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Phantom Experiment Design — Validation of Fluid-Filled Shell Model

Objective

Design a laboratory experiment to validate the analytical predictions of flexural mode frequencies and coupling efficiencies for a fluid-filled viscoelastic shell, using a tissue-mimicking phantom.

Motivation

The strongest reviewer criticism is the absence of experimental validation. Even a simple phantom experiment comparing analytical predictions with measured frequencies/amplitudes would dramatically strengthen the paper.

Phantom Design

Option A: Silicone Rubber Shell (Recommended)

Construction:

  1. Cast a hollow oblate spheroidal silicone shell using 3D-printed molds
  2. Inner mold: oblate spheroid (a=18cm, c=12cm)
  3. Outer mold: slightly larger oblate spheroid (a=19cm, c=13cm) → h=10mm
  4. Material: Ecoflex 00-30 or Dragon Skin 10 (Shore 00-30 to 10A)
  5. Fill with degassed water through a sealed port

Material Calibration:

  • Measure E of the silicone with a tensile test (Instron or similar)
  • Expected range: 0.05-0.5 MPa (Ecoflex) or 0.1-2 MPa (Dragon Skin)
  • Measure loss tangent with DMA (Dynamic Mechanical Analysis)
  • Target: E ≈ 0.1 MPa to match relaxed abdominal wall

Advantages:

  • E can be tuned by silicone formulation
  • Geometry is well-controlled and measurable
  • Water-filled → known fluid properties
  • Can inject air bubbles for gas pocket experiments
  • Can be constrained (partially clamped) to test BC effects

Disadvantages:

  • Isotropic (real tissue is anisotropic)
  • Uniform wall (real wall is multi-layer)
  • No organs or mesentery

Option B: Gelatin Shell

Construction:

  1. Double balloon technique: inflate balloon to spheroid shape in gelatin mold
  2. Multiple dipping to build up wall thickness
  3. Control E through gelatin concentration (10-20% w/v)
  4. Fill with water

Material properties:

  • E = 0.01-0.1 MPa (tunable with concentration)
  • More biologically realistic viscoelastic behavior
  • Limited shelf life (days), temperature-sensitive

Option C: Commercial Rubber Balloon (Quick Validation)

Construction:

  1. Large latex balloon (~15-20cm diameter when inflated)
  2. Fill with water to oblate spheroid shape (support in a bowl)
  3. Very thin wall (h ≈ 0.3-1mm)

Advantages: Fast, cheap, immediately available Disadvantages: Geometry not well controlled, very thin wall

Instrumentation

Frequency Measurement (Modal Analysis)

  1. Laser Doppler Vibrometer (LDV) — Gold standard

    • Polytec PSV-500 or similar
    • Scan the shell surface for mode shapes
    • Frequency resolution: 0.01 Hz
    • Non-contact → no mass loading

  2. Accelerometers — Simpler

    • Lightweight MEMS accelerometers (< 1g mass)
    • Mount to shell surface with wax
    • Multiple points for mode identification
    • Concern: mass loading on soft shell

  3. Hydrophone — Internal pressure measurement

    • B&K 8103 or similar miniature hydrophone
    • Insert through sealed port
    • Measures internal pressure oscillation

Excitation

Mechanical (WBV simulation):

  • Electrodynamic shaker (Brüel & Kjær 4809 or similar)
  • Phantom sits on rigid plate mounted to shaker
  • Sine sweep 1-20 Hz, acceleration 0.1-2.0 m/s²
  • Control with accelerometer on plate

Airborne acoustic:

  • Large subwoofer (18" or 21") in sealed enclosure
  • Or: dedicated infrasound source (Rotary Woofer, pneumatic)
  • Calibration microphone (G.R.A.S. 40AZ or similar)
  • SPL range: 90-130 dB at 2-20 Hz
  • Test in anechoic chamber or large room (λ ≈ 50m at 7 Hz!)

