RE: Rate constants..are dead?

From: "Wixley, Dick" <Dick.Wixley@solvay.com> Subject: RE: Rate constants..are dead? Date: Wed, 29 Nov 2000 10:11:48 +0100 Dear nmusers I have found the discussion very interesting. I think there are two issues *the chosen parameterization and distributional assumptions in estimation *the chosen parameterization for practical scientific inference about the model and data, etc. The former choice is important since the closer to the truth of the assumption of normality for the parameter(s), the better the performance of the estimation (e.g.FO) method, and the closer the individual predictions. (The inverse BOX-COX power transformation provides a useful and flexible parameter transformation that is also bounded >0.) The sensible scale for thinking about the real-life situation is often not the rate constant. I have made a habit of presenting half-lifes for all rate constants and am in agreement with all comments on this point. It is interesting to speculate about the true distribution of rate constants. Some years ago I did some investigation of the distribution of plasma concentration measurements in pre-clinical PK and toxicokinetics. I found that the distribution usually lay somewhere between the lognormal distribution and the gamma distribution. After intravenous dosing this gives a model: E(Y) = exp(-alpha*time+beta) or log[E(Y)] = beta - alpha*time, and, VAR(y) = V*E(Y)**2 approximately. The data could be analysed in the generalized linear model framework by a log-linear link model and Gamma errors. Alternatively the transformation Z = log(Y+const) gave a model of the mean linear in 'time' and with constant variance. i.e. E(Z)= beta -alpha*time , VAR(Z) = V` to first-order approximation. Either way the linear model assumptions seemed to apply both within and beween subjects in all the pre-clinical data-sets I investigated. The normal distribution of rate constants therefore seems natural from superficial basic considerations. But what about the constraint alpha>0. This messes up the normal assumption. Subsequently in population PK in humans I have gained the impression that the "lognormal" assumption is not bad for elimination constants. Also, it is convenient and virtually essential since it naturally bounds alpha away from zero. Also when eta becomes small, (CV<10%) the log-normal distribution tends to normality. (For the lognormal , the larger eta is, the more skew the distribution.) For absorption rate-constants in oral administration, the situation is different and the distributions between subjects arbitrary and very variable. Finally if a variable is lognormal, its inverse is lognormal. An alternative to the lognormal that is more extreme, is the (generalised) Inverse Gaussian distribution. The reciprocal Inverse gaussian or Wald distribution is potentially interesting. Time permitting it would be perhaps fruitful to explore this whole area more fully. regards Dick