a stability hypothesis of the spine can be derived from wavelet analysis and energy methods

8/9/2019 A STABILITY HYPOTHESIS OF THE SPINE CAN BE DERIVED FROM WAVELET ANALYSIS AND ENERGY METHODS

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A STABILITY HYPOTHESIS OF THE SPINE CAN BE DERIVED FROM

WAVELET ANALYSIS AND ENERGY METHODS

M.J. Conaway, Ph.D., N.D., M.D.(A.M.)

1. Introduction

It is possible to model spinal function under low back pain by EMG

wavele t ana lysis and energy methods. Since the spine is a complex

electromechanical system which behaves according to sensorimotorfeedback, the premise follows that the spinal system can be subsequentlymodeled according to principles of control theory derived from energy

methods. In addition, energy methods, which include variational or

Lagrangian/Hamiltonian mechanics, constitute a powerful form of physics

that allows any type of system to be modeled indirectly. Formulations of

problems using these methods are elegant and more easily attained thanusing Newtonian methods. This is because systems of differential equations

in all generalized variables are procedurally generated in the indirectapproach. Indirect methods can account for other factors such as metabolicand thermal activity of muscles as well as electrical activity of spinal nerves

in their solution procedures. These methods can be used in solving for

variables in a complex control system. Direct Newtonian force-dynamicmethods by design cannot account for such factors. It is the understanding

of the applicant that energy methods have no record of being applied to

spine mechanics, or for that matter, motor control, as of yet. However, thetradeoff is that solutions to problems formulated via energy methods often

require numerical methods to be generated. This is due to the nonlinearityoften inherent in the formulated differential equations.

Empirical force data generated from studies of paraspinal muscle response tosudden loads to be conducted through the Ohio Musculoskeletal andNeurolog ical Ins ti tu te will be used to validate the model. Further validationcould be done by comparing data from OMNI between high-velocity, low-amplitude (HVLA) spinal manipulation and low-velocity, low-amplitude (LVLA)spinal manipulation in treating low back pain. Nevertheless, it is the goal togenerate a more accurate framework by which to evaluate spinal function inseveral different treatment contexts.

Force data will be approximated using electromyography (EMG) recordingsof spinal muscles in response to sudden loads. However, instead of

analyzing the EMG data in traditional time or frequency domains, therecordings will be analyzed using wavelet transforms. Since the EMG signal

is by nature complex, random, and changes over time, traditional signalanalysis techniques are inadequate because much information is lost due totheir underlying assumptions of linearity, periodicity, and stationarity.


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However. wavelet analysis has been shown in the literature to provide bothfrequency information in the time domain and multiresolution features of the

signal. Thus, by providing optimal time frequency profiles, wavelet

transforms are more powerful than Fourier analysis to use with surface EMGdata. Considerable effort will be devoted to mathematically understanding

the muscle response from these wavelet analyses.

2. Background and Significance

Early spinal motor control models started with Panjabi [5], who presented

an initial conceptual basis for the idea that the stabilizing system of the

spine consists of three subsystems. The passive subsystem consists ofvertebrae, discs, and ligaments. The active subsystem consists of allmuscles and tendons surrounding the spinal column that can apply forcesto the structure. Finally, the intraspinal nerves, spinal cord, and brain

constitute the neural control subsystem. This subsystem determines therequirements for spinal stability by needed stability. A dysfunction of anycomponent of any one of the subsystems may elicit one of three

possibilities. An immediate response may be generated from other

subsystems to successfully compensate.

A long-term adaptation response of one or more subsystems could also

result. Or, an injury could occur to one or more components of any

subsystem. Thus, it was theorized that the first response results in normalfunction. The second response results in normal function but with a perturbedspinal stabilizing system. Meanwhile, the third possibility leads to general

spine dysfunction. This can result in conditions such as low back pain. Itwas also hypothesized that in situations where additional loads or complex

postures on the spine are anticipated, the neural control mechanism maychange the muscle recruitment strategy. The temporary goal of the different

strategy would be to enhance the spine stability beyond the normal

requirements, according to the hypothesis.

