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Bodyseismography (BSG): Theoretical Foundation and Numerical Modeling

WenZhan Song, Yuanyuan Zhang, Yuyan Wu, Sahas Biddappa, Yida Zhang, Jiayu Chen, Zixuan Zeng, Yixuan Li, Jiahui Li, Fei Dou, Zhenlin An, Hae Young Noh, Young Kwon

arXiv:2610.35844v2eess.SPphysics.med-ph

TL;DR

BSG vibrations reflect cardiovascular activity together with body–bed transmission, complicating their physiological interpretation. This paper combines theoretical analysis with cardiac and transmission models to relate directional observations to hemodynamic targets. Its results include waveform reconstruction across 27 participants and complementary directional information about stroke-volume changes.

  • Problem

    Relating BSG vibrations to hemodynamic features requires separating cardiac excitation from the effects of body and bed transmission.

    Source: Overview · p. 4

  • Method

    The paper develops a source–transmission–measurement framework and combines cardiac mechanics, cardiac-to-body excitation, and calibrated body–bed models.

    Source: §13 · p. 69

  • Results

    Across 27 participants, multilayer FEM achieved median PCCs of 0.865 for Z and 0.682 for X + Y; controlled simulations reached PCC 0.928 with fused features.

    Source: §13 · p. 69

  • Takeaways & Limitations

    Transmission can alter or reverse a BSG feature’s association with simulated SV, while combining directional features tracks simulated SV changes more consistently.

    Source: Overview · p. 6

    The projection imposes no coefficient bounds or prior penalty, so a small residual does not establish that its minimizing parameter changes are physiologically bounded.

    Source: §8.4 · p. 38

Abstract

from arXiv · show

Bodyseismography (BSG) records cardiovascular vibrations transmitted through the body and a supporting bed, enabling contactless assessment of cardiac mechanics. Relating these vibrations to hemodynamic features requires separating cardiac excitation from the effects of body and bed transmission. We develop a source-transmission-measurement framework that couples myocardial contraction and vascular recoil to directional BSG observations. The theory connects stroke volume, cardiac output, and blood pressure to observation and target Jacobians, establishes a local first-order target-observability criterion, and characterizes how additional sensing axes reduce ambiguities under specified source and transmission uncertainties. Numerical modeling combines four-chamber cardiac mechanics, cardiac-to-body excitation, and calibrated body-bed operators. Across 27 participants, a multilayer finite-element model with calibrated frequency correction achieves median Pearson correlation coefficients of 0.865 for the vertical channel and 0.682 for the planar-sum channel in matched-window waveform reconstruction. Controlled stroke-volume perturbations show that transmission reshapes feature sensitivity, including a reversal of the vertical feature association in 8 of 27 participant-specific transfer models. These analytical and numerical studies explain how cardiovascular sources and mechanical propagation jointly shape triaxial BSG and provide a physical basis for relating directional vibration features to hemodynamic changes.

Overview

The overview frames BSG as cardiovascular motion shaped by body–bed transmission, and develops a physical framework for interpreting its directional measurements. It connects that framework to hemodynamic observability and numerical modeling.

Overview

  • The physical account connects cardiovascular dynamics, mechanical excitation, body–bed propagation, and sensor observations to explain BSG generation and waveform interpretation.It treats source generation and transmission both individually and together.Source: Overview · pp. 4–5
  • Local cardiac deformation and whole-body recoil follow distinct mechanical paths, whose directions and timing combine in BSG.The framework links local cardiac motion and blood-momentum exchange to the recorded signal.Source: Overview · p. 5
  • Body and bed transfer functions reshape waveform amplitude, timing, and spectrum, as well as the relation between BSG features and hemodynamics.Measured waveforms show cardiac features shifting between directions and mattress-dependent changes in amplitude and timing.Source: Overview · p. 6
  • The theoretical framework links shared cardiovascular dynamics, both loading pathways, body–bed mechanics, and directional observations to local observability of SV, CO, and BP.The benefit of combining axes depends on the target and modeled uncertainty; concurrent physiological changes can imitate target-related responses.Source: Overview · p. 6
  • Horizontal and vertical BSG features retain complementary information about simulated stroke volume, and their combination tracks change direction more consistently than either alone.Measured-recording transfer models also show that body and bed mechanics can alter or reverse a feature’s association with simulated SV.Source: Overview · p. 6

Preliminary

  • The preliminary chapter establishes triaxial BSG measurement and relates its SCG and BCG components to cardiovascular mechanics and hemodynamics.Its examples motivate target-specific estimation and the later analytical and computational representations.Source: Preliminary · p. 7

1. Triaxial BSG and Its Hemodynamic Information

Triaxial BSG records cardiac vibrations shaped by both cardiovascular sources and body–bed transmission. The section relates directional observations to hemodynamic quantities and explains why target-specific inference requires physiological and mechanical constraints.

