From c322a8102a399cbb1fe38c4c4ca4485cea8bc4e8 Mon Sep 17 00:00:00 2001 From: Johannes Ranke Date: Tue, 7 May 2019 08:43:12 +0200 Subject: Fix hessian calculations Static documentation rebuilt by pkgdown --- docs/articles/web_only/FOCUS_Z.html | 219 +++++++++++++++++++----------------- 1 file changed, 118 insertions(+), 101 deletions(-) (limited to 'docs/articles/web_only/FOCUS_Z.html') diff --git a/docs/articles/web_only/FOCUS_Z.html b/docs/articles/web_only/FOCUS_Z.html index c4c69b8f..3cda0261 100644 --- a/docs/articles/web_only/FOCUS_Z.html +++ b/docs/articles/web_only/FOCUS_Z.html @@ -131,94 +131,86 @@ 
plot_sep(m.Z.2a)

summary(m.Z.2a, data = FALSE)$bpar
-
## Warning in summary.mkinfit(m.Z.2a, data = FALSE): Could not calculate
-## correlation; no covariance matrix
-
##             Estimate se_notrans t value Pr(>t) Lower Upper
-## Z0_0      9.7015e+01         NA      NA     NA    NA    NA
-## k_Z0_sink 7.2231e-10         NA      NA     NA    NA    NA
-## k_Z0_Z1   2.2360e+00         NA      NA     NA    NA    NA
-## k_Z1_sink 4.8212e-01         NA      NA     NA    NA    NA
-## sigma     4.8041e+00         NA      NA     NA    NA    NA
+
##             Estimate se_notrans    t value     Pr(>t)    Lower    Upper
+## Z0_0      9.7015e+01   3.393176 2.8591e+01 6.4352e-21 91.66556 102.3642
+## k_Z0_sink 7.2231e-10   0.225254 3.2067e-09 5.0000e-01  0.00000      Inf
+## k_Z0_Z1   2.2360e+00   0.159134 1.4051e+01 1.1369e-13  1.95303   2.5600
+## k_Z1_sink 4.8212e-01   0.065454 7.3658e+00 5.1186e-08  0.40341   0.5762
+## sigma     4.8041e+00   0.637618 7.5345e+00 3.4431e-08  3.52677   6.0815

As obvious from the parameter summary (the component of the summary), the kinetic rate constant from parent compound Z to sink is very small and the t-test for this parameter suggests that it is not significantly different from zero. This suggests, in agreement with the analysis in the FOCUS kinetics report, to simplify the model by removing the pathway to sink.

A similar result can be obtained when formation fractions are used in the model formulation:

-
Z.2a.ff <- mkinmod(Z0 = mkinsub("SFO", "Z1"),
-                   Z1 = mkinsub("SFO"),
-                   use_of_ff = "max")
+
Z.2a.ff <- mkinmod(Z0 = mkinsub("SFO", "Z1"),
+                   Z1 = mkinsub("SFO"),
+                   use_of_ff = "max")
## Successfully compiled differential equation model from auto-generated C code.
-
m.Z.2a.ff <- mkinfit(Z.2a.ff, FOCUS_2006_Z_mkin, quiet = TRUE)
+
m.Z.2a.ff <- mkinfit(Z.2a.ff, FOCUS_2006_Z_mkin, quiet = TRUE)
## Warning in mkinfit(Z.2a.ff, FOCUS_2006_Z_mkin, quiet = TRUE): Observations
 ## with value of zero were removed from the data
-
plot_sep(m.Z.2a.ff)
+
plot_sep(m.Z.2a.ff)

