# Single and Multiple Compartment Models of Drug Distribution

This chapter answers parts from Section B(i) of the 2017 CICM Primary Syllabus, which expects the exam candidate to *"Explain the concept of pharmacokinetic modeling of single and multiple compartment models". *This expectation has never materialised in the form of a written exam question, but in Viva 7 from the first paper of 2010 the unlucky candidates were asked to draw a concentration-time curve for a bolus of fentanyl, which might have led to a discussion of compartment models.

In brief:

- Compartment models simulate drug absorption distribution and elimination.
- They are a convenient oversimplification used to predict the concentration of a drug at any given time in any given body fluid or tissue.
- A single compartment model is the least accurate, as it assumes a homogeneous distribution of the drug in the body.
- A three-compartment model is the most useful for anaesthetic substances as it discriminates between fast-redistributing tissues (muscle) and slow tissues (fat)
- The net effect of having multiple compartments is that there is an initial fast distribution phase, followed by a slower elimination phase during which the concentration is maintained by redistribution from drug stores in the tissues.
- Each phase has its own halflife; usually the "overall" halflife quoted by textbooks is the halflife for slowest of the phases, which tend to massively ovepredict the "real" halflife of a highly fat-soluble drug.
- Adding more compartments may not necessarily improve the predictive value of the model
- All models have limitations, such as the assumption that clearance occurs only from the central "blood" compartment.

## Official reading material

*“Pharmacokinetics made easy”* by Birkett and Australian Prescriber is the official textbook for this topic which is listed by the college in the new curriculum. Previously no specific pharmacokinetics text was suggested, but those few SAQs from the past which offered a specific textbook reference seemed to have favoured Goodman & Gillman.

However, this textbook leaves much to be desired in terms of depth, being merely a 129-page pocket guide. For the time-poor exam candidate this length is far too great, and something rather quicker is going to be required. For the person in need of indepth understanding who has no exam stress and infinite patience, something more detailed is in order. In short, the recommended text is probably enough to make your own shortform cram summary, but it may not endow upon the reader a profound sense of oneness with pharmacokinetics.

For compartment modeling specifically, *“Pharmacokinetics made easy”* is certainly going to be somewhat pointless. If you search for "compartment", you come up with two results, neither of which is particularly informative (one happens to be the introduction chapter). The truly dedicated pharmacokinetics fiend will instead submerge themselves in the reeking bog of Peck & Hill's "*Pharmacology for Anaesthesia and Intensive Care"*. There, on page 63 of the 3rd edition after thirteen pages of hardcore calculus the authors finally arrive at a point where they feel the reader is ready to discuss compartment models. If this level of mathematical assault is insufficient for the reader, they can get their fix from *Compartment models* by Blomhøj et al (2014), which is a decent forty pages or so. Lastly, if one is still unsatisfied by this level of detail, I can only recommend *Modeling in Biopharmaceutics, Pharmacokinetics, and Pharmacodynamics* by Machera and Iliadis, which is Volume 30 of the *Interdisciplinary Applied Mathematics* collection.

## Compartments in pharmacokinetics

There is no such thing as a compartment in reality. One cannot expect to open the abdomen and find the Fat Compartment there, full of lipophilic drugs. Instead, compartments are convenient mathematical constructs which help us model drug distribution.

A handy definition of "compartment" for exam purposes might be:

"A pharmacokinetic compartment is a mathematical concept which describes a space in the body which a drug appears to occupy. It does not need to correspond to any specific anatomical space or physiological volume".

In this fashion, a compartment model is a handy mathematical construct which allows us to intuitively consider the behaviour of a substance when it is infused into a patient. The model itself appears to have actually arisen from the need to measure compartments, as in the case of this paper by Aldo Rescigno (1960) which discusses the development of a *"function which permits the fate of the labelled material in one compartment to be calculated either from the fate in another compartment or from the input of the labelled material in the system"*.

## The single compartment model

Behold, a bucket.

This bucket is your patient. Into this bucket, a drug has been added. The drug is instantly and completely dispersed to every corner of the bucket and is thereafter homogeneously distributed throughout the volume.

It is then eliminated from this volume at a constant concentration-dependent rate (i.e. for every arbitrary unit of time 50% of the drug is cleared from the compartment).

