# Mathematical Modelling question

1. Oct 4, 2011

### Boogzy

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 :

$\frac{dS}{dt} = -α.S$ ...... then using seperation of variables
$\frac{dS}{α.S}= -dt$ ...... integrate both sides to get

$\frac{ln(α.S)}{α} = -t + C$ ..... (where C is arbitrary constant)
$ln(α.S) = -αt + C$ ..... raising both sides to e, we get

$α.S = e^{-αt+C}$
$α.S = e^{-αt}.e^{C}$ .... (e$^{C}$ is an arbitrary constant)

$S = \frac{C.e^{-αt}}{α}$

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

$S(t) = \frac{M.a.e^{-αt}}{α}$
$S(t) = M.e^{-αt}$

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)

$\frac{dB}{dt}= λ + α.S(t) - β.B(t)$

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

Maybe substituting $S(t) = M.e^{-αt}$ to get

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

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

Last edited: Oct 4, 2011
2. Oct 4, 2011

### 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