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I am a grad student taking a first course in PK and have spent a
considerable amount of time discussing the concept of equilibrium
distribution. Unfortunately, the selected text for the course does not
discuss this concept. I have found two references from other texts
which introduce this concept: "Mathematical Approach to Physiological
Problems" by Gerald Riggs, MD, and "Biopharmaceutics and Clinical PK,"
4th ed., by Gibaldi. A quick search of Medline using "equilibrium
distribution" as a keyword did not produce any review articles. Any and
all suggestions for additional references are greatly appreciated.
Thanks very much.
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Philip Crooker wrote to the list asking about the concept of equilibrium
distribution. Concerning references, you might explore Lassen and Perl,
Tracer Kinetic Methods in Medical Physiology, or Jacquez's Compartmental
Analysis in Biology and Medicine.
But I think you would get a lot more from exploring the underlying physical
principles. First, learn to distinguish between equilibrium and steady
state. Many scientists, including a good number of professors lecturing on
kinetics and pharmacokinetics, treat these two terms as interchangeable.
This erroneous interchangeability has led many students astray. Steady
state requires only that nothing be changing with time. Equilibrium is a
thermodynamic concept that requires zero chemical potential gradients.
Hence, at equilibrium, there can be no net fluxes anywhere in the system,
because net fluxes are possible only when there is a nonzero chemical
potential gradient. (Three cheers for Josiah Willard Gibbs!) The difference
between steady state and equilibrium is thus the difference between life
and death. There is a more detailed discussion of this material on the
BioInformatics Services website at
Once you understand these differences, you are in a much better position to
understand textbooks and colleagues who use these terms loosely. You will
see that an open system, like a living person, cannot be in equilibrium,
only in steady state. You will also see that a closed system that is in
steady state must also be at equilibrium.
This means that the distribution of any constantly infused, metabolized
substance is actually a steady state distribution, not an equilibrium
distribution. The concept of equilibrium is, nevertheless, often loosely
attached to an exchange compartment. The reason this idea survives is that
the ratio of concentrations in the exchanging compartments in a steady
state system is the same as it would be if the two compartments were
completely isolated from the rest of the world and allowed to reach
equilibrium. This leads to the notion of equilibrium subsystems, that is,
systems exchanging with the dynamic system you are considering but
exchanging so fast that the kinetics are invisible and you only need an
equilibrium constant or partition coefficient to adequately characterize
the exchange. Ligand-receptor interactions are often treated in this way,
but there is a danger that in doing so your differential equations will
violate conservation of mass unless you write them carefully. This too is
covered in my web-published tutorial cited above.
Finally, you may want to consider the implications of extravascular
catabolism. If the extravascular exchange compartment is not a pure
exchange, but rather some of the metabolism or elimination of the compound
occurs from the extravascular exchange compartment, then the concept of
equilibrium is clearly inappropriate. In this case, the extravascular
exchange compartment can only achieve a steady state value, and the
so-called equilibrium distribution ratio will depend on metabolic clearance
from the extravascular compartment, not just on the parameters
characterizing the exchange.
I apologize if all this is obvious to you or off the point of your
question, but I know from long experience that these issues trouble many
students. I wish you all the best in your future studies, and I applaud
your search for deeper understanding.
PS The author of the classic text, The Mathematical Approach to
Physiological Problems, is Douglas not Gerald Riggs.
Robert D. Phair, Ph.D. rphair.aaa.bioinformaticsservices.com
BioInformatics Services http://www.bioinformaticsservices.com
12114 Gatewater Drive
Rockville, MD 20854 U.S.A. Phone: 1.301.315.8114
Partnering and Outsourcing for Computational Biology
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Dear Philip Crooker,
From your message it is not quite clear what you mean with the term
'distribution equilibrium', nor what you want to know about it.
