## 1. Several models to choose from

Passive
sampler calibration is often done by exposing samplers to a constant
concentration of target compounds (*C*_{w}), followed by fitting
the accumulated amounts to a kinetic model. In many cases it can be assumed
that the accumulation rate (d*N*/d*t*) is linearly proportional to
the effective concentration difference between the water and the sampler.

where *R*_{s} is the water sampling rate, *C*_{w} is the concentration in water, *m* the sorbent mass, and *K* the sorbent-water sorption coefficient. This is a first order differential equation. Sampling rate models predict that the difference between the initial and the final concentration in the sampler decays exponentially with time (at constant *C*_{w}). Some examples are shown at the right for compounds with *K* = 3×10^{3} to 1×10^{5} L/kg, with *C*_{w} = 2 ng/L, sorbent mass = 0.0003 kg, and sampling rate (*R*_{s}) = 0.15 L/d.

Several variants of passive sampling rate model fits exist.

### Model 1: *R*_{s} – *K* model

The model that is easiest to interpret has the water sampling rate (*R*_{s}) and the sorption coefficient (*K*) as adjustable parameters.

This model is sometimes expressed differently, depending on the data processing prior to statistical analysis. Some researchers model the concentration in the sampler (*C*_{s}) instead of the amount, and divide both sides of the above equation by the sorbent mass *m*

Others
first express the data as a concentration factor *CF* = *C*_{s}/*C*_{w}

In all of
these cases the accumulation data are modeled with *R*_{s} and *K* as adjustable parameters.

### Model 2: *k*_{e}* – K* model

Although
model 1 can be fitted to experimental data directly (e.g., R-project, Matlab,
and other), some statistics software requires basic transformations in order to
match the model with the built-in equations. A frequently used transformation
is to write the group *R*_{s} /(*mK*) as a rate constant *k*_{e}.

This model
can be written in terms of *C*_{s} and *CF* as above, and in
addition it can be noted that the accumulated amounts reach a plateau value of *N*_{∞} = *C*_{w}*mK* at
equilibrium

This is the ”one-phase
association” standard model in Graphpad Prism. After fitting the data, *R*_{s} is obtained from *k*_{e} by

*R*_{s} = *m* *K* *k*_{e}

and *K* is evaluated from

*K* =* N*_{∞}/(*mC*_{w})

A
difference with Model 1 is that *K* and *k*_{e} are treated
as independent adjustable parameters in the curve fitting. This causes some
difficulties because *k*_{e} is inversely proportional to *K*.

### Model 3: asymptotic regression model (ABQ model)

Townsend et
al. (2018) included an initial amount *N*_{0} in model 2, and rearranged this model to

The accumulation data could then be fitted to GenStat’s standard asymptotic regression function

*N* = *A* + *B Q*^{t}

*A *= *N*_{∞}

*B*=-(* N*_{∞} – *N*_{0})

*Q* = exp(-*k*_{e})

After fitting the data to the model, *R*_{s} is obtained from

and *K* is calculated from

*K* =* A*/(*mC*_{w})

A
difference with model 2 is that an additional parameter is needed (3 instead of
2), and that *Q* = exp(−*k*_{e}) rather than *k*_{e} is used as an adjustable parameter. The parameters *A *= *N*_{∞} and *Q* =exp(−*k*_{e})
are negatively associated in this model (higher *K* means higher *A* and lower *Q*).

## 2.* R*_{s} and *K* estimates

A challenge
with all three models is how to estimate the parameters when the uptake is
essentially linear. This happens when the sorption coefficient is very high (black
and amber lines in figure above). A further challenge with models 2 and 3 is
that *R*_{s} is obtained indirectly: from *k*_{e} and *K* for model 2, and from *A* and *Q* for model 3. This may (or
may not) result in lower accuracy of the estimated *R*_{s} values.

The performance of the models was therefore evaluated by simulating the uptake of three compounds over a time period of 21 d.

Accumulated amounts were calculated from model 1, with *K* between 3×10^{3} and 1×10^{5} L/kg, *C*_{w} = 2 ng/L, *m* = 0.0003 kg, and *R*_{s} = 0.15 L/d, for 8 exposure times up to 21 d.

A the end of the exposure period the amounts were 97, 65, 30, and 10% of their equilibrium values. For each compound 100 uptake experiments were simulated by adding 5% random noise to the amounts. See adjacent plot for some examples.

Parameters were estimated for models 1, 2, and 3 using weighted nonlinear least squares analysis (R-project, nls function). Parameters were also estimated using Excel’s Solver Add-In (Billo, 2001).

