Though this question is about medicine, the actual question has little to do with medicine and more to do with modelling ..(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

A patient is put on an intravenous drip at time t=0, the drip supplies a drug into the patients bloodstream at a constant rate λ. At the same time (t=0) the patient is given M grams of the same drug orally which immediately starts dissolving at a rate directly proportional to the mass of the drug in the stomach (co-efficient of proportionalility α) The drug in the blood stream is eliminated from the blood stream at rate directly proportional to the mass of the drug in the blood (co-efficient of proportionalility β)

Find a model for the mass of the drug in the patients stomach and bloodstream at time t in terms of λ, α, β, M and t.

2. Relevant equations

Let S = S(t) = Mass of drug in stomach at time t.

Let B = B(t) = Mass of drug in bloodstream at time t.

3. The attempt at a solution

Stomach :

[itex]\frac{dS}{dt} = -α.S [/itex] ...... then using seperation of variables

[itex]\frac{dS}{α.S}= -dt [/itex] ...... integrate both sides to get

[itex]\frac{ln(α.S)}{α} = -t + C [/itex] ..... (where C is arbitrary constant)

[itex]ln(α.S) = -αt + C [/itex] ..... raising both sides to e, we get

[itex]α.S = e^{-αt+C}[/itex]

[itex]α.S = e^{-αt}.e^{C}[/itex] .... (e[itex]^{C}[/itex] is an arbitrary constant)

[itex]S = \frac{C.e^{-αt}}{α}[/itex]

Using: at t=0, S=M, we can find that C = M.α

[itex]S(t) = \frac{M.a.e^{-αt}}{α}[/itex]

[itex]S(t) = M.e^{-αt}[/itex]

I think its right up to here, but I'm stuggling with the bloodstream part..

Here's what I tried ...

In flow = λ+α.S(t)

Out flow = β.B(t)

[itex]\frac{dB}{dt}= λ + α.S(t) - β.B(t)[/itex]

but this now has 2 dependant variables so I'm not too sure where to go.

Maybe substituting [itex]S(t) = M.e^{-αt}[/itex] to get

[itex]\frac{dB}{dt}= λ + α.M.e^{-αt} - β.B(t)[/itex]

but then I wouldn't know how to solve this differential equation

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# Homework Help: Mathematical Modelling question

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