Neurology Meets Mathematics – Calculating the Volume of Intracerebral Bleed

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:

1. 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.
2. When a bleed occurs, it occurs inside the brain. There are surrounding structures that change the shape of the bleed.
3. 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 .

The Pharmacology of Love

If love is a drug, then it must have a pharmacokinetic and pharmacodynamic properties.

I. Pharmacodynamic Profile

A. Receptors of Love

1. Words of Affirmation – Some people feel the effects of love when they hear the words “I love you.” or compliments that uplift their spirits.

2. Quality Time – Some people feel the effects of love when they spend time with their family and friends.

3. Gifts – Some people feel the effects of love when they receive the gift, or more importantly, the thought that was given in giving that gift.

4. Acts of Service – Some people feel the effects of love when someone goes through the burden of doing something for them.

5. Physical Touch – Some people feel the effects of love through a hug, a pat on the shoulder or on the head, or the touch of a hand.

B. Mechanisms of Action

1. Consistency and Faithfulness – Love is called love because it consistently binds on its receptors. In love, you always show words of affirmation, spend quality time, give gifts, do acts of service, and give a pat or a hug.

2. Perseverance – When love binds to a receptor, the cascading effects persevere throughout all circumstances and environmental factors.

3. Patience and Forgiveness – When love is denied through proteosomal degradation, or made toxic through drug-drug interactions with fear and pride, it becomes unstable. However, it stabilizes by itself towards patience and forgiveness.

4. Kindness and Gentleness – Love is kind and gentle. It binds to all its receptors in an endearing manner. It doesn’t bind too fast, too rashly.

5. Honesty and Truthfulness – Love does not try to conceal or hide its potential for being weak. It doesn’t bind to other compounds that will hide its effect. Because of its honesty, it is willing to be molded into a more potent, therapeutic, and life-changing metabolite that will induce further positive changes leading to the growth of a person.

6. Respect – Love always respects. It listens. It waits until you are stable. It doesn’t enter the cell when it is unwelcome – when there is no receptor. When the cells in your body need love, they send signals, and the love molecule willingly enters into you.

C. Drug-Drug Interactions

1. Synergism: When love goes hand-in-hand with other drugs, it becomes more potent than it normally is.

e.g. Love + Hope: If there is hope in love, then despite the lack of its consistency, there is promotion of its perseverance. Perseverance as a mechanism of action complements consistency. Hope is a drug that can either bring success or despair.

2. Addition: When love is added with other good although not too important drugs, it becomes really fancy.

e.g. Love + Cheesiness: For some mysterious reason, pick-up lines work really well. Cheesiness is a drug that has a mechanism of action in flirting that makes people fall in love. Ugh. For better or for worse.

3. Inhibition: When love is added to fear or insecurity, it cannot reach its full potential.

e.g. Love + Insecurity: Insecurity is a fear-type drug that simply prevents it from exhibiting its mechanisms of action – honesty, faithfulness, consistency. With insecurity, love cannot reach its fullest potential.

D. Indications: Being Alive. You cannot live without love.

E. Adverse Effects: The Eros isoform (conformation) of love has a tendency of becoming highly unstable, especially when it’s bound to a cofactor called “Your Crush” or “That Special Somone”. Being in love can lead to –

1. Insomnia: It occurs when you think of the other person too much.

2. Nausea: It occurs when you try to awkwardly start a conversation with that person.

3. Dizziness: It occurs when that person gives you a hug or kisses you.

4. Epigastric Pain: It occurs together with nausea, when you have butterflies in your stomach.

5. Malaise: It occurs when you know that the person that he or she likes is not you.

6. Tachycardia: It occurs practically all the time when it concerns that other party.

7. Angina: It occurs when you see that person in the arms of another, and when that person smiles because of another person that is not you.

8. Palpitations: It occurs when you are anxiously waiting for that person’s reply to your message.

9. Headache: It occurs when you are stressed by the fact that you are going to be a martyr in this fiasco of unrequited love.

10. Edema: It occurs when you suffer a first-degree burn on your finger when you accidentally touch a very hot surface on which you burned all your written hard feelings and unsaid words.

II. Pharmacokinetic Profile

A. Synthesis

1. Precursor: Love is synthesized from its precursor, which is love itself. Love only begets love.

2. Enzyme: Love synthesizes love by the enzyme lovease, because there is no other way to catalyze love but to simply love.

3. Products: Love comes from love, made to exist through love. Love exists when it begins to exist.

B. Component Processes

1. Absorption: Love is absorbed the moment it is found.

2. Distribution: Love is distributed instantly the moment it is absorbed. When love is bound and limited, it is inactive and does not grow. When love is free, it is active and does grow.

3. Metabolism: Love metabolizes to produce more love. During trials and tribulations, love proves its worth and potency. It is unaffected by CYP3A.

4. Excretion: Love is excreted as love when it overflows from somebody and inundates or floods another person.

C. Properties

1. Bioavailability: F = 0.5; Love is making sure that you are available to your loved ones – your friends and family. (Casanova, 2017) It does not have an Absolute Bioavailability because it cannot be administered via an IV line.

2. Bioequivalence: Love may appear in different forms – Agape, Storge, Philia, or Eros, but their systemic effects, measured by the Area Under the Curve or AUC (refer to reference), are equal.

3. Elimination Constant (Ke): 2.30/year [1]

4. Half-Life: 2637 hours or 110 days [2]

5. ED50 (Minimum Effective Dose): 42 mcg/mL [3]

6. TD50 (Mininum Toxic Dose): 666 mcg/mL [4]

7. Therapeutic Index: 15.86 [5]; Love is generally a safe drug.

8. The Agape isoform has a half-life of Infinity [6]. The values Ke, half-life, ED50, and TD50 cannot be calculated.

9. Dosage: 1000 mg per half-life

10. Volume of Distribution: 0.6 L/kg [7]

D. Routes of Entry: Love enters the blood through –

1. Enteral: Love is not ingested. Love is not inhaled.

2. Parenteral: If love could be administered via intravenous (IV) injection, then it would be highly potent. However, despite love having high efficacy, it is still not quite potent because it enters through dialogue and spending time with each other, and it is susceptible to misunderstanding.

Meanwhile, patience and understanding allows it to be administered parenterally, across space and time, across life and death, and through the stars (interstellar).

[Reference]

Refer to this link for the derivation of the graph and its resulting values.

1) ln(10)

2) ln(2)/ln(10) of a year

3) 42 is the answer to the ultimate question of life, the universe, and everything. Effective love is the answer to the ultimate question of life, the universe, and everything. Therefore, ED50 = 42 standard units.

4) 666 is always associated with the adversary.

5) 666 / 42 = 15.86; This relatively high therapeutic index suggests a wide therapeutic window and margin of safety.

6) Forever exists in God’s love.

7) 0.6 L/kg is the Total Body Water. Love permeates a person.

[Mathematical Derivation]