Uptake, elimination and growth dilution calculation

Uptake, elimination and growth dilution can be assessed in exposure experiments in which k1, k2 and kg can be derived (respectively rate constants for uptake, elimination and growth dilution). For this purpose, samples from organisms need to be collected at specific time points and their ENM concentrations quantified. To assess potential growth dilution, the weight of the individuals also needs to be measured (for details, see the bio-uptake assessment procedure). Accumulation experiments generally consist of an uptake phase, in which the organisms are exposed to ENMs, followed by an elimination phase in which the organisms are transferred to clean medium. Different algorithms apply to the different phases:

1. Uptake phase

\(C_{org} = C_{env} \ast k_{1}\ast SF\ast t+ C_{env}\ast\frac{k_1}{k_2+k_g} \ast (1 - e^{-(k_{2}+kg)\ast t})\ast (1-SF)\)

2. Elimination phase 

\(C_{org} = C_{env}\ast k_{1}\ast SF\ast t_m+C_{env}\ast \frac{k_1}{k_2+k_g} \ast (1 - e^{-(k_{2}+kg)\ast t})\ast e^{-(k_2+k_g)\ast (t-t_m)} )\ast (1-SF)\)

Where variables are defined as follows: Corg: concentration in organism (\(\mu g/g\)), Cenv: concentration in exposure medium (\(\mu g/g\)), SF: stored fraction (0≤SF≤1; unitless), k1: uptake rate constant (g * g-1 * day-1), k2: elimination rate constant (day-1),  kg: growth rate constant (day-1), t: days of overall experiment, tm: day of transfer from spiked soil to clean soil (i.e. from exposure phase to elimination phase).


kg can be derived by regressing the biomass of the organisms to the date (be aware that animals may also lose weight, which would result in negative kg)

\(Biomass = \alpha \ast exp({k_{g}\ast day}) \)

kg can be included in the equations, which can then be used to regress the concentrations of the ENMs in the organisms over time, depending on the phase of the experiments. If no growth dilution is apparent, kg can be removed from the algorithms.

Used in




Growth dilution


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Read also


van den Brink N.W. et al. 2019. Tools and rules for modelling uptake and bioaccumulation of nanomaterials in invertebrate organisms. Environ Sci Nano 6, 1985-2001. DOI: 10.1039/C8EN01122B

Ardestani, M.M., van Straalen, N.M., van Gestel, C.A.M., 2014. Uptake and elimination kinetics of metals in soil invertebrates: A review. Environmental Pollution 193, 277-295. DOI: 10.1016/j.envpol.2014.06.026



  Nico van den Brink

  Wageningen University

  Email: Nico.vandenbrink@wur.nl