Example evaluation of FOCUS dataset Z

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Wissenschaftlicher Berater, Kronacher -Str. 12, 79639 Grenzach-Wyhlen, Germany
Privatdozent at the -University of Bremen

The data -

The following code defines the example dataset from Appendix 7 to the -FOCUS kinetics report (FOCUS Work Group on -Degradation Kinetics 2014, 354).

-library(mkin, quietly = TRUE)
-LOD = 0.5
-FOCUS_2006_Z = data.frame(
-  t = c(0, 0.04, 0.125, 0.29, 0.54, 1, 2, 3, 4, 7, 10, 14, 21,
-        42, 61, 96, 124),
-  Z0 = c(100, 81.7, 70.4, 51.1, 41.2, 6.6, 4.6, 3.9, 4.6, 4.3, 6.8,
-         2.9, 3.5, 5.3, 4.4, 1.2, 0.7),
-  Z1 = c(0, 18.3, 29.6, 46.3, 55.1, 65.7, 39.1, 36, 15.3, 5.6, 1.1,
-         1.6, 0.6, 0.5 * LOD, NA, NA, NA),
-  Z2 = c(0, NA, 0.5 * LOD, 2.6, 3.8, 15.3, 37.2, 31.7, 35.6, 14.5,
-         0.8, 2.1, 1.9, 0.5 * LOD, NA, NA, NA),
-  Z3 = c(0, NA, NA, NA, NA, 0.5 * LOD, 9.2, 13.1, 22.3, 28.4, 32.5,
-         25.2, 17.2, 4.8, 4.5, 2.8, 4.4))
-
-FOCUS_2006_Z_mkin <- mkin_wide_to_long(FOCUS_2006_Z)

Parent and one metabolite -

The next step is to set up the models used for the kinetic analysis. -As the simultaneous fit of parent and the first metabolite is usually -straightforward, Step 1 (SFO for parent only) is skipped here. We start -with the model 2a, with formation and decline of metabolite Z1 and the -pathway from parent directly to sink included (default in mkin).

-Z.2a <- mkinmod(Z0 = mkinsub("SFO", "Z1"),
-                Z1 = mkinsub("SFO"))

## Temporary DLL for differentials generated and loaded

-m.Z.2a <- mkinfit(Z.2a, FOCUS_2006_Z_mkin, quiet = TRUE)

## Warning in mkinfit(Z.2a, FOCUS_2006_Z_mkin, quiet = TRUE): Observations with
-## value of zero were removed from the data

-plot_sep(m.Z.2a)

-summary(m.Z.2a, data = FALSE)$bpar

##            Estimate se_notrans t value     Pr(>t)    Lower    Upper
-## Z0_0       97.01488   3.301084 29.3888 3.2971e-21 91.66556 102.3642
-## k_Z0        2.23601   0.207078 10.7979 3.3309e-11  1.95303   2.5600
-## k_Z1        0.48212   0.063265  7.6207 2.8154e-08  0.40341   0.5762
-## f_Z0_to_Z1  1.00000   0.094764 10.5525 5.3560e-11  0.00000   1.0000
-## sigma       4.80411   0.635638  7.5579 3.2592e-08  3.52677   6.0815

As obvious from the parameter summary (the component of the summary), -the kinetic rate constant from parent compound Z to sink is very small -and the t-test for this parameter suggests that it is not significantly -different from zero. This suggests, in agreement with the analysis in -the FOCUS kinetics report, to simplify the model by removing the pathway -to sink.

