Validation of lumped-parameter models of the human lumbar spine using in vivo driving-point compliance data during spinal manipulation

Abstract

Previously published 5-DOF and 21-DOF lumped-parameter models of the human lumbar spine predict displacement responses to posteroanterior (PA) forces, yet neither has been validated against in vivo frequency response function (FRF) data. This study compared model-predicted driving-point compliance spectra to in vivo experimental data and estimated patient-specific mechanical properties through inverse analysis. Triaxial bone-pin accelerometers were mounted to the L2 and L4 spinous processes of two patients undergoing lumbar decompression surgery. An Impulse iQ instrument delivered 178 controlled impulses at the L3 spinous process across six conditions. Driving-point compliance FRFs were computed using the H1 spectral estimator. Patient-specific PA stiffness and damping ratios were estimated via coherence-weighted logarithmic least-squares optimization. Both patients exhibited a dominant first compliance resonance near 49 Hz, consistent with the 5-DOF prediction of 46 Hz (<7% error). The second and third modes (89–101 Hz and 107–110 Hz) agreed with model predictions of 84 and 120 Hz. Optimized stiffness converged to $52.3\pm 3.5$ kN/m (published: 50 kN/m), with damping ratios of 0.049-−0.109. Time-domain Newmark–beta simulations predicted peak L3 displacements of 1.10-−5.60 mm for a 3.2 ms impulse, consistent with cadaveric and in vivo spinal manipulation studies. These results provide the first in vivo validation of lumped-parameter spine models and demonstrate patient-specific mechanical property estimation from compliance FRF data.

Introduction

Low back pain (LBP) remains the leading cause of years lived with disability globally, affecting approximately 619 million people worldwide in 2020 with projections reaching 843 million cases by 2050 Ferreira et al. (2023). The economic burden of LBP in the United States alone exceeds $100 billion annually in direct medical costs and lost productivity (Dieleman et al. 2020). Spinal manipulation is among the most extensively studied conservative interventions for LBP, with clinical practice guidelines from the American College of Physicians (Qaseem et al. 2017) and systematic reviews (Rubinstein et al. 2019; Paige et al. 2017) recommending it as a first-line nonpharmacologic treatment option for both acute and chronic low back pain. Despite the prevalence and impact of this condition, clinical assessment of lumbar spine mechanics remains largely subjective, relying on manual palpation, visual assessment of range of motion, and patient-reported outcome measures. There exists a fundamental need for objective, quantifiable biomechanical assessment tools that can characterize the mechanical properties of the spine in individual patients (Stanton and Kawchuk 2009).

Despite this strong clinical evidence base, the biomechanical mechanisms underlying the therapeutic effects of spinal manipulative therapy (SMT) remain incompletely understood (Herzog 2010; Pickar 2002). Forces applied to the spine during SMT are complex, time-varying mechanical inputs that produce segmental and intersegmental motion responses governed by the structural dynamic properties of the spinal column (Triano 2000; Gal et al. 1997). Impulsive SMTs are defined as forces delivered over very short time durations (<5 ms) which can only be achieved by hand-held medical devices, but in addition to therapeutic action, the thrust response can be simultaneously monitored for data collection and biomechanical diagnostics (Colloca et al. 2000, 2004). Characterization of these properties requires frequency-domain analysis methods that can resolve the stiffness, damping, and inertial contributions to the mechanical response across the bandwidth of therapeutic force application.

Lumped-parameter models provide a computationally efficient framework for predicting the dynamic motion response of the lumbar spine to applied forces. Keller and Colloca (2002) developed a 5-degree-of-freedom (DOF) rigid body model representing the five lumbar vertebrae as masses connected by springs and dashpots along the posteroanterior (PA) axis. This model predicted five natural frequencies of vibration between 46 and 158 Hz and demonstrated that segmental displacements from impulsive forces were much less remarkable than those from oscillatory forces, suggesting that intersegmental motions produced by spinal manipulation may play a prominent role in eliciting therapeutic responses. The analysis was subsequently extended to a 21-DOF sagittal plane model that incorporates three coupled degrees of freedom (PA displacement, axial displacement, and flexion-extension rotation) at each of seven rigid structures (thorax, L1–L5, pelvis), enabling prediction of translation-rotation coupling during dynamic loading (Keller et al. 2002). The 21-DOF model predicted a fundamental natural frequency of 5.24 Hz with damped PA-axis frequencies of 39.3, 74.5, and 120 Hz, and provided static, steady-state, and impulsive force displacement predictions.

Both models were validated against previously published quasi-static and low-frequency oscillatory displacement data from mobilization experiments (Lee and Svensson 1990; Latimer et al. 1998; Viner et al. 1997). However, neither model has been validated against broadband frequency response function (FRF) measurements obtained in vivo during dynamic mechanical excitation. The driving-point compliance FRF, which expresses the displacement-to-force ratio as a function of frequency, is the most directly comparable metric to lumped-parameter model output and provides immediate access to the natural frequencies and stiffness properties of the system under test (Ewins 2000). Recent advances in instrument-assisted spinal manipulation devices, such as the Impulse iQ (Neuromechanical Innovations, LLC, Phoenix, AZ), deliver controlled, repeatable impulsive forces with sufficient bandwidth to excite the structural dynamic modes of the spine (Colloca et al. 2000, 2004), enabling the computation of experimental compliance FRFs from in vivo measurements.

The purpose of this study was: (1) to compute model-predicted driving-point compliance spectra from the published 5-DOF and 21-DOF lumped-parameter models; (2) to compare these predictions directly to in vivo experimental compliance FRF data; and (3) to estimate patient-specific mechanical properties (PA stiffness and modal damping ratio) through frequency-domain inverse analysis. We hypothesized that model-predicted natural frequencies and compliance spectral shapes would agree with experimental observations and that the inverse approach would yield patient-specific stiffness estimates within the physiological range reported in the literature. We analyzed in vivo compliance data from two patients enrolled in an intraoperative bone-pin accelerometry protocol.

