Mathematical models are mathematically formalized representations of a system that allow for the study of complex processes that are occurring simultaneously. In different disciplines of biology, mathematical modeling has been used to gain a deeper understanding of physiological systems and processes. For example, mathematical modeling has been used to study the rate of uptake of endogenous and exogenous compounds, to calculate enzyme kinetics, to predict pharmacological responses to drugs, and to calculate nutrient intake. For general discussions of the applications of mathematical modeling in the life sciences, see Robertson (1983), Hargrove (1998), Wastney et al. (1999), and Novotny et al. (2003).

Compartmental analysis (Foster and Boston, 1983; Green and Green, 1990a,b; Wastney et al., 1999), and in particular model-based compartmen-tal analysis, is the form of mathematical modeling that we will focus on here. Another form of compartmental analysis, empirical compartmental analysis, has also been used in the vitamin A field; see Green and Green (1990a,b) for theoretical background and Green et al. (1987) for an application. In nutrition, model-based compartmental analysis has been used to model mineral uptake and utilization (Birge et al., 1969; Pinna et al., 2001; Wastney et al., 1996), lipoprotein metabolism (Adiels et al., 2005; Berman et al., 1982), glucose homeostasis (Malmendier et al., 1974), digestion and absorption (Moore-Colyer et al., 2003), and vitamin kinetics (Coburn et al., 2003; Green et al., references cited herein); see Green and Green (1990a) for other examples.

Compartmental modeling involves the representation of a system by a finite number of homogenous states and lumped processes, called compartments, which interact by means of material exchange (DiStefano and Landaw, 1984; Green and Green, 1990a; Wastney et al., 1999, pp. 7-9) (Fig. 2). Compart-mental analysis assumes that the system under study exhibits deterministic behavior, meaning that the future state of the system may be predicted based on its current state and future input; no probabilistic effects are included (Carson et al., 1983, p. 56). Compartmental modeling provides both quantitative and predictive information about the system of interest, as well as unique insights into underlying mechanisms and metabolic processes that govern the system's kinetic behavior. The approach is unique in that it allows the researcher to investigate aspects of a system that might be difficult to study experimentally; model predictions may also provide unexpected insights into the metabolism of the compound of interest, and they may lead to the generation of new hypotheses and/or experiments (Green, 1992; Green and Green, 1990a).

The overall goal of model-based compartmental analysis is to describe and quantify the kinetics and often the dynamics of a particular system of interest (Green and Green, 1990a). Note that, although all of the work discussed here will be based on in vivo studies, model-based compartmental analysis can also be fruitfully applied to data collected from in vitro systems (see Blomhoff et al., 1989 for an example). To use model-based compartmental analysis, the investigator begins by formalizing a conceptual

FIGURE 2. A three-compartment mammillary model. Compartments are shown as circles; movement between compartments is represented by arrows and quantified by fractional transfer coefficients [L(I,J)s or the fraction of compartment J's retinol transferred to compartment I per unit time]. U(1) represents input of newly absorbed dietary retinol, the asterisk shows the site of introduction of the tracer (typically plasma), and the triangle indicates that this compartment is a site of sampling.

compartmental model based on what is known and theorized about the system under study (Fig. 2); the model includes not just the compartmental structure but also estimates of the fractional transfer coefficients (see later) that quantify movement between compartments and out of the system. Then an appropriate in vivo experiment is designed and data are collected. In the case of retinol, a stable- or radioisotopic tracer in a physiological form (i.e., as part of the plasma transport complex or in chylomicrons) is administered intravenously; alternately, the tracer may be solubilized in oil and given orally. After dose administration, tracer concentration in plasma (and perhaps in organs and excreta) is followed over time (Fig. 3). Plasma tracer data (as fraction of administered dose), along with other relevant information (e.g., initial conditions, tracee mass, and sites of input), are analyzed using appropriate software. Here we will concentrate on the Simulation, Analysis and Modeling computer program (SAAM).

First introduced in 1962, SAAM mathematically compares the proposed model to the data and provides statistical information about model solutions (Berman and Weiss, 1978; Wastney et al, 1999, pp. 95-138; www.WinSAAM. com; Stefanovski et al., 2003); thus the modeler is able to evaluate the closeness of fit for each solution. Solution results are evaluated by comparing the observed and calculated data both graphically and numerically, with adjustments being made to the model until a close fit between the observed and calculated values is obtained. During model development, the known or suspected physiology and biochemistry of the system are kept in mind so that a physiologically reasonable model is developed (Green and Green, 1990a,b).

Time (days)

FIGURE 3. Plasma retinol kinetics versus time. Hypothetical plasma tracer response curve after intravenous administration of [3H]retinol-labeled plasma to a rat with high liver vitamin A stores. Symbols represent observed data and the line is a model simulation. The inset shows the first day's data on an expanded scale.

Time (days)

FIGURE 3. Plasma retinol kinetics versus time. Hypothetical plasma tracer response curve after intravenous administration of [3H]retinol-labeled plasma to a rat with high liver vitamin A stores. Symbols represent observed data and the line is a model simulation. The inset shows the first day's data on an expanded scale.

To determine the simplest model that will provide an adequate fit to a particular data set, multiple models are tested, with model complexity being increased only when it results in a significant improvement in the weighted sum of squares (Landaw and DiStefano, 1984). Once a satisfactory fit is obtained, weighted nonlinear regression analysis is applied using the SAAM program to obtain best fit values for the fractional transfer coefficients describing movement between compartments (Fig. 2) as well as their statistical uncertainty (i.e., fractional standard deviation); then other kinetic parameters can be calculated (Table I). Typically, the modeling analysis assumes that the system is in a steady state, although the SAAM program can accommodate more complex situations (see later). Overall, the structure of the compartmental model provides a visual picture of how the compartments are linked, whereas the model-based parameters provide information about transfer between compartments, recycling, movement out of the system, and compartment masses.

In this laboratory, we have used model-based compartmental analysis to delineate the underlying mechanisms that control vitamin A homeostasis. Specifically, we have used modeling to study vitamin A kinetics in plasma and tissues under different conditions, including various levels of vitamin A status and after treatment with different exogenous factors, to better understand vitamin A utilization and whole-body vitamin A metabolism. The subsequent sections will review these studies and integrate the results to provide a better understanding of the kinetic behavior of the vitamin A system.

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Many women who have recently given birth are always interested in attempting to lose some of that extra weight that traditionally accompanies having a baby. What many of these women do not entirely realize is the fact that breast-feeding can not only help provide the baby with essential vitamins and nutrients, but can also help in the weight-loss process.

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