From 9ae42bd20bc2543a94cf1581ba9820c2f9e3afbd Mon Sep 17 00:00:00 2001 From: Johannes Ranke Date: Thu, 20 Apr 2023 19:53:28 +0200 Subject: Fix and rebuild documentation, see NEWS I had to fix the two pathway vignettes, as they did not work with the released version any more. So they and the multistart vignette which got some small fixes as well were rebuilt. Complete rebuild of the online docs with the released version. 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+
+ + + + +
+
+
+

Testing hierarchical pathway kinetics with +residue data on cyantraniliprole

+

Johannes +Ranke

+ +

Last change on 20 April 2023, +last compiled on 20 April 2023

+ + Source: vignettes/prebuilt/2022_cyan_pathway.rmd + + +
+ + + +
+

Introduction +

+

The purpose of this document is to test demonstrate how nonlinear +hierarchical models (NLHM) based on the parent degradation models SFO, +FOMC, DFOP and HS, with serial formation of two or more metabolites can +be fitted with the mkin package.

+

It was assembled in the course of work package 1.2 of Project Number +173340 (Application of nonlinear hierarchical models to the kinetic +evaluation of chemical degradation data) of the German Environment +Agency carried out in 2022 and 2023.

+

The mkin package is used in version 1.2.3 which is currently under +development. The newly introduced functionality that is used here is a +simplification of excluding random effects for a set of fits based on a +related set of fits with a reduced model, and the documentation of the +starting parameters of the fit, so that all starting parameters of +saem fits are now listed in the summary. The +saemix package is used as a backend for fitting the NLHM, +but is also loaded to make the convergence plot function available.

+

This document is processed with the knitr package, which +also provides the kable function that is used to improve +the display of tabular data in R markdown documents. For parallel +processing, the parallel package is used.

+
+library(mkin)
+library(knitr)
+library(saemix)
+library(parallel)
+n_cores <- detectCores()
+
+# We need to start a new cluster after defining a compiled model that is
+# saved as a DLL to the user directory, therefore we define a function
+# This is used again after defining the pathway model
+start_cluster <- function(n_cores) {
+  if (Sys.info()["sysname"] == "Windows") {
+    ret <- makePSOCKcluster(n_cores)
+  } else {
+    ret <- makeForkCluster(n_cores)
+  }
+  return(ret)
+}
+cl <- start_cluster(n_cores)
+
+

Test data +

+

The example data are taken from the final addendum to the DAR from +2014 and are distributed with the mkin package. Residue data and time +step normalisation factors are read in using the function +read_spreadsheet from the mkin package. This function also +performs the time step normalisation.

+
+data_file <- system.file(
+  "testdata", "cyantraniliprole_soil_efsa_2014.xlsx",
+  package = "mkin")
+cyan_ds <- read_spreadsheet(data_file, parent_only = FALSE)
+

The following tables show the covariate data and the 5 datasets that +were read in from the spreadsheet file.

+
+pH <- attr(cyan_ds, "covariates")
+kable(pH, caption = "Covariate data")
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Covariate data
pH
Nambsheim7.90
Tama6.20
Gross-Umstadt7.04
Sassafras4.62
Lleida8.05
+
+for (ds_name in names(cyan_ds)) {
+  print(
+    kable(mkin_long_to_wide(cyan_ds[[ds_name]]),
+      caption = paste("Dataset", ds_name),
+      booktabs = TRUE, row.names = FALSE))
+    cat("\n\\clearpage\n")
+}
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Nambsheim
timecyanJCZ38J9C38JSE76J9Z38
0.000000105.79NANANANA
3.21042477.267.9211.945.589.12
7.49098857.1315.4616.5812.5911.74
17.12225937.7415.9813.3626.0510.77
23.54310531.476.0514.4934.714.96
43.87578816.746.077.5740.386.52
67.4188938.8510.346.3930.718.90
107.0141165.199.611.9520.4112.93
129.4870803.456.181.3621.786.99
195.8358322.159.130.9516.297.69
254.6935961.926.920.2013.577.16
321.0423482.267.02NA11.128.66
383.110535NA5.05NA10.645.56
0.000000105.57NANANANA
3.21042478.8812.7711.945.479.12
7.49098859.9415.2716.5813.6011.74
17.12225939.6714.2613.3629.4410.77
23.54310530.2116.0714.4935.904.96
43.87578818.069.447.5742.306.52
67.4188938.545.786.3934.708.90
107.0141167.264.541.9523.3312.93
129.4870803.604.221.3623.566.99
195.8358322.843.050.9516.217.69
254.6935962.002.900.2015.537.16
321.0423481.790.94NA9.808.66
383.110535NA1.82NA9.495.56
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Tama
timecyanJCZ38J9Z38JSE76
0.000000106.14NANANA
2.40083393.476.462.85NA
5.60194388.3910.864.653.85
12.80444272.2911.974.9111.24
17.60610865.7913.116.6313.79
32.81138253.1611.248.9023.40
50.41749044.0111.349.9829.56
80.02776133.238.8211.3135.63
96.83359140.685.948.3229.09
146.45080320.654.498.7236.88
190.46607217.714.6611.1040.97
240.08328414.862.2711.6240.11
286.49938612.02NA10.7342.58
0.000000109.11NANANA
2.40083396.845.522.042.02
5.60194385.299.652.994.39
12.80444273.6812.485.0511.47
17.60610864.8912.446.2915.00
32.81138252.2710.867.6523.30
50.41749042.6110.549.3731.06
80.02776134.2910.029.0437.87
96.83359130.506.348.1433.97
146.45080319.216.298.5226.15
190.46607217.555.819.8932.08
240.08328413.225.9910.7940.66
286.49938611.096.058.8242.90
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Gross-Umstadt
timecyanJCZ38J9Z38JSE76
0.0000000103.03NANANA
2.101468187.854.793.260.62
4.903425577.358.059.891.32
10.507340469.339.7412.324.74
21.014680755.6514.5713.599.84
31.522021149.0314.6616.7112.32
42.029361541.8615.9713.6415.53
63.044042234.8818.2014.1222.02
84.058723028.2615.6414.0625.60
0.0000000104.05NANANA
2.101468185.252.687.320.69
4.903425577.227.288.371.45
10.507340465.2310.7310.934.74
21.014680757.7812.2914.809.05
31.522021154.8314.0512.0111.05
42.029361545.1712.1217.8915.71
63.044042234.8312.9015.8622.52
84.058723026.5914.2814.9128.48
0.0000000104.62NANANA
0.814522597.21NA4.00NA
1.900552589.643.595.24NA
4.072612587.904.109.58NA
8.145225186.905.969.45NA
12.217837674.747.8315.035.33
16.290450274.138.8414.415.10
24.435675365.2611.8418.336.71
32.580900457.7012.7419.939.74
0.0000000101.94NANANA
0.814522599.94NANANA
1.900552594.87NA4.56NA
4.072612586.966.756.90NA
8.145225180.5110.687.432.58
12.217837678.3810.359.463.69
16.290450270.0513.739.277.18
24.435675361.2812.5713.2813.19
32.580900452.8512.6712.9513.69
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Sassafras
timecyanJCZ38J9Z38JSE76
0.000000102.17NANANA
2.21671995.491.110.100.83
5.17234383.356.432.893.30
11.08359378.1810.005.590.81
22.16718670.4417.214.231.09
33.25077968.0020.455.861.17
44.33437159.6424.643.172.72
66.50155750.7327.506.191.27
88.66874245.6532.775.694.54
0.000000100.43NANANA
2.21671995.343.210.140.46
5.17234384.385.734.750.62
11.08359378.5011.893.990.73
22.16718671.1717.284.390.66
33.25077959.4118.7311.852.65
44.33437164.5722.935.132.01
66.50155749.0833.395.673.63
88.66874240.4139.605.936.17
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Lleida
timecyanJCZ38J9Z38JSE76
0.000000102.71NANANA
2.82105179.115.708.070.97
6.58245170.037.1711.314.72
14.10525350.9310.2514.849.95
28.21050533.4310.4014.8224.06
42.31575824.699.7516.3829.38
56.42101022.9910.0615.5129.25
84.63151614.635.6314.7431.04
112.84202112.434.1713.5333.28
0.00000099.31NANANA
2.82105182.076.555.601.12
6.58245170.657.618.013.21
14.10525353.5211.4810.8212.24
28.21050535.6011.1915.4323.53
42.31575834.2611.0913.2627.42
56.42101021.794.8018.3030.20
84.63151614.066.3016.3532.32
112.84202111.515.5712.6432.51
+
+
+
+

Parent only evaluations +

+

As the pathway fits have very long run times, evaluations of the +parent data are performed first, in order to determine for each +hierarchical parent degradation model which random effects on the +degradation model parameters are ill-defined.

+
+cyan_sep_const <- mmkin(c("SFO", "FOMC", "DFOP", "SFORB", "HS"),
+  cyan_ds, quiet = TRUE, cores = n_cores)
+cyan_sep_tc <- update(cyan_sep_const, error_model = "tc")
+cyan_saem_full <- mhmkin(list(cyan_sep_const, cyan_sep_tc))
+status(cyan_saem_full) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
consttc
SFOOKOK
FOMCOKOK
DFOPOKOK
SFORBOKOK
HSOKOK
+

All fits converged successfully.

+
+illparms(cyan_saem_full) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
consttc
SFOsd(cyan_0)sd(cyan_0)
FOMCsd(log_beta)sd(cyan_0)
DFOPsd(cyan_0)sd(cyan_0), sd(log_k1)
SFORBsd(cyan_free_0)sd(cyan_free_0), sd(log_k_cyan_free_bound)
HSsd(cyan_0)sd(cyan_0)
+

In almost all models, the random effect for the initial concentration +of the parent compound is ill-defined. For the biexponential models DFOP +and SFORB, the random effect of one additional parameter is ill-defined +when the two-component error model is used.

+
+anova(cyan_saem_full) |> kable(digits = 1)
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
nparAICBICLik
SFO const5833.9832.0-412.0
SFO tc6831.6829.3-409.8
FOMC const7709.1706.4-347.6
FOMC tc8689.2686.1-336.6
DFOP const9703.0699.5-342.5
SFORB const9701.3697.8-341.7
HS const9718.6715.1-350.3
DFOP tc10703.1699.2-341.6
SFORB tc10700.1696.2-340.1
HS tc10716.7712.8-348.3
+

Model comparison based on AIC and BIC indicates that the +two-component error model is preferable for all parent models with the +exception of DFOP. The lowest AIC and BIC values are are obtained with +the FOMC model, followed by SFORB and DFOP.

+ +
+
+

Pathway fits +

+
+

Evaluations with pathway established previously +

+

To test the technical feasibility of coupling the relevant parent +degradation models with different transformation pathway models, a list +of mkinmod models is set up below. As in the EU evaluation, +parallel formation of metabolites JCZ38 and J9Z38 and secondary +formation of metabolite JSE76 from JCZ38 is used.

+
+if (!dir.exists("cyan_dlls")) dir.create("cyan_dlls")
+cyan_path_1 <- list(
+  sfo_path_1 = mkinmod(
+    cyan = mkinsub("SFO", c("JCZ38", "J9Z38")),
+    JCZ38 = mkinsub("SFO", "JSE76"),
+    J9Z38 = mkinsub("SFO"),
+    JSE76 = mkinsub("SFO"), quiet = TRUE,
+    name = "sfo_path_1", dll_dir = "cyan_dlls", overwrite = TRUE),
+  fomc_path_1 = mkinmod(
+    cyan = mkinsub("FOMC", c("JCZ38", "J9Z38")),
+    JCZ38 = mkinsub("SFO", "JSE76"),
+    J9Z38 = mkinsub("SFO"),
+    JSE76 = mkinsub("SFO"), quiet = TRUE,
+    name = "fomc_path_1", dll_dir = "cyan_dlls", overwrite = TRUE),
+  dfop_path_1 = mkinmod(
+    cyan = mkinsub("DFOP", c("JCZ38", "J9Z38")),
+    JCZ38 = mkinsub("SFO", "JSE76"),
+    J9Z38 = mkinsub("SFO"),
+    JSE76 = mkinsub("SFO"), quiet = TRUE,
+    name = "dfop_path_1", dll_dir = "cyan_dlls", overwrite = TRUE),
+  sforb_path_1 = mkinmod(
+    cyan = mkinsub("SFORB", c("JCZ38", "J9Z38")),
+    JCZ38 = mkinsub("SFO", "JSE76"),
+    J9Z38 = mkinsub("SFO"),
+    JSE76 = mkinsub("SFO"), quiet = TRUE,
+    name = "sforb_path_1", dll_dir = "cyan_dlls", overwrite = TRUE),
+  hs_path_1 = mkinmod(
+    cyan = mkinsub("HS", c("JCZ38", "J9Z38")),
+    JCZ38 = mkinsub("SFO", "JSE76"),
+    J9Z38 = mkinsub("SFO"),
+    JSE76 = mkinsub("SFO"), quiet = TRUE,
+    name = "hs_path_1", dll_dir = "cyan_dlls", overwrite = TRUE)
+)
+cl_path_1 <- start_cluster(n_cores)
+

To obtain suitable starting values for the NLHM fits, separate +pathway fits are performed for all datasets.

+
+f_sep_1_const <- mmkin(
+  cyan_path_1,
+  cyan_ds,
+  error_model = "const",
+  cluster = cl_path_1,
+  quiet = TRUE)
+status(f_sep_1_const) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NambsheimTamaGross-UmstadtSassafrasLleida
sfo_path_1OKOKOKOKOK
fomc_path_1OKOKOKOKOK
dfop_path_1OKOKOKOKOK
sforb_path_1OKOKOKOKOK
hs_path_1CCCCC
+
+f_sep_1_tc <- update(f_sep_1_const, error_model = "tc")
+status(f_sep_1_tc) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NambsheimTamaGross-UmstadtSassafrasLleida
sfo_path_1OKOKOKOKOK
fomc_path_1OKOKOKOKC
dfop_path_1OKOKOKOKOK
sforb_path_1OKCOKOKOK
hs_path_1COKCOKOK
+

Most separate fits converged successfully. The biggest convergence +problems are seen when using the HS model with constant variance.

+

For the hierarchical pathway fits, those random effects that could +not be quantified in the corresponding parent data analyses are +excluded.

+

In the code below, the output of the illparms function +for the parent only fits is used as an argument +no_random_effect to the mhmkin function. The +possibility to do so was introduced in mkin version 1.2.2 +which is currently under development.

+
+f_saem_1 <- mhmkin(list(f_sep_1_const, f_sep_1_tc),
+  no_random_effect = illparms(cyan_saem_full),
+  cluster = cl_path_1)
+
+status(f_saem_1) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
consttc
sfo_path_1Fth, FOFth, FO
fomc_path_1OKFth, FO
dfop_path_1Fth, FOFth, FO
sforb_path_1Fth, FOFth, FO
hs_path_1Fth, FOFth, FO
+

The status information from the individual fits shows that all fits +completed successfully. The matrix entries Fth and FO indicate that the +Fisher Information Matrix could not be inverted for the fixed effects +(theta) and the random effects (Omega), respectively. For the affected +fits, ill-defined parameters cannot be determined using the +illparms function, because it relies on the Fisher +Information Matrix.

+
+illparms(f_saem_1) |> kable()
+ +++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
consttc
sfo_path_1NANA
fomc_path_1sd(log_k_J9Z38), sd(f_cyan_ilr_2), +sd(f_JCZ38_qlogis)NA
dfop_path_1NANA
sforb_path_1NANA
hs_path_1NANA
+

The model comparison below suggests that the pathway fits using DFOP +or SFORB for the parent compound provide the best fit.

+
+anova(f_saem_1) |> kable(digits = 1)
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
nparAICBICLik
sfo_path_1 const162692.82686.6-1330.4
sfo_path_1 tc172657.72651.1-1311.9
fomc_path_1 const182427.82420.8-1195.9
fomc_path_1 tc192423.42416.0-1192.7
dfop_path_1 const202403.22395.4-1181.6
sforb_path_1 const202401.42393.6-1180.7
hs_path_1 const202427.32419.5-1193.7
dfop_path_1 tc202398.02390.2-1179.0
sforb_path_1 tc202399.82392.0-1179.9
hs_path_1 tc212422.32414.1-1190.2
+

For these two parent model, successful fits are shown below. Plots of +the fits with the other parent models are shown in the Appendix.

+
+plot(f_saem_1[["dfop_path_1", "tc"]])
+
+DFOP pathway fit with two-component error

+DFOP pathway fit with two-component error +

+
+
+plot(f_saem_1[["sforb_path_1", "tc"]])
+
+SFORB pathway fit with two-component error

+SFORB pathway fit with two-component error +

+
+

A closer graphical analysis of these Figures shows that the residues +of transformation product JCZ38 in the soils Tama and Nambsheim observed +at later time points are strongly and systematically underestimated.

+
+stopCluster(cl_path_1)
+
+
+

Alternative pathway fits +

+

To improve the fit for JCZ38, a back-reaction from JSE76 to JCZ38 was +introduced in an alternative version of the transformation pathway, in +analogy to the back-reaction from K5A78 to K5A77. Both pairs of +transformation products are pairs of an organic acid with its +corresponding amide (Addendum 2014, p. 109). As FOMC provided the best +fit for the parent, and the biexponential models DFOP and SFORB provided +the best initial pathway fits, these three parent models are used in the +alternative pathway fits.

+
+cyan_path_2 <- list(
+  fomc_path_2 = mkinmod(
+    cyan = mkinsub("FOMC", c("JCZ38", "J9Z38")),
+    JCZ38 = mkinsub("SFO", "JSE76"),
+    J9Z38 = mkinsub("SFO"),
+    JSE76 = mkinsub("SFO", "JCZ38"),
+    name = "fomc_path_2", quiet = TRUE,
+    dll_dir = "cyan_dlls",
+    overwrite = TRUE
+  ),
+  dfop_path_2 = mkinmod(
+    cyan = mkinsub("DFOP", c("JCZ38", "J9Z38")),
+    JCZ38 = mkinsub("SFO", "JSE76"),
+    J9Z38 = mkinsub("SFO"),
+    JSE76 = mkinsub("SFO", "JCZ38"),
+    name = "dfop_path_2", quiet = TRUE,
+    dll_dir = "cyan_dlls",
+    overwrite = TRUE
+  ),
+  sforb_path_2 = mkinmod(
+    cyan = mkinsub("SFORB", c("JCZ38", "J9Z38")),
+    JCZ38 = mkinsub("SFO", "JSE76"),
+    J9Z38 = mkinsub("SFO"),
+    JSE76 = mkinsub("SFO", "JCZ38"),
+    name = "sforb_path_2", quiet = TRUE,
+    dll_dir = "cyan_dlls",
+    overwrite = TRUE
+  )
+)
+
+cl_path_2 <- start_cluster(n_cores)
+f_sep_2_const <- mmkin(
+  cyan_path_2,
+  cyan_ds,
+  error_model = "const",
+  cluster = cl_path_2,
+  quiet = TRUE)
+
+status(f_sep_2_const) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NambsheimTamaGross-UmstadtSassafrasLleida
fomc_path_2OKOKOKCOK
dfop_path_2OKOKOKCOK
sforb_path_2OKOKOKCOK
+

Using constant variance, separate fits converge with the exception of +the fits to the Sassafras soil data.

+
+f_sep_2_tc <- update(f_sep_2_const, error_model = "tc")
+status(f_sep_2_tc) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NambsheimTamaGross-UmstadtSassafrasLleida
fomc_path_2OKCOKCOK
dfop_path_2OKOKOKCOK
sforb_path_2OKOKOKOKOK
+

Using the two-component error model, all separate fits converge with +the exception of the alternative pathway fit with DFOP used for the +parent and the Sassafras dataset.

+
+f_saem_2 <- mhmkin(list(f_sep_2_const, f_sep_2_tc),
+  no_random_effect = illparms(cyan_saem_full[2:4, ]),
+  cluster = cl_path_2)
+
+status(f_saem_2) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + +
consttc
fomc_path_2OKFO
dfop_path_2OKOK
sforb_path_2OKOK
+

The hierarchical fits for the alternative pathway completed +successfully.

+
+illparms(f_saem_2) |> kable()
+ +++++ + + + + + + + + + + + + + + + + + + + + + + +
consttc
fomc_path_2sd(f_JCZ38_qlogis), sd(f_JSE76_qlogis)NA
dfop_path_2sd(f_JCZ38_qlogis), sd(f_JSE76_qlogis)sd(f_JCZ38_qlogis), sd(f_JSE76_qlogis)
sforb_path_2sd(f_JCZ38_qlogis), sd(f_JSE76_qlogis)sd(f_JCZ38_qlogis), sd(f_JSE76_qlogis)
+

In both fits, the random effects for the formation fractions for the +pathways from JCZ38 to JSE76, and for the reverse pathway from JSE76 to +JCZ38 are ill-defined.

+
+anova(f_saem_2) |> kable(digits = 1)
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
nparAICBICLik
fomc_path_2 const202308.32300.5-1134.2
fomc_path_2 tc212248.32240.1-1103.2
dfop_path_2 const222289.62281.0-1122.8
sforb_path_2 const222284.12275.5-1120.0
dfop_path_2 tc222234.42225.8-1095.2
sforb_path_2 tc222240.42231.8-1098.2
+

The variants using the biexponential models DFOP and SFORB for the +parent compound and the two-component error model give the lowest AIC +and BIC values and are plotted below. Compared with the original +pathway, the AIC and BIC values indicate a large improvement. This is +confirmed by the plots, which show that the metabolite JCZ38 is fitted +much better with this model.

+
+plot(f_saem_2[["fomc_path_2", "tc"]])
+
+FOMC pathway fit with two-component error, alternative pathway

+FOMC pathway fit with two-component error, alternative pathway +

+
+
+plot(f_saem_2[["dfop_path_2", "tc"]])
+
+DFOP pathway fit with two-component error, alternative pathway

+DFOP pathway fit with two-component error, alternative pathway +

+
+
+plot(f_saem_2[["sforb_path_2", "tc"]])
+
+SFORB pathway fit with two-component error, alternative pathway

+SFORB pathway fit with two-component error, alternative pathway +

+
+
+
+

Refinement of alternative pathway fits +

+

All ill-defined random effects that were identified in the parent +only fits and in the above pathway fits, are excluded for the final +evaluations below. For this purpose, a list of character vectors is +created below that can be indexed by row and column indices, and which +contains the degradation parameter names for which random effects should +be excluded for each of the hierarchical fits contained in +f_saem_2.

+
+no_ranef <- matrix(list(), nrow = 3, ncol = 2, dimnames = dimnames(f_saem_2))
+no_ranef[["fomc_path_2", "const"]] <- c("log_beta", "f_JCZ38_qlogis", "f_JSE76_qlogis")
+no_ranef[["fomc_path_2", "tc"]] <- c("cyan_0", "f_JCZ38_qlogis", "f_JSE76_qlogis")
+no_ranef[["dfop_path_2", "const"]] <- c("cyan_0", "f_JCZ38_qlogis", "f_JSE76_qlogis")
+no_ranef[["dfop_path_2", "tc"]] <- c("cyan_0", "log_k1", "f_JCZ38_qlogis", "f_JSE76_qlogis")
+no_ranef[["sforb_path_2", "const"]] <- c("cyan_free_0",
+  "f_JCZ38_qlogis", "f_JSE76_qlogis")
+no_ranef[["sforb_path_2", "tc"]] <- c("cyan_free_0", "log_k_cyan_free_bound",
+  "f_JCZ38_qlogis", "f_JSE76_qlogis")
+clusterExport(cl_path_2, "no_ranef")
+
+f_saem_3 <- update(f_saem_2,
+  no_random_effect = no_ranef,
+  cluster = cl_path_2)
+
+status(f_saem_3) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + +
consttc
fomc_path_2EFth
dfop_path_2FthFth
sforb_path_2FthFth
+

With the exception of the FOMC pathway fit with constant variance, +all updated fits completed successfully. However, the Fisher Information +Matrix for the fixed effects (Fth) could not be inverted, so no +confidence intervals for the optimised parameters are available.

+
+illparms(f_saem_3) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + +
consttc
fomc_path_2E
dfop_path_2
sforb_path_2
+
+anova(f_saem_3) |> kable(digits = 1)
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
nparAICBICLik
fomc_path_2 tc192250.92243.5-1106.5
dfop_path_2 const202281.72273.9-1120.8
sforb_path_2 const202279.52271.7-1119.7
dfop_path_2 tc202231.52223.7-1095.8
sforb_path_2 tc202235.72227.9-1097.9
+

While the AIC and BIC values of the best fit (DFOP pathway fit with +two-component error) are lower than in the previous fits with the +alternative pathway, the practical value of these refined evaluations is +limited as no confidence intervals are obtained.

+
+stopCluster(cl_path_2)
+
+
+
+

Conclusion +

+

It was demonstrated that a relatively complex transformation pathway +with parallel formation of two primary metabolites and one secondary +metabolite can be fitted even if the data in the individual datasets are +quite different and partly only cover the formation phase.

+

The run times of the pathway fits were several hours, limiting the +practical feasibility of iterative refinements based on ill-defined +parameters and of alternative checks of parameter identifiability based +on multistart runs.

+
+
+

Acknowledgements +

+

The helpful comments by Janina Wöltjen of the German Environment +Agency are gratefully acknowledged.

