- On 15 Aug 2000 at 12:38:32, Janusz Byczkowski (janusz.byczkowski.aaa.usa.net) sent the message

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On Thu, 10 Aug 2000 23:27:09 -0500, Roger Jelliffe(by way

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? ...">

Roger:

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.

Consultant

212 N. Central Ave.

Fairborn, OH 45324

phone (937)878-5531

office (614)644-3070

confidential fax (603)590-1960

e-mail januszb.-a-.AOL.com

homepage: http://members.aol.com/JanuszB/index.html

JZB Consulting web site: http://members.delphi.com/januszb/ - On 16 Aug 2000 at 22:23:16, Roger Jelliffe (jelliffe.-at-.usc.edu) sent the message

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Dear Janusz:

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 Jelliffe

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

****** - On 3 Sep 2000 at 14:11:33, "Gendron, Richard T. (GS)" (RTGendron.-at-.mar.med.navy.mil) sent the message

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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

references.

(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

correct? - On 3 Sep 2000 at 21:04:16, Stephen Day (shday.aaa.yahoo.com) sent the message

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Richard,

In a pharmacokinetic context, I agree with your two

statements...

> 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

the Km.)

However, I'm not sure this will convince your

physician friend whether the "processes" are linear or

not.

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

your friend.

Looks like you could argue both ways. Hopefully the

mathematicians among us will set this strait ;-)

=====

Stephen Day

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

the concentration.

Not to be confused with linear and non linear regression. In this

case, linear means straight line. - db] - On 4 Sep 2000 at 15:08:54, ml11439.at.goodnet.com sent the message

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Richard,

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

the system:

dC/dt= -KeC

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)

differential equation:

dC/dt= -VmaxC/[Km+C]

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:

dC/dt= -[Vmax\Km]C

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

ML11439.-at-.goodnet.com

References

1) Gibaldi, M., Perrier, D., Pharmacokinetics, New York, Marcel Dekker

1975

2) Gibaldi, M., Perrier, D., Pharmacokinetics 2nd ed, New York, Marcel Dekker

1982

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 - On 4 Sep 2000 at 15:09:41, James Wright (J.G.Wright.-a-.ncl.ac.uk) sent the message

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Dear Richard,

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,

James Wright - On 4 Sep 2000 at 16:15:14, Daniel Combs (dzc.at.gene.com) sent the message

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Dr. Gendron:

In order to answer your physician friend's question, one should

distinguish

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

t) +

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

to

dose and parameters such as CL, Volume and half-life do not change as

dose

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

levels,

most commonly with a decrease in CL due to saturation of some biological

elimination process.

Non-linear kinetics can also be observed at the same dose level when

interval exposure and PK parameters are different upon multiple dosing

than

what was predicted from single dose parameter simulations. For

instance, if

one were to determine kinetic parameters following a single dose and

simulate what steady-state would be upon pharmacokinetic accumulation,

this

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

change

in patient biology over time upon chronic dosing due to continued drug

exposure, perhaps due to liver induction, interaction with cytochrome

P-450

metabolic enzymes, or due to receptor up- (or down-) regulation. It is

also

possible that such time-dependent kinetics are really a manifestation of

a

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

saturable

process. When initial drug concentrations are low and much less than Km

the

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

linear PK-model.

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

dosing

regimens. For a drug with a narrow therapeutic window this can lead to

a

safety concern (toxicity) when higher than expected concentrations are

reached or a loss of efficacy when lower than expected concentrations

occur.

Hope this will help.

Dan Combs

dzc.at.gene.com

Genentech, Inc.

Work phone (650)225-5847

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