This is just a very simple vignette showing how to fit a degradation model for a parent compound with one transformation product using mkin. After loading the library we look a the data. We have observed concentrations in the column named value at the times specified in column time for the two observed variables named parent and m1.

library("mkin")
print(FOCUS_2006_D)
##      name time  value
## 1  parent    0  99.46
## 2  parent    0 102.04
## 3  parent    1  93.50
## 4  parent    1  92.50
## 5  parent    3  63.23
## 6  parent    3  68.99
## 7  parent    7  52.32
## 8  parent    7  55.13
## 9  parent   14  27.27
## 10 parent   14  26.64
## 11 parent   21  11.50
## 12 parent   21  11.64
## 13 parent   35   2.85
## 14 parent   35   2.91
## 15 parent   50   0.69
## 16 parent   50   0.63
## 17 parent   75   0.05
## 18 parent   75   0.06
## 19 parent  100     NA
## 20 parent  100     NA
## 21 parent  120     NA
## 22 parent  120     NA
## 23     m1    0   0.00
## 24     m1    0   0.00
## 25     m1    1   4.84
## 26     m1    1   5.64
## 27     m1    3  12.91
## 28     m1    3  12.96
## 29     m1    7  22.97
## 30     m1    7  24.47
## 31     m1   14  41.69
## 32     m1   14  33.21
## 33     m1   21  44.37
## 34     m1   21  46.44
## 35     m1   35  41.22
## 36     m1   35  37.95
## 37     m1   50  41.19
## 38     m1   50  40.01
## 39     m1   75  40.09
## 40     m1   75  33.85
## 41     m1  100  31.04
## 42     m1  100  33.13
## 43     m1  120  25.15
## 44     m1  120  33.31

Next we specify the degradation model: The parent compound degrades with simple first-order kinetics (SFO) to one metabolite named m1, which also degrades with SFO kinetics.

The call to mkinmod returns a degradation model. The differential equations represented in R code can be found in the character vector $diffs of the mkinmod object. If a C compiler (gcc) is installed and functional, the differential equation model will be compiled from auto-generated C code.

SFO_SFO <- mkinmod(parent = mkinsub("SFO", "m1"), m1 = mkinsub("SFO"))
## Successfully compiled differential equation model from auto-generated C code.
print(SFO_SFO$diffs)
##                                                       parent 
## "d_parent = - k_parent_sink * parent - k_parent_m1 * parent" 
##                                                           m1 
##             "d_m1 = + k_parent_m1 * parent - k_m1_sink * m1"

We do the fitting without progress report (quiet = TRUE).

fit <- mkinfit(SFO_SFO, FOCUS_2006_D, quiet = TRUE)

A plot of the fit including a residual plot for both observed variables is obtained using the plot method for mkinfit objects.

plot(fit, show_residuals = TRUE)

Confidence intervals for the parameter estimates are obtained using the mkinparplot function.

mkinparplot(fit)

A comprehensive report of the results is obtained using the summary method for mkinfit objects.