+
+
+

Appendix +

+
+

Plots of fits that were not refined further +

+
+plot(f_saem_1[["sfo_path_1", "tc"]])
+
+SFO pathway fit with two-component error

+SFO pathway fit with two-component error +

+
+
+plot(f_saem_1[["fomc_path_1", "tc"]])
+
+FOMC pathway fit with two-component error

+FOMC pathway fit with two-component error +

+
+
+plot(f_saem_1[["sforb_path_1", "tc"]])
+
+HS pathway fit with two-component error

+HS pathway fit with two-component error +

+
+
+
+

Hierarchical fit listings +

+
+

Pathway 1 +

+ +Hierarchical SFO path 1 fit with constant variance + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 07:44:55 2023 
+Date of summary: Thu Apr 20 20:01:30 2023 
+
+Equations:
+d_cyan/dt = - k_cyan * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * k_cyan * cyan - k_JCZ38 * JCZ38
+d_J9Z38/dt = + f_cyan_to_J9Z38 * k_cyan * cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 431.793 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+        cyan_0     log_k_cyan    log_k_JCZ38    log_k_J9Z38    log_k_JSE76 
+       95.3304        -3.8459        -3.1305        -5.0678        -5.3196 
+  f_cyan_ilr_1   f_cyan_ilr_2 f_JCZ38_qlogis 
+        0.8158        22.5404        10.4289 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_cyan log_k_JCZ38 log_k_J9Z38 log_k_JSE76
+cyan_0          4.797     0.0000       0.000       0.000      0.0000
+log_k_cyan      0.000     0.9619       0.000       0.000      0.0000
+log_k_JCZ38     0.000     0.0000       2.139       0.000      0.0000
+log_k_J9Z38     0.000     0.0000       0.000       1.639      0.0000
+log_k_JSE76     0.000     0.0000       0.000       0.000      0.7894
+f_cyan_ilr_1    0.000     0.0000       0.000       0.000      0.0000
+f_cyan_ilr_2    0.000     0.0000       0.000       0.000      0.0000
+f_JCZ38_qlogis  0.000     0.0000       0.000       0.000      0.0000
+               f_cyan_ilr_1 f_cyan_ilr_2 f_JCZ38_qlogis
+cyan_0               0.0000        0.000           0.00
+log_k_cyan           0.0000        0.000           0.00
+log_k_JCZ38          0.0000        0.000           0.00
+log_k_J9Z38          0.0000        0.000           0.00
+log_k_JSE76          0.0000        0.000           0.00
+f_cyan_ilr_1         0.7714        0.000           0.00
+f_cyan_ilr_2         0.0000        8.684           0.00
+f_JCZ38_qlogis       0.0000        0.000          13.48
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2693 2687  -1330
+
+Optimised parameters:
+                     est. lower upper
+cyan_0            95.0946    NA    NA
+log_k_cyan        -3.8544    NA    NA
+log_k_JCZ38       -3.0402    NA    NA
+log_k_J9Z38       -5.0109    NA    NA
+log_k_JSE76       -5.2857    NA    NA
+f_cyan_ilr_1       0.8069    NA    NA
+f_cyan_ilr_2      16.6623    NA    NA
+f_JCZ38_qlogis     1.3602    NA    NA
+a.1                4.8326    NA    NA
+SD.log_k_cyan      0.5842    NA    NA
+SD.log_k_JCZ38     1.2680    NA    NA
+SD.log_k_J9Z38     0.3626    NA    NA
+SD.log_k_JSE76     0.5244    NA    NA
+SD.f_cyan_ilr_1    0.2752    NA    NA
+SD.f_cyan_ilr_2    2.3556    NA    NA
+SD.f_JCZ38_qlogis  0.2400    NA    NA
+
+Correlation is not available
+
+Random effects:
+                    est. lower upper
+SD.log_k_cyan     0.5842    NA    NA
+SD.log_k_JCZ38    1.2680    NA    NA
+SD.log_k_J9Z38    0.3626    NA    NA
+SD.log_k_JSE76    0.5244    NA    NA
+SD.f_cyan_ilr_1   0.2752    NA    NA
+SD.f_cyan_ilr_2   2.3556    NA    NA
+SD.f_JCZ38_qlogis 0.2400    NA    NA
+
+Variance model:
+     est. lower upper
+a.1 4.833    NA    NA
+
+Backtransformed parameters:
+                      est. lower upper
+cyan_0           95.094581    NA    NA
+k_cyan            0.021186    NA    NA
+k_JCZ38           0.047825    NA    NA
+k_J9Z38           0.006665    NA    NA
+k_JSE76           0.005063    NA    NA
+f_cyan_to_JCZ38   0.757885    NA    NA
+f_cyan_to_J9Z38   0.242115    NA    NA
+f_JCZ38_to_JSE76  0.795792    NA    NA
+
+Resulting formation fractions:
+                   ff
+cyan_JCZ38  7.579e-01
+cyan_J9Z38  2.421e-01
+cyan_sink   5.877e-10
+JCZ38_JSE76 7.958e-01
+JCZ38_sink  2.042e-01
+
+Estimated disappearance times:
+        DT50   DT90
+cyan   32.72 108.68
+JCZ38  14.49  48.15
+J9Z38 103.99 345.46
+JSE76 136.90 454.76
+
+
+

+ +Hierarchical SFO path 1 fit with two-component error + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 07:44:53 2023 
+Date of summary: Thu Apr 20 20:01:30 2023 
+
+Equations:
+d_cyan/dt = - k_cyan * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * k_cyan * cyan - k_JCZ38 * JCZ38
+d_J9Z38/dt = + f_cyan_to_J9Z38 * k_cyan * cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 429.526 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+        cyan_0     log_k_cyan    log_k_JCZ38    log_k_J9Z38    log_k_JSE76 
+       96.0039        -3.8907        -3.1276        -5.0069        -4.9367 
+  f_cyan_ilr_1   f_cyan_ilr_2 f_JCZ38_qlogis 
+        0.7937        20.0030        15.1336 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_cyan log_k_JCZ38 log_k_J9Z38 log_k_JSE76
+cyan_0          4.859      0.000        0.00        0.00      0.0000
+log_k_cyan      0.000      0.962        0.00        0.00      0.0000
+log_k_JCZ38     0.000      0.000        2.04        0.00      0.0000
+log_k_J9Z38     0.000      0.000        0.00        1.72      0.0000
+log_k_JSE76     0.000      0.000        0.00        0.00      0.9076
+f_cyan_ilr_1    0.000      0.000        0.00        0.00      0.0000
+f_cyan_ilr_2    0.000      0.000        0.00        0.00      0.0000
+f_JCZ38_qlogis  0.000      0.000        0.00        0.00      0.0000
+               f_cyan_ilr_1 f_cyan_ilr_2 f_JCZ38_qlogis
+cyan_0               0.0000        0.000           0.00
+log_k_cyan           0.0000        0.000           0.00
+log_k_JCZ38          0.0000        0.000           0.00
+log_k_J9Z38          0.0000        0.000           0.00
+log_k_JSE76          0.0000        0.000           0.00
+f_cyan_ilr_1         0.7598        0.000           0.00
+f_cyan_ilr_2         0.0000        7.334           0.00
+f_JCZ38_qlogis       0.0000        0.000          11.78
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2658 2651  -1312
+
+Optimised parameters:
+                      est. lower upper
+cyan_0            94.72923    NA    NA
+log_k_cyan        -3.91670    NA    NA
+log_k_JCZ38       -3.12917    NA    NA
+log_k_J9Z38       -5.06070    NA    NA
+log_k_JSE76       -5.09254    NA    NA
+f_cyan_ilr_1       0.81116    NA    NA
+f_cyan_ilr_2      39.97850    NA    NA
+f_JCZ38_qlogis     3.09728    NA    NA
+a.1                3.95044    NA    NA
+b.1                0.07998    NA    NA
+SD.log_k_cyan      0.58855    NA    NA
+SD.log_k_JCZ38     1.29753    NA    NA
+SD.log_k_J9Z38     0.62851    NA    NA
+SD.log_k_JSE76     0.37235    NA    NA
+SD.f_cyan_ilr_1    0.37346    NA    NA
+SD.f_cyan_ilr_2    1.41667    NA    NA
+SD.f_JCZ38_qlogis  1.81467    NA    NA
+
+Correlation is not available
+
+Random effects:
+                    est. lower upper
+SD.log_k_cyan     0.5886    NA    NA
+SD.log_k_JCZ38    1.2975    NA    NA
+SD.log_k_J9Z38    0.6285    NA    NA
+SD.log_k_JSE76    0.3724    NA    NA
+SD.f_cyan_ilr_1   0.3735    NA    NA
+SD.f_cyan_ilr_2   1.4167    NA    NA
+SD.f_JCZ38_qlogis 1.8147    NA    NA
+
+Variance model:
+       est. lower upper
+a.1 3.95044    NA    NA
+b.1 0.07998    NA    NA
+
+Backtransformed parameters:
+                      est. lower upper
+cyan_0           94.729229    NA    NA
+k_cyan            0.019907    NA    NA
+k_JCZ38           0.043754    NA    NA
+k_J9Z38           0.006341    NA    NA
+k_JSE76           0.006142    NA    NA
+f_cyan_to_JCZ38   0.758991    NA    NA
+f_cyan_to_J9Z38   0.241009    NA    NA
+f_JCZ38_to_JSE76  0.956781    NA    NA
+
+Resulting formation fractions:
+                 ff
+cyan_JCZ38  0.75899
+cyan_J9Z38  0.24101
+cyan_sink   0.00000
+JCZ38_JSE76 0.95678
+JCZ38_sink  0.04322
+
+Estimated disappearance times:
+        DT50   DT90
+cyan   34.82 115.67
+JCZ38  15.84  52.63
+J9Z38 109.31 363.12
+JSE76 112.85 374.87
+
+
+

+ +Hierarchical FOMC path 1 fit with constant variance + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 07:45:50 2023 
+Date of summary: Thu Apr 20 20:01:30 2023 
+
+Equations:
+d_cyan/dt = - (alpha/beta) * 1/((time/beta) + 1) * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_JCZ38 * JCZ38
+d_J9Z38/dt = + f_cyan_to_J9Z38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 477.996 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      101.2314        -3.3680        -5.1108        -5.9416         0.7144 
+  f_cyan_ilr_2 f_JCZ38_qlogis      log_alpha       log_beta 
+        7.3870        15.7604        -0.1791         2.9811 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          5.416       0.000         0.0       0.000       0.0000
+log_k_JCZ38     0.000       2.439         0.0       0.000       0.0000
+log_k_J9Z38     0.000       0.000         1.7       0.000       0.0000
+log_k_JSE76     0.000       0.000         0.0       1.856       0.0000
+f_cyan_ilr_1    0.000       0.000         0.0       0.000       0.7164
+f_cyan_ilr_2    0.000       0.000         0.0       0.000       0.0000
+f_JCZ38_qlogis  0.000       0.000         0.0       0.000       0.0000
+log_alpha       0.000       0.000         0.0       0.000       0.0000
+log_beta        0.000       0.000         0.0       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis log_alpha log_beta
+cyan_0                 0.00           0.00    0.0000   0.0000
+log_k_JCZ38            0.00           0.00    0.0000   0.0000
+log_k_J9Z38            0.00           0.00    0.0000   0.0000
+log_k_JSE76            0.00           0.00    0.0000   0.0000
+f_cyan_ilr_1           0.00           0.00    0.0000   0.0000
+f_cyan_ilr_2          12.33           0.00    0.0000   0.0000
+f_JCZ38_qlogis         0.00          20.42    0.0000   0.0000
+log_alpha              0.00           0.00    0.4144   0.0000
+log_beta               0.00           0.00    0.0000   0.5077
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2428 2421  -1196
+
+Optimised parameters:
+                      est.     lower    upper
+cyan_0            101.0225 98.306270 103.7387
+log_k_JCZ38        -3.3786 -4.770657  -1.9866
+log_k_J9Z38        -5.2603 -5.902085  -4.6186
+log_k_JSE76        -6.1427 -7.318336  -4.9671
+f_cyan_ilr_1        0.7437  0.421215   1.0663
+f_cyan_ilr_2        0.9108  0.267977   1.5537
+f_JCZ38_qlogis      2.0487  0.524897   3.5724
+log_alpha          -0.2268 -0.618049   0.1644
+log_beta            2.8986  2.700701   3.0964
+a.1                 3.4058  3.169913   3.6416
+SD.cyan_0           2.5279  0.454190   4.6016
+SD.log_k_JCZ38      1.5636  0.572824   2.5543
+SD.log_k_J9Z38      0.5316 -0.004405   1.0677
+SD.log_k_JSE76      0.9903  0.106325   1.8742
+SD.f_cyan_ilr_1     0.3464  0.112066   0.5807
+SD.f_cyan_ilr_2     0.2804 -0.393900   0.9546
+SD.f_JCZ38_qlogis   0.9416 -0.152986   2.0362
+SD.log_alpha        0.4273  0.161044   0.6936
+
+Correlation: 
+               cyan_0  l__JCZ3 l__J9Z3 l__JSE7 f_cy__1 f_cy__2 f_JCZ38 log_lph
+log_k_JCZ38    -0.0156                                                        
+log_k_J9Z38    -0.0493  0.0073                                                
+log_k_JSE76    -0.0329  0.0018  0.0069                                        
+f_cyan_ilr_1   -0.0086  0.0180 -0.1406  0.0012                                
+f_cyan_ilr_2   -0.2629  0.0779  0.2826  0.0274  0.0099                        
+f_JCZ38_qlogis  0.0713 -0.0747 -0.0505  0.1169 -0.1022 -0.4893                
+log_alpha      -0.0556  0.0120  0.0336  0.0193  0.0036  0.0840 -0.0489        
+log_beta       -0.2898  0.0460  0.1305  0.0768  0.0190  0.4071 -0.1981  0.2772
+
+Random effects:
+                    est.     lower  upper
+SD.cyan_0         2.5279  0.454190 4.6016
+SD.log_k_JCZ38    1.5636  0.572824 2.5543
+SD.log_k_J9Z38    0.5316 -0.004405 1.0677
+SD.log_k_JSE76    0.9903  0.106325 1.8742
+SD.f_cyan_ilr_1   0.3464  0.112066 0.5807
+SD.f_cyan_ilr_2   0.2804 -0.393900 0.9546
+SD.f_JCZ38_qlogis 0.9416 -0.152986 2.0362
+SD.log_alpha      0.4273  0.161044 0.6936
+
+Variance model:
+     est. lower upper
+a.1 3.406  3.17 3.642
+
+Backtransformed parameters:
+                      est.     lower     upper
+cyan_0           1.010e+02 9.831e+01 1.037e+02
+k_JCZ38          3.409e-02 8.475e-03 1.372e-01
+k_J9Z38          5.194e-03 2.734e-03 9.867e-03
+k_JSE76          2.149e-03 6.633e-04 6.963e-03
+f_cyan_to_JCZ38  6.481e-01        NA        NA
+f_cyan_to_J9Z38  2.264e-01        NA        NA
+f_JCZ38_to_JSE76 8.858e-01 6.283e-01 9.727e-01
+alpha            7.971e-01 5.390e-01 1.179e+00
+beta             1.815e+01 1.489e+01 2.212e+01
+
+Resulting formation fractions:
+                ff
+cyan_JCZ38  0.6481
+cyan_J9Z38  0.2264
+cyan_sink   0.1255
+JCZ38_JSE76 0.8858
+JCZ38_sink  0.1142
+
+Estimated disappearance times:
+        DT50    DT90 DT50back
+cyan   25.15  308.01    92.72
+JCZ38  20.33   67.54       NA
+J9Z38 133.46  443.35       NA
+JSE76 322.53 1071.42       NA
+
+
+

+ +Hierarchical FOMC path 1 fit with two-component error + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 07:45:45 2023 
+Date of summary: Thu Apr 20 20:01:30 2023 
+
+Equations:
+d_cyan/dt = - (alpha/beta) * 1/((time/beta) + 1) * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_JCZ38 * JCZ38
+d_J9Z38/dt = + f_cyan_to_J9Z38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 480.648 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+     101.13827       -3.32493       -5.08921       -5.93478        0.71330 
+  f_cyan_ilr_2 f_JCZ38_qlogis      log_alpha       log_beta 
+      10.05989       12.79248       -0.09621        3.10646 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          5.643       0.000       0.000        0.00       0.0000
+log_k_JCZ38     0.000       2.319       0.000        0.00       0.0000
+log_k_J9Z38     0.000       0.000       1.731        0.00       0.0000
+log_k_JSE76     0.000       0.000       0.000        1.86       0.0000
+f_cyan_ilr_1    0.000       0.000       0.000        0.00       0.7186
+f_cyan_ilr_2    0.000       0.000       0.000        0.00       0.0000
+f_JCZ38_qlogis  0.000       0.000       0.000        0.00       0.0000
+log_alpha       0.000       0.000       0.000        0.00       0.0000
+log_beta        0.000       0.000       0.000        0.00       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis log_alpha log_beta
+cyan_0                 0.00           0.00    0.0000   0.0000
+log_k_JCZ38            0.00           0.00    0.0000   0.0000
+log_k_J9Z38            0.00           0.00    0.0000   0.0000
+log_k_JSE76            0.00           0.00    0.0000   0.0000
+f_cyan_ilr_1           0.00           0.00    0.0000   0.0000
+f_cyan_ilr_2          12.49           0.00    0.0000   0.0000
+f_JCZ38_qlogis         0.00          20.19    0.0000   0.0000
+log_alpha              0.00           0.00    0.3142   0.0000
+log_beta               0.00           0.00    0.0000   0.7331
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2423 2416  -1193
+
+Optimised parameters:
+                       est. lower upper
+cyan_0            100.57649    NA    NA
+log_k_JCZ38        -3.46250    NA    NA
+log_k_J9Z38        -5.24442    NA    NA
+log_k_JSE76        -5.75229    NA    NA
+f_cyan_ilr_1        0.68480    NA    NA
+f_cyan_ilr_2        0.61670    NA    NA
+f_JCZ38_qlogis     87.97407    NA    NA
+log_alpha          -0.15699    NA    NA
+log_beta            3.01540    NA    NA
+a.1                 3.11518    NA    NA
+b.1                 0.04445    NA    NA
+SD.log_k_JCZ38      1.40732    NA    NA
+SD.log_k_J9Z38      0.56510    NA    NA
+SD.log_k_JSE76      0.72067    NA    NA
+SD.f_cyan_ilr_1     0.31199    NA    NA
+SD.f_cyan_ilr_2     0.36894    NA    NA
+SD.f_JCZ38_qlogis   6.92892    NA    NA
+SD.log_alpha        0.25662    NA    NA
+SD.log_beta         0.35845    NA    NA
+
+Correlation is not available
+
+Random effects:
+                    est. lower upper
+SD.log_k_JCZ38    1.4073    NA    NA
+SD.log_k_J9Z38    0.5651    NA    NA
+SD.log_k_JSE76    0.7207    NA    NA
+SD.f_cyan_ilr_1   0.3120    NA    NA
+SD.f_cyan_ilr_2   0.3689    NA    NA
+SD.f_JCZ38_qlogis 6.9289    NA    NA
+SD.log_alpha      0.2566    NA    NA
+SD.log_beta       0.3585    NA    NA
+
+Variance model:
+       est. lower upper
+a.1 3.11518    NA    NA
+b.1 0.04445    NA    NA
+
+Backtransformed parameters:
+                      est. lower upper
+cyan_0           1.006e+02    NA    NA
+k_JCZ38          3.135e-02    NA    NA
+k_J9Z38          5.277e-03    NA    NA
+k_JSE76          3.175e-03    NA    NA
+f_cyan_to_JCZ38  5.991e-01    NA    NA
+f_cyan_to_J9Z38  2.275e-01    NA    NA
+f_JCZ38_to_JSE76 1.000e+00    NA    NA
+alpha            8.547e-01    NA    NA
+beta             2.040e+01    NA    NA
+
+Resulting formation fractions:
+                ff
+cyan_JCZ38  0.5991
+cyan_J9Z38  0.2275
+cyan_sink   0.1734
+JCZ38_JSE76 1.0000
+JCZ38_sink  0.0000
+
+Estimated disappearance times:
+        DT50   DT90 DT50back
+cyan   25.50 281.29    84.68
+JCZ38  22.11  73.44       NA
+J9Z38 131.36 436.35       NA
+JSE76 218.28 725.11       NA
+
+
+

+ +Hierarchical DFOP path 1 fit with constant variance + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 07:46:41 2023 
+Date of summary: Thu Apr 20 20:01:30 2023 
+
+Equations:
+d_cyan/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_JCZ38 * JCZ38
+d_J9Z38/dt = + f_cyan_to_J9Z38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 528.713 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      102.0644        -3.4008        -5.0024        -5.8613         0.6855 
+  f_cyan_ilr_2 f_JCZ38_qlogis         log_k1         log_k2       g_qlogis 
+        1.2365        13.7245        -1.8641        -4.5063        -0.6468 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          4.466       0.000       0.000       0.000       0.0000
+log_k_JCZ38     0.000       2.382       0.000       0.000       0.0000
+log_k_J9Z38     0.000       0.000       1.595       0.000       0.0000
+log_k_JSE76     0.000       0.000       0.000       1.245       0.0000
+f_cyan_ilr_1    0.000       0.000       0.000       0.000       0.6852
+f_cyan_ilr_2    0.000       0.000       0.000       0.000       0.0000
+f_JCZ38_qlogis  0.000       0.000       0.000       0.000       0.0000
+log_k1          0.000       0.000       0.000       0.000       0.0000
+log_k2          0.000       0.000       0.000       0.000       0.0000
+g_qlogis        0.000       0.000       0.000       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis log_k1 log_k2 g_qlogis
+cyan_0                 0.00           0.00 0.0000 0.0000    0.000
+log_k_JCZ38            0.00           0.00 0.0000 0.0000    0.000
+log_k_J9Z38            0.00           0.00 0.0000 0.0000    0.000
+log_k_JSE76            0.00           0.00 0.0000 0.0000    0.000
+f_cyan_ilr_1           0.00           0.00 0.0000 0.0000    0.000
+f_cyan_ilr_2           1.28           0.00 0.0000 0.0000    0.000
+f_JCZ38_qlogis         0.00          16.11 0.0000 0.0000    0.000
+log_k1                 0.00           0.00 0.9866 0.0000    0.000
+log_k2                 0.00           0.00 0.0000 0.5953    0.000
+g_qlogis               0.00           0.00 0.0000 0.0000    1.583
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2403 2395  -1182
+
+Optimised parameters:
+                      est. lower upper
+cyan_0            102.6079    NA    NA
+log_k_JCZ38        -3.4855    NA    NA
+log_k_J9Z38        -5.1686    NA    NA
+log_k_JSE76        -5.6697    NA    NA
+f_cyan_ilr_1        0.6714    NA    NA
+f_cyan_ilr_2        0.4986    NA    NA
+f_JCZ38_qlogis     55.4760    NA    NA
+log_k1             -1.8409    NA    NA
+log_k2             -4.4915    NA    NA
+g_qlogis           -0.6403    NA    NA
+a.1                 3.2387    NA    NA
+SD.log_k_JCZ38      1.4524    NA    NA
+SD.log_k_J9Z38      0.5151    NA    NA
+SD.log_k_JSE76      0.6514    NA    NA
+SD.f_cyan_ilr_1     0.3023    NA    NA
+SD.f_cyan_ilr_2     0.2959    NA    NA
+SD.f_JCZ38_qlogis   1.9984    NA    NA
+SD.log_k1           0.5188    NA    NA
+SD.log_k2           0.3894    NA    NA
+SD.g_qlogis         0.8579    NA    NA
+
+Correlation is not available
+
+Random effects:
+                    est. lower upper
+SD.log_k_JCZ38    1.4524    NA    NA
+SD.log_k_J9Z38    0.5151    NA    NA
+SD.log_k_JSE76    0.6514    NA    NA
+SD.f_cyan_ilr_1   0.3023    NA    NA
+SD.f_cyan_ilr_2   0.2959    NA    NA
+SD.f_JCZ38_qlogis 1.9984    NA    NA
+SD.log_k1         0.5188    NA    NA
+SD.log_k2         0.3894    NA    NA
+SD.g_qlogis       0.8579    NA    NA
+
+Variance model:
+     est. lower upper
+a.1 3.239    NA    NA
+
+Backtransformed parameters:
+                      est. lower upper
+cyan_0           1.026e+02    NA    NA
+k_JCZ38          3.064e-02    NA    NA
+k_J9Z38          5.692e-03    NA    NA
+k_JSE76          3.449e-03    NA    NA
+f_cyan_to_JCZ38  5.798e-01    NA    NA
+f_cyan_to_J9Z38  2.243e-01    NA    NA
+f_JCZ38_to_JSE76 1.000e+00    NA    NA
+k1               1.587e-01    NA    NA
+k2               1.120e-02    NA    NA
+g                3.452e-01    NA    NA
+
+Resulting formation fractions:
+                ff
+cyan_JCZ38  0.5798
+cyan_J9Z38  0.2243
+cyan_sink   0.1958
+JCZ38_JSE76 1.0000
+JCZ38_sink  0.0000
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_k1 DT50_k2
+cyan   25.21 167.73    50.49   4.368   61.87
+JCZ38  22.62  75.15       NA      NA      NA
+J9Z38 121.77 404.50       NA      NA      NA
+JSE76 200.98 667.64       NA      NA      NA
+
+
+

+ +Hierarchical DFOP path 1 fit with two-component error + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 07:49:05 2023 
+Date of summary: Thu Apr 20 20:01:30 2023 
+
+Equations:
+d_cyan/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_JCZ38 * JCZ38
+d_J9Z38/dt = + f_cyan_to_J9Z38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 673.139 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      101.3964        -3.3626        -4.9792        -5.8727         0.6814 
+  f_cyan_ilr_2 f_JCZ38_qlogis         log_k1         log_k2       g_qlogis 
+        6.7799        13.7245        -1.9222        -4.5035        -0.7172 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          5.317       0.000       0.000       0.000       0.0000
+log_k_JCZ38     0.000       2.272       0.000       0.000       0.0000
+log_k_J9Z38     0.000       0.000       1.633       0.000       0.0000
+log_k_JSE76     0.000       0.000       0.000       1.271       0.0000
+f_cyan_ilr_1    0.000       0.000       0.000       0.000       0.6838
+f_cyan_ilr_2    0.000       0.000       0.000       0.000       0.0000
+f_JCZ38_qlogis  0.000       0.000       0.000       0.000       0.0000
+log_k1          0.000       0.000       0.000       0.000       0.0000
+log_k2          0.000       0.000       0.000       0.000       0.0000
+g_qlogis        0.000       0.000       0.000       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis log_k1 log_k2 g_qlogis
+cyan_0                 0.00           0.00 0.0000 0.0000    0.000
+log_k_JCZ38            0.00           0.00 0.0000 0.0000    0.000
+log_k_J9Z38            0.00           0.00 0.0000 0.0000    0.000
+log_k_JSE76            0.00           0.00 0.0000 0.0000    0.000
+f_cyan_ilr_1           0.00           0.00 0.0000 0.0000    0.000
+f_cyan_ilr_2          11.77           0.00 0.0000 0.0000    0.000
+f_JCZ38_qlogis         0.00          16.11 0.0000 0.0000    0.000
+log_k1                 0.00           0.00 0.9496 0.0000    0.000
+log_k2                 0.00           0.00 0.0000 0.5846    0.000
+g_qlogis               0.00           0.00 0.0000 0.0000    1.719
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2398 2390  -1179
+
+Optimised parameters:
+                      est. lower upper
+cyan_0            100.8076    NA    NA
+log_k_JCZ38        -3.4684    NA    NA
+log_k_J9Z38        -5.0844    NA    NA
+log_k_JSE76        -5.5743    NA    NA
+f_cyan_ilr_1        0.6669    NA    NA
+f_cyan_ilr_2        0.7912    NA    NA
+f_JCZ38_qlogis     84.1825    NA    NA
+log_k1             -2.1671    NA    NA
+log_k2             -4.5447    NA    NA
+g_qlogis           -0.5631    NA    NA
+a.1                 2.9627    NA    NA
+b.1                 0.0444    NA    NA
+SD.log_k_JCZ38      1.4044    NA    NA
+SD.log_k_J9Z38      0.6410    NA    NA
+SD.log_k_JSE76      0.5391    NA    NA
+SD.f_cyan_ilr_1     0.3203    NA    NA
+SD.f_cyan_ilr_2     0.5038    NA    NA
+SD.f_JCZ38_qlogis   3.5865    NA    NA
+SD.log_k2           0.3119    NA    NA
+SD.g_qlogis         0.8276    NA    NA
+
+Correlation is not available
+
+Random effects:
+                    est. lower upper
+SD.log_k_JCZ38    1.4044    NA    NA
+SD.log_k_J9Z38    0.6410    NA    NA
+SD.log_k_JSE76    0.5391    NA    NA
+SD.f_cyan_ilr_1   0.3203    NA    NA
+SD.f_cyan_ilr_2   0.5038    NA    NA
+SD.f_JCZ38_qlogis 3.5865    NA    NA
+SD.log_k2         0.3119    NA    NA
+SD.g_qlogis       0.8276    NA    NA
+
+Variance model:
+      est. lower upper
+a.1 2.9627    NA    NA
+b.1 0.0444    NA    NA
+
+Backtransformed parameters:
+                      est. lower upper
+cyan_0           1.008e+02    NA    NA
+k_JCZ38          3.117e-02    NA    NA
+k_J9Z38          6.193e-03    NA    NA
+k_JSE76          3.794e-03    NA    NA
+f_cyan_to_JCZ38  6.149e-01    NA    NA
+f_cyan_to_J9Z38  2.395e-01    NA    NA
+f_JCZ38_to_JSE76 1.000e+00    NA    NA
+k1               1.145e-01    NA    NA
+k2               1.062e-02    NA    NA
+g                3.628e-01    NA    NA
+
+Resulting formation fractions:
+                ff
+cyan_JCZ38  0.6149
+cyan_J9Z38  0.2395
+cyan_sink   0.1456
+JCZ38_JSE76 1.0000
+JCZ38_sink  0.0000
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_k1 DT50_k2
+cyan   26.26 174.32    52.47   6.053   65.25
+JCZ38  22.24  73.88       NA      NA      NA
+J9Z38 111.93 371.82       NA      NA      NA
+JSE76 182.69 606.88       NA      NA      NA
+
+
+