1.1.1 Operating Principle and Advantages for BCG Sensing

  • 0.19 µg RMS equivalent acceleration noise is estimated for the BSG device, versus 3.1 and 30.5 µg RMS for the two MEMS models under the stated spectral assumptions.The geophone’s suspended coil converts relative motion to voltage, and its calibrated complete-device response determines retained bed-motion components.Source: §1.1.1 · pp. 9–10

1.1 Definition, Sensor Characteristics, and Measured Waveforms

  • Cardiac features shift across directions with posture, while a soft/thick mattress attenuates signals and shifts timing, so a longitudinal projection can suppress morphology.These observations motivate modeling cardiac excitation and body–bed propagation together.Source: §1.1.1 · pp. 10–11

1.2 Hemodynamic Quantities Represented by the Mechanical Signal

  • CO is HR × SV, while mean pressure depends on CO and systemic vascular resistance; recoil, volume flow, and BP therefore represent distinct hemodynamic quantities.Systolic and diastolic pressures also depend on pulsatile flow, compliance, and wave reflection.Source: §1.2 · pp. 11–12

1.3 SCG: Cardiac Timing, Stroke Volume, and Diastolic Function

  • 11.48 mL held-out SV RMSE and R2 = 0.76 were reported for combined ECG and SCG features in congenital heart disease, outperforming either modality alone.Diastolic SCG peak-to-peak amplitude correlated with e′ (r = 0.713, p < 0.001); usable data came from 38 supine and 39 tilted subjects. Timing constrains ejection duration, but SV also depends on flow magnitude and shape.Source: §1.3 · p. 13

1.4 BCG: Blood Momentum, Ejection, and Arterial Loading

  • SCG reflects local chest-wall motion from myocardial contraction, whereas BCG recoil reflects blood-momentum exchange; treating contact-force recordings as whole-body recoil conflates measurement with mechanism.The distinction matters because mattress loading and whole-body recoil are different physical pathways.Source: §1.4 · p. 13
  • I–J amplitude increases were positively associated with CO increases (Spearman rs = 0.923, p < 0.001) in 11 participants; exercise-to-rest CO ratios correlated with BCG maximum-power ratios (r = 0.65, p = 0.008), while SV-ratio relationships were nonsignificant.The reported features distinguish associations with CO from estimation of its separate HR and SV components.Source: §1.4 · p. 15

1.5 From Hemodynamic Sensitivity to Inverse Estimation

  • Physiological and transmission changes can reinforce, cancel, or reverse a feature trend, so SV recovery requires observations or constraints that distinguish source magnitude from mechanical gain.Absolute BP also requires physiological information about resistance, compliance, and wave reflection; the framework organizes complementary SCG and BCG information for target-specific estimation.Source: §1.5 · p. 16

Theoretical Foundation of BSG

The framework follows cardiovascular physiology through two mechanical pathways into bed-recorded vibration, then asks which hemodynamic targets remain distinguishable. It couples the physiology and mechanics while examining how sensing axes and uncertainty affect target recovery.

Theoretical Foundation of BSG

  • The chapter traces a shared physiological source through vibration generation and transmission before assessing which hemodynamic targets remain distinguishable.Its seven sections cover physiological parameters, targets, loading pathways, body–bed mechanics, observation, and target observability.Source: Theoretical Foundation of BSG · p. 18
  • Vascular recoil and myocardial transmission are distinct pathways from a common state, not isolated physiological sources in their measured signals.Vascular recoil is associated with BCG and myocardial transmission with SCG.Source: Theoretical Foundation of BSG · p. 18
  • The framework couples pressure, volume, thorax motion, and body–bed contact, while treating vascular recoil as a one-way mechanical input.Numerical analyses compare seven axis combinations and examine mechanical compensation and source uncertainty in target recovery.Source: Theoretical Foundation of BSG · p. 18

2. Framework and Physiological Parameters

The framework maps a coupled cardiovascular state and two mechanical loading pathways to bed-recorded vibration, then evaluates physiological targets from that state. It distinguishes evolving state variables from model parameters and makes axis selection and calibration assumptions explicit.

