Mathematical relationships between metrics of chemical bioaccumulation in fish

<jats:title>Abstract</jats:title><jats:sec><jats:label /><jats:p>Five widely used metrics of bioaccumulation in fish are defined and discussed, namely the octanol–water partition coefficient (<jats:italic>K</jats:italic><jats:sub>OW</jats:sub>), bioconcentration factor (BCF), bioaccumulation factor (BAF), biomagnification factor (BMF), and trophic magnification factor (TMF). Algebraic relationships between these metrics are developed and discussed using conventional expressions for chemical uptake from water and food and first‐order losses by respiration, egestion, biotransformation, and growth dilution. Two BCFs may be defined, namely as an equilibrium partition coefficient <jats:italic>K</jats:italic><jats:sub>FW</jats:sub> or as a nonequilibrium <jats:italic>BCF</jats:italic><jats:sub>K</jats:sub> in which egestion losses are included. Bioaccumulation factors are shown to be the product of the <jats:italic>BCF</jats:italic><jats:sub>K</jats:sub> and a novel equilibrium multiplier <jats:italic>M</jats:italic> containing 2 ratios, namely, the diet‐to‐water concentration ratio and the ratio of uptake rate constants for respiration and dietary uptake. Biomagnification factors are shown to be proportional to the lipid‐normalized ratio of the predator/prey values of <jats:italic>BCF</jats:italic><jats:sub>K</jats:sub> and the ratio of the equilibrium multipliers. Relationships with TMFs are also discussed. The effects of chemical hydrophobicity, biotransformation, and growth are evaluated by applying the relationships to a range of illustrative chemicals of varying <jats:italic>K</jats:italic><jats:sub>OW</jats:sub> in a linear 4‐trophic‐level food web with typical values for uptake and loss rate constants. The roles of respiratory and dietary intakes are demonstrated, and even slow rates of biotransformation and growth can significantly affect bioaccumulation. The <jats:italic>BCF</jats:italic><jats:sub>K</jats:sub>s and the values of <jats:italic>M</jats:italic> can be regarded as the fundamental determinants of bioaccumulation and biomagnification in aquatic food webs. Analyzing data from food webs can be enhanced by plotting logarithmic lipid‐normalized concentrations or fugacities as a linear function of trophic level to deduce TMFs. Implications for determining bioaccumulation by laboratory tests for regulatory purposes are discussed. <jats:italic>Environ Toxicol Chem</jats:italic> 2013;32:1459–1466. © 2013 SETAC</jats:p></jats:sec>