With respect to a modellistic approach, you might find some useful
information in my paper 'An Extended Pharmacokinetic/Pharmacodynamic
Model describing quantitatively the influence of plasma protein
binding, tissue binding, and receptor binding on the potency and time
course of action of drugs', in Journal of Pharmacokinetics and
Although the paper deals with PK/PD in particular, the PK may be
helpful for you (and perhaps the PK/PD part of it, since PK without
PK/PD isn't that interesting, in general, isn't it?).
Johannes H. Proost
Dept. of Pharmacokinetics and Drug Delivery
University Centre for Pharmacy
Groningen, The Netherlands
tel. 31-50 363 3292
fax 31-50 363 3247
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Distribution equilibrium (DE) can be best defined as that point in time
when the ratio of the concentration in a given compartment (or region) to
that in plasma (blood, serum) becomes a constant. For example, in a one
compartment model, one assumes instantaneous achievement of DE. So,
following iv bolus injection of a drug, the concentration ratio (say
heart/blood ) as a function of time will be line horizontal to the X or
time axis (from 0 time). Hence , theoretically, one obtains a
monoexponential decline of concentration in plasma with time (and at any
other sampling site;same slope for all sites) ) Hope this is of help. If
you need more assistance, please contact me.
Professor, Pharmaceutics and Medicine
Schools of Pharmacy and Medicine
Umiversity of Missouri-Kansas City
203B Katz Hall
Kansas City, Mo 64110
816-235-1794 (fax; 816-235-5190)
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I was interested to read the discussion on steady state and distribution
equilibrium (a term that I do not personally use) from all (especially Sri
Melethil and Robert Phair). Clearly there remains conflicting thoughts
regarding these terms. From my perspective I use "steady state" in a very
simple role to mean that the amount in = amount out. This has specific
connotations to clearance via the basic equation: dose rate = elimination
rate which is only true at "steady state". I try and avoid using the term
"steady state" outside of this definition. With respect to distribution,
which is not specifically included in the "steady state" equation above,
then I do not see a specific problem in using the term equilibrium. In
this case as Sri indicates for a one compartment model the equilibirum (net
balance) between all tissues that the drug distributes is considered
instantaneous, for a two compartment model this equilibrium is delayed
until after the so called initial distribution phase. Clearly the concept
of compartments is a function of our ability to model the time course of a
drug and is subject to various assumptions (eg it is common to use a 1 cpt
model to describe the PK of gentamicin which has been shown to display 3
cpt PK [or perhaps more]). Therefore the simplistic distribution
equilibirum approach described here, based on compartments, will always be
wrong since it is logical that there will always be a net flux of drug to
somewhere in the body (even if we can't [or don't] usually measure it).
Given an understanding of these limitations it does not seem unreasonable
to me to use the term equilibirum (despite that it maybe wrong), to
describe the situation in which the net transfer of drug between various
compartments is essentially zero (or at least very small).
"All models are wrong but some are useful"., Box
I look forward to comments.
School of Pharmacy
University of Manchester
Ph +44 161 275 2355
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Hooray for Bob Phair!
I am frequently dismayed to see models in which components of a system are
assumed to be in equilibrium for ease of mathematical representation. This
invariably leads to violation of conservation of mass and excessively rapid
metabolic clearance of an infused, inhaled, or ingested material. See my
demonstration of this problem in
M.C. Kohn, The importance of anatomical realism for validation of
physiological models of disposition of inhaled toxicants, Toxicol.
Appl. Pharmacol., 147, 448-458 (1997).
These simplified mathematical models were developed 20-30 years ago when
computer power and accessibility were limited and convenient model-construction
software did not exist. Typically, a modeler had to write a program in FORTRAN
from scratch for each new system to be described. We now have personal
computers that are more powerful than the old mainframes and modeling software
that is optimized for describing kinetic systems. So there's NO EXCUSE for not
doing things the right way!
Michael C. Kohn
Laboratory of Computational Biology and Risk Analysis
National Institute of Environmental Health Sciences
P.O. Box 12233, Mail Drop A3-06
Research Triangle Park, NC 27709-2233
WWW home page:
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