Well, I went to see what could be determined with simple arithmetic, now that I can't remember how to do integrations.
Interesting result.
My original interest, way back when, had been calculating frontloads for trenbolone acetate and testosterone propionate, where injection frequency typically matches the half-life.
In the instance where half-life is estimated at one day and injection frequency is daily, then the amount built up in the body is:
From 1 day before: 1/2 the ongoing dose, because one-half life of time has passed.
From 2 days before: 1/4 that amount
From 3 days before: 1/8 that amount
Etc.
And the sum of 1/2 plus 1/4 plus 1/8 plus.... is exactly 1. At least I do remember that much!
So therefore the frontload is exactly the amount injected per half-life, plus the ongoing dose, in this instance where injection frequency matches the half-life.
I see where this is headed indeed. It is indeed true that the sum of 1/x-->infinty will approach 1, so much so that 1 is the outcome with negligible deviation.
With regard to your formula, actually I can't figure how you would use it to determine practical examples, or rather if I try what you seem to be saying and don't go and further do the main aspect of what I posted, I wind up with a wrong answer. Could you be more specific? For example, what would you calculate for the above trenbolone acetate example?
You had written.
So for trenbolone acetate we need to substitute 24 for the 35, and we come up with (correctly) 0.5 as the value, if we skip to your following part anyway and match that.
I don't know where the log(x)=35 comes from in your explanation though. For the clenbuterol I would do it as x = .5^(24/35), if by x we mean the fraction remaining after 24 hours, relative to 24 hours previously.
Indeed. I see where you get confused, the log function is just another method of calculating the halflife per day or per hour given a half life in days or hours respectively (and then squaring it to your preferred amount of time of course). For instance, if you were to solve log(x)=35, you would get the degrading coefficient per hour, thus squaring this number 24 times, will get you the degradation per 24 hours. Mathematically it's the "cleanest" method, in practice you'll be easier off with the 0.5^(24/35) method.
If I were going by your post I would figure that then for trenbolone acetate my blood level would be 0.5 times that of yesterday's dose and that would be what I should frontload with -- or for the clenbuterol, the 0.62 times is the only number appearing in the post for me to frontload with -- but that is not right in either case. With the trenbolone the blood level is fully as much as from the entirety of yesterday's dose: because of the day before yesterday, the day before that, etc. each making fractional contributions summing exactly to one dose.
The 0.62 is the degradation coefficient per 24-hour timeframe. You then would get something like 0.62*x+daily dosage=the stable bloodvalue. Plotting this in either a table or graph, you can then experiment with any value to get the optimal frontload.
However there is more to it than this -- for clarity though I wanted to break it into parts.