Short 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. +

Johannes +Ranke

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