Neuromuscular control of spinal stability may be depicted as a feedback

control system by which the paraspinal muscle reflex acts in response toforce and motion perturbations of the trunk. The influence of anticipatory

muscle recruitment for the control of spinal stability has been previously

investigated. However, there are few reported studies that characterizeparaspinal reflex gain as feedback response. In order to understand theparaspinal muscle reflex in the context of spinal stability, the paraspinalmuscle reflex dynamics had to be quantified. Thus, Granata et al. [2] set out

to quantify the paraspinal reflex response and feedback gain as being

dependent on trunk extension and magnitude of force perturbation. An

impulse response function (IRF) was generated for this quantification. It was

thus shown that the IRF provides a robust reflex measure according to the


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linear model. According to the model, the amplitude of EMG response

increases with input perturbation. In order to quantify reflex gain, however, it

was preferable to apply a short duration impulse force because it has a wide

frequency bandwidth.

The linear model correctly predicted that trunk extension preload reducesthe gain of the reflex response. A trend was observed such that the

response gain was less under preloading of the trunk than under no loading

of the trunk. However, the reason for this phenomenon is constrained to

biomechanical issues in which the model only assumes that reflex gain isindependent of the neuromuscular state. Yet, it has been shown in other

joints that a linear model insufficiently represents reflex behavior.Furthermore, it has been shown that active muscles behave elastically

under small disturbances and viscously under large perturbation. Thus, anonlinear increase in muscle spindle stretching must be expected in

response to larger force perturbations. This would result in open-loop

nonlinear reflex dynamics. However, the linear model of Granata et al [2]did not account for the multi-segmented, nonlinear nature of the vertebral

column or independent muscle activations. However, it was suggested that

the inherent nonlinearity of the spine may enhance its own stability by

inhibiting feedback oscillations in the biomechanics. [2]

Evans, et al [1] presented a new theoretical construct, the Minimum Energy

Hypothesis, which explains structural changes observed in the spineconcomitant to spinal joint fixation resolution in initial investigations. A unified

theory of manipulative effectiveness has been proposed that integrates the

fixation and sensory tonus models of manipulation. The theory is based on

the fact that the spine will assume a position of minimum internal energywhen mechanical equilibrium is achieved. The spine was modeled as a stack

of rigid vertebrae connected with elastic springs. Individual vertebrae wereallowed to translate and rotate with respect to neighboring bones and have atotal state vector of six degrees of freedom. According to the model, the

entire spine with N vertebrae had 6(N-1) degrees of freedom. Kinetically, theforce vector was taken to be at a state of ideal functional conformation (IFC),when nominal physiological loads are present in a healthy and normal spine,and is defined as Fo=KXo.

In the Evans model, perturbations were assumed to arise from two causes:(1) changes in the spring constant matrix, K, and (2) changes in the loadingon the spine. Case 1 corresponds to the "fixation" model of the spine in

which a change in the mechanical characteristics of the spine elicits

deviations from IFC. However, Case 2 denotes the "sensory tonus" model

of the spine. According to that model, position and force sensors ofparaspinal muscles change over time which in turn causes the loading on

the spine to deviate from IFC. It has further been hypothesized that


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repeated abnormal stresses and inflammation contribute to Case 2, whichcould additionally exacerbate Case 1. [1] The authors of the study assumed that the skeletal system is linear

for their purposes. However, it was proposed that the conclusions

reached are valid for nonlinear models of the spine as well. It was

mentioned that even small fixations, be they inflammation, musclespasm, or scar tissue, are important in the model because each

increases the minimum energy level of the spine via elicitingcompensation in other segments. This phenomenon is known asthe somatosomatic reflexes. [1] These compensations are madehomeostatically to maintain static and dynamic equilibrium of the

spine. It is even hypothesized that these reflexes impair spinal

function themselves.

The Minimum Energy Hypothesis states that resolving spinal jointfixations should allow the spine to return to a state of lower internal

energy. If the optimal state of the spine is assumed to be one oflowest physiologic internal energy, then resolving fixations in the

spine would allow the system to go toward a more optimal position.Thus, this hypothesis presented a unified model in the sense that itexplains the relationship between the fixation and the sensory tonus

models of Chiropractic adjustment. This is clinically significant fordeveloping "lighter force" adjusting protocols and instruments. [1]

From the theory of analytical dynamics [3], it is known that if the total

energy of a system is conserved, then the work done on that system

must be converted to potential energy, by convention denoted by V.

Potential energy must be exclusively a function of the localcoordinates x,y,z, or equivalently a function of the generalized

coordinates X,Y,Z, and independent of time. Furthermore, since the

potential energy is solely a function of the configuration variables,independent of their rates of change, the expression T-V may be

substituted in place of the kinetic energy, T, on the right-hand sides

of these equations, so in terms of the parameter L=T-V these

equations can be written mathematically as d/dt(∂L/∂q')-∂L/∂q=FNc.