Clearly, this is not what happens in clinical reality, but as a model this is a valuable thought experiment. It illustrates some important concepts. The volume at time zero is the volume of distribution (*Vd*) - for this drug that volume is 1g/L. The rate of elimination is described by the rate constant for elimination, *k*. In this case *k*=0.5 (i.e. 50% of the drug is eliminated from the compartment per unit time). Classically, the unit of time used for *k* is one minute. The clearance (*Cl*) of the drug from this single compartment is therefore described by the equation (*k *×*Vd*).

There are whole drug classes for which pharmacokinetics are well predicted by a single compartment model. For example, highly hydrophilic drugs which are confined to body water usually have single-compartment pharmacokinetics. Aminoglycosides are an excellent example. They have barely any tissue penetration and are essentially confined to the extracellular (in fact, intravascular) fluid volume.

However, virtually none of the anaesthetic drugs which you will use can be accurately described by this model. Let us now complicate things by adding another compartment.

## The two-compartment model

Again, the same dose of drug is administered into the same compartment. Let us call it the "central" compartment. There is now also a "peripheral"compartment in the system. Though the drug is still distributed instantly and homogeneously into all of the central compartment, it now also diffuses gradually into (and out of) the peripheral compartment.

Let us assume that each compartment has the volume of 1L. Let the rate of diffusion again be something like 0.1 (i.e. 10% of the central compartment drug will have diffused into the peripheral compartment over the course of one unit of time). If there is no elimination taking place, the compartments will achieve an equilibrium. If we were to sample the central compartment, the concentration of the drug will be measured as 0.5g/L.

This is all in the absence of elimination. If the drug was being cleared from the central compartment at a concentration-dependent rate, more complexity is introduced into the model. Let us now assume that the rate of diffusion between compartments is more rapid than the rate of elimination. The concentration graph will now have two distinct phases: rapid initial distribution and slow late elimination.

This leads us to the familiar-looking graph of the bi-exponential decline, with two distinct phases of change in drug concentration:

**The distribution phase**is the initial rapid decline in serum drug concentration**The elimination phase**is the slow decline in drug concentration, sustained by redistribution of drug from tissue stores.

One can see how this modeling lends itself to ever-increasing complexity. Considering that every little lacuna of fluid in the human body can be viewed as its own individual compartment, one can go quite mad trying to accurately model pharmacokinetics. For the purpose of preserving sanity, one needs to simplify this into something easily grasped by the human mind. For example, practically speaking in anaesthetics there is usually only need to consider three compartments: blood, lean tissues and fat.

## The three-compartment model

Consider a highly fat-soluble drug. When given as a bolus, it distributes rapidly into all tissues including lean muscle and fat. However, lean muscle contains little fat and is therefore a poor storage reservoir for the drug. The drug is eliminated from that compartment at approximately the same rate as it is from the blood. The fat compartment however soaks up a large amount of the drug. After a while, much of the drug has been cleared from the circulating blood and lean tissue; at this stage the fat compartment begins to act as a source of the drug, topping up the serum levels as elimination takes place.

This, then is the three compartment model graph. In this scenario there are three distinct phases: distribution, elimination and slow tissue release.

The distribution phase finishes with the concentrations reaching their peaks in the peripheral compartments.

To call the next phase the "elimination phase" is probably incorrect, but it is a period during which the main effect on drug concentration is in fact elimination. This phase ends when the slow compartment concentration becomes higher than the tissue concentration and the total clearance is slowed down by the gradual redistribution of the drug into the blood compartment.

The last phase is called the "terminal" phase, also sometimes confusingly referred to as the elimination phase. In actual fact during this phase elimination is rather slow because the concentration of the drug in the central compartment is minimal. The major defining factor affecting drug concentration during this phase is the redistribution of stored drug out of the slow compartment.

This sort of complex graph is called a *polyexponential curve*, as the curve has multiple exponents.

A polyexponential function can be described mathematically as the sum of all its exponents:

C*(t)* = Ae* ^{−αt}* + Be

*+ Ce*

^{−βt}

^{−γt}where

*t*is the time since the bolus- C
*(t)*is the drug concentration - A, B and C are coefficients which describe the exponential functions of each phase
- α, β and γ are exponents which describe the shape of the curve for each phase

The three-phase curve has implications for predicting and describing drug half-lives.

Each of the three phases has its own distinct gradient, which describes the "half life" of the drug during that phase of distribution. Obviously early after the bolus the half-life will be very short. However, the total ("terminal") half life of the drug may be very long (i.e the distribution phase which culminates in *all* of the drug ultimately being removed by clearance mechanisms).

