Willow BioSim™: The World Model & Simulation Engine for Biomechanical Injury Prediction

Willow BioSim™: The World Model & Simulation Engine for Biomechanical Injury Prediction

 

Technical Whitepaper | Willow BioSim™ 3

Executive Summary

Traditional biomechanical analysis often treats injury risk as a binary event: either forces exceed a static limit, or they do not. This approach fails to account for two critical biological realities: muscular load sharing (the active protection of ligaments by muscle mass) and accumulated micro-trauma (the probabilistic nature of fatigue failure).

The Willow BioSim™ 3 Model represents a paradigm shift from deterministic failure thresholds to a Stochastic Probabilistic Model. By utilizing probabilistic simulations driven by computer vision-derived kinematics, BioSim™ quantifies the probability of tissue failure across Acute (single-pitch), Meso (session), and Macro (seasonal) time horizons.

This document outlines the mathematical foundation, physiological assumptions, and computational methodology of the BioSim™ engine for biomechanical engineers and medical professionals.

1. Data Acquisition & Anthropometric Modeling

1.1 Kinematic Inputs

The model ingests high-frequency pose estimation data converted into a kinematic time series. The primary drivers for the simulation are:

  • Peak Arm Angular Velocity (\(\omega\)): The maximum rotational velocity of the forearm/hand segment during the acceleration phase (deg/s).
  • Max External Rotation (MER): The maximum "layback" angle of the forearm relative to the torso plane (degrees).

1.2 Anthropometric Inertia Estimation

To convert kinematics into kinetics (Torque/Force), we must solve for the specific inertial properties of the athlete. BioSim utilizes the Zatsiorsky-De Leva parameters to estimate segment mass and radii based on patient height (\(H\)) and weight (\(W\)).

Mass Estimation:

$$ m_{forearm\_hand} = W_{kg} \times 0.022 $$ $$ m_{upper\_arm} = W_{kg} \times 0.028 $$

Radius of Gyration & Inertia:

We model the arm as a lever arm rotating around the shoulder/elbow axis. The Moment of Inertia (\(I\)) is calculated approximating the radius of gyration as 50% of segment length (\(r\)):

$$ I_{segment} = m_{segment} \times (r_{segment} \times 0.5)^2 $$ $$ I_{total\_arm} = I_{forearm} + I_{upper\_arm} $$

2. The Stochastic Engine: Probabilistic Simulation

Human movement is inherently variable; no two pitches are identical. A single data point is insufficient for determining risk. BioSim calculates the Coefficient of Variation (CV) for the session's (all repetitions in a session) velocity and geometry.

We enforce a "Biological Variance Floor" of 4% (\(CV \geq 0.04\)) to account for natural physiological noise. The engine then generates 1,000 synthetic pitches (\(N=1000\)) using a normal distribution:

$$ Velocity_{synthetic} \sim \mathcal{N}(\mu_{velocity}, \sigma_{velocity}^2) $$

This allows us to evaluate risk not just on the average mechanics, but on the volatility of the athlete’s performance, capturing the outliers where injury often occurs.

3. Kinetic Logic: The Physics of Failure

3.1 UCL Stress: The Load-Sharing Correction

Legacy models often calculate Gross Valgus Torque and compare it directly to the UCL's failure limit (\(\approx 34\) Nm in cadaveric studies). This creates a "Cliff Edge" error, where elite pitchers generating 80+ Nm of torque are flagged as "100% Risk" on every pitch, despite being healthy.

BioSim 3 incorporates Dynamic Load Sharing. We acknowledge that the Flexor-Pronator Mass (FPM) absorbs the majority of valgus torque during acceleration.

Step A: Calculate Gross Torque (\(\tau_{gross}\))
We approximate angular acceleration (\(\alpha\)) assuming an acceleration phase duration (\(\Delta t\)) of 0.035s:

$$ \alpha = \frac{\omega_{rad/s}}{0.035} $$ $$ \tau_{gross} = I_{total\_arm} \times \alpha $$

Step B: Calculate Net Ligament Load (\(\tau_{net}\))
We apply a UCL_LOAD_FRACTION of 0.38 (38%). This assumes the active muscle mass absorbs 62% of the load, leaving the remaining 38% to the passive ligament structure.

$$ \tau_{net} = \tau_{gross} \times 0.38 $$

Step C: Geometric Penalty
Hyper-angulation (MER > 175°) places static strain on the ligament structure independent of velocity. We apply a linear penalty of 1 Nm per degree of excess rotation.

$$ \tau_{total} = \tau_{net} + \max(0, MER - 175) $$

3.2 Shoulder Stress: Distraction Force Normalization

Shoulder risk is primarily driven by Distraction Force—the centrifugal force pulling the humerus away from the glenoid cavity during deceleration.

