4. Injury risk curves and their derivation

An injury risk curve is the quantitative bridge between a measurable physical response and a clinical outcome. It answers a deceptively simple question: given this value of a response variable — HIC, chest deflection, Nij, femur force, head angular velocity — what is the probability that a specific injury occurs? Every restraint performance target ultimately traces back to such a curve, so understanding how the curve is derived, what data it rests on, and where its uncertainty lies is essential for anyone setting or defending a criterion.

Anatomy of a Risk Curve

Formally, a risk curve is a cumulative distribution function of the response variable within the injured population, or equivalently a logistic or probit regression of injury outcome on response. The most common formulation in occupant safety is the logistic model:

P(injury | x) = 1 / (1 + exp(-(β₀ + β₁·x)))

Equivalent probit form:
P(injury | x) = Φ((x - μ) / σ)

Here x is the response metric (often log-transformed, e.g., ln(HIC) or ln(deflection)), β₀ and β₁ are fitted coefficients, and Φ is the standard normal CDF. The two parameters that matter most in practice are the median tolerance (the x value at P = 0.5) and the slope, which expresses how sharply risk rises with the response. A steep curve means the outcome is highly predictable from the metric; a shallow curve means substantial scatter and a weaker criterion.

Risk curves are always injury-specific. An AIS 2+ head curve, an AIS 3+ chest curve, and a rib fracture curve for the same body region are different functions with different medians and slopes. They are also population-specific: curves derived from mid-size male PMHS data do not automatically apply to small females, children, or the elderly, whose tolerance differs systematically.

Data Sources and Their Trade-offs

The credibility of a risk curve is bounded by its underlying data. Four source families dominate:

• Post-mortem human subject (PMHS) tests. Controlled impacts with known input and measured response, followed by autopsy. Gold standard for injury definition and response measurement, but limited sample sizes, elderly-skewed demographics, and ethical/scarcity constraints.
• Field accident data (e.g., NASS/CDS, GIDAS). Real-world crashes with reconstructed delta-v and occupant response, linked to injury outcomes from medical records. Large and representative, but response variables are estimated rather than measured, introducing reconstruction error.
• Animal and surrogate models. Useful for mechanisms and scaling, but require careful biomechanical scaling to human tolerance.
• Volunteer tests. Sub-injurious responses only; valuable for validating response corridors and low-severity curve segments.

In practice, robust curves are built by pooling PMHS data across studies and laboratories, then cross-checking against field data. Pooling demands harmonization: consistent injury coding (AIS version), consistent response definitions (e.g., resultant vs. component acceleration), and documented test conditions. A curve fitted to incompletely harmonized data will carry hidden bias that no statistical sophistication can remove.

Deriving a Curve: A Worked Example

Suppose we pool 48 PMHS sled tests with measured HIC₁₅ and autopsy-confirmed skull fracture (AIS 2+). Each test contributes a binary outcome (fracture / no fracture) and a continuous HIC value. The derivation proceeds as follows:

1. Define the outcome precisely — here, any skull fracture, AIS ≥ 2, verified at autopsy.
2. Transform the predictor — use ln(HIC) because tolerance is approximately log-normally distributed and this stabilizes the fit.
3. Fit a logistic regression by maximum likelihood, obtaining β₀ and β₁ with confidence intervals.
4. Compute the median and slope: median HIC = exp(−β₀/β₁); slope reported as the HIC change per unit of log-odds or as the interquartile response range.
5. Assess fit and uncertainty — plot binned observed proportions against the fitted curve, compute confidence bands (e.g., via bootstrap), and check for influential points.

For illustration only, a fit might yield a median HIC of roughly 1,500–1,800 with a relatively steep slope, meaning risk climbs from ~10% to ~90% over a modest HIC range. These numbers are illustrative; the actual values depend entirely on the pooled dataset and must never be quoted as established tolerances without reference to their source study.

From Curve to Criterion

Criteria are set by choosing an acceptable risk level and reading the corresponding response value off the curve. The choice of risk level is a policy decision, not a statistical one: a 10% AIS 2+ chest risk target and a 25% target produce very different deflection limits from the same curve. The criterion is then the response value at that probability, often rounded and expressed with a test condition (e.g., "chest deflection shall not exceed X mm at 40 km/h frontal barrier impact").

Two practical cautions apply. First, extrapolation beyond the data range is unreliable — a curve fitted to 20–60 km/h tests should not be used to justify a 90 km/h target without additional evidence. Second, population transfer requires scaling: applying a mid-size male curve to a 5th-percentile female occupant demands a biomechanical scaling factor (mass, stature, stiffness) and an explicit statement of the assumption.

Reading Uncertainty and Defending a Target

When a criterion is challenged — in a regulatory review, a design trade study, or a litigation context — the defense rests on three things: the provenance of the data, the transparency of the fit, and the explicit statement of the risk level chosen. A criterion of "chest deflection ≤ 50 mm" is weak on its own; "chest deflection ≤ 50 mm, corresponding to approximately 20% AIS 3+ chest risk based on the pooled PMHS curve of [source], with a 90% confidence band of 15–26%" is defensible. The difference is not rhetorical — it tells a reviewer exactly what assumption is being made and where it could be revised.

Finally, remember that risk curves evolve. As new PMHS data, improved field reconstructions, and refined injury definitions become available, curves are refitted and criteria revisited. A well-documented derivation is therefore not just good practice for today's target — it is the foundation for the next revision.