Dr Robert Rowland and Dr Phillipa Squires, Derriford Hospital, Plymouth
This is a review of the mathematics underpinning pharmacokinetics such as that required for the final FRCA exam. We have endeavoured to produce a continuous logical progression and to avoid any mathematical leaps of faith so often seen in other sources.
Pharmacokinetics is a system of mathematical modelling of in vivo processes particularly distribution and elimination. It is, by definition, a model and the one most often used to illustrate the pharmacokinetics of propofol , for example, is the three-compartment model. Here the body is divided into a central compartment which contains the organs that receive the bulk of the cardiac output,(heart, lungs, kidneys liver and brain) and two other compartments which are less well perfused such as skeletal muscle and fat. Of course there is no physiological or anatomical meaning to these compartments. They can be illustrated thus:
The distribution to and elimination of the drug from the compartments resolves into a negative exponential function for each compartment. Let us illustrate for simplicity a two-compartment or bi-exponential model.
The Concentration v Time Curve
When we give a single IV bolus of propofol we traditionally start by graphing the plasma concentration versus time. Time is on the x-axis since it is the independent variable. By convention the very rapid rise in propofol concentration is left out of the graph so that we see only the fall with time. The graph is the sum of negative exponentials, ie: it is two or more curves added together to form a resultant curve. There should be a distinct elbow in the curve to show the bi-exponential nature, and it has two phases labelled α and β for the redistribution phase and the elimination phase. (This is where we get β half life from).
The general equation of the graph is (for a triexponential model):
Ct = Ae-αt + Be-βt + Ge-γt
The semi-log plot
We can graph a semilog plot for linearity.
From the semi-log plot we linearize the exponential components of the graph. The concentration is plotted as the natural log, ln. The gradient, which is negative because the concentration is falling, is the rate constant, k.
EXPONENTIAL DERIVATION: Where does e come from?
We have already assigned the foregoing functions the term exponential and incorporated the term e in the equations. This is how e materialises:
In an exponential function the rate of change of the function is proportional to the dependent variable. In the case of plasma concentration of propofol for example, the rate of decline in concentration is proportional to the concentration itself. Another example would be Newton’s Law of Cooling where the rate of decrease in temperature (eg. for a hot cup of tea) is proportional to the temperature itself. Other examples are radioactivity and change in resistance in a negative temperature thermistor. So putting this mathematically:
If we write that the rate of change of y is inversely proportional to y and incorporate the necessary constant, k, and a minus sign because the function is decreasing, then we get
dy/dx = -ky this is a first order differential equation so we need to
separate and integrate→
∫1/y.dy = ∫-k.dx
Now we have to know that the integral of 1/y is a standard form and is lny or (logey), which gives us
lny = -kx and rewriting this since logay=x is equal to xa=y gives the result
y = e-kx
What is e?
e is a number, it is Euler’s number. It cannot be expressed as a fraction, nor as a recurring decimal, and is called a transcendental number, π is also transcendental. e is 2.718281828459. It is sometimes defined as the base of the natural logarithm and is the sum of the infinite Taylor series:
1 + 1/1! +1/2! +1/3! + 1/n!........
Or put another way, ∑1/k! from 0 to ∞.
In the curve y=1/x, e is the area bounded by x=1 to x=2.71828. This means ∫1/x between these limits is 1. On the graph:
Half life and the Time constant
From the concentration/time curve we can illustrate the concept of half-life, which is the time taken for the plasma concentration to fall by 50%.
Now we have, k = ln2/t1/2
Half life is not the same as the time constant, though they are frequently mistaken. The time constant depends on the rate constant which is 1/k and designated τ, and is the time taken for the plasma concentration to fall to 1/eth or to 36.7% of the starting value.
τ = t1/2/ln2 or τ = 1.44 x t1/2 [ln2 = 0.693]
thus the time constant is 44% longer than the half-life.
See graph below.
The derivation of half life is as follows.
If the initial plasma concentration at time 0 is designated C0 then at t1/2 the concentration is C0/2
The equation of the straight line in the semi-log plot is: lnC = lnC0 – k.t
[in the form y = c – mx]
So ln(C0/2) = lnC0 – k.t1/2
lnC0 – ln2 = lnC0 –k.t1/2
ln2 = k.t1/2
t1/2 = ln2/k where ln2 = 0.693
Now what about clearance?
The equation can easily be remembered from the definition if said in the right way.
Clearance is the volume of plasma completely cleared of drug per unit time.
Cl = VD x kel
Cl = VD x ln2 / t1/2 or t1/2 = VD x ln2 / Cl
These equations allow the relationship of the 4 main pharmacokinetic parameters to be manipulated easily. For example how clearance is related to half life. We know intuitively that if the clearance is high the half life is short, but these equations show the inverse nature at a glance.
Is expressed as the AUC for the oral dose divided by the AUC for the IV dose (given two equal doses).
From time zero to time infinity the concentration time curve integrates to A/α + B/β. This is the same as C0/kel. And remember C0=D/VD.
Kinetics of Infusions
This is only a little more complicated and we have all but one of the equations already. There a several named models for TCI systems ( Marsh, Minto and Schnider). They incorporate different pharmacokinetic parameters but will all produce a concentration time curve like this:
There are 3 distinct phases: the initial rapid infusion to reach a set target which is really an IV loading dose, the steady state part, and the by now familiar elimination which is exactly as before.
The initial rapid rise is described by a wash-in curve and has the same mathematical basic form as charging a capacitor or washing in a volatile agent, thus:
C = [ 1 – e-kt ] and strictly in this case: Ct = D/Cl.[ 1 – e-kt ]
At steady state the rate of drug in equals the rate out so:
Css = Rate in/ Cl
And the third phase is the negative exponential again.
THAT’S THE END OF PHARMACOKINETICS.
Now there are just a few incidentals. These are questions or parts thereof that arise every now and again and need to be seen at least once before the big day.
Dose – response curves and enzyme kinetics
These are the same. The familiar dose-response curves are based on the Michaelis – Menten model of enzyme substrate interaction. The curve looks like this, and is a rectangular hyperbola.
It is conventionally described by the MM equation and looks the same as a wash in curve. Note that there is more than one way to write an equation to fit a curve, and incidentally that taking logs does not linearize all curves.
Eqn. V = Vmax .[S] / km + [S]
Vmax is the maximal reaction rate for the enzyme and substrate and in pharmacodynamics is analogous to the maximal response.
Someone was once asked how to make this graph a straight line and the answer is to use the double reciprocal plot also known as the Lineweaver–Burke Plot.
The MM equation is turned upside down and separated into an equation of the form y=mx+c to give a straight line from which the biochemists say it is easier to look at types of enzyme inhibition.
Just for completeness:
1/V=km/Vmax.[S] + [S]/Vmax.[S]
1/V=km/Vmax.[S] + 1/Vmax
And the result is this, with the gradient of km/Vmax and a y-intercept of 1/Vmax, and an x-intercept of -1/km.
The result is this straight line, the Lineweaver Burke Plot: