• k_el = k_m + k_ex

  • where k_el = drug elimination rate constant

  • k_m = elimination rate constant due to metabolism

  • k_ex = elimination rate constant due to excretion

• t_1/2 = ln 2 /k_el = 0.693/k_el

  • where t_1/2 is the elimination half-life (units=time)

• Derivation: The time it takes to shift from one concentration to another, for example to 1/2 the initial concentration is described by:

  • f = 1 - e^-ke^t where f is the fractional shift, k_e is the elimination rate constant and t is time.

  • To get to 50% of the initial concentration, let's set f = 0.5 and rearrange the equation to give:

    • 0.5 = 1- e^-ke^t or

    • 0.5 = e^-ke^t ; now taking the natural ln of both sides,

    • ln 0.5 = -k_et which is 0.693 = -k_et

    • Therefore the time it takes to reduce the concentration by 50%, in other words the half-time, t_1/2 = 0.693/k_e.

• X_b = V_d · C

  • X_b: amount of drug in the body (units, e.g. mg)

  • V_d: apparent volume of distribution (units, e.g. mL)

  • C: plasma drug concentration (units, e.g. mg/mL)

• V_d = D_iv / C_o

  • V_d: apparent volume of distribution (units, e.g. ml/kg)

  • D_iv: i.v dose (units, e.g. mg/kg)

  • C_o: plasma drug concentration (units, e.g. mg/ml)

   

• CL = rate of elimination/C, where C is the concentration of drug in blood or plasma

• rate of elimination = CL· C

• CL = Vd x k_el where V_d = volume of distribution and k_el is the elimination rate constant

• CL = Vd · (0.693/t_1/2) where 0.693 = ln 2 and t_1/2 is the drug elimination half-life

• note that plasma clearance CL_p include renal (CL_r) and metabolic (CL_m) components

Renal Clearance

• CL_r = (U · C_ur) / C_p ; where U is urine flow (ml/min); C_ur is urinary drug concentration and C_p is plasma drug concentration.

• The calculation required to determine being steady-state drug plasma concentration illustrates the sensitivity of the plasma concentration to number of factors, in this case for a drug taken orally.

• First  look at the overall form of the equation:

equation 1: C_ss= 1/(k_e*V_d) * (F*D)/T 

• The drug elimination rate constant,k_e is related to the drug half-life ( t_1/2 = 0.693/k_e) and thus can be calculated from knowledge of the drug half-life.  

• The plasma steady-state drug levels also dependent on the dose, D, as well as a fraction of the drug that's actually absorbed following ingestion (F). 

• "T" is the dosing interval, so the once-a-day dosing would be 1 day or to keep the units consistent, 24 hours.

• The steady-state level will also be dependent on the apparent volume of distribution (V_d)

• Now let's take an example using the drug phenytoin (Dilantin) which is used to manage epilepsy.

  • The once-a-day dose is 200 mg.

  • The drug half-life is 15 hours

  • For the once-a-day dose, the dosing interval (T) is 24 hours [to keep the units the same as the drug half-life will use "hours"]

  • Let's say that about 60% of the ingested does is in fact absorbed, giving us a value of 0.6 for  "F" in equation 1 above.

  • The volume of distribution for phenytoin (Dilantin) is 40,000 mls (40 liters)

  • k_e = 0.693/15 hours = 0.0462/hr

• Let's now compute the results:

equation 1: C_ss= 1/(k_e*V_d) * (F*D)/T  or

C_ss= 1/(0.0462/hour*40000 ml) * 0.6 (200 mg)/24 hours    or

Css = 0.0027 mg/ml or 2.7 ug/ml