Anesthesiology Math
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Anesthesiology Math
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 crashcourse 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!
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 crashcourse 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
Plus K. And f(t), not f(x) [/pedantic]Syntax wrote:My brain immediately makes the jump to say that one's BAC is of the form f(x)=Ce^rt
K would be the normal amount of alcohol in the blood, which, as you stated, is nonzero.
LOWA
Re: Anesthesiology Math
oops! haha good catch.
....I should mention I've been drinking.
....I should mention I've been drinking.
Re: Anesthesiology Math
Syntax wrote:oops! haha good catch.
....I should mention I've been drinking.
Never drink and derive?
Re: Anesthesiology Math
I used to be better at math when I had had a few..
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Re: Anesthesiology Math
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.
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Re: Anesthesiology Math
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/e^{2}), and so on  my BAC decreasing by a factor of 1/e with the passage of each time τ.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.
If we introduce a "normal" amount of alcohol K, we wind up with f(t) = (CK)e^{t/τ}+K
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Re: Anesthesiology Math
ATCG wrote: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/e^{2}), and so on  my BAC decreasing by a factor of 1/e with the passage of each time τ.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.
If we introduce a "normal" amount of alcohol K, we wind up with f(t) = (CK)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.
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Re: Anesthesiology Math
mathmagic wrote:ATCG wrote: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/e^{2}), and so on  my BAC decreasing by a factor of 1/e with the passage of each time τ.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.
If we introduce a "normal" amount of alcohol K, we wind up with f(t) = (CK)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?
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Re: Anesthesiology Math
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:ATCG wrote: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/e^{2}), and so on  my BAC decreasing by a factor of 1/e with the passage of each time τ.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.
If we introduce a "normal" amount of alcohol K, we wind up with f(t) = (CK)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'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 = (CK)(1/τ)e^{t/τ}, which is proportional to f(t)K = (CK)e^{t/τ}, as desired.mathmagic wrote:I don't think the DE would be as simple as you explained [the solution that I offered was f(t) = (CK)e^{t/τ}+K], because the OP stated that dC/dt is proportional to the amount of alcohol in the body.
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Re: Anesthesiology Math
@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.
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.
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Re: Anesthesiology Math
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
The alcoholremoval system is mathematically similar to the bloodsugar/insulin control system. It's a simple control mechanism, can be used to teach math to first years bio students.
Re: Anesthesiology Math
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.
LOWA
Re: Anesthesiology Math
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?
"The age of the universe is 100 billion, if the units are dog years."  Sean Carroll
Re: Anesthesiology Math
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 34 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...

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Re: Anesthesiology Math
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/
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