summary(fit)
## mkin version:    0.9.40 
## R version:       3.2.1 
## Date of fit:     Mon Jul 20 14:51:06 2015 
## Date of summary: Mon Jul 20 14:51:06 2015 
## 
## Equations:
## d_parent = - k_parent_sink * parent - k_parent_m1 * parent
## d_m1 = + k_parent_m1 * parent - k_m1_sink * m1
## 
## Model predictions using solution type deSolve 
## 
## Fitted with method Port using 153 model solutions performed in 0.66 s
## 
## Weighting: none
## 
## Starting values for parameters to be optimised:
##                  value   type
## parent_0      100.7500  state
## k_parent_sink   0.1000 deparm
## k_parent_m1     0.1001 deparm
## k_m1_sink       0.1002 deparm
## 
## Starting values for the transformed parameters actually optimised:
##                        value lower upper
## parent_0          100.750000  -Inf   Inf
## log_k_parent_sink  -2.302585  -Inf   Inf
## log_k_parent_m1    -2.301586  -Inf   Inf
## log_k_m1_sink      -2.300587  -Inf   Inf
## 
## Fixed parameter values:
##      value  type
## m1_0     0 state
## 
## Optimised, transformed parameters with symmetric confidence intervals:
##                   Estimate Std. Error  Lower   Upper
## parent_0            99.600    1.61400 96.330 102.900
## log_k_parent_sink   -3.038    0.07826 -3.197  -2.879
## log_k_parent_m1     -2.980    0.04124 -3.064  -2.897
## log_k_m1_sink       -5.248    0.13610 -5.523  -4.972
## 
## Parameter correlation:
##                   parent_0 log_k_parent_sink log_k_parent_m1 log_k_m1_sink
## parent_0           1.00000            0.6075        -0.06625       -0.1701
## log_k_parent_sink  0.60752            1.0000        -0.08740       -0.6253
## log_k_parent_m1   -0.06625           -0.0874         1.00000        0.4716
## log_k_m1_sink     -0.17006           -0.6253         0.47163        1.0000
## 
## Residual standard error: 3.211 on 36 degrees of freedom
## 
## Backtransformed parameters:
##   Confidence intervals for internally transformed parameters are asymmetric.
##   t-test (unrealistically) based on the assumption of normal distribution
##   for estimators of untransformed parameters.
##                Estimate t value    Pr(>t)     Lower     Upper
## parent_0      99.600000  61.720 2.024e-38 96.330000 1.029e+02
## k_parent_sink  0.047920  12.780 3.050e-15  0.040890 5.616e-02
## k_parent_m1    0.050780  24.250 3.407e-24  0.046700 5.521e-02
## k_m1_sink      0.005261   7.349 5.758e-09  0.003992 6.933e-03
## 
## Chi2 error levels in percent:
##          err.min n.optim df
## All data   6.398       4 15
## parent     6.827       3  6
## m1         4.490       1  9
## 
## Resulting formation fractions:
##                 ff
## parent_sink 0.4855
## parent_m1   0.5145
## m1_sink     1.0000
## 
## Estimated disappearance times:
##           DT50   DT90
## parent   7.023  23.33
## m1     131.761 437.70
## 
## Data:
##  time variable observed predicted   residual
##     0   parent    99.46 9.960e+01 -1.385e-01
##     0   parent   102.04 9.960e+01  2.442e+00
##     1   parent    93.50 9.024e+01  3.262e+00
##     1   parent    92.50 9.024e+01  2.262e+00
##     3   parent    63.23 7.407e+01 -1.084e+01
##     3   parent    68.99 7.407e+01 -5.083e+00
##     7   parent    52.32 4.991e+01  2.408e+00
##     7   parent    55.13 4.991e+01  5.218e+00
##    14   parent    27.27 2.501e+01  2.257e+00
##    14   parent    26.64 2.501e+01  1.627e+00
##    21   parent    11.50 1.253e+01 -1.035e+00
##    21   parent    11.64 1.253e+01 -8.946e-01
##    35   parent     2.85 3.148e+00 -2.979e-01
##    35   parent     2.91 3.148e+00 -2.379e-01
##    50   parent     0.69 7.162e-01 -2.624e-02
##    50   parent     0.63 7.162e-01 -8.624e-02
##    75   parent     0.05 6.074e-02 -1.074e-02
##    75   parent     0.06 6.074e-02 -7.382e-04
##   100   parent       NA 5.151e-03         NA
##   100   parent       NA 5.151e-03         NA
##   120   parent       NA 7.155e-04         NA
##   120   parent       NA 7.155e-04         NA
##     0       m1     0.00 0.000e+00  0.000e+00
##     0       m1     0.00 0.000e+00  0.000e+00
##     1       m1     4.84 4.803e+00  3.704e-02
##     1       m1     5.64 4.803e+00  8.370e-01
##     3       m1    12.91 1.302e+01 -1.140e-01
##     3       m1    12.96 1.302e+01 -6.400e-02
##     7       m1    22.97 2.504e+01 -2.075e+00
##     7       m1    24.47 2.504e+01 -5.748e-01
##    14       m1    41.69 3.669e+01  5.000e+00
##    14       m1    33.21 3.669e+01 -3.480e+00
##    21       m1    44.37 4.165e+01  2.717e+00
##    21       m1    46.44 4.165e+01  4.787e+00
##    35       m1    41.22 4.331e+01 -2.093e+00
##    35       m1    37.95 4.331e+01 -5.363e+00
##    50       m1    41.19 4.122e+01 -2.831e-02
##    50       m1    40.01 4.122e+01 -1.208e+00
##    75       m1    40.09 3.645e+01  3.643e+00
##    75       m1    33.85 3.645e+01 -2.597e+00
##   100       m1    31.04 3.198e+01 -9.416e-01
##   100       m1    33.13 3.198e+01  1.148e+00
##   120       m1    25.15 2.879e+01 -3.640e+00
##   120       m1    33.31 2.879e+01  4.520e+00