The

expression

L

is

called

the

Lagrangian

and

is

valid

for

systems

of

any

number of particles. The equivalence between the Lagrangian equation of motion and the conservation of energy is a necessary result of the fact that

the kinetic energy of a system is strictly proportional to the square of the velocity

a

system. However, for nonconservative systems, the Lagrangian

becomes the kinetic energy and the energy losses are treated as generalized forces. [4]


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The correspondence between the conservation of energy and the

Lagrangian equations of motion suggests that there might be aconvenient variational formulation of mechanics in terms of the totalenergy H=T+V. Thus, for each generalized coordinate q there is acorresponding generalized momentum Pi· The function H(q,p) iscalled the Hamiltonian of the system. Partial differentiation of H withrespect to p and q generates the Hamiltonian equations of motion: dq/dt=∂H/∂pi and dp/dt=-∂H/∂qi.

The Lagrangian and Hamiltonian formulations of physics, which are groupedtogether in a class of analytical techniques known as energy methods, areremarkable because they express the laws of physics independent of any

specific coordinate system. Often, these methods can render an easier wayto formulate and solve models of highly complex systems. In their originalforms, however, both formulations assumed an absolute time coordinateand perfectly rigid bodies. Furthermore, it has been shown that theHamiltonian formulation of physics is a generalization of the Lagrangianformulation which, in itself, is a generalization of the Newtonian formulationof physics. [3]

Traditional signal analysis determines signal characteristics with respect to

the frequency via Fourier transform based methods. Although these

methods help to decompose a signal into its various frequency components

and to determine the relative energy of each component, Fourier transformsdo not indicate when the signal exhibits the characteristics for a particularfrequency If frequency content of a signal varies drastically between

intervals, the Fourier transform sweeps over the entire time axis and

averages out any local characteristics in the signal such as high frequencybursts, spikes, discontinuities, and transients. Thus, the Fourier transform

cannot simultaneously resolve a signal in both time and frequency domains.Nevertheless, the most important features of the wavelet transform are

computational simplicity and that the individual wavelet functions arelocalized in space. This is in contrast to the trigonometric functions used in

Fourier analysis which are global and continuous. [4]

Wavelet based signal analysis was used by Lee [4] in determining the

automatic muscle onset times from low back surface EMG signals. Three

criteria--wavelet filtering rate, width of time window, and index of scalecomparison--were used to detect muscle activation onset times from

expected sudden load experiments and various lumbar support trials. Analgorithm based on wavelet analysis demonstrated precision, consistency,


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and repeatability in both clear and noisy EMG signals. In addition, the

wavelet analysis of various data sets provided an objective measure of themuscle activity. It was found that the relationship between EMG in restingand contracting period can be analyzed by counting the number ofimpulses and the integrated electric activity in the signal. Thus, a faster,

more accurate approach to analyzing electromyographic data wasdemonstrated via wavelet analysis.

In order to understand what muscles are doing during biomechanicalactivity, a representative electromyographic (EMG) signal is typically

measured using electrodes placed on the skin and analyzed by computerusing high-level software. Traditional signal processing is based on Fourier

methods.

However, Fourier analysis has been found to be rather clumsy for handling

EMG data. Thus, Lee [4] has developed a newer method of signal analysis

based on wavelets. The underlying principle of the wavelet technique is toanalyze a signal according to scale. Similar to trigonometrically based

Fourier methods, wavelet analysis uses coefficients in a linear combination

of the wavelet functions. This means that a signal can be represented

minimally if an optimal wavelet is selected. Thus, wavelets can isolate data

into its various frequency components and detail each component with a

resolution matched to its scale. Similar to a sinusoid, a wavelet function

oscillates about zero. However, the oscillations for a wavelet decay to zero

and the function is localized in time or space.

3. Research in Progress

The energy-absorbing and energy-imparting functions of skeletal muscle arecoequally important to the modulation of joint motion, especially in the stretchreflex gain in the muscles of the spinal joints. In terms of chronic low back pain(LBP), there is wide speculation that altered trunk muscle onset timing underliesthe commonly observed increased tone in painful muscles.

Patients were fitted with kinematic markers and EMG sensors on the followingleft and right trunk and paraspinal muscles. Subjects began with their indexfinger on their hip in a "fork sensor" device and reached to a 'target sensor' anddid voluntary reaching trials to a high target and a lower target. Then, subjectswere seated in a "cage" with a pulley system attached at the front and back ofthe left and right shoulders. Volitionally, we dropped varying combinations ofweights to give an unexpected perturbation that resulted in either flexion orextension, as well as side bending or rotation. In this study, 10 controls and 10patients with chronic LBP were tested before and after a spinal manipulation. Thereaction time differences from the trunk and paraspinal muscles were determinedusing wavelet analysis in MATLAB.