2. Framework and Physiological Parameters

  • BSG records cardiovascular vibration after transmission through the body, mattress contact, and bed structure, with vascular recoil and cardiac contraction as loading pathways.SV, CO, and PP are principal targets; MAP and pressure-waveform shape can also be evaluated.Source: §2 · p. 19
  • The seven core physiological coordinates are distinct from evolving states and constitutive outputs such as pressure, flow, ventricular volume, and active tension.The table defines the coordinates; proximal stiffness and terminal impedance are represented by dimensionless scale factors and are effective model parameters, not automatically identifiable anatomical measurements.Source: §2 · pp. 19–20, Table 2.1 · p. 19
  • The free-parameter vector combines physiological and mechanical/calibration coordinates with residual-contraction parameters and a pressure-reference offset.Calibrated and fixed quantities are not free coordinates; the residual-contraction parameterization must state which kinetics vary.Source: §2 · p. 20
  • The complete state contains time-varying circulation, cardiac, and body–bed variables, whereas the parameter vector specifies one calculation.The coupled equations impose governing balances and constraints, with predicted vibration returned for the selected axes.Source: §2 · p. 21
  • Recorded axes may be stacked as single or combined x, y, and z channels, but native sensor polarity requires calibration.The model frame defines positive x leftward, y footward, and z anterior; writing a target as a function of physiology alone assumes other parameters are fixed or feedback is neglected.Source: §2 · p. 21

3. How Physiology Determines the Target Information

The section defines hemodynamic targets from the coupled circulation and explains how cardiovascular flow and cardiac motion become transmitted vibration. It also distinguishes scalar pressure measures from waveform targets and notes that physiological parameter effects interact.

3. How Physiology Determines the Target Information

  • Circulation balances link compartment pressure, flow, and volume, while ventricular pressure is solved jointly with cardiac mechanics.The storage expression assumes fixed external pressure and locally linear compliance, though nonlinear pressure–volume laws can replace it.Source: §3 · p. 21
  • SV is integrated aortic outflow between valve opening and closure, while CO combines SV and heart rate; pressure targets use the waveform at a specified site.PP is systolic minus diastolic pressure, and MAP is the time-averaged pressure; central aortic pressure and peripheral ABP require an explicit transfer relationship when compared.Source: §3 · p. 22
  • Heart rate, contractility, stressed volume, resistance, compliance, stiffness, and terminal impedance interact, so one coordinate has no generally universal monotone effect on SV or PP.Source: §3 · p. 22
  • A small CO perturbation includes both the explicit heart-rate effect and induced changes in filling and ejection.The relation PP ≈ SV/Cart requires additional assumptions and is not the target definition used here.Source: §3 · p. 22
  • Pressure-waveform targets preserve information that scalar summaries discard: one removes the mean, while range normalization also removes amplitude.Neither retains the absolute-pressure offset, and target definitions, pressure site, and time/phase alignment must stay fixed when comparing axes.Source: §3 · p. 22
  • Segment-flow derivatives generate vascular recoil, cardiac wall motion generates attachment loads, and body motion relative to the bed produces contact forces and bed velocity.The aligned simulated traces show timing, polarity, and shape changes but do not constitute a measured separation of the sources.Source: §3 · pp. 22–23, Figure 3.1 · p. 23

4. Vascular Recoil Force and Moment

The model represents vascular recoil with three cells along an assumed aortic centreline and derives force and moment from the same flow solution and geometry. Its baseline source loads show directional cancellation, while the reduction uses a one-way vascular pathway and omits several momentum contributions.

4.1 Segment Selection and Three-Dimensional Geometry

  • Three consecutive cells divide an assumed aortic centreline, with a terminal RCR load closing the reduced network.The three-cell choice is not demonstrated to be a minimum or to represent three independent vessels; the stated positions are anatomical assumptions, not subject imaging.Source: §4.1 · p. 24
  • The three-dimensional centreline uses endpoint chords for resultant recoil direction, with root and descending chords pointing in opposite longitudinal directions.The arch contributes a vertical component, and source-orientation perturbations rotate the whole centreline about the valve.Source: §4.1 · p. 24, Figure 4.1 · p. 25

4.2 Signed Force and Moment from the Same Centreline

  • Recoil force follows the change in blood momentum, and moment is derived from the distributed centreline rather than an independently assigned point arm.For curved segments, the chord—not arc length times an arbitrary tangent—sets the resultant force direction.Source: §4.2 · pp. 24–25

