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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, II (1988), 40. Google Scholar

[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. Google Scholar

[3]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

M. T. Bayliss and B. Johnstone, Biochemistry of the intervertebral disc,, in, (1992). Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[14]

D. W. L. Hukins, Disc structure and function,, in, I (1988), 1. Google Scholar

[15]

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

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[21]

R. M. H. McMinn, "Last's Anatomy: Regional and Applied,", 9th edition, (1994). Google Scholar

[22]

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

[23]

A. Nachemson, Lumbar mechanics as revealed by lumbar intradiscal pressure measurements,, in, (1992). Google Scholar

[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. Google Scholar

[25]

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

[26]

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

[27]

Matthias Ngwa, "Stress-Strain Problems in Biological Systems,", Ph.D thesis, (2003). Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

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

[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. Google Scholar

[32]

R. S. Snell, "Clinical Anatomy for Medical Students,", 6th edition, (2000), 817. Google Scholar

[33]

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

[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. Google Scholar

[35]

K. G. Vijay and J. N. Weinstein, "Biomechanics of the Spine: Clinical and Surgical Perspective,", CRC Press, (1990). Google Scholar

[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. Google Scholar

show all references

References:
[1]

M. A. Adams and W. C. Hutton, Mechanics of the intervertebral disc,, in, II (1988), 40. Google Scholar

[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. Google Scholar

[3]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

M. T. Bayliss and B. Johnstone, Biochemistry of the intervertebral disc,, in, (1992). Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[14]

D. W. L. Hukins, Disc structure and function,, in, I (1988), 1. Google Scholar

[15]

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

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[21]

R. M. H. McMinn, "Last's Anatomy: Regional and Applied,", 9th edition, (1994). Google Scholar

[22]

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

[23]

A. Nachemson, Lumbar mechanics as revealed by lumbar intradiscal pressure measurements,, in, (1992). Google Scholar

[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. Google Scholar

[25]

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

[26]

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

[27]

Matthias Ngwa, "Stress-Strain Problems in Biological Systems,", Ph.D thesis, (2003). Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

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

[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. Google Scholar

[32]

R. S. Snell, "Clinical Anatomy for Medical Students,", 6th edition, (2000), 817. Google Scholar

[33]

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

[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. Google Scholar

[35]

K. G. Vijay and J. N. Weinstein, "Biomechanics of the Spine: Clinical and Surgical Perspective,", CRC Press, (1990). Google Scholar

[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. Google Scholar

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