Which half-life is most important? Usually medical textbooks naively give you the *longest* half-life, which is usually the terminal half-life and which often massively overstates the duration of a drugs' effect in the body. At one terminal half-life after a bolus of thiopentone for example, there is barely any drug in the bloodstream, and certainly little residual drug effect.

## Extension ad absurdum

It is possible to become seduced by the mathematical and physiological elegance of multiple compartment modeling and to create increasingly more and more complex models. Here is one from Gerlowski and Jain (1983)

To be sure, that *looks* nicely authoritative. Looking at this model you might swell with confidence, thinking that it will predict the behaviour of your drug with a high degree of accuracy. Unfortunately, this is an illusion. Consider:

- You need to accurately predict the rate of diffusion to all those organs
- You need to predict the affinity of those organs and tissues for the drug
- You need to either assume a stable blood flow to them, or model changes in blood flow to accurately reflect what happens in the organism
- You need to account for what might happen under physiological stress, such as in critical illness, which affects all of the abovementioned variables
- To test your model, you will either need to collect human tissue samples (good luck convincing your healthy volunteers) or you will need to experiment on animals (in which case good luck scaling your results to human proportions)

Thus, at every step of the way the increased complexity of the model leads to increased inaccuracy, with - some might argue - minimal benefit in terms of additional information or predictive accuracy. In terms of model utility per dollar spent, the returns on your investment rapidly plateau with complex pharmacokinetic models, and a pharmacy company is certainly not going to pay for something like the Gerlowski/Jain monstrosity shown above. Outside of these crude mercenary considerations, the Effort Conservation principle dictates that you should use the simplest model which is still reasonably good at predicting the behaviour of your drug.

## Representations of the multiple compartment models

The diagram below was borrowed from Gupta and Eger, 2008. This is an example of a "hydraulic" model of distribution for anaesthetic agents. As such it is a beautiful example. The patient is represented as a series of interconnected buckets, which helps to visually appreciate the relative capacity of each compartment. As far as I can tell this diagram (used in an article from 2008) originates from Eger's chapter in Papper and Kitz's 1963 textbook. Edmond Eger must have co-authored the second paper with Gupta 45 years later in his impossibly long career (apparently the man has in excess of 500 publications). The diagram itself is so good that it begs to be reproduced in its original form. There is even a little octopus lurking in the anaesthetic reservoir.

This is an excellent example of how to effectively represent visual information, and should be held up as a model for medical graphic design. However, the time-poor exam candidate will need something simpler.

The exam candidate has two specific needs of a compartment distribution model:

- It must contain all the
*important*information, i.e. everything the examiners will need to see from a demonstration - It must be possible to draw it quickly, because it may be necessary to reproduce it in a viva
- It must be simple so that people with zero talent can reliably reproduce it in a stressful situation

The following representations are therefore in common use by the impoverished exam-sitting public:

This crude representation also has the advantage of being a historically classical emblem of pharmacokinetics. Being easy to draw by hand, it was also easy to draw on the blackboard back in the day before online self-directed learning resources and FOAM. As such, these images will evoke in the elderly examiner warm recollections of dusty auditoria and the smell of chalk dust. In short, this should be viewed as the most effective way of describing the multi-compartment model of drug distribution in the formal exam setting.

In this model, the volumes are predictably identified as V (V* _{1 }*to

_{ }V

*for the three-compartment model). If you are feeling particularly generous, you could also throw in the effect site, which is a tiny compartment but which does contribute something (it is after all the drug target) and which is conventionally represented as V*

_{3}*. The kinetics of distribution from one compartment to another are usually described as K*

_{e}*, where*

_{xy}*x*is the compartment

*from*which the drug is distributing, and

*y*is the compartment

*to*which the drug is going. In this manner, distribution of a bolus from the central compartment V

*to the peripheral compartment V*

_{1}*would be labeled as K*

_{2}*.*

_{12}## Fentanyl bolus as an example of multiple compartment modeling

Viva 7 from the first paper of 2010 asked the candidates to *"draw the concentration time curve for an intravenous bolus of fentanyl". * Though the magic words "compartment" and "model" were never used by the examiners, this was an excellent opportunity for this author to call upon fentanyl as an example of a drug which requires complex pharmacokinetic modeling.

Here is an excellent graph from a 1981 rat study by Hug and Murphy.