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Wavelet analysis protocol for LBP EMG study using the MATLAB WaveletToolbox GUI (Wavelet 1-D).

Step 1: Level 11 decomposition of EMG signal with discrete Meyer wavelet

500 1000 1500 2000 2500 3000 3500 4000 4500 5000-0.1

0

0.1

d1

-0.2

0

0.2

d2

-0.2

0

0.2

d3

-0.2

0

0.2

d4

-0.1

0

0.1

d5

-0.05

0

0.05

d6

-0.02

0

0.02

d7

-0.02

0

0.02

d8

-0.02

0

0.02

d9

-0.02

0

0.02

d10

-0.01

0

0.01

d11

00.020.04

a11

-0.2

0

0.2

s

Decomposition at level 11 : s = a11 + d11 + d10 + d9 + d8 + d7 + d6 + d5 + d4 + d3 + d2 + d1 .


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Step 2: Separate mode to compare approximations and details decompositions

of the signal

1000 2000 3000 4000 5000

-0.2

0

0.2

a1

-0.10

0.10.2

a2

-0.1

0

0.1

a3

-0.05

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a4

-0.04-0.02

00.020.04

a5

-0.020

0.020.04

a6

00.020.04

a7

00.020.04

a8

00.02

0.04

a9

-505

10

x 10-3

a10

0

0.020.04

a11

-0.2

0

0.2

s

Signal and Approximation(s)

-0.2

0

0.2

s

cfs

Coefs, Signal and Detail(s)1110 9 8 7 6 5 4 3 2 1

-0.01

0

0.01

d11

-10-505

x 10-3

d10

-0.01

0

0.01

d9

-0.01

0

0.01

d8

-0.010

0.01

d7

-0.02

00.02

d6

-0.050

0.05d5

-0.1

0

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d4

-0.050

0.050.1

d3

-0.1

0

0.1

d2

1000 2000 3000 4000 5000

-0.050

0.05d1


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Step 3: Observe DWT Wavelet Tree to determine appropriate levels of

decomposition to use to detect onset time

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Signal

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Approximation at level 11 (reconstructed).

DWT : Wavelet Tree

s

a1

d1

a2

d2

a3

d3

a4

d4

a5

d5

a6

d6

a7 d7

a8

d8

a9

d9

a10

d10

a11

d11


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Step 4: Show approximation at Level 4 and detail at Level 5 to generate most

realistic and useful reconstruction of original signal

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

1

2

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4

5

6

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8

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10

11

L e v e l n u m b e r

Details Coefficients

500 1000 1500 2000 2500 3000 3500 4000 4500 5000-0.4

-0.2

0

0.2

0.4Signal and Approximation at level 4

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Detail at level 5


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Step 5: Zoom in on Level 5 detail to get onset time

Onset time is at 800 msec. Largest peak before reaction force spike is used.

Thus, I hypothesize that patients with chronic LBP exhibit differences in theirtrunk muscle onset timing in response to a sudden perturbation. In addition, Ihypothesize that a single spinal manipulation alters the onset timing inLBP. Using wavelet analysis, interpreting the pain-spasm-pain model viaminimum energy principles can explain how manipulation ameliorates low backpain.

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

1

2

3

4

5

6

7

8

9

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L e v e l n u m b e r

Details Coefficients

500 1000 1500 2000 2500 3000 3500 4000 4500 5000-0.4

-0.2

0

0.2

0.4Signal and Approximation at level 4

2080 2100 2120 2140 2160 2180 2200 2220 2240

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Detail at level 5


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4. Literature Cited

1. Evans JM, Hill CR, Leach RA, Collins DL. (2002). The minimumenergy hypothesis: a unified model of fixation resolution. J

Manipulative Physiol Ther. 25(2):105-10. 2. Granata KP, Slota GP, Bennett BC. (2004). Paraspinal muscle reflex

dynamics. J Biomech. 37(2):241-7.

3. Langhaar, HL., (1988). Energy Methods in Applied Mechanics., secondedition. Malahar, Florida: Krieger Publishing Company.

