Pharmacokinetics

Mathematics used in pharmacokinetics

Summary of important equations

1. Rate of elimination (mg/min) = Concentration (mg/mL) x Clearance (mL/min)
2. Concentration (mg/mL) = Dose (mg) / Volume of distribution (mL)
3. Rate constant K (/min) = 1 / Time constant (min)
4. Half-life = 0.693 x Time constant
5. Clearance = Vd x K
6. Clearance = 0.693 x Vd / half-life
7. Clearance = Dose / AUC

Drug concentration vs time

Linear scale

• Drug concentration vs time
  = a exponential decrease (a wash out curve)
• i.e. Concentration = C₀ x e^-Kt

Rate of elimination

• At any time, the rate of drug elimination is proportional to the concentration at the time
  * Assuming first-order kinetics
• i.e. dC/dt = -K x C
• dC/dt is in mg/mL/min
  --> It is the change in concentration per unit time, not the change in quantity per unit time(e.g. drug eliminated rate)
  --> Different from rate of elimination

NB:

• dC/dt
  = d(C₀ x e^-Kt)/dt
  = C₀ x -K x e^-Kt
  = -K x C
• de^f(x)/dx = f'(x)e^f(x)

Log scale

• Log of drug concentration vs time
  = a straight downward slope line
  --> Gradient = K = rate constant
• i.e. ln C = ln C₀ - Kt

 

Rate constant and time constant

• Unit for rate constant = min^-1
• Time constant is the reciprocal of rate constant

Time constant

• Time constant (tau)
  = The time required for concentration to fall to 1/e of its former value (36.8%)
  = The time taken for the initial concentration to fall to ZERO if the initial rate of decline were to continue
• Unit for time constant = min
• Time constant = Vd / Cl (see below on how it is derived)

NB:

• e
  = natural number, which is the base of natural log
  = 2.718281828459...
• 1/e = 0.367879...

Time constand and half-life

• Time constant is the time required for concentration to fall to 1/e
  * See above.

Thus,

• Time constant is LONGER than half-life
• Mathematically,
  * Half-life = ln(2) x time constant
  --> Half-life = 0.693 x time constant

Derivation of half-life and time constant

C = C₀ x e^-Kt

--> C/C₀ = e^-Kt

When t = time constant, and since time constant = 1/K
--> C/C₀ = e^-1 = 1/e
* i.e. When time is one time constant, concentration is 1/e (36.8%) of the initial concentration

When t = half-time, by definition, C/C₀ = 50%
--> C/C₀ = e^-Kt = 1/2
--> ln (e^-Kt) = ln 1/2
--> -K x halflife = -ln(2)
--> Halflife = ln(2) / K

Since time constant = 1/K
--> Halflife = ln(2) x time constant

Single compartment model

• Plasma concentration (mg/mL) = Dose (mg) / Vd (mL)
  --> Vd = Dose/Concentration
• Rate of drug elimination (mg/min) = Clearance (mL/min) x Plasma concentration (mg/mL)
  --> Cl = Elimination Rate/Concentration
• Combining the above two equations
  --> Elimination rate / Dose = Cl / Vd = K (i.e. rate constant)
  --> Cl = K x Vd
• Substitute time constant into it:
  --> Cl = Vd / time constant
  --> Cl = 0.693 x Vd / Half-life
  i.e. Clearance is determined by volume of distribution and half-life

Loading dose

• Loading dose = Vd x Target plasma concentration

Maintenance dose

• Steady state is reached when rate of drug infusion = rate of drug elimination

Thus,

• Infusion rate = Cl x Plasma concentration

Steady state and half-life

• When no loading dose is given, and only an infusion is given
  --> Steady state is said to be reached after 5 half-lifes
  --> Plasma concentration is 96.875% of the steady-state

Others

Renal clearance

• Renal clearance (mL/min)
  = urine concentration (mg/mL) x urine volume per unit time (mL/min) / Plasma concentration (mg/mL)

Area under the concentration-time curve

• Area under the concentration-time curve = Initial concentration (mg/mL) / rate constant (/min)
  = Dose / Clearance
  --> Clearance = Dose / AUC

 

Derivation of the value of area under curve (AUC)

• Integration of Concentration = C₀ x e^-Kt
  --> Area under the curve = C₀ / K
• C₀ = Initial concentration = Initial dose / Vd
• AUC
  = Initial concentration / K
  = C₀ / K
  = Dose / (Vd x K)
• Cl = Vd x K

Thus,

• AUC = Dose / Cl

Thus,

• Cl = Dose / AUC