# Key Equation

By definition, at one half-life from time zero the concentration will be half that of the concentration at time zero (time zero can be chosen arbitrarily for this calculation).

U divide bivtli sidfJi try the coimnmi factor (C) U like njtunJ logarithm of both iitfcs

U Lin- -logj l-^uaIs li.ig.2, substitute to crrait ktLy Key Equation 2

The natural logarithm loge' may also be written In'. The concentration and rate of decline of concentration in Figure PK.13 fall exponentially with time. Mathematically and graphically, it is easier to work with linear relationships. Using a logarithmic scale for the concentration (or the natural logarithm of the concentration) against time on the original linear scale (Figure PK.12) produces a straight line (Figure PK.14).

Tims

Figure PK.14. Semilogarithmic plot of concentration against time

The straight line can now be extrapolated to the y-axis (t = 0) to obtain a theoretical (or apparent) value for concentration at the time of injection (C0). This is the predicted concentration if there was instantaneous uniform distribution of the drug of dose (D) throughout the compartment at the time of injection before any elimination has occurred. This single-compartment model can then be used to calculate the volume of distribution (Vd) as follows:

In practice, this simple exponential decline is masked in the clinical situation by the combined affects of absorption, distribution, redistribution and elimination. Most clinical data sets of plasma (or blood) levels of an intravenously injected drug have a pattern similar to Figure PK.15.

This two-compartment model can be seen to have two linear (graphical) components that can be separated. The initial rapid decline is the result of rapid redistribution throughout the central compartment being predominant. Later, redistribution to the peripheral compartments is predominant. Clearance is constant throughout, but elimination is proportional to concentration and is greatest earlier in the process. Each component has a half-life and volume of distribution. Elimination half-life (t1/2p) is the usual value quoted. To calculate elimination halflife using the graph, the following steps may be followed.

0 24 40 60 80 100 120 140 160 160 Tims (mifi)

0 24 40 60 80 100 120 140 160 160 Tims (mifi)

Figure PK.15. Semilogarithmic plot of plasma concentration against time

Step 1

Extrapolate the straight line of the distribution phase to the y-axis (t = 0). Call this "Co' as before. Step 2

The elimination phase has a slope of p. Extrapolate the straight line of the elimination phase to the y-axis (t = 0). Call this B'. This is the initial concentration that would have existed if there was instantaneous distribution to equilibrium on IV injection. This can be used to calculate a value for the volume of distribution following equilibration with the tissues. This is also an approximation of the volume of distribution at steady state (Vd ss). However, this is calculated and defined as the volume of distribution when giving an infusion of a drug at exactly the same rate as total clearance of the drug.

Step 3

Subtract B from C0 to create concentration A. Step 4

Subtract the P slope from the concentration plot to create a new straight line of slope a (the distribution element of the graph). Although it may seem unusual that a curve appears as the result of the addition of two straight lines in part of the graph, this is because of the logarithmic scale used for concentration. Observation of this scale shows that it does not reach zero at the x-axis.

Step 5

Use these values mathematically to calculate the following variables:

• Area under the curve (AUC) = A/a + B/p Clearance

Clearance (Cl) represents the volume of blood completely cleared of drug in unit time. The amount of drug eliminated is, therefore, dependent upon the drug concentration, which is a first-order process. Typical units for clearance are "ml of blood/kg body weight/minute' (i.e. ml/kg/min). Total clearance is the sum of all the individual clearances such as hepatic (ClH), renal (ClR) and others (ClX). Clearance can be calculated from a graph using AUC:

In other words, clearance = volume of compartment x elimination rate constant.

Referring back to the two-compartment model, the volume of distribution at steady state (VdSS) is the usual volume quoted as follows:

where k12, k21 = intercompartment rate constants VdI = initial volume of distribution.

As the plasma concentration is determined by the addition of the a- and P-profiles, Cp = Ae-at + B e-Pt. Note that at t = 0, e0 = 1, so Co = A + B).

This is a "best fit' and most drugs fit the two-compartment model; however, it should be remembered that this apparent mathematical model is derived from numerous distribution profiles and a range of tissue profiles. Some drugs such as propofol best approximate to a three-compartment model that is a tri-exponential model. Values pertaining to this are shown in Figure PK.16.

 PHARMACOKINETIC VALUES FOR PROPOFOL Compartment tl/2 (min) Central (1) 2-3 Peripheral (2) 30-60 Deep peripheral (3) 180-480 Vdl 230 ml/kg Vdss 12 l/kg

Figure PK.16 Effective Levels

### Figure PK.16 Effective Levels

Effective levels of volatile agents can be monitored by measurement of the end-tidal concentration. Blood levels of IV agents are not readily monitored, but desired concentrations can be targeted using pharmacokinetic models based on patient demographic data, and compared with plasma equivalents of MAC which are:

• MIR—minimum infusion rate that prevents response to surgical stimulus in 50% of patients

• ED50—a general term to describe the dose that is effective in 50% of subjects

• EBC—effective blood concentration (sometimes called EC)

The strategy of a target concentration is one method of delivering the anaesthetic dose. A target concentration is chosen in a similar way to volatile agent concentration and a computer-controlled pump calculates bolus and maintenance infusion rate based on patient details. The target concentration is altered according to clinical need and the pump adjusts accordingly. The concept of an effective blood concentration of the agent depends on the following conditions:

• Intensity of drug action (predictable from concentration at the receptor site)

• Concentration at the receptor site proportional to free plasma concentration

In total IV anaesthesia (TIVA) using propofol, the aim is rapidly to achieve a blood level of propofol that will induce anaesthesia, and then maintain this state until the procedure is complete. Discontinuing the infusion will then lead to a rapid recovery of consciousness as blood (and brain) concentrations of propofol fall. A constant infusion achieves a plateau level, but this is slow. A loading dose is, therefore, used, based on the volume of distribution (Vd) and the desired concentration (Cp) in the following way:

This may be followed by a maintenance dose calculated from the total clearance and the desired plasma concentration (this being equal to the amount of drug being eliminated in unit time) in the following manner:

Maintenance dose = CI x C^

Another method of maintaining a therapeutic level over a prolonged period is to administer intermittent doses so that the bolus is given before the plasma level falling below the therapeutic threshold. For maximum interval, the dose used should raise the plasma level to the top of the therapeutic range without reaching toxic levels. Repetitive dosing results in a plateau once elimination (determined by concentration) equals elimination. The dosing interval is usually approximately that of the terminal elimination half-life.