Transcalvarial brain herniation volume after decompressive craniectomy is the difference between two spherical caps

Decompressive craniectomy (DC) is a surgical procedure used to relieve severely increased intracranial pressure (ICP) by removing a portion of the skull. Following DC, the brain expands through the skull defect created by DC, resulting in transcalvarial herniation (TCH). Traditionally, people measure only changes in the ICP but not in the intracranial volume (ICV), which is equivalent to the volume of TCH (V_TCH), in patients undergoing DC.We constructed a simple model of the cerebral hemispheres, assuming the shape of the upper half of a sphere with a radius of 8cm. We hypothesized that the herniated brain following DC also conforms to the shape of a spherical cap. Considering that a circular piece of the skull with a radius of a was removed, V_TCH is the volume difference between 2 spherical caps at the operated side and the corresponding non-operated side, which represents the pre-DC volume underneath the removed skull due to the bilateral symmetry of the skull and the brain.Subsequently, we hypothesized that the maximal extent of TCH depends on a because of the biomechanical limitations imposed by the inelastic scalp. The maximum value of V_TCH is 365.0mL when a is 7.05cm and the height difference between the spherical caps (δh) at its maximum is 2.83cm. To facilitate rapid calculation of V_TCH, we proposed a simplified estimation formula, VTCH=12A2δh, where A=2a. With the a value ranging between 0 and 7cm, the ratio between VTCH and V_TCH ranges between 0.77 and 1.27, with different δh values. For elliptical skull defects with base diameters of A and C, the formula changes to VTCH=12ACδh.If our hypothesis is correct, surgeons can accurately calculate V_TCH after DC. Furthermore, this can facilitate volumetric comparisons between the effects of DCs in skulls of varying sizes, allowing quantitative comparisons between ICVs in addition to ICPs.