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.

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.

References