A similar result can be obtained when formation fractions are used in -the model formulation:

-Z.2a.ff <- mkinmod(Z0 = mkinsub("SFO", "Z1"),
-                   Z1 = mkinsub("SFO"),
-                   use_of_ff = "max")

## Temporary DLL for differentials generated and loaded

-m.Z.2a.ff <- mkinfit(Z.2a.ff, FOCUS_2006_Z_mkin, quiet = TRUE)

## Warning in mkinfit(Z.2a.ff, FOCUS_2006_Z_mkin, quiet = TRUE): Observations with
-## value of zero were removed from the data

-plot_sep(m.Z.2a.ff)

-summary(m.Z.2a.ff, data = FALSE)$bpar

##            Estimate se_notrans t value     Pr(>t)    Lower    Upper
-## Z0_0       97.01488   3.301084 29.3888 3.2971e-21 91.66556 102.3642
-## k_Z0        2.23601   0.207078 10.7979 3.3309e-11  1.95303   2.5600
-## k_Z1        0.48212   0.063265  7.6207 2.8154e-08  0.40341   0.5762
-## f_Z0_to_Z1  1.00000   0.094764 10.5525 5.3560e-11  0.00000   1.0000
-## sigma       4.80411   0.635638  7.5579 3.2592e-08  3.52677   6.0815

Here, the ilr transformed formation fraction fitted in the model -takes a very large value, and the backtransformed formation fraction -from parent Z to Z1 is practically unity. Here, the covariance matrix -used for the calculation of confidence intervals is not returned as the -model is overparameterised.

A simplified model is obtained by removing the pathway to the sink. -

In the following, we use the parameterisation with formation -fractions in order to be able to compare with the results in the FOCUS -guidance, and as it makes it easier to use parameters obtained in a -previous fit when adding a further metabolite.

-Z.3 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
-               Z1 = mkinsub("SFO"), use_of_ff = "max")

## Temporary DLL for differentials generated and loaded

-m.Z.3 <- mkinfit(Z.3, FOCUS_2006_Z_mkin, quiet = TRUE)

## Warning in mkinfit(Z.3, FOCUS_2006_Z_mkin, quiet = TRUE): Observations with
-## value of zero were removed from the data

-plot_sep(m.Z.3)

-summary(m.Z.3, data = FALSE)$bpar

##       Estimate se_notrans t value     Pr(>t)    Lower    Upper
-## Z0_0  97.01488   2.597342  37.352 2.0106e-24 91.67597 102.3538
-## k_Z0   2.23601   0.146904  15.221 9.1477e-15  1.95354   2.5593
-## k_Z1   0.48212   0.041727  11.554 4.8268e-12  0.40355   0.5760
-## sigma  4.80411   0.620208   7.746 1.6110e-08  3.52925   6.0790

As there is only one transformation product for Z0 and no pathway to -sink, the formation fraction is internally fixed to unity.

Metabolites Z2 and Z3 -

As suggested in the FOCUS report, the pathway to sink was removed for -metabolite Z1 as well in the next step. While this step appears -questionable on the basis of the above results, it is followed here for -the purpose of comparison. Also, in the FOCUS report, it is assumed that -there is additional empirical evidence that Z1 quickly and exclusively -hydrolyses to Z2.

-Z.5 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
-               Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-               Z2 = mkinsub("SFO"), use_of_ff = "max")

## Temporary DLL for differentials generated and loaded

-m.Z.5 <- mkinfit(Z.5, FOCUS_2006_Z_mkin, quiet = TRUE)

## Warning in mkinfit(Z.5, FOCUS_2006_Z_mkin, quiet = TRUE): Observations with
-## value of zero were removed from the data

-plot_sep(m.Z.5)

Finally, metabolite Z3 is added to the model. We use the optimised -differential equation parameter values from the previous fit in order to -accelerate the optimization.

-Z.FOCUS <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
-                   Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                   Z2 = mkinsub("SFO", "Z3"),
-                   Z3 = mkinsub("SFO"),
-                   use_of_ff = "max")

## Temporary DLL for differentials generated and loaded

-m.Z.FOCUS <- mkinfit(Z.FOCUS, FOCUS_2006_Z_mkin,
-                     parms.ini = m.Z.5$bparms.ode,
-                     quiet = TRUE)

## Warning in mkinfit(Z.FOCUS, FOCUS_2006_Z_mkin, parms.ini = m.Z.5$bparms.ode, :
-## Observations with value of zero were removed from the data

