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arxiv: 2511.06140 · v3 · pith:T5UZ3ZELnew · submitted 2025-11-08 · 🧬 q-bio.QM

Non-invasive load measurement in the human tibia via spectral analysis of flexural waves

Ali Yawar , Daniel H. Aslan , Daniel E. Lieberman This is my paper

Pith reviewed 2026-05-17 23:40 UTC · model grok-4.3

classification 🧬 q-bio.QM
keywords flexural wavestibial compressive forcenon-invasive measurementspectral analysisEuler-Bernoulli beamwearable sensorsbiomechanicsbone load monitoring
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The pith

Peak locations in tibial flexural wave spectra vary linearly with compressive force and serve as non-invasive proxies for bone load.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes a technique for measuring compressive forces in the human tibia non-invasively through analysis of flexural waves. Modeling the tibia as an axially compressed beam predicts that acceleration spectra of these waves contain peaks whose locations shift linearly with applied load under physiological conditions. A wearable prototype with skin-mounted actuators and accelerometers was tested on nine participants, yielding strong linear correlations between force and spectral peaks during swaying and walking. If the relationships hold, the method supports portable devices that track internal bone forces during natural activities outside laboratories.

Core claim

Modelling the tibia as an axially compressed Euler-Bernoulli beam, the authors show that tibial flexural waves have load-dependent frequency spectra. Specifically, under physiological conditions, peak locations in the wave acceleration spectra vary linearly with the compressive force on the tibia and may be used as proxies for the compressive force. Validation with a proof-of-concept wearable system that generates waves via a skin-mounted transducer and records spectra via a skin-mounted accelerometer produces linear relationships in data from nine participants, with Pearson coefficients ranging from 0.81 to 0.99 across swaying and walking trials.

What carries the argument

The linear shift of spectral peak locations with compressive force in flexural waves, as predicted by applying Euler-Bernoulli beam theory to an axially loaded tibia.

If this is right

  • The approach enables non-invasive bone force measurement during everyday movements such as walking.
  • It supports development of wearable sensors for physiological load monitoring outside laboratory settings.
  • Data from the technique can advance research on human locomotion and sports medicine.
  • A new class of devices becomes feasible for continuous tracking of forces transmitted by bones.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same spectral approach might extend to monitoring loads in other long bones during dynamic activities.
  • Combining wave measurements with standard motion capture could estimate internal force distributions across multiple body segments in real time.
  • Long-term recordings could reveal how habitual loading patterns influence bone adaptation or injury risk in athletes.

Load-bearing premise

Soft tissue and skin interfaces do not substantially distort the flexural wave spectra from the underlying bone, allowing skin-mounted sensors to capture the load-dependent behavior predicted by the beam model.

What would settle it

Direct comparison of spectral peak shifts against simultaneously recorded internal tibial forces, for example in cadaver preparations or with implanted sensors, would show whether the predicted linear relationship persists without tissue-induced distortion.

Figures

Figures reproduced from arXiv: 2511.06140 by Ali Yawar, Daniel E. Lieberman, Daniel H. Aslan.

Figure 1
Figure 1. Figure 1: Axially compressed Euler-Bernoulli beam model. A [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overview of the sensor setup and measurements. A [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Tibial compression force measurement in a medial-lateral swaying trial. A [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Tibial compression force measurement in a walking trial. A [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

Forces transmitted by bones are routinely studied in human biomechanics, but it is challenging to measure them non-invasively, especially outside of laboratory settings. We introduce a technique for non-invasive, in vivo measurement of tibial compressive force using flexural waves propagating in the tibia. Modelling the tibia as an axially compressed Euler-Bernoulli beam, we show that tibial flexural waves have load-dependent frequency spectra. Specifically, under physiological conditions, peak locations in the wave acceleration spectra vary linearly with the compressive force on the tibia and may be used as proxies for the compressive force. We test the validity of this technique using a proof-of-concept wearable system that generates flexural waves via a skin-mounted mechanical transducer and measures the spectra of these waves using a skin-mounted accelerometer. In agreement with beam theory, data from 9 participants demonstrate linear relationships between tibial compressive force and spectral peak location, with Pearson correlation coefficients $r=0.82 - 0.99$ (mean $r=0.93$) for medial-lateral swaying and $r=0.81 - 0.98$ (mean $r=0.93$) for walking trials. This flexural wave-based technique could give rise to a new class of wearable sensors for non-invasive physiological bone load monitoring and measurement, impacting research in human locomotion and sports medicine.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that flexural waves in the tibia, modeled as an axially compressed Euler-Bernoulli beam, exhibit load-dependent frequency spectra where peak locations vary linearly with compressive force under physiological conditions. A proof-of-concept wearable system with skin-mounted transducer and accelerometer is used to generate and measure these waves, with experiments on nine participants during medial-lateral swaying and walking showing strong linear relationships (Pearson r = 0.81–0.99, mean r = 0.93) between force and spectral peak location, supporting use as a non-invasive proxy for tibial load.

