A biomechanical analysis of the spine is  important for understanding its response to different loading  environments. Although substantial information exists on the dynamic  response of the spine in the axial direction, little is known about the  dynamic response to externally applied, posterior-anterior (PA) directed  forces Such as chiropractic manipulations, in this paper, a  5-degree-of-freedom (DOF), lumped equivalent model the lumbar spine is  developed. Model results are compared to quasi-static, oscillatory and  impulsive force measurements of vertebral motion associated with  mobilization [1], manual manipulation [2] and mechanical force,  manually-assisted (MFMA) adjustments [3].


Material and methods:

Five Degree-of Freedom Model A 5-DOF mass, massless-spring and damper model of the lumbar  spine is shown in Fig. 1. This model differs from that of a single-DOF  system in that it has 5 natural frequencies.

Modeling of this multi-DOF structure necessitates one governing  equation of motion for each DOF; in matrix form: [M]d2x/dt2+ [C]dx/dt +  [K]x = [F] (1) where [M] is the mass matrix, [C] is the damping matrix,  [K] is the stiffness matrix, [F] is the PA excitation force matrix, and  x = x(t) is the resulting displacement vector. Here we assume that the  system has zero mass coupling, in which case [M] is diagonal. [K] is  written in terms of the stiffness influence coefficients and is a band  matrix along the diagonal. The equations of motion are solved in modal  space using the eigensolution (i.e. the modal properties) of the  homogeneous equation of motion (free vibration without damping). The  eigenvectors (mode shapes) are then assembled into a mode shape matrix such that[M]= [I] and [K] =[frequencies 2], where {tr} denotes the transpose, [I] is the diagonal identity matrix. Given modal damping ratios for each mode shape i, the 5×5 damping

Using Matlab, the motion response of the spine was studied in  response to a 100 N static load, 100 N sinusoidal oscillation, and 100 N  impulsive force applied to each of the vertebral segments. The  following coefficients were used for the mass matrix (kg) and stiffness  (kN/m) matrix [3]: ml=m2=0.170, m3=m4=m5=0.114; kl=50, k2=40, k3=k4=30, k5=45; l,…5= 0.25 (25% of critical) resulting in damping coefficients CIJ ranging from 40-60 Ns/m.


The PA damped and undamped natural frequencies predicted by the model  were 44.6 Hz and 46. 1 Hz, respectively. Steady State Response The  steady state response to a PA sinusoidal oscillation, f= Foe is given by  the frequency response function; H(oo)=[K oo2M + iooC] (3)For PA  sinusoidal loading, the model-predicted natural frequency ranged from  39-47 Hz (Fig. 2). At resonance, segmental and inter-segmental P A  displacements were 7.1 mm and 1.7 mm, respectively, for PA thrusts on  L3. PA spine mobilization [1] and manual manipulation [2]  correspond to an oscillatory frequency of ~2 Hz. At 2 Hz segmental and  Inter-segmental displacements were predicted to be 4.0 mm (L3) and 1.5  mm (L3-L4), respectively.


Impulsive Force Response: The response to an initial displacement [X0] and velocity [V0] was derived by assuming a solution x = UeM for eq. (1): PA MFMA adjustments produce a damped sinusoidal-like  oscillation With a duration of ~5 ms (impulsive force). Hence, we used  the impulse-momentum principle to estimate V0 (1.84 m/s) for a damped  MFMA oscillation f=466e-1000sin(200(3.14)t). Model predicted L3 and  L3.L4 displacements were 1.25 mm and 0.89 mm, respectively, for PA  impulsive forces at L3.


Discussion and Conclusions:

The model predicted PA  oscillatory and impulsive resonant frequency of the lumbar spine Is  consistent with previous experimental findings [3]. Segmental  displacements were over 3-fold greater for manual and mobilization  therapies in comparison to MFMA therapy, but differences in  inter-segmental displacements were less remarkable for these three types  of spinal manipulation.


Reference: T.S. Keller and C. J. Colloca; Dynamic  Response of the Human Lumbar Spine: A 5 DOF Lumped Parameter Time and  Frequency Domain Model; Proceeding of the 2000 Meeting of the European Society of  Biomechanics, Dublin, Ireland, August 10-14.

References: [1] M. Lee and N.L. Svensson (1993) JMPT 16:  439-446. [2] I. Gal et al. (1997) JMPT 20: 30.40. [3] T.S. Keller, C.I.  Colloca, and A. W. Fuhr(1999) JMPT 22: 75-86.

Acknowledgements: National Institute of Chiropractic Research; Foundation for the Advancement of Chiropractic Education.