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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

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.
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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