Pharmacokinetic parameter estimations by minimum relative entropy method |
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Authors: | Takashi Amisaki Shinto Eguchi |
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Institution: | (1) Department of Mathematics and Computer Science, Faculty of Science and Engineering, Shimane University, 1060 Nishikawatsu, 690 Matsue, Japan;(2) The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, 106 Tokyo, Japan |
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Abstract: | For estimating pharmacokinetic parameters, we introduce the minimum relative entropy (MRE) method and compare its performance
with least squares methods. There are several variants of least squares, such as ordinary least squares (OLS), weighted least
squares, and iteratively reweighted least squares. In addition to these traditional methods, even extended least squares (ELS),
a relatively new approach to nonlinear regression analysis, can be regarded as a variant of least squares. These methods are
different from each other in their manner of handling weights. It has been recognized that least squares methods with an inadequate
weighting scheme may cause misleading results (the “choice of weights” problem). Although least squares with uniform weights,
i.e., OLS, is rarely used in pharmacokinetic analysis, it offers the principle of least squares. The objective function of
OLS can be regarded as a distance between observed and theoretical pharmacokinetic values on the Euclidean space ℝN, whereN is the number of observations. Thus OLS produces its estimates by minimizing the Euclidean distance. On the other hand, MRE
works by minimizing the relative entropy which expresses discrepancy between two probability densities. Because pharmacokinetic
functions are not density function in general, we use a particular form of the relative entropy whose domain is extended to
the space of all positive functions. MRE never assumes any distribution of errors involved in observations. Thus, it can be
a possible solution to the choice of weights problem. Moreover, since the mathematical form of the relative entropy, i.e.,
an expectation of the log-ratio of two probability density functions, is different from that of a usual Euclidean distance,
the behavior of MRE may be different from those of least squares methods. To clarify the behavior of MRE, we have compared
the performance of MRE with those of ELS and OLS by carrying out an intensive simulation study, where four pharmacokinetic
models (mono- or biexponential, Bateman, Michaelis-Menten) and several variance models for distribution of observation errors
are employed. The relative precision of each method was investigated by examining the absolute deviation of each individual
parameter estimate from the known value. OLS is the best method and MRE is not a good one when the actual observation error
magnitude conforms to the assumption of OLS, that is, error variance is constant, but OLS always behaves poorly with the other
variance models. On the other hand, MRE performs better than ELS and OLS when the variance of observation is proportional
to its mean. In contrast, ELS is superior to MRE and OLS when the standard deviation of observation is proportional to its
mean. In either case the difference between MRE and ELS is relatively small. Generally, the performance of MRE is comparable
to that of ELS. Thus MRE provides as reliable a method as ELS for estimating pharmacokinetic parameters. |
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Keywords: | least squares extended least squares maximum likelihood relative entropy parameter estimation |
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