ghosh), I, CRC Press, Inc., Boca Raton, Florida, (1988), 1-37.

[15]

M. D. Humzah and R. W. Soames, Human intervertebral disc: Structure and function, The Anatomical Record, 220 (1988), 337-356.

[16]

H. Inoue and T. Takeda, Three dimensional observation of collagen framework of intervertebral discs, Acta Orthopaedica Scandinavica, 46 (1975), 949-956.

[17]

H. Ishihara, D. S. McNally, J. P. G. Urban and A. C. Hall, Effects of hydrostatic pressure on matrix synthesis in different regions of the intervertebral disc, J. Applied Physiology, 80 (1996), 339-346.

[18]

R. F. Kulak, T. B. Belytschko and A. B. Schultz, Nonlinear behaviour of the human intervertebral disc under axial loading, J. Biomechanics, 9 (1976), 377-386.

[19]

M. Y. Lu, C. W. Hutton and M. V. Gharpuray, Can variations in intervertebral disc height affect the mechanical function of the disc?, Spine, 21 (1996), 2208-2217.

[20]

F. Marchand and A. M. Ahmed, Investigation of the laminate structure of lumbar disc annulus fibrosus, Spine, 15 (1990), 402-410.

[21]

R. M. H. McMinn, "Last's Anatomy: Regional and Applied," 9th edition, Churchill Livingstone, (1994), pp. 537.

[22]

D. S. McNally and M. A. Adams, Internal intervertebral disc mechanics as revealed by stress profilometry, Spine, 17 (1992), 66-73.

[23]

A. Nachemson, Lumbar mechanics as revealed by lumbar intradiscal pressure measurements, in "The Lumbar Spine and Back Pain" (ed. M. I. V. Jayson), Longman Group UK Limited, 1992.

[24]

A. Nachemson and G. Elfstrom, Intravital dynamic pressure measurements in lumbar discs. A study of common movements, maneuvers and exercises, Scand. J. Rehab. (Suppl.), 1 (1970), 1-40.

[25]

A. Nachemson and J. M. Morris, In vivo measurements of intradiscal pressure, J. Bone Jt Surg, 46A (1964), 1077-1092.

[26]

R. N. Natarajan, J. H. Ke and G. B. J. Andersson, A model to study the disc degeneration process, Spine, 19 (1994), 259-265.

[27]

Matthias Ngwa, "Stress-Strain Problems in Biological Systems," Ph.D thesis, The University of Manchester, UK, 2003.

[28]

N. D. Panagiotacopulos, M. H. Pope, R. Bloch and M. H. Krag, Water content in human intervertebral discs. Part II: Viscoelastic behaviour, Spine, 12 (1987), 918-924.

[29]

H. S. Ranu, R. A. Denton and A. I. King, Pressure distribution under an intervertebral disc - an experimental study, J. Biomechanics, 12 (1979), 807-812.

[30]

M. Reuber, A. Schultz, F. Denis and D. Spencer, Bulging of lumbar intervertebral disks, J. Biomech. Eng., 104 (1982), 187-192.

[31]

A. Schultz, G. Andersson, R. Ortengren, K. Haderspeck and A. Nachemson, Loads on the lumbar spine. Validation of a biomechanical analysis by measurements of intradiscal pressures and myoelectric signals, J. Bone Joint Surg., 64 (1982), 713-720.

[32]

R. S. Snell, "Clinical Anatomy for Medical Students," 6th edition, Lippincott William and Wilkins, (2000), 817-828.

[33]

R. L. Spilker, Mechanical behaviour of a simple model of an intervertebral disc under compressive loading, J. Biomechanics, 13 (1980), 895-901.

[34]

R. L. Spilker, D. M. Daugirda and A. B. Shultz, Mechanical response of a simple finite element model of the intervertebral disc under complex loading, J. Biomechanics, 17 (1984), 103-112.

[35]

K. G. Vijay and J. N. Weinstein, "Biomechanics of the Spine: Clinical and Surgical Perspective," CRC Press, Boca Raton, 1990.

[36]

H. J. Wilke, P. Neef, T. Hoogland and L. E. Claes, New In vivo measurements of pressures in the intervertebral disc in daily life, Spine, 24 (1999), 775-762.

show all references

References:
[1]

M. A. Adams and W. C. Hutton, Mechanics of the intervertebral disc, in "The Biology of the Intervertebral Disc" (ed. P. Ghosh), II, CRC Press, Inc., Boca Raton, Florida, (1988), 40-69.

[2]

I. Althoff, P. Brinckmann, W. Frobin, J. Sandover and K. Burton, An improved method of stature measurement for quantitative determination of spinal loading. Application of sitting postures and whole body vibration, Spine, 17 (1992), 682-693.

[3]

G. B. J. Andersson and A. B. Schultz, Effects of fluid injection on mechanical properties of intervertebral discs, J. Biomechanics, 12 (1979), 453-458.

[4]

G. B. J. Andersson and A. Nachemson, Intradiskal pressure, intra-abdominal pressure and myoelectric back muscle activity related to posture and loading, Clin. Orthop., 129 (1977), 156-164.

[5]

B. A. Best, F. Guilak, L. A. Setton, W. Zhu, F. Saed-Nejad, A. Ratcliffe, M. Weidenbaum and V. C. Mow, Compressive mechanical properties of the human annulus fibrosus and their relationship to biochemical composition, Spine, 19 (1994), 212-221.

[6]

M. T. Bayliss, B. Johnstone and J. P. O'brien, Poteoglycan systhesis in the human intervertebral disc: Variation with age, region and pathology, Spine, 13 (1988), 972-981.