Impact (for modal identification):

  • Instrumented impact hammer with soft tip
  • Impulse response → FFT → natural frequencies
  • Quick screening method before detailed sweeps

Experimental Protocol

Phase 1: Modal Identification (Impact Test)

  1. Fill phantom, seal, equilibrate temperature (22°C)
  2. Mount phantom on vibration-isolated stand
  3. Impact with instrumented hammer at 10 locations
  4. Record response at 3-5 locations (accelerometers or LDV)
  5. Compute FRFs → identify natural frequencies and mode shapes
  6. Compare with analytical predictions

Phase 2: Mechanical Excitation (Shaker)

  1. Mount phantom on shaker platform
  2. Sine sweep: 1-20 Hz, 0.1 m/s² to 2.0 m/s²
  3. Record:
    • Base acceleration (control accelerometer)
    • Shell surface displacement (LDV)
    • Internal pressure (hydrophone)
  4. Compute transfer functions
  5. Compare with model predictions:
    • Resonant frequency
    • Transmissibility T(f)
    • Internal pressure amplitude

Phase 3: Airborne Excitation

  1. Place phantom in anechoic environment (or large room)
  2. Expose to infrasound: 2-20 Hz, 90-130 dB
  3. Record:
    • Incident pressure (reference microphone)
    • Shell surface displacement (LDV)
    • Internal pressure (hydrophone)
  4. Measure coupling coefficient: ξ_measured / ξ_predicted
  5. KEY TEST: Verify (ka)^n coupling penalty

Phase 4: Gas Pocket Experiments

  1. Inject known volumes of air (1, 5, 10, 20, 50 mL)
  2. Repeat Phase 3 at each air volume
  3. Measure:
    • Change in natural frequencies
    • Change in airborne coupling
    • Local displacement near gas pocket (LDV pointed at bubble)
  4. Compare with gas pocket resonance model predictions

Phase 5: Boundary Condition Effects

  1. Partially clamp phantom (rigid supports simulating spine/pelvis)
  2. Repeat Phase 1 with different constraint configurations
  3. Measure frequency shift vs. constraint geometry
  4. Compare with BC multiplier estimates

Expected Results

What We Expect to Confirm

  • n=2 flexural mode at 4-10 Hz (depending on silicone E)
  • Breathing mode at much higher frequency (>100 Hz for silicone)
  • Mechanical coupling >> airborne coupling (by 10³ factor)
  • Gas injection increases airborne coupling

What Might Surprise Us

  • Non-linear effects at large amplitude (WBV)
  • Mode coupling between flexural and sloshing modes
  • The breathing mode might be measurable as internal pressure oscillation
  • Gas pocket resonance might be at unexpected frequency (not Minnaert)

Budget Estimate

Item Cost Range
Silicone (Ecoflex 00-30, 2 gallons) $80-120
3D-printed molds (resin, ~500g) $50-100
Accelerometers (3× MEMS) $50-200
Hydrophone (miniature) $200-500
Impact hammer (PCB 086C03) ~$1500 (may borrow)
Shaker + amplifier (existing lab) $0 (shared facility)
Subwoofer (18") $200-500
Calibration microphone $500-1000
Data acquisition system $0 (existing lab)
Total (excluding shared equipment) $600-2500

Timeline

  • Phantom fabrication: 1-2 weeks
  • Material characterization: 1 week
  • Modal testing: 1 week
  • Mechanical excitation: 1 week
  • Acoustic excitation: 1-2 weeks
  • Gas pocket experiments: 1 week
  • Data analysis and comparison: 2 weeks
  • Total: 8-10 weeks

Key Measurements for Paper

The minimum set needed for a compelling validation section:

  1. Measured f₂ vs. predicted f₂ — the core frequency validation
  2. Transmissibility T(f) curve — compare with model and ISO 2631
  3. Airborne displacement vs. SPL — confirm weak coupling
  4. Mechanical displacement vs. acceleration — confirm strong coupling
  5. Coupling ratio: ξ_mech / ξ_air — the key novel result

If time permits: 6. Mode shapes from scanning LDV 7. Gas pocket coupling enhancement 8. BC sensitivity