+ +Hierarchical SFORB path 1 fit with constant variance + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 07:46:35 2023 
+Date of summary: Thu Apr 20 20:01:30 2023 
+
+Equations:
+d_cyan_free/dt = - k_cyan_free * cyan_free - k_cyan_free_bound *
+           cyan_free + k_cyan_bound_free * cyan_bound
+d_cyan_bound/dt = + k_cyan_free_bound * cyan_free - k_cyan_bound_free *
+           cyan_bound
+d_JCZ38/dt = + f_cyan_free_to_JCZ38 * k_cyan_free * cyan_free - k_JCZ38
+           * JCZ38
+d_J9Z38/dt = + f_cyan_free_to_J9Z38 * k_cyan_free * cyan_free - k_J9Z38
+           * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 531.17 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+          cyan_free_0       log_k_cyan_free log_k_cyan_free_bound 
+             102.0643               -2.8987               -2.7077 
+log_k_cyan_bound_free           log_k_JCZ38           log_k_J9Z38 
+              -3.4717               -3.4008               -5.0024 
+          log_k_JSE76          f_cyan_ilr_1          f_cyan_ilr_2 
+              -5.8613                0.6855                1.2366 
+       f_JCZ38_qlogis 
+              13.7418 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+                      cyan_free_0 log_k_cyan_free log_k_cyan_free_bound
+cyan_free_0                 4.466          0.0000                 0.000
+log_k_cyan_free             0.000          0.6158                 0.000
+log_k_cyan_free_bound       0.000          0.0000                 1.463
+log_k_cyan_bound_free       0.000          0.0000                 0.000
+log_k_JCZ38                 0.000          0.0000                 0.000
+log_k_J9Z38                 0.000          0.0000                 0.000
+log_k_JSE76                 0.000          0.0000                 0.000
+f_cyan_ilr_1                0.000          0.0000                 0.000
+f_cyan_ilr_2                0.000          0.0000                 0.000
+f_JCZ38_qlogis              0.000          0.0000                 0.000
+                      log_k_cyan_bound_free log_k_JCZ38 log_k_J9Z38 log_k_JSE76
+cyan_free_0                           0.000       0.000       0.000       0.000
+log_k_cyan_free                       0.000       0.000       0.000       0.000
+log_k_cyan_free_bound                 0.000       0.000       0.000       0.000
+log_k_cyan_bound_free                 1.058       0.000       0.000       0.000
+log_k_JCZ38                           0.000       2.382       0.000       0.000
+log_k_J9Z38                           0.000       0.000       1.595       0.000
+log_k_JSE76                           0.000       0.000       0.000       1.245
+f_cyan_ilr_1                          0.000       0.000       0.000       0.000
+f_cyan_ilr_2                          0.000       0.000       0.000       0.000
+f_JCZ38_qlogis                        0.000       0.000       0.000       0.000
+                      f_cyan_ilr_1 f_cyan_ilr_2 f_JCZ38_qlogis
+cyan_free_0                 0.0000         0.00           0.00
+log_k_cyan_free             0.0000         0.00           0.00
+log_k_cyan_free_bound       0.0000         0.00           0.00
+log_k_cyan_bound_free       0.0000         0.00           0.00
+log_k_JCZ38                 0.0000         0.00           0.00
+log_k_J9Z38                 0.0000         0.00           0.00
+log_k_JSE76                 0.0000         0.00           0.00
+f_cyan_ilr_1                0.6852         0.00           0.00
+f_cyan_ilr_2                0.0000         1.28           0.00
+f_JCZ38_qlogis              0.0000         0.00          16.14
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2401 2394  -1181
+
+Optimised parameters:
+                             est. lower upper
+cyan_free_0              102.7803    NA    NA
+log_k_cyan_free           -2.8068    NA    NA
+log_k_cyan_free_bound     -2.5714    NA    NA
+log_k_cyan_bound_free     -3.4426    NA    NA
+log_k_JCZ38               -3.4994    NA    NA
+log_k_J9Z38               -5.1148    NA    NA
+log_k_JSE76               -5.6335    NA    NA
+f_cyan_ilr_1               0.6597    NA    NA
+f_cyan_ilr_2               0.5132    NA    NA
+f_JCZ38_qlogis            37.2090    NA    NA
+a.1                        3.2367    NA    NA
+SD.log_k_cyan_free         0.3161    NA    NA
+SD.log_k_cyan_free_bound   0.8103    NA    NA
+SD.log_k_cyan_bound_free   0.5554    NA    NA
+SD.log_k_JCZ38             1.4858    NA    NA
+SD.log_k_J9Z38             0.5859    NA    NA
+SD.log_k_JSE76             0.6195    NA    NA
+SD.f_cyan_ilr_1            0.3118    NA    NA
+SD.f_cyan_ilr_2            0.3344    NA    NA
+SD.f_JCZ38_qlogis          0.5518    NA    NA
+
+Correlation is not available
+
+Random effects:
+                           est. lower upper
+SD.log_k_cyan_free       0.3161    NA    NA
+SD.log_k_cyan_free_bound 0.8103    NA    NA
+SD.log_k_cyan_bound_free 0.5554    NA    NA
+SD.log_k_JCZ38           1.4858    NA    NA
+SD.log_k_J9Z38           0.5859    NA    NA
+SD.log_k_JSE76           0.6195    NA    NA
+SD.f_cyan_ilr_1          0.3118    NA    NA
+SD.f_cyan_ilr_2          0.3344    NA    NA
+SD.f_JCZ38_qlogis        0.5518    NA    NA
+
+Variance model:
+     est. lower upper
+a.1 3.237    NA    NA
+
+Backtransformed parameters:
+                          est. lower upper
+cyan_free_0          1.028e+02    NA    NA
+k_cyan_free          6.040e-02    NA    NA
+k_cyan_free_bound    7.643e-02    NA    NA
+k_cyan_bound_free    3.198e-02    NA    NA
+k_JCZ38              3.022e-02    NA    NA
+k_J9Z38              6.007e-03    NA    NA
+k_JSE76              3.576e-03    NA    NA
+f_cyan_free_to_JCZ38 5.787e-01    NA    NA
+f_cyan_free_to_J9Z38 2.277e-01    NA    NA
+f_JCZ38_to_JSE76     1.000e+00    NA    NA
+
+Estimated Eigenvalues of SFORB model(s):
+cyan_b1 cyan_b2  cyan_g 
+0.15646 0.01235 0.33341 
+
+Resulting formation fractions:
+                    ff
+cyan_free_JCZ38 0.5787
+cyan_free_J9Z38 0.2277
+cyan_free_sink  0.1936
+cyan_free       1.0000
+JCZ38_JSE76     1.0000
+JCZ38_sink      0.0000
+
+Estimated disappearance times:
+        DT50  DT90 DT50back DT50_cyan_b1 DT50_cyan_b2
+cyan   24.48 153.7    46.26         4.43        56.15
+JCZ38  22.94  76.2       NA           NA           NA
+J9Z38 115.39 383.3       NA           NA           NA
+JSE76 193.84 643.9       NA           NA           NA
+
+
+

+ +Hierarchical SFORB path 1 fit with two-component error + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 07:49:08 2023 
+Date of summary: Thu Apr 20 20:01:30 2023 
+
+Equations:
+d_cyan_free/dt = - k_cyan_free * cyan_free - k_cyan_free_bound *
+           cyan_free + k_cyan_bound_free * cyan_bound
+d_cyan_bound/dt = + k_cyan_free_bound * cyan_free - k_cyan_bound_free *
+           cyan_bound
+d_JCZ38/dt = + f_cyan_free_to_JCZ38 * k_cyan_free * cyan_free - k_JCZ38
+           * JCZ38
+d_J9Z38/dt = + f_cyan_free_to_J9Z38 * k_cyan_free * cyan_free - k_J9Z38
+           * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 675.659 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+          cyan_free_0       log_k_cyan_free log_k_cyan_free_bound 
+             101.3964               -2.9881               -2.7949 
+log_k_cyan_bound_free           log_k_JCZ38           log_k_J9Z38 
+              -3.4376               -3.3626               -4.9792 
+          log_k_JSE76          f_cyan_ilr_1          f_cyan_ilr_2 
+              -5.8727                0.6814                6.8139 
+       f_JCZ38_qlogis 
+              13.7419 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+                      cyan_free_0 log_k_cyan_free log_k_cyan_free_bound
+cyan_free_0                 5.317          0.0000                 0.000
+log_k_cyan_free             0.000          0.7301                 0.000
+log_k_cyan_free_bound       0.000          0.0000                 1.384
+log_k_cyan_bound_free       0.000          0.0000                 0.000
+log_k_JCZ38                 0.000          0.0000                 0.000
+log_k_J9Z38                 0.000          0.0000                 0.000
+log_k_JSE76                 0.000          0.0000                 0.000
+f_cyan_ilr_1                0.000          0.0000                 0.000
+f_cyan_ilr_2                0.000          0.0000                 0.000
+f_JCZ38_qlogis              0.000          0.0000                 0.000
+                      log_k_cyan_bound_free log_k_JCZ38 log_k_J9Z38 log_k_JSE76
+cyan_free_0                           0.000       0.000       0.000       0.000
+log_k_cyan_free                       0.000       0.000       0.000       0.000
+log_k_cyan_free_bound                 0.000       0.000       0.000       0.000
+log_k_cyan_bound_free                 1.109       0.000       0.000       0.000
+log_k_JCZ38                           0.000       2.272       0.000       0.000
+log_k_J9Z38                           0.000       0.000       1.633       0.000
+log_k_JSE76                           0.000       0.000       0.000       1.271
+f_cyan_ilr_1                          0.000       0.000       0.000       0.000
+f_cyan_ilr_2                          0.000       0.000       0.000       0.000
+f_JCZ38_qlogis                        0.000       0.000       0.000       0.000
+                      f_cyan_ilr_1 f_cyan_ilr_2 f_JCZ38_qlogis
+cyan_free_0                 0.0000         0.00           0.00
+log_k_cyan_free             0.0000         0.00           0.00
+log_k_cyan_free_bound       0.0000         0.00           0.00
+log_k_cyan_bound_free       0.0000         0.00           0.00
+log_k_JCZ38                 0.0000         0.00           0.00
+log_k_J9Z38                 0.0000         0.00           0.00
+log_k_JSE76                 0.0000         0.00           0.00
+f_cyan_ilr_1                0.6838         0.00           0.00
+f_cyan_ilr_2                0.0000        11.84           0.00
+f_JCZ38_qlogis              0.0000         0.00          16.14
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2400 2392  -1180
+
+Optimised parameters:
+                              est. lower upper
+cyan_free_0              100.69983    NA    NA
+log_k_cyan_free           -3.11584    NA    NA
+log_k_cyan_free_bound     -3.15216    NA    NA
+log_k_cyan_bound_free     -3.65986    NA    NA
+log_k_JCZ38               -3.47811    NA    NA
+log_k_J9Z38               -5.08835    NA    NA
+log_k_JSE76               -5.55514    NA    NA
+f_cyan_ilr_1               0.66764    NA    NA
+f_cyan_ilr_2               0.78329    NA    NA
+f_JCZ38_qlogis            25.35245    NA    NA
+a.1                        2.99088    NA    NA
+b.1                        0.04346    NA    NA
+SD.log_k_cyan_free         0.48797    NA    NA
+SD.log_k_cyan_bound_free   0.27243    NA    NA
+SD.log_k_JCZ38             1.42450    NA    NA
+SD.log_k_J9Z38             0.63496    NA    NA
+SD.log_k_JSE76             0.55951    NA    NA
+SD.f_cyan_ilr_1            0.32687    NA    NA
+SD.f_cyan_ilr_2            0.48056    NA    NA
+SD.f_JCZ38_qlogis          0.43818    NA    NA
+
+Correlation is not available
+
+Random effects:
+                           est. lower upper
+SD.log_k_cyan_free       0.4880    NA    NA
+SD.log_k_cyan_bound_free 0.2724    NA    NA
+SD.log_k_JCZ38           1.4245    NA    NA
+SD.log_k_J9Z38           0.6350    NA    NA
+SD.log_k_JSE76           0.5595    NA    NA
+SD.f_cyan_ilr_1          0.3269    NA    NA
+SD.f_cyan_ilr_2          0.4806    NA    NA
+SD.f_JCZ38_qlogis        0.4382    NA    NA
+
+Variance model:
+       est. lower upper
+a.1 2.99088    NA    NA
+b.1 0.04346    NA    NA
+
+Backtransformed parameters:
+                          est. lower upper
+cyan_free_0          1.007e+02    NA    NA
+k_cyan_free          4.434e-02    NA    NA
+k_cyan_free_bound    4.276e-02    NA    NA
+k_cyan_bound_free    2.574e-02    NA    NA
+k_JCZ38              3.087e-02    NA    NA
+k_J9Z38              6.168e-03    NA    NA
+k_JSE76              3.868e-03    NA    NA
+f_cyan_free_to_JCZ38 6.143e-01    NA    NA
+f_cyan_free_to_J9Z38 2.389e-01    NA    NA
+f_JCZ38_to_JSE76     1.000e+00    NA    NA
+
+Estimated Eigenvalues of SFORB model(s):
+cyan_b1 cyan_b2  cyan_g 
+0.10161 0.01123 0.36636 
+
+Resulting formation fractions:
+                       ff
+cyan_free_JCZ38 6.143e-01
+cyan_free_J9Z38 2.389e-01
+cyan_free_sink  1.468e-01
+cyan_free       1.000e+00
+JCZ38_JSE76     1.000e+00
+JCZ38_sink      9.763e-12
+
+Estimated disappearance times:
+        DT50  DT90 DT50back DT50_cyan_b1 DT50_cyan_b2
+cyan   25.91 164.4    49.49        6.822        61.72
+JCZ38  22.46  74.6       NA           NA           NA
+J9Z38 112.37 373.3       NA           NA           NA
+JSE76 179.22 595.4       NA           NA           NA
+
+
+

+ +Hierarchical HS path 1 fit with constant variance + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 07:46:30 2023 
+Date of summary: Thu Apr 20 20:01:30 2023 
+
+Equations:
+d_cyan/dt = - ifelse(time <= tb, k1, k2) * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * ifelse(time <= tb, k1, k2) * cyan -
+           k_JCZ38 * JCZ38
+d_J9Z38/dt = + f_cyan_to_J9Z38 * ifelse(time <= tb, k1, k2) * cyan -
+           k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 525.846 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      102.8738        -3.4490        -4.9348        -5.5989         0.6469 
+  f_cyan_ilr_2 f_JCZ38_qlogis         log_k1         log_k2         log_tb 
+        1.2854         9.7193        -2.9084        -4.1810         1.7813 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          5.409        0.00        0.00       0.000       0.0000
+log_k_JCZ38     0.000        2.33        0.00       0.000       0.0000
+log_k_J9Z38     0.000        0.00        1.59       0.000       0.0000
+log_k_JSE76     0.000        0.00        0.00       1.006       0.0000
+f_cyan_ilr_1    0.000        0.00        0.00       0.000       0.6371
+f_cyan_ilr_2    0.000        0.00        0.00       0.000       0.0000
+f_JCZ38_qlogis  0.000        0.00        0.00       0.000       0.0000
+log_k1          0.000        0.00        0.00       0.000       0.0000
+log_k2          0.000        0.00        0.00       0.000       0.0000
+log_tb          0.000        0.00        0.00       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis log_k1 log_k2 log_tb
+cyan_0                0.000           0.00 0.0000 0.0000 0.0000
+log_k_JCZ38           0.000           0.00 0.0000 0.0000 0.0000
+log_k_J9Z38           0.000           0.00 0.0000 0.0000 0.0000
+log_k_JSE76           0.000           0.00 0.0000 0.0000 0.0000
+f_cyan_ilr_1          0.000           0.00 0.0000 0.0000 0.0000
+f_cyan_ilr_2          2.167           0.00 0.0000 0.0000 0.0000
+f_JCZ38_qlogis        0.000          10.22 0.0000 0.0000 0.0000
+log_k1                0.000           0.00 0.7003 0.0000 0.0000
+log_k2                0.000           0.00 0.0000 0.8928 0.0000
+log_tb                0.000           0.00 0.0000 0.0000 0.6774
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2427 2420  -1194
+
+Optimised parameters:
+                       est. lower upper
+cyan_0            101.84849    NA    NA
+log_k_JCZ38        -3.47365    NA    NA
+log_k_J9Z38        -5.10562    NA    NA
+log_k_JSE76        -5.60318    NA    NA
+f_cyan_ilr_1        0.66127    NA    NA
+f_cyan_ilr_2        0.60283    NA    NA
+f_JCZ38_qlogis     45.06408    NA    NA
+log_k1             -3.10124    NA    NA
+log_k2             -4.39028    NA    NA
+log_tb              2.32256    NA    NA
+a.1                 3.32683    NA    NA
+SD.log_k_JCZ38      1.41427    NA    NA
+SD.log_k_J9Z38      0.54767    NA    NA
+SD.log_k_JSE76      0.62147    NA    NA
+SD.f_cyan_ilr_1     0.30189    NA    NA
+SD.f_cyan_ilr_2     0.34960    NA    NA
+SD.f_JCZ38_qlogis   0.04644    NA    NA
+SD.log_k1           0.39534    NA    NA
+SD.log_k2           0.43468    NA    NA
+SD.log_tb           0.60781    NA    NA
+
+Correlation is not available
+
+Random effects:
+                     est. lower upper
+SD.log_k_JCZ38    1.41427    NA    NA
+SD.log_k_J9Z38    0.54767    NA    NA
+SD.log_k_JSE76    0.62147    NA    NA
+SD.f_cyan_ilr_1   0.30189    NA    NA
+SD.f_cyan_ilr_2   0.34960    NA    NA
+SD.f_JCZ38_qlogis 0.04644    NA    NA
+SD.log_k1         0.39534    NA    NA
+SD.log_k2         0.43468    NA    NA
+SD.log_tb         0.60781    NA    NA
+
+Variance model:
+     est. lower upper
+a.1 3.327    NA    NA
+
+Backtransformed parameters:
+                      est. lower upper
+cyan_0           1.018e+02    NA    NA
+k_JCZ38          3.100e-02    NA    NA
+k_J9Z38          6.063e-03    NA    NA
+k_JSE76          3.686e-03    NA    NA
+f_cyan_to_JCZ38  5.910e-01    NA    NA
+f_cyan_to_J9Z38  2.320e-01    NA    NA
+f_JCZ38_to_JSE76 1.000e+00    NA    NA
+k1               4.499e-02    NA    NA
+k2               1.240e-02    NA    NA
+tb               1.020e+01    NA    NA
+
+Resulting formation fractions:
+               ff
+cyan_JCZ38  0.591
+cyan_J9Z38  0.232
+cyan_sink   0.177
+JCZ38_JSE76 1.000
+JCZ38_sink  0.000
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_k1 DT50_k2
+cyan   29.09 158.91    47.84   15.41   55.91
+JCZ38  22.36  74.27       NA      NA      NA
+J9Z38 114.33 379.80       NA      NA      NA
+JSE76 188.04 624.66       NA      NA      NA
+
+
+

+ +Hierarchical HS path 1 fit with two-component error + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 07:46:19 2023 
+Date of summary: Thu Apr 20 20:01:30 2023 
+
+Equations:
+d_cyan/dt = - ifelse(time <= tb, k1, k2) * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * ifelse(time <= tb, k1, k2) * cyan -
+           k_JCZ38 * JCZ38
+d_J9Z38/dt = + f_cyan_to_J9Z38 * ifelse(time <= tb, k1, k2) * cyan -
+           k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 514.968 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+       101.168         -3.358         -4.941         -5.794          0.676 
+  f_cyan_ilr_2 f_JCZ38_qlogis         log_k1         log_k2         log_tb 
+         5.740         13.863         -3.147         -4.262          2.173 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0           5.79       0.000       0.000       0.000       0.0000
+log_k_JCZ38      0.00       2.271       0.000       0.000       0.0000
+log_k_J9Z38      0.00       0.000       1.614       0.000       0.0000
+log_k_JSE76      0.00       0.000       0.000       1.264       0.0000
+f_cyan_ilr_1     0.00       0.000       0.000       0.000       0.6761
+f_cyan_ilr_2     0.00       0.000       0.000       0.000       0.0000
+f_JCZ38_qlogis   0.00       0.000       0.000       0.000       0.0000
+log_k1           0.00       0.000       0.000       0.000       0.0000
+log_k2           0.00       0.000       0.000       0.000       0.0000
+log_tb           0.00       0.000       0.000       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis log_k1 log_k2 log_tb
+cyan_0                0.000           0.00 0.0000 0.0000  0.000
+log_k_JCZ38           0.000           0.00 0.0000 0.0000  0.000
+log_k_J9Z38           0.000           0.00 0.0000 0.0000  0.000
+log_k_JSE76           0.000           0.00 0.0000 0.0000  0.000
+f_cyan_ilr_1          0.000           0.00 0.0000 0.0000  0.000
+f_cyan_ilr_2          9.572           0.00 0.0000 0.0000  0.000
+f_JCZ38_qlogis        0.000          19.19 0.0000 0.0000  0.000
+log_k1                0.000           0.00 0.8705 0.0000  0.000
+log_k2                0.000           0.00 0.0000 0.9288  0.000
+log_tb                0.000           0.00 0.0000 0.0000  1.065
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2422 2414  -1190
+
+Optimised parameters:
+                      est. lower upper
+cyan_0            100.9521    NA    NA
+log_k_JCZ38        -3.4629    NA    NA
+log_k_J9Z38        -5.0346    NA    NA
+log_k_JSE76        -5.5722    NA    NA
+f_cyan_ilr_1        0.6560    NA    NA
+f_cyan_ilr_2        0.7983    NA    NA
+f_JCZ38_qlogis     42.7949    NA    NA
+log_k1             -3.1721    NA    NA
+log_k2             -4.4039    NA    NA
+log_tb              2.3994    NA    NA
+a.1                 3.0586    NA    NA
+b.1                 0.0380    NA    NA
+SD.log_k_JCZ38      1.3754    NA    NA
+SD.log_k_J9Z38      0.6703    NA    NA
+SD.log_k_JSE76      0.5876    NA    NA
+SD.f_cyan_ilr_1     0.3272    NA    NA
+SD.f_cyan_ilr_2     0.5300    NA    NA
+SD.f_JCZ38_qlogis   6.4465    NA    NA
+SD.log_k1           0.4135    NA    NA
+SD.log_k2           0.4182    NA    NA
+SD.log_tb           0.6035    NA    NA
+
+Correlation is not available
+
+Random effects:
+                    est. lower upper
+SD.log_k_JCZ38    1.3754    NA    NA
+SD.log_k_J9Z38    0.6703    NA    NA
+SD.log_k_JSE76    0.5876    NA    NA
+SD.f_cyan_ilr_1   0.3272    NA    NA
+SD.f_cyan_ilr_2   0.5300    NA    NA
+SD.f_JCZ38_qlogis 6.4465    NA    NA
+SD.log_k1         0.4135    NA    NA
+SD.log_k2         0.4182    NA    NA
+SD.log_tb         0.6035    NA    NA
+
+Variance model:
+     est. lower upper
+a.1 3.059    NA    NA
+b.1 0.038    NA    NA
+
+Backtransformed parameters:
+                      est. lower upper
+cyan_0           1.010e+02    NA    NA
+k_JCZ38          3.134e-02    NA    NA
+k_J9Z38          6.509e-03    NA    NA
+k_JSE76          3.802e-03    NA    NA
+f_cyan_to_JCZ38  6.127e-01    NA    NA
+f_cyan_to_J9Z38  2.423e-01    NA    NA
+f_JCZ38_to_JSE76 1.000e+00    NA    NA
+k1               4.191e-02    NA    NA
+k2               1.223e-02    NA    NA
+tb               1.102e+01    NA    NA
+
+Resulting formation fractions:
+                ff
+cyan_JCZ38  0.6127
+cyan_J9Z38  0.2423
+cyan_sink   0.1449
+JCZ38_JSE76 1.0000
+JCZ38_sink  0.0000
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_k1 DT50_k2
+cyan   29.94 161.54    48.63   16.54   56.68
+JCZ38  22.12  73.47       NA      NA      NA
+J9Z38 106.50 353.77       NA      NA      NA
+JSE76 182.30 605.60       NA      NA      NA
+
+
+

+
+
+

Pathway 2 +

+ +Hierarchical FOMC path 2 fit with constant variance + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 07:58:00 2023 
+Date of summary: Thu Apr 20 20:01:30 2023 
+
+Equations:
+d_cyan/dt = - (alpha/beta) * 1/((time/beta) + 1) * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_JCZ38 * JCZ38 + f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_to_J9Z38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 522.351 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      101.8173        -1.8998        -5.1449        -2.5415         0.6705 
+  f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis      log_alpha       log_beta 
+        4.4669        16.1281        13.3327        -0.2314         2.8738 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          5.742       0.000       0.000        0.00       0.0000
+log_k_JCZ38     0.000       1.402       0.000        0.00       0.0000
+log_k_J9Z38     0.000       0.000       1.718        0.00       0.0000
+log_k_JSE76     0.000       0.000       0.000        3.57       0.0000
+f_cyan_ilr_1    0.000       0.000       0.000        0.00       0.5926
+f_cyan_ilr_2    0.000       0.000       0.000        0.00       0.0000
+f_JCZ38_qlogis  0.000       0.000       0.000        0.00       0.0000
+f_JSE76_qlogis  0.000       0.000       0.000        0.00       0.0000
+log_alpha       0.000       0.000       0.000        0.00       0.0000
+log_beta        0.000       0.000       0.000        0.00       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis log_alpha log_beta
+cyan_0                 0.00           0.00           0.00    0.0000   0.0000
+log_k_JCZ38            0.00           0.00           0.00    0.0000   0.0000
+log_k_J9Z38            0.00           0.00           0.00    0.0000   0.0000
+log_k_JSE76            0.00           0.00           0.00    0.0000   0.0000
+f_cyan_ilr_1           0.00           0.00           0.00    0.0000   0.0000
+f_cyan_ilr_2          10.56           0.00           0.00    0.0000   0.0000
+f_JCZ38_qlogis         0.00          12.04           0.00    0.0000   0.0000
+f_JSE76_qlogis         0.00           0.00          15.26    0.0000   0.0000
+log_alpha              0.00           0.00           0.00    0.4708   0.0000
+log_beta               0.00           0.00           0.00    0.0000   0.4432
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2308 2301  -1134
+
+Optimised parameters:
+                      est.    lower     upper
+cyan_0            101.9586 99.22024 104.69700
+log_k_JCZ38        -2.4861 -3.17661  -1.79560
+log_k_J9Z38        -5.3926 -6.08842  -4.69684
+log_k_JSE76        -3.1193 -4.12904  -2.10962
+f_cyan_ilr_1        0.7368  0.42085   1.05276
+f_cyan_ilr_2        0.6196  0.06052   1.17861
+f_JCZ38_qlogis      4.8970 -4.68003  14.47398
+f_JSE76_qlogis      4.4066 -1.02087   9.83398
+log_alpha          -0.3021 -0.68264   0.07838
+log_beta            2.7438  2.57970   2.90786
+a.1                 2.9008  2.69920   3.10245
+SD.cyan_0           2.7081  0.64216   4.77401
+SD.log_k_JCZ38      0.7043  0.19951   1.20907
+SD.log_k_J9Z38      0.6248  0.05790   1.19180
+SD.log_k_JSE76      1.0750  0.33157   1.81839
+SD.f_cyan_ilr_1     0.3429  0.11688   0.56892
+SD.f_cyan_ilr_2     0.4774  0.09381   0.86097
+SD.f_JCZ38_qlogis   1.5565 -7.83970  10.95279
+SD.f_JSE76_qlogis   1.6871 -1.25577   4.63000
+SD.log_alpha        0.4216  0.15913   0.68405
+
+Correlation: 
+               cyan_0  l__JCZ3 l__J9Z3 l__JSE7 f_cy__1 f_cy__2 f_JCZ38 f_JSE76
+log_k_JCZ38    -0.0167                                                        
+log_k_J9Z38    -0.0307  0.0057                                                
+log_k_JSE76    -0.0032  0.1358  0.0009                                        
+f_cyan_ilr_1   -0.0087  0.0206 -0.1158 -0.0009                                
+f_cyan_ilr_2   -0.1598  0.0690  0.1770  0.0002 -0.0007                        
+f_JCZ38_qlogis  0.0966 -0.1132 -0.0440  0.0182 -0.1385 -0.4583                
+f_JSE76_qlogis -0.0647  0.1157  0.0333 -0.0026  0.1110  0.3620 -0.8586        
+log_alpha      -0.0389  0.0113  0.0209  0.0021  0.0041  0.0451 -0.0605  0.0412
+log_beta       -0.2508  0.0533  0.0977  0.0098  0.0220  0.2741 -0.2934  0.1999
+               log_lph
+log_k_JCZ38           
+log_k_J9Z38           
+log_k_JSE76           
+f_cyan_ilr_1          
+f_cyan_ilr_2          
+f_JCZ38_qlogis        
+f_JSE76_qlogis        
+log_alpha             
+log_beta        0.2281
+
+Random effects:
+                    est.    lower   upper
+SD.cyan_0         2.7081  0.64216  4.7740
+SD.log_k_JCZ38    0.7043  0.19951  1.2091
+SD.log_k_J9Z38    0.6248  0.05790  1.1918
+SD.log_k_JSE76    1.0750  0.33157  1.8184
+SD.f_cyan_ilr_1   0.3429  0.11688  0.5689
+SD.f_cyan_ilr_2   0.4774  0.09381  0.8610
+SD.f_JCZ38_qlogis 1.5565 -7.83970 10.9528
+SD.f_JSE76_qlogis 1.6871 -1.25577  4.6300
+SD.log_alpha      0.4216  0.15913  0.6840
+
+Variance model:
+     est. lower upper
+a.1 2.901 2.699 3.102
+
+Backtransformed parameters:
+                      est.     lower     upper
+cyan_0           101.95862 99.220240 1.047e+02
+k_JCZ38            0.08323  0.041727 1.660e-01
+k_J9Z38            0.00455  0.002269 9.124e-03
+k_JSE76            0.04419  0.016098 1.213e-01
+f_cyan_to_JCZ38    0.61318        NA        NA
+f_cyan_to_J9Z38    0.21630        NA        NA
+f_JCZ38_to_JSE76   0.99259  0.009193 1.000e+00
+f_JSE76_to_JCZ38   0.98795  0.264857 9.999e-01
+alpha              0.73924  0.505281 1.082e+00
+beta              15.54568 13.193194 1.832e+01
+
+Resulting formation fractions:
+                  ff
+cyan_JCZ38  0.613182
+cyan_J9Z38  0.216298
+cyan_sink   0.170519
+JCZ38_JSE76 0.992586
+JCZ38_sink  0.007414
+JSE76_JCZ38 0.987950
+JSE76_sink  0.012050
+
+Estimated disappearance times:
+         DT50   DT90 DT50back
+cyan   24.157 334.68    100.7
+JCZ38   8.328  27.66       NA
+J9Z38 152.341 506.06       NA
+JSE76  15.687  52.11       NA
+
+
+