Methods

Experimental data acquisition

The experimental data utilized in this study were obtained from an intraoperative biomechanical investigation approved by the institutional ethics committee (Colloca et al. 2007). Two patients (Patient 03 and Patient 04; from a four-patient cohort: 3 female, 1 male; age range 54–70 years, mean $61.5\pm 5.7$ years) undergoing elective posterior lumbar decompressive surgery provided written informed consent (Colloca et al. 2026). Prior to the surgical procedure, triaxial accelerometers (Model CXL10HF3, Crossbow Technology, Inc., San Jose, CA; bandwidth 0.3–10,000 Hz, noise 300 $\mu$g rms, nonlinearity <1%) were attached to intraosseous Steinmann pins rigidly fixed to the L2 and L4 spinous processes under fluoroscopic guidance, providing direct measurement of vertebral acceleration along the medial-lateral (x), posteroanterior (PA/z), and cephalad-caudal (y) axes. Mechanical impulses were delivered through the intact skin overlying the L3 spinous process using an electromagnetic servo-controlled, force-controlled Impulse iQ Adjusting Instrument (Neuromechanical Innovations, LLC, Phoenix, AZ) at three force settings (Low, Medium, High), producing six experimental conditions (2 patients $\times$ 3 force settings). A total of 178 thrusts were recorded across all conditions. Mean peak forces ranged from 57.7 N (Patient 03, Low setting) to 295.9 N (Patient 03, High setting). Force was measured by the instrument’s integrated force transducer (Model 201A03, PCB Piezotronics, Inc., Depew, NY; calibration: 434.78 N/V), and a device accelerometer (Model 305A04, PCB Piezotronics, Inc.; calibration: 1025.64 g/V) at the stylus tip provided the driving-point acceleration measurement. Force and device accelerometer signals were sampled at 25,000 Hz, and bone-pin accelerometer signals were sampled at 3,125 Hz, using a 16-channel MP150 data acquisition system (Biopac Systems, Inc., Goleta, CA). The 25,000 Hz rate provided adequate temporal resolution for the short-duration ($\sim$2 ms) force pulse, while the 3,125 Hz rate for the bone-pin channels (Nyquist frequency 1,562.5 Hz) far exceeded the 131 Hz bone-pin resonance limit; the per-channel rates were determined by the MP150 time-division multiplexing architecture across 12 simultaneously active channels. Four patients were enrolled in the parent study; Patient 01 was excluded due to force transducer malfunction (recording approximately 5–9 N across all settings), and Patient 02 was excluded because the compliance FRF exhibited a dominant resonance near 150 Hz with poor model fits, leaving Patients 03 and 04 for the present analysis.

Driving-point compliance computation

The driving-point compliance FRF was computed using the H1 spectral estimator, which minimizes the effect of uncorrelated output noise (Ewins 2000). For each thrust, a 300 ms Hanning-windowed segment centered on the force peak was extracted from both the force and instrument tip accelerometer signals. Cross-spectral and auto-spectral density estimates were computed using a segment length of 1,024 samples (frequency resolution $\mathrm{\Delta }f\approx 3.1$ Hz) and averaged across all thrusts within each condition. The accelerance FRF was obtained as

$$\begin{array}{c}\hfill H_{\text{acc}}(f)=\frac{G_{\mathit{xy}}(f)}{G_{\mathit{xx}}(f)},\end{array}$$ 1

where $G_{\mathit{xy}}(f)$ is the cross-spectral density between the force input x and acceleration output y, and $G_{\mathit{xx}}(f)$ is the auto-spectral density of the force. The compliance FRF was then obtained by double integration in the frequency domain:

$$\begin{array}{c}\hfill H_{\text{comp}}(f)=\frac{H_{\text{acc}}(f)}{-{(2\pi f)}^2}.\end{array}$$ 2

The ordinary coherence function was computed as

$$\begin{array}{c}\hfill {\gamma }^2(f)=\frac{|G_{\mathit{xy}}{(f)|}^2}{G_{\mathit{xx}}(f)G_{\mathit{yy}}(f)},\end{array}$$ 3

where ${\gamma }^2\in [0,1]$, with values above 0.5 considered acceptable for model comparison.

Transfer accelerance and mode identification

In addition to the driving-point FRF at L3 ($H_{33}$), transfer accelerance FRFs were computed from the bone-pin accelerometer signals at L2 and L4 to aid in experimental mode identification. The transfer FRF $H_{23}$ represents the PA-axis acceleration at L2 per unit force applied at L3, and $H_{43}$ represents the PA-axis acceleration at L4 per unit force at L3. These transfer functions were computed using the same H1 spectral estimation procedure described above, with the bone-pin PA-axis acceleration channel substituted as the output signal. Mean coherence values for the transfer channels were 0.91 (L2 PA-axis) and 0.69 (L4 PA-axis).

Experimental modes were identified by examining local maxima in the PA-axis accelerance magnitude spectra from all three channels ($H_{23}$, $H_{33}$, $H_{43}$) across all force settings for each patient. A mode was assigned only when a peak appeared consistently across multiple settings and channels. The analysis was restricted to frequencies below the identified bone-pin resonance at 131 Hz, above which the transfer FRF estimates are unreliable.

Five-degree-of-freedom model

The 5-DOF model of Keller and Colloca (2002) represents the five lumbar vertebrae (L1–L5) as rigid masses connected in series by linear springs and viscous dashpots along the PA axis. The equation of motion is

$$\begin{array}{c}\hfill \mathbf{M}\ddot{\mathbf{q}}(t)+\mathbf{C}\dot{\mathbf{q}}(t)+\mathbf{K}\mathbf{q}(t)=\mathbf{f}(t),\end{array}$$ 4

where $\mathbf{q}(t)\in {\mathbb{R}}^5$ is the vector of PA displacements, $\mathbf{M}$ is the diagonal mass matrix, $\mathbf{C}$ is the damping matrix, $\mathbf{K}$ is the symmetric tridiagonal stiffness matrix, and $\mathbf{f}(t)$ is the applied force vector.

The mass matrix is

$$\begin{array}{c}\hfill \mathbf{M}=\text{diag}(m_1,m_2,m_3,m_4,m_5),\end{array}$$ 5

with segment masses $m_1=m_2=0.170$ kg (L1, L2) and $m_3=m_4=m_5=0.114$ kg (L3–L5), based on a 70 kg, 170 cm reference subject (Keller and Colloca 2002).