4.3 Scope of the Reduction

  • 0.02, 0.35, and 0.16 N are the baseline recoil-force RMS values on x, y, and z, respectively, with a predominant pitching moment about x.Segment timing and opposing chord directions cause cancellation in the resultant; these are source loads before body–bed transmission.Source: §4.3 · p. 26
  • The fixed, coarse-centreline reduction omits intracardiac, branch, peripheral, venous, boundary-flux, and vessel-motion contributions.It also requires a spatial projection to represent deformation-mode forcing.Source: §4.3 · p. 26
  • The vascular pathway is one-way, omitting body/vessel acceleration feedback into flow while retaining cardiac–thorax feedback.The reported small pressure-term ratios are a diagnostic, not a demonstrated bound on target-recovery or axis-ranking error.Source: §4.3 · p. 26

5. Cardiac-Contraction Loading

Cardiac contraction reaches the thorax as a generalized mechanical load produced by active contraction, passive deformation, and attachment motion. Because shape changes can alter attachment loads without changing cavity volume, stroke volume, cardiac output, or pressure alone does not uniquely determine that load.

5. Cardiac-Contraction Loading

  • The reduced wall-mechanics system combines passive restoring load, cavity-pressure load, active generalized load, and a flow balance that sets cavity-volume change.The active generalized cardiac load is distinct from scalar blood flow Q.Source: §5 · p. 27
  • Cardiac and thorax coordinates map to attachment displacements, whose relative motion and link stiffness and damping generate forces and a generalized thorax load.The heart receives the paired reaction, allowing thorax/contact motion to affect the cardiac state.Source: §5 · p. 27
  • The transmitted load includes resultant forces, moments, and local deformation loads projected from attachment forces according to attachment geometry and the stiffness/damping law.Force directions are not fixed by the vascular chords.Source: §5 · p. 27
  • Active stress is fibre-directed, while the modeled wall mechanics use calcium-dependent contraction, reduced fibre geometry, passive wall mechanics, and mediastinal, chest, and diaphragm links.Pressure follows the wall/flow balance rather than an added elastance pressure generator.Source: §5 · pp. 27–28
  • SV, CO, or a pressure waveform alone does not uniquely specify the transmitted cardiac-contraction force.Isovolumic shape changes can alter attachment loads without directly changing cavity volume.Source: §5 · p. 28

6. From the Two Loads to Body and Bed Vibration

Cardiac-contraction and vascular-recoil pathways force the retained body–bed system, with contact mechanics coupling body motion to bed motion. The bed-mounted sensor records calibrated directional velocity, but a bed axis does not identify the source contributing to it.

6. From the Two Loads to Body and Bed Vibration

  • The retained body–bed system is forced by the two cardiovascular loading pathways, with disjoint loading boundaries required to avoid counting attachment tractions twice.The modeled forcing closes once spatial projections are specified, but does not represent every physiological or environmental excitation.Source: §6 · p. 28
  • At each contact, relative body–bed displacement drives contact force, which acts oppositely on body and bed and contributes to assembled stiffness and damping.The bed response also changes contact force through the coupled mechanics.Source: §6 · pp. 28–29
  • Cardiac attachment forces are evaluated from coupled wall mechanics and relative attachment motion, rather than replaced by a fixed transfer function with prescribed loads.Their stiffness and damping contributions are already included through the cardiac-contraction load.Source: §6 · p. 29
  • The sensor observes calibrated bed velocity along transverse x, longitudinal y, and vertical z, with gain, filtering, polarity, timing, and noise affecting the measurement.In the periodic model, velocity is obtained from acceleration by inverse-frequency weighting, so the two can rank axes differently.Source: §6 · p. 29
  • A bed axis mixes both loading pathways, so its name does not identify a source fraction.The contact illustration resolves body translation, rotation, local thorax displacement, bed displacement, and contact forces as distinct components.Source: §6 · p. 29, Figure 6.1 · p. 30

7. BSG, SCG, and BCG as Observations of One Coupled Motion

BCG, SCG, and BSG observe different locations and motions of one coupled mechanical solution, in which vascular recoil and cardiac contraction both contribute to BSG. The source decomposition holds coupled load histories fixed, and its assumptions and practical recovery implications remain limited.

