Mathematical modeling has been used to interpret anatomical and physiological data obtained from metabolic and hemodynamic studies aimed at investigating structure-function relationships in the vasculature of the lung, and how these relationships are affected by lung injury and disease. The indicator dilution method was used to study the activity of redox processes within the lung. A steady-state model of the data was constructed and used to show that pulmonary endothelial cells may play an important role in reducing redox active compounds and that those reduction rates can be altered with oxidative stress induced by exposure to high oxygen environments. In addition, a morphometric model of the pulmonary vasculature was described and used to detect, describe,and predict changes in vascular morphology that occur in response to chronic exposure to low-oxygen environments, a common model of pulmonary hypertension. Finally, the model was used to construct simulated circulatory networks designed to aid in evaluation of competing hypotheses regarding the relative contribution of various morphological and biomechanical changes observed with hypoxia. These examples illustrate the role of mathematical modeling in the integration of the emerging metabolic, hemodynamic, and morphometric databases.

We present an overview of two approaches to validated one-dimensional indefinite integration. The first approach is to find an inclusion of the integrand, then integrate this inclusion to obtain an inclusion of the indefinite integral. Inclusions for the integrand may be obtained from Taylor polynomials, Tschebyscheff polynomials, or other approximating forms which have a known error term. The second approach finds an inclusion of the indefinite integral directly as a linear combination of function evaluations plus an interval-valued error term. This requires a self-validating form of a quadrature formula such as Gaussian quadrature. In either approach, composite formulae improve the accuracy of the inclusion. The result of the validated indefinite integration is an algorithm which may be represented as a character string, a subroutine in a high-level programming language such as Pascal-SC or Fortran, or as a collection of data. An example is given showing the application of validated indefinite integration in constructing a validated inclusion of the error function, erf( x ).