The stiffness matrix is assembled from six PA spring stiffness coefficients $k_1$ through $k_6$, where $k_1$ connects the thorax (fixed) to L1, $k_2$ connects L1 to L2, and so on, with $k_6$ connecting L5 to the sacrum (fixed). The individual spring stiffnesses follow the profile

$$\begin{array}{cc}& k_1=1.0k,k_2=0.8k,k_3=0.7k,\hfill \\ \hfill & k_4=k_5=0.6k,k_6=0.9k,\hfill \end{array}$$ 6

where k is the baseline PA stiffness parameter (published median: $k=50$ kN/m). The global stiffness matrix takes the standard assembly form for a chain of spring elements:

$$\begin{array}{c}\hfill \mathbf{K}=\left[\begin{array}{ccccc}{k}_1+k_2& -k_2& 0& 0& 0\\ -k_2& k_2+k_3& -k_3& 0& 0\\ 0& -k_3& k_3+k_4& -k_4& 0\\ 0& 0& -k_4& k_4+k_5& -k_5\\ 0& 0& 0& -k_5& k_5+k_6\end{array}\right].\end{array}$$ 7

The undamped natural frequencies ${\omega }_n$ and mode shapes ${\mathit{\varphi }}_n$ are obtained from the generalized eigenvalue problem:

$$\begin{array}{c}\hfill \mathbf{K}{\mathit{\varphi }}_n={\omega }_n^2\mathbf{M}{\mathit{\varphi }}_n,n=1,\dots ,5.\end{array}$$ 8

The mass-normalized mode shape matrix $\mathbf{\Phi }=[{\mathit{\varphi }}_1,\dots ,{\mathit{\varphi }}_5]$ satisfies ${\mathbf{\Phi }}^T\mathbf{M}\mathbf{\Phi }=\mathbf{I}$ and ${\mathbf{\Phi }}^T\mathbf{K}\mathbf{\Phi }={\mathbf{\Omega }}^2=\text{diag}({\omega }_1^2,\dots ,{\omega }_5^2)$.

The damping matrix was constructed using modal damping with a uniform damping ratio $\zeta$ applied to all modes:

$$\begin{array}{c}\hfill \mathbf{C}=\mathbf{M}\mathbf{\Phi }[2\zeta \mathbf{\Omega }]{\mathbf{\Phi }}^T\mathbf{M},\end{array}$$ 9

where $\mathbf{\Omega }=\text{diag}({\omega }_1,\dots ,{\omega }_5)$ is the diagonal matrix of natural frequencies (published: $\zeta =0.15$).

The driving-point compliance at L3 was computed in the frequency domain as the (3,3) element of the receptance matrix:

$$\begin{array}{c}\hfill H_{33}(\omega )={\left[\mathbf{K},-,{\omega }^2,\mathbf{M},+,\text{i},\omega ,\mathbf{C}\right]}_{33}^{-1},\end{array}$$ 10

where $\omega =2\pi f$ is the circular frequency. Equivalently, exploiting modal decomposition:

$$\begin{array}{c}\hfill H_{33}(\omega )=\sum _{n=1}^5\frac{{\varphi }_{3n}^2}{{\omega }_n^2-{\omega }^2+2\text{i}\zeta {\omega }_n\omega },\end{array}$$ 11

where ${\varphi }_{3n}$ is the third component (L3) of the nth mass-normalized mode shape.

Twenty-one-degree-of-freedom model

The 21-DOF sagittal plane model of Keller et al. (2002) extends the 5-DOF framework by incorporating three coupled degrees of freedom per segment: PA displacement (u), axial displacement (w), and flexion-extension rotation ($\theta$), at each of seven rigid structures (thorax, L1–L5, pelvis). The generalized coordinate vector is $\mathbf{q}\in {\mathbb{R}}^{21}$, ordered as $(u_1,w_1,{\theta }_1,\dots ,u_7,w_7,{\theta }_7)$.

The global stiffness matrix $\mathbf{K}\in {\mathbb{R}}^{21\times 21}$ is assembled from $3\times 3$ element stiffness submatrices at each of eight flexible joint structures (FJSs). Each element submatrix incorporates PA stiffness ($k_y$), axial stiffness ($k_x$), and torsional stiffness ($k_c$), with translation-rotation coupling arising through a spring axis offset parameter $a=20$ mm and segment link dimensions. The element stiffness matrix for the jth FJS, expressed in the local frame, takes the form

$$\begin{array}{c}\hfill {\mathbf{k}}^{(j)}=\left[\begin{array}{ccc}{k}_y^{(j)}& 0& k_y^{(j)}a\\ 0& k_x^{(j)}& 0\\ k_y^{(j)}a& 0& k_c^{(j)}+k_y^{(j)}{a}^2\end{array}\right],\end{array}$$ 12

where $k_y^{(j)}$, $k_x^{(j)}$, and $k_c^{(j)}$ are the PA, axial, and torsional stiffness values at FJS j, respectively. The global stiffness matrix is assembled using standard finite element procedures with appropriate geometric transformations.

The mass matrix $\mathbf{M}\in {\mathbb{R}}^{21\times 21}$ is diagonal, with segment masses and rotational moments of inertia from published anthropometric data (Keller et al. 2002): thorax mass 22.85 kg, pelvis mass 10.42 kg, and vertebral masses of 0.114-−0.170 kg. The published modal damping ratio was $\zeta =0.25$ for this model. The driving-point compliance at L3 was computed identically to Eq. (10) using the PA displacement DOF corresponding to L3.

Patient-specific parameter estimation

Patient-specific mechanical properties were estimated by minimizing the coherence-weighted logarithmic least-squares error between model-predicted and experimental driving-point compliance magnitude spectra at L3. The cost function was defined as

$$\begin{array}{cc}& J(k,\zeta )=\sum _n{\gamma }^2(f_n)[{log}_{10}|H_{33}^{\text{mod}}(f_n;k,\zeta )|\hfill \\ \hfill & -{log}_{10}|H_{33}^{\text{exp}}(f_n)|{]}^2,\hfill \end{array}$$ 13

where ${\gamma }^2(f_n)$ is the coherence at frequency bin $f_n$, summed over the 5–120 Hz bandwidth. The upper frequency limit of 120 Hz was chosen to remain below the identified bone-pin resonance at 131 Hz.