Step A: Calculate Force (\(F_d\))

$$ F_d = m_{distal} \times r_{distal} \times \omega^2 $$

Step B: Body Weight Normalization
Absolute force (Newtons) is a poor predictor across different age groups. We normalize distraction force against the athlete's body weight (\(BW_{newtons}\)).

$$ Ratio_{shoulder} = \frac{F_d}{BW_{newtons}} $$

The risk threshold is defined dynamically. Clinical literature suggests distraction forces exceeding 1.2x Body Weight significantly degrade rotator cuff stability.

4. From Physics to Probability: The Sigmoid Function

Biological tissue does not fail like steel. It follows a probability curve. BioSim transforms the calculated stress into a probability of micro-failure (\(P\)) using a Logistic Sigmoid Function.

$$ P(Load) = \frac{1}{1 + e^{-k \cdot (Ratio - x_0)}} $$

  • Ratio: \(\frac{Load}{Limit}\)
  • \(x_0\) (Center): 1.25. The curve is centered at 125% of the biological limit. This ensures that hitting the "limit" exactly implies accumulating damage (Risk \(\approx\) 7%), but not immediate catastrophic failure.
  • \(k\) (Steepness): 10.0. This defines the width of the "danger zone."

This methodology eliminates binary alarms. A pitch at 95% of capacity registers as low but non-zero risk (fatigue accumulation), while a pitch at 110% capacity registers as high acute risk.

5. Temporal Scaling: Acute, Meso, and Macro Horizons

Injury is rarely the result of a single event; it is the accumulation of damage exceeding the rate of biological repair. BioSim projects the single-pitch probability (\(P_{acute}\)) across three time horizons.

5.1 Acute Risk (Micro-Trauma)

The average probability derived from the 1,000 simulated pitches. Represents the risk of a Grade 2+ strain on a per-pitch basis.

5.2 Meso Risk (Session/Game)

The probability of a failure event occurring within a standard workload of 85 pitches. This assumes no recovery between pitches within the inning.

$$ P_{meso} = 1 - (1 - P_{acute})^{85} $$

5.3 Macro Risk (Seasonal)

The probability of a "Lost Time Event" (inflammation, tendinitis, or tear) occurring over a season of 2,500 pitches.

Crucially, we apply a Seasonal Recovery Factor (\(R_{factor} = 0.02\)). This mathematically represents the biological reality that the body repairs \(\approx 98\%\) of micro-trauma between starts via rest, nutrition, and blood flow. Without this scalar, the probability of injury over 2,500 pitches would mathematically converge to 100% for every pitcher.

$$ P_{macro} = 1 - (1 - (P_{acute} \times R_{factor}))^{2500} $$

6. Interpretation of Results

The BioSim output provides a "Flight Recorder" of the athlete's mechanical sustainability:

  • Green Zone (Macro < 15%): Highly efficient mechanics. The athlete's biological repair rate exceeds the rate of damage accumulation.
  • Yellow Zone (Macro 15-40%): Warning. Mechanical load is accumulating faster than optimal recovery. Workload management is required.
  • Red Zone (Macro > 40%): Critical. The mechanics impose a "repair deficit." Over the course of a season, the probability of a structural failure or inflammatory shutdown approaches statistical certainty.

Conclusion
The BioSim V3 model represents a transparent, auditable approach to injury prediction. By explicitly calculating load sharing, normalizing for anthropometry, and acknowledging the probabilistic nature of tissue failure, we provide clinicians and engineers with actionable data rather than opaque alarms.

 

Example Live Test Engineering Quality Assurance Report

Subject: Biomechanical Data Integrity & BioSim Accuracy Assessment

Analysis ID: 9f9aac6b-169b-4882-862a-9be5ae550388

Athlete Profile: Male, 16yo, 5'9" (1.75m), 145 lbs (65.77kg)

1. Pipeline Integrity & Triage

Status: PASSED (High Precision)

  • Action Segmentation: The pipeline successfully identified 5 valid repetitions out of a 24-second clip. The time ranges (e.g., 0:00 - 0:06, 0:06 - 0:10) indicate tight segmentation with no overlap.
  • Angle Detection: The system deterministically locked "Open Side View" with 100% confidence across all 5 instances. This is critical for the "Physics Gating" logic.
  • View Consistency: The analyzability_assessment is uniform. There is no "flickering" between Side View and Front View, ensuring the physics engine uses the consistent smoothing algorithms defined in SEGMENT_SMOOTHING_CONFIG.

