**The derivation of half life is as follows**.

If the initial plasma concentration at time 0 is designated C_{0} then at t_{1/2} the concentration is C_{0}/2

The equation of the straight line in the semi-log plot is: lnC = lnC_{0 }– k.t

[in the form y = c – mx]

So ln(C_{0}/2) = lnC_{0 }– k.t_{1/2}

lnC_{0} – ln2 = lnC_{0} –k.t_{1/2}

_{ }

_{ }ln2 = k.t_{1/2}

_{ }

t_{1/2} = 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.

So:

Cl = V_{D} x k_{el}

And rearranging:

Cl = V_{D } x ln2 / t_{1/2 }or t_{1/2} = V_{D} 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.

** **

** **

**Bioavailability**

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 C_{0}/k_{el. } And remember C_{0}=D/V_{D}.

**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: C_{t} = D/Cl.[ 1 – e^{-kt} ]

At steady state the rate of drug in equals the rate out so:

C^{ss }= 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 = V_{max .}[S] / k_{m} + [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:

V=Vmax.[S]/Km+[S]

1/V=km+[S]/Vmax.[S]

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: