What a GUTS analysis is?

When following survival over time, toxicity test data consist in exposing an initial number \(N_{0i}\) of organisms at concentrations \(c_i(t)\) of a given contaminant (constant or variable over time), and following the number \(N_i^k\) of survivors at time \(t_k\) (with \(t_0 < t_1 < \ldots < t_m\) and \(t_0 = 0\)), thus providing a collection \(D = (c_i, t_k, N_i^k)_{i,k}\) of experimental data.

The number of survivors at time \(t_k\) given the number of survivors at time \(t_{k-1}\) is assumed to follow a binomial distribution: \[ N_i^k \sim \mathcal{B}(N_i^{k-1}, f_i(t_{k-1}, t_k)) \] where \(f_i\) is the conditional probability of survival at time \(t_k\) given survival at time \(t_{k-1}\) under concentration \(c_i(t)\). Denoting \(S_i(t)\) the probability of survival at time \(t\), we have: \[ f_i(t_{k-1}, t_k) = \frac{S_i(t_k)}{S_i(t_{k-1})} \] The formulation of the survival probability \(S_i(t)\) is based on the scaled internal concentration, denoted \(D_w(t)\) and described by the TK part of both GUTS-RED models: \[ \frac{dD_w(t)}{dt} = k_d \left(C_w(t) - D_w(t) \right) \] where \(C_w(t)\) corresponds to the exposure concentration and \(k_d\) [time\(^{-1}\)] to the dominant rate constant, that is to the slowest compensating process dominating the overall dynamics of toxicity.

Under the GUTS-RED-SD model, all organisms are supposed to have the same internal threshold concentration, denoted \(z\) [e.g., mol.L\(^{-1}\)], and, once this threshold exceeded, the instantaneous probability to die, denoted \(h(t)\), increases linearly with the scaled internal concentration: \[ h(t) = b_w \mathop {\max }\limits_{0 \leqslant \tau \leqslant t} \left( {{D_w}\left( t \right) - z,0} \right) + h_b \] where \(b_W\) is [e.g., L.mol.time\(^{-1}\)] is the killing rate and \(h_b\) [time\(^{-1}\)] is the background mortality rate. Then, the survival probability over time under the GUTS-RED-SD model is given by: \[ {S_{SD}}\left( t \right) = \exp \left( { - \int\limits_0^t {h\left( \tau \right)d\tau} } \right) \] Under the GUTS-RED-IT model, the threshold concentration is assumed to be distributed among organisms and for each organism the death is immediate as soon as this threshold is reached. The probability to die at the maximal scaled internal concentration with background mortality \(h_b\) is then given by: \[ {S_{IT}}\left( t \right) = \exp \left( { - {h_b}t} \right)\left( {1 - F\left( {\mathop {\max }\limits_{0 < \tau < t} {D_w}\left( \tau \right)} \right)} \right) \] Assuming a log-logistic probability distribution for the threshold, we get: \[ F\left( {\mathop {\max }\limits_{0 < \tau < t} {D_w}\left( \tau \right)} \right) = \frac{1}{{1 + {{\left( {\frac{{\mathop {\max }\limits_{0 < \tau < t} {D_w}\left( \tau \right)}}{{{m_w}}}} \right)}^\beta }}} \] with \(m_w\) [e.g., mol.L\(^{-1}\)] the median and \(\beta\) the shape of the threshold distribution.

With MOSAIC\(_{GUTS-fit}\), either model GUTS-RED-SD or model GUTS-RED-IT can be fitted to survival toxicity data, either under constant or time-variable exposure concentration.

Step 1: Data uploading

When using MOSAIC\(_{GUTS-fit}\), the first step is to upload input data.

Step 2: Results and interpretation

After choosing cadmium-2 data set, after choosing model SD and clicking on Run, you immediately see a summary of the experimental design corresponding to the data.

You also get plots of the survival raw data:

• The number of survivors over time for each replicate and each concentration.

