American Institute of Mathematical Sciences

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

Modeling and simulation for toxicity assessment

Cristina Anton 1, , Jian Deng 2, , Yau Shu Wong 2, , Yile Zhang 2, , Weiping Zhang 3, , Stephan Gabos 4, , Dorothy Yu Huang 5, and  Can Jin 6,

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.

Keywords: Mathematical model, cytotoxicity, parameter estimation, persistence.
Mathematics Subject Classification: Primary: 93A30, 37N25; Secondary: 60G35.
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
References:
[1]

C. Biernacki, G. Celeux and G. Govaert, Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate gaussian mixture models, Comput. Statist. Data Anal., 41 (2003), 561-575.  doi: 10.1016/S0167-9473(02)00163-9.  Google Scholar

[2]

F. Cannavó, Sensitivity analysis for volcanic source modeling quality assessment and model selection, Computers & Geosciences,, 44 (2012), 52-59.   Google Scholar

[3]

Z. Ghahramani and S. Roweis, Learning nonlinear dynamical systems using an EM algorithm, in Advances in Neural Information Processing Systems (eds. M. Kearns, S. Solla and C. D. A.), MIT Press, 1999,599-605. Google Scholar

[4]

T. Hallam, C. Clark and G. Jordan, Effects of toxicants on populations: A qualitative approach Ⅱ. First order kinetics, J. Math. Biology, 18 (1983), 25-37.   Google Scholar

[5]

J. He and K. Wang, The survival analysis for a population in a polluted environment, Nonlinear Analysis: Real World Applications, 10 (2009), 1555-1571.  doi: 10.1016/j.nonrwa.2008.01.027.  Google Scholar

[6]

B. Huang and J. Xing, Dynamic modeling and prediction of cytotoxicity on microelectronic cell sensor array, The Canadian Journal of Chemical Engineering, 84 (2006), 393-405.   Google Scholar

[7]

Q. Huang, L. Parshotam, H. Wang, C. Bampfylde and M. Lewis, A model for the impact of contaminants on fish population dynamics, Journal of Theoretical Biology, 334 (2013), 71-79.  doi: 10.1016/j.jtbi.2013.05.018.  Google Scholar

[8]

F. Ibrahim, B. Huang, J. Xing and S. Gabos, Early determination of toxicant concentration in water supply using MHE, Water Research, 44 (2010), 3252-3260.   Google Scholar

[9]

A.M. Jarrett, Y. Liu, N. Cogan and M.Y. Hussaini, Global sensitivity analysis used to interpret biological experimental results, Journal of Mathematical Biology, 71 (2015), 151-170.  doi: 10.1007/s00285-014-0818-3.  Google Scholar

[10]

J. Jiao, W. Long and L. Chen, A single stage-structured population model with mature individuals in a polluted environment and pulse input of environmental toxin, Nonlinear Analysis: Real World Applications, 10 (2009), 3073-3081.  doi: 10.1016/j.nonrwa.2008.10.007.  Google Scholar

[11]

S. Julier, J. Uhlmann and H. Durrant-White, A new method for the nonlinear transformation of means and covariances in filters and estimators, IEEE Trans. Aut. Control, 45 (2000), 477-482.  doi: 10.1109/9.847726.  Google Scholar

[12]

S. Julier, J. Uhlmann and H. Durrant-Whyte, A new approach for filtering nonlinear systems, in American Control Conference, Seattle, Washington, 1995,1628–1632. doi: 10.1109/ACC.1995.529783.  Google Scholar

[13]

A. Kiparissides, S. Kucherenko, A. Mantalaris and E.N. Pistikopoulos, Global sensitivity analysis challenges in biological systems modeling, Industrial & Engineering Chemistry Research, 48 (2009), 7168-7180.  doi: 10.1021/ie900139x.  Google Scholar

[14]

K. Kothawad, A. Pathan and M. Logad, Evaluation of in vitro anti-cancer activity of fruit lagenaria siceraria against MCF7, HOP62 and DU145 cell line, Int. J. Pharm. & Technol, 4 (2012), 3909-4392.   Google Scholar

[15]

M. Liu and K. Wang, Survival analysis of stochastic single-species population models in polluted environments, Ecological Modeling, 220 (2009), 1347-1357.   Google Scholar

[16]

