Overview

The central concept within the cvasi package is the scenario. It encapsulates all key properties to run a simulation such as initial state and model parameters. The scenario can be thought of as the collection of data and parameters necessary to reproduce a certain study condition or situation. The set of available scenario properties depends on the model type but may include:

• Initial state,
• Model parameters,
• Assessment parameters,
• Exposure time-series, and
• Environmental factor time-series

Scenario properties can be modified using intuitively named tidyr verbs: in general, each verb accepts a scenario as an argument and returns a modified scenario as the result. For instance, the set_param() function will update a scenario’s parameters to new values:

library(cvasi)

# Create a new GUTS-RED-IT scenario and set its model parameters
GUTS_RED_IT() %>%
  set_param(c(kd=1.2, hb=0, alpha=9.2, beta=4.3))
#> 'GUTS-RED-IT' scenario
#> param: kd=1.2, hb=0, alpha=9.2, beta=4.3
#> init : D=0, H=0
#> endpt: L
#> times: none
#> forcs: none
#> expsr:
#> >> exposure series is empty

If a scenario is properly defined, it can be passed to dedicated functions that, for instance, run a simulation or derive assessment endpoints. These higher-level functions do not return a scenario object but a table of e.g. simulation results for that particular scenario. As an example, simulating the sample scenario minnow_it will run a GUTS-RED-IT model using substance-specific parameters:

# Example GUTS-RED-IT scenario derived from an acute fish toxicity study
# of the fathead minnow and Chlorpyrifos (Geiger et al. 1988)
minnow_it %>%
  simulate()
#>   time         D      H         S
#> 1    0 0.0000000 0.0000 1.0000000
#> 2    1 0.7075891 0.0017 0.9982869
#> 3    2 0.9144958 0.0034 0.9965618
#> 4    3 0.9749972 0.0051 0.9948550
#> 5    4 0.9926889 0.0068 0.9931605

The table returned by simulate() contains the value of each state variable at the requested output time points. In case of the GUTS-RED-IT model, this comprises the state variables D (scaled damage) and H (cumulative hazard). Effect levels, effect profiles, and dose-response graphs can be derived in a similar manner. Please refer to EFSA PPR Panel (2018) or the package help for more information on GUTS-RED type models:

# Access the package help on GUTS-RED type models
?"GUTS-RED-models"

Setting up a scenario

A scenario is created by first calling a constructor function that is dedicated to a particular model. The cvasi package supports multiple TKTD models, for instance:

• Reduced General Unified Threshold models of Survival (GUTS-RED)
  • GUTS_RED_SD()
  • GUTS_RED_IT()
• Macrophyte models
  • Lemna_Schmitt(), published by Schmitt et al. (2013)
  • Lemna_SETAC(), published by Klein et al. (2021)
  • Myrio(), an adaption of Klein et al. (2021) to Myriophyllum at Tier 2C
• Algae models
  • Algae_Weber(), published by Weber et al. (2012)
  • Algae_TKTD(), based on Weber et al. (2012), but with scaled damage
  • Algae_Simple(), simplified growth model without external forcing variables
• Dynamic Energy Budget (DEB) models
  • DEB_abj(), the abj model with type M acceleration
  • DEBtox(), as implemented in the DEBtox2019 module of BYOM

The listed functions create objects that will carry all data and settings to fully describe a scenario. In case of a GUTS-RED-IT scenario, one needs to provide at least the four model parameters and an exposure time-series to fully define a scenario:

# Define an exposure time-series
myexposure <- data.frame(time=c(0, 1, 1.01, 5),
                         conc=c(10, 10, 0, 0))

# Create and parameterize a scenario
GUTS_RED_IT() %>%
  set_param(c(kd=1.2, hb=0, alpha=9.2, beta=4.3)) %>%
  set_exposure(myexposure) -> myscenario

One can examine the main scenario properties by passing the scenario to the console like in the next example. The console will print the model name, model parameters (param), initial state (init), enabled effect endpoints (endpt), output times (times), environmental forcings (forcs), and exposure series (expsr). Depending on model type, the scenario may carry additional properties that are not necessarily displayed in the console output.

