- On 4 Oct 2001 at 10:46:08, "Dr. Simon Whyte" (sdwhyte.-at-.liverpool.ac.uk) sent the message

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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 - On 5 Oct 2001 at 15:00:44, Nick Holford (n.holford.at.auckland.ac.nz) sent the message

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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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