+ +Hierarchical FOMC path 2 fit with two-component error + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 07:57:52 2023 
+Date of summary: Thu Apr 20 20:01:30 2023 
+
+Equations:
+d_cyan/dt = - (alpha/beta) * 1/((time/beta) + 1) * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_JCZ38 * JCZ38 + f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_to_J9Z38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 514.301 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      101.9028        -1.9055        -5.0249        -2.5646         0.6807 
+  f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis      log_alpha       log_beta 
+        4.8883        16.0676         9.3923        -0.1346         3.0364 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          6.321       0.000       0.000       0.000       0.0000
+log_k_JCZ38     0.000       1.392       0.000       0.000       0.0000
+log_k_J9Z38     0.000       0.000       1.561       0.000       0.0000
+log_k_JSE76     0.000       0.000       0.000       3.614       0.0000
+f_cyan_ilr_1    0.000       0.000       0.000       0.000       0.6339
+f_cyan_ilr_2    0.000       0.000       0.000       0.000       0.0000
+f_JCZ38_qlogis  0.000       0.000       0.000       0.000       0.0000
+f_JSE76_qlogis  0.000       0.000       0.000       0.000       0.0000
+log_alpha       0.000       0.000       0.000       0.000       0.0000
+log_beta        0.000       0.000       0.000       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis log_alpha log_beta
+cyan_0                 0.00           0.00           0.00    0.0000   0.0000
+log_k_JCZ38            0.00           0.00           0.00    0.0000   0.0000
+log_k_J9Z38            0.00           0.00           0.00    0.0000   0.0000
+log_k_JSE76            0.00           0.00           0.00    0.0000   0.0000
+f_cyan_ilr_1           0.00           0.00           0.00    0.0000   0.0000
+f_cyan_ilr_2          10.41           0.00           0.00    0.0000   0.0000
+f_JCZ38_qlogis         0.00          12.24           0.00    0.0000   0.0000
+f_JSE76_qlogis         0.00           0.00          15.13    0.0000   0.0000
+log_alpha              0.00           0.00           0.00    0.3701   0.0000
+log_beta               0.00           0.00           0.00    0.0000   0.5662
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2248 2240  -1103
+
+Optimised parameters:
+                       est.      lower      upper
+cyan_0            101.55545  9.920e+01  1.039e+02
+log_k_JCZ38        -2.37354 -2.928e+00 -1.819e+00
+log_k_J9Z38        -5.14736 -5.960e+00 -4.335e+00
+log_k_JSE76        -3.07802 -4.243e+00 -1.913e+00
+f_cyan_ilr_1        0.71263  3.655e-01  1.060e+00
+f_cyan_ilr_2        0.95202  2.701e-01  1.634e+00
+f_JCZ38_qlogis      3.58473  1.251e+00  5.919e+00
+f_JSE76_qlogis     19.03623 -1.037e+07  1.037e+07
+log_alpha          -0.15297 -4.490e-01  1.431e-01
+log_beta            2.99230  2.706e+00  3.278e+00
+a.1                 2.04816         NA         NA
+b.1                 0.06886         NA         NA
+SD.log_k_JCZ38      0.56174         NA         NA
+SD.log_k_J9Z38      0.86509         NA         NA
+SD.log_k_JSE76      1.28450         NA         NA
+SD.f_cyan_ilr_1     0.38705         NA         NA
+SD.f_cyan_ilr_2     0.54153         NA         NA
+SD.f_JCZ38_qlogis   1.65311         NA         NA
+SD.f_JSE76_qlogis   7.51468         NA         NA
+SD.log_alpha        0.31586         NA         NA
+SD.log_beta         0.24696         NA         NA
+
+Correlation is not available
+
+Random effects:
+                    est. lower upper
+SD.log_k_JCZ38    0.5617    NA    NA
+SD.log_k_J9Z38    0.8651    NA    NA
+SD.log_k_JSE76    1.2845    NA    NA
+SD.f_cyan_ilr_1   0.3870    NA    NA
+SD.f_cyan_ilr_2   0.5415    NA    NA
+SD.f_JCZ38_qlogis 1.6531    NA    NA
+SD.f_JSE76_qlogis 7.5147    NA    NA
+SD.log_alpha      0.3159    NA    NA
+SD.log_beta       0.2470    NA    NA
+
+Variance model:
+       est. lower upper
+a.1 2.04816    NA    NA
+b.1 0.06886    NA    NA
+
+Backtransformed parameters:
+                      est.    lower    upper
+cyan_0           1.016e+02 99.20301 103.9079
+k_JCZ38          9.315e-02  0.05349   0.1622
+k_J9Z38          5.815e-03  0.00258   0.0131
+k_JSE76          4.605e-02  0.01436   0.1477
+f_cyan_to_JCZ38  6.438e-01       NA       NA
+f_cyan_to_J9Z38  2.350e-01       NA       NA
+f_JCZ38_to_JSE76 9.730e-01  0.77745   0.9973
+f_JSE76_to_JCZ38 1.000e+00  0.00000   1.0000
+alpha            8.582e-01  0.63824   1.1538
+beta             1.993e+01 14.97621  26.5262
+
+Resulting formation fractions:
+                   ff
+cyan_JCZ38  6.438e-01
+cyan_J9Z38  2.350e-01
+cyan_sink   1.212e-01
+JCZ38_JSE76 9.730e-01
+JCZ38_sink  2.700e-02
+JSE76_JCZ38 1.000e+00
+JSE76_sink  5.403e-09
+
+Estimated disappearance times:
+         DT50   DT90 DT50back
+cyan   24.771 271.70    81.79
+JCZ38   7.441  24.72       NA
+J9Z38 119.205 395.99       NA
+JSE76  15.052  50.00       NA
+
+
+

+ +Hierarchical DFOP path 2 fit with constant variance + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 07:58:43 2023 
+Date of summary: Thu Apr 20 20:01:30 2023 
+
+Equations:
+d_cyan/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_JCZ38 * JCZ38 +
+           f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_to_J9Z38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 565.562 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      102.4358        -2.3107        -5.3123        -3.7120         0.6753 
+  f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis         log_k1         log_k2 
+        1.1462        12.4095        12.3630        -1.9317        -4.4557 
+      g_qlogis 
+       -0.5648 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          4.594      0.0000       0.000         0.0       0.0000
+log_k_JCZ38     0.000      0.7966       0.000         0.0       0.0000
+log_k_J9Z38     0.000      0.0000       1.561         0.0       0.0000
+log_k_JSE76     0.000      0.0000       0.000         0.8       0.0000
+f_cyan_ilr_1    0.000      0.0000       0.000         0.0       0.6349
+f_cyan_ilr_2    0.000      0.0000       0.000         0.0       0.0000
+f_JCZ38_qlogis  0.000      0.0000       0.000         0.0       0.0000
+f_JSE76_qlogis  0.000      0.0000       0.000         0.0       0.0000
+log_k1          0.000      0.0000       0.000         0.0       0.0000
+log_k2          0.000      0.0000       0.000         0.0       0.0000
+g_qlogis        0.000      0.0000       0.000         0.0       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis log_k1 log_k2
+cyan_0                0.000           0.00            0.0  0.000 0.0000
+log_k_JCZ38           0.000           0.00            0.0  0.000 0.0000
+log_k_J9Z38           0.000           0.00            0.0  0.000 0.0000
+log_k_JSE76           0.000           0.00            0.0  0.000 0.0000
+f_cyan_ilr_1          0.000           0.00            0.0  0.000 0.0000
+f_cyan_ilr_2          1.797           0.00            0.0  0.000 0.0000
+f_JCZ38_qlogis        0.000          13.85            0.0  0.000 0.0000
+f_JSE76_qlogis        0.000           0.00           14.1  0.000 0.0000
+log_k1                0.000           0.00            0.0  1.106 0.0000
+log_k2                0.000           0.00            0.0  0.000 0.6141
+g_qlogis              0.000           0.00            0.0  0.000 0.0000
+               g_qlogis
+cyan_0            0.000
+log_k_JCZ38       0.000
+log_k_J9Z38       0.000
+log_k_JSE76       0.000
+f_cyan_ilr_1      0.000
+f_cyan_ilr_2      0.000
+f_JCZ38_qlogis    0.000
+f_JSE76_qlogis    0.000
+log_k1            0.000
+log_k2            0.000
+g_qlogis          1.595
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2290 2281  -1123
+
+Optimised parameters:
+                      est.     lower    upper
+cyan_0            102.6903 101.44420 103.9365
+log_k_JCZ38        -2.4018  -2.98058  -1.8230
+log_k_J9Z38        -5.1865  -5.92931  -4.4437
+log_k_JSE76        -3.0784  -4.25226  -1.9045
+f_cyan_ilr_1        0.7157   0.37625   1.0551
+f_cyan_ilr_2        0.7073   0.20136   1.2132
+f_JCZ38_qlogis      4.6797   0.43240   8.9269
+f_JSE76_qlogis      5.0080  -1.01380  11.0299
+log_k1             -1.9620  -2.62909  -1.2949
+log_k2             -4.4894  -4.94958  -4.0292
+g_qlogis           -0.4658  -1.34443   0.4129
+a.1                 2.7158   2.52576   2.9059
+SD.log_k_JCZ38      0.5818   0.15679   1.0067
+SD.log_k_J9Z38      0.7421   0.16751   1.3167
+SD.log_k_JSE76      1.2841   0.43247   2.1356
+SD.f_cyan_ilr_1     0.3748   0.13040   0.6192
+SD.f_cyan_ilr_2     0.4550   0.08396   0.8261
+SD.f_JCZ38_qlogis   2.0862  -0.73390   4.9062
+SD.f_JSE76_qlogis   1.9585  -3.14773   7.0647
+SD.log_k1           0.7389   0.25761   1.2201
+SD.log_k2           0.5132   0.18143   0.8450
+SD.g_qlogis         0.9870   0.35773   1.6164
+
+Correlation: 
+               cyan_0  l__JCZ3 l__J9Z3 l__JSE7 f_cy__1 f_cy__2 f_JCZ38 f_JSE76
+log_k_JCZ38    -0.0170                                                        
+log_k_J9Z38    -0.0457  0.0016                                                
+log_k_JSE76    -0.0046  0.1183  0.0005                                        
+f_cyan_ilr_1    0.0079  0.0072 -0.0909  0.0003                                
+f_cyan_ilr_2   -0.3114  0.0343  0.1542  0.0023 -0.0519                        
+f_JCZ38_qlogis  0.0777 -0.0601 -0.0152  0.0080 -0.0520 -0.2524                
+f_JSE76_qlogis -0.0356  0.0817  0.0073  0.0051  0.0388  0.1959 -0.6236        
+log_k1          0.0848 -0.0028  0.0010 -0.0010 -0.0014 -0.0245  0.0121 -0.0177
+log_k2          0.0274 -0.0001  0.0075  0.0000 -0.0023 -0.0060  0.0000 -0.0130
+g_qlogis        0.0159  0.0002 -0.0095  0.0002  0.0029 -0.0140 -0.0001  0.0149
+               log_k1  log_k2 
+log_k_JCZ38                   
+log_k_J9Z38                   
+log_k_JSE76                   
+f_cyan_ilr_1                  
+f_cyan_ilr_2                  
+f_JCZ38_qlogis                
+f_JSE76_qlogis                
+log_k1                        
+log_k2          0.0280        
+g_qlogis       -0.0278 -0.0310
+
+Random effects:
+                    est.    lower  upper
+SD.log_k_JCZ38    0.5818  0.15679 1.0067
+SD.log_k_J9Z38    0.7421  0.16751 1.3167
+SD.log_k_JSE76    1.2841  0.43247 2.1356
+SD.f_cyan_ilr_1   0.3748  0.13040 0.6192
+SD.f_cyan_ilr_2   0.4550  0.08396 0.8261
+SD.f_JCZ38_qlogis 2.0862 -0.73390 4.9062
+SD.f_JSE76_qlogis 1.9585 -3.14773 7.0647
+SD.log_k1         0.7389  0.25761 1.2201
+SD.log_k2         0.5132  0.18143 0.8450
+SD.g_qlogis       0.9870  0.35773 1.6164
+
+Variance model:
+     est. lower upper
+a.1 2.716 2.526 2.906
+
+Backtransformed parameters:
+                      est.     lower     upper
+cyan_0           1.027e+02 1.014e+02 103.93649
+k_JCZ38          9.056e-02 5.076e-02   0.16154
+k_J9Z38          5.591e-03 2.660e-03   0.01175
+k_JSE76          4.603e-02 1.423e-02   0.14890
+f_cyan_to_JCZ38  6.184e-01        NA        NA
+f_cyan_to_J9Z38  2.248e-01        NA        NA
+f_JCZ38_to_JSE76 9.908e-01 6.064e-01   0.99987
+f_JSE76_to_JCZ38 9.934e-01 2.662e-01   0.99998
+k1               1.406e-01 7.214e-02   0.27393
+k2               1.123e-02 7.086e-03   0.01779
+g                3.856e-01 2.068e-01   0.60177
+
+Resulting formation fractions:
+                  ff
+cyan_JCZ38  0.618443
+cyan_J9Z38  0.224770
+cyan_sink   0.156787
+JCZ38_JSE76 0.990803
+JCZ38_sink  0.009197
+JSE76_JCZ38 0.993360
+JSE76_sink  0.006640
+
+Estimated disappearance times:
+         DT50   DT90 DT50back DT50_k1 DT50_k2
+cyan   21.674 161.70    48.68   4.931   61.74
+JCZ38   7.654  25.43       NA      NA      NA
+J9Z38 123.966 411.81       NA      NA      NA
+JSE76  15.057  50.02       NA      NA      NA
+
+
+

+ +Hierarchical DFOP path 2 fit with two-component error + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 08:01:24 2023 
+Date of summary: Thu Apr 20 20:01:30 2023 
+
+Equations:
+d_cyan/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_JCZ38 * JCZ38 +
+           f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_to_J9Z38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 726.501 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      101.7523        -1.5948        -5.0119        -2.2723         0.6719 
+  f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis         log_k1         log_k2 
+        5.1681        12.8238        12.4130        -2.0057        -4.5526 
+      g_qlogis 
+       -0.5805 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          5.627       0.000       0.000       0.000       0.0000
+log_k_JCZ38     0.000       2.327       0.000       0.000       0.0000
+log_k_J9Z38     0.000       0.000       1.664       0.000       0.0000
+log_k_JSE76     0.000       0.000       0.000       4.566       0.0000
+f_cyan_ilr_1    0.000       0.000       0.000       0.000       0.6519
+f_cyan_ilr_2    0.000       0.000       0.000       0.000       0.0000
+f_JCZ38_qlogis  0.000       0.000       0.000       0.000       0.0000
+f_JSE76_qlogis  0.000       0.000       0.000       0.000       0.0000
+log_k1          0.000       0.000       0.000       0.000       0.0000
+log_k2          0.000       0.000       0.000       0.000       0.0000
+g_qlogis        0.000       0.000       0.000       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis log_k1 log_k2
+cyan_0                  0.0           0.00           0.00 0.0000 0.0000
+log_k_JCZ38             0.0           0.00           0.00 0.0000 0.0000
+log_k_J9Z38             0.0           0.00           0.00 0.0000 0.0000
+log_k_JSE76             0.0           0.00           0.00 0.0000 0.0000
+f_cyan_ilr_1            0.0           0.00           0.00 0.0000 0.0000
+f_cyan_ilr_2           10.1           0.00           0.00 0.0000 0.0000
+f_JCZ38_qlogis          0.0          13.99           0.00 0.0000 0.0000
+f_JSE76_qlogis          0.0           0.00          14.15 0.0000 0.0000
+log_k1                  0.0           0.00           0.00 0.8452 0.0000
+log_k2                  0.0           0.00           0.00 0.0000 0.5968
+g_qlogis                0.0           0.00           0.00 0.0000 0.0000
+               g_qlogis
+cyan_0            0.000
+log_k_JCZ38       0.000
+log_k_J9Z38       0.000
+log_k_JSE76       0.000
+f_cyan_ilr_1      0.000
+f_cyan_ilr_2      0.000
+f_JCZ38_qlogis    0.000
+f_JSE76_qlogis    0.000
+log_k1            0.000
+log_k2            0.000
+g_qlogis          1.691
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2234 2226  -1095
+
+Optimised parameters:
+                       est.      lower      upper
+cyan_0            101.10667  9.903e+01  103.18265
+log_k_JCZ38        -2.49437 -3.297e+00   -1.69221
+log_k_J9Z38        -5.08171 -5.875e+00   -4.28846
+log_k_JSE76        -3.20072 -4.180e+00   -2.22163
+f_cyan_ilr_1        0.71059  3.639e-01    1.05727
+f_cyan_ilr_2        1.15398  2.981e-01    2.00984
+f_JCZ38_qlogis      3.18027  1.056e+00    5.30452
+f_JSE76_qlogis      5.61578 -2.505e+01   36.28077
+log_k1             -2.38875 -2.517e+00   -2.26045
+log_k2             -4.67246 -4.928e+00   -4.41715
+g_qlogis           -0.28231 -1.135e+00    0.57058
+a.1                 2.08190  1.856e+00    2.30785
+b.1                 0.06114  5.015e-02    0.07214
+SD.log_k_JCZ38      0.84622  2.637e-01    1.42873
+SD.log_k_J9Z38      0.84564  2.566e-01    1.43464
+SD.log_k_JSE76      1.04385  3.242e-01    1.76351
+SD.f_cyan_ilr_1     0.38568  1.362e-01    0.63514
+SD.f_cyan_ilr_2     0.68046  7.166e-02    1.28925
+SD.f_JCZ38_qlogis   1.25244 -4.213e-02    2.54700
+SD.f_JSE76_qlogis   0.28202 -1.515e+03 1515.87968
+SD.log_k2           0.25749  7.655e-02    0.43843
+SD.g_qlogis         0.94535  3.490e-01    1.54174
+
+Correlation: 
+               cyan_0  l__JCZ3 l__J9Z3 l__JSE7 f_cy__1 f_cy__2 f_JCZ38 f_JSE76
+log_k_JCZ38    -0.0086                                                        
+log_k_J9Z38    -0.0363 -0.0007                                                
+log_k_JSE76     0.0015  0.1210 -0.0017                                        
+f_cyan_ilr_1   -0.0048  0.0095 -0.0572  0.0030                                
+f_cyan_ilr_2   -0.4788  0.0328  0.1143  0.0027 -0.0316                        
+f_JCZ38_qlogis  0.0736 -0.0664 -0.0137  0.0145 -0.0444 -0.2175                
+f_JSE76_qlogis -0.0137  0.0971  0.0035  0.0009  0.0293  0.1333 -0.6767        
+log_k1          0.2345 -0.0350 -0.0099 -0.0113 -0.0126 -0.1652  0.1756 -0.2161
+log_k2          0.0440 -0.0133  0.0199 -0.0040 -0.0097 -0.0119  0.0604 -0.1306
+g_qlogis        0.0438  0.0078 -0.0123  0.0029  0.0046 -0.0363 -0.0318  0.0736
+               log_k1  log_k2 
+log_k_JCZ38                   
+log_k_J9Z38                   
+log_k_JSE76                   
+f_cyan_ilr_1                  
+f_cyan_ilr_2                  
+f_JCZ38_qlogis                
+f_JSE76_qlogis                
+log_k1                        
+log_k2          0.3198        
+g_qlogis       -0.1666 -0.0954
+
+Random effects:
+                    est.      lower     upper
+SD.log_k_JCZ38    0.8462  2.637e-01    1.4287
+SD.log_k_J9Z38    0.8456  2.566e-01    1.4346
+SD.log_k_JSE76    1.0439  3.242e-01    1.7635
+SD.f_cyan_ilr_1   0.3857  1.362e-01    0.6351
+SD.f_cyan_ilr_2   0.6805  7.166e-02    1.2893
+SD.f_JCZ38_qlogis 1.2524 -4.213e-02    2.5470
+SD.f_JSE76_qlogis 0.2820 -1.515e+03 1515.8797
+SD.log_k2         0.2575  7.655e-02    0.4384
+SD.g_qlogis       0.9453  3.490e-01    1.5417
+
+Variance model:
+       est.   lower   upper
+a.1 2.08190 1.85595 2.30785
+b.1 0.06114 0.05015 0.07214
+
+Backtransformed parameters:
+                      est.     lower     upper
+cyan_0           1.011e+02 9.903e+01 103.18265
+k_JCZ38          8.255e-02 3.701e-02   0.18411
+k_J9Z38          6.209e-03 2.809e-03   0.01373
+k_JSE76          4.073e-02 1.530e-02   0.10843
+f_cyan_to_JCZ38  6.608e-01        NA        NA
+f_cyan_to_J9Z38  2.419e-01        NA        NA
+f_JCZ38_to_JSE76 9.601e-01 7.419e-01   0.99506
+f_JSE76_to_JCZ38 9.964e-01 1.322e-11   1.00000
+k1               9.174e-02 8.070e-02   0.10430
+k2               9.349e-03 7.243e-03   0.01207
+g                4.299e-01 2.432e-01   0.63890
+
+Resulting formation fractions:
+                  ff
+cyan_JCZ38  0.660808
+cyan_J9Z38  0.241904
+cyan_sink   0.097288
+JCZ38_JSE76 0.960085
+JCZ38_sink  0.039915
+JSE76_JCZ38 0.996373
+JSE76_sink  0.003627
+
+Estimated disappearance times:
+         DT50   DT90 DT50back DT50_k1 DT50_k2
+cyan   24.359 186.18    56.05   7.555   74.14
+JCZ38   8.397  27.89       NA      NA      NA
+J9Z38 111.631 370.83       NA      NA      NA
+JSE76  17.017  56.53       NA      NA      NA
+
+
+