M. Liu, K. Wang and X. Liu, Long term behaviors of stochastic single-species growth models in a polluted environment, Applied Mathematical Modelling, 35 (2011), 752-762.  doi: 10.1016/j.apm.2010.07.031.  Google Scholar

[17]

X. Meng and D. Van Dyk, The EM algorithm -an old folk-song to a fast new tune, J.R. Statist. Soc.B, 59 (1997), 511-567.  doi: 10.1111/1467-9868.00082.  Google Scholar

[18]

R. Neal and G. Hinton, A view of the EM algorithm that justifies incremental, sparse, an other variants, in Learning in Graphical Models (ed. M. Jordan), 89 (1998), 355-368. doi: 10.1007/978-94-011-5014-9_12.  Google Scholar

[19]

T. Pan, B. Huang, W. Zhang, S. Gabos, D. Huang and V. Devendran, Cytotoxicity assessment based on the AUC50 using multi-concentration time-dependent cellular response curves, Anal. Chim. Acta, 764 (2013), 44-52.   Google Scholar

[20]

T. Pan, S. Khare, F. Ackah, B. Huang, W. Zhang, S. Gabos, C. Jin and M. Stampfl, In vitro cytotoxicity assessment based on KC50 with real-time cell analyzer (RTCA) assay, Comp. Biol. Chem., 47 (2013), 113-120.   Google Scholar

[21]

L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 2001. doi: 10.1007/978-1-4613-0003-8.  Google Scholar

[22]

R. Shumway and D. Stoffer, An approach to time series smoothing and forecasting using the EM algorithm, J. Time Ser. Anal., 3 (1982), 253-264.  doi: 10.1111/j.1467-9892.1982.tb00349.x.  Google Scholar

[23]

I.M. Sobol, Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates, Mathematics and Computers in Simulation, 55 (2001), 271-280.  doi: 10.1016/S0378-4754(00)00270-6.  Google Scholar

[24]

H. Thieme, Mathematics in Population Biology, Princeton Series in theoretical and Computational Biology., 2003  Google Scholar

[25]

E. A. Wan, R. Van der Merwe and A. T. Nelson, Dual estimation and the unscented transformation, in Advances in Neural Information Processing Systems (ed. M. I. J. et al.), MIT Press, 2000. Google Scholar

[26]

C. Wu, On the convergence properties of the EM algorithm, The Annals of Statistics, 11 (1983), 95-103.  doi: 10.1214/aos/1176346060.  Google Scholar

[27]

Z. Xi, S. Khare, A. Cheung, B. Huang, T. Pan, W. Zhang, F. Ibrahim, C. Jin and S. Gabos, Mode of action classification of chemicals using multi-concentration time-dependent cellular response profiles, Comp. Biol. Chem., 49 (2014), 23-35.  doi: 10.1016/j.compbiolchem.2013.12.004.  Google Scholar

[28]

J. Xing, L. Zhu, S. Gabos and L. Xie, Microelectronic cell sensor assay for detection of cytotoxicity and prediction of acute toxicity, Toxicology in Vitro, 20 (2006), 995-1004.  doi: 10.1016/j.tiv.2005.12.008.  Google Scholar

[29]

M. Zhang, D. Aguilera, C. Das, H. Vasquez, P. Zage, V. Gopalakrishnan and J. Wolff, Measuring cytotoxicity: A new perspective on LC50, Anticancer Res., 27 (2007), 35-38.   Google Scholar

show all references

References:
[1]

C. Biernacki, G. Celeux and G. Govaert, Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate gaussian mixture models, Comput. Statist. Data Anal., 41 (2003), 561-575.  doi: 10.1016/S0167-9473(02)00163-9.  Google Scholar

[2]

F. Cannavó, Sensitivity analysis for volcanic source modeling quality assessment and model selection, Computers & Geosciences,, 44 (2012), 52-59.   Google Scholar

[3]

Z. Ghahramani and S. Roweis, Learning nonlinear dynamical systems using an EM algorithm, in Advances in Neural Information Processing Systems (eds. M. Kearns, S. Solla and C. D. A.), MIT Press, 1999,599-605. Google Scholar

[4]

T. Hallam, C. Clark and G. Jordan, Effects of toxicants on populations: A qualitative approach Ⅱ. First order kinetics, J. Math. Biology, 18 (1983), 25-37.   Google Scholar

[5]

