Calculus in Medicine

When I thought of calculus, I used to think of maths. And physics. And engineering. Which is why I found these lines in Steven Strogatz’s Infinite Powers very, very  surprising:

“Consider the surprising role that differentials (calculus) played in the understanding and treatment of HIV.”

Back then, it was known that HIV went through 3 stages:

1)      In the first stage, an infected person has flu-like symptoms;

2)     Next, a 10-year period of no symptoms;

3)     And finally, AIDS sets in, weakening the immune system to a point where other infections overwhelm the system and kill the patient.

Tests had shown that the number of virus particles in the bloodstream was high in the first stage (hence the symptoms); was low but not gone altogether (the “set point”) in the second stage; and was overwhelming in the last stage.

With no cure in sight, all kinds of ideas were tried. One such team was of Dr. David Ho and Alan Perelson. The former was a former physics major, and the latter was a mathematical immunologist. Note how both had a maths background and were probably guys “comfortable with calculus”, writes Strogatz.

In 1995, the team gave patients a protease inhibitor as a probe, not as a treatment. This moved the “set point” and allowed them to “track the dynamics of the immune system as it battled HIV”:

“The rate of decay was incredible; half of all the virus particles in the bloodstream were cleared by the immune system every two days.”

This was an exponential decay, exactly the kind of stuff that calculus is used for. Ergo, the two realized if V = concentration of the virus, then:

That’s maths-speak for saying that the rate of change of V (virus concentration) is a function of the virus concentration itself. The negative sign signifies a decrease.

Solving the equation, they realized that the supposedly “dormant” 10-year period was anything but. Instead, it was a period of pitched battle where the two sides were dead-locked:

“The immune system was in a furious, all-out war with the virus and fighting it to a near standstill.”

The casualty count in this standstill was staggering: a “billion virus particles” were being killed every day in this supposedly dormant phase. Obviously, the same billion were also being replaced every day, hence the deadlock.

The researchers then collected data right after the protease inhibitor was injected: “every two hours until the sixth hour, then every six hours until day two, and then once a day until day seven”. They also adjusted for other aspects not factored in the first time around. The numbers this time were even more stunning:

“Ten billion virus particles were being produced and then cleared from the bloodstream every day.”

Moreover, they also found that the lifespan of the infected cells was only two days. That short life cycle would also a prove key learning.

So how did all this help fight HIV/AIDS? Until then, doctors prescribed anti-viral drugs only after, not during, the 10-year supposedly hibernation period. Why? It wasn’t doing any harm until then, they reasoned, and besides, hit it too early and it would inevitably mutate and become immune to those drugs by the time you really needed them.

Ho and Perelson’s work upended the entire approach to treatment:

1)      It showed there what was thought of as a hibernation period was really a period of pitched battle every second, every day;

2)     It was clear that the key was to help the immune system in that 10-year period when it was still functioning. Conversely, the last stage was the period where the virus had the upper hand. Reinforcements then were too little, too late;

3)     It also explained why no single medication ever worked:

“The virus replicated so rapidly and mutated so quickly, it could find a way to escape almost any therapeutic drug.”

But wait, the duo wasn’t done yet. Perelson proved mathematically why no one drug stood a chance:

“By taking into account the measured mutation rate of HIV, the size of its genome, and the newly estimated number of virus particles that were produced daily, he demonstrated mathematically that HIV was generating every possible mutation at every base of its genome many times a day.”

He went on to prove that a two drug-combo wasn’t good enough either, but a three-drug combo had a ten million to one odds of success.

And that is why HIV has been a triple-combination therapy ever since. But even that combo doesn’t eradicate the virus altogether. Yes, their numbers become undetectable within two weeks, but they are there nonetheless. And they rebound if the medication stops. That’s why HIV-positive people need to keep taking their meds lifelong: it’s a chronic condition.

A chronic condition is still a huge improvement over the pre-calculus analysis days outcome of inevitable death, so let’s treat this as a win.