As a complete outsider to medical sciences, biology, chemistry, etc, I was being curious about the usual prescriptions, when taking antibiotics, of spacing the pills by 8 hours, which at least in my personal experience is quite standard. Now, without getting too technical, I guess this has something to do with how the concentration (in blood?) of some molecules evolve in time, C(t), and the "8 hours rule" (or any other similar rule) is based on when this function reaches a certain threshold or something of the like. Is that so? Now well, I was wondering how different can this C(t) function be for different persons for a given substance? Let's say a physician prescribes taking amoxicillin to treat an ordinary infection. If we take a population of N persons, and call C1(t),C2(t),...,CN(t) to the corresponding concentrations of the substance of interest in time, how different can these curves expected to be? I guess the "8 hours" type of rule is a sort of average, but what about deviations from this average? And on what other factors can these curves depend (assuming reasonable healthy individuals, nothing extreme)? Are there detailed statistics for this?
In professional terms, you're asking about pharmacokinetics - what fate does a drug have once administered and how fast does it get there.
Yes, for some drugs, the goal is to keep a roughly steady concentration of the drug. For example, if you were taking a medication to regulate high blood pressure, you'd want it acting fairly constantly all the time, otherwise you'd have spikes in blood pressure when the drug wears off.
For an antibiotic, that's not necessarily the case. You're trying to kill bacteria, and it may not matter much if you reduce the bacterial population by the same amount after 24 hours whether that's because all the activity happened in the first hour or if it happened gradually. You also don't necessarily need the drug to be in the blood: for example, if you are treating a urinary tract infection, you might only care about the drug concentration in the urinary tract. However, there are other limits like side effects that you want to limit that might be worse if the serum concentration is too high, and you might get diminishing returns if the drug is administered all at once versus over a long time.
For amoxicillin itself, I found this paper:
Arancibia, A., Guttmann, J., Gonzalez, G., & Gonzalez, C. (1980). Absorption and disposition kinetics of amoxicillin in normal human subjects. Antimicrobial agents and chemotherapy, 17(2), 199-202.
among several others. For this paper, they gave 500 mg amoxicillin to 9 healthy humans.
Intravenous amoxicillin has bi-exponential kinetics - that is, the concentration is well-described by the sum of two exponential decay functions. You can measure the parameters for this equation by sampling the serum concentration of the drug over time and then look at variance in the population.
Since it's a bi-exponential, there are two different "half-lifes", alpha and beta. Arancibia measured the beta half life as mean = 3.40 hours, standard deviation = 1.88; alpha half life as mean = 0.27 hours, standard deviation = 0.15 hours. So yes, there's quite a bit of variability among subjects.
Another measurement is the AUC, "area under the curve". If you imagine the drug concentration plotted as a function of time, this is literally the area under that curve, which provides a nice summary of how much drug is available and for how long. The mean AUC was 37.0, standard deviation 9.7 (units are in mg/liter/hour). So there's a ballpark of ~25% variability in how much drug is "around and available" among the population.