## Warning in mkinfit(Z.FOCUS, FOCUS_2006_Z_mkin, parms.ini = m.Z.5$bparms.ode, : Optimisation did not converge:
-## false convergence (8)

-plot_sep(m.Z.FOCUS)

-summary(m.Z.FOCUS, data = FALSE)$bpar

##             Estimate se_notrans t value     Pr(>t)     Lower      Upper
-## Z0_0       96.838822   1.994274 48.5584 4.0280e-42 92.826981 100.850664
-## k_Z0        2.215393   0.118458 18.7019 1.0413e-23  1.989456   2.466989
-## k_Z1        0.478305   0.028258 16.9266 6.2418e-22  0.424708   0.538666
-## k_Z2        0.451627   0.042139 10.7176 1.6314e-14  0.374339   0.544872
-## k_Z3        0.058692   0.015245  3.8499 1.7803e-04  0.034808   0.098965
-## f_Z2_to_Z3  0.471502   0.058351  8.0805 9.6608e-11  0.357769   0.588274
-## sigma       3.984431   0.383402 10.3923 4.5575e-14  3.213126   4.755736

-endpoints(m.Z.FOCUS)

## $ff
-##   Z2_Z3 Z2_sink 
-##  0.4715  0.5285 
-## 
-## $distimes
-##        DT50    DT90
-## Z0  0.31288  1.0394
-## Z1  1.44917  4.8141
-## Z2  1.53478  5.0984
-## Z3 11.80986 39.2315

This fit corresponds to the final result chosen in Appendix 7 of the -FOCUS report. Confidence intervals returned by mkin are based on -internally transformed parameters, however.

Using the SFORB model -

As the FOCUS report states, there is a certain tailing of the time -course of metabolite Z3. Also, the time course of the parent compound is -not fitted very well using the SFO model, as residues at a certain low -level remain.

Therefore, an additional model is offered here, using the single -first-order reversible binding (SFORB) model for metabolite Z3. As -expected, the \(\chi^2\) error level is -lower for metabolite Z3 using this model and the graphical fit for Z3 is -improved. However, the covariance matrix is not returned.

-Z.mkin.1 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
-                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                    Z2 = mkinsub("SFO", "Z3"),
-                    Z3 = mkinsub("SFORB"))

## Temporary DLL for differentials generated and loaded

-m.Z.mkin.1 <- mkinfit(Z.mkin.1, FOCUS_2006_Z_mkin, quiet = TRUE)

## Warning in mkinfit(Z.mkin.1, FOCUS_2006_Z_mkin, quiet = TRUE): Observations
-## with value of zero were removed from the data

-plot_sep(m.Z.mkin.1)

-summary(m.Z.mkin.1, data = FALSE)$cov.unscaled

## NULL

Therefore, a further stepwise model building is performed starting -from the stage of parent and two metabolites, starting from the -assumption that the model fit for the parent compound can be improved by -using the SFORB model.

-Z.mkin.3 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
-                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                    Z2 = mkinsub("SFO"))

## Temporary DLL for differentials generated and loaded

-m.Z.mkin.3 <- mkinfit(Z.mkin.3, FOCUS_2006_Z_mkin, quiet = TRUE)

## Warning in mkinfit(Z.mkin.3, FOCUS_2006_Z_mkin, quiet = TRUE): Observations
-## with value of zero were removed from the data

-plot_sep(m.Z.mkin.3)

This results in a much better representation of the behaviour of the -parent compound Z0.

Finally, Z3 is added as well. These models appear overparameterised -(no covariance matrix returned) if the sink for Z1 is left in the -models.

