Food Web Model Philosophy

One of the key challenges in ecotoxicology is to understand the interaction between the chemistry of the contaminant and the biology (including the physiology and ecology) of the organism. Food-web bioaccumulation models are among the important tools to document our knowledge of this interaction and to apply our knowledge to practical problems. These models incorporate a number of fundamental chemical and biological principles. In addition, to apply the model in an acceptable fashion, the model calculations have to be conducted according to a set of modeling and management principles. These fundamental principles need to be understood to recognize the application domain of the models and the strengths and limitations of the model outcomes. This section discusses these principles briefly. Current food-web bioaccumulation models show good agreement on the main principles included in the models. To avoid duplication, this section refers to the Gobas models for specific examples.

Chemical Equilibrium

One of the most important chemical principles embedded in food-web bioaccumulation is the natural tendency of chemicals to involve in net transport with the goal to achieve a chemical equilibrium. Equilibrium is thermo-dynamically defined as a situation where the chemical potential p or the fugacity f (Pa) of the chemical in two or more media (e.g., water and fish) is the same, that is,

— fw where and pw and fB and fw are the chemical potentials and fugacities in the organism or biota and water, respectively. The fugacity of a chemical, f (in units of pascal), in a given phase is related to the molar concentration C (in mol m-3) by the fugacity capacity Z (in mol m-3 Pa-1) of the phase in which the chemical is solubilized:

The fugacity capacity Z is compound and phase specific, and it represents the capacity of that phase to sorb and retain a given chemical within its matrix. Net passive chemical transport occurs from the medium of high fuga-city (e.g., water) to the medium of low fugacity (e.g., an organism) until the fugacities in both media are equal (Figure 1), at which the chemical concentrations in both media (e.g., in the organism, CB, and the water, Cw (mol m-3)) are related by the chemical's partition coefficient Kbw (unitless), that is,

The application of this principle was the foundation of some of the first bioaccumulation models for organic contaminants, in which the BCF (typically expressed in units of l kg-1) of the chemical in fish was found to be

Fugacity format Dw Dw


Mass balance



Fugacity format Dw Dw

Concentration format k1 k2

k1 k2

Figure 1 Conceptual diagrams illustrating key chemical principles (equilibrium, mass balance, magnification, and integration of processes) at the organism level.

related to the octanol-water partition coefficient (KoW); for example, one such model estimates the BCF in fish as 0.048KOW, where 0.048 is an estimate of the average lipid content of fish.

Mass Balance

The second principle that is embedded in the food-web bioaccumulation models is that physical transport processes (e.g., gill ventilation flows, ingestion rates) and reaction processes (e.g., metabolic transformation) may interfere with the chemical's natural tendency to achieve a chemical equilibrium. This is captured by the mass balance equation where the net flux of chemical NB (in units of mol day- ) into an organism is the sum of the chemical fluxes into and out of the organism. When expressed in fugacity format, the flux is the product of the transport parameter, D, and the fugacity of the chemical in the medium, f, in which the transport occurs. The transport parameter can represent transport by molecular diffusion, in which case D is the product of the mass transfer coefficient S (in m day-1), the area of diffusion A (m2), and the Z in which the diffusion occurs, that is, D equals S-A-Z. D can also represent an advective transport process (e.g., gill ventilation rate or ingestion) described by a flow rate G (m day- ) in which case D equals G- Z. D can further describe a transformation process (e.g., metabolic transformation), often described by a transformation rate constant (day-1), in which case D equals k- V- Z, where V is the volume of the medium in which transformation occurs. One of the attractive features of the fugacity approach is that it separates biological variables such as the flow rate (e.g., gill flow rate, ingestion rate, urine excretion rate) and chemical variables such as Z which control the physical-chemical partitioning of the chemical.

