The model is a mechanistic combined toxicokinetic-toxicodynamic (TK/TD) and growth model for the aquatic macrophytes Lemna spp. The model simulates the development of Lemna biomass under laboratory and environmental conditions and was developed by Schmitt et al. (2013). Growth of the Lemna population is simulated on basis of photosynthesis and respiration rates which are functions of environmental conditions. The toxicodynamic sub-model describes the effects of growth-inhibiting substances by a respective reduction in the photosynthesis rate based on internal concentrations. This is the historical version of the Lemna model. For current uses, we recommend the Lemna (SETAC) model, which is a more recent version of the Schmitt model.
Lemna_Schmitt(param, init)Lemna_SchmittThold(param, init)
an S4 object of type LemnaSchmitt
optional named list or vector of model parameters
optional named numeric vector of initial state values
Lemna_SchmittThold(): model variant with cumulative exposure threshold
The following list describes the default names and standard units of the model's state variables:
BM, g_dw/m2, dry weight biomass per square meter
E, -, effect [0,1]
M_int, ug, internal toxicant mass
AUC, ug/L, cumulative exposure (only for LemnaThreshold model)
Biomass (BM) and internal toxicant mass (M_int) are initialized to zero by
default. See set_init() on how to set the initial states.
The following model parameters are required:
Fate and biomass
k_phot_fix, logical, TRUE then k_phot_max is not changed by environmental factors, else FALSE
k_phot_max, 1/d, maximum photosynthesis rate
k_resp, 1/d, respiration rate
k_loss, 1/d, rate of loss (e.g. flow rate)
mass_per_frond, g_dw/frond, dry weight per frond
BMw2BMd, g_fw/g_dw, Fresh weight/dry weight
Effect
Emax, -, maximum effect [0,1]
EC50, ug/L, midpoint of effect curve
b, -, slope of effect curve
Toxicokinetics
P_up, cm/d, Permeability for uptake
AperBM, cm2/g_dw, A_leaf / d_leaf = 1/d_leaf (for circular disc, d=0.05 cm)
Kbm, -, Biomass(fw) : water partition coefficient
P_Temp, logical, TRUE to enable temperature dependence of cuticle permeability, else FALSE
MolWeight, g/mol, Molmass of molecule (determines Q10_permeability)
Temperature dependence
Tmin, deg C, minimum temperature for growth
Tmax, deg C, maximum temperature for growth
Topt, deg C, optimal temperature for growth
t_ref, deg C, reference temperature for respiration rate
Q10, -, temperature dependence factor for respiration rate
Light dependence
k_0, 1/d, light dependence: intercept of linear part
a_k, (1/d)/(kJ/m2.d), light dependence: slope of linear part
Phosphorus dependence (Hill like dep.)
C_P, mg/L, phosphorus concentration in water
CP50, mg/L, phosphorus conc. where growth rate is halfed
a_p, -, Hill coefficient
KiP, mg/L, p-inhibition constant for very high p-conc.
Nitrogen dependence (Hill like dep.)
C_N, mg/L, nitrogen concentration in water
CN50, mg/L, n-conc. where growth rate is halfed
a_N, -, Hill coefficient
KiN, mg/L, n-inhibition constant for very high p-conc.
Density dependence
BM50, g_dw/m2, cut off BM
The Lemna_SchmittThold model requires the following additional parameter:
threshold, ug/L, cumulative exposure threshold
Besides exposure, the Lemna model requires two environmental properties as
time-series input: global radiation (rad, kJ/m2.d) and temperature (temp, deg C).
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.
Supported effect endpoints include BM (biomass) and r (average growth rate during simulation). The effect on biomass is calculated from the last state of a simulation. Be aware that endpoint r is incompatible with frond transfers.
Default values for parameter boundaries are set for all parameters by expert judgement, for calibration purposes. Values can be access from the object, and defaults overwritten.
Simulation results will contain two additional columns besides state variables:
C_int, ug/L, internal concentration of toxicant
FrondNo, -, number of fronds
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(), see examples below. nout=1
enables reporting of internal concentration (C_int), nout=14 enables all
additional outputs, and nout=0 will disable additional outputs.
The available output levels are as follows:
nout >= 1: C_int, internal concentration (ug/L)
nout >= 2: FrondNo, number of fronds (-)
nout >= 3: C_int_u, unbound internal concentration (ug/l)
Growth and TK/TD
nout >= 4: BM_fresh, fresh weight biomass (g_fw/m2)
nout >= 5: k_photo_eff, current photosynthesis rate (1/d)
nout >= 6: k_resp_eff, current respiration rate (1/d)
nout >= 7: f_Eff, toxic effect factor (-)
nout >= 8: P_up_eff, current permeability for uptake (cm/d)
Environmental variables
nout >= 9: actConc, current toxicant concentration in surrounding medium (ug/L)
nout >= 10: actTemp, current environmental temperature (deg C)
nout >= 11: actRad, current environmental radiation (kJ/m2.d)
Derivatives
nout >= 12: d BM/dt, current change in state variable BM
nout >= 13: d E/dt, current change in effect
nout >= 14: d M_int/dt, current change in internal toxicant mass
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.
Models supporting biomass transfer can be instructed to move a fixed amount of biomass to a new medium after a period of time. This feature replicates a procedure occurring in e.g. Lemna effect studies and may be necessary to recreate study results.
The biomass transfer feature assumes that always a fixed amount of
biomass is transferred. Transfers can occur at any fixed point in time or
in regular intervals. During a transfer, the biomass is reset to the
transferred amount and additional compartments can be scaled 1:1 accordingly,
to e.g. reflect the change in internal toxicant mass when biomass is modified.
Transfer settings can be modified using set_transfer().
If a transfer occurs, simulation results of that time point will report the model state
before the transfer. Be aware that if transfers are defined using the
interval argument, the transfers will always occur relative to time point
zero (t = 0). As an example, setting a regular transfer of seven days,
interval = 7, will result at transfers occurring at time points which are
integer multiplicates of seven, such as t=0, t=7, t=14 and so forth.
The starting and end times of a scenario do not influece when a regular
transfer occurs, only if it occurs.
Constructors to ease creation of scenarios based on the Lemna model by
Schmitt et al. (2013).
A variant of this Lemna model, Lemna_SchmittThold(), provides an additional
cumulative exposure threshold parameter. The Lemna biomass stops growing
if the integral of exposure over time exceeds the threshold. The integral
of exposure is internally accounted for by an additional state variable
AUC (Area Under Curve).
Schmitt W., Bruns E., Dollinger M., and Sowig P., 2013: Mechanistic TK/TD-model simulating the effect of growth inhibitors on Lemna populations. Ecol Model 255, pp. 1-10. tools:::Rd_expr_doi("10.1016/j.ecolmodel.2013.01.017")
Lemna-models, Macrophyte-models, Transferable, Scenarios
Other Lemna models:
Lemna-models,
Lemna_SETAC()
Other macrophyte models:
Lemna_SETAC(),
Macrophyte-models,
Myrio(),
Myrio_log()