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Nick Holford (Department of Pharmacology & Clinical Pharmacology at the
University of Auckland in New Zealand
> What do you mean by ....?
Fair enough. Through PharmPK, I asked you and Hans Proost (Department of
Pharmacokinetics and Drug Delivery at the University Centre for Pharmacy in
the Netherlands) to define and help me understand the STS (the =93standard
two step=94) model, which turned out to be more of a procedure or method than
a model, but is an acronym used in pharmacokinetic analysis. You did. In
return, I will try to answer your questions, as follows:
> 1. =93Risk assessment=94
To me, a risk assessment consists of the process of (or the procedures
involved in) estimating risk. A model usually accomplishes a risk analysis.
(a) I (and some of my colleagues) define risk as =93the probability of a
future loss,=94 which is, in part, Kaplan's original definition and thus,
consistent with it.
(b) Our definition has the advantage of being axiomatic. See:
P.C. Fishburn, =93Foundations of risk measurement I: Risk as probable loss.=94
Management Science 30: 296-306 (1984). AND P.C. Fishburn, =93Foundations of
risk measurement II: Effects of gain on risk.=94 J. Mathematical Psychol. 25:
(c) To me, risk assessment also is the first part of a three-part process
called risk analysis. The other (following) parts are risk management and
risk communication. (Before you try to manage or explain a risk, first try
to understand the risk.)
(d) Some portion of risk assessments predict measurable outcomes. In these
instances, risk assessments resemble experimental hypotheses. The
estimates are subject to validation.
(e) When a risk assessment predicts some number of cases, and the
prediction is subject to validation or verification, feelings about the
risk (risk perceptions) do not change the measured outcome. Thus, many
risk assessors wonder how the incorporation of risk perception could ever
contribute to estimation. If so, risk perception is properly part of the
(f) Other definitions of risk exist.
> 2. =93Poisson model=94
The Poisson equation motivates a Poisson model.
(a) The (fundamental) Poisson equation is 1-e^-x, where x is the average
number of hits per cell. Imagine that a hit is a discrete event, like an
irreversible change. E.g., the equation estimates the probability that
some number of hits will fall into a cell in a grid, given an average
number, x, of hits per cell.
(b) The (discrete) probability density function (PDF) of a Poisson model is
e^p =AD (A)Ax / x!, where x is the average number of hits per cell and x! is
the factorial of x.
(c) These (and similar) probabilistic models depend on the results of
imaginary experiments, even conceptual mathematical experiments (Gedanken
experiments), involving tossing coins, rolling dice, or analogous =93random=94
events. The assumption that observers cannot distinguish between outcomes
will condition these models. If some way exists to distinguish between the
outcomes of experiments (e.g., rolls of the dice), the model provides an
incomplete description of the process.
(d) Try =93Statistical Distributions=94 by Merran Evans, Nicholas Hastings and
Brian Peacock [ISBN: 0471371246] Wiley-Interscience, third edition (2000)
pp. 221, OR download the compendium of probability distributions available at
http://www.causascientia.org/math_stat/Dists/Compendium.pdf. (P.S. The
file illustrates Poisson distributions on pp. 101 and 103.)
Augusto Sanabria, Ph.D., sent the internet address for =93Compendium=94 to me
via RiskAnal, another listserver. He also reads PharmPK. Augusto works as
a modeler at the Risk Research Group in the Geohazards Division of the
Australian government [Geoscience Australia (www.ga.gov.au)] at
Jerrabomberra Avenue and Hindmarsh Drive in Symonston
> 3. =93Filtered Poisson model=94
I feel certain that a competent mathematician can provide a better
definition. However, ....
(a) To me, a filtered Poisson model uses the same (or additional) values
(parameters, variables) to estimate the probability of some outcome through
a standard Poisson model. However, even the same values that an unfiltered
Poisson model would use, get preprocessed through other equations, so their
outcomes (or their probability density functions) differ. Thus, the
Poisson model delivers =93filtered=94 estimates.
(b) The U.S. Environmental Protection Agency (EPA) based their regulatory
model of potency on the carcinogenic process, or =93mutagenic hits.=94 The
somatic cell theory of carcinogenesis holds that cancer cells result from
somatic cell mutations. This model also was a filtered Poisson model. It
modeled the multistage carcinogenic process, using exposure (dose) as an
additional variable. It processed information from toxicological or
epidemiological observations to generate a carcinogenic
=93potency.=94 Unfortunately, the Agency=92s model made some untenable
assumptions. Among these were the ideas that the number of stages in the
model was a function of the different exposures (doses) used, and that a
carcinogen only altered one stage in one (increased risk) direction.
(c) EPA used this altered multistage model to estimate an upper bound to
risk for regulatory purposes, not as a model of the biology of carcinogenesis.
> 4. =93Suresh Moolgavkar's =91two-stage=92 model=94
Suresh Moolgavkar is a physician-mathematician, who currently works at the
University of Washington in Seattle.
(a) Moolgavkar=92s "two-stage" model, sometimes described as an MVK model
(Moolgavkar-Knudson-Venzon), is an exposure (dose) independent model of the
(biological) carcinogenic process. It allows for the expansion and
contraction of target cells. This =93two-stage=94 model incorporates more of
the biology of carcinogenesis into estimates of potency, but it requires an
external model of the relationship between carcinogen exposure (dose) and
mutagenic potency for untransformed cells and transformed (or initiated) cells.
