Solving Drug Modeling: 130 micrograms of Thyroxine Needed

[linyen416]

Could someone please help me urgently?
The problem is: 130 micrograms of thyroxine is needed in her bloodstream. Her doctor prescribed one 100 microgram ,thyroxine tablet to be taken every day. If the half-time of thyroxine is 6.7 days, either justify that the uptake of thyroxine by the body and clearance through the liver fits a model similar to that for renal clearance stated above, or modify the mathematical model (do not alter the dosage or prescribe other drugs) so that it fits the given conditions.


I know already that to model the renal clearance, it would be a geometric progression with each term having exponential decay:
so
amount in body = De^(-0.10nT) + De^(-0.10 (n-1)T) +De^(-0.10 (n-2)T) +...+ D
because k = -0.10 So this is a geometric progression, defining variables: T = time interval between intakes, D = initial dose, n = nth number of dosage
But I know it is not as simple as this becasue this only modesl the renal clearance and not the absorption of the drug by the body. Could you please help me out? I don't know what I can do to the model to make it fit those numbers (100 per dose , resulting in 130 all the time in the body)

I also don't understand how you found long term amount of drug in the body using the initial dosages using a spreadsheet. If possible could you explain to me how you set out the spreadsheet? Becasue what I did to find the long-term saturation level was to take the sum to inifinity of the terms of that geometric progression.


This problem is quite urgentso appreciate ANY feedback! Could yopu please help me out with the developemtns of the model to fit those numbers?

-thank you kindly!

 

[fresh_42]

$S(n)= 100k \left(n +2 - \dfrac{1-q^{n+1}}{1-q}\right)$ is the total amount of substance, where $k$ is the daily absorption rate and $q=2^{-\frac{1}{6.7}} \approx 0.9$.

Hence there is no stable amount, because the decay rate is much slower than the intake rate.

 

[hilbert2]

Actually, there's always a stable amount if the drug eliminates with some half-life (1st order kinetics). The "half-time" with which the steady-state amount is approached is the same as the elimination half-life. With a 100 mcg tablet every day and $t_{1/2}$ of over 6 days , the steady state amount (taken to be either the maximum or minimum amount during each 24 hours between doses) will be much more than 130 mcg. Are you sure that the required final amont is in micrograms and not in milligrams?

Of course, the human body itself produces thyroxine, which complicates this calculation, but let's assume that it's a patient whose thyroid gland has become completely inactive because of some disease.