-
summary(m.Z.2a.ff, data = FALSE)$bpar
-
## Warning in summary.mkinfit(m.Z.2a.ff, data = FALSE): Could not calculate
-## correlation; no covariance matrix
-
##            Estimate se_notrans t value Pr(>t) Lower Upper
-## Z0_0       97.01488         NA      NA     NA    NA    NA
-## k_Z0        2.23601         NA      NA     NA    NA    NA
-## k_Z1        0.48212         NA      NA     NA    NA    NA
-## f_Z0_to_Z1  1.00000         NA      NA     NA    NA    NA
-## sigma       4.80411         NA      NA     NA    NA    NA
+
summary(m.Z.2a.ff, data = FALSE)$bpar
+
##            Estimate se_notrans t value     Pr(>t)    Lower    Upper
+## Z0_0       97.01488   3.301084 29.3888 3.2971e-21 91.66556 102.3642
+## k_Z0        2.23601   0.207078 10.7979 3.3309e-11  1.95303   2.5600
+## k_Z1        0.48212   0.063265  7.6207 2.8155e-08  0.40341   0.5762
+## f_Z0_to_Z1  1.00000   0.094764 10.5525 5.3560e-11  0.00000   1.0000
+## sigma       4.80411   0.635638  7.5579 3.2592e-08  3.52677   6.0815

Here, the ilr transformed formation fraction fitted in the model takes a very large value, and the backtransformed formation fraction from parent Z to Z1 is practically unity. Here, the covariance matrix used for the calculation of confidence intervals is not returned as the model is overparameterised.

A simplified model is obtained by removing the pathway to the sink.

In the following, we use the parameterisation with formation fractions in order to be able to compare with the results in the FOCUS guidance, and as it makes it easier to use parameters obtained in a previous fit when adding a further metabolite.

-
Z.3 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
-               Z1 = mkinsub("SFO"), use_of_ff = "max")
+
Z.3 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
+               Z1 = mkinsub("SFO"), use_of_ff = "max")
## Successfully compiled differential equation model from auto-generated C code.
-
m.Z.3 <- mkinfit(Z.3, FOCUS_2006_Z_mkin, quiet = TRUE)
+
m.Z.3 <- mkinfit(Z.3, FOCUS_2006_Z_mkin, quiet = TRUE)
## Warning in mkinfit(Z.3, FOCUS_2006_Z_mkin, quiet = TRUE): Observations with
 ## value of zero were removed from the data
-
plot_sep(m.Z.3)
+
plot_sep(m.Z.3)

-
summary(m.Z.3, data = FALSE)$bpar
-
## Warning in summary.mkinfit(m.Z.3, data = FALSE): Could not calculate
-## correlation; no covariance matrix
-
##       Estimate se_notrans t value Pr(>t) Lower Upper
-## Z0_0  97.01488         NA      NA     NA    NA    NA
-## k_Z0   2.23601         NA      NA     NA    NA    NA
-## k_Z1   0.48212         NA      NA     NA    NA    NA
-## sigma  4.80411         NA      NA     NA    NA    NA
+
summary(m.Z.3, data = FALSE)$bpar
+
##       Estimate se_notrans t value     Pr(>t)    Lower    Upper
+## Z0_0  97.01488   2.597342  37.352 2.0106e-24 91.67597 102.3538
+## k_Z0   2.23601   0.146904  15.221 9.1477e-15  1.95354   2.5593
+## k_Z1   0.48212   0.041727  11.554 4.8268e-12  0.40355   0.5760
+## sigma  4.80411   0.620208   7.746 1.6110e-08  3.52925   6.0790

As there is only one transformation product for Z0 and no pathway to sink, the formation fraction is internally fixed to unity.

Metabolites Z2 and Z3

As suggested in the FOCUS report, the pathway to sink was removed for metabolite Z1 as well in the next step. While this step appears questionable on the basis of the above results, it is followed here for the purpose of comparison. Also, in the FOCUS report, it is assumed that there is additional empirical evidence that Z1 quickly and exclusively hydrolyses to Z2.