Optimization was performed using a Nelder–Mead simplex algorithm (Nelder and Mead 1965) with multi-start initialization across a grid of $k\in [20,150]$ kN/m and $\zeta \in [0.05,0.50]$. The logarithmic cost function weights the compliance spectrum uniformly across the multiple orders of magnitude spanned by the data. Goodness of fit was assessed using the coherence-weighted coefficient of determination ($R^2$) in log-compliance space. Optimization was performed independently for each of the six experimental conditions.

Time-domain displacement analysis

To evaluate the models’ ability to predict vertebral displacement magnitudes, time-domain simulations were performed by direct integration of Eq. (4) using the Newmark–beta average acceleration method (Newmark 1959). This unconditionally stable implicit scheme uses the parameters $\gamma =1/2$ and $\beta =1/4$. At each time step $\mathrm{\Delta }t$, the displacement and velocity are updated as

$$\begin{array}{cc}\hfill {\mathbf{q}}_{n+1}& ={\mathbf{q}}_n+\mathrm{\Delta }t{\dot{\mathbf{q}}}_n\hfill \\ \hfill & +\frac{\mathrm{\Delta }{t}^2}{4}({\ddot{\mathbf{q}}}_n+{\ddot{\mathbf{q}}}_{n+1}),\hfill \end{array}$$ 14

$$\begin{array}{cc}\hfill {\dot{\mathbf{q}}}_{n+1}& ={\dot{\mathbf{q}}}_n+\frac{\mathrm{\Delta }t}{2}({\ddot{\mathbf{q}}}_n+{\ddot{\mathbf{q}}}_{n+1}),\hfill \end{array}$$ 15

with ${\ddot{\mathbf{q}}}_{n+1}$ obtained by solving the effective stiffness system:

$$\begin{array}{c}\hfill \widehat{\mathbf{K}}{\mathbf{q}}_{n+1}={\widehat{\mathbf{f}}}_{n+1},\end{array}$$ 16

where $\widehat{\mathbf{K}}=\mathbf{K}+\frac{4}{\mathrm{\Delta }{t}^2}\mathbf{M}+\frac{2}{\mathrm{\Delta }t}\mathbf{C}$ is the effective stiffness matrix and ${\widehat{\mathbf{f}}}_{n+1}$ is the effective force vector incorporating the current state.

Time to peak force and impulse duration were characterized from the raw 25 kHz force recordings. The mean contact force pulse durations (0.42-−0.52 ms across settings) were shorter than the known impulse duration of the Impulse iQ instrument (Colloca et al. 2000, 2004) due to mechanical impedance coupling at the spine contact point. For each of the six experimental conditions, a half-sine force pulse with a 3.2 ms duration and amplitude equal to the measured mean peak force was applied at the L3 DOF. Simulations were run for 50 ms at a time step of $\mathrm{\Delta }t=0.05$ ms using the patient-specific optimized parameters from the inverse analysis.

Results

Model natural frequencies

The 5-DOF model with published parameters ($k=50$ kN/m, $\zeta =0.15$) predicted five undamped natural frequencies of 46.1, 84.8, 121.1, 141.3, and 158.3 Hz, with corresponding damped natural frequencies of 45.6, 83.8, 119.8, 139.7, and 156.5 Hz (Table 1). These values are consistent with those reported in the original publication (Keller and Colloca 2002). The 21-DOF model predicted 21 natural frequencies spanning 6.5–637 Hz, with the fundamental PA-axis modes at approximately 41, 91, and 122 Hz, consistent with the published values (Keller et al. 2002).

Table 1.

Natural frequencies of vibration (Hz) predicted by the 5-DOF and 21-DOF lumped-parameter models with published parameters

Mode	5-DOF Undamped	5-DOF Damped	21-DOF PA	P03 Exp. (Hz)	P04 Exp. (Hz)
1	46.1	45.6	41.1	48.8	48.8
2	84.8	83.8	91.0	88.5	100.7
3	121.1	119.8	121.9	106.8	109.9

Experimental peak frequencies were identified from combined driving-point ($H_{33}$) and transfer ($H_{23}$, $H_{43}$) PA-axis accelerance spectra. Only modes below the bone-pin resonance at 131 Hz are reported

Three experimental modes were identified below the bone-pin resonance at 131 Hz from the combined driving-point ($H_{33}$) and transfer ($H_{23}$, $H_{43}$) PA-axis accelerance spectra (Table 1). Model modes 4 and 5 (141–158 Hz) could not be validated experimentally because they fall above the pin resonance.

The first experimental mode at 48.8 Hz was clearly identified in both patients across all force settings and all PA-axis channels. This frequency is within 7% of the 5-DOF model prediction of 45.6 Hz and within 16% of the 21-DOF model’s first PA mode at 41 Hz. The second mode appeared at 88.5 Hz for Patient 03 (observed in 4 spectra across 3 settings in both $H_{33}$ and $H_{23}$ channels) and at 100.7 Hz for Patient 04 (observed in 4 spectra across 3 settings). The third mode was identified at 106.8 Hz for Patient 03 (4 observations across 2 settings) and 109.9 Hz for Patient 04 (3 observations across 2 settings).

Driving-point compliance characteristics

Both patients exhibited a dominant first resonance in the driving-point compliance spectrum near 49 Hz at the L3 spinous process contact (Table 2). Five of six conditions showed a compliance peak at 48.8 Hz, while Patient 03 at the High force setting exhibited a peak at 36.6 Hz. Patient 03 compliance magnitudes at the peak frequency ranged from 0.224 to 0.378 mm/N. Patient 04 exhibited compliance magnitudes of 0.549 to 1.762 mm/N. Coherence at the peak frequency was at least 0.79 for all Medium and High force settings, confirming reliable FRF estimation at therapeutic force levels. At Low force settings, coherence was lower (0.28 and 0.36 for Patients 03 and 04, respectively), reflecting the reduced signal-to-noise ratio.

Table 2.