(b) Moolgavkar and his colleagues published =93complete=94 or =93closed form=94
versions of this =93two-stage=94 model. See: W.F. Heidenreich, E.G. Luebeck
and S.H. Moolgavkar, Some properties of the hazard function of the
two-mutation clonal expansion model. Risk Anal. 17(3): 391-399 (1997). For
practical applications, see the following two citations.
(c) S.H. Moolgavkar, E.G. Luebeck and E.L. Anderson, Estimation of unit
risk for coke oven emissions. Risk Anal. 18(6): 813-825 (1998).
(d) S.H. Moolgavkar, E.G. Luebeck, J. Turim and L. Hanna, Quantitative
assessment of the risk of lung cancer associated with occupational exposure
to refractory ceramic fibers. Risk Anal. 19(4): 599-611 (1999).
(e) For several years now, James D. Wilson and I have tried to understand
the implications of expanding and contracting target cells for the mode of
action in carcinogen risk assessment and for the toxicology of an
interesting substance, dioxin (2,4,5,6-tetrachlorodibenzo-p-dioxin). We
published several abstracts at Society for Risk Analysis meetings, and Jim
gave a talk about dioxin at a meeting of SRA=92s Dose-Response section.
(f) The necessity of understanding exposures to carcinogenic substances and
converting these exposures into doses, explains why I and some of my
colleagues (e.g., Tony Cox or Paul Price) follow developments in
5. Your use of the term STS refers to population pharmacokinetics.
I have some problems with the STS model (or procedure), as used by many
kineticists. To refresh everyone=92s memory, the STS model requires that the
(a) Estimate parameters for each individual, e.g., clearance (CL) and
volume of distribution (Vd).
(b) Calculate the average and standard deviation for each parameter across
the sample of individuals. [Measured volumes of distribution and clearances
should generate a half-life for the substance in question for this population.]
(c) The cited averages estimate the mean clearances and the associated
standard deviations of a population. These average values relate to each
other through an equation that describes a hypothetical, average
individual=92s handling of a chemical substance. Cl = kel x Vd, AND kel
How do you know the PDF distributes normally? If the measurements
distribute log-normally, an application of average and standard deviation
calculations may yield aberrant values.
What is the referent population? (How do I understand the
representativeness of the data?) Unless the investigator defines the
population carefully (but narrowly, e.g., male, sophomore medical students
at Oxford), I do not know what the average and standard deviation
represent. In the U.S., we try to reference populations to the
census. Thus, the sampling frame might consist of random phone calls to
persons residing in the U.S. with secondary tests to convince me that the
sample of subjects resembles the census (e.g., same heights, weights, ages,
I lack confidence that unselected pharmacokinetic data distribute normally,
as expected in an =93overall uncertainty=94 model. In theory, I could measure
the values of elimination rate and volume of distribution for a
representative group. Then, I could propagate the distributions through
the above equation to derive a distribution equal to the measured
distributions of the population. Currently, most risk assessors use Monte
Carlo techniques to accomplish this task. Charles Yoe, an economist at the
College of Notre Dame in Baltimore, MD, is perhaps the best teacher of
Daniel M. Byrd III, Ph.D., D.A.B.T.
Not an infectious disease expert
8370 Greensboro Drive
McLean, VA 22102-3500
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The following message was posted to: PharmPK
You made a few comments with respect to STS, and
population pharmacokinetics in general:
> How do you know the PDF distributes normally? If the
>measurements distribute log-normally, an application of
>average and standard deviation calculations may yield
This is quite often a difficult question. Either one has
only a low number of subjects (e.g. 10), with fairly
precise individual parameter estimates, or a larger number
of subjects with only a few measurements per subjects,
thus with imprecise individual parameter estimations. None
of these cases allows a clear discrimination between
statistical distributions. In general I prefer a
log-normal distribution, based on theoretical arguments,
unless there is clear evidence for a different
In case of assuming a log-normal distribution, the mean
and sd values are of course the geometric mean and sd.
> What is the referent population? (How do I understand
>the representativeness of the data?) Unless the
>investigator defines the population carefully (but
>narrowly, e.g., male, sophomore medical students at
>Oxford), I do not know what the average and standard
>deviation represent. In the U.S., we try to reference
>populations to the census. Thus, the sampling frame
>might consist of random phone calls to persons residing
>in the U.S. with secondary tests to convince me that the
>sample of subjects resembles the census (e.g., same
>heights, weights, ages, genders, etc.).
In general, population pharmacokinetics does not refer to
a 'general population', but to a specific population, as
defined by the inclusion and exclusion criteria in the
experimental protocol. This implies that the conclusions
refer to this population only.
> I lack confidence that unselected pharmacokinetic data
>distribute normally, as expected in an =93overall
>uncertainty=94 model. In theory, I could measure the
>values of elimination rate and volume of distribution for
>a representative group. Then, I could propagate the
>distributions through the above equation to derive a
>distribution equal to the measured distributions of the
>population. Currently, most risk assessors use Monte
>Carlo techniques to accomplish this task.
I agree. In this case, I would prefer to 'measure' (taking
into account the relative large standard error, I would
say 'estimate') clearance and volume of distribution
rather than elimination rate constant and volume of
distribution, since the latter are by definition
correlated. In case one assumes a log-normal distribution,
the estimation of standard errors of derived parameters
(e.g., elimination rate constant, half-life, mean
residence time) can be performed easily, even in case of
correlations between the parameters.
Johannes H. Proost
Dept. of Pharmacokinetics and Drug Delivery
University of Groningen
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Copyright 1995-2010 David W. A. Bourne (firstname.lastname@example.org)