For example, for a chemical that is absorbed by a fish with a volume VB (in m3) from the water (via the gills) and eliminated to the water (via the gills) as well as being metabolized in the organisms, the mass balance equation is

where t is time (day). When expressed in terms of concentrations and kinetic rate constants, the same balance is given by

where i1, i2, and iM are rate constants (in units of day-1) for chemical uptake from water, elimination to the water, and metabolic transformation, respectively (Figure 1). The units of the rate constant can vary depending on the units selected for the concentration in the media involved. The rate constants can also be expressed in terms of

fugacity-based transport parameters, for example, k1 equals DW/(VB? ZW), which allows one to use both concepts and benefit from the fugacity-based approach to distinguish between biological and chemical variables in the model.

Equations [4] and [5] illustrate that after a long-term exposure, when a steady state is reached (i.e., N = 0), the chemical fugacities in water and organism are no longer equal, that is, fB/fW = DW/(DW + Dm), with the fugacity in the organism being smaller than that in the water. In concentration format, this equates to CB/CW = kx/(k2 + kM).


One of the key principles in a food-web bioaccumulation model is the biomagnification effect, which causes the fugacity and concentration of the chemical to increase with increasing trophic level. This process can lead to food-web magnification of the chemical when this process occurs at each predator-prey interaction in the food web. Food-web biomagnification alone can produce a 10 000100000-fold increase in lipid-normalized concentration of a bioaccumulative substance. Biomagnification is of ecotoxicological significance because it can cause organisms at higher trophic levels to be exposed to high concentrations, which can produce toxicological effects or high risk levels. In our models and those of Mackay and co-authors, this occurs as a result of food absorption and digestion and can be described by the following mass balance equation:

where fD is the chemical fugacity in the diet, Dd is the dietary ingestion rate, which is the product of the dietary ingestion rate GD (m3 day-1) and the fugacity capacity of the diet ZD (i.e., GD-ZD), and DF is the fecal egestion rate of the chemical (Figure 1). Hence, at steady state (Nb = 0), it follows that

illustrating that the fugacity in the organism fB) exceeds that in its diet fD) as a result of dietary uptake because the feeding rate GD exceeds the fecal excretion rate GF due to food absorption and ZD exceeds ZF because of food digestion which leaves the feces depleted of lipids, proteins, and other food constituents that give the food its high fugacity capacity. This magnification effect is approximately 8 times in fish, but much higher in mammals, birds, and humans with a more efficient digestive system. A graphical presentation of the magnification effect is presented in Figure 2. However, for the gastrointestinal magnification effect to cause biomagnification, it is key that the combined rate of chemical elimination due to metabolic transformation and excretion in the organisms is slow. However, if the combined rate of

Figure 2 Conceptual diagram depicting the mechanism of biomagnification. Top panel: Predator with a chemical fugacity of 1 Pa in water with a chemical fugacity of 1 Pa consuming prey with a chemical fugacity of 1 Pa. Middle panel: Because predator and prey are at the same fugacity (i.e., no net passive uptake of chemical), food absorption (which reduces the volume of food in the gastrointestinal tract) and food digestion (which reduces the fugacity capacity of the intestinal contents) produce an increase in chemical fugacity in the gastrointestinal tract which leads to net uptake of chemical in the predator Bottom panel: Net chemical uptake will cause a fugacity in the predator that exceeds the fugacity in its prey (i.e., biomagnification) as long as the combined chemical elimination rate by metabolic transformation, elimination to water, and growth dilution are slow.

Figure 2 Conceptual diagram depicting the mechanism of biomagnification. Top panel: Predator with a chemical fugacity of 1 Pa in water with a chemical fugacity of 1 Pa consuming prey with a chemical fugacity of 1 Pa. Middle panel: Because predator and prey are at the same fugacity (i.e., no net passive uptake of chemical), food absorption (which reduces the volume of food in the gastrointestinal tract) and food digestion (which reduces the fugacity capacity of the intestinal contents) produce an increase in chemical fugacity in the gastrointestinal tract which leads to net uptake of chemical in the predator Bottom panel: Net chemical uptake will cause a fugacity in the predator that exceeds the fugacity in its prey (i.e., biomagnification) as long as the combined chemical elimination rate by metabolic transformation, elimination to water, and growth dilution are slow.

chemical elimination is high, then a high chemical concentration in the organism cannot be maintained and the chemical will not biomagnify. Chemicals which are predominantly absorbed via the diet and subject to a high rate of chemical elimination due to metabolic transformation and other excretion rates, will exhibit concentrations in the organisms of a food chain that decline with increasing trophic level. This phenomenon is sometimes referred to as trophic dilution, which is the opposite of biomagnification.


