RE: Code to avoid flip-flop kinetics

Dear Nick, Dear Paolo, Dear Leonid, Dear All, Depending on the application, I think there are the following reasons for constraining ka and ke: 1) Standard-two-stage (STS) approach: As discussed previously, calculating average and SD from STS estimates is likely to yield biased averages and SDs and this bias may have a great impact when these descriptive statistics are used as means and variances in a Monte Carlo simulation. Thus, manually switching of individual ka & ke (if needed) makes perfect sense for STS. 2) Nonparametric population PK with nonparametric simulation: As the interindividual distribution of PK parameters can take any shape, it will make no difference for a non-parametric simulation based on the list of support points whether ka and ke are constrained or not. Even if a support point (with e.g. 10% probability and ka>ke) splits into two equivalent support points with 5% probability each (5% for ka>ke and 5% for ke>ka) this will not affect the results of a nonparametric simulation. Thus for a nonparametric simulation, constraining is not needed. 3) Nonparametric population PK with parametric simulation: Once one computes averages and SDs (etc.) from the list of support points, the same issue of biased averages and SDs arises as for STS, if ka and ke are not constrained. As averages and SDs are commonly calculated for nonparametric pop PK models, constraining at the individual parameter estimate level seems most reasonable, as proposed by Vladimir Piotrovsky (option 2 in Paolo's email). I think this was the approach taken at the 2005 PAGE software comparison. I am not aware of a method of constraining a nonparametric model at the population level. 4) Parametric EM-algorithms: Assume true mean ka and true mean ke are within 10% of each other and BSV has CV of 30% each. Clearly, there must be substantial overlap in the true individual estimates of ka and ke. 4.1) As one EM algorithm, the MC-PEM algorithm assumes that the interindividual distribution of individual model parameters is normal (or lognormal, or can be transformed to be normal). Otherwise, the population mean and population variance-covariance matrix obtained from the conditional means and condition variance matrices of each subject (see eq. 23 and 27 in [1] and ref. [2]) does not necessarily minimize the overall log-likelihood. Truncation of the interindividual distribution of PK parameters, for example by use of the EXIT command (option 3 in Paolo's email), seems to violate the normal distribution assumption. 4.2) The conditional means calculated in the expectation-step can be biased due to e.g. a bi-modal distribution caused by the flip-flop (see also Serge's comment at the end of the discussion in 2003). This might be a more severe concern compared to 4.1). If such bias in the conditional means and in the conditional variance matrices occurs, the computation of the population means and population variance-covariance matrix via eq. (21) and (22) (see Bauer et al. [1]) does no longer guarantee to minimize the objective function, I think. Constraining ka and ke at the population level (option 1 in Paolo's email) is unlikely to solve this problem, if mean ka and mean ke are close. If one does not constrain ka and ke at the individual parameter level, bias of conditional means is likely to occur, if: a) the mean ka and mean ke are close given their interindividual variability b) the data are sparse and one therefore uses a wide envelope function to approximate an individual's conditional density (this problem is more likely to occur if one uses low values of gefficiency [please see S-ADAPT manual] or if one uses e.g. a t-distribution with small degrees of freedom as envelope function) Therefore, constraining at the individual parameter level via estimation of CL, V, and (ka-kel) (i.e. the Vladimir Piotrovsky method) and use of a full var-cov matrix (as proposed by Leonid) seems the best choice for the MC-PEM algorithm. I think this also applies to the SAEM and MCMC algorithm. These algorithms usually have no problem with estimating a full var-cov matrix and do not require excessively longer runtimes to estimate a full var-cov matrix, as opposed to FOCE. SUMMARY: 1) There is no need to constrain ka and ke for a nonparametric estimation with nonparametric simulation. However, as constraining does not hurt, I would still constrain ka and ke, since interpretation of mean parameters is easier. 2) For nonparametric estimation, ka and ke should be constrained at the individual level, if means and SDs (etc.) are to be computed from the list of support points (as commonly done). 3) Constraining ka and ke at the individual level and estimation of a full var-cov matrix seems most appropriate for parametric EM algorithms and for nonparametric estimation with subsequent parametric Monte Carlo simulations. Hope these comments are helpful despite their length. Nick, I would agree that in many cases, fortunately the issues mentioned above only have a minor practical impact. Best wishes Juergen References: [1] Bauer RJ et al. AAPS Journal, 2007, 9(1): E60-E83. See bottom left and right column on page E64. [2] Schumitzky A . EM algorithms and two stage methods in pharmacokinetics population analysis. In: D'Argenio DZ , ed. Advanced Methods of Pharmacokinetic and Pharmacodynamic Systems Analysis. vol. 2. Boston, MA : Kluwer Academic Publishers ; 1995 :145- 160. ------------------------------------------------------ Jurgen Bulitta, Ph.D., Senior Scientist Ordway Research Institute, Albany, NY, USA ------------------------------------------------------