-
Z.5 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
-               Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-               Z2 = mkinsub("SFO"), use_of_ff = "max")
+
Z.5 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
+               Z1 = mkinsub("SFO", "Z2", sink = FALSE),
+               Z2 = mkinsub("SFO"), use_of_ff = "max")
## Successfully compiled differential equation model from auto-generated C code.
-
m.Z.5 <- mkinfit(Z.5, FOCUS_2006_Z_mkin, quiet = TRUE)
+
m.Z.5 <- mkinfit(Z.5, FOCUS_2006_Z_mkin, quiet = TRUE)
## Warning in mkinfit(Z.5, FOCUS_2006_Z_mkin, quiet = TRUE): Observations with
 ## value of zero were removed from the data
-
plot_sep(m.Z.5)
+
plot_sep(m.Z.5)

Finally, metabolite Z3 is added to the model. We use the optimised differential equation parameter values from the previous fit in order to accelerate the optimization.

-
Z.FOCUS <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
-                   Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                   Z2 = mkinsub("SFO", "Z3"),
-                   Z3 = mkinsub("SFO"),
-                   use_of_ff = "max")
+
Z.FOCUS <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
+                   Z1 = mkinsub("SFO", "Z2", sink = FALSE),
+                   Z2 = mkinsub("SFO", "Z3"),
+                   Z3 = mkinsub("SFO"),
+                   use_of_ff = "max")
## Successfully compiled differential equation model from auto-generated C code.
- + 
## Warning in mkinfit(Z.FOCUS, FOCUS_2006_Z_mkin, parms.ini = m.Z.
 ## 5$bparms.ode, : Observations with value of zero were removed from the data
-
plot_sep(m.Z.FOCUS)
+
plot_sep(m.Z.FOCUS)

-
summary(m.Z.FOCUS, data = FALSE)$bpar
-
## Warning in summary.mkinfit(m.Z.FOCUS, data = FALSE): Could not calculate
-## correlation; no covariance matrix
-
##             Estimate se_notrans t value Pr(>t) Lower Upper
-## Z0_0       96.838607         NA      NA     NA    NA    NA
-## k_Z0        2.215405         NA      NA     NA    NA    NA
-## k_Z1        0.478300         NA      NA     NA    NA    NA
-## k_Z2        0.451618         NA      NA     NA    NA    NA
-## k_Z3        0.058693         NA      NA     NA    NA    NA
-## f_Z2_to_Z3  0.471508         NA      NA     NA    NA    NA
-## sigma       3.984431         NA      NA     NA    NA    NA
-
endpoints(m.Z.FOCUS)
+
summary(m.Z.FOCUS, data = FALSE)$bpar
+
##             Estimate se_notrans t value     Pr(>t)     Lower      Upper
+## Z0_0       96.838607   1.994273 48.5584 4.0283e-42 92.826626 100.850589
+## k_Z0        2.215405   0.118459 18.7018 1.0415e-23  1.989465   2.467003
+## k_Z1        0.478300   0.028257 16.9267 6.2408e-22  0.424701   0.538662
+## k_Z2        0.451618   0.042138 10.7177 1.6308e-14  0.374328   0.544867
+## k_Z3        0.058693   0.015246  3.8498 1.7806e-04  0.034805   0.098978
+## f_Z2_to_Z3  0.471508   0.058352  8.0804 9.6648e-11  0.357735   0.588320
+## sigma       3.984431   0.383402 10.3923 4.5575e-14  3.213126   4.755736
+
endpoints(m.Z.FOCUS)
## $ff
 ##   Z2_Z3 Z2_sink 
 ## 0.47151 0.52849 
@@ -239,77 +231,102 @@
 Using the SFORB model
 
As the FOCUS report states, there is a certain tailing of the time course of metabolite Z3. Also, the time course of the parent compound is not fitted very well using the SFO model, as residues at a certain low level remain.

 
Therefore, an additional model is offered here, using the single first-order reversible binding (SFORB) model for metabolite Z3. As expected, the \(\chi^2\) error level is lower for metabolite Z3 using this model and the graphical fit for Z3 is improved. However, the covariance matrix is not returned.