It is important to recognize that body burdens of contaminants in animals are the combined result of a number of chemical uptake and elimination processes acting together. In water-respiring organisms (e.g., fish), the most important processes are respiratory uptake via the gills and body surface area, dietary uptake and elimination via the respiratory surface, fecal egestion, and metabolic transformation. Growth of the animal is also often viewed as an elimination process and referred to as 'growth dilution' although no chemical is actually excreted or transformed. An increase in body mass has a 'diluting' effect on the chemical mass in the organisms. Hence, growth is often treated as an elimination process in bioaccumulation models. A model for bioaccumulation in fish can therefore be formulated as:

- [kj + kE + kM + ¿g)'Vb'Cb where k1, kD, k2, kE, and ¿G are the rate constants (in units of day-1 if concentrations are in mol m-3) for chemical uptake via the respiratory area (k1), uptake via food ingestion (kD) and elimination via the respiratory area (k2), excretion into egested feces (kE), metabolic transformation (kM) and growth dilution (£g); Pi is the fraction of the diet consisting of prey item ;, CD,; is the concentration of chemical (gkg-1) in prey item i, k2 is the rate constant (day- ) for chemical elimination via the respiratory area (i.e., gills and skin), kE is the rate constant (day- ) for chemical elimination via excretion into egested feces, and kM is the rate constant (day-1) for metabolic transformation of the chemical (Figure 1). A similar approach can be followed to develop models for other species. For example, for mammals (Figure 3), we have used

- [ko + kE + ku + kG + kp + kL + ¿m)'Vb'Cb where CA is the gaseous chemical concentration in the air and kA, ko, ku, kp, and kL are the rate constants (in units of day-1 if concentrations are in molm-3) for chemical

Growth dilution

Growth dilution

Figure 3 Chemical uptake and elimination processes included in the bioaccumulation model for organic contaminants in mammals.

uptake via the respiratory area (k1), elimination via the respiratory area (ko), excretion into urine (ku), production in female animals (kp), and lactation in female animals (kL).

This modeling approach is based on several key assumptions. First, it is assumed that the chemical is homogeneously distributed within the organism as long as differences in tissue composition and phase partitioning are taken into account. There is considerable evidence, especially for poorly metabolizable substances after long exposure periods, that supports this assumption. However, since the model is not designed to estimate concentrations in specific organs, the model is best applied in situations where the mass or concentration of the chemical in the whole organism is of interest. Internal physiological based pharmacokinetic (PBPK) models are more suitable to estimate concentration differences between various parts of the organism. Second, it is assumed that the organism can be described as a single compartment in its exchange with its surrounding environment. The one-compartment model for an organism is best applied in situations where variations in concentration over time are relatively slow or of secondary concern. A third assumption of the model concerns chemical elimination associated with sexual reproduction and offspring production. Examples are egg deposition or sperm ejection in fish and parturition in mammals. Studies in fish have shown that lipid-normalized concentrations of many persistent organic chemicals in eggs and adult female fish are often approximately equal. This implies that while egg deposition transfers a significant fraction of the chemical body burden from the adult female fish into the eggs, the lipid-equivalent concentration within the organism remains the same. The mechanism in the model by which egg deposition can lower the internal concentration in the organism compared to fish that do not produce eggs (e.g., male fish) is through growth dilution associated with the formation of eggs in the fish. Formation of eggs produces extra tissue in which the chemical resides, hence reducing the chemical's concentration. offspring production in female mammals and birds follow a similar mechanism. Equations [8] and [9] illustrate that this growth dilution effect is counteracted by uptake of chemical from water and the diet and that the balance of these processes controls the ultimate concentration in the organism.