Male rats (approximately 50 of them) were given a 50μg/kg dose of fentanyl through the tail vein, and then six at a time were sacrificed at each time interval. Over this short time frame, the plasma concentration demonstrates a classic two-phase distribution pattern, with a very short distribution half life (α = 7.9 min) and a much longer elimination half life (β = 44.5 min). The tissue concentrations clearly demonstrate and early and rapid distribution into fat, followed by a slower redistribution and elimination which occurs at approximately the same rate in every compartment.

To conclude, this final image from Hug and Murphy would be the ideal quick graph to reproduce at the viva, as it also incorporates the incredibly slow terminal γ-half life of fentanyl, which is due to its redistribution out of the fat compartment and which can take hours.

One can bring out this graph to illustrate the three-compartment model very easily, as there is a clear three-phase distribution process. Under ideal circumstances, one should not take any longer than 5 minutes with their explanation.

## Limitations of compartment modeling

The representation of drugs distribution using mathematical compartment modeling has various limitations, which are partly pragmatic and partly the consequences of various assumptions we make about pharmacokinetics.

For instance, some of the common assumptions and fallacies are as follows:

1) That the central compartment is the only compartment from which the drug is eliminated. Of course that is not the case, as for example in the case of cisatracurium where the drug degrades spontaneously no matteer where it is in the body, i.e. elimination takes place in numerous compartments simultaneously.

2) That the multicompartment model is more accurate the more compartments it has. Of course it is not, and often the plasma concentration data derived from a two-compartment model matches the empiric data at least as well as the multicompartment prediction. For instance, Levy et al (1969) found that the plasma concentration of LSD was predicted very well by using only two compartments, and that it was *"virtually impossible, in most instances, to distinguish between a two-compartment and more complex pharmacokinetic system on the basis of plasma concentrations alone"*. For the record, the authors gave 2μg/kg of LSD to five volunteers and then measured their ability to solve maths problems.

3) The mathematical model may include various compartments (eg. brain), but it may be highly incovenient to verify the concentration of drug in that compartment by sampling it (eg. by brain biopsy). We end up relying on samples from more easily accessible compartments (eg. blood plasma) but as mentioned above the multicompartment models and the two and three compartment models are virtually indistinguishable when you are sampling only plasma, which renders modeling of specific organs highly unreliable.

## References

Mould, D. R., and Richard Neil Upton. "Basic Concepts in Population Modeling, Simulation, and Model‐Based Drug Development—Part 2: Introduction to Pharmacokinetic Modeling Methods." *CPT: pharmacometrics & systems pharmacology* 2.4 (2013): 1-14.

Nikkelen, Eric, Willem L. van Meurs, and Maria AK Öhrn. "Hydraulic analog for simultaneous representation of pharmacokinetics and pharmacodynamics: application to vecuronium." *Journal of clinical monitoring and computing* 14.5 (1998): 329-337.

Gupta, D. K., and E. I. Eger. "Inhaled Anesthesia: The Original Closed‐Loop Drug Administration Paradigm." *Clinical Pharmacology & Therapeutics* 84.1 (2008): 15-18.

Eger, E. I. "A mathematical model of uptake and distribution" *Uptake and Distribution of Anesthetic Agents,* ed EM Papper and RJ Kitz." (1963).

Peng, Philip WH, and Alan N. Sandler. "A review of the use of fentanyl analgesia in the management of acute pain in adults." *The Journal of the American Society of Anesthesiologists* 90.2 (1999): 576-599.

Hug Jr, C. C., and Michael R. Murphy. "Tissue redistribution of fentanyl and termination of its effects in rats." *Anesthesiology* 55.4 (1981): 369-375.

Gerlowski, Leonard E., and Rakesh K. Jain. "Physiologically based pharmacokinetic modeling: principles and applications." *Journal of pharmaceutical sciences* 72.10 (1983): 1103-1127.

Levy, Gerhard, Milo Gibaldi, and William J. Jusko. "Multicompartment pharmacokinetic models and pharmacologic effects." *Journal of pharmaceutical sciences* 58.4 (1969): 422-424.

Rescigno, Aldo. "Synthesis of a multicompartmented biological model." *Biochimica et biophysica acta* 37.3 (1960): 463-468.

Nestorov, Ivan A., et al. "Lumping of whole-body physiologically based pharmacokinetic models." *Journal of pharmacokinetics and biopharmaceutics*26.1 (1998): 21-46.

Gerlowski, Leonard E., and Rakesh K. Jain. "Physiologically based pharmacokinetic modeling: principles and applications." *Journal of pharmaceutical sciences* 72.10 (1983): 1103-1127.