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### Anesthesiology Math

Posted: Fri Oct 19, 2007 2:23 am UTC
Hello and good evening!

I recently heard that alcohol is removed from the blood stream at a rate which is proportional to the deviation from its normal amount(yes, even the most innocent of you have even a *little* bit in there!). My brain immediately makes the jump to say that one's BAC is of the form f(x)=Ce^rt.

I'd be very interested in taking a crash-course in anesthesiology, but I'd be content to know how to compute the constants in this particular case(without experimentation, of course!), based on physical characteristics such as weight. Any info is well appreciated!

### Re: Anesthesiology Math

Posted: Fri Oct 19, 2007 2:37 am UTC
Syntax wrote:My brain immediately makes the jump to say that one's BAC is of the form f(x)=Ce^rt
Plus K. And f(t), not f(x) [/pedantic]
K would be the normal amount of alcohol in the blood, which, as you stated, is non-zero.

### Re: Anesthesiology Math

Posted: Fri Oct 19, 2007 2:40 am UTC
oops! haha good catch.

....I should mention I've been drinking.

### Re: Anesthesiology Math

Posted: Fri Oct 19, 2007 4:51 am UTC
Syntax wrote:oops! haha good catch.

....I should mention I've been drinking.

Never drink and derive? ### Re: Anesthesiology Math

Posted: Fri Oct 19, 2007 5:01 am UTC
I used to be better at math when I had had a few..

Biolution can attest to this.

### Re: Anesthesiology Math

Posted: Fri Oct 19, 2007 5:05 am UTC
It would definitely be a differential equation. It reminds me of a restoring force in physics, where the force it exerts is proportional to the deviation from the equilibrium position.

### Re: Anesthesiology Math

Posted: Fri Oct 19, 2007 6:34 am UTC
mathmagic wrote:It would definitely be a differential equation. It reminds me of a restoring force in physics, where the force it exerts is proportional to the deviation from the equilibrium position.
Most definitely. Also a simple series RC circuit. The equation can be rewritten slightly as f(t) = Ce-t/τ, where τ is what an electrical engineer would recognize as a time constant (the product of R and C in the case of an RC circuit). If at t=0 I chug down some quantity of adult beverage that instantaneously brings my BAC up to C, then f(0)=C, f(τ)=C(1/e), f(2τ)=C(1/e2), and so on - my BAC decreasing by a factor of 1/e with the passage of each time τ.

If we introduce a "normal" amount of alcohol K, we wind up with f(t) = (C-K)e-t/τ+K

### Re: Anesthesiology Math

Posted: Fri Oct 19, 2007 1:41 pm UTC
ATCG wrote:
mathmagic wrote:It would definitely be a differential equation. It reminds me of a restoring force in physics, where the force it exerts is proportional to the deviation from the equilibrium position.
Most definitely. Also a simple series RC circuit. The equation can be rewritten slightly as f(t) = Ce-t/τ, where τ is what an electrical engineer would recognize as a time constant (the product of R and C in the case of an RC circuit). If at t=0 I chug down some quantity of adult beverage that instantaneously brings my BAC up to C, then f(0)=C, f(τ)=C(1/e), f(2τ)=C(1/e2), and so on - my BAC decreasing by a factor of 1/e with the passage of each time τ.

If we introduce a "normal" amount of alcohol K, we wind up with f(t) = (C-K)e-t/τ+K

Well, just about anybody in the sciences would recognize it as the characteristic time, which is equal to the amount of time that needs to pass for the original amount of whatever is decaying, to decay to 1/e worth.

I don't think the DE would be as simple as you explained, because the OP stated that dC/dt is proportional to the amount of alcohol in the body.

### Re: Anesthesiology Math

Posted: Fri Oct 19, 2007 1:53 pm UTC
mathmagic wrote:
ATCG wrote:
mathmagic wrote:It would definitely be a differential equation. It reminds me of a restoring force in physics, where the force it exerts is proportional to the deviation from the equilibrium position.
Most definitely. Also a simple series RC circuit. The equation can be rewritten slightly as f(t) = Ce-t/τ, where τ is what an electrical engineer would recognize as a time constant (the product of R and C in the case of an RC circuit). If at t=0 I chug down some quantity of adult beverage that instantaneously brings my BAC up to C, then f(0)=C, f(τ)=C(1/e), f(2τ)=C(1/e2), and so on - my BAC decreasing by a factor of 1/e with the passage of each time τ.

If we introduce a "normal" amount of alcohol K, we wind up with f(t) = (C-K)e-t/τ+K

Well, just about anybody in the sciences would recognize it as the characteristic time, which is equal to the amount of time that needs to pass for the original amount of whatever is decaying, to decay to 1/e worth.

I don't think the DE would be as simple as you explained, because the OP stated that dC/dt is proportional to the amount of alcohol in the body.