# Print information about the scenario
myscenario
#> 'GUTS-RED-IT' scenario
#> param: kd=1.2, hb=0, alpha=9.2, beta=4.3
#> init : D=0, H=0
#> endpt: L
#> times: [0,5] n=4, regular
#> forcs: none
#> expsr:
#>   time conc
#> 1 0.00   10
#> 2 1.00   10
#> 3 1.01    0
#> 4 5.00    0

Several set_*() verbs are available to modify scenario properties, such as:

• set_tag(), to assign a custom tag to identify the scenario
• set_init(), to set the initial state
• set_param(), to modify model parameters
• set_times(), to define output time points
• set_forcings(), to set environmental factors (i.e. forcings)
• set_exposure(), to set an exposure time-series
• set_endpoints(), to enable/disable assessment endpoints
• set_transfer(), to define transfer intervals of Lemna fronds
• set_moa(), to select a model’s mode of action
• set_window(), to control moving exposure windows

Please note that set_exposure() will, by default, use the exposure series’ time points also as output time points. If this behavior is undesired, it can be disabled by setting the argument reset_times=FALSE:

# Update the exposure time-series but keep former output time points
myscenario %>%
  set_noexposure()
#> 'GUTS-RED-IT' scenario
#> param: kd=1.2, hb=0, alpha=9.2, beta=4.3
#> init : D=0, H=0
#> endpt: L
#> times: [0,5] n=4, regular
#> forcs: none
#> expsr: no exposure
#>   time conc
#> 1    0    0

Note that times still has the original four elements although the assigned exposure series has only a single entry. Output time points can be explicitly set with set_times(). For deriving effect endpoints, the simulated time frame can be split up into moving exposure windows of a certain length using set_window(). A more complex scenario setup may look like the following workflow:

# Selected model parameters
myparam <- c(p_M=3211, v=0.023, k_J=0.63)
# Initial body length L
myinit <- c(L=0.02)
# Constant non-zero exposure
myexposure <- data.frame(time=0, conc=1.72)

DEB_abj() %>%
  set_param(myparam) %>%
  set_init(myinit) %>%
  set_exposure(myexposure, reset_times=FALSE) %>%
  set_times(0:10) %>%              # Output times 0,1,2,...,10
  set_mode_of_action(4) %>%        # Method of Action #4 to be activated
  set_window(length=3, interval=1) # Using moving exposure windows of length 3 days
#> 'DEB_abj' scenario
#> param: p_M=3211, v=0.023, k_J=0.63, MoA=4
#> init : L=0.02, E=0, H=0, R=0, cV=0, Lmax=0
#> endpt: L, R
#> times: [0,10] n=11, regular
#> forcs: none
#> expsr:
#>   time conc
#> 1    0 1.72

Running a simulation

After a scenario is fully set up, it can be passed to simulate() to return model results:

# Example scenario of the Lemna TKTD model
metsulfuron %>%
  set_times(0:7) %>%
  simulate()
#>   time       BM E    M_int     C_int  FrondNo
#> 1    0 50.00000 1   0.0000 0.0000000 500000.0
#> 2    1 52.15860 1 223.0746 0.2560990 521586.0
#> 3    2 52.43488 1 369.6343 0.4221196 524348.8
#> 4    3 52.33910 1 463.0257 0.5297395 523391.0
#> 5    4 52.17916 1 521.8922 0.5989178 521791.6
#> 6    5 52.00010 1 558.6262 0.6432809 520001.0
#> 7    6 51.81396 1 581.2167 0.6716992 518139.6
#> 8    7 51.63733 1 474.0708 0.5497470 516373.3

simulate() returns the model state for each output time point. In this example, it returns results for the period from zero to seven with an equi-distant time step length of one. The temporal resolution of results can be increased by specifying additional output times:

# Same simulation period, but with a smaller step length of 0.1
metsulfuron %>%
  set_times(seq(0, 7, 0.1)) %>%
  simulate() %>%
  tail()
#>    time       BM E    M_int     C_int  FrondNo
#> 66  6.5 51.72105 1 556.4461 0.6442276 517210.5
#> 67  6.6 51.70312 1 544.1607 0.6302225 517031.2
#> 68  6.7 51.68563 1 529.6839 0.6136637 516856.3
#> 69  6.8 51.66871 1 513.1144 0.5946620 516687.1
#> 70  6.9 51.65253 1 494.5465 0.5733227 516525.3
#> 71  7.0 51.63733 1 474.0708 0.5497470 516373.3

The resulting table now contains ten times as many rows because we decreased the step length by a factor of ten but simulated the same period. The temporal resolution of simulations can have an influence on results, i.e. the numerical values of the returned variables. These differences usually originate from small numerical errors introduced by the solver of the model’s Ordinary Differential Equations. Generally, parameters such as the step-length in time can have influence on the precision of simulation results.

