Seminar 3

Euler buckling as a model for the curvature and flexion of the human lumbar spine

Presented byM.Karthikeyan

11ME63R33

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Contents Brief description of the human anatomy Introduction The static model The dynamic model Estimating EI with existing knowledge Implementing the new model Implications of the new model Discussion of the results Conclusion

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The human body can be studied in three planes – coronal plane, sagittal plane and the transverse plane

The human spine can be divided into 3 portions – the cervical vertebrae, the thoracic vertebrae and the lumbar vertebrae

When viewed from the side (Sagittal Plane) the mature spine has four distinct curves. These curves are described as being either kyphotic or lordotic.

Brief description of human anatomy

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Introduction This study describes the application of Euler buckling theory

and the Euler pendulum to the curvature and dynamics of the human lumbar spine

In the coronal plane the spine is a vertical column but, it is curved in the sagittal plane, resulting in the lumbar ‘lordosis’

It resembles a vertical column which has buckled under the application of an axial load

This approach is to develop a simple model which does not incorporate all of the anatomical features of the spine

The purpose is to investigate both the static structure and bending in the sagittal plane

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The static model Consider a vertical coulumn of height -L, supporting a weight –Mg.

If the mass M is below the critical value, the column is stable in the vertical position

If the height of column or the mass being supported increases or the bending stiffness (EI) decreases, then the column will buckle into a conformation that is determined by the inequality

L(Mg/EI)1/2 ≥ Π (2n-1)/2 ----- (1) n=1,2,….∞

The integer n describes the mode of buckling: n=1 represents the first buckled mode, n=2 the second buckled mode etc

Comparison of the two figures shows that the curvature of the thoraco-lumbar spine resembles that of an n=2 buckled column. (A buckled column is still statically stable)

If the inequality is not satisfied the column is stable and unbuckled and can behave dynamically as a Euler pendulum

Modes of buckling characterized by values for n

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The dynamic model The dynamic model implies the spine along with the spinal musculature

Consider the Euler pendulum consisting of a vertical column supporting a mass.

A small lateral force applied to the column will cause it to deflect. In the absence of

damping, it will oscillate about its equilibrium position when the force is removed.

The natural frequency of oscillation is given by

vo = (1/2Π)(Sg/(L(tanS-S)))1/2 ----- (2)

where S is defined by S = L(Mg/EI)1/2 ------(3)

According to the inequality in eqn (1), the limiting condition for a real value of vo is that S

has a value of Π/2, when vo becomes zero. When S exceeds Π/2 the oscillating column

becomes unstable since the vo becomes a positive real number

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The dynamic model Forward flexion of the spine will cause an active increase of EI, because greater forces

are required to bend the column when the action of antagonist muscles have to be

overcome

According to eqn (1) increasing EI sufficiently will unbuckle the column and its dynamic

behaviour can then be represented by Euler pendulum.

Movement of this pendulum is initiated by the flexor muscles and is controlled by the

active damping of the antagonistic muscles until required position is achieved.

Then the muscles can relax to decrease EI sufficiently for the n=1 ( a flexed posture)

column to be stable

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Estimating EI with existing knowledge The bending stiffness of the isolated lumbar spine can be estimated from the results

published by Crisco et al (1992). They determined the upper bounds of 78N and 98N for the loads required to buckle two cadaveric lumbar spines in the coronal plane.

Intervertebral discs are roughly elliptical with an aspect ratio of 1.6 so sagittal plane bending should be easier than coronal plane (Farfan 1973)

However buckling in the sagittal plane is expected to be resisted by tension in the ligaments on the anterior and posterior aspects of the spine which are absent in coronal plane

Both EI values are considerably less than estimates of 10, 16 and 15 Nm2 for the spine in the living body (Scholten & Veldhuizen 1986; Smeathers 1987)

The reason proposed here for the discrepancy is that because in living people EI will be increased by the adaptive damping of the musculature.

Load (Mg), N n Length of lumbar spine, m EI, Nm2 Remarks

88 1 0.22 1.7 (Crisco & Panjabi 1992)

88 1 0.16 0.9 (Pooni 1983)

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The model shown here is used to calculate the COM of the body supported by the lumbar spine.

