Splines and compartment models. An introduction (Q2865720)

This book presents methods of mathematical modeling where different kinds of splines provide a useful tool. In the first part, splines are considered from the origins, analyzing their different degrees and main properties. In particular, cubic interpolating spline with different end conditions are proposed with several numerical and graphical examples. Spline least squares approximation and different problems of optimization are formulated with techniques to detect suitable smoothing parameters.NEWLINENEWLINE In the second part, mathematical aid to physics, medicine and in biological sciences is considered, with particular reference to mathematical applications in pharmacokinetics, i.e., the description of distribution in space and time of intake, elimination and metabolism of a quantity of a substance in an organism, which is characterized by a specific kinetics of transformation or transport processes, by using radioactively marked substances. It is described as a compartment. Compartments can interact by exchanging substances.NEWLINENEWLINE Then on one hand discussions of model validation, involving initial value problems, linear differential equations (compartment models), nonlinear systems of differential equations, differential equations with delay, as well as model construction and interpretation are carried out. On the other hand fundamental methods are borrowed from kinetics of chemical reactions. Such methods involve both deterministic and stochastic modeling of kinetic processes. Indeed the calculability and identifiability of a pharmacokinetical model have to be examined with regard to the stochastic model underlying the parameter calculation and the stochastic process of the distribution of a set of substance molecules in a system of compartments.NEWLINENEWLINE Spline functions come into play in methods of least squares parameter calculations and they are suitable to approximate the observed concentration-time function.NEWLINENEWLINE The goal of this treatise is to recommend an adequate method for the evaluation of individual kinetics with the calculation of suitable model parameter estimates and to study its qualities.NEWLINENEWLINE It is both a comprehensive introduction to the subject and a valuable resource to those already well versed in the area.NEWLINENEWLINE Finally, the text contains Mathematica programs for selected problems and a rich reference section, that is widely discussed all along the book.