There are several ways to increase the precision of simulation results without modifying a scenario’s output times. One option is to decrease the solver’s maximum step length in time by setting the optional argument hmax. The smaller hmax, the more precise the results:

# Original simulation period, but with a maximum solver step length of hmax=0.01
metsulfuron %>%
  set_times(0:7) %>%
  simulate(hmax=0.01)
#>   time       BM E    M_int     C_int  FrondNo
#> 1    0 50.00000 1   0.0000 0.0000000 500000.0
#> 2    1 52.15858 1 223.0745 0.2560989 521585.8
#> 3    2 52.43495 1 369.6344 0.4221192 524349.5
#> 4    3 52.33917 1 463.0261 0.5297392 523391.7
#> 5    4 52.17923 1 521.8927 0.5989176 521792.3
#> 6    5 52.00016 1 558.6268 0.6432808 520001.6
#> 7    6 51.81403 1 581.2174 0.6716992 518140.3
#> 8    7 51.63740 1 474.0725 0.5497483 516374.0

Although very effective, reducing hmax to small values may lead to inefficiently long runtimes of simulations. If numerical precision is an issue, it is advisable to also test other solver settings such as absolute and relative error tolerances (atol and rtol) or to switch to another solver method. Numerical precision and stability is a complex and advanced topic, please refer to the manual of simulate() for additional information:

?cvasi::simulate

Deriving assessment endpoints

Simulation results are the basis for deriving assessment endpoints such as effect levels and effect profiles. The cvasi package implements routines to derive these endpoints efficiently independent of the actual model that is being assessed. Hence, if a model is available within the package framework, it can be used to derive assessment endpoints. As an example, the following statement derives effect levels for a sample GUTS-RED-IT scenario:

# GUTS-RED-IT scenario of the fathead minnow and chlorpyrifos
minnow_it %>% effect()
#> # A tibble: 1 × 4
#>   scenario           L L.dat.start L.dat.end
#>   <list>         <dbl>       <dbl>     <dbl>
#> 1 <GutsRdIt> 0.0000630           0         4

The lethality endpoint L has a value of 6.3e-5 which means that lethality has only marginally increased by 0.0063% compared to a control scenario without exposure to the contaminant. The columns L.dat.start and L.dat.end denote the period for which the effect level was observed. In case of GUTS-RED models, this usually refers to the whole simulation period. For certain assessments it may be necessary to evaluate all exposure periods of a certain length within the original series separately. This approach is referred to as a moving exposure window:

# Setting up a custom scenario with a total simulation period of 14 days and
# an exposure window length of 7 to assess a trapezoidal exposure pattern
americamysis %>%
  set_window(7) %>%
  set_exposure(data.frame(t=c(0,3,4,7,8), c=c(0,0,3,3,0))) %>%
  set_times(0:14) -> mydeb

# Derive maximum effect level of all exposure windows
mydeb %>% effect()
#> # A tibble: 1 × 7
#>   scenario      L L.dat.start L.dat.end     R R.dat.start R.dat.end
#>   <list>    <dbl>       <dbl>     <dbl> <dbl>       <dbl>     <dbl>
#> 1 <DebAbj> 0.0521           2         9     0           0         7

The americamysis scenario provides two effect endpoints: structural length (L) and reproduction (R). Exposure to the simulated toxicant results at maximum in a 5.21% decrease in body length compared to control scenarios. The maximum effect was observed in the exposure window starting at time 2.0 and ending at 9.0, cf. columns L.dat.start and L.dat.end. The toxicant had no effect on reproduction due to the simulation being too short for reproduction to occur.

