When Half Life of a Drug is greater than uptake time: What happen?

Commonly Drugs are given at a rate of daily uptake (1,2,3…Time/ daily). However sometimes overlooked by patients and practitioners when the half life of the Drug is greater than the uptake , drugs accumulate as decay is at slower rate of preceedings intake doses.

Lets see by an R simulation with the pdf of an exponential law how the drugs accumulate in the body under these kinetics before a steady state:

Theory of pharmaco-kinetics such as compartment model, physiology behaviour,volumetry…for sake of simplicity.

THEORY

The rate of decay “lambda is related to an exponential Law with formula [like Nuclear decay law]:

C(t) = C(t=0)* explambda*t

Here the simulation is based on a psycho-active drugs (old benzodiazepine like: “Tranxilium CH”):

Given :

Half Life T1/2: 40 Hours

Uptake: 1 Unit every 24H .

Dose: 5mg

With a little bit of “log tech” Lambda is given by ln2 / T1/2 = 0.01733

## [1] 0.01732868

This plot show you the exponential decay after lim(t)<48 Hours that is before the 3rd^ uptake.

Time Matrix building and doses:

The problem arise when constructing this time dose matrix in R (Code adv.).

One have to construct 24 Hour columns time intervals, the calculate the doses in iteration form (n+[n-1] mg) by summing the rows: However setting time t 0-40 H = 0 i.e. cause a problem in the exponential for Doses calculation as it result of exp^0=1 in the column doses [mg] that is wrong:

Corrective Solution: ifelse command set Dose at t0-40 H = 0 [mg] hence make it right the iterative calculation as graphed below:

The matrix hence possible is hard to construct with easy coding and become as large as days under study.

##  [1] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
## [17] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
## [25] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
## [33] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
## [41] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
## [49] 4.914103 4.829682

Plot up to 6 half life

ISSUE WITH MODELING IN CLINICAL TRIAL

One question might arise? Does this phenomena is accounted in RCT and associated model?

Well let is state clearly: If treatment is evaluated on a daily basis by scoring endpoint i.e. treatment patient might perceived usually as IMPROVING until the 6 half life time (Steady state) resulting in a bias of parameters estimate and treatment effect. A lillte bit of similitude of Cross Over trials we could call it:

The carry over decay effect

Note: However some drugs might show the reverse effect : A degrading score during the first 15 days (with a lot of between subject variability: Anti-depressive drugs i.e).

How to include this effect in model

Well if score is recorded in time (daily i.e) then constructing a dummy variable for t0 to t6T1/2 might be a solution (R use contrast treatment Therefore no significant difference between dummies ensure that no layover decay effect take place in treatment efficiency (at least statistically).

In my next blog will show you how to proceed with modelling this phase up steady state including covariates.

CONCLUSIONS

It is now demonstrated that giving a DRUG when her half life is greater than the uptake result as an accumulation and rise of initial doses. In the present study it approximate 6 half life for a steady state for more than doubling the initial doses (Stuntly up to exp^5mg of that dose!?).

Practitioners, clinicians has to remind this effect if there Patient report a too high effect especially with elderly where the enzymatic kinetics is reduced.