# American Institute of Mathematical Sciences

June  2017, 14(3): 581-606. doi: 10.3934/mbe.2017034

## Modeling and simulation for toxicity assessment

 1 Department of Mathematics and Statistics, Grant MacEwan University, Edmonton, Alberta, T5P2P7, Canada 2 Department of Mathematical and statistical Sciences, University of Alberta, Edmonton, Alberta, T6G2G1, Canada 3 Alberta Health, Edmonton, Alberta, T5J1S6, Canada 4 Department of Laboratory Medicine and Pathology, University of Alberta, Edmonton, Alberta, T6G2B7, Canada 5 Alberta Centre for Toxicology, University of Calgary, Calgary, Alberta, T2N4N1, Canada 6 ACEA Biosciences Inc, San Diego, California, 92121, USA

Received  February 29, 2016 Accepted  October 17, 2016 Published  December 2016

The effect of various toxicants on growth/death and morphology of human cells is investigated using the xCELLigence Real-Time Cell Analysis High Troughput in vitro assay. The cell index is measured as a proxy for the number of cells, and for each test substance in each cell line, time-dependent concentration response curves (TCRCs) are generated. In this paper we propose a mathematical model to study the effect of toxicants with various initial concentrations on the cell index. This model is based on the logistic equation and linear kinetics. We consider a three dimensional system of differential equations with variables corresponding to the cell index, the intracellular concentration of toxicant, and the extracellular concentration of toxicant. To efficiently estimate the model's parameters, we design an Expectation Maximization algorithm. The model is validated by showing that it accurately represents the information provided by the TCRCs recorded after the experiments. Using stability analysis and numerical simulations, we determine the lowest concentration of toxin that can kill the cells. This information can be used to better design experimental studies for cytotoxicity profiling assessment.