-
Z.mkin.1 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
-                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                    Z2 = mkinsub("SFO", "Z3"),
-                    Z3 = mkinsub("SFORB"))

+
Z.mkin.1 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
+                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
+                    Z2 = mkinsub("SFO", "Z3"),
+                    Z3 = mkinsub("SFORB"))

 
## Successfully compiled differential equation model from auto-generated C code.

-
m.Z.mkin.1 <- mkinfit(Z.mkin.1, FOCUS_2006_Z_mkin, quiet = TRUE)

+
m.Z.mkin.1 <- mkinfit(Z.mkin.1, FOCUS_2006_Z_mkin, quiet = TRUE)

 
## Warning in mkinfit(Z.mkin.1, FOCUS_2006_Z_mkin, quiet = TRUE): Observations
 ## with value of zero were removed from the data

-
plot_sep(m.Z.mkin.1)

+
plot_sep(m.Z.mkin.1)

 

-
summary(m.Z.mkin.1, data = FALSE)$cov.unscaled

-
## Warning in summary.mkinfit(m.Z.mkin.1, data = FALSE): Could not calculate
-## correlation; no covariance matrix

-
## NULL

+
summary(m.Z.mkin.1, data = FALSE)$cov.unscaled

+
##                            Z0_0 log_k_Z0_Z1 log_k_Z1_Z2 log_k_Z2_sink
+## Z0_0                 3.8375e+00  5.4918e-03  3.0584e-02    1.2969e-01
+## log_k_Z0_Z1          5.4918e-03  2.7613e-03 -1.8820e-04    2.6634e-04
+## log_k_Z1_Z2          3.0584e-02 -1.8820e-04  3.3807e-03    3.2177e-03
+## log_k_Z2_sink        1.2969e-01  2.6634e-04  3.2177e-03    3.4256e-02
+## log_k_Z2_Z3_free    -2.4223e-02 -2.6169e-04 -1.1845e-03   -8.1134e-03
+## log_k_Z3_free_sink  -6.5467e-02 -4.0815e-04 -3.2978e-03   -3.6010e-02
+## log_k_Z3_free_bound -6.0659e-02 -4.4768e-04 -3.0588e-03   -3.9074e-02
+## log_k_Z3_bound_free  5.2844e-01  4.5458e-03  7.9800e-03    4.6274e-02
+## sigma                2.0366e-10 -3.4658e-10  8.9910e-11   -2.5946e-10
+##                     log_k_Z2_Z3_free log_k_Z3_free_sink
+## Z0_0                     -2.4223e-02        -6.5467e-02
+## log_k_Z0_Z1              -2.6169e-04        -4.0815e-04
+## log_k_Z1_Z2              -1.1845e-03        -3.2978e-03
+## log_k_Z2_sink            -8.1134e-03        -3.6010e-02
+## log_k_Z2_Z3_free          1.5500e-02         2.1583e-02
+## log_k_Z3_free_sink        2.1583e-02         7.5705e-02
+## log_k_Z3_free_bound       2.5836e-02         1.1964e-01
+## log_k_Z3_bound_free       5.2534e-02         2.9441e-01
+## sigma                     1.3063e-10         3.4170e-10
+##                     log_k_Z3_free_bound log_k_Z3_bound_free       sigma
+## Z0_0                        -6.0659e-02          5.2844e-01  2.0366e-10
+## log_k_Z0_Z1                 -4.4768e-04          4.5458e-03 -3.4658e-10
+## log_k_Z1_Z2                 -3.0588e-03          7.9800e-03  8.9910e-11
+## log_k_Z2_sink               -3.9074e-02          4.6274e-02 -2.5946e-10
+## log_k_Z2_Z3_free             2.5836e-02          5.2534e-02  1.3063e-10
+## log_k_Z3_free_sink           1.1964e-01          2.9441e-01  3.4170e-10
+## log_k_Z3_free_bound          6.5902e-01          5.4737e+00 -6.7704e-10
+## log_k_Z3_bound_free          5.4737e+00          2.8722e+08  7.2421e-02
+## sigma                       -6.7704e-10          7.2421e-02  1.4170e-01

 
Therefore, a further stepwise model building is performed starting from the stage of parent and two metabolites, starting from the assumption that the model fit for the parent compound can be improved by using the SFORB model.