Experimental driving-point compliance characteristics at the L3 spinous process contact

Patient	Setting	Force (N)	n Thrusts	Peak Hz	$|H_{33}|$ at Peak (mm/N)	${\gamma }^2$ at Peak
P03	Low	57.7	32	48.8	0.224	0.28
P03	Med	171.5	21	48.8	0.378	0.79
P03	Hi	295.9	34	36.6	0.279	0.91
P04	Low	85.7	25	48.8	1.762	0.36
P04	Med	208.6	34	48.8	0.549	0.98
P04	Hi	240.9	32	48.8	0.570	0.97

Model–experiment compliance comparison

Figure 1 illustrates the 5-DOF model dynamic response to a representative impulse (Patient 03, High setting, 296 N, 3.2 ms), showing the temporal sequence of vertebral displacements alongside the displacement time history at all five lumbar levels. Figure 2 shows the comparison between the 5-DOF model compliance and the experimental driving-point compliance at L3 for both patients. The model with published parameters captured the overall spectral shape, including the approximate location of the first resonance peak. The optimized model provided substantially improved agreement, particularly in the 20–100 Hz bandwidth where the compliance magnitude and resonance peak amplitude were better matched. The 21-DOF model (Fig. 3) showed similar trends in the PA compliance response, with the additional capability of predicting coupled axial and rotational responses. The experimental compliance spectra exhibited fluctuations at frequencies below approximately 30 Hz. These arise from: (i) the AC-coupled piezoelectric force transducer (discharge time constant $\approx$1.5 s), which introduces spectral distortion at low frequencies; (ii) the frequency resolution ($\mathrm{\Delta }f\approx 3.1$ Hz) being a significant fraction of the frequency itself at low frequencies, increasing spectral leakage from the Hanning window; and (iii) the impulsive excitation concentrating energy primarily in the 20–120 Hz bandwidth, yielding reduced signal-to-noise ratio and lower coherence below 20 Hz.

Fig. 1.

Dynamic response of the 5-DOF lumped-parameter model to a 3.2 ms half-sine force pulse (Patient 03, High setting: 296 N peak) applied at L3. Top panels (A–F): temporal sequence of vertebral positions at six time points showing wave propagation through the lumbar chain. Bottom panel: displacement time histories at all five lumbar levels (L1–L5), with L3 exhibiting the largest peak displacement (5.60 mm at 3.8 ms)

Fig. 2.

Driving-point compliance magnitude at the L3 spinous process: comparison of 5-DOF lumped-parameter model predictions (published parameters: $k=50$ kN/m, $\zeta =0.15$; optimized parameters) with experimental data for Patient 03 (top) and Patient 04 (bottom) at three force settings (Low, Medium, High). Shaded regions indicate coherence below 0.5

Fig. 3.

Driving-point compliance magnitude at the L3 spinous process: comparison of 21-DOF sagittal plane model predictions (published parameters: $\zeta =0.25$) with experimental data for Patient 03 (top) and Patient 04 (bottom) at three force settings

Inverse analysis

Table 3 presents the patient-specific stiffness and damping estimates from the inverse analysis. For Patient 03, the optimized PA stiffness ranged from 52.5 kN/m (High force setting, 296 N) to 56.5 kN/m (Medium setting, 172 N), with damping ratios between 0.071 and 0.109 and $R^2$ values of 0.68-−0.83. For Patient 04, stiffness ranged from 46.7 kN/m (Low setting, 86 N) to 53.9 kN/m (Medium setting, 209 N), with damping ratios of 0.049-−0.085 and $R^2$ values of 0.57-−0.76. Across all six conditions, the mean optimized stiffness was $52.3\pm 3.5$ kN/m (range: 46.7-−56.5 kN/m), in close agreement with the published model parameter of 50 kN/m. The mean damping ratio was $0.079\pm 0.020$ (range: 0.049-−0.109), lower than the published value of 0.15 but within the physiological range.

Table 3.

Patient-specific PA stiffness (k) and modal damping ratio ($\zeta$) estimates from frequency-domain inverse analysis at L3

Patient	Setting	Force (N)	$k_{\text{opt}}$ (kN/m)	${\zeta }_{\text{opt}}$	$f_1$ (Hz)	$R^2$
P03	Low	57.7	54.2	0.082	35.5	0.68
P03	Med	171.5	56.5	0.071	36.2	0.83
P03	Hi	295.9	52.5	0.109	34.9	0.80
P04	Low	85.7	46.7	0.085	32.9	0.57
P04	Med	208.6	53.9	0.078	35.4	0.76
P04	Hi	240.9	49.9	0.049	34.1	0.71

Patient 03 showed stiffness values of 54.2, 56.5, and 52.5 kN/m at Low, Medium, and High force settings, respectively. The lowest stiffness at the highest force amplitude is consistent with expected nonlinear softening behavior at higher loading amplitudes (Keller and Spengler 1987). Patient 04 showed stiffness values of 46.7, 53.9, and 49.9 kN/m at Low, Medium, and High settings, respectively. The lower stiffness at Patient 04’s Low setting coincided with the lowest coherence (0.36), suggesting that measurement noise at low force levels may have influenced the parameter estimation.

Sensitivity analysis

The sensitivity analysis (Fig. 4) demonstrated that the model-predicted first resonance frequency shifts from 32 Hz ($k=25$ kN/m) to 56 Hz ($k=75$ kN/m), spanning the experimental resonance at 48.8 Hz. The damping ratio primarily affected the peak amplitude: $\zeta =0.05$ produced a sharp resonance peak, while $\zeta =0.40$ produced a heavily damped response with minimal amplification. The experimental compliance spectra for both patients fell within the envelope bounded by $k\in [25,75]$ kN/m and $\zeta \in [0.05,0.40]$.

Fig. 4.

Sensitivity of the 5-DOF model driving-point compliance spectrum at L3 to variations in PA stiffness ($k=25$, 50, 75 kN/m at fixed $\zeta =0.15$; left panel) and modal damping ratio ($\zeta =0.05$, 0.15, 0.40 at fixed $k=50$ kN/m; right panel). Experimental compliance spectra for Patients 03 and 04 (Medium force setting) are shown for comparison

Time-domain displacement predictions

Table 4 presents the model-predicted peak PA displacements at L1–L5 for all six conditions using the patient-specific optimized parameters and a 3.2 ms impulse duration. The 5-DOF model predicted L3 displacements ranging from 1.10 mm (Patient 03, Low setting, 58 N) to 5.60 mm (Patient 03, High setting, 296 N). The contact point (L3) exhibited the largest displacement in all conditions, with adjacent segments L2 and L4 reaching 78–88% of the L3 displacement, and the more distal segments L1 and L5 reaching 48–57%. Time-to-peak displacement at L3 ranged from 3.8 to 4.0 ms, exceeding the 3.2 ms impulse duration as the system continues to accelerate during the free vibration phase following force cessation. A representative displacement waveform (Fig. 5) illustrates the oscillatory nature of the response and the wave propagation pattern through the multi-mass chain.

