{"id":3605,"date":"2000-08-10T18:39:31","date_gmt":"2000-08-11T01:39:31","guid":{"rendered":"https:\/\/www.activator.com\/?p=3605"},"modified":"2023-06-07T08:32:01","modified_gmt":"2023-06-07T15:32:01","slug":"dynamic-response-of-the-human-lumbar-spine-a-5-dof-lumped-parameter-time-and-frequency-domain-model","status":"publish","type":"post","link":"https:\/\/activator.com\/dynamic-response-of-the-human-lumbar-spine-a-5-dof-lumped-parameter-time-and-frequency-domain-model\/","title":{"rendered":"Dynamic Response of the Human Lumbar Spine: A 5 DOF Lumped Parameter Time and Frequency Domain Model"},"content":{"rendered":"

Introduction:<\/h4>\n

A biomechanical analysis of the spine is\u00a0 important for understanding its response to different loading\u00a0 environments. Although substantial information exists on the dynamic\u00a0 response of the spine in the axial direction, little is known about the\u00a0 dynamic response to externally applied, posterior-anterior (PA) directed\u00a0 forces Such as chiropractic manipulations, in this paper, a\u00a0 5-degree-of-freedom (DOF), lumped equivalent model the lumbar spine is\u00a0 developed. Model results are compared to quasi-static, oscillatory and\u00a0 impulsive force measurements of vertebral motion associated with\u00a0 mobilization [1], manual manipulation [2] and mechanical force,\u00a0 manually-assisted (MFMA) adjustments [3].<\/p>\n

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Material and methods:<\/h4>\n

Five Degree-of Freedom Model A 5-DOF mass, massless-spring and damper model of the lumbar\u00a0 spine is shown in Fig. 1. This model differs from that of a single-DOF\u00a0 system in that it has 5 natural frequencies.<\/p>\n

\"\"<\/a> Modeling of this multi-DOF structure necessitates one governing\u00a0 equation of motion for each DOF; in matrix form: [M]d2x\/dt2+ [C]dx\/dt +\u00a0 [K]x = [F] (1) where [M] is the mass matrix, [C] is the damping matrix,\u00a0 [K] is the stiffness matrix, [F] is the PA excitation force matrix, and\u00a0 x = x(t) is the resulting displacement vector. Here we assume that the\u00a0 system has zero mass coupling, in which case [M] is diagonal. [K] is\u00a0 written in terms of the stiffness influence coefficients and is a band\u00a0 matrix along the diagonal. The equations of motion are solved in modal\u00a0 space using the eigensolution (i.e. the modal properties) of the\u00a0 homogeneous equation of motion (free vibration without damping). The\u00a0 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\u00d75 damping<\/p>\n

\"\"<\/a> Using Matlab, the motion response of the spine was studied in\u00a0 response to a 100 N static load, 100 N sinusoidal oscillation, and 100 N\u00a0 impulsive force applied to each of the vertebral segments. The\u00a0 following coefficients were used for the mass matrix (kg) and stiffness\u00a0 (kN\/m) matrix [3]: ml=m2=0.170, m3=m4=m5=0.114; kl=50, k2=40, k3=k4=30, k5=45; \"\"l,\u20265= 0.25 (25% of critical) resulting in damping coefficients CIJ ranging from 40-60 Ns\/m.<\/p>\n

Results:<\/h4>\n

The PA damped and undamped natural frequencies predicted by the model\u00a0 were 44.6 Hz and 46. 1 Hz, respectively. Steady State Response The\u00a0 steady state response to a PA sinusoidal oscillation, f= Foe is given by\u00a0 the frequency response function; H(oo)=[K oo2M + iooC] (3)For PA\u00a0 sinusoidal loading, the model-predicted natural frequency ranged from\u00a0 39-47 Hz (Fig. 2). At resonance, segmental and inter-segmental P A\u00a0 displacements were 7.1 mm and 1.7 mm, respectively, for PA thrusts on\u00a0 L3. \"\"<\/a> PA spine mobilization [1] and manual manipulation [2]\u00a0 correspond to an oscillatory frequency of ~2 Hz. At 2 Hz segmental and\u00a0 Inter-segmental displacements were predicted to be 4.0 mm (L3) and 1.5\u00a0 mm (L3-L4), respectively.<\/p>\n

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Impulsive Force Response:<\/em> The response to an initial displacement [X0] and velocity [V0] was derived by assuming a solution x = UeM for eq. (1): \"\"<\/a> PA MFMA adjustments produce a damped sinusoidal-like\u00a0 oscillation With a duration of ~5 ms (impulsive force). Hence, we used\u00a0 the impulse-momentum principle to estimate V0 (1.84 m\/s) for a damped\u00a0 MFMA oscillation f=466e-1000sin(200(3.14)t). Model predicted L3 and\u00a0 L3.L4 displacements were 1.25 mm and 0.89 mm, respectively, for PA\u00a0 impulsive forces at L3.<\/p>\n

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Discussion and Conclusions:<\/h4>\n

The model predicted PA\u00a0 oscillatory and impulsive resonant frequency of the lumbar spine Is\u00a0 consistent with previous experimental findings [3]. Segmental\u00a0 displacements were over 3-fold greater for manual and mobilization\u00a0 therapies in comparison to MFMA therapy, but differences in\u00a0 inter-segmental displacements were less remarkable for these three types\u00a0 of spinal manipulation.<\/p>\n

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Reference: <\/strong>T.S. Keller and C. J. Colloca; Dynamic\u00a0 Response of the Human Lumbar Spine: A 5 DOF Lumped Parameter Time and\u00a0 Frequency Domain Model; Proceeding of the 2000 Meeting of the European Society of\u00a0 Biomechanics, Dublin, Ireland, August 10-14.<\/p>\n

References:<\/strong> [1] M. Lee and N.L. Svensson (1993) JMPT 16:\u00a0 439-446. [2] I. Gal et al. (1997) JMPT 20: 30.40. [3] T.S. Keller, C.I.\u00a0 Colloca, and A. W. Fuhr(1999) JMPT 22: 75-86.<\/p>\n

Acknowledgements:<\/strong> National Institute of Chiropractic Research; Foundation for the Advancement of Chiropractic Education.<\/p>\n