+ +Hierarchical SFORB path 2 fit with constant variance + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 07:58:46 2023 
+Date of summary: Thu Apr 20 20:01:30 2023 
+
+Equations:
+d_cyan_free/dt = - k_cyan_free * cyan_free - k_cyan_free_bound *
+           cyan_free + k_cyan_bound_free * cyan_bound
+d_cyan_bound/dt = + k_cyan_free_bound * cyan_free - k_cyan_bound_free *
+           cyan_bound
+d_JCZ38/dt = + f_cyan_free_to_JCZ38 * k_cyan_free * cyan_free - k_JCZ38
+           * JCZ38 + f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_free_to_J9Z38 * k_cyan_free * cyan_free - k_J9Z38
+           * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 568.562 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+          cyan_free_0       log_k_cyan_free log_k_cyan_free_bound 
+             102.4394               -2.7673               -2.8942 
+log_k_cyan_bound_free           log_k_JCZ38           log_k_J9Z38 
+              -3.6201               -2.3107               -5.3123 
+          log_k_JSE76          f_cyan_ilr_1          f_cyan_ilr_2 
+              -3.7120                0.6754                1.1448 
+       f_JCZ38_qlogis        f_JSE76_qlogis 
+              13.2672               13.3538 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+                      cyan_free_0 log_k_cyan_free log_k_cyan_free_bound
+cyan_free_0                 4.589          0.0000                  0.00
+log_k_cyan_free             0.000          0.4849                  0.00
+log_k_cyan_free_bound       0.000          0.0000                  1.62
+log_k_cyan_bound_free       0.000          0.0000                  0.00
+log_k_JCZ38                 0.000          0.0000                  0.00
+log_k_J9Z38                 0.000          0.0000                  0.00
+log_k_JSE76                 0.000          0.0000                  0.00
+f_cyan_ilr_1                0.000          0.0000                  0.00
+f_cyan_ilr_2                0.000          0.0000                  0.00
+f_JCZ38_qlogis              0.000          0.0000                  0.00
+f_JSE76_qlogis              0.000          0.0000                  0.00
+                      log_k_cyan_bound_free log_k_JCZ38 log_k_J9Z38 log_k_JSE76
+cyan_free_0                           0.000      0.0000       0.000         0.0
+log_k_cyan_free                       0.000      0.0000       0.000         0.0
+log_k_cyan_free_bound                 0.000      0.0000       0.000         0.0
+log_k_cyan_bound_free                 1.197      0.0000       0.000         0.0
+log_k_JCZ38                           0.000      0.7966       0.000         0.0
+log_k_J9Z38                           0.000      0.0000       1.561         0.0
+log_k_JSE76                           0.000      0.0000       0.000         0.8
+f_cyan_ilr_1                          0.000      0.0000       0.000         0.0
+f_cyan_ilr_2                          0.000      0.0000       0.000         0.0
+f_JCZ38_qlogis                        0.000      0.0000       0.000         0.0
+f_JSE76_qlogis                        0.000      0.0000       0.000         0.0
+                      f_cyan_ilr_1 f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis
+cyan_free_0                 0.0000        0.000           0.00           0.00
+log_k_cyan_free             0.0000        0.000           0.00           0.00
+log_k_cyan_free_bound       0.0000        0.000           0.00           0.00
+log_k_cyan_bound_free       0.0000        0.000           0.00           0.00
+log_k_JCZ38                 0.0000        0.000           0.00           0.00
+log_k_J9Z38                 0.0000        0.000           0.00           0.00
+log_k_JSE76                 0.0000        0.000           0.00           0.00
+f_cyan_ilr_1                0.6349        0.000           0.00           0.00
+f_cyan_ilr_2                0.0000        1.797           0.00           0.00
+f_JCZ38_qlogis              0.0000        0.000          13.84           0.00
+f_JSE76_qlogis              0.0000        0.000           0.00          14.66
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2284 2275  -1120
+
+Optimised parameters:
+                             est.      lower      upper
+cyan_free_0              102.7730  1.015e+02  1.041e+02
+log_k_cyan_free           -2.8530 -3.167e+00 -2.539e+00
+log_k_cyan_free_bound     -2.7326 -3.543e+00 -1.922e+00
+log_k_cyan_bound_free     -3.5582 -4.126e+00 -2.990e+00
+log_k_JCZ38               -2.3810 -2.921e+00 -1.841e+00
+log_k_J9Z38               -5.2301 -5.963e+00 -4.497e+00
+log_k_JSE76               -3.0286 -4.286e+00 -1.771e+00
+f_cyan_ilr_1               0.7081  3.733e-01  1.043e+00
+f_cyan_ilr_2               0.5847  7.846e-03  1.162e+00
+f_JCZ38_qlogis             9.5676 -1.323e+03  1.342e+03
+f_JSE76_qlogis             3.7042  7.254e-02  7.336e+00
+a.1                        2.7222  2.532e+00  2.913e+00
+SD.log_k_cyan_free         0.3338  1.086e-01  5.589e-01
+SD.log_k_cyan_free_bound   0.8888  3.023e-01  1.475e+00
+SD.log_k_cyan_bound_free   0.6220  2.063e-01  1.038e+00
+SD.log_k_JCZ38             0.5221  1.334e-01  9.108e-01
+SD.log_k_J9Z38             0.7104  1.371e-01  1.284e+00
+SD.log_k_JSE76             1.3837  4.753e-01  2.292e+00
+SD.f_cyan_ilr_1            0.3620  1.248e-01  5.992e-01
+SD.f_cyan_ilr_2            0.4259  8.145e-02  7.704e-01
+SD.f_JCZ38_qlogis          3.5332 -1.037e+05  1.037e+05
+SD.f_JSE76_qlogis          1.6990 -2.771e-01  3.675e+00
+
+Correlation: 
+                      cyn_f_0 lg_k_c_ lg_k_cyn_f_ lg_k_cyn_b_ l__JCZ3 l__J9Z3
+log_k_cyan_free        0.2126                                                
+log_k_cyan_free_bound  0.0894  0.0871                                        
+log_k_cyan_bound_free  0.0033  0.0410  0.0583                                
+log_k_JCZ38           -0.0708 -0.0280 -0.0147      0.0019                    
+log_k_J9Z38           -0.0535 -0.0138  0.0012      0.0148      0.0085        
+log_k_JSE76           -0.0066 -0.0030 -0.0021     -0.0005      0.1090  0.0010
+f_cyan_ilr_1          -0.0364 -0.0157 -0.0095     -0.0015      0.0458 -0.0960
+f_cyan_ilr_2          -0.3814 -0.1104 -0.0423      0.0146      0.1540  0.1526
+f_JCZ38_qlogis         0.2507  0.0969  0.0482     -0.0097     -0.2282 -0.0363
+f_JSE76_qlogis        -0.1648 -0.0710 -0.0443     -0.0087      0.2002  0.0226
+                      l__JSE7 f_cy__1 f_cy__2 f_JCZ38
+log_k_cyan_free                                      
+log_k_cyan_free_bound                                
+log_k_cyan_bound_free                                
+log_k_JCZ38                                          
+log_k_J9Z38                                          
+log_k_JSE76                                          
+f_cyan_ilr_1           0.0001                        
+f_cyan_ilr_2           0.0031  0.0586                
+f_JCZ38_qlogis         0.0023 -0.1867 -0.6255        
+f_JSE76_qlogis         0.0082  0.1356  0.4519 -0.7951
+
+Random effects:
+                           est.      lower     upper
+SD.log_k_cyan_free       0.3338  1.086e-01 5.589e-01
+SD.log_k_cyan_free_bound 0.8888  3.023e-01 1.475e+00
+SD.log_k_cyan_bound_free 0.6220  2.063e-01 1.038e+00
+SD.log_k_JCZ38           0.5221  1.334e-01 9.108e-01
+SD.log_k_J9Z38           0.7104  1.371e-01 1.284e+00
+SD.log_k_JSE76           1.3837  4.753e-01 2.292e+00
+SD.f_cyan_ilr_1          0.3620  1.248e-01 5.992e-01
+SD.f_cyan_ilr_2          0.4259  8.145e-02 7.704e-01
+SD.f_JCZ38_qlogis        3.5332 -1.037e+05 1.037e+05
+SD.f_JSE76_qlogis        1.6990 -2.771e-01 3.675e+00
+
+Variance model:
+     est. lower upper
+a.1 2.722 2.532 2.913
+
+Backtransformed parameters:
+                          est.     lower     upper
+cyan_free_0          1.028e+02 1.015e+02 104.06475
+k_cyan_free          5.767e-02 4.213e-02   0.07894
+k_cyan_free_bound    6.505e-02 2.892e-02   0.14633
+k_cyan_bound_free    2.849e-02 1.614e-02   0.05028
+k_JCZ38              9.246e-02 5.390e-02   0.15859
+k_J9Z38              5.353e-03 2.572e-03   0.01114
+k_JSE76              4.838e-02 1.376e-02   0.17009
+f_cyan_free_to_JCZ38 6.011e-01 5.028e-01   0.83792
+f_cyan_free_to_J9Z38 2.208e-01 5.028e-01   0.83792
+f_JCZ38_to_JSE76     9.999e-01 0.000e+00   1.00000
+f_JSE76_to_JCZ38     9.760e-01 5.181e-01   0.99935
+
+Estimated Eigenvalues of SFORB model(s):
+cyan_b1 cyan_b2  cyan_g 
+0.13942 0.01178 0.35948 
+
+Resulting formation fractions:
+                       ff
+cyan_free_JCZ38 6.011e-01
+cyan_free_J9Z38 2.208e-01
+cyan_free_sink  1.780e-01
+cyan_free       1.000e+00
+JCZ38_JSE76     9.999e-01
+JCZ38_sink      6.996e-05
+JSE76_JCZ38     9.760e-01
+JSE76_sink      2.403e-02
+
+Estimated disappearance times:
+         DT50   DT90 DT50back DT50_cyan_b1 DT50_cyan_b2
+cyan   23.390 157.60    47.44        4.971        58.82
+JCZ38   7.497  24.90       NA           NA           NA
+J9Z38 129.482 430.13       NA           NA           NA
+JSE76  14.326  47.59       NA           NA           NA
+
+
+

+ +Hierarchical SFORB path 2 fit with two-component error + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 08:01:30 2023 
+Date of summary: Thu Apr 20 20:01:30 2023 
+
+Equations:
+d_cyan_free/dt = - k_cyan_free * cyan_free - k_cyan_free_bound *
+           cyan_free + k_cyan_bound_free * cyan_bound
+d_cyan_bound/dt = + k_cyan_free_bound * cyan_free - k_cyan_bound_free *
+           cyan_bound
+d_JCZ38/dt = + f_cyan_free_to_JCZ38 * k_cyan_free * cyan_free - k_JCZ38
+           * JCZ38 + f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_free_to_J9Z38 * k_cyan_free * cyan_free - k_J9Z38
+           * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 732.212 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+          cyan_free_0       log_k_cyan_free log_k_cyan_free_bound 
+              101.751                -2.837                -3.016 
+log_k_cyan_bound_free           log_k_JCZ38           log_k_J9Z38 
+               -3.660                -2.299                -5.313 
+          log_k_JSE76          f_cyan_ilr_1          f_cyan_ilr_2 
+               -3.699                 0.672                 5.873 
+       f_JCZ38_qlogis        f_JSE76_qlogis 
+               13.216                13.338 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+                      cyan_free_0 log_k_cyan_free log_k_cyan_free_bound
+cyan_free_0                 5.629           0.000                 0.000
+log_k_cyan_free             0.000           0.446                 0.000
+log_k_cyan_free_bound       0.000           0.000                 1.449
+log_k_cyan_bound_free       0.000           0.000                 0.000
+log_k_JCZ38                 0.000           0.000                 0.000
+log_k_J9Z38                 0.000           0.000                 0.000
+log_k_JSE76                 0.000           0.000                 0.000
+f_cyan_ilr_1                0.000           0.000                 0.000
+f_cyan_ilr_2                0.000           0.000                 0.000
+f_JCZ38_qlogis              0.000           0.000                 0.000
+f_JSE76_qlogis              0.000           0.000                 0.000
+                      log_k_cyan_bound_free log_k_JCZ38 log_k_J9Z38 log_k_JSE76
+cyan_free_0                           0.000      0.0000       0.000      0.0000
+log_k_cyan_free                       0.000      0.0000       0.000      0.0000
+log_k_cyan_free_bound                 0.000      0.0000       0.000      0.0000
+log_k_cyan_bound_free                 1.213      0.0000       0.000      0.0000
+log_k_JCZ38                           0.000      0.7801       0.000      0.0000
+log_k_J9Z38                           0.000      0.0000       1.575      0.0000
+log_k_JSE76                           0.000      0.0000       0.000      0.8078
+f_cyan_ilr_1                          0.000      0.0000       0.000      0.0000
+f_cyan_ilr_2                          0.000      0.0000       0.000      0.0000
+f_JCZ38_qlogis                        0.000      0.0000       0.000      0.0000
+f_JSE76_qlogis                        0.000      0.0000       0.000      0.0000
+                      f_cyan_ilr_1 f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis
+cyan_free_0                 0.0000         0.00           0.00           0.00
+log_k_cyan_free             0.0000         0.00           0.00           0.00
+log_k_cyan_free_bound       0.0000         0.00           0.00           0.00
+log_k_cyan_bound_free       0.0000         0.00           0.00           0.00
+log_k_JCZ38                 0.0000         0.00           0.00           0.00
+log_k_J9Z38                 0.0000         0.00           0.00           0.00
+log_k_JSE76                 0.0000         0.00           0.00           0.00
+f_cyan_ilr_1                0.6519         0.00           0.00           0.00
+f_cyan_ilr_2                0.0000        10.78           0.00           0.00
+f_JCZ38_qlogis              0.0000         0.00          13.96           0.00
+f_JSE76_qlogis              0.0000         0.00           0.00          14.69
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2240 2232  -1098
+
+Optimised parameters:
+                              est.     lower    upper
+cyan_free_0              101.10205  98.99221 103.2119
+log_k_cyan_free           -3.16929  -3.61395  -2.7246
+log_k_cyan_free_bound     -3.38259  -3.63022  -3.1350
+log_k_cyan_bound_free     -3.81075  -4.13888  -3.4826
+log_k_JCZ38               -2.42057  -3.00756  -1.8336
+log_k_J9Z38               -5.07501  -5.85138  -4.2986
+log_k_JSE76               -3.12442  -4.21277  -2.0361
+f_cyan_ilr_1               0.70577   0.35788   1.0537
+f_cyan_ilr_2               1.14824   0.15810   2.1384
+f_JCZ38_qlogis             3.52245   0.43257   6.6123
+f_JSE76_qlogis             5.65140 -21.22295  32.5257
+a.1                        2.07062   1.84329   2.2980
+b.1                        0.06227   0.05124   0.0733
+SD.log_k_cyan_free         0.49468   0.18566   0.8037
+SD.log_k_cyan_bound_free   0.28972   0.07188   0.5076
+SD.log_k_JCZ38             0.58852   0.16800   1.0090
+SD.log_k_J9Z38             0.82500   0.24730   1.4027
+SD.log_k_JSE76             1.19201   0.40313   1.9809
+SD.f_cyan_ilr_1            0.38534   0.13640   0.6343
+SD.f_cyan_ilr_2            0.72463   0.10076   1.3485
+SD.f_JCZ38_qlogis          1.38223  -0.20997   2.9744
+SD.f_JSE76_qlogis          2.07989 -72.53027  76.6901
+
+Correlation: 
+                      cyn_f_0 lg_k_c_ lg_k_cyn_f_ lg_k_cyn_b_ l__JCZ3 l__J9Z3
+log_k_cyan_free        0.1117                                                
+log_k_cyan_free_bound  0.1763  0.1828                                        
+log_k_cyan_bound_free  0.0120  0.0593  0.5030                                
+log_k_JCZ38           -0.0459 -0.0230 -0.0931     -0.0337                    
+log_k_J9Z38           -0.0381 -0.0123 -0.0139      0.0237      0.0063        
+log_k_JSE76           -0.0044 -0.0038 -0.0175     -0.0072      0.1120  0.0003
+f_cyan_ilr_1          -0.0199 -0.0087 -0.0407     -0.0233      0.0268 -0.0552
+f_cyan_ilr_2          -0.4806 -0.1015 -0.2291     -0.0269      0.1156  0.1113
+f_JCZ38_qlogis         0.1805  0.0825  0.3085      0.0963     -0.1674 -0.0314
+f_JSE76_qlogis        -0.1586 -0.0810 -0.3560     -0.1563      0.2025  0.0278
+                      l__JSE7 f_cy__1 f_cy__2 f_JCZ38
+log_k_cyan_free                                      
+log_k_cyan_free_bound                                
+log_k_cyan_bound_free                                
+log_k_JCZ38                                          
+log_k_J9Z38                                          
+log_k_JSE76                                          
+f_cyan_ilr_1           0.0024                        
+f_cyan_ilr_2           0.0087  0.0172                
+f_JCZ38_qlogis        -0.0016 -0.1047 -0.4656        
+f_JSE76_qlogis         0.0119  0.1034  0.4584 -0.8137
+
+Random effects:
+                           est.     lower   upper
+SD.log_k_cyan_free       0.4947   0.18566  0.8037
+SD.log_k_cyan_bound_free 0.2897   0.07188  0.5076
+SD.log_k_JCZ38           0.5885   0.16800  1.0090
+SD.log_k_J9Z38           0.8250   0.24730  1.4027
+SD.log_k_JSE76           1.1920   0.40313  1.9809
+SD.f_cyan_ilr_1          0.3853   0.13640  0.6343
+SD.f_cyan_ilr_2          0.7246   0.10076  1.3485
+SD.f_JCZ38_qlogis        1.3822  -0.20997  2.9744
+SD.f_JSE76_qlogis        2.0799 -72.53027 76.6901
+
+Variance model:
+       est.   lower  upper
+a.1 2.07062 1.84329 2.2980
+b.1 0.06227 0.05124 0.0733
+
+Backtransformed parameters:
+                          est.     lower     upper
+cyan_free_0          1.011e+02 9.899e+01 103.21190
+k_cyan_free          4.203e-02 2.695e-02   0.06557
+k_cyan_free_bound    3.396e-02 2.651e-02   0.04350
+k_cyan_bound_free    2.213e-02 1.594e-02   0.03073
+k_JCZ38              8.887e-02 4.941e-02   0.15984
+k_J9Z38              6.251e-03 2.876e-03   0.01359
+k_JSE76              4.396e-02 1.481e-02   0.13054
+f_cyan_free_to_JCZ38 6.590e-01 5.557e-01   0.95365
+f_cyan_free_to_J9Z38 2.429e-01 5.557e-01   0.95365
+f_JCZ38_to_JSE76     9.713e-01 6.065e-01   0.99866
+f_JSE76_to_JCZ38     9.965e-01 6.067e-10   1.00000
+
+Estimated Eigenvalues of SFORB model(s):
+cyan_b1 cyan_b2  cyan_g 
+0.08749 0.01063 0.40855 
+
+Resulting formation fractions:
+                     ff
+cyan_free_JCZ38 0.65905
+cyan_free_J9Z38 0.24291
+cyan_free_sink  0.09805
+cyan_free       1.00000
+JCZ38_JSE76     0.97132
+JCZ38_sink      0.02868
+JSE76_JCZ38     0.99650
+JSE76_sink      0.00350
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_cyan_b1 DT50_cyan_b2
+cyan   24.91 167.16    50.32        7.922        65.19
+JCZ38   7.80  25.91       NA           NA           NA
+J9Z38 110.89 368.36       NA           NA           NA
+JSE76  15.77  52.38       NA           NA           NA
+
+
+

+
+
+

Pathway 2, refined fits +

+ +Hierarchical FOMC path 2 fit with reduced random effects, two-component +error + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 08:15:01 2023 
+Date of summary: Thu Apr 20 20:01:31 2023 
+
+Equations:
+d_cyan/dt = - (alpha/beta) * 1/((time/beta) + 1) * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_JCZ38 * JCZ38 + f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_to_J9Z38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 808.728 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      101.9028        -1.9055        -5.0249        -2.5646         0.6807 
+  f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis      log_alpha       log_beta 
+        4.8883        16.0676         9.3923        -0.1346         3.0364 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          6.321       0.000       0.000       0.000       0.0000
+log_k_JCZ38     0.000       1.392       0.000       0.000       0.0000
+log_k_J9Z38     0.000       0.000       1.561       0.000       0.0000
+log_k_JSE76     0.000       0.000       0.000       3.614       0.0000
+f_cyan_ilr_1    0.000       0.000       0.000       0.000       0.6339
+f_cyan_ilr_2    0.000       0.000       0.000       0.000       0.0000
+f_JCZ38_qlogis  0.000       0.000       0.000       0.000       0.0000
+f_JSE76_qlogis  0.000       0.000       0.000       0.000       0.0000
+log_alpha       0.000       0.000       0.000       0.000       0.0000
+log_beta        0.000       0.000       0.000       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis log_alpha log_beta
+cyan_0                 0.00           0.00           0.00    0.0000   0.0000
+log_k_JCZ38            0.00           0.00           0.00    0.0000   0.0000
+log_k_J9Z38            0.00           0.00           0.00    0.0000   0.0000
+log_k_JSE76            0.00           0.00           0.00    0.0000   0.0000
+f_cyan_ilr_1           0.00           0.00           0.00    0.0000   0.0000
+f_cyan_ilr_2          10.41           0.00           0.00    0.0000   0.0000
+f_JCZ38_qlogis         0.00          12.24           0.00    0.0000   0.0000
+f_JSE76_qlogis         0.00           0.00          15.13    0.0000   0.0000
+log_alpha              0.00           0.00           0.00    0.3701   0.0000
+log_beta               0.00           0.00           0.00    0.0000   0.5662
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2251 2244  -1106
+
+Optimised parameters:
+                      est.   lower   upper
+cyan_0           101.05768      NA      NA
+log_k_JCZ38       -2.73252      NA      NA
+log_k_J9Z38       -5.07399      NA      NA
+log_k_JSE76       -3.52863      NA      NA
+f_cyan_ilr_1       0.72176      NA      NA
+f_cyan_ilr_2       1.34610      NA      NA
+f_JCZ38_qlogis     2.08337      NA      NA
+f_JSE76_qlogis  1590.31880      NA      NA
+log_alpha         -0.09336      NA      NA
+log_beta           3.10191      NA      NA
+a.1                2.08557 1.85439 2.31675
+b.1                0.06998 0.05800 0.08197
+SD.log_k_JCZ38     1.20053 0.43329 1.96777
+SD.log_k_J9Z38     0.85854 0.26708 1.45000
+SD.log_k_JSE76     0.62528 0.16061 1.08995
+SD.f_cyan_ilr_1    0.35190 0.12340 0.58039
+SD.f_cyan_ilr_2    0.85385 0.15391 1.55378
+SD.log_alpha       0.28971 0.08718 0.49225
+SD.log_beta        0.31614 0.05938 0.57290
+
+Correlation is not available
+
+Random effects:
+                  est.   lower  upper
+SD.log_k_JCZ38  1.2005 0.43329 1.9678
+SD.log_k_J9Z38  0.8585 0.26708 1.4500
+SD.log_k_JSE76  0.6253 0.16061 1.0900
+SD.f_cyan_ilr_1 0.3519 0.12340 0.5804
+SD.f_cyan_ilr_2 0.8538 0.15391 1.5538
+SD.log_alpha    0.2897 0.08718 0.4923
+SD.log_beta     0.3161 0.05938 0.5729
+
+Variance model:
+       est. lower   upper
+a.1 2.08557 1.854 2.31675
+b.1 0.06998 0.058 0.08197
+
+Backtransformed parameters:
+                      est. lower upper
+cyan_0           1.011e+02    NA    NA
+k_JCZ38          6.506e-02    NA    NA
+k_J9Z38          6.257e-03    NA    NA
+k_JSE76          2.935e-02    NA    NA
+f_cyan_to_JCZ38  6.776e-01    NA    NA
+f_cyan_to_J9Z38  2.442e-01    NA    NA
+f_JCZ38_to_JSE76 8.893e-01    NA    NA
+f_JSE76_to_JCZ38 1.000e+00    NA    NA
+alpha            9.109e-01    NA    NA
+beta             2.224e+01    NA    NA
+
+Resulting formation fractions:
+                 ff
+cyan_JCZ38  0.67761
+cyan_J9Z38  0.24417
+cyan_sink   0.07822
+JCZ38_JSE76 0.88928
+JCZ38_sink  0.11072
+JSE76_JCZ38 1.00000
+JSE76_sink  0.00000
+
+Estimated disappearance times:
+        DT50   DT90 DT50back
+cyan   25.36 256.37    77.18
+JCZ38  10.65  35.39       NA
+J9Z38 110.77 367.98       NA
+JSE76  23.62  78.47       NA
+
+
+

+ +Hierarchical DFOP path 2 fit with reduced random effects, constant +variance + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 08:16:32 2023 
+Date of summary: Thu Apr 20 20:01:31 2023 
+
+Equations:
+d_cyan/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_JCZ38 * JCZ38 +
+           f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_to_J9Z38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 900.061 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      102.4358        -2.3107        -5.3123        -3.7120         0.6753 
+  f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis         log_k1         log_k2 
+        1.1462        12.4095        12.3630        -1.9317        -4.4557 
+      g_qlogis 
+       -0.5648 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          4.594      0.0000       0.000         0.0       0.0000
+log_k_JCZ38     0.000      0.7966       0.000         0.0       0.0000
+log_k_J9Z38     0.000      0.0000       1.561         0.0       0.0000
+log_k_JSE76     0.000      0.0000       0.000         0.8       0.0000
+f_cyan_ilr_1    0.000      0.0000       0.000         0.0       0.6349
+f_cyan_ilr_2    0.000      0.0000       0.000         0.0       0.0000
+f_JCZ38_qlogis  0.000      0.0000       0.000         0.0       0.0000
+f_JSE76_qlogis  0.000      0.0000       0.000         0.0       0.0000
+log_k1          0.000      0.0000       0.000         0.0       0.0000
+log_k2          0.000      0.0000       0.000         0.0       0.0000
+g_qlogis        0.000      0.0000       0.000         0.0       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis log_k1 log_k2
+cyan_0                0.000           0.00            0.0  0.000 0.0000
+log_k_JCZ38           0.000           0.00            0.0  0.000 0.0000
+log_k_J9Z38           0.000           0.00            0.0  0.000 0.0000
+log_k_JSE76           0.000           0.00            0.0  0.000 0.0000
+f_cyan_ilr_1          0.000           0.00            0.0  0.000 0.0000
+f_cyan_ilr_2          1.797           0.00            0.0  0.000 0.0000
+f_JCZ38_qlogis        0.000          13.85            0.0  0.000 0.0000
+f_JSE76_qlogis        0.000           0.00           14.1  0.000 0.0000
+log_k1                0.000           0.00            0.0  1.106 0.0000
+log_k2                0.000           0.00            0.0  0.000 0.6141
+g_qlogis              0.000           0.00            0.0  0.000 0.0000
+               g_qlogis
+cyan_0            0.000
+log_k_JCZ38       0.000
+log_k_J9Z38       0.000
+log_k_JSE76       0.000
+f_cyan_ilr_1      0.000
+f_cyan_ilr_2      0.000
+f_JCZ38_qlogis    0.000
+f_JSE76_qlogis    0.000
+log_k1            0.000
+log_k2            0.000
+g_qlogis          1.595
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2282 2274  -1121
+
+Optimised parameters:
+                     est.   lower  upper
+cyan_0           102.5254      NA     NA
+log_k_JCZ38       -2.9358      NA     NA
+log_k_J9Z38       -5.1424      NA     NA
+log_k_JSE76       -3.6458      NA     NA
+f_cyan_ilr_1       0.6957      NA     NA
+f_cyan_ilr_2       0.6635      NA     NA
+f_JCZ38_qlogis  4984.8163      NA     NA
+f_JSE76_qlogis     1.9415      NA     NA
+log_k1            -1.9456      NA     NA
+log_k2            -4.4705      NA     NA
+g_qlogis          -0.5117      NA     NA
+a.1                2.7455 2.55392 2.9370
+SD.log_k_JCZ38     1.3163 0.47635 2.1563
+SD.log_k_J9Z38     0.7162 0.16133 1.2711
+SD.log_k_JSE76     0.6457 0.15249 1.1390
+SD.f_cyan_ilr_1    0.3424 0.11714 0.5677
+SD.f_cyan_ilr_2    0.4524 0.09709 0.8077
+SD.log_k1          0.7353 0.25445 1.2161
+SD.log_k2          0.5137 0.18206 0.8453
+SD.g_qlogis        0.9857 0.35651 1.6148
+
+Correlation is not available
+
+Random effects:
+                  est.   lower  upper
+SD.log_k_JCZ38  1.3163 0.47635 2.1563
+SD.log_k_J9Z38  0.7162 0.16133 1.2711
+SD.log_k_JSE76  0.6457 0.15249 1.1390
+SD.f_cyan_ilr_1 0.3424 0.11714 0.5677
+SD.f_cyan_ilr_2 0.4524 0.09709 0.8077
+SD.log_k1       0.7353 0.25445 1.2161
+SD.log_k2       0.5137 0.18206 0.8453
+SD.g_qlogis     0.9857 0.35651 1.6148
+
+Variance model:
+     est. lower upper
+a.1 2.745 2.554 2.937
+
+Backtransformed parameters:
+                      est. lower upper
+cyan_0           1.025e+02    NA    NA
+k_JCZ38          5.309e-02    NA    NA
+k_J9Z38          5.844e-03    NA    NA
+k_JSE76          2.610e-02    NA    NA
+f_cyan_to_JCZ38  6.079e-01    NA    NA
+f_cyan_to_J9Z38  2.272e-01    NA    NA
+f_JCZ38_to_JSE76 1.000e+00    NA    NA
+f_JSE76_to_JCZ38 8.745e-01    NA    NA
+k1               1.429e-01    NA    NA
+k2               1.144e-02    NA    NA
+g                3.748e-01    NA    NA
+
+Resulting formation fractions:
+                ff
+cyan_JCZ38  0.6079
+cyan_J9Z38  0.2272
+cyan_sink   0.1649
+JCZ38_JSE76 1.0000
+JCZ38_sink  0.0000
+JSE76_JCZ38 0.8745
+JSE76_sink  0.1255
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_k1 DT50_k2
+cyan   22.29 160.20    48.22    4.85   60.58
+JCZ38  13.06  43.37       NA      NA      NA
+J9Z38 118.61 394.02       NA      NA      NA
+JSE76  26.56  88.22       NA      NA      NA
+
+
+

