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The following equation represents the stimulation of output of
response (R) in the family of indirect response models proposed by
Jusko:
dR/dt=kin - kout * (1+((Emax*Cp)/(EC50 +Cp)))* R
where kin is defined as the zero order input rate constant, kout is
the first output rate constant, Emax is the maximal effect, EC50 is
the concentration producing 50% of maximal effect, and Cp is plasma
concentration.
Although this may sound silly, Emax should be unitless in order to
balance the equation. If so, what is the physiological meaning of
Emax and is there a constraint in its direction (+ or - ) and
magnitude?
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Ronald,
In the parametrization proposed by Jusko :
dR/dt=kin - kout * (1+((Emax*Cp)/(EC50 +Cp)))* R
one plus the expression for Effect ((Emax*Cp)/(EC50 +Cp))is a
multiplicator of kout. This means that in the absence of any effect
(at Cp=0), kout is at its basal value (kout*1), and at maximal effect
(at Cp >> EC50), kout reaches the value kout*(1+Emax). So Emax is the
maximal percent increase attainable with the drug. Therefore it is
unitless : an Emax of 0.7 simply means that the drug is able, at very
high concentration, to increase the elimination of the Response
marker by 70%. A negative Emax is equivalent to the indirect model of
inhibition of the output response.
Hope this helps
Thierry BUCLIN, MD
Division of Clinical Pharmacology
University Hospital CHUV - Beaumont 633
CH 1011 Lausanne - SWITZERLAND
Tel: +41 21 314 42 61 - Fax: +41 21 314 42 66
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Hello,
1. You are right, Emax in the equation you specified is unitless. It
is define as the maximal 'FRACTION' stimulation in the bio-flux (kin or
kout). For eg: Emax=0.6 implies 60% increase in the kout from basal
value.
2. The constraint in the direction is imposed by the mechanistic basis
of the model. If you believe that insulin indeed stimulates glucose
metabolism then you would constrain the value of Emax to be greater than
zero. But for this particular model there is no upper bound. The
equation (or the modeler!) fails otherwise, in the sense that it means a
different mechanism (inhibition of kout).
3. To avoid confusion, some researchers prefer to use the terms Smax
and Imax for maximal stimulation and inhibition, respectively.
Regards,
Joga Gobburu
Pharmacometrics,
CDER, FDA.
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This is a good question. I am currently working on something similar
and have struck the same problem.
Intuitively, you would think changing the ..... kout(1+ ... to
......kout(1-.... might account for negative changes from baseline,
but it doesn't solve the units problem.
Would the equation work if you convert your data to ratios - ie
Rt/Rt=0? This would also solve the units problem and as ratios the
data would be readily interpreted. You could also model responses
that fall below baseline which would be a fraction of Rt=0. The
problem comes with PD responses that start at 0.
I hope you get some useful responses
Best wishes
Dave
Dave Boulton, Ph.D. M.P.S.
Assistant Director
Laboratory of Drug Disposition and Pharmacogenetics
Department of Psychiatry and Behavioral Science
Medical University of South Carolina
67 President Street, P.O. Box 250861
Charleston, SC 29425
Ph: (843) 792 5589 Fax: (843) 792 7260
--F77C9A9.B4D5F55C--
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Hello,
1. Data transformation is not recommended. It confounds the
interpretation of the results and further, modeling all the data, as is,
is more efficient. Particularly data transformation affects aspects
related to understanding of disease progression, time course of drug
effects, and handling missing data (and others).
2. The physiologic interpretation power of the formation and
dissipation rates is lost if the response modeled is a ratio (unitless).
3. The fact that some individuals may have lower than baseline values
after drug treatment, even though mechanism dictates otherwise, has to
do with factors such as possible biorhythms and/or just variability! It
is case-specific. But this is not a unique problem with the indirect
response models. The way we should/handle the data is identical
irrespective of the PD model/mechanism.
Regards,
Joga Gobburu
Pharmacometrics,
CDER, FDA.
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Ron,
I can see that this indirect pharmacologic response model
is completely different from the direct Emax model, as defined
in textbooks:
dR/dt=kin - kout * (1+((Emax*Cp)/(EC50 +Cp)))* R
The (1+E) term changes the mathematical meaning of the Emax
equation(E).
In the kinetics of direct pharmacologic response, the
Emax term is simply the maximum response when C approaches
infinity(1-4). So that in the Emax model, the response would
approach Emax as the concentration greatly exceeds EC50:
E= [EmaxC]/[EC50 +C]
The units of Emax would be determined by the nature
of the pharmacologic response. For example, in one study
regarding midazolam, the pharmacologic response was the
change over baseline in the EEG activity in the 13-30HZ
range. The Emax in this study was calculated to be a
21.2% change in baseline EEG activity(4). However, the Emax
could also be fraction in this case as well, if the
response is modeled as a measurable fraction of something.
Mike Leibold, PharmD, RPh
ML11439.aaa.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) Wagner, J.G., Pharmacokinetics for the Pharmaceutical Scientist, Lancaster,
Technomic Publishing Co 1993
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Ron,
The indirect pharmacologic response model of Jusko
is similar to a Emax model with a baseline(5):
E= EmaxC/[EC50+ C] + Eo
The inhibitory version is:
E= Eo - EmaxC/[IC50+C]
Eo= baseline effect
The form of the above equation can be changed to:
E= Eo*[Emax'C/[EC50+C] +1]
Where Emax'= Emax/Eo
In this case, the Emax' term would be a multiple of Eo, the
basline response. This form is similar to the Jusko equation, and
might explain the units of Emax:
dR/dt=kin - kout * (1+((Emax*Cp)/(EC50 +Cp)))* R
The Emax baseline model could also be modeled as occurring in an
effect compartment, in which case the concentration term would
represent the concentration in the effect compartment(1-5). This
differential equation is:
dCe/dt= K1eCp - KeoCe
Mike Leibold, PharmD, RPh
ML11439.aaa.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) Wagner, J.G., Pharmacokinetics for the Pharmaceutical Scientist, Lancaster,
Technomic Publishing Co 1993
5) Holford, H.G., Sheiner, L.B., Understanding the dose-effect relationship:
clinical application of pharmacokinetic-pharmacodynamic models, Clinical
Pharmcokinetics 1981;6:429-453
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"David W Boulton"
Re: Indirect Response Model
Wrote:
>"...Would the equation work if you convert your data to ratios - ie
Rt/Rt=0? This would also solve the units problem and as ratios the
data would be readily interpreted...">
David:
As far as I know,
Rt/Rt=1
always!
Am I missing something?
Janusz Z. Byczkowski, Ph.D.,D.Sc.,D.A.B.T.
Consultant
212 N. Central Ave.
Fairborn, OH 45324
voice (937)878-5531
office (614)644-3070
confidential fax (603)590-1960
e-mail januszb.aaa.AOL.com
homepage: http://members.aol.com/JanuszB/index.html
JZB Consulting web site: http://members.delphi.com/januszb/
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