Table 4.

Model-predicted peak posteroanterior (PA) vertebral displacements (mm) from the 5-DOF model using patient-specific optimized parameters

Patient	Setting	Force (N)	TTP (ms)	Dur (ms)	L1	L2	L3	L4	L5
P03	Low	57.7	1.04	3.2	0.59	0.88	1.10	0.95	0.56
P03	Med	171.5	0.98	3.2	1.76	2.61	3.23	2.81	1.69
P03	Hi	295.9	1.00	3.2	2.84	4.38	5.60	4.70	2.71
P04	Low	85.7	1.04	3.2	0.94	1.41	1.78	1.52	0.90
P04	Med	208.6	0.98	3.2	2.15	3.21	4.02	3.46	2.06
P04	Hi	240.9	1.00	3.2	2.84	4.09	5.02	4.42	2.73

TTP, time to peak force; Dur, impulse duration

Fig. 5.

Model-predicted PA displacement waveforms at L1–L5 from the 5-DOF model (optimized parameters) in response to a 3.2 ms half-sine force pulse (Patient 03, High setting: 296 N peak) applied at L3. The L3 contact point shows the largest displacement (5.60 mm peak at 3.8 ms), with adjacent segments L2 and L4 reaching 4.38 and 4.70 mm, respectively. Displacements diminish with distance from the contact, and time-to-peak increases for more distal segments due to wave propagation through the multi-mass chain

Model fit quality ($R^2$) was positively correlated with measurement coherence at the compliance peak frequency ($r=0.69$), confirming that the coherence-weighted inverse approach yields better parameter estimates when the signal-to-noise ratio is higher. The optimized stiffness showed no significant dependence on applied force amplitude ($r=0.11$), indicating that the linear model assumption was adequate across the 58–296 N force range.

Discussion

This study presents the first in vivo experimental validation of lumped-parameter models of the human lumbar spine using driving-point compliance FRF methodology combined with transfer accelerance data from adjacent vertebral segments. By computing FRFs at the L3 driving point ($H_{33}$) and at the L2 and L4 bone-pin accelerometers ($H_{23}$, $H_{43}$), the analysis identifies three experimental PA-axis modes below the bone-pin resonance limit and directly tests the models’ ability to predict the observed resonance structure. The results demonstrate that the 5-DOF model predicts natural frequencies that agree with the three experimentally identified modes to within 6–17%, and that patient-specific PA stiffness can be estimated from the compliance data through frequency-domain inverse analysis.

The dominant experimental compliance resonance at approximately 49 Hz is consistent with the 5-DOF model’s first damped natural frequency of 45.6 Hz and the 21-DOF model’s first PA mode at approximately 41 Hz. This frequency also agrees with the mechanical impedance resonance of 40–50 Hz reported by Colloca et al. (2000) for in vivo lumbar spine assessment during instrument-assisted manipulation and with the 46 Hz resonance predicted by the analytical model of Solinger (2000). The consistency across models, experiments, and prior literature provides strong evidence that the fundamental PA-axis natural frequency of the human lumbar spine lies in the 40–50 Hz range.

The second experimental mode (89–101 Hz) and third mode (107–110 Hz) were identified through combined analysis of the driving-point and transfer PA-axis accelerance spectra. These higher modes were not consistently visible in the driving-point FRF alone, because the 5-DOF mode shapes predict displacement nodes near L3 for modes 2 and 3, which would suppress the driving-point response at those frequencies while remaining detectable in the transfer measurements at L2 and L4.

The patient-specific stiffness estimates of 52.5-−56.5 kN/m for Patient 03 and 46.7-−53.9 kN/m for Patient 04 are in close agreement with the published model parameter of 50 kN/m and within the 25–75 kN/m range examined in both original publications (Keller and Colloca 2002; Keller et al. 2002). The mean stiffness across all conditions ($52.3\pm 3.5$ kN/m) differs from the published value by only 5%. These stiffness values are notably higher than the 10–20 kN/m segmental stiffness values typically obtained from low-frequency mobilization experiments (Lee and Svensson 1990; Edmondston et al. 1998; Kawchuk and Fauvel 2001), which is consistent with the frequency-dependent stiffening behavior expected from the viscoelastic properties of spinal tissues. At the higher loading rates produced by impulsive mechanical excitation, the effective stiffness increases due to the rate-dependent behavior of the intervertebral disc, facet joints, and ligamentous structures (Keller and Spengler 1987; Brown et al. 2008; Panjabi et al. 1977). This distinction highlights the importance of characterizing spinal mechanics across the full frequency bandwidth of clinical loading conditions, rather than relying solely on quasi-static assessments.

The optimized damping ratios (mean: $0.079\pm 0.020$) were lower than the published value of 0.15 for the 5-DOF model and 0.25 for the 21-DOF model. This difference may reflect the nature of the excitation source; the impulsive forces from instrument-assisted manipulation produce predominantly elastic responses, whereas the published damping ratios were estimated from quasi-static load-deformation data that include greater viscoelastic contributions. The relatively low damping ratios are consistent with the sharp compliance resonance peaks observed experimentally, which indicate a lightly damped system at the frequencies of interest.

The choice to focus on the L3 spinous process contact was motivated by the model topology. In the 5-DOF chain model, L3 (DOF 3) occupies the central position with two internal springs on each side, providing the most symmetric and well-constrained driving point. The L5 contact (DOF 5), by contrast, is an end mass with only one internal spring connection to L4 and a boundary spring to the sacrum, making the model less sensitive to interior spring parameters. Preliminary analysis of L5 contact data confirmed that the 5-DOF model provided poor fits at L5 (damping ratio estimates hitting constraint boundaries), consistent with the asymmetric boundary conditions at the chain terminus.