7. BSG, SCG, and BCG as Observations of One Coupled Motion

  • BCG is torso rigid-body translational acceleration, SCG is precordial chest-wall acceleration, and BSG is bed-sensor velocity.They are distinct observation modalities, not names for three independent forces.Source: §7 · pp. 30–31
  • In the unrestrained limit, BCG reflects momentum exchange among body, heart, and represented blood; SCG also contains rotation and local chest deformation.Ventricular volume and shape affect SCG motion, and neither volume acceleration alone nor force amplitude fully specifies it.Source: §7 · p. 31
  • 150 versus 98 µm/s on y, 45 versus 91 on z, and 10 versus 33 on x are the BSG RMS velocities for vascular recoil versus cardiac-contraction contributions.Opposing contributions can cancel, so these magnitudes are not additive energy fractions.Source: §7 · p. 31, Figure 7.1 · p. 32
  • Vascular recoil dominates BCG on y and z, while cardiac contraction dominates local SCG.The figure separates total, vascular-recoil, and cardiac-contraction responses at body, chest, and bed locations; sums reproduce totals to approximately 0.1%, 1.1%, and 0.2% for BCG, SCG, and BSG, respectively.Source: §7 · p. 31, Figure 7.1 · p. 32
  • The decomposition holds coupled load histories fixed; removing a source and re-solving would change feedback and need not yield additive responses.Chest-mode frequencies, sensor location, apex attachment, and chest-wall participation are unvalidated assumptions; practical recovery improvement from combining modalities is not established.Source: §7 · p. 31

8. Observability and the Value of Multiple Axes

The section examines local observability of target recovery and how sensing-axis combinations constrain target-changing responses under specified models and uncertainty. It shows that coverage and mechanical compensation rank axes differently, and that multiple axes can improve separation from specified mechanical variation without guaranteeing stable, uncalibrated recovery.

8.1 Concepts and Target Recovery

  • Noiseless first-order target recovery requires every target-Jacobian row to lie in the observation-Jacobian row space; adding axes intersects their invisible-change null spaces.Three axes do not imply three independent physiological degrees of freedom, nor does a larger score show that every axis pair is inadequate.Source: §8.1 · pp. 32–33

8.2 Soft Coverage and Target-Preserving Compensation

  • The scale-dependent soft coverage score and compensation residual diagnose different properties, and neither alone establishes stable recovery in noise.A small normalized residual means allowed target-preserving directions can imitate much of the target-changing observation response; enlarging the compensation family cannot increase that residual under fixed normalization.Source: §8.2 · p. 34

8.3 Tolerance, Pressure Reference, and Finite Uncertainty

  • Tightening the target cutoff raises all-direction waveform residuals from 0.01–0.15 to 0.35–0.79, while a mechanically silent pressure-reference offset obstructs absolute-pressure recovery.The offset does not enter dynamics but changes reported MAP; pulse pressure and centered or normalized waveforms remove this particular ambiguity.Source: §8.3 · p. 35

8.4 How the Numerical Indicators Are Calculated

  • The numerical indicators use total derivatives of the coupled model and fixed scaling across axis combinations, but the local compensation projection imposes no coefficient bounds or prior penalty.A small residual can reveal a local confounding mechanism without showing that the minimizing changes are physiologically feasible.Source: §8.4 · pp. 37–38

8.5 Results across Scalar and Waveform Targets

  • xyz coverage is 0.976 for SV, 0.965 for CO, and 0.983 for PP, while the best mechanical-compensation pair varies by target.The best pair is xz for SV, PP, and both waveform targets, xy for CO, and yz for MAP; adding the third axis raises mechanical residuals by 0.049–0.172 across the six targets.Source: Figure 8.1 · p. 39, §8.5 · pp. 39–40
  • Only 0.014 of the SV and 0.034 of the PP response norms remain under xyz all-direction compensation, indicating weak local separation, not estimation-error percentages.The result uses the broader family of target-preserving latent changes; it does not establish finite-amplitude ambiguity or clinical recovery error.Source: Table 8.3 · p. 40, §8.5 · p. 40
  • Bounded searches retain 3.7–4.5% all-axis waveform mismatch for imposed 10% SV/PP changes; measured-data held-out NRMSE is 1.12 versus 1.04 with baseline mechanics.The finite-range searches differ from the primary linear projection, and the measured-data fit did not validate source-to-bed transfer or demonstrate SV/PP recovery.Source: §8.5 · p. 40

8.6 Conclusions Supported by the Theory and Figures

  • 0.014 for SV and 0.034 for PP are the xyz all-direction residuals, revealing strong local confounding despite three-axis BSG’s complementary spatial constraints.The longitudinal bed-velocity response is strongest at baseline but easiest for mechanical variation to imitate; the largest separation gain comes from adding a second axis, with further target-dependent gains from a third. Practical sufficiency requires justified uncertainty bounds, calibrated observation gain, polarity, timing and noise, independent validation of the coupled mechanical transfer, and comparison with the best fixed dual-axis alternative under the same target-error criterion.Source: §8.6 · p. 41

Numerical Modeling and Validation of BSG

The BSG digital twin links cardiovascular dynamics to body–bed transmission and sensor response. This chapter describes computational realizations of that source-to-measurement chain.