The practical application of eqns [8] and [9] to environmental pollution problems is often limited by access to time-dependent model input parameter values. Hence, for the model to become useful, it is often further simplified by applying a steady-state assumption (WB = 0). The steady-state assumption transforms eqn [9] into

Cb = (ki-Cw + kD.i ^ (P,-Cd,,))/[kj + ¿e + ¿m + kG) [10]

The steady-state assumption is reasonable for applications to field situations where organisms have been exposed to the chemical over a long period of time, often throughout their entire life. It applies best to chemicals that are subject to relatively fast exchange kinetics (e.g., lower-if0w substances, small organisms), as steady state is achieved rapidly in these situations. It should be used with caution in situations where the exchange kinetics are very slow (e.g., slowly metabolizable chemicals of high KoW (i.e., larger than 107 5) in large, lipid-rich organisms), because steady state takes a long time to achieve. In cases where changes in concentrations with the age of the organism are of interest, it is possible to introduce various age classes of the species and apply the steady-state model to each age class independently. One of the implications of applying a steady-state assumption is that the growth of the organism needs to be expressed as a growth rate constant ¿g, which is dWB/(WB?dt) and assumes that over the period of time that the model applies, the growth of the organism can be represented by a constant fraction of the organism's body weight WB. The main driving forces of the kinetic bioaccumulation model are: (1) the chemical partitioning of chemical between water and the organism, represented by k1/k2; (2) the dietary magnification of chemical, represented by kD/kE; and (3) the combined rate of chemical elimination via metabolic transformation, growth dilution. It is noteworthy that any error in the estimation of the respiration rate, which affects both k1 and k2, has a tendency to cancel out, causing the ratio of k1 and k2 not to be affected. The same argument applies to the ingestion rate, which is related to the egestion rate. This gives the model some remarkable robustness. However, errors in respiration and ingestion rates do affect the relative contribution of the various uptake and elimination pathways and hence the model outcome.

Trophic Interactions

Describing the transfer of contaminants between organisms due to feeding interactions is challenging as food webs are typically complex and vary as a function of time and space. The role of food-web models is to simplify this process to a level that is understandable and can provide useful information. Since the scope of the model is typically limited to several key species, it is often sufficient to include only the most relevant trophic interactions relating to these species of interest (Figure 4). Also, because chemical concentrations in organism lipid tissues tend to increase significantly between trophic positions but considerably less between organisms that occupy a similar trophic level, it is often possible to 'lump' species of comparable trophic guilds. The latter should be done with caution as some species exhibit specific feeding behaviors that cannot be generalized to other organisms.

In the development of a food-web structure for modeling the bioaccumulation of contaminants, some basic rules of thumb can be suggested:

1. Include species of primary management interest.

2. Include species that can be considered residents of the area of interest unless migrant species are of importance.

3. Include species representing trophic guilds that are of key relevance to the food-web transfer and accumulation of polychlorinated biphenyls (PCBs) in the species of interest. For example, phytoplankton and algae, zooplankton, filter-feeding invertebrates, benthic det-ritovores, juvenile and adult fish, male and female fish-eating birds, and male, female, and juvenile marine mammals.

4. Minimize the number of species included in the model by representing key trophic guilds by one or two species. This is done to simplify the model and make the calculations more transparent.

5. Include species for which empirical concentration data are available. This provides the opportunity to test and ground-truth the model's calculations.

The structure of the food web represented in food-web models is typically subject to uncertainty. As a test to check whether the trophic structure of the model is adequate for chemical bioaccumulation modeling, it is often useful to explore the relationship between the trophic level of the species of the model as assigned by the feeding relationships used in the model, and stable nitrogen isotope ratios measured in samples of the species included in the model (Figure 5). Stable isotope ratios provide an empirical measure of the trophic status of the organism, with stable isotope ratios increasing with increasing trophic level. A good correlation between trophic level and stable nitrogen isotope ratios provides confidence in the structure of the food web for the purpose of food-web bioaccumulation modeling of contaminants.

