Introduction to mkin

- - - - -

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

Abstract -

In the regulatory evaluation of chemical substances like plant -protection products (pesticides), biocides and other chemicals, -degradation data play an important role. For the evaluation of pesticide -degradation experiments, detailed guidance has been developed, based on -nonlinear optimisation. The R add-on package -mkin implements fitting some of the models recommended in -this guidance from within R and calculates some statistical measures for -data series within one or more compartments, for parent and -metabolites.

-library("mkin", quietly = TRUE)
-# Define the kinetic model
-m_SFO_SFO_SFO <- mkinmod(parent = mkinsub("SFO", "M1"),
-                         M1 = mkinsub("SFO", "M2"),
-                         M2 = mkinsub("SFO"),
-                         use_of_ff = "max", quiet = TRUE)
-
-
-# Produce model predictions using some arbitrary parameters
-sampling_times = c(0, 1, 3, 7, 14, 28, 60, 90, 120)
-d_SFO_SFO_SFO <- mkinpredict(m_SFO_SFO_SFO,
-  c(k_parent = 0.03,
-    f_parent_to_M1 = 0.5, k_M1 = log(2)/100,
-    f_M1_to_M2 = 0.9, k_M2 = log(2)/50),
-  c(parent = 100, M1 = 0, M2 = 0),
-  sampling_times)
-
-# Generate a dataset by adding normally distributed errors with
-# standard deviation 3, for two replicates at each sampling time
-d_SFO_SFO_SFO_err <- add_err(d_SFO_SFO_SFO, reps = 2,
-                             sdfunc = function(x) 3,
-                             n = 1, seed = 123456789 )
-
-# Fit the model to the dataset
-f_SFO_SFO_SFO <- mkinfit(m_SFO_SFO_SFO, d_SFO_SFO_SFO_err[[1]], quiet = TRUE)
-
-# Plot the results separately for parent and metabolites
-plot_sep(f_SFO_SFO_SFO, lpos = c("topright", "bottomright", "bottomright"))

Background -

The mkin package (J. Ranke -2021) implements the approach to degradation kinetics recommended -in the kinetics report provided by the FOrum for Co-ordination of -pesticide fate models and their USe (FOCUS Work -Group on Degradation Kinetics 2006, 2014). It covers data series -describing the decline of one compound, data series with transformation -products (commonly termed metabolites) and data series for more than one -compartment. It is possible to include back reactions. Therefore, -equilibrium reactions and equilibrium partitioning can be specified, -although this often leads to an overparameterisation of the model.

When the first mkin code was published in 2010, the most -commonly used tools for fitting more complex kinetic degradation models -to experimental data were KinGUI (Schäfer et al. -2007), a MATLAB based tool with a graphical user interface that -was specifically tailored to the task and included some output as -proposed by the FOCUS Kinetics Workgroup, and ModelMaker, a general -purpose compartment based tool providing infrastructure for fitting -dynamic simulation models based on differential equations to data.

The ‘mkin’ code was first uploaded to the BerliOS development -platform. When this was taken down, the version control history was -imported into the R-Forge site (see e.g. the -initial commit on 11 May 2010), where the code is still being -updated.

At that time, the R package FME (Flexible Modelling -Environment) (Soetaert and Petzoldt 2010) -was already available, and provided a good basis for developing a -package specifically tailored to the task. The remaining challenge was -to make it as easy as possible for the users (including the author of -this vignette) to specify the system of differential equations and to -include the output requested by the FOCUS guidance, such as the \(\chi^2\) error level as defined in this -guidance.

Also, mkin introduced using analytical solutions for -parent only kinetics for improved optimization speed. Later, Eigenvalue -based solutions were introduced to mkin for the case of -linear differential equations (i.e. where the FOMC or DFOP -models were not used for the parent compound), greatly improving the -optimization speed for these cases. This, has become somehow obsolete, -as the use of compiled code described below gives even faster execution -times.

The possibility to specify back-reactions and a biphasic model -(SFORB) for metabolites were present in mkin from the very -beginning.