4. Lee, JS. (1998). "The Analysis of Electromyography Using Wavelets."Doctoral dissertation. The University of Iowa.

5. Panjabi MM. (1992). The stabilizing system of the spine. Part I. Function,dysfunction, adaptation, and enhancement. J Spinal Disord. 5(4):383-9;discussion 397.

6. Powers, WT and Kennaway, JR, (1998). “A Revised Muscle Model.”

Available online.


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APPENDIX: LAGRANGIAN DERIVATION OF THE EQUATIONS OF MOTION

FOR A GENRALIZED VERTEBRAL SEGMENT

Energy functions are determined to yield model equations based on generalized

coordinates for the various electromechanical elements of the system. The

model of the system is to be generated via the Euler-Lagrange equation (5),

where x is the generalized coordinate..

The system of interest is the generalized vertebral segment. (6) Its constitutive

electromechanical elements are defined and have the following kinematic

functions.

Generalized vertebra A: f(r A, θ A)

Generalized vertebra B: f(r B, θB)

Spinal nerve: f(i)

Left short rotator: f(r 1,θ1)

Left long rotator: f(r 2,θ2)

Right short rotator: f(r 3,θ3)

Right long rotator: f(r 4,θ4)

Rostral interspinalis: f(r 5,θ5)

Caudal interspinalis: f(r 6,θ6)

Left rostral intertransversarius: f(r 7,θ7)

Left caudal intertransversarius: f(r 8,θ8)

Right rostral intertransversarius: f(r 9,θ9)

Right caudal intertransversarius: f(r 10,θ10)

Hence, the generalized coordinates are as follows:

{(r A, θ A), (r B, θB), (i), (r 1,θ1),(r 2,θ2),(r 3,θ3),(r 4,θ4),(r 5),(r 6),(r 7,θ7),(r 8,θ8),(r 9,θ9),(r 10,θ10)}


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General Energy Relationships:

T=½mGENERALIZEDr’2+½IGENERALIZEDθ’

2+½Li2, VGENERALIZED=-mgr

Vnerve=½CV2

Vnonlinspring=[ekr ]/k-r

Vtorspring=-½cθ2

Vtransdamp=[ebr ]/b-r, b=1/(1+k*Ftransdamp)

Assumptions:

The system is conservative, or L=T-V, as well as holonomic and under gravity.

Vertebral motions of small magnitude are primarily effected by the deep back

muscles such as the rotatores, interspinales, etc. when subjected to sudden

loads. Motion is restricted to the x-y coordinate plane. Paraspinal muscles have

their origins on points fixed to an inertial frame of reference. Also, from a revised

Hill muscle model, the interspinalis muscles have both series (nonlinear spring)

and parallel (nonlinear dashpot) elastic components to them. Meanwhile, the

intertransversarii and rotatores are considered to be torsional springs.

From the assumptions, the Lagrangian, L=T-V, can be formed from computing

the energy expressions for each electromechanical element of the system.

L(q,q’)=½mGENERALIZEDr’2+½IGENERALIZEDθ’

2+mGENERALIZEDgr+½c1θ12+½c2θ2

2+½c3

θ32+½c4θ4

2+…

[ek5r5]/k5-r 5-[eb5r5]/b5-r 5+[e

k6r6]/k6-r 6-[eb6r6]/b6-r 6+½c7θ7

2+…

½c8θ82+½c9θ9

2+½c10θ102-½Li2-½CV2

The equations of motion for the electromechanical system can be procedurally

generated from the Lagrangian. This is done for each generalized coordinate via

the Euler-Lagrange equation as indicated above.


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Equations of motion for generalized vertebral segment:

r: mGENERALIZEDr”+IGENERALIZEDθ”-mGENERALIZEDg=F

i: -Li-(i’)CR2i=0

r 1: c1θ1”=0

r 2: c2θ2”=0

r 3: c3θ3”=0

r 4: c4θ4”=0

r 5: r 5’[ek5r5-eb5r5-2]=0

r 6: r 6’[ek6r6-eb6r6-2]=0

r 7: c7θ7”=0

r 8: c8θ8”=0

r 9: c9θ9”=0

r 10: c10θ10”=0

For each unit of time, the entire system of ordinary differential equations must be

solved using numerical methods. Runge-Kutta or Levenburg-Marquardt methods

are suggested as preferable for solution of the system.

Thus, it is shown that a complicated biological system, such as a generalized

vertebral segment, can in fact be conceptually set up for numerical solution using

the method of Lagrange.

a stability hypothesis of the spine can be derived from wavelet analysis and energy methods

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