Mixing Half Lives In Medicine Is A Mistake

Many drug manufacturers are guilty of a mathematical error. They mistakenly believe that mixing together drugs with different half lives can improve their stability when actually it achieves the exact opposite. For example, Adderall contains dextroamphetamine saccharate and dextroamphetamine sulfate. These both do exactly the same thing but their half lives are different. That is, it takes different amounts of time for the body to processes these chemicals.

In a hypothetical scenario, Sach and Eno are drugs which do the same thing with different half lives. Sach has a half life of 2 hours and Eno has a half life of 1 hour. The subject will take 10mg of each in the first table, then 20mg of Sach in the second table, and 20mg of Eno in the final table.

1st Table:

Hour Active Amount Sach Active Amount Eno Total Active
0 10.00 10.00 20.00
0.5 8.41 7.07 15.48
1 7.07 5.00 12.07
1.5 5.95 3.54 9.48
2 5.00 2.50 7.50
2.5 4.20 1.77 5.97
3 3.54 1.25 4.79
3.5 2.97 0.88 3.86
4 2.50 0.63 3.12
4.5 2.10 0.44 2.54
5 1.77 0.31 2.08
5.5 1.49 0.22 1.71
6 1.25 0.16 1.41
6.5 1.05 0.11 1.16
7 0.88 0.08 0.96
7.5 0.74 0.06 0.80
8 0.62 0.04 0.66
8.5 0.53 0.03 0.55
9 0.44 0.02 0.46
9.5 0.37 0.01 0.39
10 0.31 0.01 0.32
10.5 0.26 0.01 0.27
11 0.22 0.00 0.23
11.5 0.19 0.00 0.19
12 0.16 0.00 0.16
12.5 0.13 0.00 0.13

2nd Table:

Hour Active Amount Sach Active Amount Eno Total Active
0 10.00 10.00 20.00
0.5 8.41 7.07 15.48
1 7.07 5.00 12.07
1.5 5.95 3.54 9.48
2 5.00 2.50 7.50
2.5 4.20 1.77 5.97
3 3.54 1.25 4.79
3.5 2.97 0.88 3.86
4 2.50 0.63 3.12
4.5 2.10 0.44 2.54
5 1.77 0.31 2.08
5.5 1.49 0.22 1.71
6 1.25 0.16 1.41
6.5 1.05 0.11 1.16
7 0.88 0.08 0.96
7.5 0.74 0.06 0.80
8 0.62 0.04 0.66
8.5 0.53 0.03 0.55
9 0.44 0.02 0.46
9.5 0.37 0.01 0.39
10 0.31 0.01 0.32
10.5 0.26 0.01 0.27
11 0.22 0.00 0.23
11.5 0.19 0.00 0.19
12 0.16 0.00 0.16
12.5 0.13 0.00 0.13

3rd Table:

Hour Active Amount Eno
0 20.00
0.5 14.14
1 10.00
1.5 7.07
2 5.00
2.5 3.54
3 2.50
3.5 1.77
4 1.25
4.5 0.88
5 0.63
5.5 0.44
6 0.31
6.5 0.22
7 0.16
7.5 0.11
8 0.08
8.5 0.06
9 0.04
9.5 0.03
10 0.02
10.5 0.01
11 0.01
11.5 0.01
12 0.00
12.5 0.00

The 2nd and 3rd tables have no Total Active column because there is nothing to add up.

My point is basically this: it is never more effective to mix two drugs together (at the initial starting point) than to use all of one type of drug. Intuitively this is obvious to me but proving it is another matter.

What is efficacy when it comes to drugs? Good question. The ideal drug would work as follows: once ingested, it activates at a constant rate until x hours later where it suddenly stops activating. So a child on adderall version of this would be fully concentrated for an hour, then once that time is up the effect of the drug completely disappears altogether. This gives the drug user the most control over how much is active in their blood levels.

If one part of the drug has a lower half life than another part of the drug, its effects will be uneven – it must be more stable to change the half lives to be closer together. The mathematical proof of this would basically consist of how to minimize the total distance (as an area) from the ideal scenario (where the drug’s graph is basically a straight line up to the active amount then a straight downwards line to 0).

The clinical results backing up Adderall are good but if math can disprove it then the math takes precedence. I do believe they’re wrong.

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