# Uncertainty Quantification for Toxicity Assessment

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Toxicity assessment is of great importance to environmental and human health, since toxic chemical produces an adverse effect to our health and could be harmful to natural environment. To study the impact of a toxicant and to obtain a quantitative assessment, we propose a stochastic mathematical model and we carry out an uncertainty quantification for toxicity assessment.\\

For a toxic environment, we consider the following model:

\begin{align}

\frac{dn(t)}{dt}&=\beta n(t)(1-\frac{n(t)}{K})-\alpha C_0(t)n(t)\nonumber\\

\frac{dC_0(t)}{dt}&=\lambda_1^2CE(t)-\eta_1^2C_0(t)\nonumber\\

\frac{dCE(t)}{dt}&=\lambda_2^2C_0(t)n(t)-\eta_2^2CE(t)n(t)

\end{align}

where $C_0(t)$ and $CE(t)$ denote the intracellular and the extracellular concentrations of toxicant at time t, respectively.

\begin{table}[h!]

\center

\begin{tabular}{c c}

\hline

symbol & definition\\

\hline\hline

n(t)&cell index at time t\\

$\beta$&cell growth rate\\

$\alpha$&coefficient of toxicant on the cell's growth\\

K&capacity volume\\

$\lambda_1^2$&the uptake rate of the toxicant from environment\\

$\lambda_2^2$&the toxicant uptake rate from cells\\

$\eta_1^2$&the toxicant input rate to the environment\\

$\eta_2^2$&the losses rate of toxicant absorbed by cells\\

\hline

\end{tabular}

\end{table}

Utilizing the real time cell analysis data for a given chemical compound, we can determine whether or not cells will survive under certain toxic conditions. The parameter set $\Theta=\lbrace\beta,K, \alpha, \lambda_1, \lambda_2, \eta_1, \eta_2\rbrace$ in the model is estimated using the information from the real data. A nonlinear least square method is first employed to find $\beta$ and $K$. The Expectation Maximization algorithm based on the the Unscented Filter is then applied to estimate the rest of the parameters $\lbrace\alpha, \lambda_1, \lambda_2, \eta_1, \eta_2\rbrace$.\\

For the model validation, we divide the data into training set and test set. The trainging set consisting of 70\% of the data is used for parameter estimation. The remaining 30\% data is to preform model validation.\\

Since small changes in the input may lead to large variations in the model response, it is of interact to investigate the uncertainty effects due to the parameters. Consider the parameters $\beta$ and $\gamma=\frac{\beta}{K}$ are replaced by random variables, the deterministic model (1) is now replaced by a stochastic differential equation

\begin{align}

dn(t)&=n(t)(\beta-\gamma n(t)-\alpha C_0(t))dt+\sigma_1n(t)dB_1(t)-\sigma_2n^2(t)dB_2(t)\nonumber\\

dC_0(t)&=(\lambda_1^2CE(t)-\eta_1^2C_0(t))dt\nonumber\\

dCE(t)&=(\lambda_2^2C_0(t)n(t)-\eta_2^2CE(t)n(t))dt

\end{align}

where $\sigma_i, i=1,2$ are probability spaces and $B_i(t), i=1,2$ are independent standard Brownian motions.\\

The uncertainty effect will be examined by studying the solutions obtained by the deterministic and stochastic models.

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