This article is part of the monograph on Mathematical Modeling in Environmental Health Studies.
Address correspondence to M.C. Kohn, Laboratory of Computational Biology and Risk Analysis, NIEHS, PO Box 12233, MD A3-06, Research Triangle Park, NC 27709 USA. Telephone: (919) 541-4929. Fax: (919) 541-1470. E-mail: kohn@valiant.niehs.nih.gov
Received 25 August 2000; accepted 31 August 2000.
Characterization of the risks to human health associated with exposure to environmental pollutants has historically followed the model of toxicology, namely, observing the incidence of pathology in experimental animals consequent to exposure at several doses. The incidence of the response was then (usually linearly) extrapolated from the experimental doses to exposures typical in the environment and occupational settings. Regulatory agencies estimate doses in humans that would produce the extrapolated response by scaling the extrapolated dose for animals by the two-thirds power of body weight. Owing to lack of knowledge and in order to protect human health, humans are assumed to be as sensitive to the toxicant as is the most sensitive animal species.
In 1937 the mathematical basis for physiologically based pharmacokinetic modeling was established by Torsten Teorell (1,2), but solution of the equations was too onerous before the invention of the digital computer. Kenneth Bischoff (3) pioneered automatic solution of a physiologically realistic mathematical description of the uptake, distribution, and clearance of a chemical agent. At the time, limitations in computer memory and computational speed necessitated simplification of the general formulation. One such simplification, the assumption that distribution of the chemical between tissue and blood is instantaneously at equilibrium, led to physiological models with blood flow-limited delivery of chemicals to tissues. The Ramsey-Andersen model (4) for inhalation pharmacokinetics is a well-known example.
Originally, the model's differential equations had to be coded in a programming language such as Fortran and linked to commercial subroutines for their solution. Few environmental scientists had such skills, and computer modeling was used by only a small number of specialists. Most environmental health scientists did not evaluate the quantitative consequences of their inferences to ensure that their hypotheses were consistent with contemporary knowledge (5). They formulated hypotheses intuitively and directly performed experiments to confirm or falsify those assumptions. Such a strategy forfeits the opportunity to formally validate hypotheses (6).
As computer hardware became faster and cheaper and new software made it easier to construct models, modeling became accessible to a wider community of researchers. Guides for construction and use of more advanced physiologically based dosimetry models appeared in the literature (7). The increased research activity led to the recognition of several factors that influence the utility of physiological models. Among these are physiological realism in the model's structure (6,8), biochemical (9) and pharmacodynamic (5) consequences of delivery of the agent to tissues, and the influence of interindividual variability in estimating model parameters (10-12). Inclusion of greater detail in physiological modeling is continuing and is the motivation for producing this monograph.
Diffusional barriers to hepatic uptake have been identified both experimentally and theoretically (13), and models with different representations of the liver anatomy predict different behaviors (14). Models of the uptake of butadiene from closed chambers (8) give different fits to the observed clearance of the gas, depending on the level of anatomical detail included. A model that uses standard physiological parameters and biochemical constants measured in vitro and assumes flow limitation of delivery of the chemical to tissues overestimates gas uptake. It has been suggested (15) that to compensate for this defect the ventilation rate should be reduced from its measured value, perhaps due to adsorption of inhaled gas on the surface of the airway. However, distribution of the blood among compartments for arterial, venous, and capillary spaces and accounting for tissue permeability toward the gas permits reproduction of the observations without altering measured parameter values (8). The report by Schwab and Pang (16) is a further advance in the characterization of diffusional barriers to transport of an agent between blood and tissue.
Physiological models have typically included equations for uptake of a toxicant, its distribution among tissues, metabolism, and elimination from the body. Many biological effects are the result of consecutive biochemical processes occurring in several tissues and may involve feedback regulation. The complex relationships among fatty acid metabolism, production of chylomicrons, intestinal absorption of lipophilic compounds, and distribution of the chemical via lymph and blood have been incorporated into physiological models (17,18). Renal accumulation of ligands of 
2u-globulin may be the net consequence of hepatic synthesis of this binding protein, its filtration at the glomerulus and uptake by the kidney, and degradation of the protein by renal cathepsins (19). A model of 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD)-stimulated hepatic metabolism of thyroxine predicted reduced feedback inhibition of thyrotropin release from the pituitary (20,21). The physiological interactions among testosterone, luteinizing hormone, and follicle-stimulating hormone have been simulated (22). The investigation by Schlosser and Selgrade (23) of the regulation of pituitary gonadotropin production by ovarian hormones during the human menstrual cycle continues progress in modeling such distributed processes.