+ +Hierarchical DFOP path 2 fit with reduced random effects, two-component +error + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 08:16:47 2023 
+Date of summary: Thu Apr 20 20:01:31 2023 
+
+Equations:
+d_cyan/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_JCZ38 * JCZ38 +
+           f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_to_J9Z38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 914.763 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      101.7523        -1.5948        -5.0119        -2.2723         0.6719 
+  f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis         log_k1         log_k2 
+        5.1681        12.8238        12.4130        -2.0057        -4.5526 
+      g_qlogis 
+       -0.5805 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          5.627       0.000       0.000       0.000       0.0000
+log_k_JCZ38     0.000       2.327       0.000       0.000       0.0000
+log_k_J9Z38     0.000       0.000       1.664       0.000       0.0000
+log_k_JSE76     0.000       0.000       0.000       4.566       0.0000
+f_cyan_ilr_1    0.000       0.000       0.000       0.000       0.6519
+f_cyan_ilr_2    0.000       0.000       0.000       0.000       0.0000
+f_JCZ38_qlogis  0.000       0.000       0.000       0.000       0.0000
+f_JSE76_qlogis  0.000       0.000       0.000       0.000       0.0000
+log_k1          0.000       0.000       0.000       0.000       0.0000
+log_k2          0.000       0.000       0.000       0.000       0.0000
+g_qlogis        0.000       0.000       0.000       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis log_k1 log_k2
+cyan_0                  0.0           0.00           0.00 0.0000 0.0000
+log_k_JCZ38             0.0           0.00           0.00 0.0000 0.0000
+log_k_J9Z38             0.0           0.00           0.00 0.0000 0.0000
+log_k_JSE76             0.0           0.00           0.00 0.0000 0.0000
+f_cyan_ilr_1            0.0           0.00           0.00 0.0000 0.0000
+f_cyan_ilr_2           10.1           0.00           0.00 0.0000 0.0000
+f_JCZ38_qlogis          0.0          13.99           0.00 0.0000 0.0000
+f_JSE76_qlogis          0.0           0.00          14.15 0.0000 0.0000
+log_k1                  0.0           0.00           0.00 0.8452 0.0000
+log_k2                  0.0           0.00           0.00 0.0000 0.5968
+g_qlogis                0.0           0.00           0.00 0.0000 0.0000
+               g_qlogis
+cyan_0            0.000
+log_k_JCZ38       0.000
+log_k_J9Z38       0.000
+log_k_JSE76       0.000
+f_cyan_ilr_1      0.000
+f_cyan_ilr_2      0.000
+f_JCZ38_qlogis    0.000
+f_JSE76_qlogis    0.000
+log_k1            0.000
+log_k2            0.000
+g_qlogis          1.691
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2232 2224  -1096
+
+Optimised parameters:
+                     est.   lower   upper
+cyan_0          101.20051      NA      NA
+log_k_JCZ38      -2.93542      NA      NA
+log_k_J9Z38      -5.03151      NA      NA
+log_k_JSE76      -3.67679      NA      NA
+f_cyan_ilr_1      0.67290      NA      NA
+f_cyan_ilr_2      0.99787      NA      NA
+f_JCZ38_qlogis  348.32484      NA      NA
+f_JSE76_qlogis    1.87846      NA      NA
+log_k1           -2.32738      NA      NA
+log_k2           -4.61295      NA      NA
+g_qlogis         -0.38342      NA      NA
+a.1               2.06184 1.83746 2.28622
+b.1               0.06329 0.05211 0.07447
+SD.log_k_JCZ38    1.29042 0.47468 2.10617
+SD.log_k_J9Z38    0.84235 0.25903 1.42566
+SD.log_k_JSE76    0.56930 0.13934 0.99926
+SD.f_cyan_ilr_1   0.35183 0.12298 0.58068
+SD.f_cyan_ilr_2   0.77269 0.17908 1.36631
+SD.log_k2         0.28549 0.09210 0.47888
+SD.g_qlogis       0.93830 0.34568 1.53093
+
+Correlation is not available
+
+Random effects:
+                  est.  lower  upper
+SD.log_k_JCZ38  1.2904 0.4747 2.1062
+SD.log_k_J9Z38  0.8423 0.2590 1.4257
+SD.log_k_JSE76  0.5693 0.1393 0.9993
+SD.f_cyan_ilr_1 0.3518 0.1230 0.5807
+SD.f_cyan_ilr_2 0.7727 0.1791 1.3663
+SD.log_k2       0.2855 0.0921 0.4789
+SD.g_qlogis     0.9383 0.3457 1.5309
+
+Variance model:
+       est.   lower   upper
+a.1 2.06184 1.83746 2.28622
+b.1 0.06329 0.05211 0.07447
+
+Backtransformed parameters:
+                      est. lower upper
+cyan_0           1.012e+02    NA    NA
+k_JCZ38          5.311e-02    NA    NA
+k_J9Z38          6.529e-03    NA    NA
+k_JSE76          2.530e-02    NA    NA
+f_cyan_to_JCZ38  6.373e-01    NA    NA
+f_cyan_to_J9Z38  2.461e-01    NA    NA
+f_JCZ38_to_JSE76 1.000e+00    NA    NA
+f_JSE76_to_JCZ38 8.674e-01    NA    NA
+k1               9.755e-02    NA    NA
+k2               9.922e-03    NA    NA
+g                4.053e-01    NA    NA
+
+Resulting formation fractions:
+                ff
+cyan_JCZ38  0.6373
+cyan_J9Z38  0.2461
+cyan_sink   0.1167
+JCZ38_JSE76 1.0000
+JCZ38_sink  0.0000
+JSE76_JCZ38 0.8674
+JSE76_sink  0.1326
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_k1 DT50_k2
+cyan   24.93 179.68    54.09   7.105   69.86
+JCZ38  13.05  43.36       NA      NA      NA
+J9Z38 106.16 352.67       NA      NA      NA
+JSE76  27.39  91.00       NA      NA      NA
+
+
+

+ +Hierarchical SFORB path 2 fit with reduced random effects, constant +variance + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 08:16:33 2023 
+Date of summary: Thu Apr 20 20:01:31 2023 
+
+Equations:
+d_cyan_free/dt = - k_cyan_free * cyan_free - k_cyan_free_bound *
+           cyan_free + k_cyan_bound_free * cyan_bound
+d_cyan_bound/dt = + k_cyan_free_bound * cyan_free - k_cyan_bound_free *
+           cyan_bound
+d_JCZ38/dt = + f_cyan_free_to_JCZ38 * k_cyan_free * cyan_free - k_JCZ38
+           * JCZ38 + f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_free_to_J9Z38 * k_cyan_free * cyan_free - k_J9Z38
+           * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 901.179 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+          cyan_free_0       log_k_cyan_free log_k_cyan_free_bound 
+             102.4394               -2.7673               -2.8942 
+log_k_cyan_bound_free           log_k_JCZ38           log_k_J9Z38 
+              -3.6201               -2.3107               -5.3123 
+          log_k_JSE76          f_cyan_ilr_1          f_cyan_ilr_2 
+              -3.7120                0.6754                1.1448 
+       f_JCZ38_qlogis        f_JSE76_qlogis 
+              13.2672               13.3538 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+                      cyan_free_0 log_k_cyan_free log_k_cyan_free_bound
+cyan_free_0                 4.589          0.0000                  0.00
+log_k_cyan_free             0.000          0.4849                  0.00
+log_k_cyan_free_bound       0.000          0.0000                  1.62
+log_k_cyan_bound_free       0.000          0.0000                  0.00
+log_k_JCZ38                 0.000          0.0000                  0.00
+log_k_J9Z38                 0.000          0.0000                  0.00
+log_k_JSE76                 0.000          0.0000                  0.00
+f_cyan_ilr_1                0.000          0.0000                  0.00
+f_cyan_ilr_2                0.000          0.0000                  0.00
+f_JCZ38_qlogis              0.000          0.0000                  0.00
+f_JSE76_qlogis              0.000          0.0000                  0.00
+                      log_k_cyan_bound_free log_k_JCZ38 log_k_J9Z38 log_k_JSE76
+cyan_free_0                           0.000      0.0000       0.000         0.0
+log_k_cyan_free                       0.000      0.0000       0.000         0.0
+log_k_cyan_free_bound                 0.000      0.0000       0.000         0.0
+log_k_cyan_bound_free                 1.197      0.0000       0.000         0.0
+log_k_JCZ38                           0.000      0.7966       0.000         0.0
+log_k_J9Z38                           0.000      0.0000       1.561         0.0
+log_k_JSE76                           0.000      0.0000       0.000         0.8
+f_cyan_ilr_1                          0.000      0.0000       0.000         0.0
+f_cyan_ilr_2                          0.000      0.0000       0.000         0.0
+f_JCZ38_qlogis                        0.000      0.0000       0.000         0.0
+f_JSE76_qlogis                        0.000      0.0000       0.000         0.0
+                      f_cyan_ilr_1 f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis
+cyan_free_0                 0.0000        0.000           0.00           0.00
+log_k_cyan_free             0.0000        0.000           0.00           0.00
+log_k_cyan_free_bound       0.0000        0.000           0.00           0.00
+log_k_cyan_bound_free       0.0000        0.000           0.00           0.00
+log_k_JCZ38                 0.0000        0.000           0.00           0.00
+log_k_J9Z38                 0.0000        0.000           0.00           0.00
+log_k_JSE76                 0.0000        0.000           0.00           0.00
+f_cyan_ilr_1                0.6349        0.000           0.00           0.00
+f_cyan_ilr_2                0.0000        1.797           0.00           0.00
+f_JCZ38_qlogis              0.0000        0.000          13.84           0.00
+f_JSE76_qlogis              0.0000        0.000           0.00          14.66
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2279 2272  -1120
+
+Optimised parameters:
+                              est.   lower  upper
+cyan_free_0               102.5621      NA     NA
+log_k_cyan_free            -2.8531      NA     NA
+log_k_cyan_free_bound      -2.6916      NA     NA
+log_k_cyan_bound_free      -3.5032      NA     NA
+log_k_JCZ38                -2.9436      NA     NA
+log_k_J9Z38                -5.1140      NA     NA
+log_k_JSE76                -3.6472      NA     NA
+f_cyan_ilr_1                0.6887      NA     NA
+f_cyan_ilr_2                0.6874      NA     NA
+f_JCZ38_qlogis           4063.6389      NA     NA
+f_JSE76_qlogis              1.9556      NA     NA
+a.1                         2.7460 2.55451 2.9376
+SD.log_k_cyan_free          0.3131 0.09841 0.5277
+SD.log_k_cyan_free_bound    0.8850 0.29909 1.4710
+SD.log_k_cyan_bound_free    0.6167 0.20391 1.0295
+SD.log_k_JCZ38              1.3555 0.49101 2.2200
+SD.log_k_J9Z38              0.7200 0.16166 1.2783
+SD.log_k_JSE76              0.6252 0.14619 1.1042
+SD.f_cyan_ilr_1             0.3386 0.11447 0.5627
+SD.f_cyan_ilr_2             0.4699 0.09810 0.8417
+
+Correlation is not available
+
+Random effects:
+                           est.   lower  upper
+SD.log_k_cyan_free       0.3131 0.09841 0.5277
+SD.log_k_cyan_free_bound 0.8850 0.29909 1.4710
+SD.log_k_cyan_bound_free 0.6167 0.20391 1.0295
+SD.log_k_JCZ38           1.3555 0.49101 2.2200
+SD.log_k_J9Z38           0.7200 0.16166 1.2783
+SD.log_k_JSE76           0.6252 0.14619 1.1042
+SD.f_cyan_ilr_1          0.3386 0.11447 0.5627
+SD.f_cyan_ilr_2          0.4699 0.09810 0.8417
+
+Variance model:
+     est. lower upper
+a.1 2.746 2.555 2.938
+
+Backtransformed parameters:
+                          est. lower upper
+cyan_free_0          1.026e+02    NA    NA
+k_cyan_free          5.767e-02    NA    NA
+k_cyan_free_bound    6.777e-02    NA    NA
+k_cyan_bound_free    3.010e-02    NA    NA
+k_JCZ38              5.267e-02    NA    NA
+k_J9Z38              6.012e-03    NA    NA
+k_JSE76              2.606e-02    NA    NA
+f_cyan_free_to_JCZ38 6.089e-01    NA    NA
+f_cyan_free_to_J9Z38 2.299e-01    NA    NA
+f_JCZ38_to_JSE76     1.000e+00    NA    NA
+f_JSE76_to_JCZ38     8.761e-01    NA    NA
+
+Estimated Eigenvalues of SFORB model(s):
+cyan_b1 cyan_b2  cyan_g 
+ 0.1434  0.0121  0.3469 
+
+Resulting formation fractions:
+                    ff
+cyan_free_JCZ38 0.6089
+cyan_free_J9Z38 0.2299
+cyan_free_sink  0.1612
+cyan_free       1.0000
+JCZ38_JSE76     1.0000
+JCZ38_sink      0.0000
+JSE76_JCZ38     0.8761
+JSE76_sink      0.1239
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_cyan_b1 DT50_cyan_b2
+cyan   23.94 155.06    46.68        4.832        57.28
+JCZ38  13.16  43.71       NA           NA           NA
+J9Z38 115.30 383.02       NA           NA           NA
+JSE76  26.59  88.35       NA           NA           NA
+
+
+

+ +Hierarchical SFORB path 2 fit with reduced random effects, two-component +error + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 08:16:19 2023 
+Date of summary: Thu Apr 20 20:01:31 2023 
+
+Equations:
+d_cyan_free/dt = - k_cyan_free * cyan_free - k_cyan_free_bound *
+           cyan_free + k_cyan_bound_free * cyan_bound
+d_cyan_bound/dt = + k_cyan_free_bound * cyan_free - k_cyan_bound_free *
+           cyan_bound
+d_JCZ38/dt = + f_cyan_free_to_JCZ38 * k_cyan_free * cyan_free - k_JCZ38
+           * JCZ38 + f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_free_to_J9Z38 * k_cyan_free * cyan_free - k_J9Z38
+           * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 887.343 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+          cyan_free_0       log_k_cyan_free log_k_cyan_free_bound 
+              101.751                -2.837                -3.016 
+log_k_cyan_bound_free           log_k_JCZ38           log_k_J9Z38 
+               -3.660                -2.299                -5.313 
+          log_k_JSE76          f_cyan_ilr_1          f_cyan_ilr_2 
+               -3.699                 0.672                 5.873 
+       f_JCZ38_qlogis        f_JSE76_qlogis 
+               13.216                13.338 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+                      cyan_free_0 log_k_cyan_free log_k_cyan_free_bound
+cyan_free_0                 5.629           0.000                 0.000
+log_k_cyan_free             0.000           0.446                 0.000
+log_k_cyan_free_bound       0.000           0.000                 1.449
+log_k_cyan_bound_free       0.000           0.000                 0.000
+log_k_JCZ38                 0.000           0.000                 0.000
+log_k_J9Z38                 0.000           0.000                 0.000
+log_k_JSE76                 0.000           0.000                 0.000
+f_cyan_ilr_1                0.000           0.000                 0.000
+f_cyan_ilr_2                0.000           0.000                 0.000
+f_JCZ38_qlogis              0.000           0.000                 0.000
+f_JSE76_qlogis              0.000           0.000                 0.000
+                      log_k_cyan_bound_free log_k_JCZ38 log_k_J9Z38 log_k_JSE76
+cyan_free_0                           0.000      0.0000       0.000      0.0000
+log_k_cyan_free                       0.000      0.0000       0.000      0.0000
+log_k_cyan_free_bound                 0.000      0.0000       0.000      0.0000
+log_k_cyan_bound_free                 1.213      0.0000       0.000      0.0000
+log_k_JCZ38                           0.000      0.7801       0.000      0.0000
+log_k_J9Z38                           0.000      0.0000       1.575      0.0000
+log_k_JSE76                           0.000      0.0000       0.000      0.8078
+f_cyan_ilr_1                          0.000      0.0000       0.000      0.0000
+f_cyan_ilr_2                          0.000      0.0000       0.000      0.0000
+f_JCZ38_qlogis                        0.000      0.0000       0.000      0.0000
+f_JSE76_qlogis                        0.000      0.0000       0.000      0.0000
+                      f_cyan_ilr_1 f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis
+cyan_free_0                 0.0000         0.00           0.00           0.00
+log_k_cyan_free             0.0000         0.00           0.00           0.00
+log_k_cyan_free_bound       0.0000         0.00           0.00           0.00
+log_k_cyan_bound_free       0.0000         0.00           0.00           0.00
+log_k_JCZ38                 0.0000         0.00           0.00           0.00
+log_k_J9Z38                 0.0000         0.00           0.00           0.00
+log_k_JSE76                 0.0000         0.00           0.00           0.00
+f_cyan_ilr_1                0.6519         0.00           0.00           0.00
+f_cyan_ilr_2                0.0000        10.78           0.00           0.00
+f_JCZ38_qlogis              0.0000         0.00          13.96           0.00
+f_JSE76_qlogis              0.0000         0.00           0.00          14.69
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2236 2228  -1098
+
+Optimised parameters:
+                              est.   lower   upper
+cyan_free_0              100.72760      NA      NA
+log_k_cyan_free           -3.18281      NA      NA
+log_k_cyan_free_bound     -3.37924      NA      NA
+log_k_cyan_bound_free     -3.77107      NA      NA
+log_k_JCZ38               -2.92811      NA      NA
+log_k_J9Z38               -5.02759      NA      NA
+log_k_JSE76               -3.65835      NA      NA
+f_cyan_ilr_1               0.67390      NA      NA
+f_cyan_ilr_2               1.15106      NA      NA
+f_JCZ38_qlogis           827.82299      NA      NA
+f_JSE76_qlogis             1.83064      NA      NA
+a.1                        2.06921 1.84443 2.29399
+b.1                        0.06391 0.05267 0.07515
+SD.log_k_cyan_free         0.50518 0.18962 0.82075
+SD.log_k_cyan_bound_free   0.30991 0.08170 0.53813
+SD.log_k_JCZ38             1.26661 0.46578 2.06744
+SD.log_k_J9Z38             0.88272 0.27813 1.48730
+SD.log_k_JSE76             0.53050 0.12561 0.93538
+SD.f_cyan_ilr_1            0.35547 0.12461 0.58633
+SD.f_cyan_ilr_2            0.91446 0.20131 1.62761
+
+Correlation is not available
+
+Random effects:
+                           est.  lower  upper
+SD.log_k_cyan_free       0.5052 0.1896 0.8207
+SD.log_k_cyan_bound_free 0.3099 0.0817 0.5381
+SD.log_k_JCZ38           1.2666 0.4658 2.0674
+SD.log_k_J9Z38           0.8827 0.2781 1.4873
+SD.log_k_JSE76           0.5305 0.1256 0.9354
+SD.f_cyan_ilr_1          0.3555 0.1246 0.5863
+SD.f_cyan_ilr_2          0.9145 0.2013 1.6276
+
+Variance model:
+       est.   lower   upper
+a.1 2.06921 1.84443 2.29399
+b.1 0.06391 0.05267 0.07515
+
+Backtransformed parameters:
+                          est. lower upper
+cyan_free_0          1.007e+02    NA    NA
+k_cyan_free          4.147e-02    NA    NA
+k_cyan_free_bound    3.407e-02    NA    NA
+k_cyan_bound_free    2.303e-02    NA    NA
+k_JCZ38              5.350e-02    NA    NA
+k_J9Z38              6.555e-03    NA    NA
+k_JSE76              2.578e-02    NA    NA
+f_cyan_free_to_JCZ38 6.505e-01    NA    NA
+f_cyan_free_to_J9Z38 2.508e-01    NA    NA
+f_JCZ38_to_JSE76     1.000e+00    NA    NA
+f_JSE76_to_JCZ38     8.618e-01    NA    NA
+
+Estimated Eigenvalues of SFORB model(s):
+cyan_b1 cyan_b2  cyan_g 
+0.08768 0.01089 0.39821 
+
+Resulting formation fractions:
+                     ff
+cyan_free_JCZ38 0.65053
+cyan_free_J9Z38 0.25082
+cyan_free_sink  0.09864
+cyan_free       1.00000
+JCZ38_JSE76     1.00000
+JCZ38_sink      0.00000
+JSE76_JCZ38     0.86184
+JSE76_sink      0.13816
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_cyan_b1 DT50_cyan_b2
+cyan   25.32 164.79    49.61        7.906        63.64
+JCZ38  12.96  43.04       NA           NA           NA
+J9Z38 105.75 351.29       NA           NA           NA
+JSE76  26.89  89.33       NA           NA           NA
+
+
+

+
+
+
+

Session info +

+
R version 4.2.3 (2023-03-15)
+Platform: x86_64-pc-linux-gnu (64-bit)
+Running under: Debian GNU/Linux 12 (bookworm)
+
+Matrix products: default
+BLAS:   /usr/lib/x86_64-linux-gnu/openblas-serial/libblas.so.3
+LAPACK: /usr/lib/x86_64-linux-gnu/openblas-serial/libopenblas-r0.3.21.so
+
+locale:
+ [1] LC_CTYPE=de_DE.UTF-8       LC_NUMERIC=C              
+ [3] LC_TIME=de_DE.UTF-8        LC_COLLATE=de_DE.UTF-8    
+ [5] LC_MONETARY=de_DE.UTF-8    LC_MESSAGES=de_DE.UTF-8   
+ [7] LC_PAPER=de_DE.UTF-8       LC_NAME=C                 
+ [9] LC_ADDRESS=C               LC_TELEPHONE=C            
+[11] LC_MEASUREMENT=de_DE.UTF-8 LC_IDENTIFICATION=C       
+
+attached base packages:
+[1] parallel  stats     graphics  grDevices utils     datasets  methods  
+[8] base     
+
+other attached packages:
+[1] saemix_3.2 npde_3.3   knitr_1.42 mkin_1.2.3
+
+loaded via a namespace (and not attached):
+ [1] pillar_1.9.0      bslib_0.4.2       compiler_4.2.3    jquerylib_0.1.4  
+ [5] tools_4.2.3       mclust_6.0.0      digest_0.6.31     tibble_3.2.1     
+ [9] jsonlite_1.8.4    evaluate_0.20     memoise_2.0.1     lifecycle_1.0.3  
+[13] nlme_3.1-162      gtable_0.3.3      lattice_0.21-8    pkgconfig_2.0.3  
+[17] rlang_1.1.0       DBI_1.1.3         cli_3.6.1         yaml_2.3.7       
+[21] pkgdown_2.0.7     xfun_0.38         fastmap_1.1.1     gridExtra_2.3    
+[25] dplyr_1.1.1       stringr_1.5.0     generics_0.1.3    desc_1.4.2       
+[29] fs_1.6.1          vctrs_0.6.1       sass_0.4.5        systemfonts_1.0.4
+[33] tidyselect_1.2.0  rprojroot_2.0.3   lmtest_0.9-40     grid_4.2.3       
+[37] inline_0.3.19     glue_1.6.2        R6_2.5.1          textshaping_0.3.6
+[41] fansi_1.0.4       rmarkdown_2.21    purrr_1.0.1       ggplot2_3.4.2    
+[45] magrittr_2.0.3    scales_1.2.1      htmltools_0.5.5   colorspace_2.1-0 
+[49] ragg_1.2.5        utf8_1.2.3        stringi_1.7.12    munsell_0.5.0    
+[53] cachem_1.0.7      zoo_1.8-12
+
+
+

Hardware info +

+
CPU model: AMD Ryzen 9 7950X 16-Core Processor
+
MemTotal:       64936316 kB
+
+
+
+ + + +
+ + + +
+

+

Developed by Johannes Ranke.

+
+ +
+

+

Site built with pkgdown 2.0.7.

+
+ +
+
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+
+ + + + +
+
+
+

Testing hierarchical parent degradation kinetics +with residue data on dimethenamid and dimethenamid-P

+

Johannes +Ranke

+ +

Last change on 5 January +2023, last compiled on 20 April 2023

+ + Source: vignettes/prebuilt/2022_dmta_parent.rmd + + +
+ + + +
+

Introduction +

+

The purpose of this document is to demonstrate how nonlinear +hierarchical models (NLHM) based on the parent degradation models SFO, +FOMC, DFOP and HS can be fitted with the mkin package.

+

It was assembled in the course of work package 1.1 of Project Number +173340 (Application of nonlinear hierarchical models to the kinetic +evaluation of chemical degradation data) of the German Environment +Agency carried out in 2022 and 2023.

+

The mkin package is used in version 1.2.3. It contains the test data +and the functions used in the evaluations. The saemix +package is used as a backend for fitting the NLHM, but is also loaded to +make the convergence plot function available.

+

This document is processed with the knitr package, which +also provides the kable function that is used to improve +the display of tabular data in R markdown documents. For parallel +processing, the parallel package is used.

+
+library(mkin)
+library(knitr)
+library(saemix)
+library(parallel)
+n_cores <- detectCores()
+if (Sys.info()["sysname"] == "Windows") {
+  cl <- makePSOCKcluster(n_cores)
+} else {
+  cl <- makeForkCluster(n_cores)
+}
+
+
+

Data +

+

The test data are available in the mkin package as an object of class +mkindsg (mkin dataset group) under the identifier +dimethenamid_2018. The following preprocessing steps are +still necessary:

+
    +
  • The data available for the enantiomer dimethenamid-P (DMTAP) are +renamed to have the same substance name as the data for the racemic +mixture dimethenamid (DMTA). The reason for this is that no difference +between their degradation behaviour was identified in the EU risk +assessment.
    • +
  • The data for transformation products and unnecessary columns are +discarded
    • +
  • The observation times of each dataset are multiplied with the +corresponding normalisation factor also available in the dataset, in +order to make it possible to describe all datasets with a single set of +parameters that are independent of temperature
    • +
  • Finally, datasets observed in the same soil (Elliot 1 +and Elliot 2) are combined, resulting in dimethenamid +(DMTA) data from six soils.
    • +
+

The following commented R code performs this preprocessing.

+
+# Apply a function to each of the seven datasets in the mkindsg object to create a list
+dmta_ds <- lapply(1:7, function(i) {
+  ds_i <- dimethenamid_2018$ds[[i]]$data                     # Get a dataset
+  ds_i[ds_i$name == "DMTAP", "name"] <-  "DMTA"              # Rename DMTAP to DMTA
+  ds_i <- subset(ds_i, name == "DMTA", c("name", "time", "value")) # Select data
+  ds_i$time <- ds_i$time * dimethenamid_2018$f_time_norm[i]  # Normalise time
+  ds_i                                                       # Return the dataset
+})
+
+# Use dataset titles as names for the list elements
+names(dmta_ds) <- sapply(dimethenamid_2018$ds, function(ds) ds$title)
+
+# Combine data for Elliot soil to obtain a named list with six elements
+dmta_ds[["Elliot"]] <- rbind(dmta_ds[["Elliot 1"]], dmta_ds[["Elliot 2"]]) #
+dmta_ds[["Elliot 1"]] <- NULL
+dmta_ds[["Elliot 2"]] <- NULL
+

The following tables show the 6 datasets.