Scenarios can also be assessed regarding to so-called effect profiles, or EPx values for short. An EPx value quantifies the multiplication factor that is necessary to produce an effect level of x% for that particular scenario. As an example, an EP50 of 17.0 would mean that the scenario’s exposure series needs to be multiplied by a factor of 17 to observe a 50% effect. The function epx() can derive effect profiles efficiently with arbitrary precision:

# Restrict assessed endpoints to structural length (L)
mydeb %>% 
  set_endpoints("L") %>%
  epx()
#> # A tibble: 1 × 3
#>   scenario L.EP10 L.EP50
#>   <list>    <dbl>  <dbl>
#> 1 <DebAbj>   1.16   1.71

In this example, a factor of 1.16 resulted in 10% effect on structural length and a factor of 1.71 in 50% effect. epx() provides means to control the precision of derived EPx values, the range of tested multiplication factors, factor cutoff thresholds, and debugging capabilities. To examine how a particular EPx value was derived, simply set the argument verbose=TRUE:

# Examine how the EP20 value is derived
minnow_it %>% epx(level=20, verbose=TRUE)
#> epx: screening multiplication factors
#>   start: 10
#>   L.EP20: 1 <<<
#>   L.EP20: 5.5
#>   L.EP20: 7.75
#>   L.EP20: 6.625
#>   L.EP20: 7.1875
#>   L.EP20: 6.90625
#>   L.EP20: 6.765625
#>   L.EP20: 6.8359375
#>   L.EP20: 6.80078125
#> # A tibble: 1 × 2
#>   scenario   L.EP20
#>   <list>      <dbl>
#> 1 <GutsRdIt>   6.80

In this example, the algorithm of epx() first tests a multiplication factor of 10 and then continues testing additional factors using a binary search algorithm until a factor is found that causes 20% effect.

Please note: Depending on model and chosen parameters, it may be impossible to calculate EPx values (for selected effect levels), because the model or organism cannot reach the required state. As an example: If the simulated period of a DEB_abj scenario is too short, reproduction cannot occur. Therefore, effects on reproduction will always be zero independent of the exposure level. Consequently, EPx values for the reproduction endpoint will not be available.

Simulation results

The return value of simulate() is a data.frame containing the value of each state variable at each requested output time. The first column represents time, followed by columns for each state variable:

## Display selected scenario properties
metsulfuron@times  # Output times
#>  [1]  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14
metsulfuron@init   # Initial state
#>    BM     E M_int 
#>    50     1     0

## Simulate the sample scenario
metsulfuron %>% simulate()
#>    time       BM E     M_int      C_int  FrondNo
#> 1     0 50.00000 1   0.00000 0.00000000 500000.0
#> 2     1 52.15860 1 223.07463 0.25609895 521586.0
#> 3     2 52.43488 1 369.63428 0.42211960 524348.8
#> 4     3 52.33910 1 463.02568 0.52973949 523391.0
#> 5     4 52.17916 1 521.89216 0.59891777 521791.6
#> 6     5 52.00010 1 558.62619 0.64328091 520001.0
#> 7     6 51.81396 1 581.21672 0.67169918 518139.6
#> 8     7 51.63733 1 474.07075 0.54974702 516373.3
#> 9     8 51.61800 1 301.84473 0.35015952 516180.0
#> 10    9 52.32258 1 192.18697 0.21994713 523225.8
#> 11   10 54.34844 1 122.36700 0.13482199 543484.4
#> 12   11 57.08420 1  77.91205 0.08172826 570842.0
#> 13   12 59.99453 1  49.60723 0.04951273 599945.3
#> 14   13 62.96107 1  31.58532 0.03003978 629610.7
#> 15   14 65.95873 1  20.11063 0.01825732 659587.3

The model state in the first row is, by definition, equal to the initial state of the scenario. The simulation result may contain additional columns that are derived variables or intermediate model variables. In case of the metsulfuron scenario, the internal toxicant concentration (C_int) was derived from BM, M_int, and model parameters. The number of fronds (FrondNo) was derived from BM and model parameters. Additional outputs such as C_int and FrondNo are included in results for user convenience and to support model understanding. Additional outputs can be disabled by setting the optional argument nout=0:

metsulfuron %>%
  simulate(nout=0) %>%
  head(5)
#>   time       BM E    M_int
#> 1    0 50.00000 1   0.0000
#> 2    1 52.15860 1 223.0746
#> 3    2 52.43488 1 369.6343
#> 4    3 52.33910 1 463.0257
#> 5    4 52.17916 1 521.8922

