Have you ever wondered about what happens when you swallow a tablet of ibuprofen? How does it get from your gut to the circulation? What happens to it in your blood and tissues? How does it know where to go to ‘kill’ your pain? More importantly, in case you took it to combat the pain, why does it take so long to work?! These are some of the questions that pharmacokinetics, i.e. the study of the kinetics of drugs within the human body, is concerned with, and the aim of my next few articles is to guide you through how the pharmacokinetic field attempts to answer them.

The first article of this series will be concerned with introducing one the most basic, but extremely useful, parameters known as the *apparent volume of distribution* (V_{D}). The apparent volume of distribution gives us not only an estimate for the extent of drug distribution in the body (more specifically, the volume ‘taken up by the drug’), but also relates to the physicochemical properties of that drug. The concept does sound very simple; however, as the word ‘apparent’ suggests, V_{D} is not just a regular volume in the classic sense of the word, but rather a more abstract term. Therefore, before I can explain *how* we can measure the volume of distribution and what it means, I first need to define the boundaries of that distribution by reviewing the fundamental assumptions used in its determination.

#### Volume of distribution in the single-compartment model

It is no secret that the human body is a very complex system. Abiding to the laws of physiology and biochemistry, the various organs and tissues, bound together by anatomical structures, work together to provide a constant internal environment into which a given drug is administered. This complexity should therefore be reflected in pharmacokinetic calculations, in order to approximate an accurate description of drug kinetics within the body. This is certainly attempted at a more advanced level but, for the purposes of this article, I will be using a more simplified model.

This is called the *single compartment open model*. Its scheme is represented in Figure 1, and shows that the model essentially works on the **assumption** that the drug can enter (administration) or leave (elimination) the body, and that the body **acts as a single, uniform compartment**. It is important to note that it certainly uses assumptions that are much too simplistic; however, it is well suited for the introduction of key pharmacokinetic concepts such as the volume of distribution, or clearance.

To make things even simpler, I will only consider drugs given by intravenous administration, as it is the simplest route of administration from modelling point of view and, as the drug enters the bloodstream immediately after injection, the drug absorption process is considered to be** instantaneous** and **homogeneous **throughout the compartment. As this model does not distinguish between different tissues, which are indeed experiencing different blood flows, and certainly a different binding affinity for the drug based on what receptors they are expressing, it is again assumed that all tissues take up the drug at the same rate, and thus that the drug **equilibrates** rapidly between the bloodstream and tissues.

In reality, the uptake of the drug by tissues takes place at varying rates and, on top of the factors described above, its uptake is also characterised by its physicochemical properties, such as lipophilicity and molecular weight. Thus, the only strictly kinetic parameter that can be calculated from the single compartment model is the **rate of elimination**, which is performed by determining the *elimination rate constant*, *k*.

To summarise the single compartment model, it thinks of the body as a single tank of uniform fluid into which a single dose of drug is injected and gets instantaneously absorbed (as it would be injected directly into the bloodstream) and assumed to be uniformly distributed throughout the bloodstream and tissues. If this were the case, then the only parameter that determines the *amount of the drug in that compartment* (D_{B}) would be the rate of elimination (by the kidneys or liver) denoted by the constant *k*. The volume of distribution is defined as the volume in which *the drug appears to be distributed in*.

**Figure 1: Single-compartment open model with IV administration**. *In this rather simplistic model, the drug is administered straight into the bloodstream thus is instantaneously absorbed and is assumed to be uniformly distributed throughout the body, which is modelled as a single compartment. The amount of drug in the body (D _{B}) is governed by the rate of elimination, represented by the elimination constant k (mainly through the kidneys and liver). The apparent volume of distribution (V_{D}) represents the volume of that compartment in which the drug appears to be distributed in, and C_{p} represents the concentration of the drug in plasma.*

#### Why is the volume of distribution called ‘apparent’?

The single compartment model assumes a uniform distribution with an almost instantaneous equilibration between the bloodstream and tissues. Therefore, the model assumes that changes drug levels in the plasma will result in proportional changes in drug levels in tissues. This means that getting a sample of the drug concentration within accessible bodily fluids, such as the plasma, may be used to provide an indirect measure for the amount of drug in the body at a given time. Similarly, if the dose of the drug given by injection is known (and in most healthcare settings it certainly should be!), sampling the plasma for the concentration of that drug (C_{p}) will enable the calculation of the apparent volume of distribution. This relationship between drug amount, concentration in plasma and volume is given in Figure 2.

**Figure 2: Relationship of plasma concentration, volume of distribution and amount of drug in the body**. *If the pre-injected amount of a given drug is known (D _{B}), and the plasma concentration of that drug is sampled from the plasma after the injection (C_{p}), one is able to determine the volume of distribution (V_{D}). This relationship can be rearranged to get an expression for any of the other two variables.*

However, the calculation cannot be just performed as is shown in Figure 2. As the single compartment model predicts, the drug also gets eliminated from the body, which is governed both by the rate of metabolism in the liver and rate of elimination in the kidneys, as summarised by the variable *k* – the *elimination rate constant*. This means that as soon as the drug gets injected, it also gets eliminated with time. Therefore, the longer the delay between drug injection and plasma concentration sampling, the smaller the estimate for drug’s volume of distribution would be because the drug would have been subjected to elimination in that time. For this reason, it is important to use the *concentration of a given drug at time 0 *(C_{p}^{0}) rather than plasma concentration at any time.

