Amount of drug in body and absorbed in bloodstream

[xicor]

Homework Statement

Determine the behavior of a drug in the bloodstream that enters the brain and the amount of this drug in the bloodstream that is delivered to the body at a given time as it decays. The amount of drug in the body is eliminated as described by an exponential decay half life equation. Plot the behavior to show an graphical representation.

Homework Equations

A = Ai*e^((-log(2)*(t/H)) Amount of drug in body, t is only variable
dA/dt = (-log(2)/H)*Ai*e^((-log(2)*(t/H)) t is only variable
dB/dt = -k1*B + (k2*A)
B is amount of drug in bloodstream
H = Ci*(280-W)/26 Half life Equation
k1 = 2.5 Half life constant
k2 = 2.7 Half life constant

The Attempt at a Solution

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I know what the final graph is suppose to look like where the amount of drug in the bloodstream is represented by starting at an initial value and exponentially decaying. The amount of drug in the body starts at zero and exponentially increases as the drug from bloodstream enters the body until it reaches a maximum value and then decays. Logically thinking through the problem, I would think that the amount of drug in body that decays is determined by two different decaying rates from body and bloodstream where so far I have created the equation
g(t) = A(o)*(e^((-log10(2)*(t/H))) - e^((-k1*t)))
but am not sure if this is correct and not sure what to plot for the amount of drug in the bloodstream. I have tried plotting this equation against A but they end up both decaying to the same value whereas the decay for the bloodstream should produce values that are a little smaller after the intersection of the graphs takes place. I did try solving for B and got B = (k2*A)/k1 + exp(-k2*t) but when I tried plotting this equation it wasn't the correct behavior. Does this have to do something with the two differential equations being coupled so I would solve for B in a different way then using integrating factors?

 

[Bystander]

The way your post reads, the drug is moving from the body to the bloodstream, and back to the body, with maybe a stop in the brain along the way, and out of the body.

Are you talking, for instance, intramuscular injection, exponential decay of intramuscular concentration into bloodstream, and exponential decay of bloodstream concentration via renal system and/or brain metabolism. "In body" is not clear.

If you could clarify the problem statement vis a vis the number of "decay" processes, it might be all you need, and it will ease efforts to help you.

 

[epenguin]

A few suggestions:

That you just lump all the constants appearing in your first two equations so that the first becomes just

A = A_ie^-k₃t

(Now but also as a general habit). All the complication is probably helping prevent you seeing wood for trees, can even intimidate you (as well as anyone trying to follow)).

You might notice that your second equation follows from the first, so it may not be necessary.

Try (always!) to see, as it were, the physiological mechanism through the equations (and vice versa). Avoiding unnecessary complication in the equations will help this.

When you try to see this some things sound a bit discrepant, as bystander has picked up. Does 'body' mean body including or excluding the bloodstream?

I question k₁ and k₂ being called "half-life constants" - they are plain physiological rate constants, related to certain half-lives.

I think to say anthing profitable it would be better for us to have the original problem verbatim.