Pharmacokinetic Modelling Principles

Mathematical models allow us to predict how plasma concentration of a drug changes with time
This is important because of the relationship between plasma concentration of a drug and its pharmacodynamic effects
They may be used to programme algorithms that will deliver variable infusion rates in order to achieve predetermined plasma concentration levels (and thus therapeutic effects)
Compartmental models make assumptions based on virtual volumes, without attempting to model 'real' volumes such as plasma volume or extracellular fluid volume

C = dC/dt

In general this is an exponential relationship, which can be plotted on a concentration-time graph

Positive exponential curves

If the rate at which concentration changes increases as time increases it is a positive exponential e.g. exponential growth curve
Relevant examples of exponential growth curves:

Negative exponential curves

If the rate at which concentration changes decreases as time increases it is a negative exponential e.g. exponential decay curve
Relevant examples of exponential decay curves:

C = C₀.e^-kt

C = drug concentration and is the dependent variable
C₀ = the initial drug concentration (concentration at t = 0) and is the intercept on the y-axis
t = time and is the independent variable
k = the rate constant for elimination and determines the 'steepness' of the curve

This relationship is commonly referred to as a drug wash-out curve (curve A below)
The wash-out curve starts at C₀ and is asymptotic with zero

Drug wash-in curve

The way that plasma concentration increases with time during an infusion of constant rate is termed a wash-in curve (curve B above)

It is also a negative exponential curve (because the rate of change of concentration decreases with time)
A wash-in curve starts at the origin and rises in a negative exponential fashion
It is asymptotic with the concentration at steady state (C_ss)
It has the equation: C = C_ss.(1 - e^-kt)
For both the wash-in and wash-out curves, the rate constant for elimination is k

Euler's number: 'e'

Semi-logarithmic transformation of the concentration-time curve to a log(concentration)-time curve makes it a linear decay
This makes it easier to mathematically model and derive certain information:

Initial concentration (C₀) - intercept with the y-axis
Rate constant (k) - gradient of the line
Time constant (τ) - reciprocal of k (i.e. τ = 1/k)
Half-life (t_1/2) - time taken for concentration to fall to half the original concentration

Volume of distribution (V_D) - the theoretical volume into which the initial dose distributes in order to produce the measured plasma concentration (V_D = dose/C₀)
Clearance (Cl) - the volume of plasma from which the drug is completely cleared per unit time (Cl = V_D x k)

For an exponential process, the time constant (τ) is the time taken for the plasma concentration of a drug to drop to zero if the initial rate of elimination continued
One time constant is the time taken for the plasma concentration to fall by a factor of e (i.e. to 37% of its initial value)
The time constant (τ) is the inverse of the rate constant (k):

τ = 1/k

The consequence of this is that the concentration of drug in the plasma falls by the same proportion over equal time periods
After one time constant, the concentration falls from C to C/e

After three time constants, the process is >99% complete
As clearance is the product of the rate constant 'k' and the volume of distribution, the time constant can also be expressed in these terms

The half life (t_1/2) is the time taken for the plasma concentration to fall by a factor of a half
The half life is shorter than the time constant:

t_1/2 = 0.693 x τ