Numerical Modeling and Validation of BSG

  • The framework represents cardiovascular dynamics, body–bed transmission, and sensor response in one computational representation.Its parameters or state are conditioned on measurements of the physical system.Source: Numerical Modeling and Validation of BSG · p. 42
  • The digital-twin framework combines four-chamber cardiac mechanics, cardiac-to-body excitation, and reduced body–bed transmission.The following sections examine waveform fitting and stroke-volume-related feature transfer.Source: Numerical Modeling and Validation of BSG · p. 42

9. A Common Architecture for the BSG Digital Twin

The digital-twin architecture separates cardiovascular source generation from source-to-measurement transmission while linking both through shared physiology. Its computational representations connect cardiac state and interface loads to directional sensor observations.

9. A Common Architecture for the BSG Digital Twin

  • Cardiac physiology supplies mechanical source histories, while the source-to-measurement operator combines excitation mapping with body–bed propagation and sensor observation.The hierarchy separates source generation G from transmission H; four-region FEM and participant-conditioned FIR models are complementary realizations.Source: §9 · p. 43
  • A force-driven branch supplies body force, whereas an alternative branch prescribes interface motion; both begin from cardiac source histories.The four-chamber representation replaces reduced cardiac coordinates with spatial tissue mechanics and four chamber states.Source: §9 · pp. 43–44
  • The architecture assigns cardiac source calculation, body excitation, effective transfer realization, and numerical conclusions to successive model stages.The four-region FEM realization includes excitation mapping, while HBB alone begins at body excitation.Source: §9 · p. 43
  • The coupled representation uses cardiac coordinates, body–thorax–bed coordinates, hemodynamic states, and algebraic variables in a differential–algebraic system.The calibrated acquisition operator maps the system output, and valve states make the governing equations piecewise defined.Source: §9 · pp. 43–44
  • The model couples LV pressure and aortic-root pressure feedback with cardiac–thorax attachment forces, connecting activation, pressure, flow, recoil, and structural vibration.The shared state also provides consistent recoil timing alongside pressure and flow.Source: §9 · p. 44
  • The framework interprets BSG amplitude and morphology through source dynamics, directional propagation, and measurement response together.Physiological interpretation uses the specified source, propagation, and measurement operators with their parameters.Source: §9 · p. 44

10. Four-Chamber Cardiac Mechanics and Wall Motion

The cardiac model couples four-chamber mechanics to circulation and extracts directional local wall-motion signals for comparison with a reference simulation. It reproduces pressure–volume trajectories and local displacement and acceleration with distinct qualifications on scope and processing.

10. Four-Chamber Cardiac Mechanics and Wall Motion

  • The four-chamber model couples three-dimensional cardiac mechanics to lumped circulation, using upstream geometry, material state, and active-stress histories as inputs.The loaded geometry carries deformation and stress, so its transfer requires both shape and initial stress.Source: §10 · p. 45, §10.1 · p. 46

10.2 Fiber Activation, Active Stress, and Prestress

  • Fiber orientation sets active-tension direction, while upstream conduction timing and calcium-sensitive contraction determine activation and tension histories.The mechanical input uses stored active stress; subtracting its initial value preserves imported prestress as subsequent deformation and tension change stress.Source: §10.2 · pp. 47–48

10.3 Cavity Loading, Finite Supports, and Nearly Incompressible Mechanics

  • Moving cavity pressure loads and finite spring–damper supports couple wall motion to chamber loading, while nodal pressure averaging modifies volumetric stress in nearly incompressible tissue.The JAX averaging pool spans all 331,365 cells, including material interfaces; discrete equivalence with the upstream operator has not been established.Source: §10.3 · pp. 49–50

10.4 Coupled Three-Dimensional Mechanics and Zero-Dimensional Circulation

  • The semi-implicit mechanics–circulation coupling updates structure and solves four-cavity pressures using trial estimates of the compliance matrix.It uses a fixed number of pressure iterations without an outer iteration to converge the full interface.Source: §10.4 · pp. 50–51

10.5 Four-Chamber Pressure–Volume Agreement

  • Pressure RMSEs were 4.92, 1.73, 2.27, and 1.11 mmHg for LV, RV, LA, and RA, respectively, in the adjusted simulation comparison.Corresponding volume RMSEs were 4.07, 5.64, 3.94, and 4.10 mL; multipliers were selected from the same reference pressure–volume histories.Source: §10.5 · p. 51, Figure 10.7 · p. 52