Is that equation not the one that is always derived when a function's derivative is proportional to the difference between the function and a constant? Why would it be more complicated?

### Re: Anesthesiology Math

Posted: Fri Oct 19, 2007 4:52 pm UTC
mathmagic wrote:
ATCG wrote:
mathmagic wrote:It would definitely be a differential equation. It reminds me of a restoring force in physics, where the force it exerts is proportional to the deviation from the equilibrium position.
Most definitely. Also a simple series RC circuit. The equation can be rewritten slightly as f(t) = Ce-t/τ, where τ is what an electrical engineer would recognize as a time constant (the product of R and C in the case of an RC circuit). If at t=0 I chug down some quantity of adult beverage that instantaneously brings my BAC up to C, then f(0)=C, f(τ)=C(1/e), f(2τ)=C(1/e2), and so on - my BAC decreasing by a factor of 1/e with the passage of each time τ.

If we introduce a "normal" amount of alcohol K, we wind up with f(t) = (C-K)e-t/τ+K

Well, just about anybody in the sciences would recognize it as the characteristic time, which is equal to the amount of time that needs to pass for the original amount of whatever is decaying, to decay to 1/e worth.
I agree completely. It's about the simplest DE imaginable and pops up all over the place. My intention wasn't to claim it as the exclusive domain of electrical engineering, but to offer one more application (in addition to physiology and physics) of the DE to a specific problem and show how it adds to our physical intuition of the original problem.

mathmagic wrote:I don't think the DE would be as simple as you explained [the solution that I offered was f(t) = (C-K)e-t/τ+K], because the OP stated that dC/dt is proportional to the amount of alcohol in the body.
I'm prepared to be proved wrong (it would hardly be the first time), but I'm pretty confident that this fits the OP's criteria. I would note that f(0)=C and f(t)→K as t→∞, as we would expect. Also, df(t)/dt = (C-K)(-1/τ)e-t/τ, which is proportional to f(t)-K = (C-K)e-t/τ, as desired.

### Re: Anesthesiology Math

Posted: Fri Oct 19, 2007 11:47 pm UTC
@Token, ATCG

After thinking over it last night and for today, I realize it's a lot simpler than I originally thought. After working it out for myself, I got pretty much the same thing as you did, ATCG.

### Re: Anesthesiology Math

Posted: Sat Oct 20, 2007 8:02 am UTC
Anesthesiology... Well, you need to get comfortable with modular arithmetic. Get comfortable with solving some basic Diophantine equations, and then learn about the Euler totient function and the Legendre symbol. Yes, I'd say those are the basics of number theory. *rimshot, followed by uncomfortable silence*

### Re: Anesthesiology Math

Posted: Tue Oct 23, 2007 10:45 am UTC
The alcohol-removal system is mathematically similar to the blood-sugar/insulin control system. It's a simple control mechanism, can be used to teach math to first years bio students.

### Re: Anesthesiology Math

Posted: Tue Oct 23, 2007 7:26 pm UTC
ATCG wrote:If at t=0 I chug down some quantity of adult beverage that instantaneously brings my BAC up to C

I wonder about the assumption that BAC instantaneously goes up to C when you chug down some quantity of adult beverage. My naive guess is that if the alcohol removal process didn't function, the BAC would climb logistically when said beverage is so consumed.

### Re: Anesthesiology Math

Posted: Thu Oct 25, 2007 10:54 pm UTC
Nimz wrote:
ATCG wrote:If at t=0 I chug down some quantity of adult beverage that instantaneously brings my BAC up to C

I wonder about the assumption that BAC instantaneously goes up to C when you chug down some quantity of adult beverage. My naive guess is that if the alcohol removal process didn't function, the BAC would climb logistically when said beverage is so consumed.

A hypodermic syringe loaded with ethanol and saline would yield a better approximation to a step function, but then how do you order that from a bartender? ### Re: Anesthesiology Math

Posted: Fri Oct 26, 2007 8:09 am UTC
ATCG wrote:A hypodermic syringe loaded with ethanol and saline would yield a better approximation to a step function, but then how do you order that from a bartender? It's... reasonable. Burning up a unit of alcohol takes an hour, emptying one can be done in a second; any sip taken arrives in your stomach spread over less than a second.

So a 1:3600 difference in timescale, yes, step functions are the reasonable assumption. More than integrating a "steady input" during 1 sec, that just screws up your equations (a bitsy) and leads to the same first 3-4 digits in your solution. (I've taught these things in 1st year math for biologists.)

Incidentally, that syringe is filled with insulin in the analogous example I stated before...

### Re: Anesthesiology Math

Posted: Wed Oct 31, 2007 12:43 am UTC
Gordon wrote:I used to be better at math when I had had a few..

Biolution can attest to this.

Just reading through thread, and I remembered this when I came across your reponse: http://xkcd.com/323/