Stroke occurs when a blood vessel that carries oxygen and nutrients to the brain is either blocked by a clot or bursts (or ruptures) [American Stroke Association].
If a blood vessel gets blocked by a clot, it’s called “Ischemic Stroke”.
If a blood vessel ruptures, it’s called a “Hemorrhagic Stroke” or “Intracerebral Bleed” (blood inside the brain).
Traditionally, if stroke occurs, patients undergo a CT Scan of the head. Afterwards, neurologists (doctors who specialize on diseases of the nervous system) determine if it is one type or the other. If it is a bleed, then they calculate the volume of the bleed using a formula called “Kothari Formula”. Why is this important? The prognosis (chance of a patient living or dying) can be predicted from the volume of the bleed!
During my Neurology rotation, we used a guidebook titled “Ictus” which contains much of the pertinent information used to manage the most common neurological diseases seen in the hospital. Stroke is the most common one among them. The guidebook elucidated that the volume of an intracerebral bleed using the Kothari Formula is as follows:
When I saw this formula, I could not accept such a simplistic ‘estimate’ formula, especially with the denominator of 4. The volume of a bleed has spherical elements, so I believe that the Greek letter π should at least be part of the formula.
Consequently, I tried to derive the formula of the bleed using calculus.
Let there be an ellipsoid with dimensions A/2, B/2, C/2 satisfying the equation:
First, we have to use generalized spherical coordinates. Let:
Since the absolute value of the Jacobian for transformation of Cartesian coordinates into generalized spherical coordinates is:
Hence,
The volume of the ellipsoid is expressed through the triple integral:
By symmetry, we can find the volume of the 1/8 part of the ellipsoid lying in the first octant (x ≥ 0, y ≥ 0, z ≥ 0) and then multiple the result by 8. The generalized spherical coordinates will range within the limits:
Then the volume of the ellipsoid is:
When I saw the result, I was even more confused. How can the calculated theoretical volume be almost twice as large as the one shown in the guidebook? Were we overestimating the volume of the bleed?
I started comparing it with other methods of getting volume of solids:
| Volume of Prism = ABC | Volume of Elliptical Cylinder = πABC |
| Volume of Pyramid = ABC/3 | Volume of Elliptical Cone = πABC/3 |
I started researching for journals online. Among the references I noted [1][2], none of them pointed to ABC/4. My derivation is closest to ABC/2, with π being approximately 3 – this is more consistent with the original research [1] made by Kothari himself. Another paper [2] found out that the true volume is closer to ABC/3 – the volume of a pyramid.
Again, none of the papers I’ve seen so far show ABC/4.
I cannot accept this formula yet.
Maybe, the discrepancies can be explained by the following points:
- Perhaps, just as the papers indicated, the reason behind this formula lies within the radiologic and technical aspects of the CT Scan used – the resolution and the size of the slices.
- When a bleed occurs, it occurs inside the brain. There are surrounding structures that change the shape of the bleed.
- The hemodynamics and pressure dynamics inside the brain are defined by the Monro-Kellie Doctrine – The total pressure inside the cranium is fixed; an increase in one of the three components of the brain, namely, brain tissue, blood, and cerebrospinal fluid (CSF) would occur at the expense of another. A bleed can occur at first, but the edema (brain swelling) that occurs later can therefore further decrease the volume.
This begs the research question: “What if we construct a formula – a multivariate function – of the volume of intracerebral bleed as a function of time as well as blood pressure?”
References
[1] Kothari RU, Brott T, Broderick JP, Barsan WG, Sauerbeck LR, Zuccarello M, Khoury J. The ABCs of measuring intracerebral hemorrhage volumes. Stroke. 27 (8): 1304-5. Pubmed
[2] Huttner HB, Steiner T, Hartmann M, Köhrmann M, Juettler E, Mueller S, Wikner J, Meyding-Lamade U, Schramm P, Schwab S, Schellinger PD. Comparison of ABC/2 estimation technique to computer-assisted planimetric analysis in warfarin-related intracerebral parenchymal hemorrhage. Stroke. 37 (2): 404-8. doi:10.1161/01.STR.0000198806.67472.5c – Pubmed .
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