The estimation of this parameter is now more complicated, as one has to precisely describe the kinetics of drug elimination because it is physically impossible to obtain a plasma sample at the time of injection. If the elimination kinetics are known and plotted graphically, C_{p}^{0} can be found by **extrapolation** from the elimination curve. In practice, many samples are taken after injection at particular time points and a curve of drug plasma concentration versus time is plotted. As most elimination and metabolism processes are **first-order**, that is, their rate depends on the concentration of the drug present; drug excretion will also follow first-order kinetics. This enables the linearization of the plasma concentration versus time curve, and thus an easy determination of C_{p}^{0} by extrapolation (Figure 3). Please note that I will only offer the description of how this is performed here. The full explanation of elimination calculations is not the focus of this article but will certainly be discussed in the series of articles on pharmacokinetics.

**Figure 3: A method to determine plasma concentration of a given drug at time 0**. *One way to determine C _{p}^{0 } is to collect plasma samples at set time intervals to obtain a graph of plasma concentration of the drug versus time. If the drug elimination kinetics follow a first-order direction, the plasma concentration at a given is described by the decay equation Cp = C_{p}^{0}e^{-kt}, which can be converted to a linear graph by taking the natural logarithm of both sides of the equation and plotting ln Cp versus time. One can then extrapolate the value of C_{p}^{0} from the obtained straight line.*

As soon as C_{p}^{0} is found, we can then re-visit the equation for volume of distribution given in Figure 2 and re-formulate it to give a better estimate for the value of V_{D}. This is given in Figure 4.

**Figure 4: Formula for the volume of distribution**. *This formula gives us a better estimate for the value of V _{D} calculated from a single compartment model after a bolus intravenous injection of a given drug. D_{B0} and C_{p}^{0} signify the amount of drug in the body at time 0 (the dose given in the injection) and plasma concentration at time 0 as determined by the method outlined in Figure 3, respectively.*

Now all that remains is to answer the question stated in the title of this subsection. Why do we give the V_{D} estimate the label ‘apparent’ even though we have described a method for its more precise estimation? The reason is because V_{D} does not have a *real physiological meaning, in terms of anatomical space*, as the model we derived it from (the single compartment model) only regards the body as a simple, fluid-filled tank where the drug distributes. The standard 70-kg male is estimated to have a total body volume of 42 Litres. It is common for V_{D} values, which are effectively based on the ratio of dose given and concentration sampled at time 0, to be much larger than that. The drug is being injected into the plasma but because the model represents a single compartment of an unknown volume, the determination of that volume will give us an idea of the extent of the distribution of the given drug dose within the compartment. In another words, the smaller the concentration of drug in plasma at time 0 compared to the dose given (i.e. the larger the ratio between them), the greater the volume of distribution will be, and the greater the extent of drug distribution in the peripheral tissues of the body.

#### The significance of the apparent volume of distribution

So if the volume of distribution can give us a volume that is physiologically impossible, why is it a commonly used pharmacokinetic parameter? Volume of distribution is useful because it can tell us something about *where* in the body the drug is distributed. Drugs with a large V_{D} are more likely to be concentrated in extravascular tissues and less concentrated intravascularly. Similarly, if the value of V_{D} is small, such as 3 Litres, it tells us that the drug likes to stay within plasma only, most likely bound to plasma proteins. If the value of V_{D} is larger than total body water, it would suggest the drug likes to concentrate in peripheral tissues (probably a specific ones where the drug has a large affinity).

In addition, it can tell us something about the physicochemical properties of the drug. If the V_{D} suggests distribution in plasma only, then it the drug is more likely to be charged and thus has difficulty penetrating cell membranes and diffuse into tissues. If the value of V_{D} rather suggests extensive distribution in peripheral tissues, it is more likely that the compound is lipophilic and has no trouble diffusing through cell membranes.

Lastly, since the value of apparent volume of distribution is determined by the properties of the drug, it has a constant value (apart from certain physiological states such as oedema or changes in total body weight, commonly in old age). This therefore enables us to estimate the amount of drug in the body at a particular time if the concentration of the drug is sampled, using formula in Figure 2.

#### Conclusion

This article aimed to review how the commonly used, basic pharmacokinetic parameter known as the volume of distribution is estimated from single compartment models of drug distribution. I outlined the basics of this model, the method used to derive the apparent volume of distribution from it and the meaning of V_{D} and its significance for clinical pharmacology. In my next article I will focus on the concept of clearance and half-life and their obvious relationship to the volume of distribution.

*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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