+
+for (ds_name in names(dmta_ds)) {
+    print(kable(mkin_long_to_wide(dmta_ds[[ds_name]]),
+      caption = paste("Dataset", ds_name),
+      label = paste0("tab:", ds_name), booktabs = TRUE))
+    cat("\n\\clearpage\n")
+}
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Calke
timeDMTA
095.8
098.7
1460.5
3039.1
5915.2
1204.8
1204.6
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Borstel
timeDMTA
0.000000100.5
0.00000099.6
1.94129591.9
1.94129591.3
6.79453481.8
6.79453482.1
13.58906769.1
13.58906768.0
27.17813551.4
27.17813551.4
56.29756527.6
56.29756526.8
86.38764315.7
86.38764315.3
115.5070737.9
115.5070738.1
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Flaach
timeDMTA
0.000000096.5
0.000000096.8
0.000000097.0
0.623385682.9
0.623385686.7
0.623385687.4
1.870156772.8
1.870156769.9
1.870156771.9
4.363698951.4
4.363698952.9
4.363698948.6
8.727397928.5
8.727397927.3
8.727397927.5
13.091096814.8
13.091096813.4
13.091096814.4
17.45479577.7
17.45479577.3
17.45479578.1
26.18219362.0
26.18219361.5
26.18219361.9
34.90959151.3
34.90959151.0
34.90959151.1
43.63698930.9
43.63698930.7
43.63698930.7
52.36438720.6
52.36438720.4
52.36438720.5
74.80626740.4
74.80626740.3
74.80626740.3
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset BBA 2.2
timeDMTA
0.000000098.09
0.000000098.77
0.767892293.52
0.767892292.03
2.303676588.39
2.303676587.18
5.375245269.38
5.375245271.06
10.750490445.21
10.750490446.81
16.125735530.54
16.125735530.07
21.500980721.60
21.500980720.41
32.25147119.10
32.25147119.70
43.00196146.58
43.00196146.31
53.75245183.47
53.75245183.52
64.50294213.40
64.50294213.67
91.37916801.62
91.37916801.62
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset BBA 2.3
timeDMTA
0.000000099.33
0.000000097.44
0.673393893.73
0.673393893.77
2.020181487.84
2.020181489.82
4.713756571.61
4.713756571.42
9.427513145.60
9.427513145.42
14.141269631.12
14.141269631.68
18.855026223.20
18.855026224.13
28.28253939.43
28.28253939.82
37.71005237.08
37.71005238.64
47.13756544.41
47.13756544.78
56.56507854.92
56.56507855.08
80.13386122.13
80.13386122.23
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Elliot
timeDMTA
0.00000097.5
0.000000100.7
1.22847886.4
1.22847888.5
3.68543569.8
3.68543577.1
8.59934959.0
8.59934954.2
17.19869731.3
17.19869733.5
25.79804619.6
25.79804620.9
34.39739513.3
34.39739515.8
51.5960926.7
51.5960928.7
68.7947898.8
68.7947898.7
103.1921846.0
103.1921844.4
146.1889283.3
146.1889282.8
223.5830661.4
223.5830661.8
0.00000093.4
0.000000103.2
1.22847889.2
1.22847886.6
3.68543578.2
3.68543578.1
8.59934955.6
8.59934953.0
17.19869733.7
17.19869733.2
25.79804620.9
25.79804619.9
34.39739518.2
34.39739512.7
51.5960927.8
51.5960929.0
68.79478911.4
68.7947899.0
103.1921843.9
103.1921844.4
146.1889282.6
146.1889283.4
223.5830662.0
223.5830661.7
+
+
+

Separate evaluations +

+

In order to obtain suitable starting parameters for the NLHM fits, +separate fits of the four models to the data for each soil are generated +using the mmkin function from the mkin +package. In a first step, constant variance is assumed. Convergence is +checked with the status function.

+
+deg_mods <- c("SFO", "FOMC", "DFOP", "HS")
+f_sep_const <- mmkin(
+  deg_mods,
+  dmta_ds,
+  error_model = "const",
+  quiet = TRUE)
+
+status(f_sep_const) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
CalkeBorstelFlaachBBA 2.2BBA 2.3Elliot
SFOOKOKOKOKOKOK
FOMCOKOKOKOKOKOK
DFOPOKOKOKOKOKOK
HSOKOKOKCOKOK
+

In the table above, OK indicates convergence, and C indicates failure +to converge. All separate fits with constant variance converged, with +the sole exception of the HS fit to the BBA 2.2 data. To prepare for +fitting NLHM using the two-component error model, the separate fits are +updated assuming two-component error.

+
+f_sep_tc <- update(f_sep_const, error_model = "tc")
+status(f_sep_tc) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
CalkeBorstelFlaachBBA 2.2BBA 2.3Elliot
SFOOKOKOKOKOKOK
FOMCOKOKOKOKCOK
DFOPOKOKCOKCOK
HSOKCOKOKOKOK
+

Using the two-component error model, the one fit that did not +converge with constant variance did converge, but other non-SFO fits +failed to converge.

+
+
+

Hierarchichal model fits +

+

The following code fits eight versions of hierarchical models to the +data, using SFO, FOMC, DFOP and HS for the parent compound, and using +either constant variance or two-component error for the error model. The +default parameter distribution model in mkin allows for variation of all +degradation parameters across the assumed population of soils. In other +words, each degradation parameter is associated with a random effect as +a first step. The mhmkin function makes it possible to fit +all eight versions in parallel (given a sufficient number of computing +cores being available) to save execution time.

+

Convergence plots and summaries for these fits are shown in the +appendix.

+
+f_saem <- mhmkin(list(f_sep_const, f_sep_tc), transformations = "saemix")
+

The output of the status function shows that all fits +terminated successfully.

+
+status(f_saem) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
consttc
SFOOKOK
FOMCOKOK
DFOPOKOK
HSOKOK
+

The AIC and BIC values show that the biphasic models DFOP and HS give +the best fits.

+
+anova(f_saem) |> kable(digits = 1)
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
nparAICBICLik
SFO const5796.3795.3-393.2
SFO tc6798.3797.1-393.2
FOMC const7734.2732.7-360.1
FOMC tc8720.4718.8-352.2
DFOP const9711.8710.0-346.9
HS const9714.0712.1-348.0
DFOP tc10665.5663.4-322.8
HS tc10667.1665.0-323.6
+

The DFOP model is preferred here, as it has a better mechanistic +basis for batch experiments with constant incubation conditions. Also, +it shows the lowest AIC and BIC values in the first set of fits when +combined with the two-component error model. Therefore, the DFOP model +was selected for further refinements of the fits with the aim to make +the model fully identifiable.

+
+

Parameter identifiability based on the Fisher Information +Matrix +

+

Using the illparms function, ill-defined statistical +model parameters such as standard deviations of the degradation +parameters in the population and error model parameters can be +found.

+
+illparms(f_saem) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
consttc
SFOb.1
FOMCsd(DMTA_0)
DFOPsd(k2)sd(k2)
HSsd(tb)
+

According to the illparms function, the fitted standard +deviation of the second kinetic rate constant k2 is +ill-defined in both DFOP fits. This suggests that different values would +be obtained for this standard deviation when using different starting +values.

+

The thus identified overparameterisation is addressed by removing the +random effect for k2 from the parameter model.

+
+f_saem_dfop_tc_no_ranef_k2 <- update(f_saem[["DFOP", "tc"]],
+  no_random_effect = "k2")
+

For the resulting fit, it is checked whether there are still +ill-defined parameters,

+
+illparms(f_saem_dfop_tc_no_ranef_k2)
+

which is not the case. Below, the refined model is compared with the +previous best model. The model without random effect for k2 +is a reduced version of the previous model. Therefore, the models are +nested and can be compared using the likelihood ratio test. This is +achieved with the argument test = TRUE to the +anova function.

+
+anova(f_saem[["DFOP", "tc"]], f_saem_dfop_tc_no_ranef_k2, test = TRUE) |>
+  kable(format.args = list(digits = 4))
+ ++++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
nparAICBICLikChisqDfPr(>Chisq)
f_saem_dfop_tc_no_ranef_k29663.8661.9-322.9NANANA
f_saem[[“DFOP”, “tc”]]10665.5663.4-322.80.280910.5961
+

The AIC and BIC criteria are lower after removal of the ill-defined +random effect for k2. The p value of the likelihood ratio +test is much greater than 0.05, indicating that the model with the +higher likelihood (here the model with random effects for all +degradation parameters f_saem[["DFOP", "tc"]]) does not fit +significantly better than the model with the lower likelihood (the +reduced model f_saem_dfop_tc_no_ranef_k2).

+

Therefore, AIC, BIC and likelihood ratio test suggest the use of the +reduced model.

+

The convergence of the fit is checked visually.

+
+Convergence plot for the NLHM DFOP fit with two-component error and without a random effect on 'k2'

+Convergence plot for the NLHM DFOP fit with two-component error and +without a random effect on ‘k2’ +

+
+

All parameters appear to have converged to a satisfactory degree. The +final fit is plotted using the plot method from the mkin package.

+
+plot(f_saem_dfop_tc_no_ranef_k2)
+
+Plot of the final NLHM DFOP fit

+Plot of the final NLHM DFOP fit +

+
+

Finally, a summary report of the fit is produced.

+
+summary(f_saem_dfop_tc_no_ranef_k2)
+
saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:09 2023 
+Date of summary: Thu Apr 20 14:07:10 2023 
+
+Equations:
+d_DMTA/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 4.175 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+   DMTA_0        k1        k2         g 
+98.759266  0.087034  0.009933  0.930827 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k1 k2 g
+DMTA_0  98.76  0  0 0
+k1       0.00  1  0 0
+k2       0.00  0  1 0
+g        0.00  0  0 1
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  663.8 661.9 -322.9
+
+Optimised parameters:
+               est.     lower     upper
+DMTA_0    98.228939 96.285869 100.17201
+k1         0.064063  0.033477   0.09465
+k2         0.008297  0.005824   0.01077
+g          0.953821  0.914328   0.99331
+a.1        1.068479  0.869538   1.26742
+b.1        0.029424  0.022406   0.03644
+SD.DMTA_0  2.030437  0.404824   3.65605
+SD.k1      0.594692  0.256660   0.93272
+SD.g       1.006754  0.361327   1.65218
+
+Correlation: 
+   DMTA_0  k1      k2     
+k1  0.0218                
+k2  0.0556  0.0355        
+g  -0.0516 -0.0284 -0.2800
+
+Random effects:
+            est.  lower  upper
+SD.DMTA_0 2.0304 0.4048 3.6560
+SD.k1     0.5947 0.2567 0.9327
+SD.g      1.0068 0.3613 1.6522
+
+Variance model:
+       est.   lower   upper
+a.1 1.06848 0.86954 1.26742
+b.1 0.02942 0.02241 0.03644
+
+Estimated disappearance times:
+      DT50 DT90 DT50back DT50_k1 DT50_k2
+DMTA 11.45 41.4    12.46   10.82   83.54
+
+
+

Alternative check of parameter identifiability +

+

The parameter check used in the illparms function is +based on a quadratic approximation of the likelihood surface near its +optimum, which is calculated using the Fisher Information Matrix (FIM). +An alternative way to check parameter identifiability (Duchesne et al. 2021) based on a multistart +approach has recently been implemented in mkin.

+

The graph below shows boxplots of the parameters obtained in 50 runs +of the saem algorithm with different parameter combinations, sampled +from the range of the parameters obtained for the individual datasets +fitted separately using nonlinear regression.

+
+f_saem_dfop_tc_multi <- multistart(f_saem[["DFOP", "tc"]], n = 50, cores = 15)
+
+par(mar = c(6.1, 4.1, 2.1, 2.1))
+parplot(f_saem_dfop_tc_multi, lpos = "bottomright", ylim = c(0.3, 10), las = 2)
+
+Scaled parameters from the multistart runs, full model

+Scaled parameters from the multistart runs, full model +

+
+

The graph clearly confirms the lack of identifiability of the +variance of k2 in the full model. The overparameterisation +of the model also indicates a lack of identifiability of the variance of +parameter g.

+

The parameter boxplots of the multistart runs with the reduced model +shown below indicate that all runs give similar results, regardless of +the starting parameters.

+
+f_saem_dfop_tc_no_ranef_k2_multi <- multistart(f_saem_dfop_tc_no_ranef_k2,
+  n = 50, cores = 15)
+
+par(mar = c(6.1, 4.1, 2.1, 2.1))
+parplot(f_saem_dfop_tc_no_ranef_k2_multi, ylim = c(0.5, 2), las = 2,
+  lpos = "bottomright")
+
+Scaled parameters from the multistart runs, reduced model

+Scaled parameters from the multistart runs, reduced model +

+
+

When only the parameters of the top 25% of the fits are shown (based +on a feature introduced in mkin 1.2.2 currently under development), the +scatter is even less as shown below.

+
+par(mar = c(6.1, 4.1, 2.1, 2.1))
+parplot(f_saem_dfop_tc_no_ranef_k2_multi, ylim = c(0.5, 2), las = 2, llquant = 0.25,
+  lpos = "bottomright")
+
+Scaled parameters from the multistart runs, reduced model, fits with the top 25\% likelihood values

+Scaled parameters from the multistart runs, reduced model, fits with the +top 25% likelihood values +

+
+
+
+
+

Conclusions +

+

Fitting the four parent degradation models SFO, FOMC, DFOP and HS as +part of hierarchical model fits with two different error models and +normal distributions of the transformed degradation parameters works +without technical problems. The biphasic models DFOP and HS gave the +best fit to the data, but the default parameter distribution model was +not fully identifiable. Removing the random effect for the second +kinetic rate constant of the DFOP model resulted in a reduced model that +was fully identifiable and showed the lowest values for the model +selection criteria AIC and BIC. The reliability of the identification of +all model parameters was confirmed using multiple starting values.

+
+
+

Acknowledgements +

+

The helpful comments by Janina Wöltjen of the German Environment +Agency are gratefully acknowledged.

+
+
+

References +

+
+
+Duchesne, Ronan, Anissa Guillemin, Olivier Gandrillon, and Fabien +Crauste. 2021. “Practical Identifiability in the Frame of +Nonlinear Mixed Effects Models: The Example of the in Vitro +Erythropoiesis.” BMC Bioinformatics 22 (478). https://doi.org/10.1186/s12859-021-04373-4. +
+
+
+
+

Appendix +

+
+

Hierarchical model fit listings +

+ +Hierarchical mkin fit of the SFO model with error model const + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:02 2023 
+Date of summary: Thu Apr 20 14:08:16 2023 
+
+Equations:
+d_DMTA/dt = - k_DMTA * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 0.982 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+ DMTA_0  k_DMTA 
+97.2953  0.0566 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k_DMTA
+DMTA_0   97.3      0
+k_DMTA    0.0      1
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  796.3 795.3 -393.2
+
+Optimised parameters:
+              est.    lower   upper
+DMTA_0    97.28130 95.71113 98.8515
+k_DMTA     0.05665  0.02909  0.0842
+a.1        2.66442  2.35579  2.9731
+SD.DMTA_0  1.54776  0.15447  2.9411
+SD.k_DMTA  0.60690  0.26248  0.9513
+
+Correlation: 
+       DMTA_0
+k_DMTA 0.0168
+
+Random effects:
+            est.  lower  upper
+SD.DMTA_0 1.5478 0.1545 2.9411
+SD.k_DMTA 0.6069 0.2625 0.9513
+
+Variance model:
+     est. lower upper
+a.1 2.664 2.356 2.973
+
+Estimated disappearance times:
+      DT50  DT90
+DMTA 12.24 40.65
+
+
+

+ +Hierarchical mkin fit of the SFO model with error model tc + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:03 2023 
+Date of summary: Thu Apr 20 14:08:16 2023 
+
+Equations:
+d_DMTA/dt = - k_DMTA * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 2.398 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+  DMTA_0   k_DMTA 
+96.99175  0.05603 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k_DMTA
+DMTA_0  96.99      0
+k_DMTA   0.00      1
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  798.3 797.1 -393.2
+
+Optimised parameters:
+               est.     lower    upper
+DMTA_0    97.271822 95.703157 98.84049
+k_DMTA     0.056638  0.029110  0.08417
+a.1        2.660081  2.230398  3.08976
+b.1        0.001665 -0.006911  0.01024
+SD.DMTA_0  1.545520  0.145035  2.94601
+SD.k_DMTA  0.606422  0.262274  0.95057
+
+Correlation: 
+       DMTA_0
+k_DMTA 0.0169
+
+Random effects:
+            est.  lower  upper
+SD.DMTA_0 1.5455 0.1450 2.9460
+SD.k_DMTA 0.6064 0.2623 0.9506
+
+Variance model:
+        est.     lower   upper
+a.1 2.660081  2.230398 3.08976
+b.1 0.001665 -0.006911 0.01024
+
+Estimated disappearance times:
+      DT50  DT90
+DMTA 12.24 40.65
+
+
+

+ +Hierarchical mkin fit of the FOMC model with error model const + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:02 2023 
+Date of summary: Thu Apr 20 14:08:16 2023 
+
+Equations:
+d_DMTA/dt = - (alpha/beta) * 1/((time/beta) + 1) * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 1.398 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+ DMTA_0   alpha    beta 
+ 98.292   9.909 156.341 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 alpha beta
+DMTA_0  98.29     0    0
+alpha    0.00     1    0
+beta     0.00     0    1
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  734.2 732.7 -360.1
+
+Optimised parameters:
+              est.   lower   upper
+DMTA_0     98.3435 96.9033  99.784
+alpha       7.2007  2.5889  11.812
+beta      112.8746 34.8816 190.868
+a.1         2.0459  1.8054   2.286
+SD.DMTA_0   1.4795  0.2717   2.687
+SD.alpha    0.6396  0.1509   1.128
+SD.beta     0.6874  0.1587   1.216
+
+Correlation: 
+      DMTA_0  alpha  
+alpha -0.1125        
+beta  -0.1227  0.3632
+
+Random effects:
+            est.  lower upper
+SD.DMTA_0 1.4795 0.2717 2.687
+SD.alpha  0.6396 0.1509 1.128
+SD.beta   0.6874 0.1587 1.216
+
+Variance model:
+     est. lower upper
+a.1 2.046 1.805 2.286
+
+Estimated disappearance times:
+      DT50  DT90 DT50back
+DMTA 11.41 42.53     12.8
+
+
+

+ +Hierarchical mkin fit of the FOMC model with error model tc + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:04 2023 
+Date of summary: Thu Apr 20 14:08:16 2023 
+
+Equations:
+d_DMTA/dt = - (alpha/beta) * 1/((time/beta) + 1) * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 3.044 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+DMTA_0  alpha   beta 
+98.772  4.663 92.597 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 alpha beta
+DMTA_0  98.77     0    0
+alpha    0.00     1    0
+beta     0.00     0    1
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  720.4 718.8 -352.2
+
+Optimised parameters:
+              est.    lower     upper
+DMTA_0    98.99136 97.26011 100.72261
+alpha      5.86312  2.57485   9.15138
+beta      88.55571 29.20889 147.90254
+a.1        1.51063  1.24384   1.77741
+b.1        0.02824  0.02040   0.03609
+SD.DMTA_0  1.57436 -0.04867   3.19739
+SD.alpha   0.59871  0.17132   1.02611
+SD.beta    0.72994  0.22849   1.23139
+
+Correlation: 
+      DMTA_0  alpha  
+alpha -0.1363        
+beta  -0.1414  0.2542
+
+Random effects:
+            est.    lower upper
+SD.DMTA_0 1.5744 -0.04867 3.197
+SD.alpha  0.5987  0.17132 1.026
+SD.beta   0.7299  0.22849 1.231
+
+Variance model:
+       est.  lower   upper
+a.1 1.51063 1.2438 1.77741
+b.1 0.02824 0.0204 0.03609
+
+Estimated disappearance times:
+      DT50 DT90 DT50back
+DMTA 11.11 42.6    12.82
+
+
+

+ +Hierarchical mkin fit of the DFOP model with error model const + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:02 2023 
+Date of summary: Thu Apr 20 14:08:16 2023 
+
+Equations:
+d_DMTA/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 1.838 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+  DMTA_0       k1       k2        g 
+98.64383  0.09211  0.02999  0.76814 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k1 k2 g
+DMTA_0  98.64  0  0 0
+k1       0.00  1  0 0
+k2       0.00  0  1 0
+g        0.00  0  0 1
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC BIC logLik
+  711.8 710 -346.9
+
+Optimised parameters:
+               est.     lower    upper
+DMTA_0    98.092481 96.573898 99.61106
+k1         0.062499  0.030336  0.09466
+k2         0.009065 -0.005133  0.02326
+g          0.948967  0.862079  1.03586
+a.1        1.821671  1.604774  2.03857
+SD.DMTA_0  1.677785  0.472066  2.88350
+SD.k1      0.634962  0.270788  0.99914
+SD.k2      1.033498 -0.205994  2.27299
+SD.g       1.710046  0.428642  2.99145
+
+Correlation: 
+   DMTA_0  k1      k2     
+k1  0.0246                
+k2  0.0491  0.0953        
+g  -0.0552 -0.0889 -0.4795
+
+Random effects:
+           est.   lower  upper
+SD.DMTA_0 1.678  0.4721 2.8835
+SD.k1     0.635  0.2708 0.9991
+SD.k2     1.033 -0.2060 2.2730
+SD.g      1.710  0.4286 2.9914
+
+Variance model:
+     est. lower upper
+a.1 1.822 1.605 2.039
+
+Estimated disappearance times:
+      DT50 DT90 DT50back DT50_k1 DT50_k2
+DMTA 11.79 42.8    12.88   11.09   76.46
+
+
+

+ +Hierarchical mkin fit of the DFOP model with error model tc + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:04 2023 
+Date of summary: Thu Apr 20 14:08:16 2023 
+
+Equations:
+d_DMTA/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 3.297 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+   DMTA_0        k1        k2         g 
+98.759266  0.087034  0.009933  0.930827 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k1 k2 g
+DMTA_0  98.76  0  0 0
+k1       0.00  1  0 0
+k2       0.00  0  1 0
+g        0.00  0  0 1
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  665.5 663.4 -322.8
+
+Optimised parameters:
+               est.     lower     upper
+DMTA_0    98.377019 96.447952 100.30609
+k1         0.064843  0.034607   0.09508
+k2         0.008895  0.006368   0.01142
+g          0.949696  0.903815   0.99558
+a.1        1.065241  0.865754   1.26473
+b.1        0.029340  0.022336   0.03634
+SD.DMTA_0  2.007754  0.387982   3.62753
+SD.k1      0.580473  0.250286   0.91066
+SD.k2      0.006105 -4.920337   4.93255
+SD.g       1.097149  0.412779   1.78152
+
+Correlation: 
+   DMTA_0  k1      k2     
+k1  0.0235                
+k2  0.0595  0.0424        
+g  -0.0470 -0.0278 -0.2731
+
+Random effects:
+              est.   lower  upper
+SD.DMTA_0 2.007754  0.3880 3.6275
+SD.k1     0.580473  0.2503 0.9107
+SD.k2     0.006105 -4.9203 4.9325
+SD.g      1.097149  0.4128 1.7815
+
+Variance model:
+       est.   lower   upper
+a.1 1.06524 0.86575 1.26473
+b.1 0.02934 0.02234 0.03634
+
+Estimated disappearance times:
+      DT50  DT90 DT50back DT50_k1 DT50_k2
+DMTA 11.36 41.32    12.44   10.69   77.92
+
+
+

+ +Hierarchical mkin fit of the HS model with error model const + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:03 2023 
+Date of summary: Thu Apr 20 14:08:16 2023 
+
+Equations:
+d_DMTA/dt = - ifelse(time <= tb, k1, k2) * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 1.972 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+  DMTA_0       k1       k2       tb 
+97.82176  0.06931  0.02997 11.13945 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k1 k2 tb
+DMTA_0  97.82  0  0  0
+k1       0.00  1  0  0
+k2       0.00  0  1  0
+tb       0.00  0  0  1
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+  AIC   BIC logLik
+  714 712.1   -348
+
+Optimised parameters:
+              est.    lower    upper
+DMTA_0    98.16102 96.47747 99.84456
+k1         0.07876  0.05261  0.10491
+k2         0.02227  0.01706  0.02747
+tb        13.99089 -7.40049 35.38228
+a.1        1.82305  1.60700  2.03910
+SD.DMTA_0  1.88413  0.56204  3.20622
+SD.k1      0.34292  0.10482  0.58102
+SD.k2      0.19851  0.01718  0.37985
+SD.tb      1.68168  0.58064  2.78272
+
+Correlation: 
+   DMTA_0  k1      k2     
+k1  0.0142                
+k2  0.0001 -0.0025        
+tb  0.0165 -0.1256 -0.0301
+
+Random effects:
+            est.   lower  upper
+SD.DMTA_0 1.8841 0.56204 3.2062
+SD.k1     0.3429 0.10482 0.5810
+SD.k2     0.1985 0.01718 0.3798
+SD.tb     1.6817 0.58064 2.7827
+
+Variance model:
+     est. lower upper
+a.1 1.823 1.607 2.039
+
+Estimated disappearance times:
+      DT50  DT90 DT50back DT50_k1 DT50_k2
+DMTA 8.801 67.91    20.44   8.801   31.13
+
+
+

+ +Hierarchical mkin fit of the HS model with error model tc + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:04 2023 
+Date of summary: Thu Apr 20 14:08:16 2023 
+
+Equations:
+d_DMTA/dt = - ifelse(time <= tb, k1, k2) * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 3.378 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+  DMTA_0       k1       k2       tb 
+98.45190  0.07525  0.02576 19.19375 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k1 k2 tb
+DMTA_0  98.45  0  0  0
+k1       0.00  1  0  0
+k2       0.00  0  1  0
+tb       0.00  0  0  1
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC BIC logLik
+  667.1 665 -323.6
+
+Optimised parameters:
+              est.    lower    upper
+DMTA_0    97.76570 95.81350 99.71791
+k1         0.05855  0.03080  0.08630
+k2         0.02337  0.01664  0.03010
+tb        31.09638 29.38289 32.80987
+a.1        1.08835  0.88590  1.29080
+b.1        0.02964  0.02257  0.03671
+SD.DMTA_0  2.04877  0.42607  3.67147
+SD.k1      0.59166  0.25621  0.92711
+SD.k2      0.30698  0.09561  0.51835
+SD.tb      0.01274 -0.10914  0.13462
+
+Correlation: 
+   DMTA_0  k1      k2     
+k1  0.0160                
+k2 -0.0070 -0.0024        
+tb -0.0668 -0.0103 -0.2013
+
+Random effects:
+             est.    lower  upper
+SD.DMTA_0 2.04877  0.42607 3.6715
+SD.k1     0.59166  0.25621 0.9271
+SD.k2     0.30698  0.09561 0.5183
+SD.tb     0.01274 -0.10914 0.1346
+
+Variance model:
+       est.   lower   upper
+a.1 1.08835 0.88590 1.29080
+b.1 0.02964 0.02257 0.03671
+
+Estimated disappearance times:
+      DT50  DT90 DT50back DT50_k1 DT50_k2
+DMTA 11.84 51.71    15.57   11.84   29.66
+
+
+

+
+
+

Hierarchical model convergence plots +

+
+Convergence plot for the NLHM SFO fit with constant variance

+Convergence plot for the NLHM SFO fit with constant variance +

+
+
+Convergence plot for the NLHM SFO fit with two-component error

+Convergence plot for the NLHM SFO fit with two-component error +

+
+
+Convergence plot for the NLHM FOMC fit with constant variance

+Convergence plot for the NLHM FOMC fit with constant variance +

+
+
+Convergence plot for the NLHM FOMC fit with two-component error

+Convergence plot for the NLHM FOMC fit with two-component error +

+
+
+Convergence plot for the NLHM DFOP fit with constant variance

+Convergence plot for the NLHM DFOP fit with constant variance +

+
+
+Convergence plot for the NLHM DFOP fit with two-component error

+Convergence plot for the NLHM DFOP fit with two-component error +

+
+
+Convergence plot for the NLHM HS fit with constant variance

+Convergence plot for the NLHM HS fit with constant variance +

+
+
+Convergence plot for the NLHM HS fit with two-component error

+Convergence plot for the NLHM HS fit with two-component error +

+
+
+
+

Session info +

+
R version 4.2.3 (2023-03-15)
+Platform: x86_64-pc-linux-gnu (64-bit)
+Running under: Debian GNU/Linux 12 (bookworm)
+
+Matrix products: default
+BLAS:   /usr/lib/x86_64-linux-gnu/openblas-serial/libblas.so.3
+LAPACK: /usr/lib/x86_64-linux-gnu/openblas-serial/libopenblas-r0.3.21.so
+
+locale:
+ [1] LC_CTYPE=de_DE.UTF-8       LC_NUMERIC=C              
+ [3] LC_TIME=de_DE.UTF-8        LC_COLLATE=de_DE.UTF-8    
+ [5] LC_MONETARY=de_DE.UTF-8    LC_MESSAGES=de_DE.UTF-8   
+ [7] LC_PAPER=de_DE.UTF-8       LC_NAME=C                 
+ [9] LC_ADDRESS=C               LC_TELEPHONE=C            
+[11] LC_MEASUREMENT=de_DE.UTF-8 LC_IDENTIFICATION=C       
+
+attached base packages:
+[1] parallel  stats     graphics  grDevices utils     datasets  methods  
+[8] base     
+
+other attached packages:
+[1] saemix_3.2 npde_3.3   knitr_1.42 mkin_1.2.3
+
+loaded via a namespace (and not attached):
+ [1] highr_0.10        pillar_1.9.0      bslib_0.4.2       compiler_4.2.3   
+ [5] jquerylib_0.1.4   tools_4.2.3       mclust_6.0.0      digest_0.6.31    
+ [9] tibble_3.2.1      jsonlite_1.8.4    evaluate_0.20     memoise_2.0.1    
+[13] lifecycle_1.0.3   nlme_3.1-162      gtable_0.3.3      lattice_0.21-8   
+[17] pkgconfig_2.0.3   rlang_1.1.0       DBI_1.1.3         cli_3.6.1        
+[21] yaml_2.3.7        pkgdown_2.0.7     xfun_0.38         fastmap_1.1.1    
+[25] gridExtra_2.3     dplyr_1.1.1       stringr_1.5.0     generics_0.1.3   
+[29] desc_1.4.2        fs_1.6.1          vctrs_0.6.1       sass_0.4.5       
+[33] systemfonts_1.0.4 tidyselect_1.2.0  rprojroot_2.0.3   lmtest_0.9-40    
+[37] grid_4.2.3        glue_1.6.2        R6_2.5.1          textshaping_0.3.6
+[41] fansi_1.0.4       rmarkdown_2.21    purrr_1.0.1       ggplot2_3.4.2    
+[45] magrittr_2.0.3    codetools_0.2-19  scales_1.2.1      htmltools_0.5.5  
+[49] colorspace_2.1-0  ragg_1.2.5        utf8_1.2.3        stringi_1.7.12   
+[53] munsell_0.5.0     cachem_1.0.7      zoo_1.8-12
+
+
+

Hardware info +

+
CPU model: AMD Ryzen 9 7950X 16-Core Processor
+
MemTotal:       64936316 kB
+
+
+
+ + + +
+ + + +
+

+

Developed by Johannes Ranke.

