A distributed parameter liver model of benzene transport and metabolism in human and mice: Developmental, theoretical, and numerical considerations

In the Clean Air Act of 1970, the U.S. Congress names benzene a hazardous air pollutant and directs certain government agencies to regulate public exposure. Court battles over subsequent regulations have led to the need for quantitative risk assessment techniques. Models for human exposure to various chemicals exist, but most current models assume the liver is well-mixed. This assumption does not recognize---most significantly---the spatial distribution of enzymes involved in benzene metabolism.; The development of a distributed parameter liver model that accounts for benzene transport and metabolism is presented. The mathematical model consists of a parabolic system of nonlinear partial differential equations and enables the modeling of convection, diffusion, and reaction within the liver. Unlike the commonly used well-mixed model, this distributed parameter model has the capacity to accommodate spatial variations in enzyme distribution.; The system of partial differential equations is formulated in a weak or variational setting that provides natural means for the mathematical and numerical analysis. In particular, general well-posedness results of Banks and Musante for a class of abstract nonlinear parabolic systems are applied to establish well-posedness for the benzene distributed liver model. Banks and Musante also presented theoretical results for a general least squares parameter estimation problem. They included a convergence result for the Galerkin approximation scheme used in our numerical simulations as a special case.; Preliminary investigations on the qualitative behavior of the distributed liver model have included simulations with orthograde and retrograde bloodflow through mouse liver tissue. Simulation of human exposure with the partial differential equation and the existing ordinary differential equation model are presented and compared. Finally, the dependence of the solution on model parameters is explored.