---
title: Introduction to mkin
author: Johannes Ranke
date: "`r Sys.Date()`"
output:
  html_document:
    toc: true
    toc_float: true
    code_folding: hide
    fig_retina: null
bibliography: references.bib
vignette: >
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteIndexEntry{mkin - Kinetic evaluation of chemical degradation data}
  %\VignetteEncoding{UTF-8}
---

[Wissenschaftlicher Berater, Kronacher Str. 12, 79639 Grenzach-Wyhlen, Germany](https://www.jrwb.de)<br />
[Privatdozent at the University of Bremen](http://chem.uft.uni-bremen.de/ranke/)

```{r, include = FALSE}
require(knitr)
opts_chunk$set(engine='R', tidy=FALSE)
```

# 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.

```{r, echo = TRUE, cache = TRUE, fig = TRUE, fig.width = 8, fig.height = 7}
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

Many approaches are possible regarding the evaluation of chemical degradation
data.

The `mkin` package [@pkg:mkin] implements the approach recommended in the
kinetics report provided by the FOrum for Co-ordination of pesticide fate
models and their USe [@FOCUS2006; -@FOCUSkinetics2014]
for simple decline data series, data series with transformation products,
commonly termed metabolites, and for data series for more than one compartment.
It is also possible to include back reactions, so equilibrium reactions and
equilibrium partitioning can be specified, although this oftentimes 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 [@schaefer2007], 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 code was first uploaded to the BerliOS 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](https://cgit.jrwb.de/mkin/commit/?id=30cbb4092f6d2d3beff5800603374a0d009ad770)),
where the code is still occasionally updated.

At that time, the R package `FME` (Flexible Modelling Environment)
[@soetaert2010] 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 relative standard deviation
that has to be assumed for the residuals, such that the $\chi^2$
goodness-of-fit test as defined by the FOCUS kinetics workgroup would pass
using an significance level $\alpha$ of 0.05. This relative error, expressed
as a percentage, is often termed $\chi^2$ error level or similar.

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, however, has become
somehow obsolete, as the use of compiled code described below gives even
smaller 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 (available from 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 3.3 of CAKE release in March 2016 uses a basic scheme for
up to six metabolites in a flexible arrangement, but does not support
back-reactions (non-instantaneous equilibria) or biphasic kinetics for metabolites.

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](https://pkgdown.jrwb.de/gmkin/) and
[manual](https://pkgdown.jrwb.de/gmkin/articles/gmkin_manual.html)
for further information.

A comparison of scope, usability and numerical results obtained with these
tools has been recently been published by @ranke2018.

## Recent developments

Currently (July 2019), the main features available in `mkin` which are not
present in KinGUII or CAKE, are the speed increase by using compiled code when
a compiler is present, parallel model fitting on multicore machines using the
`mmkin` function, and the estimation of parameter confidence intervals based on
transformed parameters.

In addition, the possibility to use two alternative error models to constant
variance have been integrated. The variance by variable error model introduced
by @gao11 has been available via an iteratively reweighted least squares (IRLS)
procedure since mkin
[version 0.9-22](https://pkgdown.jrwb.de/mkin/news/index.html#mkin-0-9-22-2013-10-26).
With [release 0.9.49.5](https://pkgdown.jrwb.de/mkin/news/index.html#mkin-0-9-49-5-2019-07-04),
the IRLS algorithm has been replaced 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](https://pkgdown.jrwb.de/mkin/reference/sigma_twocomp.html)
inspired by error models developed in analytical chemistry.

# Internal parameter transformations

For rate constants, the log
transformation is used, as proposed by Bates and Watts [-@bates1988, p. 77,
149]. Approximate intervals are constructed for the transformed rate
constants [compare @bates1988, p. 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 [@ranke2012], 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.

## 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
only one formation fraction is to be estimated, directly corresponding to one
component of the ilr transformed parameter.

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 [@FOCUSkinetics2014, p.  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
[@ranke2015]. 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

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