• The observed fraction of surviving animals at final time at each tested contaminant concentration (black dots) together with the 95% binomial confidence intervals (black vertical segments).

Then you get the fitting results:

• The fitting plot which shows the mean survival rate as a function of time for each concentration (orange plain line). The 95% credible interval for this mean is represented by the light grey zone. Black dots correspond to observed data while black vertical segments stand for the between-replicate variability (95% binomial confidence intervals).

• Model parameter estimates: the table on the right of the fitting results displays the inferred values for the model parameters. We report the median (as the point estimated value) and the 2.5% and 97.5% quantiles of the posterior (as a measure of uncertainty, a.k.a. 95% credible intervals). Parameter \(z\) corresponds to the concentration threshold in model GUTS-RED-SD, that is to the No-Effect-Concentration for all organisms.

  parameters    median      Q2.5     Q97.5
1         kd 2.246e-01 1.466e-01 3.957e-01
2         hb 1.291e-03 5.315e-04 2.527e-03
3          z 9.564e+01 7.379e+01 1.084e+02
4         kk 5.955e-04 4.080e-04 8.375e-04

• \(LC_x\) estimates: we first provide the concentration-response curve as predicted by the chosen GUTS model, by default at the latest observed time point pour \(x=50\%\). From this predicted curve, some \(LC_x\) estimates are provided with medians (as the point estimated value) and 2.5% and 97.5% quantiles (as measures of uncertainty, a.k.a. 95% credible intervals) for \(x = 5, 10, 20, 50\%\).

LC5 =  97.21637  [ NA  ;  110.0498 ] 

LC10 =  100.1212  [ 80.05984  ;  112.16 ] 

LC20 =  104.7718  [ 86.24588  ;  116.1564 ] 

LC50 =  121.5519  [ 106.6044  ;  131.9035 ] 

• The posterior predictive check (PPC): we plot the observed data against the corresponding predicted value from the fitted model as a 95% credible interval. If the fit is correct, we expect to see 95% of the observed values fall inside the credible intervals. Each black dot corresponds to one observation from the dataset, and the corresponding observed value can be read on the x-axis, while the y-axis reports the value predicted by the fitted model, as well as the 95% credible interval. The interval is depicted in green if it contains the observed value and in red otherwise.

• R script: a gateway to perform further calculations directly within the R software [4].

# This script was generated by MOSAIC, a web application dedicated to
# ecotoxicology. It is available at http://pbil.univ-lyon1.fr/software/mosaic/

# For any further question, please contact us at mosaic@univ-lyon1.fr

library(morse)

# Your input file 'cadmium-2' should be present in the current working directory
dat <- read.table(file='cadmium-2',sep='\t',header=T)
# or directly downloaded from pacakge 'morse'
# data("cadmium2")
# dat <- cadmium2

sdat <- survData(dat)
plot(sdat)
plotDoseResponse(sdat)
plotDoseResponse(sdat, log.scale = T)
fit <- survFit(sdat, model_type = 'SD')
summary(fit)
plot(fit, xlab = "Time", adddata = TRUE)
plot(LCx(fit, X=50))
ppc(fit)

References

[1] Jager T, Albert C, Preuss T and Ashauer R (2011). General unified threshold model of survival - a toxicokinetic-toxicodynamic framework for ecotoxicology. Environmental Science & Technology, 45(7): 2529–2540.

[2] Jager T and Ashauer R (2018). Modelling survival under chemical stress. A comprehensive guide to the GUTS framework. Version 1.0. Leanpub (eds). .

[3] Baudrot V, Charles S, Delignette-Muller ML, Duchemin W, Kon-Kam-King G, Lopes C, Ruiz P and Veber P (2018). morse: Modelling Tools for Reproduction and Survival Data in Ecotoxicology. R package version 3.1.1. https://CRAN.R-project.org/package=morse.

[4] R Core Team (2018). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/.