+
+ +
+

+

Site built with pkgdown 2.0.7.

+
+ +
+
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00000000..c8323add --- /dev/null +++ b/docs/articles/prebuilt/2022_dmta_pathway.html @@ -0,0 +1,2053 @@ + + + + + + + +Testing hierarchical pathway kinetics with residue data on dimethenamid and dimethenamid-P • mkin + + + + + + + + + + + + +
+
+ + + + +
+
+
+

Testing hierarchical pathway kinetics with +residue data on dimethenamid and dimethenamid-P

+

Johannes +Ranke

+ +

Last change on 20 April 2023, +last compiled on 20 April 2023

+ + Source: vignettes/prebuilt/2022_dmta_pathway.rmd + + +
+ + + +
+

Introduction +

+

The purpose of this document is to test demonstrate how nonlinear +hierarchical models (NLHM) based on the parent degradation models SFO, +FOMC, DFOP and HS, with parallel formation of two or more metabolites +can be fitted with the mkin package.

+

It was assembled in the course of work package 1.2 of Project Number +173340 (Application of nonlinear hierarchical models to the kinetic +evaluation of chemical degradation data) of the German Environment +Agency carried out in 2022 and 2023.

+

The mkin package is used in version 1.2.3, which is currently under +development. It contains the test data, and the functions used in the +evaluations. The saemix package is used as a backend for +fitting the NLHM, but is also loaded to make the convergence plot +function available.

+

This document is processed with the knitr package, which +also provides the kable function that is used to improve +the display of tabular data in R markdown documents. For parallel +processing, the parallel package is used.

+
+library(mkin)
+library(knitr)
+library(saemix)
+library(parallel)
+n_cores <- detectCores()
+
+# We need to start a new cluster after defining a compiled model that is
+# saved as a DLL to the user directory, therefore we define a function
+# This is used again after defining the pathway model
+start_cluster <- function(n_cores) {
+  if (Sys.info()["sysname"] == "Windows") {
+    ret <- makePSOCKcluster(n_cores)
+  } else {
+    ret <- makeForkCluster(n_cores)
+  }
+  return(ret)
+}
+
+
+

Data +

+

The test data are available in the mkin package as an object of class +mkindsg (mkin dataset group) under the identifier +dimethenamid_2018. The following preprocessing steps are +done in this document.

+
    +
  • The data available for the enantiomer dimethenamid-P (DMTAP) are +renamed to have the same substance name as the data for the racemic +mixture dimethenamid (DMTA). The reason for this is that no difference +between their degradation behaviour was identified in the EU risk +assessment.
    • +
  • Unnecessary columns are discarded
    • +
  • The observation times of each dataset are multiplied with the +corresponding normalisation factor also available in the dataset, in +order to make it possible to describe all datasets with a single set of +parameters that are independent of temperature
    • +
  • Finally, datasets observed in the same soil (Elliot 1 +and Elliot 2) are combined, resulting in dimethenamid +(DMTA) data from six soils.
    • +
+

The following commented R code performs this preprocessing.

+
+# Apply a function to each of the seven datasets in the mkindsg object to create a list
+dmta_ds <- lapply(1:7, function(i) {
+  ds_i <- dimethenamid_2018$ds[[i]]$data                     # Get a dataset
+  ds_i[ds_i$name == "DMTAP", "name"] <-  "DMTA"              # Rename DMTAP to DMTA
+  ds_i <- subset(ds_i, select = c("name", "time", "value")) # Select data
+  ds_i$time <- ds_i$time * dimethenamid_2018$f_time_norm[i]  # Normalise time
+  ds_i                                                       # Return the dataset
+})
+
+# Use dataset titles as names for the list elements
+names(dmta_ds) <- sapply(dimethenamid_2018$ds, function(ds) ds$title)
+
+# Combine data for Elliot soil to obtain a named list with six elements
+dmta_ds[["Elliot"]] <- rbind(dmta_ds[["Elliot 1"]], dmta_ds[["Elliot 2"]]) #
+dmta_ds[["Elliot 1"]] <- NULL
+dmta_ds[["Elliot 2"]] <- NULL
+

The following tables show the 6 datasets.

+
+for (ds_name in names(dmta_ds)) {
+  print(
+    kable(mkin_long_to_wide(dmta_ds[[ds_name]]),
+      caption = paste("Dataset", ds_name),
+      booktabs = TRUE, row.names = FALSE))
+    cat("\n\\clearpage\n")
+}
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Calke
timeDMTAM23M27M31
095.8NANANA
098.7NANANA
1460.54.11.52.0
3039.15.32.42.1
5915.26.03.22.2
1204.84.33.81.8
1204.64.13.72.1
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Borstel
timeDMTAM23M27M31
0.000000100.5NANANA
0.00000099.6NANANA
1.94129591.90.4NANA
1.94129591.30.50.30.1
6.79453481.81.20.81.0
6.79453482.11.30.90.9
13.58906769.12.81.42.0
13.58906768.02.01.42.5
27.17813551.42.92.74.3
27.17813551.44.92.63.2
56.29756527.612.24.44.3
56.29756526.812.24.74.8
86.38764315.712.25.45.0
86.38764315.312.05.25.1
115.5070737.910.45.44.3
115.5070738.111.65.44.4
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Flaach
timeDMTAM23M27M31
0.000000096.5NANANA
0.000000096.8NANANA
0.000000097.0NANANA
0.623385682.90.71.10.3
0.623385686.70.71.10.3
0.623385687.40.20.30.1
1.870156772.82.22.60.7
1.870156769.91.82.40.6
1.870156771.91.62.30.7
4.363698951.44.15.01.3
4.363698952.94.25.91.2
4.363698948.64.24.81.4
8.727397928.57.58.52.4
8.727397927.37.18.52.1
8.727397927.57.58.32.3
13.091096814.88.49.33.3
13.091096813.46.88.72.4
13.091096814.48.09.12.6
17.45479577.77.28.64.0
17.45479577.37.28.53.6
17.45479578.16.98.93.3
26.18219362.04.98.12.1
26.18219361.54.37.71.7
26.18219361.94.57.41.8
34.90959151.33.85.91.6
34.90959151.03.16.01.6
34.90959151.13.15.91.4
43.63698930.92.75.61.8
43.63698930.72.35.21.5
43.63698930.72.15.61.3
52.36438720.61.64.31.2
52.36438720.41.13.70.9
52.36438720.51.33.91.1
74.80626740.40.42.50.5
74.80626740.30.42.40.5
74.80626740.30.32.20.3
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset BBA 2.2
timeDMTAM23M27M31
0.000000098.09NANANA
0.000000098.77NANANA
0.767892293.520.360.420.36
0.767892292.030.400.470.33
2.303676588.391.030.710.55
2.303676587.181.070.820.64
5.375245269.383.602.191.94
5.375245271.063.662.281.62
10.750490445.216.975.454.22
10.750490446.817.225.194.37
16.125735530.548.658.816.31
16.125735530.078.387.936.85
21.500980721.609.1010.257.05
21.500980720.418.6310.776.84
32.25147119.107.6310.896.53
32.25147119.708.0110.857.11
43.00196146.586.4010.416.06
43.00196146.316.3510.356.05
53.75245183.475.359.925.50
53.75245183.525.069.425.07
64.50294213.405.149.154.94
64.50294213.675.919.254.39
91.37916801.623.357.143.64
91.37916801.622.877.133.55
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset BBA 2.3
timeDMTAM23M27M31
0.000000099.33NANANA
0.000000097.44NANANA
0.673393893.730.180.500.47
0.673393893.770.180.830.34
2.020181487.840.521.251.00
2.020181489.820.431.090.89
4.713756571.611.193.283.58
4.713756571.421.113.243.41
9.427513145.602.267.178.74
9.427513145.421.997.918.28
14.141269631.122.8110.159.67
14.141269631.682.839.558.95
18.855026223.203.3912.0910.34
18.855026224.133.5611.8910.00
28.28253939.433.4913.327.89
28.28253939.823.2812.058.13
37.71005237.082.8010.045.06
37.71005238.642.9710.785.54
47.13756544.412.429.323.79
47.13756544.782.519.624.11
56.56507854.922.228.003.11
56.56507855.081.958.452.98
80.13386122.131.285.711.78
80.13386122.230.993.331.55
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Elliot
timeDMTAM23M27M31
0.00000097.5NANANA
0.000000100.7NANANA
1.22847886.4NANANA
1.22847888.5NANA1.5
3.68543569.82.82.35.0
3.68543577.11.72.12.4
8.59934959.04.34.04.3
8.59934954.25.83.45.0
17.19869731.38.26.68.0
17.19869733.55.26.97.7
25.79804619.65.18.27.8
25.79804620.96.18.86.5
34.39739513.36.09.78.0
34.39739515.86.08.87.4
51.5960926.75.08.36.9
51.5960928.74.29.29.0
68.7947898.83.99.35.5
68.7947898.72.98.56.1
103.1921846.01.98.66.1
103.1921844.41.56.04.0
146.1889283.32.05.63.1
146.1889282.82.34.52.9
223.5830661.41.24.11.8
223.5830661.81.93.92.6
0.00000093.4NANANA
0.000000103.2NANANA
1.22847889.2NANA1.3
1.22847886.6NANANA
3.68543578.22.61.03.1
3.68543578.12.42.62.3
8.59934955.65.54.53.4
8.59934953.05.64.64.3
17.19869733.77.37.67.8
17.19869733.26.56.78.7
25.79804620.95.88.77.7
25.79804619.97.77.66.5
34.39739518.27.88.06.3
34.39739512.77.38.68.7
51.5960927.87.07.45.7
51.5960929.06.37.24.2
68.79478911.44.310.33.2
68.7947899.03.89.44.2
103.1921843.92.66.53.8
103.1921844.42.86.94.0
146.1889282.61.64.64.5
146.1889283.41.14.54.5
223.5830662.01.44.33.8
223.5830661.71.34.22.3
+
+
+

Separate evaluations +

+

As a first step to obtain suitable starting parameters for the NLHM +fits, we do separate fits of several variants of the pathway model used +previously (Ranke et al. 2021), varying +the kinetic model for the parent compound. Because the SFORB model often +provides faster convergence than the DFOP model, and can sometimes be +fitted where the DFOP model results in errors, it is included in the set +of parent models tested here.

+
+if (!dir.exists("dmta_dlls")) dir.create("dmta_dlls")
+m_sfo_path_1 <- mkinmod(
+  DMTA = mkinsub("SFO", c("M23", "M27", "M31")),
+  M23 = mkinsub("SFO"),
+  M27 = mkinsub("SFO"),
+  M31 = mkinsub("SFO", "M27", sink = FALSE),
+  name = "m_sfo_path", dll_dir = "dmta_dlls",
+  unload = TRUE, overwrite = TRUE,
+  quiet = TRUE
+)
+m_fomc_path_1 <- mkinmod(
+  DMTA = mkinsub("FOMC", c("M23", "M27", "M31")),
+  M23 = mkinsub("SFO"),
+  M27 = mkinsub("SFO"),
+  M31 = mkinsub("SFO", "M27", sink = FALSE),
+  name = "m_fomc_path", dll_dir = "dmta_dlls",
+  unload = TRUE, overwrite = TRUE,
+  quiet = TRUE
+)
+m_dfop_path_1 <- mkinmod(
+  DMTA = mkinsub("DFOP", c("M23", "M27", "M31")),
+  M23 = mkinsub("SFO"),
+  M27 = mkinsub("SFO"),
+  M31 = mkinsub("SFO", "M27", sink = FALSE),
+  name = "m_dfop_path", dll_dir = "dmta_dlls",
+  unload = TRUE, overwrite = TRUE,
+  quiet = TRUE
+)
+m_sforb_path_1 <- mkinmod(
+  DMTA = mkinsub("SFORB", c("M23", "M27", "M31")),
+  M23 = mkinsub("SFO"),
+  M27 = mkinsub("SFO"),
+  M31 = mkinsub("SFO", "M27", sink = FALSE),
+  name = "m_sforb_path", dll_dir = "dmta_dlls",
+  unload = TRUE, overwrite = TRUE,
+  quiet = TRUE
+)
+m_hs_path_1 <- mkinmod(
+  DMTA = mkinsub("HS", c("M23", "M27", "M31")),
+  M23 = mkinsub("SFO"),
+  M27 = mkinsub("SFO"),
+  M31 = mkinsub("SFO", "M27", sink = FALSE),
+  name = "m_hs_path", dll_dir = "dmta_dlls",
+  unload = TRUE, overwrite = TRUE,
+  quiet = TRUE
+)
+cl <- start_cluster(n_cores)
+
+deg_mods_1 <- list(
+  sfo_path_1 = m_sfo_path_1,
+  fomc_path_1 = m_fomc_path_1,
+  dfop_path_1 = m_dfop_path_1,
+  sforb_path_1 = m_sforb_path_1,
+  hs_path_1 = m_hs_path_1)
+
+sep_1_const <- mmkin(
+  deg_mods_1,
+  dmta_ds,
+  error_model = "const",
+  quiet = TRUE)
+
+status(sep_1_const) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
CalkeBorstelFlaachBBA 2.2BBA 2.3Elliot
sfo_path_1OKOKOKOKOKOK
fomc_path_1OKOKOKOKOKOK
dfop_path_1OKOKCOKOKOK
sforb_path_1OKOKCOKOKOK
hs_path_1CCCCCC
+

All separate pathway fits with SFO or FOMC for the parent and +constant variance converged (status OK). Most fits with DFOP or SFORB +for the parent converged as well. The fits with HS for the parent did +not converge with default settings.

+
+sep_1_tc <- update(sep_1_const, error_model = "tc")
+status(sep_1_tc) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
CalkeBorstelFlaachBBA 2.2BBA 2.3Elliot
sfo_path_1OKOKOKOKOKOK
fomc_path_1OKOKCOKOKC
dfop_path_1OKCOKOKOKOK
sforb_path_1OKCOKOKOKOK
hs_path_1CCCCCOK
+

With the two-component error model, the set of fits with convergence +problems is slightly different, with convergence problems appearing for +different data sets when applying the DFOP and SFORB model and some +additional convergence problems when using the FOMC model for the +parent.

+
+
+

Hierarchichal model fits +

+

The following code fits two sets of the corresponding hierarchical +models to the data, one assuming constant variance, and one assuming +two-component error.

+
+saem_1 <- mhmkin(list(sep_1_const, sep_1_tc))
+

The run time for these fits was around two hours on five year old +hardware. After a recent hardware upgrade these fits complete in less +than twenty minutes.

+
+status(saem_1) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
consttc
sfo_path_1OKOK
fomc_path_1OKOK
dfop_path_1OKOK
sforb_path_1OKOK
hs_path_1OKOK
+

According to the status function, all fits terminated +successfully.

+
+anova(saem_1) |> kable(digits = 1)
+
Warning in FUN(X[[i]], ...): Could not obtain log likelihood with 'is' method
+for sforb_path_1 const
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
nparAICBICLik
sfo_path_1 const172291.82288.3-1128.9
sfo_path_1 tc182276.32272.5-1120.1
fomc_path_1 const192099.02095.0-1030.5
fomc_path_1 tc201939.61935.5-949.8
dfop_path_1 const212038.82034.4-998.4
hs_path_1 const212024.22019.8-991.1
dfop_path_1 tc221879.81875.2-917.9
sforb_path_1 tc221832.91828.3-894.4
hs_path_1 tc221831.41826.8-893.7
+

When the goodness-of-fit of the models is compared, a warning is +obtained, indicating that the likelihood of the pathway fit with SFORB +for the parent compound and constant variance could not be calculated +with importance sampling (method ‘is’). As this is the default method on +which all AIC and BIC comparisons are based, this variant is not +included in the model comparison table. Comparing the goodness-of-fit of +the remaining models, HS model model with two-component error provides +the best fit. However, for batch experiments performed with constant +conditions such as the experiments evaluated here, there is no reason to +assume a discontinuity, so the SFORB model is preferable from a +mechanistic viewpoint. In addition, the information criteria AIC and BIC +are very similar for HS and SFORB. Therefore, the SFORB model is +selected here for further refinements.

+
+

Parameter identifiability based on the Fisher Information +Matrix +

+

Using the illparms function, ill-defined statistical +model parameters such as standard deviations of the degradation +parameters in the population and error model parameters can be +found.

+
+illparms(saem_1) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
consttc
sfo_path_1sd(DMTA_0)
fomc_path_1sd(DMTA_0)
dfop_path_1
sforb_path_1sd(log_k_DMTA_bound_free)
hs_path_1sd(log_tb)
+

When using constant variance, no ill-defined variance parameters are +identified with the illparms function in any of the +degradation models. When using the two-component error model, there is +one ill-defined variance parameter in all variants except for the +variant using DFOP for the parent compound.

+

For the selected combination of the SFORB pathway model with +two-component error, the random effect for the rate constant from +reversibly bound DMTA to the free DMTA (k_DMTA_bound_free) +is not well-defined. Therefore, the fit is updated without assuming a +random effect for this parameter.

+
+saem_sforb_path_1_tc_reduced <- update(saem_1[["sforb_path_1", "tc"]],
+  no_random_effect = "log_k_DMTA_bound_free")
+illparms(saem_sforb_path_1_tc_reduced)
+

As expected, no ill-defined parameters remain. The model comparison +below shows that the reduced model is preferable.

+
+anova(saem_1[["sforb_path_1", "tc"]], saem_sforb_path_1_tc_reduced) |> kable(digits = 1)
+ + + + + + + + + + + + + + + + + + + + + + + + +
nparAICBICLik
saem_sforb_path_1_tc_reduced211830.31825.9-894.2
saem_1[[“sforb_path_1”, “tc”]]221832.91828.3-894.4
+

The convergence plot of the refined fit is shown below.

+
+plot(saem_sforb_path_1_tc_reduced$so, plot.type = "convergence")
+

+

For some parameters, for example for f_DMTA_ilr_1 and +f_DMTA_ilr_2, i.e. for two of the parameters determining +the formation fractions of the parallel formation of the three +metabolites, some movement of the parameters is still visible in the +second phase of the algorithm. However, the amplitude of this movement +is in the range of the amplitude towards the end of the first phase. +Therefore, it is likely that an increase in iterations would not improve +the parameter estimates very much, and it is proposed that the fit is +acceptable. No numeric convergence criterion is implemented in +saemix.

+
+
+

Alternative check of parameter identifiability +

+

As an alternative check of parameter identifiability (Duchesne et al. 2021), multistart runs were +performed on the basis of the refined fit shown above.

+
+saem_sforb_path_1_tc_reduced_multi <- multistart(saem_sforb_path_1_tc_reduced,
+  n = 32, cores = 10)
+
+print(saem_sforb_path_1_tc_reduced_multi)
+
<multistart> object with 32 fits:
+ E OK 
+15 17 
+OK: Fit terminated successfully
+E: Error
+

Out of the 32 fits that were initiated, only 17 terminated without an +error. The reason for this is that the wide variation of starting +parameters in combination with the parameter variation that is used in +the SAEM algorithm leads to parameter combinations for the degradation +model that the numerical integration routine cannot cope with. Because +of this variation of initial parameters, some of the model fits take up +to two times more time than the original fit.

+
+par(mar = c(12.1, 4.1, 2.1, 2.1))
+parplot(saem_sforb_path_1_tc_reduced_multi, ylim = c(0.5, 2), las = 2)
+
+Parameter boxplots for the multistart runs that succeeded

+Parameter boxplots for the multistart runs that succeeded +

+
+

However, visual analysis of the boxplot of the parameters obtained in +the successful fits confirms that the results are sufficiently +independent of the starting parameters, and there are no remaining +ill-defined parameters.

+
+
+
+

Plots of selected fits +

+

The SFORB pathway fits with full and reduced parameter distribution +model are shown below.

+
+plot(saem_1[["sforb_path_1", "tc"]])
+
+SFORB pathway fit with two-component error

+SFORB pathway fit with two-component error +

+
+
+plot(saem_sforb_path_1_tc_reduced)
+
+SFORB pathway fit with two-component error, reduced parameter model

+SFORB pathway fit with two-component error, reduced parameter model +

+
+

Plots of the remaining fits and listings for all successful fits are +shown in the Appendix.

+ +
+
+

Conclusions +

+

Pathway fits with SFO, FOMC, DFOP, SFORB and HS models for the parent +compound could be successfully performed.

+
+
+

Acknowledgements +

+

The helpful comments by Janina Wöltjen of the German Environment +Agency on earlier versions of this document are gratefully +acknowledged.

+
+
+

References +

+
+
+Duchesne, Ronan, Anissa Guillemin, Olivier Gandrillon, and Fabien +Crauste. 2021. “Practical Identifiability in the Frame of +Nonlinear Mixed Effects Models: The Example of the in Vitro +Erythropoiesis.” BMC Bioinformatics 22 (478). https://doi.org/10.1186/s12859-021-04373-4. +
+
+Ranke, Johannes, Janina Wöltjen, Jana Schmidt, and Emmanuelle Comets. +2021. “Taking Kinetic Evaluations of Degradation Data to the Next +Level with Nonlinear Mixed-Effects Models.” Environments +8 (8). https://doi.org/10.3390/environments8080071. +
+
+
+
+

Appendix +

+
+

Plots of hierarchical fits not selected for refinement +

+
+plot(saem_1[["sfo_path_1", "tc"]])
+
+SFO pathway fit with two-component error

+SFO pathway fit with two-component error +

+
+
+plot(saem_1[["fomc_path_1", "tc"]])
+
+FOMC pathway fit with two-component error

+FOMC pathway fit with two-component error +

+
+
+plot(saem_1[["sforb_path_1", "tc"]])
+
+HS pathway fit with two-component error

+HS pathway fit with two-component error +

+
+
+
+

Hierarchical model fit listings +

+
+

Fits with random effects for all degradation parameters +

+ +
+
+

Improved fit of the SFORB pathway model with two-component +error +

+ +
+
+
+

Session info +

+
R version 4.2.3 (2023-03-15)
+Platform: x86_64-pc-linux-gnu (64-bit)
+Running under: Debian GNU/Linux 12 (bookworm)
+
+Matrix products: default
+BLAS:   /usr/lib/x86_64-linux-gnu/openblas-serial/libblas.so.3
+LAPACK: /usr/lib/x86_64-linux-gnu/openblas-serial/libopenblas-r0.3.21.so
+
+locale:
+ [1] LC_CTYPE=de_DE.UTF-8       LC_NUMERIC=C              
+ [3] LC_TIME=de_DE.UTF-8        LC_COLLATE=de_DE.UTF-8    
+ [5] LC_MONETARY=de_DE.UTF-8    LC_MESSAGES=de_DE.UTF-8   
+ [7] LC_PAPER=de_DE.UTF-8       LC_NAME=C                 
+ [9] LC_ADDRESS=C               LC_TELEPHONE=C            
+[11] LC_MEASUREMENT=de_DE.UTF-8 LC_IDENTIFICATION=C       
+
+attached base packages:
+[1] parallel  stats     graphics  grDevices utils     datasets  methods  
+[8] base     
+
+other attached packages:
+[1] saemix_3.2 npde_3.3   knitr_1.42 mkin_1.2.3
+
+loaded via a namespace (and not attached):
+ [1] deSolve_1.35      zoo_1.8-12        tidyselect_1.2.0  xfun_0.38        
+ [5] bslib_0.4.2       purrr_1.0.1       lattice_0.21-8    colorspace_2.1-0 
+ [9] vctrs_0.6.1       generics_0.1.3    htmltools_0.5.5   yaml_2.3.7       
+[13] utf8_1.2.3        rlang_1.1.0       pkgbuild_1.4.0    pkgdown_2.0.7    
+[17] jquerylib_0.1.4   pillar_1.9.0      glue_1.6.2        DBI_1.1.3        
+[21] lifecycle_1.0.3   stringr_1.5.0     munsell_0.5.0     gtable_0.3.3     
+[25] ragg_1.2.5        codetools_0.2-19  memoise_2.0.1     evaluate_0.20    
+[29] inline_0.3.19     callr_3.7.3       fastmap_1.1.1     ps_1.7.4         
+[33] lmtest_0.9-40     fansi_1.0.4       highr_0.10        scales_1.2.1     
+[37] cachem_1.0.7      desc_1.4.2        jsonlite_1.8.4    systemfonts_1.0.4
+[41] fs_1.6.1          textshaping_0.3.6 gridExtra_2.3     ggplot2_3.4.2    
+[45] digest_0.6.31     stringi_1.7.12    processx_3.8.0    dplyr_1.1.1      
+[49] grid_4.2.3        rprojroot_2.0.3   cli_3.6.1         tools_4.2.3      
+[53] magrittr_2.0.3    sass_0.4.5        tibble_3.2.1      crayon_1.5.2     
+[57] pkgconfig_2.0.3   prettyunits_1.1.1 rmarkdown_2.21    R6_2.5.1         
+[61] mclust_6.0.0      nlme_3.1-162      compiler_4.2.3
+
+
+

Hardware info +

+
CPU model: AMD Ryzen 9 7950X 16-Core Processor
+
MemTotal:       64936316 kB
+
+
+
+ + + +
+ + + + +
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