Intermediate model variables can be enabled in the same manner:

metsulfuron %>%
  simulate(nout=8) %>%
  head(5)
#>   time       BM E    M_int     C_int  FrondNo   C_int_u BM_fresh k_phot_eff
#> 1    0 50.00000 1   0.0000 0.0000000 500000.0 0.0000000 835.0000 0.07548561
#> 2    1 52.15860 1 223.0746 0.2560990 521586.0 0.3414653 871.0486 0.03746125
#> 3    2 52.43488 1 369.6343 0.4221196 524348.8 0.5628261 875.6624 0.01993781
#> 4    3 52.33910 1 463.0257 0.5297395 523391.0 0.7063193 874.0630 0.01760497
#> 5    4 52.17916 1 521.8922 0.5989178 521791.6 0.7985570 871.3920 0.01699677
#>   k_resp_eff     f_Eff P_up_eff
#> 1 0.02030631 1.0000000   0.0054
#> 2 0.02030631 0.5049202   0.0054
#> 3 0.02030631 0.2693320   0.0054
#> 4 0.02030631 0.2376343   0.0054
#> 5 0.02030631 0.2291285   0.0054

Effect levels

The return value of effect() is a data.frame containing the effect level of each selected endpoint. The first column, scenario, contains the original scenario which was the basis of the assessment. Followed by the effect level and the exposure window in which the maximum effect was observed:

metsulfuron %>%
  set_times(0:7) %>%  # restrict scenario to the period [0,7]
  effect()
#> # A tibble: 1 × 7
#>   scenario      BM BM.dat.start BM.dat.end     r r.dat.start r.dat.end
#>   <list>     <dbl>        <dbl>      <dbl> <dbl>       <dbl>     <dbl>
#> 1 <LmnSchmS> 0.268            0          7 0.906           0         7

Effect levels are reported as a fraction relative to a control scenario. Effects can be converted to percentages by multiplying the value with 100 (%). For scenarios with moving exposure windows disabled, the reported window will cover the entire simulation period. If multiple windows result in identical maximum effect levels, the first matching window will be reported. The table’s content can be reduced to endpoints only by setting the argument ep_only=TRUE.

effect() can also be instructed to return the effect levels of all assessed exposure windows by setting the argument max_only=FALSE:

metsulfuron %>%
  set_window(length=7, interval=1) %>%   # enable moving exposure windows
  effect(max_only=FALSE)                 # return effects of all windows
#> # A tibble: 8 × 5
#>   scenario              BM            r dat.start dat.end
#>   <list>             <dbl>        <dbl>     <dbl>   <dbl>
#> 1 <LmnSchmS>  0.268         0.906               0       7
#> 2 <LmnSchmS>  0.265         0.895               1       8
#> 3 <LmnSchmS>  0.248         0.829               2       9
#> 4 <LmnSchmS>  0.205         0.669               3      10
#> 5 <LmnSchmS>  0.147         0.463               4      11
#> 6 <LmnSchmS>  0.0739        0.223               5      12
#> 7 <LmnSchmS>  0.00315       0.00918             6      13
#> 8 <LmnSchmS> -0.0000000571 -0.000000166         7      14

In few cases, effect() may report spurious non-zero effect levels that are very close to zero. These can either be avoided by increasing the numerical precision of simulations or by setting a marginal effect threshold which instructs effect() to zero any effect below that threshold. A warning will be shown if a marginal effect threshold greater than 1% is chosen. Applying a threshold of 1% to the previous example results in zeroing the effect in the last and second to last exposure window but leaving the remaining results unchanged:

metsulfuron %>%
  set_window(length=7, interval=1) %>%         # enable moving exposure windows
  effect(max_only=FALSE, marginal_effect=0.01) # return effects of all windows
#> # A tibble: 8 × 5
#>   scenario       BM     r dat.start dat.end
#>   <list>      <dbl> <dbl>     <dbl>   <dbl>
#> 1 <LmnSchmS> 0.268  0.906         0       7
#> 2 <LmnSchmS> 0.265  0.895         1       8
#> 3 <LmnSchmS> 0.248  0.829         2       9
#> 4 <LmnSchmS> 0.205  0.669         3      10
#> 5 <LmnSchmS> 0.147  0.463         4      11
#> 6 <LmnSchmS> 0.0739 0.223         5      12
#> 7 <LmnSchmS> 0      0             6      13
#> 8 <LmnSchmS> 0      0             7      14

