Algae_TKTD: Algae model variant (Weber et al. 2012) with scaled damage

In cvasi: Calibration, Validation, and Simulation of TKTD Models

Algae model variant (Weber et al. 2012) with scaled damage

Description

The model is a mechanistic combined toxicokinetic-toxicodynamic (TK/TD) and growth model for algae. The model simulates the development of algal biomass under laboratory and environmental conditions. The growth of the algae population is simulated on the basis of growth rates, which are dependent on environmental conditions (radiation, temperature and phosphorus). The model is a variant of the Algae_Weber() model (Weber 2012) as cited in EFSA TKTD opinion (2018). This Algae model, Algae_TKTD(), provides an additional possibility (probit) to simulate the dose-response curve and considers a scaled internal damage instead of the external concentration.

Usage

Algae_TKTD()

Value

an S4 object of type AlgaeTKTD

State variables

The model has four state variables:

• A, Biomass (µg fresh wt/mL, cells/mL *10^4)

• Q, Mass of phosphorous internal (µg P/µg fresh wt)

• P, Mass of phosphorous external (µg P/L)

• Dw, Damage concentration (µg/L)

Model parameters

• Growth model

  • mu_max, Maximum growth rate (d-1)

  • Q_min, Minimum intracellular P (µg P/µg fresh wt)

  • Q_max, Maximum intracellular P (µg P/µg fresh wt)

  • v_max, Maximum P-uptake rate at non-limited growth (µg P/µg fresh wt/d)

  • k_s, Half-saturation constant for extracellular P (mg P/L)

  • m_max, Natural mortality rate (1/d)

  • I_opt, Optimum light intensity for growth (µE/m²/s)

  • T_opt, Optimum temperature for growth (°C)

  • T_max, Maximum temperature for growth (°C)

  • T_min, Minimum temperature for growth (°C)

  • D, Dilution rate (1/d)

  • R_0, Influx concentration of P (mg P/L)

• Concentration response (Toxicodynamics)

  • EC_50, Effect concentration of 50% inhibition of growth rate (µg L-1)

  • b, slope of concentration effect curve at EC_50 (-)

  • dose_resp, shape of the dose response curve (0 = logit, 1 = probit)

• External concentration (Toxicokinetics)

  • kD, dominant rate constant (d-1)

Forcings

The Weber model variant requires two environmental properties as time-series input:

• T_act, temperature (°C), and

• I, irradiance (uE/m²/s).

The following constant default values are used for these properties:

• T_act = 23 °C

• I = 100 uE/m²/s

Forcings time-series are represented by data.frame objects consisting of two columns. The first for time and the second for the environmental factor in question.

Entries of the data.frame need to be ordered chronologically. A time-series can consist of only a single row; in this case it will represent constant environmental conditions. See scenarios for more details.

Simulation output

Simulation results will contain the state variables Biomass (A), mass of internal phosphorous (Q), mass of external phosphorous (P) and the damage concentration (Dw).

It is possible to amend the output of simulate() with additional model quantities that are not state variables, for e.g. debugging purposes or to analyze model behavior. To enable or disable additional outputs, use the optional argument nout of simulate(). As an example, set nout=2 to enable reporting of model derivatives dA and dQ. Set nout=0 to disable additional outputs (default).

The available output levels are as follows:

• nout >= 1: C, external concentration (µg/L)

• nout >= 2: f(T), temperature dependence (-)

• nout >= 3: f(I), light dependence (-)

• nout >= 4: f(Q), nutrient dependence (-)

• nout >= 5: f(Q, P), uptake flow reduction (-)

• nout >= 6: f(C), effect of chemical stressor (-)

• nout >= 7: dA, biomass derivative (µg)

• nout >= 8: dQ, internal phosphorous derivative (mg P/µg fresh wt)

• nout >= 9: dP, external phosphorous derivative (mg P L-1)

• nout >= 10: dDw, damage concentration derivative (µg L-1)

Solver settings

The arguments to ODE solver deSolve::ode() control how model equations are numerically integrated. The settings influence stability of the numerical integration scheme as well as numerical precision of model outputs. Generally, the default settings as defined by deSolve are used, but all deSolve settings can be modified in cvasi workflows by the user, if needed. Please refer to e.g. simulate() on how to pass arguments to deSolve in cvasi workflows.

Some default settings of deSolve were adapted for this model by expert judgement to enable precise, but also computationally efficient, simulations for most model parameters. These settings can be modified by the user, if needed:

• hmax = 0.1
  Maximum step length in time suitable for most simulations.

Model history and changes

• cvasi v1.5.0

  • Support for simulating flow-through conditions by introducing new parameters D and R_0 and adapting the ODEs according to the Algae_Weber model.

  • ODE of external phosphorous concentration P corrected, which contained an erroneous growth term before.

  • Response functions added to optional simulation outputs, order of output levels modified.

References

Weber D, Schaefer D, Dorgerloh M, Bruns E, Goerlitz G, Hammel K, Preuss TG and Ratte HT, 2012. Combination of a higher-tier flow-through system and population modeling to assess the effects of time-variable exposure of isoproturon on the green algae Desmodesmus subspictatus and Pseudokirchneriella subcapitata. Environmental Toxicology and Chemistry, 31(4), 899-908. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/etc.1765")}

See Also

Scenarios, Transferable

Other algae models: Algae-models, Algae_Simple(), Algae_Weber()

cvasi documentation built on Sept. 11, 2025, 5:11 p.m.