Energy Balance

While the mass balance principle ensures the conservation of mass in the model, it does not necessarily ensure that an energy balance is maintained. The latter is important to avoid implausible scenarios such as an animal growing faster than it eats. It is therefore important to consider the relationships among the growth rate, the fecal egestion rate, and the feeding rate GD as well as the sorptive capacity of fecal matter ZF and the diet ZD. This can be done by applying a general energy budget, that is,

Immigrant organisms

Immigrant organisms

Figure 4 Illustrative example of a food-web structure used to describe the dynamics of contaminants in a West Coast marine food web that includes resident and migrant species.






Figure 4 Illustrative example of a food-web structure used to describe the dynamics of contaminants in a West Coast marine food web that includes resident and migrant species.

E. Sole


E. Sole


Trophic position

Figure 5 Illustrative example of the application of the relationship between trophic position, derived from gut content studies and stable 15N/14N isotope ratios (<515N) measured in organism tissues, to test the applicability of the trophic structure of the model's food web. A high degree of correlation is indicative of an appropriate food-web structure.

where I is energy ingestion, L is the sum of fecal and urinary losses, P is production, and R is respiration, all expressed in units of energy flux (kJ day-1). These can be converted to mass fluxes (gday-1) by energy-

biomass interconversion ratios. The application of the energy mass balance in the food-web bioaccumulation model makes it possible to include bioenergetic efficiencies in the food-web model. This provides the opportunity to apply the model to a large variety of species for which bioenergetic efficiencies are known. This is an interesting application of the model for ecotoxicological evaluations as it makes it possible to assess which species are most likely to receive highest body burdens of particular contaminants and are potentially at the highest risk.

Model Parametrization

Upon verification of the model's logic and correctness, the model's key state variables are parametrized, by compiling data on chemical properties, environmental conditions, and biological variables. An important consideration is to use data that represent the system of interest as best as possible. This ensures that the concentrations of contaminants calculated by the model can be compared to observed concentrations.

Sensitivity analysis

The purpose of the sensitivity analysis is to provide insight into the relative importance of the various state variables to the outcome of the model. This is useful in the analysis of the internal mechanics of the model. It can be used to characterize potential errors in the model and to develop a better understanding of the interaction of the processes that control the behavior of the contaminant in the food web. In multiparameter models like the food-web model, it is important to keep in mind that the sensitivity of each model variable is a function of other state variables of the model. There are various methods that can be used to conduct sensitivity analysis. The simplest technique involves the variation of a particular parameter and recording its effect on the model output; for example, the contaminant concentration in a target organism. Other techniques use stochastic procedures such as Monte Carlo simulation (MCS) to express the relative importance of various state variables and parameters of the model.

Calibration and hypothesis testing

Model calibration is a technique in which the model parametrization and/or the model structure are altered to produce a better agreement between observations and model predictions. Calibration methods include 'fine-tuning' model parameters and the application of concentration observations in the basic structure of the model. While the application of model calibration depends on the objective of the model, the model strategy, and the state of knowledge and information about the contamination problem at hand, we found that for estimating hydrophobic organic chemical concentrations in organisms of food webs, there is rarely a need for model calibration as long as the model is used within its application domain. The food-web modeling approach described in this article is a mechanistic model and can be used without including empirical chemical concentration data. In many cases though, there is a need to check whether the model calculations are consistent with available empirical data such that confidence in the model is gained. This can be done in a 'hypothesis-testing' approach where the model outcomes are compared with independent data (data not used in the construction of the model). For example, if the preferred model outcome is the chemical concentration in a particular species (e.g., fish), this involves the comparison of observed and predicted concentrations in that fish species. Various measures can be used to express the degree to which the model outcomes match the observations. In the past, we have used the mean model bias MB to express the central tendency of the model:

where CP,¿ and Co,¿ are the concentrations of the chemical in a particular species that are respectively predicted by the model and observed in the field and i can refer to the number of observations (i.e., for a particular chemical in a particular species), number of chemicals (in a particular species), or a number of species (for a particular chemical substance). In essence, MB is the geometric mean (assuming a log-normal distribution) of the ratio of predicted and observed concentrations. MB is a measure of the systematic over- (MB>1) or underprediction (MB<1) of the model. It should be stressed that in the calculation of MB, over- and underestimations have a tendency to cancel out. Hence, it describes the central tendency of the model outcome. Variability in the over- and underestimation of measured values can be represented by the 95% confidence interval of MB. Due to the lognormal distribution of the ratio of predicted and observed BSAFs, this variability can be expressed as a factor (rather than a term) of the geometric mean. For example, if the 95% confidence interval of the MB is 3, it means that 95% of the predicted/observed concentration ratios are found between MB/3 and MB x 3.

Uncertainty analysis

The role of the 'uncertainty analysis' is to assess the error in the model calculations. The uncertainty analysis is important because the magnitude of the model needs to be considered when interpreting the results of the model calculations for management purposes. One of the most established techniques for conducting uncertainty analysis of models is MCS, which calculates the effect of inherent error in the model state variables and parameters on the model outcome. This methodology is based on the representation of the model state variables by statistical distributions rather than point estimates. The distribution represents the uncertainty in the value of the model state variable used in the model. The distribution expresses how the state variables may vary due to geographical location, time of the year, differences in behavior among individuals of a species, and other factors. In MCS, these distributions are repeatedly sampled and the sampled values are used in the model to produce a distribution of model outcomes (e.g., chemical concentrations in fish). This distribution of model results represents the variability in the model outcome due to variability and error in the model's state variables (temperature, organic carbon content, lipid contents, etc.). It represents the model uncertainty. The uncertainties in all state variables contribute to the magnitude of the range of model outcomes; however, the contributions are not necessarily additive. The uncertainty calculated through MCS has a strong theoretical foundation. However, it is subject to difficulties associated with the characterization of errors in model parameters and it cannot include errors in model architecture.

One of the key requirements for a meaningful MCS is that the model state variables included in the MCS are independent. In a food-web bioaccumulation, this requirement can pose difficulties. For example, for a predator with three prey items, uncertainty in the proportion of prey item 1 consumed by the predator has direct consequence for the proportion of the other two diet items in the predator's prey. Hence, diet item proportions or feeding preferences are typically not independent. Other examples of related state variables are animal size, growth rate, lipid content, and feeding rate. Also, regression coefficients in regression equations introduce co-dependency. Before attempting to apply MCS to food-web bioaccumulation models, it is important to ensure that in the model structure chosen the model state variables are indeed independent since lack of dependence creates biologically implausible model outcomes that should not be considered. A second requirement for MCS is a realistic characterization of the uncertainty in each state variable. The latter is not always available for all model state variables and parameters. For example, feeding preferences derived from gut content studies contain considerable uncertainty that is often not characterized.

An alternative method for determining model uncertainty is to use calculated differences between observed and predicted chemical concentrations. This method applies the mean model bias and its 95% confidence interval to the model outcome (e.g., concentration) to predict a distribution of concentrations that includes 95% of the observed data. This method requires that observed concentration data are available. If this is not the case, it is sometimes possible to use uncertainty calculations from application of the model to other systems as a measure of model uncertainty. The application of available site-specific concentration data to characterize model uncertainty has the advantage that estimates of model uncertainty are grounded in empirical observations. However, this method is subject to the limitations of the sampling programs used to obtain the contaminant concentrations. For example, in larger systems, monitoring programs may only have collected samples from a subpopulation of the larger population of a particular species. In that case, it is possible that the distribution of the concentrations in the sampled organisms does not accurately represent the actual distribution of chemical concentrations in the population of the system. In such cases, the model uncertainty may be underestimated.

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