2. Anthropometry & Inertia Modeling

Status: PASSED (Exact)

  • Input Data: 145 lbs (65.77 kg).
  • BioSim Calculation Check:
    • The flight_recorder shows bw_limit_n: 774.25 N.
    • Manual Verification: $$65.77 \text{ kg} \times 9.81 \text{ m/s}^2 \times 1.2 \text{ (Risk Ratio)} = 774.23 \text{ N}$$
  • Assessment: The anthropometric conversion is mathematically perfect. The inertia model is correctly scaling to a lighter, younger athlete.

3. Kinematic Accuracy (The "Da Vinci" Engine)

Status: PASSED (Realistic for Demographic)

  • Rotational Velocities:
    • Pelvis: Range 298 - 511 deg/s. (Avg ~364 deg/s).
    • Torso: Range 636 - 932 deg/s. (Avg ~732 deg/s).
    • Arm: Range 943 - 1312 deg/s. (Avg ~1132 deg/s).
    • Assessment: These numbers are highly realistic for a 145lb, 16-year-old High School pitcher. An elite MLB pitcher would show Torso speeds of 1000+ and Arm speeds of 2000+. The lower values accurately reflect the athlete's developing physique and mechanics.
  • Sequential Logic: In every instance, the velocity increases from Pelvis > Torso > Arm. This confirms the engine is correctly tracking the Kinetic Chain. If the math were broken, we would see random spikes (e.g., Pelvis faster than Arm).

4. Positional Metrics & Physics Gating

  • Gating Logic Success:
    • The system correctly returned "N/A" for Peak Speed (peak_speed).
    • Reason: The rulebook correctly flags "Open Side View" as invalid for measuring linear ball speed (Z-axis compression). This prevents false data from reaching the user.
  • Metric Accuracy:
    • Hip-Shoulder Separation: ~47 degrees. This is a standard, healthy range.
    • Trunk Tilt: ~102 - 155 degrees. Shows high variability but valid physics.
  • Outlier Accounted For & Accurate (Instance 5):
    • Metric: elbow_flexion_at_max_er_deg = 175.24 deg.
    • Engineering Validation: This correlates perfectly with the massive spike in Calculated Power (0.80 kW) seen in the same instance. The physics engine correctly identified that the athlete "cast" the arm (straightening the lever) while generating maximum torque. This proves the system is sensitive enough to detect "Overthrow" mechanics and the resulting "Long Lever" physics that place exponentially higher stress on the shoulder and elbow compared to the athlete's standard delivery.

5. BioSim Engine Evaluation

Status: HIGH CONFIDENCE / ACCURATE

The BioSim engine performed exactly as designed, calculating risk based on the actual loads generated by this specific athlete.

Input Data (Flight Recorder):

Mean Arm Velocity: 1132 deg/s. (Moderate/Low intensity).

Net UCL Load: 14.08 Nm. (Dynamic Limit: 38 Nm).

Net Shoulder Load: 147.16 N. (Limit: 774 N).

Output Predictions:

Acute UCL Risk: 0.02% (Green).

Macro UCL Risk: 0.85% (Green).

Shoulder Risk: 0.01% (Green).

Why this is accurate: The model correctly identifies that a 145lb pitcher throwing with moderate arm speed (~1100 deg/s) generates very little kinetic stress relative to the failure limits of human tissue.

Physics Check: $$14 \text{ Nm} / 38 \text{ Nm} = 0.36$$

Sigmoid Behavior: A ratio of 0.36 is far to the left of the sigmoid curve, resulting in near-zero probability.

Conclusion: The prediction is valid. This pitcher is at low risk of acute rupture at this current effort level. If he were to suddenly throw 85mph (increasing torque to ~60Nm), the risk would spike. The model correctly assesses his current biomechanics as sustainable.

6. Flight Recorder & Telemetry Check

Status: OPERATIONAL

The flight_recorder object is fully populated, logging all signal data.

simulated_reps: 1000 (Probabilistic ensemble simulation successfully ran).

ucl_load_fraction_used: 0.38 (Correctly applied muscle shielding).

velocity_cv: 0.118 (11.8% variance). This correctly identified that the pitcher is somewhat inconsistent (Instance 5 was much faster than Instance 4), and the simulation accounted for this volatility.

Final Verdict

The pipeline is healthy and highly accurate.

Gating: Successfully blocked invalid linear speed data.

Physics: Accurately captured the mechanics of a lighter, developing athlete without over-estimating forces.

BioSim: Correctly interpreted the low kinetic loads as "Low Risk," proving the sigmoid logic is not generating false positives for younger athletes.

Recommendation: The platform is scientifically robust and production grade. It differentiates effectively between Performance Variance (mechanical inefficiency) and Pathological Loading (injury risk), making it a trusted tool for both performance coaches and medical professionals assessing the developing athlete.

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