Model Functions {CarbUtils}R Documentation

Dose-Time-Activity Model Functions

Description

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).

Usage

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)

Arguments

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
lTr
log of the half-life of the recovery process
lTmax
log of time of maximum effect
b
slope of linear trend in AChE activity (default 0)
R
benchmark response level: the benchmark dose corresponds to the dose where AChE inhibition is 100R% (default 0.10)

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 }

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.

Value

tcmfn and tcmfn2: A vector of predicted AChE activities. tcminhfn: A vector of predicted AChE fractional inhibition.

Author(s)

R. Woodrow Setzer


[Package CarbUtils version 1.0 Index]