This model consists of

(i) the trunk above the lumbosacral joint (accounts for 36.6% of total body mass)

(ii) the head and neck (accounts for 8.4% of total body mass)

(iii) both arms (accounts for 10.2% of total body mass) The lumbar spine has a length of 0.16m while the entire

thoraco-lumbar spine is about 0.45m long (Pooni 1983)

Therefore the proportion of the trunk weight being supported

by the lumbar spine is about 36.6x(0.45-0.16)/0.45 = 23.6%

of the body weight. Thus the total proportion of the body mass above the lumbar spine is about 42%

Geometric model used to calculate the COM of the body (Pooni 1983)

Implementing the new model

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Implementing the new model The COM of the proportion of the body mass above the lumbar spine is used to calculate

a value for the length L. This length is defined as the distance between this COM and the lumbosacral joint (L1)

The COM of that part of the trunk which lies above the lumbar spine is taken to be at the centre of that part of the cylinder i.e., (0.45-0.16)/2 above L1

The combined centre of mass of the two arms is assumed to act at the same point as the COM of the trunk. This can be achieved by bending the arms at the elbows so that the hands touch the shoulder

The mass of the head and neck is considered to be centred about 0.44m above L1. Taking moments then yield the result that the required COM

is 0.36m above the lumbosacral joint (L=0.36m)

Geometric model used to calculate the COM of the body (Pooni 1983)

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The representation of the problem can be simplified

by considering a column of constant length of value

0.36m.

The figure shows the values of the body mass (42%

of this mass equals M) and bending stiffness, EI,

which lead to stability for n=1 and n=2 buckled

columns

For a typical person with a body mass of 70kg, this

figure shows that an n=1 column cannot be stable

for EI in the expected range of about 1-2 Nm2

However the n=2 column lies close to the boundary

between the regions of stability for an n=1 and n=2

column

Implications of the new model

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Discussion of results The n=2 buckled column has considerable advantages for load carriage

Load carriage increases the value of M and hence the effective body mass

An unbuckled column corresponding to a vertical spine would be in danger of buckling in

the n=1 mode i.e. spontaneously flexing as soon as an additional load was supported. An

n=1 buckled column would flex further under the same conditions

Thus load carriage would be inconsistent with upright stance, unless EI were increased

by muscle action

Application of additional load to an n=2 buckled column would increase its curvature,

unless EI were increased, but the midline of this curved spine would remain vertical.

Thus load carriage with an n=2 buckled spine reduces the energy required to maintain

the normal posture while supporting a load.

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Discussion of results The bending stiffness of the spine predicted by the model is consistent with the range of

values observed experimentally.

According to inequality (1), equation (2) and the definition of S in equation (3), the spine is

dynamically stable when its natural frequency has a negative real value i.e when S ≤ Π/2

This limiting value for S can be substituted into equation (3) to calculate the minimum

bending stiffness, EI, needed to stabilize the spine

For an average person with body mass of 70kg and with L=0.36m, EI is found to be 15Nm2.

This value is considerably greater than the value developed earlier (0.9 – 1.7 Nm2) which

considered only an isolated spine.

However it is in excellent agreement with the experimental values of 16.2 and 10 Nm2

found by Smeathers (1987) for male and female groups respectively and of 15.3 Nm2 found

by Scholten & Veldhuizen (1986)

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Discussion of results The model also makes predictions concerning the natural frequency for flexion which are

consistent with experimental observations

Assuming that the L=0.36m and body mass =70kg (its assumed that 42% of this mass is

supported by the lumbar spine), equation (2) and (3) can be used to relate the natural

frequency, vo, to the bending stiffness, EI as given in the figure

The maximum bending stiffness of the trunk in a living person is 97.7 Nm2 for males and

56.1 Nm2 for females ( Smeathers 1987)

This value gives a natural frequency of order 2 Hz.

This implies a comfortable time of 0.5s to move from an

upright position to a flexed posture and back again, which is

reasonable for a healthy person

Measurements of the natural frequency for flexion of the spine have yielded results in the

range of 1.89 – 2.41 Hz (Smeathers 1995)

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Conclusion

The main conclusion is that it is reasonable to consider the lumbar spine as being part of

a buckled column for which the integer n has a value of 2 in eqn (1)

Such a column is expected to be stable for a spine of typical length and bending stiffness

supporting a reasonable body weight. The bending stiffness of the spine can be

controlled by the musculature

A column buckled in the n=2 mode has the advantage that increasing the weight

supported does not lead to a change in posture

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Conclusion

The second main conclusion is concerned directly with movement of the spine. Forward

flexion of the spine can be accounted for by changing its shape from an n=2 to an

unbuckled mode. The column then behaves as an Euler pendulum which is brought to

rest, at the required position in an n=1 buckled mode by the adaptive control exerted by

the spinal muscles

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References

• J.R.Meakin, D.W.L.Hukins and R.M.Aspden, 1996, Euler buckling as a model for the curvature and flexion of the human lumbar spine, Proceedings: Biological Sciences, Vol. 263, No. 1375, pp.1383-1387

• www.spinesurgeon.com.au• www.wikipedia.org

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Thank You