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].
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.
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.
