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On Thu, 10 Aug 2000 23:27:09 -0500, Roger Jelliffe
of David_Bourne) wrote about
Subject: Re: Log-linear or linear trapezoid
<"... So, instead of having to ask, as we must in toxicological work, if the
drug is PRESENT OR NOT, and having therefore to develop a LOQ, we know the
drug is present. The question being asked is not the same as in toxicology. It
is instead - HOW MUCH drug is present? ...">
I do not understand your generalization about toxicology.
Since the times of Paracelsus we know that "this is the dose that makes a
poison". Thus, contemporary toxicology, and a related discipline of health
risk assessment, are utilizing quantitative pharmacokinetic modeling tools
(including PBPK models) to estimate with the best accuracy available the
internal doses of potentially toxic chemicals. In other words, for example,
for a potentially hepatotoxic chemical, the relevant dose metrics will be its
concentration in the liver over time, under the given exposure scenario (often
chronic or even life time exposures). All kinetic elements must be
considered, including bioavailability, tissue distribution/partitioning,
metabolism and other means of clearance. Moreover, most of the U.S. agencies
and institutions involved in toxicology and health risk assessment require now
explicit quantification of uncertainty.
So, the question being asked in modern toxicology is not only "HOW MUCH drug
[chemical] is present" but also "in which tissue", "how long and at what
time", and "how accurate and how precise your estimates are".
With the best wishes.
Janusz Z. Byczkowski, Ph.D.,D.Sc.,D.A.B.T.
212 N. Central Ave.
Fairborn, OH 45324
confidential fax (603)590-1960
JZB Consulting web site: http://members.delphi.com/januszb/
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I don't understand your problem. I agree that in modern
toxicology we like to know drug concentrations not only in serum, but
also in many tissues if possible. I also agree wholeheartedly that we
should seek explicit quantification of uncertainty.
I have not made a generalization about toxicology. I have
tried to show that pharmacokinetic work is often different from
toxicological work. In toxicology we often do not have any
information as to when the sample was obtained in relation to the
last dose. This is why one has to ask if the drug is present in the
sample or not., and why there is a lower LOQ, that is some
significant value above the blank.
The point is that in PK work, we are more fortunate, in that
we usually know then the sample was obtained in relation to the
dosage history. Because of this, and because most drugs have
half-times, we usually know that the drug is still present. This is
why we do not need to ask, as we must in toxicology, whether the drug
is present or not, but rather, since we know it is still present, we
ask instead HOW MUCH drug is still present. Because of this, if we
determine the assay error pattern carefully over its entire range, we
can take this all the way down to the blank, with no lower LOQ. Does
this help? Look at the article in Therapeutic rug Monitoring 15:
380-393, 1993. Look at the figures of the various assay error
patterns. See what you think.
Very best regards,
Roger W. Jelliffe, M.D. Professor of Medicine, USC
USC Laboratory of Applied Pharmacokinetics
2250 Alcazar St, Los Angeles CA 90033, USA
Phone (323)442-1300, fax (323)442-1302, email= jelliffe.-at-.hsc.usc.edu
Our web site= http://www.usc.edu/hsc/lab_apk
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Please confirm the following relationships:
Michaelis-Menten = non-linear kinetics
First order processes = linear kinetics
I can't find these terms (linear and non-linear) clearly defined in standard
(One of our physicians feels that since first order relationships do not
plot linear, they are non-linear processes. I contend that the ln(conc) v.
time is linear, and Michalis-Menten processes are non-linear. Who is
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In a pharmacokinetic context, I agree with your two
> Michaelis-Menten = non-linear kinetics
> First order processes = linear kinetics
In fact, the terms "Michaelis-Menton kinetics" and
"non-linear kinetics" seem to be used almost
interchangeably in pk literature. (I personally prefer
the term "non-linear kinetics" because, strictly
speaking, the Michaelis-Menton equation also describes
"linear kinetics" when drug concentrations well below
However, I'm not sure this will convince your
physician friend whether the "processes" are linear or
The first order decline of drug concentration as a
function of time is described by a *linear*
differential equation (dC/dt = -k*C). Does this make
the process linear?
On the other hand, the solution of the above
differential equation gives a function that is
nonlinear (C = Co*exp(-k*t)). By nonlinear I mean that
it is not of the form y = mx + b, as pointed out by
Looks like you could argue both ways. Hopefully the
mathematicians among us will set this strait ;-)
Merck-Frosst Centre for Therapeutic Research
Kirkland, QC CANADA
[In this context, linear means concentrations increase linearly with
dose. Double dose, double the concentrations.
MM kinetics, non-linear as double dose and you get more than double
Not to be confused with linear and non linear regression. In this
case, linear means straight line. - db]
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The terms "nonlinear" and "linear" refer to the description of
the system by differential equations. If the equation is described
by linear differential equations, then the system is considered
linear. An example of the simplest linear system is the one
compartment model where one "linear" differential equation describes
Linear differential equations have constants describing the decline
in compartmental quantities with time, which in the one compartment
case is Ke. Such that, that the rate of decline in compartmental
quantities is a linear (straight line), constant (i.e. Ke) function of
the amount of drug in the compartment.