The greater variability observed in Patient 04’s parameter estimates, particularly at the Low force setting ($k=46.7$ kN/m, $R^2=0.57$), may reflect the influence of measurement noise on the inverse optimization. The Low setting coherence of 0.36 at the resonance frequency indicates a marginal signal-to-noise ratio. In contrast, the Medium and High settings for Patient 04 (coherence 0.97-−0.98) yielded stiffness estimates of 49.9-−53.9 kN/m with higher $R^2$ values (0.71-−0.76). This observation suggests that force amplitudes above approximately 100 N are needed to reliably estimate mechanical properties from instrument-assisted manipulation FRF data.

The model-predicted vertebral displacements of 1.10-−5.60 mm at L3 across the force range of 58–296 N are consistent with the cadaveric literature on vertebral motion during spinal manipulation. Gal et al. (1997) measured vertebral displacements of 1–5 mm during PA manipulative thrusts delivered to unembalmed human cadavers using manual techniques with longer thrust durations (100–200 ms). The smaller displacements predicted for the lower force settings are consistent with the force- and duration-dependent displacement relationship demonstrated by Colloca et al. (2006) in an ovine model, where shorter-duration thrusts (10 ms) produced approximately fivefold lower displacements compared to longer-duration thrusts (100–200 ms) at the same force amplitude. Maigne and Guillon (2000) measured intervertebral movements and intradiscal pressure changes during lumbar manipulation in unembalmed cadavers, providing further evidence that manipulation-induced displacements are in the millimeter range. In vivo human studies using the Impulse instrument have reported three-dimensional vertebral motion patterns consistent with the present model predictions. Keller et al. (2006) measured PA displacements of 1–4 mm at L3–L5 in live subjects during instrument-assisted SMT using intraosseous pins, and Colloca et al. (2004) demonstrated that biomechanical and neurophysiological responses to impulsive SMT are force-amplitude dependent in patients with lumbar disc herniation. Keller and Colloca (2003) provided further neuromechanical characterization of the in vivo lumbar spine response, reporting segmental motion patterns consistent with the multi-mass wave propagation predicted by the 5-DOF model. The agreement between our model predictions and these experimental observations provides confidence in the 5-DOF model as a tool for estimating vertebral motion during instrument-assisted spinal manipulation.

The consistency of the dominant resonant frequency ($\sim$49 Hz) across both patients warrants discussion in light of known age-related changes in intervertebral disc composition. At the high loading rates produced by the Impulse iQ ($\sim$2 ms duration, corresponding to loading rates of 50–150 kN/s), the spine’s mechanical response is dominated by the instantaneous stiffness rather than the equilibrium stiffness that characterizes quasi-static behavior. The frequency-dependent stiffening of intervertebral discs is well documented (Keller and Spengler 1987), and at the frequencies of interest (30–120 Hz), the effective stiffness may be less variable across ages than at quasi-static rates. Furthermore, the PA stiffness parameter k captures the combined contribution of the intervertebral disc, bilateral facet joints, posterior ligaments (interspinous, supraspinous, ligamentum flavum), and joint capsules. While the disc’s contribution changes with age, these other posterior elements may show different age-related trajectories. Both patients in this study were in a relatively narrow age range (within the 54–70 year cohort), so we cannot rule out that younger patients would exhibit different resonant frequencies. Future studies spanning a wider age range are needed to investigate age-dependent variation in spinal resonant frequency.

The mechanical properties estimated in this study reflect the response of the anesthetized spine. The patients were under general anesthesia with neuromuscular blockade allowed to dissipate prior to data collection to preserve reflexive EMG responses. The impulse duration ($\sim$2 ms) is far shorter than the fastest neuromuscular reflex latency ($\sim$30–50 ms), so reflex-mediated muscle activation could not have altered the spine’s stiffness during the impulse itself, though it could influence the damping characteristics of the subsequent free vibration. In awake patients, active muscle co-contraction would increase effective stiffness, which would shift resonant frequencies upward and decrease displacement amplitudes for a given force input, because higher stiffness reduces compliance. The present model therefore captures passive structural mechanics and should be interpreted accordingly. Mörl et al. (2020) demonstrated significant muscle contributions to spinal load distribution during passive flexion, suggesting that muscle contributions to effective joint stiffness can be substantial in clinical contexts.

The 21-DOF model introduces axial displacement and flexion-extension rotation at each level, enabling prediction of coupled motions that the 5-DOF PA-only model cannot capture. The first PA mode shifts from 46.1 Hz (5-DOF) to approximately 41 Hz (21-DOF) because the additional degrees of freedom allow energy redistribution through translation-rotation coupling, mediated by the spring axis offset parameter ($a=20$ mm). The 21-DOF model also predicts that impulsive PA forces produce concurrent flexion-extension rotations between adjacent vertebrae, controlled by the ratio of PA to torsional stiffness. This coupling is clinically relevant because intervertebral rotation may contribute to facet joint gapping and the therapeutic mechanisms proposed for spinal manipulation.

The clinical implications of this work extend beyond model validation. The ability to estimate patient-specific spinal stiffness and damping from compliance FRF measurements has the potential to provide clinicians with objective biomechanical metrics for diagnostic assessment, treatment planning, and outcomes evaluation. Compliance is a particularly intuitive metric for clinicians, as it directly quantifies how much a spinal segment displaces per unit of applied force. Establishing normative ranges for stiffness and damping across age, sex, and pathological conditions would enable evidence-based clinical decision-making.