Derived software tools -

Soon after the publication of mkin, two derived tools -were published, namely KinGUII (developed at Bayer Crop Science) and -CAKE (commissioned to Tessella by Syngenta), which added a graphical -user interface (GUI), and added fitting by iteratively reweighted least -squares (IRLS) and characterisation of likely parameter distributions by -Markov Chain Monte Carlo (MCMC) sampling.

CAKE focuses on a smooth use experience, sacrificing some flexibility -in the model definition, originally allowing only two primary -metabolites in parallel. The current version 3.4 of CAKE released in May -2020 uses a scheme for up to six metabolites in a flexible arrangement -and supports biphasic modelling of metabolites, but does not support -back-reactions (non-instantaneous equilibria).

KinGUI offers an even more flexible widget for specifying complex -kinetic models. Back-reactions (non-instantaneous equilibria) were -supported early on, but until 2014, only simple first-order models could -be specified for transformation products. Starting with KinGUII version -2.1, biphasic modelling of metabolites was also available in -KinGUII.

A further graphical user interface (GUI) that has recently been -brought to a decent degree of maturity is the browser based GUI named -gmkin. Please see its documentation page and manual -for further information.

A comparison of scope, usability and numerical results obtained with -these tools has been recently been published by Johannes Ranke, Wöltjen, and Meinecke -(2018).

Unique features -

Currently, the main unique features available in mkin -are

the speed -increase by using compiled code when a compiler is present,
parallel model fitting on multicore machines using the mmkin -function,
the estimation of parameter confidence intervals based on -transformed parameters (see below) and
the possibility to use the two-component -error model -

The iteratively reweighted least squares fitting of different -variances for each variable as introduced by Gao -et al. (2011) has been available in mkin since version -0.9-22. With release -0.9.49.5, the IRLS algorithm has been complemented by direct or -step-wise maximisation of the likelihood function, which makes it -possible not only to fit the variance by variable error model but also a -two-component -error model inspired by error models developed in analytical -chemistry (Johannes Ranke and Meinecke -2019).

Internal parameter transformations -

For rate constants, the log transformation is used, as proposed by -Bates and Watts (1988, 77, 149). -Approximate intervals are constructed for the transformed rate constants -(compare Bates and Watts 1988, 135), -i.e. for their logarithms. Confidence intervals for the rate -constants are then obtained using the appropriate backtransformation -using the exponential function.

In the first version of mkin allowing for specifying -models using formation fractions, a home-made reparameterisation was -used in order to ensure that the sum of formation fractions would not -exceed unity.

This method is still used in the current version of KinGUII (v2.1 -from April 2014), with a modification that allows for fixing the pathway -to sink to zero. CAKE uses penalties in the objective function in order -to enforce this constraint.

In 2012, an alternative reparameterisation of the formation fractions -was proposed together with René Lehmann (J. Ranke -and Lehmann 2012), based on isometric logratio transformation -(ILR). The aim was to improve the validity of the linear approximation -of the objective function during the parameter estimation procedure as -well as in the subsequent calculation of parameter confidence intervals. -In the current version of mkin, a logit transformation is used for -parameters that are bound between 0 and 1, such as the g parameter of -the DFOP model.

Confidence intervals based on transformed parameters -

In the first attempt at providing improved parameter confidence -intervals introduced to mkin in 2013, confidence intervals -obtained from FME on the transformed parameters were simply all -backtransformed one by one to yield asymmetric confidence intervals for -the backtransformed parameters.

However, while there is a 1:1 relation between the rate constants in -the model and the transformed parameters fitted in the model, the -parameters obtained by the isometric logratio transformation are -calculated from the set of formation fractions that quantify the paths -to each of the compounds formed from a specific parent compound, and no -such 1:1 relation exists.

Therefore, parameter confidence intervals for formation fractions -obtained with this method only appear valid for the case of a single -transformation product, where currently the logit transformation is used -for the formation fraction.

