# Volume of distribution

This chapter answers parts from Section B(iii) of the 2023 CICM Primary Syllabus, which expects the exam candidate to "describe the distribution of drugs and factors that influence this". Though this learning objective does not explicitly name "volume of distribution" as an expectation, the understanding of this fundamental concept is implied by the fact that it has appeared in the past papers. Question 12 from the second paper of 2015 specifically asked the candidates to "define volume of distribution and describe the factors that influence it"; Question 5 from the first paper of 2019 also asks for methods of its measurement.  This chapter aims to offer a sensible and memorable definition, and then discuss some of the patient factors and drug factors which influence a volume of distribution (or the measurements used to determine it).

In summary, volume of distribution is a pharmacokinetic concept which is used to describe the distribution of drugs in the body as relative to the measured concentration. In brief, it is the apparent volume into which the drug appears to be distributed when only the sample concentration is considered. It is a purely theoretical volume, which can substantially exceed the total body volume, or potentially even be infinite in size. Among many other uses, the volume of distribution (VD or Vd) plugs into loading dose calculations. It can also help you decide instantly whether your drug is going to be easily cleared by dialysis, and it can be used to retrospectively estimate the magnitude of a drug overdose.

Or, if we like boxes,

Volume of distribution (Vd) is the apparent volume into which a drug disperses in order to produce the observed plasma concentration.

Vd is used to calculate loading doses, much as clearance is used to calculate maintenance dose.

Vd = dose / plasma concentration

It can be expressed as litres, or indexed to body mass in L/kg.

Plasma concentration can be observed at different times, giving rise to several different possible strategies of calculating the volume of distribution.

Vinitial  = Vd of the central compartment (from the rapid distribution phase)

Vextrap = Vd of the tissue compartment (from the elimination phase)

Varea = Vd extrapolated from the AUC of the concentration curve

The major determinants of Vd are drug properties which affect protein binding and tissue binding. These consist of molecule size, charge, pKa, and the lipid/water partition coefficient.

Patient factors which affect Vd include age, gender, body muscle/fat proportion, level of hydration, water distribution (oedema, effusions, ascites, pregnancy) and extracorporeal sites of distribution (circuit, filters, oxygenator etc)

Published literature which discusses volume of distribution is hardly rare. In fact the time-poor exam candidate can safely limit their reading to any of the primary texts recommended by the college, or any educational institution for that matter. It is such a fundamental concept that usually the pharmacology textbook will tackle it in the first couple of chapters. For example, Birkett (2009) has it in Chapter 2 (p.24). For those unwilling to pay \$49.00AUD to McGraw Hill, the 1968 article by Riegelman and Rowland is a classic. Or, you could turn to veterinary literature and read Toutaine et al (2004), given that pharmacokinetic theory transcends the barriers of species and phyla. For literature specific to critically ill humans, Peck and Hill have a section in their Chapter 5 (3rd ed, p. 72-73) which is both detailed and concise. No specific single source was used to create this revision chapter and the trainee reader is reminded that pretty much every serious resource will give a satisfactory explanation

## Definition of volume of distribution

Birkett's Pharmacokinetics Made Easy (2009) defines volume of distribution as

"...not a "real volume"... It is the parameter relating the concentration of a drug in the plasma to the total amount of the drug in the body"

This is the definitive text for the CICM primary and when questioned every CICM exam candidate probably ought to regurgitate this phrase in undigested form. Other definitions also exist. Peck and Hill give a particularly good one:

"(Vd) is defined as the apparent volume into which a drug disperses in order to produce the observed plasma concentration"

The concept is relatively easy to explain. Say, having given 1 gram of something, one is presented with a sample of blood in which there is only 1mg per litre of this drug. If the drug were dispersed in the blood alone, one would be forced to conclude that it has been dispersed in a truly ridiculous 1000 litre bloodstream, in some sort of enormous fluid-filled patient.

So, it is an apparent volume. Going by the college answer to Question 12 from the second paper of 2015, it is important to conclude one's definition with a statement emphasising the ephemeral and dream-like nature of this concept. All authors feel the need to patiently remind us about Vd being a purely imaginary construct, and how it does not correspond to any real physical volume. Because without this sort of defence people would totally lose their minds when confronted with a drug like chloroquine, the Vd of which is in the thousands of litres.

## Calculating the volume of distribution

For a simple concept, one might expect a simple equation. In fact, this one certainly is:

Straightforward. Until you try to apply it to real life situations. Say, you have a bunch of healthy volunteers in your lab, your fifty bucks still warm in their pockets. You just gave each of them a brightly coloured gelatin capsule. You want to know the volume of distribution. You need to take a sample.

When do you sample them?

## Different volumes of distribution

It is clear that timing plays a major role in this, because the measured drug concentration will vary depending on the rate and extent of absorption. Even with IV administration there is going to be some delay.  A discussion of the volume of distribution as dose divided by concentration meets with a limitation of the multi-compartment model, which is the assumption that the drug is distributed equally and instantly throughout the compartment. In reality, drug concentration in the sample will vary over time because it takes time for the drug to distribute around the body, and a concentration taken within minutes of administration will be very different to the concentration taken many hours later. Clearly these will produce completely different Vd values. So, which concentration value should we use to calculate the "gospel" Vd?

