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Mathematical Modelling question

  1. Oct 4, 2011 #1
    Though this question is about medicine, the actual question has little to do with medicine and more to do with modelling ..

    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
     
    Last edited: Oct 4, 2011
  2. jcsd
  3. Oct 4, 2011 #2

    gneill

    User Avatar

    Staff: Mentor

    Your S(t) yields how much mass remains in the stomach. Ergo, the rate at which the mass goes into the bloodstream from the stomach is -dS(t)/dt.

    You might find that you can solve the resulting differential equation fairly easily using Laplace transforms.
     
    Last edited: Oct 4, 2011
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