Now that we have covered what is meant by terms volume of distribution and clearance, we can now turn out attention to yet another key pharmacokinetic parameter – the elimination constant, k_{el}. The elimination constant is closely linked to the ‘half-life’ of a drug in the body, therefore this article attempts to explain what both of those concepts mean, how they are estimated, and their importance in pharmacokinetic profiling of medicines.

### Elimination and half-life all in one compartment

When the study of pharmacokinetics emerged, it was clear that modeling the drug’s journey through the body would be complex.

Many factors needed to be considered, such as the mechanism and kinetics of the drug’s absorption into general circulation; its propensity to bind plasma proteins; and its physical and chemical properties that govern its movement from the bloodstream into tissues where it acts on the target receptor. To simplify things, the body was assumed to be a single, well-stirred compartment. This assumption is known in pharmacokinetics as **one-compartment model**, a concept used in this article to explain the concept of the elimination constant and its relationship with half-life, clearance and the volume of distribution.

Another decision that needs to be considered, before carrying out calculations to estimate pharmacokinetic parameters, is whether a drug’s elimination is **zero-order** or **first-order**.

**Put simply, if the elimination rate of a drug is dependent on its plasma concentration, the elimination is said to be first-order. Conversely, if the elimination rate is independent of drug’s plasma concentration, the elimination is said to be zero-order. **

A classic example of the latter is the elimination of ethanol from the body, which is dependent on its metabolism by the enzyme alcohol dehydrogenase, whose catalytic activity is the limiting factor for alcohol elimination from the body. Zero-order kinetics therefore represent a process which is saturable, whereas first-order kinetics represent a process where the limiting factor for the elimination of the drug, or its metabolites, is its delivery to the eliminating organs (in most cases, that means the kidneys), which, in turn, is dependent on the concentration of the drug in plasma. Plasma concentration of a drug will therefore decline **exponentially**, as the drug elimination rate will be highest at the highest concentration in plasma (Figure 1).

In my previous article about the apparent volume of distribution (V_{D}), I outlined how this parameter also gives us clues about the extent of the drug distribution in tissues. Without making any calculations, I hope that you can now appreciate that the greater the V_{D}, the easier the extent to which the drug distributes into tissues, and the longer it stays in the body. This is precisely why V_{D}, clearance and the elimination constants are closely related parameters.

**Figure 1: Exponential drug elimination.** A) Shows the typical shape of a curve on plasma concentration – time graph, demonstrating the exponential elimination of a drug, if one-compartment first-order elimination model is employed. B) Shows that a straight line is obtained if a semi-logarithmic plot is used instead.

### Determining the elimination constant after a single dose administration

**Half-life** is a basic pharmacokinetic parameter that can be easily obtained from the plasma concentration – time curve, like the one shown in Figure 1a. It can be defined as the time taken for the initial drug concentration to halve. Thus, after two half-lives the original concentration of the drug in plasma would have fallen by 75%, as the decline is exponential for a first-order elimination process. Half-life does not only determine the time course of drug ‘removal’ from the plasma but it also predicts the time course of drug accumulation in the body to a steady-state concentration when the drug is first given, which is important clinically, especially in situations where rapid drug action is required. As such, half-life is a very useful parameter; however, it is **not** a direct measure of drug elimination but rather a product of other parameters such as clearance, which tells us the efficiency of drug elimination, or V_{D}, which tells us the extent of drug distribution.

The process of elimination begins immediately after the single dose of a drug is administered, and it is governed by the total systemic clearance of the drug, **Cl _{s}**, which can be determined by several different ways. One of them is by determining the

**elimination constant (k**first from the semi-logarithmic plot of plasma concentration – time curve, and using the relationship between the elimination constant, V

_{el})_{D}and Cl

_{s}(Figure 2). Indeed, as the elimination constant is, in fact, the ratio of clearance of the drug to its volume of distribution, the greater the elimination constant, the quicker the elimination of the drug from the body.

**Figure 2: Relationship between clearance, volume of distribution and the elimination constant.** This relationship clearly shows that, in fact, the elimination constant is a ratio of total systemic clearance to the volume of distribution of the drug. The higher the elimination constant, the greater the elimination of the drug, as the drug would likely have a small V_{D} (distributes only in plasma) and a high Cl_{s} (is efficiently cleared by the eliminating organs).

So, how can we determine k_{el} and V_{D} to estimate clearance, arguably the most important pharmacokinetic parameter? There are two ways to determine k_{el}: graphically or by simple arithmetic if the half-life is known. Both of those ways are illustrated in Figure 3 below.

**Figure 3: Estimation of the elimination constant.** There are two ways in which we can estimate the elimination constant. First, it can be estimated graphically: the exponential plasma concentration-time curve (A) is plotted in a semi-logarithmic plot, which is converting the expression describing plasma concentration to an equation for a straight line (B) where the slope is the negative of the elimination constant. To then determine clearance using the expression in Figure 2, this straight line can be extrapolated to x=0, which allows you to calculate the concentration at time 0 (C_{0}), and thus the volume of distribution at time = 0 because the dose given is known (V_{D} = D/C_{0}). Another way to determine the elimination constant is to use the relationship shown in C, where the half-life is estimated directly from the plasma concentration – time graph. 0.693/t_{1/2 }would then give the elimination constant (this article does not go into details about why we use 0.693 but it comes from substitution of half-life into the expression that describes plasma concentration of the drug).

Lastly, combining the expressions for half-life in Figure 3C and that for the elimination constant in Figure 2, one can directly obtain the clearance if both half-life and volume of distribution are known (Figure 4).

**Figure 4: Determining clearance from half-life and volume of distribution.** This expression can be used to directly determine clearance of a drug if the V_{D} and half-life are known.

### Summary

The aim of this article was to introduce the reader to the concepts of half-life and elimination constant – two pharmacokinetic parameters which can be used to estimate clearance, thus the efficiency of elimination of drug from the body. It is important to stress that these methods only apply to drugs, which do behave according to the one-compartment model. In more complicated situations, clearance is determined using a method that uses a so-called area-under-curve (AUC) parameter, a subject under discussion in the next article of this series.

*Michal Barabas is currently a medical student at the University of Cambridge, where he also teaches pharmacology to undergraduates in small group seminars. Michal obtained a BSc. degree in Pharmacology from UCL in 2013 and an M. Phil degree in Translational Medicine from the University of Cambridge in 2014.*

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