3. Old stuff
          3.1. Old pharm stuff (pre 2009)
              3.1.3. Pharmacology
                  3.1.3.1. Pharmacology principles
 3.1.3.1.1. Pharmacokinetics 

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
    * (Cl = Vd / Time constant)
  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
  • def(x)/dx = f'(x)ef(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