Citation: Cristina Anton, Jian Deng, Yau Shu Wong, Yile Zhang, Weiping Zhang, Stephan Gabos, Dorothy Yu Huang, Can Jin. Modeling and simulation for toxicity assessment. Mathematical Biosciences & Engineering, 2017, 14 (3) : 581-606. doi: 10.3934/mbe.2017034
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##### References:
TCRCs for (a) PF431396 and (b) monastrol
Trajectories corresponding to monastrol and initial values $0<n(0)<K$, $C_0(0)=0$, and (a) $CE(0)<\frac{\beta\eta_1^2}{\alpha\lambda_1^2}=6.51$.(b) $CE(0)>\frac{\beta\eta_1^2}{\alpha\lambda_1^2}=6.51$
The separation between persistence and extinction according to the initial values $n(0)$ and $CE(0)$, red $*$: persistence; blue $\circ$: extinction
Negative control data fitted by logistic model, dot: experimental data, line: logistic model
Smooth spline approximation, dot: experimental data, line: smooth spline
Estimation results for PF431396, dot: experimental data, line: filtered or predicted observations; (a) CE(0)=5.00uM, (b) CE(0)= 1.67uM, (c) CE(0)=0.56uM, (d) CE(0)=0.19uM, (e) CE(0)=61.73nM, (f) CE(0)= 20.58nM, (g) CE(0)= 6.86nM, (h) CE(0)=2.29nM
Estimation results for monastrol, dot: experimental data, line: filtered or predicted observations; (a) CE(0)=100.00uM, (b) CE(0)=33.33uM, (c) CE(0)=11.11uM, (d) CE(0)= 3.70uM, (e) CE(0)=1.23uM, (f) CE(0)= 0.41uM, (g) CE(0)=0.14uM, (h) CE(0)=45.72nM
Estimation results for ABT888, dot: experimental data, line: filtered or predicted observations; (a) CE(0)=308.00uM, (b) CE(0)=102.67uM, (c) CE(0)=34.22uM, (d) CE(0)=11.41uM, (e) CE(0)=3.80uM, (f) CE(0)=1.27uM, (g) CE(0)=0.42uM, (h) CE(0)=0.14uM
(a) Experimental TCRCs for PF431396 for CE(0)=5uM, 1.67uM, 0.56uM (b) Expected cell index and probability of extinction for different concentrations for PF431396
(a) Experimental TCRCs for ABT888 for CE(0)=308uM, 103uM, 34uM (b) Expected cell index and probability of extinction for different concentrations for ABT888
Estimation results for HA1100 hydrochloride, dot: experimental data, line: filtered or predicted observations; (a) CE(0)=1.00mM, (b) CE(0)=0.33mM, (c) CE(0)=0.11mM, (d) CE(0)= 37.04uM, (e) CE(0)=12.35uM, (f) CE(0)=4.12uM, (g) CE(0)=1.37uM, (h) CE(0)= 0.46uM
The first order GSA indices ranking for PF431396 (higher rank means more sensitive)
The first order GSA indices ranking for ABT888 (higher rank means more sensitive)
Network graph visualizing the second order GSA indices for (a) PF431396 with CE(0)=10uM (b) ABT888 with CE(0)=400uM
List of Variables and Parameters
 Symbol Definition $n(t)$ cell index ≈ cell population $C_0(t)$ toxicant concentration inside the cell $CE(t)$ toxicant concentration outside the cell $\beta$ cell growth rate in the absence of toxicant $K$ capacity volume $\alpha$ effect coefficient of toxicant on the cell's growth $\lambda_1^2$ 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
 Symbol Definition $n(t)$ cell index ≈ cell population $C_0(t)$ toxicant concentration inside the cell $CE(t)$ toxicant concentration outside the cell $\beta$ cell growth rate in the absence of toxicant $K$ capacity volume $\alpha$ effect coefficient of toxicant on the cell's growth $\lambda_1^2$ 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
The EM algorithm
 Initialize the model parameters $\Theta=\{Q, R,\alpha, \lambda_1,\lambda_2, \eta_1, \eta_2\}$ Repeat until the log likelihood has converged The E step For k=1 to N Run the UF filter to compute $\bar{x}_{k+1}$, $\bar{P}_{k+1}$, $\hat{x}_{k+1}$, $\hat{P}_{k+1}$ and $\bar{P}_{x_kx_{k+1}}$ For k=N to 1 Calculate the smoothed values $x_{k|N}$, and $P_{k|N}$ using (13), (14) The M step Update the values of the parameters $\Theta$ to maximize $\hat{E}$
 Initialize the model parameters $\Theta=\{Q, R,\alpha, \lambda_1,\lambda_2, \eta_1, \eta_2\}$ Repeat until the log likelihood has converged The E step For k=1 to N Run the UF filter to compute $\bar{x}_{k+1}$, $\bar{P}_{k+1}$, $\hat{x}_{k+1}$, $\hat{P}_{k+1}$ and $\bar{P}_{x_kx_{k+1}}$ For k=N to 1 Calculate the smoothed values $x_{k|N}$, and $P_{k|N}$ using (13), (14) The M step Update the values of the parameters $\Theta$ to maximize $\hat{E}$
Estimated Values of Parameters
 Toxicant Cluster β K $\eta_1$ $\lambda_1$ $\lambda_2$ $\eta_2$ $\alpha$ PF431396 Ⅹ 0.077 21.912 0.273 0.058 0 0.008 0.238 monastrol Ⅹ 0.074 18.17 0.209 0.177 0.204 0.5 0.016 ABT888 Ⅰ 0.083 17.543 0.079 0.177 0.205 0.5 0.005 HA1100 hydrochloride Ⅰ 0.077 21.913 0.143 0.0098 0.0786 0.147 0.351
 Toxicant Cluster β K $\eta_1$ $\lambda_1$ $\lambda_2$ $\eta_2$ $\alpha$ PF431396 Ⅹ 0.077 21.912 0.273 0.058 0 0.008 0.238 monastrol Ⅹ 0.074 18.17 0.209 0.177 0.204 0.5 0.016 ABT888 Ⅰ 0.083 17.543 0.079 0.177 0.205 0.5 0.005 HA1100 hydrochloride Ⅰ 0.077 21.913 0.143 0.0098 0.0786 0.147 0.351
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