Several limitations should be acknowledged. First, the sample size of two patients limits the generalizability of the parameter estimates and precludes statistical inference. Four patients were enrolled, but two were excluded (force transducer malfunction in Patient 01; anomalous compliance resonance near 150 Hz in Patient 02). This study should be viewed as a methodological proof-of-concept demonstrating the feasibility of the approach. Second, the bone-pin accelerometers were attached to the spinous processes rather than the vertebral bodies, introducing a lever arm that may affect the measured PA acceleration. Third, the lumped-parameter models assume linear spring behavior, homogeneous material properties, and simplified geometry that do not fully capture the complex anatomy of the spine. The linear stiffness assumption is reasonable for the small perturbation displacements (1–6 mm) produced about the preloaded operating point, analogous to small-signal linearization in structural dynamics. The lack of significant force-amplitude dependence in the optimized stiffness ($r=0.11$) provides partial support for the linear assumption within the tested force range. However, at larger amplitudes, nonlinear tissue behavior may become significant. Recent work by Hammer et al. (2024) has demonstrated energy-conserving surrogate models for the coupled, nonlinear responses of intervertebral discs using learned kernel functions. Incorporating such nonlinear coupling formulations into the lumped-parameter framework would require iterative frequency-domain solution or exclusively time-domain integration and substantially more experimental data, but represents a promising direction for extending the current approach. Fourth, the level-dependent stiffness profile ($k_1=1.0k$ through $k_6=0.9k$) reflects anatomical variation in posterior element properties across lumbar levels, but uniform damping was adopted because a single driving-point FRF provides insufficient information to independently estimate level-specific damping parameters. Fifth, while the transfer accelerance data from L2 and L4 bone-pin accelerometers were used to identify experimental modes, the inverse optimization was performed using driving-point compliance only. Future work should incorporate the transfer compliance FRFs ($H_{23}$, $H_{43}$) as additional objective functions to simultaneously constrain multiple model parameters and improve the estimation of inter-segmental spring stiffness ratios. Sixth, the bone-pin resonance at 131 Hz restricted the validated frequency range to the first three model modes; the fourth and fifth predicted modes (141–158 Hz) could not be assessed. Seventh, differences in underlying pathology among patients (e.g., spondylolisthesis, spinal stenosis, disc herniation) may alter segmental stiffness differently; larger cohorts with documented diagnoses are needed to determine how pathology influences the estimated mechanical properties.

Conclusions

This study demonstrates for the first time that previously published 5-DOF and 21-DOF lumped-parameter models of the human lumbar spine can be validated against in vivo FRF data obtained during instrument-assisted spinal manipulation at the L3 spinous process. Three experimental PA-axis modes were identified below the bone-pin resonance limit using combined driving-point and transfer accelerance spectra from L2, L3, and L4. The first mode at 49 Hz agreed with the 5-DOF model prediction to within 7%, and the second and third modes (89–101 Hz and 107–110 Hz) corresponded to the model’s second and third predicted frequencies. Patient-specific PA stiffness estimates (mean: $52.3\pm 3.5$ kN/m) were consistent with the published model parameter of 50 kN/m, and the coherence-weighted inverse analysis approach provided $R^2$ values of 0.57-−0.83 in log-compliance space. Model-predicted vertebral displacements of 1.10-−5.60 mm were consistent with cadaveric and in vivo spinal manipulation literature, providing a biomechanical framework for quantifying the mechanical dose delivered during SMT. Future multi-center studies with larger, demographically diverse cohorts spanning a wider age range are needed to establish normative ranges for spinal stiffness and damping, to investigate age- and pathology-dependent variation in resonant frequency, and to incorporate transfer compliance and nonlinear coupling formulations into the inverse optimization for improved multi-parameter estimation.

Modal decomposition of the 5-DOF compliance

The receptance (compliance) matrix for the 5-DOF system can be expressed in closed form using the modal decomposition. Let $\mathbf{\Phi }=[{\mathit{\varphi }}_1,\dots ,{\mathit{\varphi }}_5]$ denote the mass-normalized mode shape matrix satisfying ${\mathbf{\Phi }}^T\mathbf{M}\mathbf{\Phi }=\mathbf{I}$ and ${\mathbf{\Phi }}^T\mathbf{K}\mathbf{\Phi }={\mathbf{\Omega }}^2$. The receptance matrix is

$$\begin{array}{c}\hfill \mathbf{H}(\omega )=\mathbf{\Phi }\text{diag}\left(\frac{1}{{\omega }_n^2-{\omega }^2+2\text{i}{\zeta }_n{\omega }_n\omega }\right){\mathbf{\Phi }}^T.\end{array}$$ A1

Each diagonal element $H_{\mathit{jj}}(\omega )$ represents the driving-point compliance at DOF j:

$$\begin{array}{c}\hfill H_{\mathit{jj}}(\omega )=\sum _{n=1}^5\frac{{\varphi }_{\mathit{jn}}^2}{{\omega }_n^2-{\omega }^2+2\text{i}{\zeta }_n{\omega }_n\omega }.\end{array}$$ A2

The off-diagonal elements $H_{\mathit{jk}}(\omega )$ with $j\ne k$ represent transfer compliance:

$$\begin{array}{c}\hfill H_{\mathit{jk}}(\omega )=\sum _{n=1}^5\frac{{\varphi }_{\mathit{jn}}{\varphi }_{\mathit{kn}}}{{\omega }_n^2-{\omega }^2+2\text{i}{\zeta }_n{\omega }_n\omega }.\end{array}$$ A3

These transfer compliance functions can be directly compared to transfer FRF measurements obtained from bone-pin accelerometers. The (2,3) element $H_{23}(\omega )$ corresponds to the PA displacement at L2 per unit force at L3, and $H_{43}(\omega )$ to the displacement at L4 per unit force at L3.

For uniform damping (${\zeta }_n=\zeta$ for all n), the modal damping matrix simplifies to $\mathbf{C}=\mathbf{M}\mathbf{\Phi }[2\zeta \mathbf{\Omega }]{\mathbf{\Phi }}^T\mathbf{M}$, and each mode contributes a second-order resonance with quality factor $Q_n=1/(2\zeta )$.

Author contributions

C.J.C. conceived the study, developed the instrumentation, performed the data acquisition and analysis, and wrote the manuscript. R.G. provided surgical access, performed the bone-pin implantation procedures, and wrote and reviewed the manuscript.

Funding

The International Spine Research (INSPIRE) Foundation; CBP Nonprofit, Inc.; Foundation for the Advancement of Chiropractic Education.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Declarations

Conflict of interests

C.J.C. is the developer, CEO and majority shareholder of the company that manufactures  the Impulse iQ Adjusting Instrument (Neuromechanical Innovations, LLC, Phoenix, AZ) used in this study. R.G. declares no conflict of interest.

Ethical approval

This study was approved by the institutional ethics committee (Authorization Belgium 2009). Written informed consent was obtained from all participants prior to the surgical procedure.