Systems described by nonlinear differential equations are considered
nonlinear systems. In the case of pharmacokinetics, the nonlinear
system is one chararacterized by the Michaels-Menten (enzyme saturable)
In the case of this nonlinear differenntial equation, the rate of
decline in compartmental quantity with time is not a linear function
of the amount of the drug in the compartment. The rate of decline is a
nonlinear function where the rate of removal increases with increases
with increasing concentration (as in the linear case), but approaches
a constant, maximum rate of elimination with increasing concentrations
as the eliminating enzyme system becomes saturated with drug.
Pharmacokinetics texts suggest that all drugs are susceptible to
Michaelis-Menten, nonlinear pharmacokinetics, but only certain drugs
obey nonlinear pharmacokinetics when dosed in the therapeutic range
(e.g. phenytoin, theophylline). Other drugs obey linear pharmacokinetics
when dosed in the therapeutic range, but exhibit nonlinear kinetics
when dosed above the therapeutic range such as in overdoses. When
drug concentrations are well below Km, the Michaelis-Menten equation
reduces to the linear version:
Linear pharmacokinetic systems have certain observable properties
as mathematical consequences of the linear differetial equations
describing the system. Linear properties include serum levels proportional
to doses, AUC's proportional to doses, bioavailability equal to AUCoral/
AUCiv, and the superposition principle where the concentrations resulting
from one dose can be added to that of another to determine to the total
concentration (basis of all multiple dose equations).
Mike Leibold, PharmD, RPh
1) Gibaldi, M., Perrier, D., Pharmacokinetics, New York, Marcel Dekker
2) Gibaldi, M., Perrier, D., Pharmacokinetics 2nd ed, New York, Marcel Dekker
3) Wagner, J., Fundamentals of Clinical Pharmacokinetics, Hamilton, Drug
Intelligence Publications 1975
4) Godfrey, Keith, Compartment Models and Their Application, New York,
Academic Press 1983
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The words linear and non-linear as commonly used in pharmacokinetics refer
to the underlying diffrential equations.
In terms of differential equations:
Linear= no products or powers of derivative terms. This also means no
explicit time-dependency (ie no t *dC/dt or dC/dt + t type terms), although
in mathematical parlance, one would specify this by stating "linear with
constant coefficients". A linear differential equation can, in technical
terms, include derivatives to any degree.
Non-linear = not linear. This includes Michaelis-Menten kinetics because
they are concentration-dependent and concentration is a function of time
(as well as some other stuff).
The use of the word linear is aimed generally at clearance processes which
are decribed by equations of the form
dC/dt = -CL.C where clearance is a constant coeffient.
Equations that a first-order, in mathematical parlance, can include
derivative of the first-order or lower (ie they can include zero-order
terms). In pharmacokinetics however this term has the more specific (and
more useful) meaning of only first-order terms.
So, in pharmacokinetic terms, you are correct although the statement
first-order process=linear kinetics will have some very famous
mathematicians spinning in their graves. However, your colleague is also
correct in that the solution to a first-order pharmacokinetic equation is
an exponential term which is not linear on the ordinary scale, but is
linear on the log scale. In a sense, one can always find a scale on which
something is nonlinear (although sadly one cannot always find a scale on
which something is linear).
Curiously, zero-order processes (ie fixed amount per unit time) are linear
on the ordinary scale.
Hope this clarifies things,
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In order to answer your physician friend's question, one should
between linear plots on linear-scale graphs and linear pharmacokinetics.
Linear plots are of the sort y = mx + b rather than Cp = A exp(-alpha
B exp (-beta t). Note that a biexponential equation will NOT plot as a
straight line on a semilog graph. However, this question of plotting is
an entirely different meaning from linear PK.
Linear kinetics mean that the exposure (AUC) and Cmax are proportional
dose and parameters such as CL, Volume and half-life do not change as
is increased. Non-Linear kinetics occur when the change in exposure is
greater (or less) than proportional with dose level changes. Parameters
such as CL, Volume, half-life will be observed to differ across dose
most commonly with a decrease in CL due to saturation of some biological
Non-linear kinetics can also be observed at the same dose level when
interval exposure and PK parameters are different upon multiple dosing
what was predicted from single dose parameter simulations. For
one were to determine kinetic parameters following a single dose and
simulate what steady-state would be upon pharmacokinetic accumulation,
might be quite different than what is actually observed, due to
non-linearities over time. One might see a markedly higher steady-state
concentration than predicted. Such non-linearities might be due to a
in patient biology over time upon chronic dosing due to continued drug
exposure, perhaps due to liver induction, interaction with cytochrome
metabolic enzymes, or due to receptor up- (or down-) regulation. It is
possible that such time-dependent kinetics are really a manifestation of
missing element in the PK-model assummed from single-dose data and this
manifestation only shows up upon multiple dosing.
Michaelis-Menton kinetics are the classical model for describing a
process. When initial drug concentrations are low and much less than Km
process is concentration-dependent (first order). As concentration
increases and greatly exceeds Km, the process becomes
concentration-independent (zero order). This saturation effect leads to
exposures and PK parameters that are different for low and high doses or
yields a plasma profile at a single dose that cannot be described with a
The big deal about non-linear kinetics is that they make it difficult to
predict patient exposure when dose levels are changed or in chronic
regimens. For a drug with a narrow therapeutic window this can lead to
safety concern (toxicity) when higher than expected concentrations are
reached or a loss of efficacy when lower than expected concentrations
Hope this will help.
Work phone (650)225-5847
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Copyright 1995-2010 David W. A. Bourne (firstname.lastname@example.org)