M. Costa*, E. Berno#, G. P. Zara#, C. Della Pepa#, R. Canaparo, M. Eandi#
*Dept. of Electronics, Polytechnic of Torino #Inst. of Pharmacology and Experimental Therapy, School of Medicine, Univ. of Torino
Population PK/PD modeling can provide useful insight into the complex input-output relationships of pharmacological and therapeutic processes in naturalistic settings. From a clinical standpoint it should be basically considered as a tool for the estimation of relevant quantities from sets of measurements, in order to provide optimal strategies of adaptive control of clinical outcomes. Nowadays computer programs can assist the clinicians in the analysis of drug concentrations and their relationships with pharmacological effects. However, full- fledged PK/PD applications remain challenging, especially when only poor data are available because of ethical issues or technical constraints. Moreover, the well- established compartmental modeling is quite often inadequate or insufficient, or very complex to manage, when dealing with highly non-linear input-output relationships. To overcome, at least in part, some of these difficulties we developed a completely data- driven, model-independent, non-parametric approach to population PK/PD problems, which is based on the well-known Parzen methods[1]
Let: X≡{xj} (j= 1,..,n) be a collection of n random variables; D≡{ xij } (i= 1,..,m; j= 1,..,n} be a set of (perhaps partial) realizations of X obtained through measurements taken on a population of m individuals. By partial realization we mean that for each individual only some of the n quantities involved may have been actually measured. For the sake of simplicity, we will hereafter consider X as a real-valued random vector. However, we point out that whatever kind of random variables (reals, integers, booleans, mutually exclusive events, and so on) can be intermixed with minor changes in the following formulas, thus accounting also for the non-parametric analysis of co-variates into a unified framework.
The true joint Probability Density Function (PDF) p(X) of X retains the most detailed information about the system under study as it is described by the statistical relationships among the n quantities involved. Once p(X) is known, any conceivable model can be applied and validated by starting a suitable bayesian inference chain. For instance, given a complete parametric model M equipped with a set of parameters Ω, a fitting model p(X/Ω,M) and a prior p(Ω/M), then the expected posterior PDF of Ω based on a single observation of X randomly drawn from p(X) can be expressed as follows:
Of course, p(X) is not available. However, the Parzen Method produces an estimate pD(X) based on the experimental evidence provided by D. Such an estimate is expressed as a superposition of m (customarily gaussian) kernels Ki(x), each one centered around a different sample. Possible missing components are reconstructed through the EM (Expectation-Maximization) algorithm[2]
Now that we got useful information from our population, let us consider a new set S of (perhaps partial) realizations of X. The sample points along with their corresponding uncertainties induce a PDF ps(X). Especially in case of very poor data, pS(X) might in fact not depend on some of the n variables. These are in a sense “filled in” by our knowledge about the population through the restriction of pD(X) to S, that gives rise to a new PDF pR(X) defined as:
In conclusion our algorithm is almost completely data-driven methods, i.e. it can directly rely upon the experimental evidence alone, without requiring that any C explanatory model of the problem of interest be defined in advance.
This approach was implemented as a new algorithm of a computer program suitable to manage population PK and/or PK/PD data. The program must be “trained” by the experimental data, then it calculates the PDF of all the variables of the population, and, finally it can be used to predict, in a specific patient, the probability distribution of a given variable starting from the measurement of few covariates. Our algorithm has been applied to analyse some very difficult PK/PD relationships, because the high non-linearity. A very complex example is the PK/PD of tolcapone, a new selective inhibitors of peripheral and central COMT, developed as adjunctive drug to 1-DOPA + AAD inhibitors for the treatment of Parkinson disease. In particular we studied the relationships between the dose and concentration of tolcapone (input) and the plasma concentration of L-DOPA and/or 3-OMD, the main catabolic product of L- DOPA via COMT (as output). Plasma levels of L-DOPA show a highly inter- and intra- individual variability that depends on many factors: oral dosage schedule, pharmaceutical formulation, type of AAD inhibitors (carbidopa or benserazide), circadian rhythms, etc. Tolcapone, inhibiting the second main metabolic pathway of L- DOPA, induces an increase, almost proportionally linear, of the AUC and half-life of L- DOPA, without significantly modifying Cmax and tmax. Moreover, the plasma concentrations of 3-OMD are drastically decreased by tolcapone. We used data obtained by different pharmacokinetic trials on healthy volunteers. After training the program with a set of data, we were able to estimate the “missing values” of new “test subjects” starting from few measured or known data. In particular we were able to estimate the PDF of the steady-state plasma concentrations of L-DOPA and 3-OMD from the dose of tolcapone and/or few plasma concentrations and AUC value of tolcapone measured after the first dose.
References
[1] Parzen E (1962). An estimation of a probability density function and mode. Ann. Math. Stat., 33, 1065-1076.
[2] Dempster AP, Laird NM and Rubin DB (1977). Maximum Likelihood from Incomplete Data via the EM Algorithm (with Discussion). Journal of the Royal Statistical Society B, 39, 1-38.
Reference: PAGE 7 (1998) Abstr 690 [www.page-meeting.org/?abstract=690]
Poster: poster