-Z.mkin.4 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
-                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                    Z2 = mkinsub("SFO", "Z3"),
-                    Z3 = mkinsub("SFO"))

## Temporary DLL for differentials generated and loaded

-m.Z.mkin.4 <- mkinfit(Z.mkin.4, FOCUS_2006_Z_mkin,
-                      parms.ini = m.Z.mkin.3$bparms.ode,
-                      quiet = TRUE)

## Warning in mkinfit(Z.mkin.4, FOCUS_2006_Z_mkin, parms.ini =
-## m.Z.mkin.3$bparms.ode, : Observations with value of zero were removed from the
-## data

-plot_sep(m.Z.mkin.4)

The error level of the fit, but especially of metabolite Z3, can be -improved if the SFORB model is chosen for this metabolite, as this model -is capable of representing the tailing of the metabolite decline -phase.

-Z.mkin.5 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
-                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                    Z2 = mkinsub("SFO", "Z3"),
-                    Z3 = mkinsub("SFORB"))

## Temporary DLL for differentials generated and loaded

-m.Z.mkin.5 <- mkinfit(Z.mkin.5, FOCUS_2006_Z_mkin,
-                      parms.ini = m.Z.mkin.4$bparms.ode[1:4],
-                      quiet = TRUE)

## Warning in mkinfit(Z.mkin.5, FOCUS_2006_Z_mkin, parms.ini =
-## m.Z.mkin.4$bparms.ode[1:4], : Observations with value of zero were removed from
-## the data

-plot_sep(m.Z.mkin.5)

The summary view of the backtransformed parameters shows that we get -no confidence intervals due to overparameterisation. As the optimized is -excessively small, it seems reasonable to fix it to zero.

-m.Z.mkin.5a <- mkinfit(Z.mkin.5, FOCUS_2006_Z_mkin,
-                       parms.ini = c(m.Z.mkin.5$bparms.ode[1:7],
-                                     k_Z3_bound_free = 0),
-                       fixed_parms = "k_Z3_bound_free",
-                       quiet = TRUE)

## Warning in mkinfit(Z.mkin.5, FOCUS_2006_Z_mkin, parms.ini =
-## c(m.Z.mkin.5$bparms.ode[1:7], : Observations with value of zero were removed
-## from the data

-plot_sep(m.Z.mkin.5a)

As expected, the residual plots for Z0 and Z3 are more random than in -the case of the all SFO model for which they were shown above. In -conclusion, the model is proposed as the best-fit model for the dataset -from Appendix 7 of the FOCUS report.

A graphical representation of the confidence intervals can finally be -obtained.

-mkinparplot(m.Z.mkin.5a)

The endpoints obtained with this model are

-endpoints(m.Z.mkin.5a)

## $ff
-## Z0_free   Z2_Z3 Z2_sink Z3_free 
-## 1.00000 0.53656 0.46344 1.00000 
-## 
-## $SFORB
-##     Z0_b1     Z0_b2      Z0_g     Z3_b1     Z3_b2      Z3_g 
-## 2.4471322 0.0075125 0.9519862 0.0800069 0.0000000 0.9347820 
-## 
-## $distimes
-##      DT50   DT90 DT50back DT50_Z0_b1 DT50_Z0_b2 DT50_Z3_b1 DT50_Z3_b2
-## Z0 0.3043 1.1848  0.35666    0.28325     92.266         NA         NA
-## Z1 1.5148 5.0320       NA         NA         NA         NA         NA
-## Z2 1.6414 5.4526       NA         NA         NA         NA         NA
-## Z3     NA     NA       NA         NA         NA     8.6636        Inf

It is clear the degradation rate of Z3 towards the end of the -experiment is very low as DT50_Z3_b2 (the second Eigenvalue of the -system of two differential equations representing the SFORB system for -Z3, corresponding to the slower rate constant of the DFOP model) is -reported to be infinity. However, this appears to be a feature of the -data.

References -

- -

-FOCUS Work Group on Degradation Kinetics. 2014. Generic Guidance for -Estimating Persistence and Degradation Kinetics from Environmental Fate -Studies on Pesticides in EU Registration. 1.1 ed. http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics. -

- - - -

Johannes -Ranke

Last change 16 January 2018 -(rebuilt 2023-04-16)

The data -

Parent and one metabolite -

Metabolites Z2 and Z3 -

Using the SFORB model -

References -