Option 1: bolus or plug injection of a set amount of the drug at fixed time intervals.
Use impulse forcing to model this situation.
Drug in the blood stream is similar to a heavily damped oscillator being forced by the Dirac Delta function. Equilibrium is when there is no drug present in the blood stream.
The equation we can use to model the amount of drug in the blood stream is where is the rate at which the drug leaves the blood stream and is frequency which is a little awkward in this sense. To be heavily damped, the relation must be true.
My model: Note* I am using arbitrary values for and
is a piecewise function in which a delta function of amplitude occurs every half hour. This represents the injection of a set amount of the drug at fixed time intervals. So
, To get the equation
We have initial condition of
To Solve:
I'm using the laplace transform technique to solve this.
Then
Calling the above expression , we can now form the analytical expression for drugs in the blood stream.
From the graph above, I can tell that the model is working, however, it is obvious that the coefficients I am using do not allow each does of the drug to remain in the blood for very long.
Adjusted coefficients:
I adjusted the coefficients so the model is now In this model, the drug takes longer to leave the blood.
The cell above shows that administering 3mg of the drug every 30 minutes for a patient whose anatomy matches the equation works very well to keep the drug in blood level within a reasonable range. In this model, the drugs in the blood disappear rather quickly. I'm not sure if this is the case in a real world scenario; regardless, once you adjust the coefficients to match the behavior of a real body, from then is just a simple matter of experimenting with dosage so that the drugs in the blood value reaches and stays within the accepatble range. It the case of the graph above, this is achieved with a dosage of 3mg applied every 30 minutes to a person whose drug in blood equation is The drug in blood of the patient above quickly settles into an equilibrium range between roughly 3.5 and 2 mg of the drug. In a body that holds onto the drugs longer, this value of the drugs in the blood would fluctuate less and the doses would be smaller than 3mg.
Option 2: a continuous drip administration of the drug over specified time intervals
For this scenario I will use the same patient as before () but will a different type of forcing function. I'm going to assume that at the top of every hour, the patient is administered drugs at rate for 30 minutes. To model a drip administration forcing function, I will have to use a piece wise function were basically f(t) = 0 for 30 minutes then for 30 minutes.
The lapalce tansform of the first 30 minutes of the forcing function is So, the first 30 minutes of the model will follow this:
As you can see from the model above, using a rate of to administer drugs to the patient for 30 minutes at the top of every hour works very well. After a few cycles of administering the drug, there is constantly between 3.0 and 2.0 mg of the drug int he patient's blood.
From a modeling standpoint, the impulse function based model was way easier to manipulate and then plot. But from a medical standpoint, the drip based model is the better option for keeping a constant amount of some drug in the patient's bloodstream. The impulse model causes rapid spikes in the drug in blood value which then taper off while the drip model has less aggressive spikes in value. With the impulse method, I think that there is a greater risk of giving too much of the drug to a patient than with the iv drip. One of the spikes could easily rocket right past the 4mg threshold. The iv drip would allow the doctors to keep the patient at the higher end of the acceptable range because the spikes are not as aggressive and thus are less likely to go past the 4 mg threshold. The oscillations of the iv drip method are 1mg, while the oscillations of the plug injection are 1.5 mg.
Despite my suggestion to use the iv drip method, the graphs above show that either the drip of the plug injection are both valid methods to maintain between 2 and 4 mg of drugs in a patients blood stream.