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A patient diagnosed with cancer takes antiproliferative drugs. Suppose the intake of drug(dose) is 250mg and excreted amount 125 mg. Amount of excreted drug as metabolites 1 and 2 are 75mg and 50mg. T1/2 is 4 hours. Calculate its elimination rate constant (K).

My idea is : amount actually used from the body is 250(dose) - 125mg amount excreted= 125mg. Which means that we are at steady state. Steady state is reached after 4 half lives (t1/2). Taking this into consideration we calculate 4 k respectively: K=0.693/4 ; K=0.693/8 ; K=0.693/12 ; K=0.693/16

You can use this equation to *very generally* describe the one-compartment pharmacokinetics of a drug:

## Cplasma = (Dose*F/Vd)e^(-kt)

Where F is the fraction that is bioavailable and Vd is the volume of distribution of the drug. These values can be obtained from the Lexi-comp monograph. If a drug is administered i.v., F = 1. k in this equation is your elimination rate constant.

Your description of steady state is not correct. Steady state is reached with multiple administrations of the drug after 3-5 half lives. We usually assume steady state is reached in 5 half lives. Here you are only dealing with one administration of the drug and a steady state is never reached. The drug will be continuously cleared from plasma until Cplasma = 0. The time that this takes is usually 5 half lives (50% of initial plasma concentration excreted after 1 half life, 75% after 2 half lives, 87.5% after 3 half lives, 93.75% after 4 half lives and 96.875% after 5 half lives).

Without any plasma concentration data, a route of administration, volume of distribution and bioavailability, a proper elimination rate constant cannot be calculated. Consider the case where an oral drug is only 75% bioavailable. This means that 75% of the administered dose reaches the plasma and systemic circulation while 25% is cleared initially by stomach acid, digestion, poor absorption from the GI tract, etc. The elimination rate constant for this drug will be different than for one that is 100% orally bioavailable.