-
Z.mkin.3 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
-                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                    Z2 = mkinsub("SFO"))

+
Z.mkin.3 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
+                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
+                    Z2 = mkinsub("SFO"))

 
## Successfully compiled differential equation model from auto-generated C code.

-
m.Z.mkin.3 <- mkinfit(Z.mkin.3, FOCUS_2006_Z_mkin, quiet = TRUE)

+
m.Z.mkin.3 <- mkinfit(Z.mkin.3, FOCUS_2006_Z_mkin, quiet = TRUE)

 
## Warning in mkinfit(Z.mkin.3, FOCUS_2006_Z_mkin, quiet = TRUE): Observations
 ## with value of zero were removed from the data

-
plot_sep(m.Z.mkin.3)

+
plot_sep(m.Z.mkin.3)

 

 
This results in a much better representation of the behaviour of the parent compound Z0.

 
Finally, Z3 is added as well. These models appear overparameterised (no covariance matrix returned) if the sink for Z1 is left in the models.

-
Z.mkin.4 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
-                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                    Z2 = mkinsub("SFO", "Z3"),
-                    Z3 = mkinsub("SFO"))

+
Z.mkin.4 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
+                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
+                    Z2 = mkinsub("SFO", "Z3"),
+                    Z3 = mkinsub("SFO"))

 
## Successfully compiled differential equation model from auto-generated C code.

-
+
 
## Warning in mkinfit(Z.mkin.4, FOCUS_2006_Z_mkin, parms.ini = m.Z.mkin.
 ## 3$bparms.ode, : Observations with value of zero were removed from the data

-
plot_sep(m.Z.mkin.4)

+
plot_sep(m.Z.mkin.4)

 

 
The error level of the fit, but especially of metabolite Z3, can be improved if the SFORB model is chosen for this metabolite, as this model is capable of representing the tailing of the metabolite decline phase.

-
Z.mkin.5 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
-                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                    Z2 = mkinsub("SFO", "Z3"),
-                    Z3 = mkinsub("SFORB"))

+
Z.mkin.5 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
+                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
+                    Z2 = mkinsub("SFO", "Z3"),
+                    Z3 = mkinsub("SFORB"))

 
## Successfully compiled differential equation model from auto-generated C code.

-
+
 
## Warning in mkinfit(Z.mkin.5, FOCUS_2006_Z_mkin, parms.ini = m.Z.mkin.
 ## 4$bparms.ode[1:4], : Observations with value of zero were removed from the
 ## data

-
plot_sep(m.Z.mkin.5)

+
plot_sep(m.Z.mkin.5)

 

 
The summary view of the backtransformed parameters shows that we get no confidence intervals due to overparameterisation. As the optimized  is excessively small, it seems reasonable to fix it to zero.

-
+
 
## Warning in mkinfit(Z.mkin.5, FOCUS_2006_Z_mkin, parms.ini = c(m.Z.mkin.
 ## 5$bparms.ode[1:7], : Observations with value of zero were removed from the
 ## data

-
plot_sep(m.Z.mkin.5a)

+
plot_sep(m.Z.mkin.5a)

 

 
As expected, the residual plots for Z0 and Z3 are more random than in the case of the all SFO model for which they were shown above. In conclusion, the model  is proposed as the best-fit model for the dataset from Appendix 7 of the FOCUS report.

 
A graphical representation of the confidence intervals can finally be obtained.

-
mkinparplot(m.Z.mkin.5a)

-
## Warning in summary.mkinfit(object): Could not calculate correlation; no
-## covariance matrix

+
mkinparplot(m.Z.mkin.5a)

 

 
The endpoints obtained with this model are

-
endpoints(m.Z.mkin.5a)

+
endpoints(m.Z.mkin.5a)

 
## $ff
 ##   Z0_free_Z1        Z1_Z2      Z2_sink   Z2_Z3_free Z3_free_sink 
 ##      1.00000      1.00000      0.46344      0.53656      1.00000 
-- 
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