10.6.1 Estimating the Body Axes from the Heart Mesh; 10.6.2 Selecting the Cardiac Surface Sampling Patches

  • The body frame is estimated from cardiac anatomy, then three fixed 500-node surface patches provide X, Y, and Z local acceleration traces.The curves share systolic timing but differ in amplitude and polarity; they describe myocardial motion before chest-wall and body–bed transmission.Source: §10.6.1 · p. 53, §10.6.2 · pp. 53–54, Figure 10.8 · p. 54

10.6.3 From Displacement Snapshots to the Z-Axis Proxy SCG

  • The Z-axis proxy SCG is computed by twice differentiating sampled displacement and projecting onto the estimated body frame; its peak absolute amplitude was 5.15 m/s2.Snapshot spacing sets a nominal sampling ceiling near 100 Hz, and numerical differentiation changes frequency response and amplifies displacement errors.Source: §10.6.3 · p. 55

10.6.4 Reference Agreement in Local Wall Motion

  • Z-axis PCC was 0.959 for local wall displacement and 0.715 for acceleration over 30–1000 ms in the reference comparison.Displacement RMSE was 2.30 mm; excursion was 9.45 mm versus 7.13 mm in the reference. JAX and filtered-reference peak absolute accelerations were 5.15 and 6.02 m/s2, respectively.Source: §10.6.4 · p. 55

11. Cardiac-to-Body Mechanical Excitation

Cardiac wall motion and flow histories are converted into force-driven or prescribed-motion inputs for body–bed models. The section defines the source mappings and vascular recoil construction, with the evaluated prescribed-source implementation using one-way coupling.

11. Cardiac-to-Body Mechanical Excitation; 11.1 Mechanical Source Interface

  • The section converts cardiac wall motion, acceleration, inertial force proxy, and flow histories into body excitation with specified spatial support, coordinates, and units.The source interface can combine body force and prescribed interface displacement as alternative loading branches.Source: §11 · p. 56, §11.1 · p. 56

11.2 Prescribed-Motion Excitation; 11.2.1 Displacement Mapping and Mechanical Work

  • Prescribed interface displacement is mapped across coordinates and meshes, with the transpose serving as the work-conjugate force map.Preserving rigid translations and rotations supports conservation of resultant force and moment; prescribing displacement and traction arbitrarily on the same degree of freedom generally overconstrains the interface.Source: §11.2 · pp. 56–57, §11.2.1 · p. 57, §11.2.2 · p. 57

11.2.2 Equivalent Excitation on the Unconstrained Coordinates

  • Prescribed motion excites unconstrained body coordinates through inertial, damping, and stiffness coupling terms.Applying the constrained reaction again as an independent force would double-count the excitation.Source: §11.2.2 · p. 57

11.3 Force-Driven Excitation; 11.3.1 Cardiac Inertial Force as a Body–Bed Excitation

  • The whole-heart inertial force proxy sums nodal acceleration using imported lumped masses, while local SCG acceleration is a distinct motion observable.The proxy does not separate support, pressure, or numerical stabilization forces and requires a transfer operator with matching input units.Source: §11.3.1 · pp. 57–58

11.3.2 Vascular Recoil as a Distributed Body–Bed Excitation

  • The vascular-recoil source distributes flow-related loads along an approximated aortic path with 16 points and prescribed segment geometry.The participant’s complete aorta is outside the available anatomical reconstruction.Source: §11.3.2 · p. 58, Figure 11.1 · p. 58
  • The prescribed-flow approximation leaves pulsatile fluid–structure interaction unresolved, and combining momentum-derived and pressure-based reactions requires control-volume accounting to avoid including overlapping mechanisms twice.The implemented LV outflow reduction assumes negligible inflow during ejection; filtering and clipping yielded 62.82 mL versus a 63.93-mL volume excursion.Source: §11.3.2 · p. 59
  • The distributed recoil loads combine temporal, convective, and curvature contributions, producing signed loads that vary across ascending, arch, and descending regions.The decompositions sum to the same longitudinal force.Source: §11.3.2 · p. 59, Figure 11.2 · p. 59

11.4 Coupling Assumption and Excitation Handoff

  • The evaluated prescribed-source implementation uses one-way coupling, with cardiac trajectories driving body deformation but no return of body traction to the cardiac solve.The section hands off an excitation with defined support, coordinates, polarity, units, and time origin.Source: §11.4 · p. 60

12. Simulated Body–Bed System

The section evaluates alternative body–bed transmission models, then tests how transmission and fusion affect stroke-volume feature relationships and held-out waveform reconstruction. Across these experiments, results depend on calibration, channel direction, and whether the endpoint is waveform fit or SV association.

