Splines and Compartment Models

The following sections are included:

• Preface

• Contents

In this part, a general method for the mathematicel modeling of real processes is developed using a deterministic approach. Three aspects are important here: • the data, • the model and • the criterion of goodness of fit…

The fundamental starting point of a curve fitting problem using spline functions with one variable is a finite number of points (x_i, y_i), i = 1, 2, … ,m, which describe the interesting but otherwise unknown functional dependency y = g(x) at chosen knots x_i. The system of knots consists of increasingly ordered pairwise different values x₁ < x₂ < x₃ < … < x_m−1 < x_m…

It should be noted again here that a natural cubic spline s(x) on the knots x₁ < x₂ < … < x_m is a twice continuously differentiable function on ℝ that on the intervals (−∞, x₁] and [x_m,+∞), and on [x_i, x_i+1], i = 1, 2, … ,m − 1, can be described as a straight line and a polynomial to a maximum degree of 3, respectively…

Up to this point tasks of interpolation for spline functions were discussed. Given are the points (x_i, y_i), i = 1, 2, … ,m, that can be understood as points of a function y = g(x) that is to be described, but is otherwise unknown…

The class of quadratic splines is interesting because it does not contain natural splines and because the numeric calculation of the interpolating problem requires little effort. Interpolating quadratic splines with minimal total curvature are developed. Further optimization criteria for the construction of interpolating quadratic splines are observed. In addition, the following question is posed: when is a polynomial of the degree 2 exactly reconstructed by a quadratic spline? More interesting results can be seen here than for cubic splines…

Observed was an interpolating task using quadratic splines. This has a unique solution when an additional condition is fulfilled (Theorem 5.1). Parabolas are special quadratic spline functions. In this chapter the question of how a given parabola can be reconstructed by interpolating quadratic splines is examined. Initial possibilities become apparent when the necessary additional condition, due to Theorem 5.1, is suitable formulated. Another approach is offered by the different attempts at optimization. All of these problems of optimization can be solved by the realization of an initial condition. The examination of this condition under the assumption that the interpolation points stem from a parabola produces additional possibilities for reconstruction. These can be optimization conditions in connection with the number of points or conditions for the knots of the points of interpolation. If the formulation of a special optimization problem is foregone, it is still possible to find initial conditions that can, using the given points, reproduce every quadratic polynomial…

The following sections are included:

• Smoothing quadratic splines and the integral of the quadratic first derivative

• Quadratic splines smoothing the predefined first derivatives

The following sections are included:

• Averaged splines in the case of common knots

• Averaged kinetics and reference ranges

• Growth curves and averaged splines without common knots

To aid in physics, mathematics came into contact with medicine. Blood flow models were already studied in the 1850’s by Helmholtz, who was a doctor and physicist (Kac 1979). The classic compartment and flow models of pharmacokinetics also follow this tradition. Stochastic methods actually introduced new ways of thought in chemistry and physics in the 19th century. In biological sciences, genetics was the first to take part in such a development. The in those days seemingly forgotten works of Mendel (1866) bring a theoretical probability model into relation to heredity. None other than R.A.Fisher (1936) made a reproach that the results of Mendel’s experiments were “too exact”. Later inquiries lead to the impression that experimental results were only “good” due to the typical but statistical undependable selection that was used at the time, and because Mendel’s integrity was undoubtedly not in question (Hagemann 1984)…

The following sections are included:

• One-compartment model

• Two-compartment models

• More-compartment models

The following sections are included:

• Compartment models with delay

• Nonlinear kinetics

As indicated at the end of Definition 10.3, the structural indentifiability of linear compartment models are of interest in the following sense: A linear more-compartment model is predefined. The inhomogeneities, or input functions, or the initial conditions imply in which way and into which compartment the drug is applied. The time courses of concentrations of this drug are observable in certain compartments. Curve fitting to the measurements delivers a set of system parameters…

The following sections are included:

• Unbounded residence times

• Properties of distributions of unbounded residence times

• Truncation

The following sections are included:

• Stochastic differential equations

• Stochastic processes

• Regression attempts

The following sections are included:

• Method of least squares parameter calculations

• Statistical parameter estimation for an individual kinetics
  • Varied minimum-χ²-estimation
  • Qualities of the varied minimum-χ²-estimator

• The varied minimum-χ²-method applied to population kinetics

If a mathematical model is formulated for the timely change of the concentrations of drugs in an organism, then it is to be biochemically, physiologically, etc. verified in a technical context. The a-posteriori selection of a model cannot do without a mathematical characterization of the attempts of the formula and the method for model parameter calculation regarding given data. Well-founded rules for decisions should be formulated. The judgement of the agreement of measurements with a function chosen for the description of them is a generally formulated task. It requires the appropriate choice of respective applications and assesment of distance concepts. For experimental evaluations, statistical methods are preferred in which experimental planning (in the statistical sense), parameter estimation and tests of goodness of fit infer the ideal problem solution. There is an extensive literature to this problem area concerning linear statistical models…

If the time courses of concentrations of active agents are repeatedly observed for an individual, sequences of individual kinetics are of interest. Such problems appear, for example, in the modeling of urea kinetics in dialysis patients (Lopot 1990, JÄger 1989, Sargent and Gotch 1980)…

The following sections are included:

• Program list
  • Cubic splines
  • Quadratic splines

The following sections are included:

• Bibliography

• Index