This is example 3.7 on page 108 of Quantitative Ecotoxicology - reproduced with R. This example is about accumulation in mosquitofish (Gambusia holbrooki ).

Get the data from here and read it into R:

require ( RCurl )
url <- getURL ( "https://raw.github.com/EDiLD/r-ed/master/quantitative_ecotoxicology/data/p108.csv" ,
ssl.verifypeer = FALSE , .opts = curlOptions ( followlocation = TRUE ))
MERCURY <- read.table ( text = url , header = TRUE , sep = ";" )

head ( MERCURY )

```
## DAY HG
## 1 0 0
## 2 1 380
## 3 2 540
## 4 3 570
## 5 4 670
## 6 6 780
```

This is pretty much like the previous examples:

We fit a nonlinear model to our data
.
The model is given in equation 3.42 of the book:

plot ( MERCURY )

We can specify the model as follows:

mod <- nls ( HG ~ KU / KE * 0.24 * ( 1 - exp ( - KE * DAY )),
data = MERCURY ,
start = list ( KU = 1000 , KE = 0.5 ))

This equals to equation 3.42:

$HG = C_t$
$KU = k_u$
$KE = k_e$
$0.24 = C_1$
$DAY = t$
Unlike in the book I did not specify bounds here (see the previous posts how to do this).

This results in:

summary ( mod )

```
##
## Formula: HG ~ KU/KE * 0.24 * (1 - exp(-KE * DAY))
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## KU 1866.700 241.784 7.72 0.0015 **
## KE 0.589 0.106 5.55 0.0051 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 43.7 on 4 degrees of freedom
##
## Number of iterations to convergence: 7
## Achieved convergence tolerance: 2.07e-06
```

So the parameter estimates are:

$k_e = 0.589 \pm 0.106$
$k_u = 1866.7 \pm 241.784$
The BCF is given as $BCF = \frac{k_u}{k_e} = 3171.4$

BCF = coef ( mod )[ 1 ] / coef ( mod )[ 2 ]
BCF

```
## KU
## 3171.4
```

From this we can predict the fish concentration as

BCF * 0.24

```
## KU
## 761.14
```

Finally we plot the data and our model:

DAY_pred <- seq ( 0 , 6 , by = 0.1 )
# Raw data
plot ( MERCURY )
# add model
lines ( DAY_pred , predict ( mod , newdata = data.frame ( DAY = DAY_pred )))
# add model-equation
text ( 3 , 100 , bquote ( HG == . ( BCF * 0.24 ) %.% ( 1 - exp ( - . ( coef ( mod )[ 2 ]) %.% DAY ))))

Once again we reproduced the results as in the book using R :)
The differences for BCF and $C_{fish}$ are due to rounding errors.

Code and data are available on my github-repo under file name ‘p108’.