2. Head injury criteria and biomechanical limits

In a pole impact test, the accelerometers and angular-rate sensors mounted in the headform produce time histories that must be reduced to a small set of scalar metrics before any pass/fail judgment can be made. Those metrics are not interchangeable: each one encodes a different physical hypothesis about what injures the brain, and each carries its own threshold, its own window of validity, and its own failure modes when the underlying signal is noisy or the impact is oblique. This section explains what each metric actually computes, how to compute it from raw data, and how to interpret it against published risk curves.

Linear acceleration and the HIC integral

Peak linear acceleration is the simplest metric: the maximum magnitude of the resultant translational acceleration at the headform center of gravity, usually expressed in g. It is computed from the three orthogonal channels as a_res(t) = sqrt(ax² + ay² + az²). Peak resultant acceleration correlates reasonably well with skull fracture and focal contusion, and many standards still cite values in the 200–300 g range for rigid-surface head impacts. Its weakness is that it ignores duration entirely — a 250 g spike lasting 1 ms is treated identically to a 250 g plateau lasting 15 ms, although the latter is far more injurious.

The Head Injury Criterion repairs that omission by integrating acceleration over a sliding time window:

HIC = max over (t1, t2) of [ (t2 - t1) * ( (1/(t2-t1)) * ∫ a_res(t) dt )^2.5 ]
subject to: t2 - t1 ≤ 15 ms

In practice, the integral is evaluated on the sampled signal. With a 10 kHz sample rate, dt = 0.1 ms, and the search runs over all window lengths from roughly 1 ms up to 15 ms. A common implementation accumulates the acceleration integral in a running sum, then for each candidate t1 scans forward to find the window that maximizes the expression. Two details matter for reproducibility: the acceleration must be expressed in g (not m/s²), and the window duration must be in seconds. Mixing units is the single most common source of HIC values that are off by orders of magnitude.

HIC is conventionally reported with its window, e.g. HIC36 = 812 (Δt = 9.4 ms). The 36 ms variant (HIC36) and the 15 ms variant (HIC15) are both in use; HIC15 is the more common choice for hard, short-duration pole and helmet impacts because the 36 ms window rarely binds on such events. A frequently cited automotive limit is HIC15 = 700, while helmet standards and sports-surrogate research often discuss values between 1000 and 1500 for concussive risk. These numbers are not universal truths — they are tied to the population, the impact site, and the criterion variant, and must be quoted together with those conditions.

Rotational metrics

Linear metrics treat the head as a translating point mass, but the brain is a soft body suspended in the skull, and oblique impacts produce shear strains that linear acceleration does not capture. Three rotational families are in routine use:

• Peak angular acceleration (rad/s²) and peak angular velocity (rad/s) — direct sensor outputs, easy to compute, but sensitive to sensor mounting and to differentiation noise.
• HIC-style rotational variants such as RIC (Rotational Injury Criterion), which applies the HIC integral form to the angular acceleration signal with its own exponent and window.
• Brain strain surrogates such as BrIC (Brain Injury Criterion), which combines peak angular velocity components along the three anatomical axes, normalized by axis-specific critical values, and takes the maximum. BrIC is attractive because it is computed directly from angular velocity without finite-difference differentiation.

For BrIC, the standard form is BrIC = max( |ωx|/ωxC , |ωy|/ωyC , |ωz|/ωzC ), where the critical angular velocities are direction-dependent (commonly around 66.3, 53.1, and 41.5 rad/s for the x, y, and z axes in the published formulation). A BrIC of 1.0 corresponds to the 50% risk level in the underlying model. Because the axes are anatomical, the headform must be instrumented and oriented consistently — a sign error or a swapped axis silently produces a plausible but wrong BrIC.

Injury risk curves

Risk curves convert a metric value into a probability of a specified injury outcome. They are almost always logistic in form:

P(injury) = 1 / (1 + exp( -α * (metric - β) ) )

where β is the metric value at 50% risk and α controls the steepness. Published curves exist for AIS 2+ and AIS 3+ head injury against HIC, against peak linear acceleration, and against BrIC or peak angular velocity. The practical consequence is that a single threshold is a fiction: HIC15 = 700 might correspond to roughly 30–40% risk of AIS 2+ in one dataset and a different value in another, depending on the population and the injury definition.

Worked example: reducing a pole impact record

Consider a synthetic headform record sampled at 10 kHz for 40 ms, with a resultant acceleration that rises to a 180 g peak at 6 ms and decays by 14 ms. Filtering with a CFC 1000 filter and running the HIC15 search yields, for illustration, HIC15 ≈ 640 with a window of about 8.5 ms. The same record gives a peak resultant of 180 g and, from the angular-rate channels, a peak angular velocity of 38 rad/s about the sagittal axis. Substituting into the BrIC formulation with the axis critical values above gives BrIC ≈ 0.72. Read against a representative AIS 2+ curve, HIC15 = 640 sits in a moderate-risk band while BrIC = 0.72 sits below the 50% point — a pattern typical of impacts that are linear-dominated but not purely translational. These numbers are illustrative only and are not measured results.

The figure below shows how the same time history feeds the different metric branches and where each one is evaluated.

Practical interpretation and common pitfalls

When the metrics disagree — for example, a high HIC but a low BrIC — the impact is predominantly translational, and the linear criteria should drive the assessment. The reverse pattern points to an oblique impact where rotational shear is the dominant mechanism, and a linear-only pass would be misleading. Reporting all four families together, with their computation conditions, is the only defensible practice.

Three failure modes recur in pole impact work. First, unit and window errors: HIC computed in m/s², or with a 36 ms window on a 15 ms event, produces values that look plausible but are wrong. Second, axis and sign errors in the rotational channels, which corrupt BrIC without any visible symptom in the linear metrics. Third, curve mismatch: applying an automotive HIC curve to a helmeted pole impact, or a BrIC curve derived for a different head model, without acknowledging the extrapolation. A short validation step — recomputing HIC on a known synthetic pulse and confirming the expected value — catches the first two before any test data is interpreted.

The metrics are tools for translating a noisy time history into a defensible statement about injury probability. Their value depends entirely on the discipline with which the computation conditions are recorded and the honesty with which the risk curve's provenance is stated.