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The following message was posted to: PharmPK
I am brand new to the list & have not read much about pharmacodynamic
modelling since I left medical school, hence I am after some advice.
I am designing an anaesthetic study to examine the 'dose response'
relationship between end-tidal anaesthetic agent concentration & a
measure of depth of anaesthesia, the bispectral index. Previous
workers have modelled this relationship using an inhibitory sigmoid
Emax model, with a scaling factor of 1, but none have referenced the
model, or explained their choice of scaling factor. If I understand
correctly, this scaling factor determines the Hill slope.
What I'm after, therefore, is a good reference for understanding
inhibitory sigmoid Emax models (preferably online). Also, should I
not just plot end tidal concentration versus bispectral index &
calculate the resulting slope of the graph, rather than
predetermining it?
Thanks to anyone who can help resolve my ignorance & confusion!!
Dr. Simon Whyte
Clinical Lecturer in Paediatric Anaesthesia
Dept. of Anaesthesia
Royal Liverpool Children's Hospital
Liverpool
L12 2AP
Tel: +44 (0)151 252 5701
Fax: +44 (0)151 293 3692
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The following message was posted to: PharmPK
"Dr. Simon Whyte (by way of David Bourne)" wrote:
> relationship between end-tidal anaesthetic agent concentration & a
> measure of depth of anaesthesia, the bispectral index. Previous
> workers have modelled this relationship using an inhibitory sigmoid
> Emax model, with a scaling factor of 1, but none have referenced the
> model, or explained their choice of scaling factor. If I understand
> correctly, this scaling factor determines the Hill slope.
It is not clear what you mean by the "scaling factor" so let me
define some terms. The Emax model looks like this:
E = E0 + Emax * C / ( EC50 + C )
The sigmoid Emax model looks like this:
E = E0 + Emax * C^Hill / (EC50^Hill + C^Hill)
E= Effect at concentration C
E0=Response predicted when C is 0
Emax=Maximum possible effect (at infinite C)
EC50=C producing 50% of Emax
Hill=Hill coefficient
The operator "^" means "raised to the power"
The inhibitory sigmoid Emax model simply has a negative value for
Emax but can also be written like this (the inhibitory fractional
sigmoid Emax model):
E = E0*(1 - Emax * C^Hill / (EC50^Hill + C^Hill))
where Emax is now a (non-negative) fraction between 0 and 1. If
Emax=1 then the response will be 0 at infinite C i.e. the drug
produces complete inhibition of the response.
The "scale" factors of these models are E0 and Emax (scale for
response) and EC50 (scale for concentration). The Hill coefficient is
dimensionless i.e. it has no scale. The Hill coefficient is sometimes
referred to as the "steepness" factor because concentration-effect
curves look "steeper" in the 20 to 80% of maximum effect range as
Hill gets larger. When Hill=1 then the sigmoid Emax model is the same
as the Emax model. Note that the Hill coefficent is NOT the slope of
the concentration-effect curve. The slope i.e. the derivative of E
with respect to C is a continuously changing value for these models.
> Also, should I
> not just plot end tidal concentration versus bispectral index &
> calculate the resulting slope of the graph, rather than
> predetermining it?
Plotting the measured conc versus the observed effect is an excellent
idea. It can give you some initial idea of Emax and EC50 (and Hill).
However, better estimates of these parameters and application of more
sophisticated models linking plasma concentration to the time course
of effect require non-linear regression. NLR can be used to estimate
all the parameters, including the Hill coefficient. It can also be
used to test hypotheses such as "Is Hill=1?" or
"Is Emax=1?" in the inhibitory fractional sigmoid Emax model.
--
Nick Holford, Divn Pharmacology & Clinical Pharmacology
University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand
email:n.holford.at.auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556
http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.htm
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Copyright 1995-2010 David W. A. Bourne (david@boomer.org)