Well. That depends on which compartment you are interested in.

## Initial volume of distribution (Vinitial)

Consider if you measure the Vd of the central (intravascular) compartment. It is possible to calculate this soon after a drug is administered intravenously, by extrapolating an imaginary line from plasma concentration measurements, extended to time zero. You need to extrapolate this line because under no realistic circumstances could you ever actually measure the concentration at time zero.

So, by this method, you measure the volume of initial dispersion of the drug. This volume is usually called either Vinitial, or Vc, and it represents the behaviour of the drug during the first rapid phase of distribution through the central compartment.  It is generally determined by the degree of protein binding. Drugs which are highly protein-bound will have a larger Vinitial  if you intend to measure free drug levels.

What is the point of this variable? Well, for one - if you know that the drug will not bind any protein (eg. if your drug is a protein) you can use this relationship to estimate the volume of the central compartment. This is one of the ways of determining the total blood volume (classically, 53Cr labeled red cells are used). For therapeutic purposes, it can be used to estimate high peak plasma concentrations so that - if need be- you can divide your loading dose to avoid toxicity.

## Extrapolated volume of distribution (Vextrap)

Obviously if you completely ignore the distribution of the drug into the tissues your volume of distribution estimate is going to be inaccurate for the purpose of determining such things as loading doses. The alternative approach is to ignore everything but the tissue distribution. This method takes the slow late stages along the concentration/time curve (the terminal elimination phase) and extrapolates a line of best fit from them.

Obviously this is going to be a massive overestimate for many drugs, particularly if they are drugs which disperse extensively into the tissues. Your (time=0) concentration estimate will potentially be a very low value, producing an unreasonably large Vd estimate.

Again, what is the point of all this? Good question. Hilariously, the otherwise sober ADMET for Medicinal Chemists (Tsaioun & Kates, 2011) reports that the term "does not have any scientific utility" and exists purely as a means of scaring pharmacokinetically-inclined children at bedtime ("...it was sometimes calculated to inform "budding" pharmacokineticists of the perils of estimating erroneous parameters", p. 221).  This could actually be true. Even when one can conceive of a situation where the tissue distribution of the drug plays a major role in the dosing regimen (eg. in the context of something like amiodarone), one still cannot come up with any way of using this value at the bedside. One could potentially use the Vextrap value  to identify drugs which have so much tissue distribution that clearance by dialysis is near-impossible.

## Non-compartmental volume of distribution (Varea)

The Vinitial value and the Vextrap value both focus on the drug distributing into some compartment volume (be it central or peripheral). Neither give a good estimate for the "ideal" volume of distribution, one which you could reliably use to calculate your loading doses. Varea is an attempt to get around the errors of focusing on just one compartment at a time. It uses a non-compartmental pragmatic model, easily calculated from serial concentration measurements.

where AUC is the area under the concentration-time curve and the "β" terminal elimination time constant is the slow exponential rate of decline at the latter stages of a drug's tenure in the body.

You take the whole concentration/time curve, integrate the area under it (AUC) and use this to establish the "true" volume. This gives a better (smaller) Vd estimate than Vextrap but is still frequently incorrect if there is significant distribution around compartments.  The Varea equation assumes that the rate of the concentration decline during the terminal elimination phase is the average rate of clearance for the entire duration of the dose, and that this rate remains constant. Practically, clearance is almost never constant and is usually concentration-dependent ("first order") which means that using the "β" terminal elimination time constant will always yield an underestimate of the "time=0" intercept and therefore an overestimate of the Vd.

This problem also limits the utility of Varea in altered clearance states. For instance, for a renally cleared drug Varea measured in a patient with renal failure will always be smaller (because the slope of the β terminal elimination rate will be near-horisontal). But this will not represent any sort of change in the drug's distribution.

## Steady-state volume of distribution (Vss)

As mentioned above, the Varea method assumes some sort of linear rate of drug clearance. So, it's clearly going to be useless in situations where the clearance rate is zero, or appears to be zero- for example, in renal failure, in the context of an intravenous drug infusion or when the drug has long-term regular administration.  Fortunately the real or apparent absence of clearance makes Vd calculations much easier:

The point of intersection hardly matters any more. Nobody needs to draw any intercept lines.

Vss describes the volume of distribution during steady state conditions, i.e. when there is a stable drug concentration. It is always going to be slightly lower than Varea because of the effect of clearance on the β terminal elimination time constant.

Of all the volumes of distribution, Vss is probably the most useful for calculating the loading dose. The loading dose, after all, is the dose you wish to give in order to achieve a desired (steady state) drug concentration. With the simplicity of the steady state model, the dose is calculated as (Vss × Css) where Css  is the desired steady-state concentration.