12.1 Reduced Transmission and Multilayer FEM

  • The reduced FIR operator combines excitation and transmission, with axis-specific gains, delays, and source weights representing effective response under a fixed operating condition.Its 121 taps act on a periodic 100-sample cycle as a morphology operator, not as 121 independent physical delays or causal transit times.Source: §12 · p. 61, §12.1 · p. 61
  • The multilayer FEM uses four conforming body, interface, mattress/contact, and bed-board regions, while matching horizontal and vertical coupling in the waveform experiment.The reduced model has 190 quadrilateral elements and 223 nodes; its reported channels are vertical Z and planar sum XY, not three independently validated anatomical axes.Source: §12.1 · pp. 61–62, §12.2 · p. 62

12.2 Matched-Window Comparison of Analytical, FEM, and FIR Models

  • 0.865 for Z and 0.682 for XY are the multilayer FEM’s median PCCs after frequency correction, versus 0.878 and 0.699 for the analytical model and 0.870 and 0.678 for FIR.Uncorrected FEM and analytical medians are 0.838 and 0.644, and 0.807 and 0.501, respectively; paired intervals versus FIR span both signs, leaving model ordering unresolved without establishing equivalence.Source: §12.2 · p. 63, Figure 12.2 · p. 63, Table 12.1 · p. 63

12.3 Transmission Shapes Feature Sensitivity to Stroke Volume

  • +0.928 PCC and 18/18 correctly directed SV changes result from equal-weight fusion of the two bed-level features, versus PCCs +0.644 and +0.536 and 12/18 and 10/18 directions for the individual features.The combined low-contractility/low-resistance state is incorrectly directed under another fusion analysis, and the participant-specific vertical association reverses relative to input in 8 of 27 models.Source: Figure 12.3 · p. 65, §12.3 · p. 66

12.4 Calibration Determines the Benefit of Feature Fusion

  • Across 27 transfer models, fusion increases |r| over Z in 21 and over the stronger single channel in 11; median gains are +0.153 and −0.019, respectively.Fusion has a median of 15/18 correctly signed changes, matching XY and exceeding Z’s 8/18; signed channel responses can either complement or reverse under the shared weight.Source: §12.4 · p. 66, Figure 12.4 · p. 67
  • Two labeled calibration states improve test PCC over XY alone in four of five participants, but the fifth decreases from 0.821 to 0.671.The selected calibration states influence how the coefficient carries feature complementarity into the remaining perturbations.Source: §12.4 · pp. 66–67

12.5.1 Preserving Source Locations Through the Observation Model

  • The temporal experiment preserves separate vascular source locations because summing vascular sources before transmission can impose a common path and discard spatial information.The heart-plus-three-segment source model outperforms timing control on held-out data under both acquisition splits, with lower correlations on a separate acquisition set.Source: §12.5.1 · p. 68

12.5 Cardiovascular Source Models Increase Held-Out Waveform Correlation

  • 0.374 median held-out PCC for the heart-plus-three-segment source model exceeds the timing control’s 0.317 on same-acquisition splits; on a withheld acquisition set, the medians are 0.210 and 0.179.The measured chest-SCG reference reaches 0.399 on same-acquisition splits; correlations are lower when transferring to a separate acquisition set.Source: §12.5.1 · p. 68

13. Conclusions and Key Takeaways

The conclusions frame BSG as a coupled cardiovascular-source and body–bed-transmission measurement problem. They emphasize complementary directional information and the role of transmission in shaping both waveform morphology and feature sensitivity.

13. Conclusions and Key Takeaways

  • Across 27 participants, the frequency-corrected multilayer FEM reaches median PCCs of 0.865 for Z and 0.682 for X + Y in waveform reconstruction.The framework links cardiovascular dynamics, mechanical excitation, body–bed transmission, and sensor observation.Source: §13 · p. 69
  • In the illustrated bed, combining horizontal spectral-centroid and vertical peak-width features reaches PCC 0.928 and recovers all 18 simulated SV-change directions, versus 12 and 10 for the individual features.The conclusion relates these results to the SV, CO, and BP observability analysis in Chapter 2.Source: §13 · p. 69
  • Across 27 recording-derived transfer models, identical simulated cardiac perturbations yield different feature–SV relationships, including vertical-association reversals in 8 models relative to source input.The conclusion identifies body–bed transfer functions as part of the physiological measurement model.Source: §13 · p. 69

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