The confidence intervals obtained by backtransformation for the cases -where a 1:1 relation between transformed and original parameter exist -are considered by the author of this vignette to be more accurate than -those obtained using a re-estimation of the Hessian matrix after -backtransformation, as implemented in the FME package.

Parameter t-test based on untransformed parameters -

The standard output of many nonlinear regression software packages -includes the results from a test for significant difference from zero -for all parameters. Such a test is also recommended to check the -validity of rate constants in the FOCUS guidance (FOCUS Work Group on Degradation Kinetics 2014, -96ff).

It has been argued that the precondition for this test, i.e. -normal distribution of the estimator for the parameters, is not -fulfilled in the case of nonlinear regression (J. -Ranke and Lehmann 2015). However, this test is commonly used by -industry, consultants and national authorities in order to decide on the -reliability of parameter estimates, based on the FOCUS guidance -mentioned above. Therefore, the results of this one-sided t-test are -included in the summary output from mkin.

As it is not reasonable to test for significant difference of the -transformed parameters (e.g. \(log(k)\)) from zero, the t-test is -calculated based on the model definition before parameter -transformation, i.e. in a similar way as in packages that do -not apply such an internal parameter transformation. A note is included -in the mkin output, pointing to the fact that the t-test is -based on the unjustified assumption of normal distribution of the -parameter estimators.

References -

- -

-Bates, D., and D. Watts. 1988. Nonlinear Regression and Its -Applications. Wiley-Interscience. -

-FOCUS Work Group on Degradation Kinetics. 2006. Guidance Document on -Estimating Persistence and Degradation Kinetics from Environmental Fate -Studies on Pesticides in EU Registration. Report of the FOCUS Work Group -on Degradation Kinetics. http://esdac.jrc.ec.europa.eu/projects/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. -

-Gao, Z., J. W. Green, J. Vanderborght, and W. Schmitt. 2011. -“Improving Uncertainty Analysis in Kinetic Evaluations Using -Iteratively Reweighted Least Squares.” Journal. Environmental -Science and Technology 45: 4429–37. -

-Ranke, J. 2021. ‘mkin‘: -Kinetic Evaluation of Chemical Degradation Data. https://CRAN.R-project.org/package=mkin. -

-Ranke, J., and R. Lehmann. 2012. “Parameter Reliability in Kinetic -Evaluation of Environmental Metabolism Data - Assessment and the -Influence of Model Specification.” In SETAC World 20-24 -May. Berlin. https://jrwb.de/posters/Poster_SETAC_2012_Kinetic_parameter_uncertainty_model_parameterization_Lehmann_Ranke.pdf. -

-———. 2015. “To t-Test or Not to t-Test, That Is the -Question.” In XV Symposium on Pesticide Chemistry 2-4 -September 2015. Piacenza. https://jrwb.de/posters/piacenza_2015.pdf. -

-Ranke, Johannes, and Stefan Meinecke. 2019. “Error Models for the -Kinetic Evaluation of Chemical Degradation Data.” -Environments 6 (12). https://doi.org/10.3390/environments6120124. -

-Ranke, Johannes, Janina Wöltjen, and Stefan Meinecke. 2018. -“Comparison of Software Tools for Kinetic Evaluation of Chemical -Degradation Data.” Environmental Sciences Europe 30 (1): -17. https://doi.org/10.1186/s12302-018-0145-1. -

-Schäfer, D., B. Mikolasch, P. Rainbird, and B. Harvey. 2007. -“KinGUI: A New Kinetic Software Tool for Evaluations -According to FOCUS Degradation Kinetics.” In -Proceedings of the XIII Symposium Pesticide Chemistry, edited -by Del Re A. A. M., Capri E., Fragoulis G., and Trevisan M., 916–23. -Piacenza. -

-Soetaert, Karline, and Thomas Petzoldt. 2010. “Inverse Modelling, -Sensitivity and Monte Carlo Analysis in R Using Package -FME.” Journal of Statistical Software 33 -(3): 1–28. https://doi.org/10.18637/jss.v033.i03. -

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Johannes -Ranke

Last change 15 February 2021 -(rebuilt 2023-04-16)