[7]

M. T. Bayliss and B. Johnstone, Biochemistry of the intervertebral disc, in "The Lumbar Spine and Back Pain" (ed. M.I.V. Jayson), Longman Group UK Limited, 1992.

[8]

T. B. Belytschko, R. F. Kulak, A. B. Schultz and J. D. Galante, Finite element stress analysis of an intervertebral disc, J. Biomechanics, 7 (1974), 277-285.

[9]

S. Bernick and R. Caillet, Vertebral end-plate changes with ageing of human vertebrae, Spine, 7 (1982), 97-102.

[10]

P. Brinckmann and H. Grootenboer, Change of disc height, radial disc bulge, and intradiscal pressure from discectomy: An in vitro investigation on human lumbar discs, Spine, 16 (1991), 641-646.

[11]

P. Dolan, M. Earley and M. A. Adams, Bending and compressive stresses on the lumbar spine during lifting activities, J. Biomechanics, 27 (1994), 1237-1248.

[12]

Wu Han-Chin and Yao Ren-Feng, Mechanical behaviour of the human annulus fibrosus, J. Biomechanics, 9 (1976), 1-7.

[13]

A. D. Holmes, D. W. L. Hukins and A. J. Freemont, End-plate displacement during compression of lumbar vertebra-disc-vertebra segments and the mechanism of failure, Spine, 18 (1993), 128-135.

[14]

D. W. L. Hukins, Disc structure and function, in "The Biology of the Intervertebral Disc" (ed. P Ghosh), I, CRC Press, Inc., Boca Raton, Florida, (1988), 1-37.

[15]

M. D. Humzah and R. W. Soames, Human intervertebral disc: Structure and function, The Anatomical Record, 220 (1988), 337-356.

[16]

H. Inoue and T. Takeda, Three dimensional observation of collagen framework of intervertebral discs, Acta Orthopaedica Scandinavica, 46 (1975), 949-956.

[17]

H. Ishihara, D. S. McNally, J. P. G. Urban and A. C. Hall, Effects of hydrostatic pressure on matrix synthesis in different regions of the intervertebral disc, J. Applied Physiology, 80 (1996), 339-346.

[18]

R. F. Kulak, T. B. Belytschko and A. B. Schultz, Nonlinear behaviour of the human intervertebral disc under axial loading, J. Biomechanics, 9 (1976), 377-386.

[19]

M. Y. Lu, C. W. Hutton and M. V. Gharpuray, Can variations in intervertebral disc height affect the mechanical function of the disc?, Spine, 21 (1996), 2208-2217.

[20]

F. Marchand and A. M. Ahmed, Investigation of the laminate structure of lumbar disc annulus fibrosus, Spine, 15 (1990), 402-410.

[21]

R. M. H. McMinn, "Last's Anatomy: Regional and Applied," 9th edition, Churchill Livingstone, (1994), pp. 537.

[22]

D. S. McNally and M. A. Adams, Internal intervertebral disc mechanics as revealed by stress profilometry, Spine, 17 (1992), 66-73.

[23]

A. Nachemson, Lumbar mechanics as revealed by lumbar intradiscal pressure measurements, in "The Lumbar Spine and Back Pain" (ed. M. I. V. Jayson), Longman Group UK Limited, 1992.

[24]

A. Nachemson and G. Elfstrom, Intravital dynamic pressure measurements in lumbar discs. A study of common movements, maneuvers and exercises, Scand. J. Rehab. (Suppl.), 1 (1970), 1-40.

[25]

A. Nachemson and J. M. Morris, In vivo measurements of intradiscal pressure, J. Bone Jt Surg, 46A (1964), 1077-1092.

[26]

R. N. Natarajan, J. H. Ke and G. B. J. Andersson, A model to study the disc degeneration process, Spine, 19 (1994), 259-265.

[27]

Matthias Ngwa, "Stress-Strain Problems in Biological Systems," Ph.D thesis, The University of Manchester, UK, 2003.

[28]

N. D. Panagiotacopulos, M. H. Pope, R. Bloch and M. H. Krag, Water content in human intervertebral discs. Part II: Viscoelastic behaviour, Spine, 12 (1987), 918-924.

[29]

H. S. Ranu, R. A. Denton and A. I. King, Pressure distribution under an intervertebral disc - an experimental study, J. Biomechanics, 12 (1979), 807-812.

[30]

M. Reuber, A. Schultz, F. Denis and D. Spencer, Bulging of lumbar intervertebral disks, J. Biomech. Eng., 104 (1982), 187-192.

[31]

A. Schultz, G. Andersson, R. Ortengren, K. Haderspeck and A. Nachemson, Loads on the lumbar spine. Validation of a biomechanical analysis by measurements of intradiscal pressures and myoelectric signals, J. Bone Joint Surg., 64 (1982), 713-720.

[32]

R. S. Snell, "Clinical Anatomy for Medical Students," 6th edition, Lippincott William and Wilkins, (2000), 817-828.

[33]

R. L. Spilker, Mechanical behaviour of a simple model of an intervertebral disc under compressive loading, J. Biomechanics, 13 (1980), 895-901.

[34]

R. L. Spilker, D. M. Daugirda and A. B. Shultz, Mechanical response of a simple finite element model of the intervertebral disc under complex loading, J. Biomechanics, 17 (1984), 103-112.

[35]

K. G. Vijay and J. N. Weinstein, "Biomechanics of the Spine: Clinical and Surgical Perspective," CRC Press, Boca Raton, 1990.

[36]

H. J. Wilke, P. Neef, T. Hoogland and L. E. Claes, New In vivo measurements of pressures in the intervertebral disc in daily life, Spine, 24 (1999), 775-762.

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