title: Mathematical formulae to estimate chronic subdural haematoma volume. Flawed assumption regarding ellipsoid morphology.
abstract: 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.
  
  
  CONCLUSIONS
  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.
authors.affiliation: Department of Neurosurgery, The Townsville Hospital, Douglas, Townsville 4810, Queensland, Australia.
authors.forename: Appukutty
authors.initials: A
authors.lastname: Manickam
authors.affiliation: Department of Neurosurgery, The Townsville Hospital, Douglas, Townsville 4810, Queensland, Australia; School of Medicine and Dentistry, James Cook University, Douglas, Townsville 4810, Queensland, Australia. Electronic address: l.a.g.marshman@btinternet.com.
authors.forename: Laurence A G
authors.initials: LAG
authors.lastname: Marshman
authors.affiliation: Department of Marine & Tropical Biology, James Cook University, Douglas, Townsville 4810, Queensland, Australia.
authors.forename: Ross
authors.initials: R
authors.lastname: Johnston
authors.affiliation: Department of Neurosurgery, The Townsville Hospital, Douglas, Townsville 4810, Queensland, Australia; School of Medicine and Dentistry, James Cook University, Douglas, Townsville 4810, Queensland, Australia.
authors.forename: Piers A W
authors.initials: PAW
authors.lastname: Thomas
mesh_terms: D000465:Algorithms; D020200:Hematoma, Subdural, Chronic / Q000000981:diagnostic imaging*; D006801:Humans; D011857:Radiographic Image Interpretation, Computer-Assisted / Q000379:methods* / Q000592:standards; D014057:Tomography, X-Ray Computed / Q000379:methods* / Q000592:standards
keywords: Chronic; Mathematical formula; Subdural haematoma