Help with a formula for half life

[Varta]

Everyday of the week i consume 15mg but one day i consume only 10mg, the half life of the substance is 3 days. I want to know how much of it is in my body after 5 weeks but that would take ages to calculate because i would have to calculate day by day with this formula:

N(t) = N(0) * 0.5^(t/T)

where N(0) is the initial amount, t is elapsed time (in days) and T is the half life (in days). I would input 15 then get 11.9 and add 15 to get 26,9 then that would calculate to 21,35 , that would take ages to do 35 times so i came here for help for a formula

 

[Dr.Peterson]

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Second, what you describe appears to be ignoring the one day with a low dose; is that intentional? I would have recommended doing this first anyway, to see what the "normal" amount in the body is. (But is that formula really what you meant?)

Third, it isn't nearly so hard to do what you describe if you use a spreadsheet; that is what I would do.

Only after doing that would I try to see if there is a simple formula, first for the steady-state amount, and then for the amount 5 weeks after the 10 mg day.

 

[blamocur]

The answer will also depend on which day of the week you take the 10mg dose. The difference won't be large though.

 

[JeffM]

Where in the world did you get that formula? Was that what was in your textbook? What are you studying? That formula makes no sense.

 

[Dr.Peterson]

Everyday of the week i consume 15mg but one day i consume only 10mg, the half life of the substance is 3 days. I want to know how much of it is in my body after 5 weeks but that would take ages to calculate because i would have to calculate day by day with this formula:

N(t) = N(0) * 0.5(t/T)

where N(0) is the initial amount, t is elapsed time (in days) and T is the half life (in days). I would input 15 then get 11.9 and add 15 to get 26,9 then that would calculate to 21,35 , that would take ages to do 35 times so i came here for help for a formula

I think the formula you are using is actually N(t) = N(0) * 0.5^(t/T), and you are using it with t=1 to get the amount remaining each day from what was there the previous day, and adding the new amount, 15. That makes sense. Here is part of my spreadsheet, which does that:

Half-life:​	3​	​
Initial:​	15​	​
Day​	input​	level​
0​	15​	15.00​
1​	15​	26.91​
2​	15​	36.35​
3​	15​	43.85​
4​	15​	49.81​
5​	15​	54.53​
6​	15​	58.28​
7​	15​	61.26​
8​	15​	63.62​
9​	15​	65.50​
10​	15​	66.98​
11​	15​	68.17​
12​	15​	69.10​
13​	15​	69.85​
14​	15​	70.44​
15​	15​	70.91​
16​	15​	71.28​
17​	15​	71.57​
18​	15​	71.81​
19​	15​	71.99​
20​	15​	72.14​
21​	15​	72.26​
22​	15​	72.35​
23​	15​	72.43​
24​	15​	72.48​
25​	15​	72.53​
26​	15​	72.57​
27​	15​	72.60​
28​	15​	72.62​
29​	15​	72.64​
30​	15​	72.65​
31​	15​	72.67​
32​	15​	72.67​
33​	15​	72.68​
34​	15​	72.69​
35​	15​	72.69​

If I change one of the 15s to 10, it has no effect after 30 days or so.

 

[Cubist]

If I change one of the 15s to 10, it has no effect after 30 days or so.

It could be 10mg one day per week, otherwise there wouldn't be any need to mention the word "week".

Everyday of the week i consume 15mg but one day i consume only 10mg

I guess we really need OP to verify this. Let's hope @Varta will respond.

 

[mmm4444bot]

That formula makes no sense.

That's my fault, Jeff. I didn't notice lost formatting from a copy-and-paste while editing the op.

[imath]\;[/imath]

 

[Cubist]

That's my fault, Jeff. I didn't notice lost formatting from a copy-and-paste while editing the op.

Lost power? ?

 

[Varta]

It could be 10mg one day per week, otherwise there wouldn't be any need to mention the word "week".

I guess we really need OP to verify this. Let's hope @Varta will respond.

Yeah i meant one day of the week i would take 10mg instead of the regural 15mg , so is there a way to automate it in excell or something?

 

[Dr.Peterson]

It could be 10mg one day per week, otherwise there wouldn't be any need to mention the word "week".

Yeah i meant one day of the week i would take 10mg instead of the regular 15mg , so is there a way to automate it in excel or something?

Yes; or you can just manually put 10 in every 7th day, as I did here:

​

Here is a graph:

​

Here is the content of cell C5: =C4*0.5^(1/$B$1)+B5

You could also change the starting value to, say, 71, to see how it behaves after a month or so.

 

[BigBeachBanana]

Alternatively, you can use a formula to input 10mg every 7th day.

=IF(MOD(A4,7)=0,10,15)

 

[Varta]

Yes; or you can just manually put 10 in every 7th day, as I did here:

View attachment 33527​

Here is a graph:

View attachment 33526​

Here is the content of cell C5: =C4*0.5^(1/$B$1)+B5

You could also change the starting value to, say, 71, to see how it behaves after a month or so.

Bro thank you saves me alot of time

 

[Cubist]

An answer has been found above, but some people may be interested in the steady state, very long term, solution. This can be found by writing...

\(\displaystyle d_1 = d_0*(1/2)^{1/3} + \red{10}\)
\(\displaystyle d_2 = d_1*(1/2)^{1/3} + 15\)
...
\(\displaystyle d_6 = d_5*(1/2)^{1/3} + 15\)
\(\displaystyle d_7 = d_6*(1/2)^{1/3} + 15\)

Then set [imath]d_0=d_7[/imath] and solve. The answer, for the first day, is...

\(\displaystyle d_1=\frac{5\left(18\left(2^{-\frac{1}{3}}+4^{-\frac{1}{3}}\right)+17\right)}{4-2^{-\frac{1}{3}}} \approx 66.4721110149\)