## Factors which influence the volume of distribution

Question 12 from the second paper of 2015 asked the candidates to describe factors which influence the volume of distribution. The college suggested these be divided into drug factors and patient factors. Following from the discussion of many different Vd models, one may also add "scientist factors" to the list of categories.

"Patient factors could include age, gender, muscle mass, fat mass and abnormal fluid distribution (oedema, ascites, pleural effusion). The drug factors would include tissue binding, plasma protein binding and physicochemical properties of drug (size, charge, pKa, lipid solubility, water solubility)."

Birkett (2009) at the end of Chapter 2 lists protein and tissue binding as the "major physiological determinant" of Vd. There are also a whole pile of possible influences which would probably be relevant. It might be more effective to tabulate these into "factor" and "influence" columns.

 Measurement and pharmacokinetic modelling of Vd Timing of measurements Depending on when the measurements are taken, the Vd will be different (i.e. it will correspond to Vinitial if the measurements are taken too early, and Vextrap if they are taken during the elimination phase). Pharmacokinetic model Vinitial, Vextrap, Varea and Vss are various ways to estimate the Vd of a drug from empirical measurements. All of these methods will yield slightly different results - or, occasionally completely different results. Free vs. total drug levels In highly protein bound drugs, the calculated volume of distribution for the "total" drug levels will be totally different to the Vd calculated for the free drug. Total Vd will correspond to the Vd of the binding protein rather than the drug itself. Properties of the drug Molecule size The larger the molecule, the harder it will be for it to passively diffuse out of the central compartment, and therefore the smaller the Vd. Molecule charge Highly ionised charged molecules will have higher water solubility, and may even be trapped in the central compartment by electrostatic factors which keep them bound to proteins with corresponding charge. pKa pKa determines the degree of ionisation and therefore influences lipid solubility Lipid solubility Lipid solubility is one of the major determinants of Vd; highly lipid-soluble drugs will have the highest Vd values because of the low fat content of the bloodstream. Water solubility Highly water-soluble drugs will have difficulty penetrating lipid bilayer membranes and generally tent do have smaller volumes of distribution, essentially being limited to extracellular water. Properties of the patient's body fluids pH pH interactes with the drug's pKa to influence the degree of lipid solubility. pH also influences the degree of protein binding (a good exmaple of this is ionised calcium) Body water volume Dehydrated patients will have drug levels concentrated in the plasma just as all dissolved substances are concentrated by loss of water. Protein levels For highly protein-bound drugs, lower serum protein levels will result in a higher free (unbound) drug fraction. This may have little effect on the Vd as calculated from total drug concentration, but if you are measuring free drug levels it will make the Vd appear smaller. Displacement Drugs may be displaced from their protein and tissue binding sites by the effects of pH or by competition from other drugs/substances (eg. urea). Displaced drugs mayl redistribute into plasma, decreasing the calculated Vd. Effects of physiology and pathological states Age As an old professor of mine had put it, babies are grapes and the elderly are raisins. As you age, body water content decreases, shrinking the Vd of water-soluble drugs. Muscle mass also decreases, and so tissue binding diminishes. Gender Female Vds tend to be lower than male Vds due to the generally lower  body water content (Soldin & Mattison, 2009) Pregnancy Both the body water and the body fat content increases, and therefore the Vd increases for most drugs. Not to speak of the possible distribution into amniotic fluid and foetus. Oedema Oedema represents increased body water and this influences water-soluble substances; Vd for these will increase Ascites / effusions Just as in oedema, large fluid collections may sequester water soluble drugs and act as reservoirs. Effects of apparatus Adsorption on to apparatus Dialysis filters and ECMO circuits tend to adsorb drugs in an unpredictable fashion, resulting in an apparent increase in the volume of distribution. Volume expansion In the context of bypass circuits and other large extracorporeal machinery, there may be 2000-2500ml of additional extracorporeal fluid, which will change the volume of distribution (particularly for drugs which are largely confined to the central compartment)

## References

Gibaldi, M., and P. J. McNamara. "Apparent volumes of distribution and drug binding to plasma proteins and tissues." European journal of clinical pharmacology 13.5 (1978): 373-378.

Toutain, Pierre-Louis, and Alain BOUSQUET‐MÉLOU. "Volumes of distribution." Journal of veterinary pharmacology and therapeutics 27.6 (2004): 441-453.

Riegelman, S., J. Loo, and M. Rowland. "Concept of a volume of distribution and possible errors in evaluation of this parameter." Journal of Pharmaceutical Sciences 57.1 (1968): 128-133.

Krishna, Sanjeev, and Nicholas J. White. "Pharmacokinetics of quinine, chloroquine and amodiaquine." Clinical pharmacokinetics30.4 (1996): 263-299.

Wagner, John G. "Significance of ratios of different volumes of distribution in pharmacokinetics." Biopharmaceutics & drug disposition 4.3 (1983): 263-270.

Soldin, Offie P., and Donald R. Mattison. "Sex differences in pharmacokinetics and pharmacodynamics.Clinical pharmacokinetics 48.3 (2009): 143-157.