Model Functions {CarbUtils} | R Documentation |
Several versions of a model that accounts for both time course and dose-response of AChE inhibition by carbamates that differ among each other in the way the time course is parametrized (see details).
tcmfn(x, Tpd, Tc, lA, tz, lD, lg, lTr, lTmax, b, R ) tcmfn2(x, Tpd, Tc, lA, tz, lD, lg, ldT, lTmax, b, R) tcmfn3(x, Tpd, Tc, lA, tz, lD, lg, ldT.alpha, ldT.beta, lTmax, b, R) tcminhfn(x, Tpd, tz, lD, lg, lTr, lTmax, R)
x |
dose (mg/kg/day) |
Tpd |
time post dosing (hours) |
Tc |
time since study began (may be negative if there are control values that predate the dose) |
lA |
log of background (control) AChE activity |
tz |
transform of limiting high-dose AChE activity (see details) |
lD |
log of benchmark dose (see argument R ) |
lg |
log of power parameter (see details) |
ldT |
log(log(ratio of half life of recovery to half life of absorption)) (see details) |
ldT.alpha, ldT.beta |
intercept and slope of the modeled
relationship between ldT and dose in tcmfn3() :
ldT = ldT.alpha + ldT.beta * x
|
value{
tcmfn
and tcmfn2
: A vector of predicted AChE
activities. tcminhfn
: A vector of predicted AChE fractional
inhibition.
}
details{
The dose-time-response model used for the N-methyl carbamates is
y(x,t) = A - f(x) g(t),
where:
f(x) = A*[P + (1 - P) * exp(-(m*x)^g)]
and:
g(t) = (exp(-ln(2) * t / Ta) - exp(-ln(2) * t / Tr))/Cmax
and where Cmax is chosen so that the maximum value of g(t) is 1. That is,
Cmax = exp(-ln(2) * Tmax / T_a) - exp(-ln(2) * Tmax / T_r)
and
Tmax = Ta * Tr * (ln(Tr) - ln(Ta))/(ln(2) * (Tr - Ta))
is the time to maximum effect. In this formulation Ta and
Tr are half-lives for an “absorption” phase and
“recovery”
phase, respectively.
This parametrization of the time-course part of the model leads to two
problems. First of all, g(t) is symmetric in Tr and
Ta. Thus the two parameters are not identifiable (and,
indeed, designating one as representing “absorption” and the
other as “recovery” phase is therefore inappropriate). This
causes problems for optimization (to say the least). From the
underlying pharmacokinetic and pharmacodynamics of this response, it
is reasonable to characterize the inhibition as due to rapid
absorption and subsequent inhibition, followed by more-or-less slow
restoration of cholinesterase activity. It may well be that the rate
of recovery is governed largely by the de-carbamylation of the AChE
molecule, as the carbamate parent and active metabolites are cleared
very quickly. Further data analysis in this project should provide
evidence for this one way or another. In any case, since g(t)
is symmetric, it makes sense to identify Tr as the larger
of the two half-lives. Then, for purposes of estimating parameters,
reparametrize the model in terms of the ratio of Tr to
Ta. Since we want that ratio to be greater than 1, we work
with ldT = log(log(Tr/Ta)).
Secondly, many or most of the toxicology studies that
generated data on the recovery time course identified the time of
maximum inhibition, and only looked at cholinesterase activity at
times after that point. It would greatly simplify fitting such data
sets if the time of maximum effect could be fixed at the value
asserted by the investigators, not estimated from
the data. This suggests reparametrizing the models so that
Tmax appears explicitly as a parameter, to be estimated
when possible, and otherwise to be specified. To force the parameter
to be positive, work with lTmax =
log(Tmax).
So, the time course portion of the model for the fitting functions is
parametrized throught ldT and lTmax. For functions used
solely for making predictions, ldT is replaced by lTr, the
log of the recovery half-life.
The parametrization for the dose-response portion of the model is
driven by the need to bound P in the interval [0,1), to keep
g positive, and to include the BMD explicitly as a parameter.
Thus, parameters are lg = log(g),
tz = log(P/(1 - P)),
or
P = 1/(1 + exp(-tz))
and m in the function f(x) is replaced by
m = frac{[-log(frac{1 - R - P}{1 - P})]^{1/g}}{exp(lD)}
Here lD is the log of the dose that results in 100R% cholinesterase inhibition.
}
author{R. Woodrow Setzer}
keyword{ ~kwd1 }
The dose-time-response model used for the N-methyl carbamates is
y(x,t) = A - f(x) g(t),
where:
f(x) = A*[P + (1 - P) * exp(-(m*x)^g)]
and:
g(t) = (exp(-ln(2) * t / Ta) - exp(-ln(2) * t / Tr))/Cmax
and where Cmax is chosen so that the maximum value of g(t) is 1. That is,
Cmax = exp(-ln(2) * Tmax / T_a) - exp(-ln(2) * Tmax / T_r)
and
Tmax = Ta * Tr * (ln(Tr) - ln(Ta))/(ln(2) * (Tr - Ta))
is the time to maximum effect. In this formulation Ta and Tr are half-lives for an “absorption” phase and “recovery” phase, respectively.
This parametrization of the time-course part of the model leads to two problems. First of all, g(t) is symmetric in Tr and Ta. Thus the two parameters are not identifiable (and, indeed, designating one as representing “absorption” and the other as “recovery” phase is therefore inappropriate). This causes problems for optimization (to say the least). From the underlying pharmacokinetic and pharmacodynamics of this response, it is reasonable to characterize the inhibition as due to rapid absorption and subsequent inhibition, followed by more-or-less slow restoration of cholinesterase activity. It may well be that the rate of recovery is governed largely by the de-carbamylation of the AChE molecule, as the carbamate parent and active metabolites are cleared very quickly. Further data analysis in this project should provide evidence for this one way or another. In any case, since g(t) is symmetric, it makes sense to identify Tr as the larger of the two half-lives. Then, for purposes of estimating parameters, reparametrize the model in terms of the ratio of Tr to Ta. Since we want that ratio to be greater than 1, we work with ldT = log(log(Tr/Ta)).
Secondly, many or most of the toxicology studies that generated data on the recovery time course identified the time of maximum inhibition, and only looked at cholinesterase activity at times after that point. It would greatly simplify fitting such data sets if the time of maximum effect could be fixed at the value asserted by the investigators, not estimated from the data. This suggests reparametrizing the models so that Tmax appears explicitly as a parameter, to be estimated when possible, and otherwise to be specified. To force the parameter to be positive, work with lTmax = log(Tmax).
So, the time course portion of the model for the fitting functions is parametrized throught ldT and lTmax. For functions used solely for making predictions, ldT is replaced by lTr, the log of the recovery half-life.
The parametrization for the dose-response portion of the model is driven by the need to bound P in the interval [0,1), to keep g positive, and to include the BMD explicitly as a parameter. Thus, parameters are lg = log(g),
tz = log(P/(1 - P)),
or
P = 1/(1 + exp(-tz))
and m in the function f(x) is replaced by
m = frac{[-log(frac{1 - R - P}{1 - P})]^{1/g}}{exp(lD)}
Here lD is the log of the dose that results in 100R% cholinesterase inhibition.
tcmfn
and tcmfn2
: A vector of predicted AChE
activities. tcminhfn
: A vector of predicted AChE fractional
inhibition.
R. Woodrow Setzer