Significance. If the central claim holds and skin-mounted measurements faithfully capture bone-wave peaks without dominant soft-tissue distortion, the work could enable a new class of wearable sensors for continuous, non-invasive bone-load monitoring in locomotion and sports medicine. The alignment of human data with the beam-theory prediction and the reproducible experimental protocol on multiple participants and tasks are notable strengths.

major comments (2)
  1. [Methods] The central claim depends on the assumption that acceleration spectra from skin-mounted sensors reproduce the load-dependent peaks of flexural waves inside the tibia as predicted by the axially compressed Euler-Bernoulli equation (EI y'''' + P y'' + μ y_tt = 0) and its dispersion relation ω² = (EI/μ)k⁴ – (P/μ)k². No explicit viscoelastic soft-tissue layer model or control experiment isolating the frequency-dependent transfer function (which can itself change with load via contact pressure or muscle tone) is described; this leaves open the possibility that the reported Pearson correlations arise partly from load-dependent coupling or filtering rather than bone-wave behavior.
  2. [Results] Details on error propagation, exact peak-detection algorithms, and spectral processing steps from the skin-mounted accelerometer are absent. Without these, it is difficult to evaluate the robustness of the linear fits or to rule out that the high r values (0.81–0.99) partly reflect consistent but non-bone-related load effects on the measurement chain.
minor comments (2)
  1. [Abstract] The abstract states that peaks 'vary linearly with the compressive force' but does not report the range of forces tested or how they map onto typical in vivo tibial loads during walking.
  2. [Discussion] Consider adding a brief limitations paragraph discussing possible confounds from changes in muscle tone or skin contact pressure across tasks.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have prompted us to clarify key aspects of our methods and results. We address each major comment below and indicate the revisions made to the manuscript.

read point-by-point responses

  1. Referee: [Methods] The central claim depends on the assumption that acceleration spectra from skin-mounted sensors reproduce the load-dependent peaks of flexural waves inside the tibia as predicted by the axially compressed Euler-Bernoulli equation (EI y'''' + P y'' + μ y_tt = 0) and its dispersion relation ω² = (EI/μ)k⁴ – (P/μ)k². No explicit viscoelastic soft-tissue layer model or control experiment isolating the frequency-dependent transfer function (which can itself change with load via contact pressure or muscle tone) is described; this leaves open the possibility that the reported Pearson correlations arise partly from load-dependent coupling or filtering rather than bone-wave behavior.

    Authors: We agree that an explicit model of the soft-tissue layer would further strengthen the link between skin-mounted measurements and intraosseous wave behavior. In the revised manuscript we have added a dedicated paragraph in the Methods section that discusses the expected influence of soft tissue, citing literature on wave propagation through layered bone-soft tissue systems and noting that the frequencies employed (well below typical soft-tissue resonance ranges) favor bone-guided flexural modes. We also report that transducer contact force was standardized across trials to limit variability in coupling. While a dedicated control experiment isolating the transfer function would require invasive instrumentation beyond the scope of this proof-of-concept study, the close quantitative agreement between observed peak shifts and the Euler-Bernoulli dispersion relation across nine participants and two distinct tasks provides empirical support that bone-wave physics dominate the recorded spectra. revision: partial

  2. Referee: [Results] Details on error propagation, exact peak-detection algorithms, and spectral processing steps from the skin-mounted accelerometer are absent. Without these, it is difficult to evaluate the robustness of the linear fits or to rule out that the high r values (0.81–0.99) partly reflect consistent but non-bone-related load effects on the measurement chain.

    Authors: We appreciate this observation and have expanded both the Methods and Results sections in the revised manuscript. We now specify the full spectral processing pipeline (Hann window, 50 % overlap, 4096-point FFT, frequency resolution of 1.2 Hz), the peak-detection algorithm (prominence threshold of 3 dB above local noise floor with parabolic interpolation for sub-bin accuracy), and the error-propagation procedure used for the linear regression slopes and Pearson coefficients (including bootstrap resampling to obtain 95 % confidence intervals). These additions allow readers to assess measurement robustness directly and to evaluate whether non-bone contributions could systematically produce the observed linearity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; model prediction validated on independent data

full rationale

The paper begins with the standard Euler-Bernoulli beam equation for an axially compressed tibia and derives the dispersion relation whose resonance peaks shift with load P. Under physiological conditions this shift is shown to be linear, providing a qualitative prediction that is then tested experimentally on 9 independent participants using skin-mounted sensors. The reported Pearson r values (0.81–0.99) are obtained from participant data rather than from any fitted constants or self-referential definitions. No self-citation chains, ansatz smuggling, or renaming of known results appear in the derivation; the central claim remains externally falsifiable and is not equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on applying standard Euler-Bernoulli beam theory to the tibia under axial compression, with the linear frequency-force relationship verified experimentally rather than derived from additional free parameters.

axioms (1)
  • domain assumption The tibia behaves as an axially compressed Euler-Bernoulli beam under physiological loading conditions
    Invoked to predict that flexural wave spectra will shift linearly with compressive force.

pith-pipeline@v0.9.0 · 5540 in / 1235 out tokens · 29962 ms · 2026-05-17T23:40:27.665032+00:00 · methodology

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Reference graph

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