Physiological models typically use mean values for their physiological and biochemical parameters. The observed values include both uncertainties in measurement and variation due to genetic differences among individuals. Human populations may exhibit large interindividual variability, and a subpopulation may be significantly more sensitive than the average individual. A simplified physiological model of the disposition of benzene in men (24) was fit to chamber uptake data under the assumption that parameter values were log-normally distributed in the human population. Parameter values were obtained for individual subjects by formal optimization, generating posterior distributions of the values in the tested group. The use of prior information about the distribution of parameter values in the population from which the tested group is a sample reduces the uncertainty in the optimized values. Bernillon and Bois (25) continue these investigations and delineate how application of population toxicokinetics can increase confidence in estimates of risks to human health.
The proximate toxicant from environmental exposure may not be the chemical to which a subject is exposed. The active agent may be produced by metabolic activation, and the toxicant itself may undergo enzymatic detoxication. Early in the development of physiological dosimetry models, the importance to risk assessment of representing the several steps in the metabolic pathway was illustrated (26). Depletion of cofactors required by the metabolic enzymes (27) also exerts profound effects on metabolism and must be represented appropriately (28). However, the enzymatic kinetics are often not determined in sufficient detail, which introduces considerable uncertainty about the model's reliability. In an effort to obviate this problem, biochemical systems theory represents the unknown kinetics by a generalized law of mass action and estimates rate constants and kinetic orders to reproduce observed time courses. Although this approximation cannot be proved valid beyond the range of the kinetic data, the method is often applicable over a broad range of concentrations. Voit (29) describes the use of this technique and demonstrates its utility for representing nonlinear behavior, solving the resulting equations, and analyzing the stability and sensitivity of the solution.
Biochemical processes generally occur in discrete regions of the cells of metabolizing tissues, and it may be important to distinguish among these regions in modeling such responses. Early physiological models of the disposition of butadiene (30-32) predicted much higher blood concentrations of the toxic metabolite epoxybutene than were subsequently measured in exposed mice and rats (33). The metabolic parameters in these models, obtained from in vitro measurements, that reproduce the uptake of epoxybutene from closed chambers led to underprediction of the rate of metabolism of epoxybutene produced in situ. Endogenously produced epoxide may have enhanced access to a detoxifying enzyme compared to exogenous epoxide, as enzymes that produce and detoxify this intermediate are colocalized in the cell on the endoplasmic reticulum (34,35). In vitro measurements would not detect such a proximity effect. Kohn and Melnick (36) extend the earlier models to demonstrate the consistency of this hypothesis with data for both clearance from chambers and concentration of circulating epoxybutene.
Anatomical structure can have a profound influence on the development of toxicity resulting from exposure to a pollutant. A model of the lobular structure of the liver (37) predicted regional differences in the induction of enzymes by TCDD (38). The extent of nasal lesions consequent to inhalation of formaldehyde was shown to be dependent on the species-specific structure of nasal membranes (39,40). Conolly et al. (41) use finite element modeling of fluid flow in the upper airway to predict localized deposition of inhaled formaldehyde and production of DNA-protein cross-links in several species.
Modeling is an iterative pr ocess in which experimental data are used to specify a set of equations. The predictions of the model suggest new experiments to address unresolved issues, and the model is refined based on the new data. A model becomes more detailed and realistic with each iteration of this cycle, and confirmation of the model's predictions builds confidence in the use of the model to estimate responses of individuals exposed to a toxicant. Mechanistic detail was accumulated through three iterations of a model of enzyme induction by TCDD (20,21,42,43). In the process, the model produced reliable estimates of tissue accumulation of TCDD, alterations in gene expression, and modulation of hormonal status in treated animals.
Modern computational tools facilitate the construction and interpretation of more realistic toxicological models. The contributions to this monograph exemplify the potential of today's hardware and software to identify factors that determine observed responses of subjects exposed to toxicants. Continuing this trend toward more detailed representations of biochemical, physiological, and anatomical structures to predict the effects of exposure to toxic agents will enhance the credibility of mathematical models, inspire greater confidence in the models' predictions, and produce more reliable estimates of health risks (6). Such advances can only improve the scientific basis for estimation of effects of xenobiotic agents on the health of human populations.
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Last Updated: October 10, 2000
