Elimination rate constant

The elimination rate constant K is a value used in pharmacokinetics to describe the rate at which a drug is removed from the system.[1]

It is often abbreviated K or Ke. It is equivalent to the fraction of a substance that is removed per unit time measured at any particular instant and has units of T−1. This can be expressed mathematically with the differential equation

${\displaystyle C_{t+dt}=C_{t}-C_{t}\cdot K\cdot dt}$,

where ${\displaystyle C_{t}}$ is the blood plasma concentration of drug in the system at a given point in time ${\displaystyle t}$, ${\displaystyle dt}$ is an infinitely small change in time, and ${\displaystyle C_{t+dt}}$ is the concentration of drug in the system after the infinitely small change in time.

The solution of this differential equation is useful in calculating the concentration after the administration of a single dose of drug via IV bolus injection:

${\displaystyle C_{t}=C_{0}\cdot e^{-Kt}\,}$
• Ct is concentration after time t
• C0 is the initial concentration (t=0)
• K is the elimination rate constant

Derivation

In first-order (linear) kinetics, the plasma concentration of a drug at a given time t ${\displaystyle C_{t}}$ after single dose administration via IV bolus injection is given by;

${\displaystyle C_{t}={\frac {C_{0}}{2^{\frac {t}{t_{1/2}}}}}\,}$

where:

• C0 is the initial concentration (at t=0)
• t1/2 is the half-life time of the drug, which is the time needed for the plasma drug concentration to drop to its half

Therefore, the amount of drug present in the body at time t ${\displaystyle A_{t}}$ is;

${\displaystyle A_{t}=V_{d}\cdot C_{t}=V_{d}\cdot {\frac {C_{0}}{2^{\frac {t}{t_{1/2}}}}}\,}$

where Vd is the apparent volume of distribution

Then, the amount eliminated from the body after time t ${\displaystyle E_{t}}$ is;

${\displaystyle E_{t}=V_{d}\cdot {C_{0}}{\Biggl (}1-{\frac {1}{2^{\frac {t}{t_{1/2}}}}}{\Biggr )}\,}$

Then, the rate of elimination at time t is given by the derivative of this function with respect to t;

${\displaystyle {dE_{t} \over dx}={{\frac {\ln 2\cdot {V_{d}\cdot {C_{0}}}}{2^{\frac {t}{t_{1/2}}}\cdot {t_{1/2}}}}\,}}$

And since ${\displaystyle K}$ is fraction of the drug that is removed per unit time measured at any particular instant, then if we divide the rate of elimination by the amount of drug in the body at time t, we get;

${\displaystyle K={dE_{t} \over dx}\div A_{t}={\frac {\ln 2}{t_{1/2}}}\approx {\frac {0.693}{t_{1/2}}}}$

Sample values and equations

Characteristic Description Example value Symbol Formula
Dose Amount of drug administered. 500 mg ${\displaystyle D}$ Design parameter
Dosing interval Time between drug dose administrations. 24 h ${\displaystyle \tau }$ Design parameter
Cmax The peak plasma concentration of a drug after administration. 60.9 mg/L ${\displaystyle C_{\text{max}}}$ Direct measurement
tmax Time to reach Cmax. 3.9 h ${\displaystyle t_{\text{max}}}$ Direct measurement
Cmin The lowest (trough) concentration that a drug reaches before the next dose is administered. 27.7 mg/L ${\displaystyle C_{{\text{min}},{\text{ss}}}}$ Direct measurement
Volume of distribution The apparent volume in which a drug is distributed (i.e., the parameter relating drug concentration to drug amount in the body). 6.0 L ${\displaystyle V_{\text{d}}}$ ${\displaystyle ={\frac {D}{C_{0}}}}$
Concentration Amount of drug in a given volume of plasma. 83.3 mg/L ${\displaystyle C_{0},C_{\text{ss}}}$ ${\displaystyle ={\frac {D}{V_{\text{d}}}}}$
Elimination half-life The time required for the concentration of the drug to reach half of its original value. 12 h ${\displaystyle t_{\frac {1}{2}}}$ ${\displaystyle ={\frac {\ln(2)}{k_{\text{e}}}}}$
Elimination rate constant The rate at which a drug is removed from the body. 0.0578 h−1 ${\displaystyle k_{\text{e}}}$ ${\displaystyle ={\frac {\ln(2)}{t_{\frac {1}{2}}}}={\frac {CL}{V_{\text{d}}}}}$
Infusion rate Rate of infusion required to balance elimination. 50 mg/h ${\displaystyle k_{\text{in}}}$ ${\displaystyle =C_{\text{ss}}\cdot CL}$
Area under the curve The integral of the concentration-time curve (after a single dose or in steady state). 1,320 mg/L·h ${\displaystyle AUC_{0-\infty }}$ ${\displaystyle =\int _{0}^{\infty }C\,\operatorname {d} t}$
${\displaystyle AUC_{\tau ,{\text{ss}}}}$ ${\displaystyle =\int _{t}^{t+\tau }C\,\operatorname {d} t}$
Clearance The volume of plasma cleared of the drug per unit time. 0.38 L/h ${\displaystyle CL}$ ${\displaystyle =V_{\text{d}}\cdot k_{\text{e}}={\frac {D}{AUC}}}$
Bioavailability The systemically available fraction of a drug. 0.8 ${\displaystyle f}$ ${\displaystyle ={\frac {AUC_{\text{po}}\cdot D_{\text{iv}}}{AUC_{\text{iv}}\cdot D_{\text{po}}}}}$
Fluctuation Peak trough fluctuation within one dosing interval at steady state 41.8 % ${\displaystyle \%PTF}$ ${\displaystyle ={\frac {C_{{\text{max}},{\text{ss}}}-C_{{\text{min}},{\text{ss}}}}{C_{{\text{av}},{\text{ss}}}}}\cdot 100}$
where
${\displaystyle C_{{\text{av}},{\text{ss}}}={\frac {1}{\tau }}AUC_{\tau ,{\text{ss}}}}$
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