Mathematical formulae to estimate chronic subdural haematoma volume. Flawed assumption regarding ellipsoid morphology

Mathematical formulae are commonly used to estimate intra-cranial haematoma volume. Such formulae tacitly assume an ellipsoid geometrical morphology. Recently, the 'XYZ/2' formula has been validated and recommended for chronic subdural haematoma (CSDH) volumetric estimation. We aimed to assess the precision and accuracy of mathematical formulae specifically in estimating CSDH volume, and to determine typical CSDH 3-D morphology. Three extant formulae ('XYZ/2', 'π/6·XYZ' and '2/3S·h') were compared against computer-assisted 3D volumetric analysis as Gold standard in CTs where CSDH sufficiently contrasted with brain. Scatter-plots (n=45) indicated that, in contrast to prior reports, all formulae most commonly over-estimated CSDH volume against 3-D Gold standard ('2/3S·h': 44.4%, 'XYZ/2': 48.84% and 'π/6·XYZ': 55.6%). With all formulae, imprecision increased with increased CSDH volume: in particular, with clinically-relevant CSDH volumes (i.e. >50ml). Deviations >10% of equivalence were observed in 60% of estimates for 2/3S·h, 77.8% for 'XYZ/2' and 84.4% for 'π/6·XYZ'. The maximum error for 'XYZ/2' was 142.3% of a clinically-relevant volume. Three-D simulations revealed that only 4/45 (9%) CSDH remotely conformed to ellipsoid geometrical morphology. Most (41/45, 91%) demonstrated highly irregular morphology neither recognisable as ellipsoid, nor as any other regular/non-regular geometric solid.

Mathematical formulae, including 'XYZ/2', most commonly proved inaccurate and imprecise when applied to CSDH. In contrast to prior studies, all most commonly over-estimated CSDH volume. Imprecision increased with CSDH volume, and was maximal with clinically-relevant CSDH volumes. Errors most commonly related to a flawed assumption regarding ellipsoid 3-D CSDH morphology. The validity of mean comparisons, or correlation analyses, used in prior studies is questioned.