Effect profiles

The return value of epx() is a data.frame containing the effect profile (EPx) of each selected endpoint and effect level. The first column, scenario, contains the original scenario which was the basis of the assessment. Followed by the derived effect profiles, i.e. the multiplication factors to achieve x% effect:

metsulfuron %>% epx()
#> Warning in epx(.): Some scenarios have failed (n=1), result will include NAs
#>   **  multiplication factor out of range: BM.EP50 > 1e+30
#> # A tibble: 1 × 6
#>   scenario   BM.EP10 r.EP10 BM.EP50 r.EP50 error                                
#>   <list>       <dbl>  <dbl>   <dbl>  <dbl> <chr>                                
#> 1 <LmnSchmS>   0.395  0.325      NA  0.880 " multiplication factor out of range…

For the sample metsulfuron scenario, a factor of 0.395 would need to be applied to the exposure series to achieve an effect level of 10% on biomass (BM). An effect level of 50% on biomass is unattainable with the tested factors ranging from 1e-30 to 1e30 which is demonstrated by the warning message and the NA value for BM.EP50.

The table’s content can be reduced to endpoints only by setting the argument ep_only=TRUE. The range of tested factors can be controlled by setting the arguments min_factor and max_factor. Multiplication factors exceeding this range will raise a warning and NA will be returned for that EPx. Finding multiplication factors can be cut short if a certain margin of safety is reached by using the argument factor_cutoff: if the algorithm identifies that an EPx greater or equal to factor_cutoff is required, then factor_cutoff is reported as the result. This approach may be used to save computational resources for long running models if the exact EPx value above the cutoff is not of interest.

The precision of EPx values is controlled by the argument effect_tolerance and is given as the upper absolute error threshold of effects that is deemed acceptable. The default value of 0.001 ensures that a derived EPx will result in an effect of x% ± 0.1. Decreasing the effect_tolerance will result in additional model iterations and longer runtime.

The algorithm’s inner workings can be examined by setting the argument verbose=TRUE:

metsulfuron %>%
  set_endpoints("BM") %>%
  epx(verbose=TRUE)
#> epx: screening multiplication factors
#>   start: 10
#>   BM.EP10: 1 <<<
#>   BM.EP10: 0.1 <<<
#>   BM.EP10: 0.55
#>   BM.EP10: 0.325
#>   BM.EP10: 0.4375
#>   BM.EP10: 0.38125
#>   BM.EP10: 0.409375
#>   BM.EP10: 0.3953125
#>   BM.EP50: 100 >>>
#>   BM.EP50: 1000 >>>
#>   BM.EP50: 10000 >>>
#>   BM.EP50: 1e+05 >>>
#>   BM.EP50: 1e+06 >>>
#>   BM.EP50: 1e+07 >>>
#>   BM.EP50: 1e+08 >>>
#>   BM.EP50: 1e+09 >>>
#>   BM.EP50: 1e+10 >>>
#>   BM.EP50: 1e+11 >>>
#>   BM.EP50: 1e+12 >>>
#>   BM.EP50: 1e+13 >>>
#>   BM.EP50: 1e+14 >>>
#>   BM.EP50: 1e+15 >>>
#>   BM.EP50: 1e+16 >>>
#>   BM.EP50: 1e+17 >>>
#>   BM.EP50: 1e+18 >>>
#>   BM.EP50: 1e+19 >>>
#>   BM.EP50: 1e+20 >>>
#>   BM.EP50: 1e+21 >>>
#>   BM.EP50: 1e+22 >>>
#>   BM.EP50: 1e+23 >>>
#>   BM.EP50: 1e+24 >>>
#>   BM.EP50: 1e+25 >>>
#>   BM.EP50: 1e+26 >>>
#>   BM.EP50: 1e+27 >>>
#>   BM.EP50: 1e+28 >>>
#>   BM.EP50: 1e+29 >>>
#>   BM.EP50: 1e+30 >>>
#>   BM.EP50: 1e+31 >>>
#> Warning in epx(., verbose = TRUE): Some scenarios have failed (n=1), result will include NAs
#>   **  multiplication factor out of range: BM.EP50 > 1e+30
#> # A tibble: 1 × 4
#>   scenario   BM.EP10 BM.EP50 error                                              
#>   <list>       <dbl>   <dbl> <chr>                                              
#> 1 <LmnSchmS>   0.395      NA " multiplication factor out of range: BM.EP50 > 1e…

In this example, the binary search finds a BM.EP10 with acceptable precision in less than ten iterations. For the BM.EP50, the algorithm re-uses the knowledge gained from previously performed calculations but is ultimately unable to find a suitable multiplication factor within the allowed range of multiplication factors.