Ratios of
estimated and true *R*_{s} were close to 1 for all models and all
model compounds, even though the *R*_{s} estimates from models 2
and 3 were obtained indirectly. The *R*_{s} estimates from model 3
show a slightly larger scatter.

Median
ratios of estimated and true *K* values are also close to 1 for all three
models. The scatter in the *K*/*K*_{true} ratios is larger
for the more linear uptake data, as expected.

## 3. Uncertainty estimates in *R*_{s} and *K*

Nonlinear
least squares analysis includes estimation of the standard errors in the
adjustable parameters. *R*_{s} is estimated directly with model 1,
but is calculated from two other parameters with model 2 (*R*_{s} = *m* *K* *k*_{e} ) and model 3 (*R*_{s} = -[A lnQ]/*C*_{w}). The usual method for estimating standard
errors in calculated values is error propagation under the assumption that the estimated parameters are uncorrelated. The
standard error *s*_{Rs} is then given for model 2 as

where *s*_{ke} and *s*_{K} are the standard errors in *k*_{e} and *K*.

Assuming
that the error in *C*_{w} can be neglected, *s*_{Rs} for
model 3 is obtained from

where* s*_{A} and *s*_{Q} are the standard errors in *A* and *Q*.

Standard
errors in the sorption coefficient (*s*_{K}) are obtained directly
from the nonlinear least squares output of model 2, and for model 3 *s*_{K} is given by

To test the
validity of these error estimates, the predicted standard errors in *R*_{s} and *K* for individual model runs (*s*_{predicted}) were
compared with the observed standard errors within each set of 100 runs (*s*_{observed}).

Standard
errors in *R*_{s} are well-predicted with model 1, but are
overestimated with models 2 and 3, also for the compound that reaches 97%
equilibrium. Standard errors in *K* are on average well predicted by all
three models, but (naturally) show a large scatter for the compounds that reach
a small degree of equilibrium during the exposure.

Better estimates
of *s*_{Rs} from models 2 and 3 can be obtained by accounting for
the covariance between *k*_{e} and *K* (model 2) or *A* and *Q* (model 3), but this increases the computational burden, and using
model 1 is a more straightforward approach.

## 4. Choosing nonlinear versus linear least-squares

Nonlinear least
squares analysis yields the correct *R*_{s}, even when the uptake
is essentially linear (see *R*_{s}/*R*_{s,true} plot
above). A practical approach for selecting linear versus nonlinear least squares
is to select the method that yields the smallest residual errors. A visual
inspection of plots with experimental values and model fit, and a comparison of *R*_{s} estimates and standard errors from linear and nonlinear
least squares suffices. Alternatively, a partial *F*-test can
be used to check if the nonlinear model (two parameters) gives a significantly
better fit than the linear model (one parameter).

Nonlinear least squares estimations do not always converge when the uptake is linear. In those cases the linear model can be selected, again after checking the plots with measured and modeled data.

## 5. Optimal passive sampling rate model fit

- All
three nonlinear models yield similar values of
*R*_{s}and*K* - Only
model 1 gives a realistic estimate of the standard error in
*R*_{s}. - Choosing
between nonlinear and linear model is not very critical. The choice can be
based on visual inspection of plots of modeled and experimental data, or on a partial
*F*-test.

## 6. Free template for the *R*_{s} –* K* model

An excel template for estimation of *R*_{s} and *K* can be downloaded for free. It is easy to operate, and comes with a manual including screenshots.

Basic features are

- estimates of
*R*_{s}and*K*for 10 data sets in one run - standard errors of
*R*_{s}and*K* - comparison with linear regression results
- partial
*F*-test for choosing nonlinear versus linear modeling - plots of residual errors
- plots of data + model fit
- weighted and unweighted nonlinear regression
- no macro’s; no security issues
- up to 20 data points per experiment

## 7. Need to test a different model?

PaSOC is happy to adapt the free passive sampling rate model fit template according you your needs.

- Include a lag phase
- Change the dependent variable from amount to concentration in the sorbent (
*C*_{s}=*N*/*m*) or concentration factor (*CF*=*C*_{s}/*C*_{w}) - Optimize log
*K*instead of*K* - Any other modification

Click How we work for further details.

## References

Billo, E.J., 2001. Non-linear regression using the solver. In: Excel for Chemists: A Comprehensive Guide. John Wiley & Sons, Inc., New York, pp. 223–238.

Townsend, I., Jones, L., Broom, M., Gravell, A., Schumacher, M., Fones, G.R., Greenwood, R., Mills, G.A., 2018. Calibration and application of the Chemcatcher® passive sampler for monitoring acidic herbicides in the River Exe, UK catchment. Environ. Sci. Pollut. Res. 25, 25130–25142. https://doi.org/10.1007/s11356-018-2556-3

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