J. He and K. Wang, The survival analysis for a population in a polluted environment, Nonlinear Analysis: Real World Applications, 10 (2009), 1555-1571.  doi: 10.1016/j.nonrwa.2008.01.027.  Google Scholar

[6]

B. Huang and J. Xing, Dynamic modeling and prediction of cytotoxicity on microelectronic cell sensor array, The Canadian Journal of Chemical Engineering, 84 (2006), 393-405.   Google Scholar

[7]

Q. Huang, L. Parshotam, H. Wang, C. Bampfylde and M. Lewis, A model for the impact of contaminants on fish population dynamics, Journal of Theoretical Biology, 334 (2013), 71-79.  doi: 10.1016/j.jtbi.2013.05.018.  Google Scholar

[8]

F. Ibrahim, B. Huang, J. Xing and S. Gabos, Early determination of toxicant concentration in water supply using MHE, Water Research, 44 (2010), 3252-3260.   Google Scholar

[9]

A.M. Jarrett, Y. Liu, N. Cogan and M.Y. Hussaini, Global sensitivity analysis used to interpret biological experimental results, Journal of Mathematical Biology, 71 (2015), 151-170.  doi: 10.1007/s00285-014-0818-3.  Google Scholar

[10]

J. Jiao, W. Long and L. Chen, A single stage-structured population model with mature individuals in a polluted environment and pulse input of environmental toxin, Nonlinear Analysis: Real World Applications, 10 (2009), 3073-3081.  doi: 10.1016/j.nonrwa.2008.10.007.  Google Scholar

[11]

S. Julier, J. Uhlmann and H. Durrant-White, A new method for the nonlinear transformation of means and covariances in filters and estimators, IEEE Trans. Aut. Control, 45 (2000), 477-482.  doi: 10.1109/9.847726.  Google Scholar

[12]

S. Julier, J. Uhlmann and H. Durrant-Whyte, A new approach for filtering nonlinear systems, in American Control Conference, Seattle, Washington, 1995,1628–1632. doi: 10.1109/ACC.1995.529783.  Google Scholar

[13]

A. Kiparissides, S. Kucherenko, A. Mantalaris and E.N. Pistikopoulos, Global sensitivity analysis challenges in biological systems modeling, Industrial & Engineering Chemistry Research, 48 (2009), 7168-7180.  doi: 10.1021/ie900139x.  Google Scholar

[14]

K. Kothawad, A. Pathan and M. Logad, Evaluation of in vitro anti-cancer activity of fruit lagenaria siceraria against MCF7, HOP62 and DU145 cell line, Int. J. Pharm. & Technol, 4 (2012), 3909-4392.   Google Scholar

[15]

M. Liu and K. Wang, Survival analysis of stochastic single-species population models in polluted environments, Ecological Modeling, 220 (2009), 1347-1357.   Google Scholar

[16]

M. Liu, K. Wang and X. Liu, Long term behaviors of stochastic single-species growth models in a polluted environment, Applied Mathematical Modelling, 35 (2011), 752-762.  doi: 10.1016/j.apm.2010.07.031.  Google Scholar

[17]

X. Meng and D. Van Dyk, The EM algorithm -an old folk-song to a fast new tune, J.R. Statist. Soc.B, 59 (1997), 511-567.  doi: 10.1111/1467-9868.00082.  Google Scholar

[18]

R. Neal and G. Hinton, A view of the EM algorithm that justifies incremental, sparse, an other variants, in Learning in Graphical Models (ed. M. Jordan), 89 (1998), 355-368. doi: 10.1007/978-94-011-5014-9_12.  Google Scholar

[19]

T. Pan, B. Huang, W. Zhang, S. Gabos, D. Huang and V. Devendran, Cytotoxicity assessment based on the AUC50 using multi-concentration time-dependent cellular response curves, Anal. Chim. Acta, 764 (2013), 44-52.   Google Scholar

[20]

T. Pan, S. Khare, F. Ackah, B. Huang, W. Zhang, S. Gabos, C. Jin and M. Stampfl, In vitro cytotoxicity assessment based on KC50 with real-time cell analyzer (RTCA) assay, Comp. Biol. Chem., 47 (2013), 113-120.   Google Scholar

[21]

L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 2001. doi: 10.1007/978-1-4613-0003-8.  Google Scholar

[22]

R. Shumway and D. Stoffer, An approach to time series smoothing and forecasting using the EM algorithm, J. Time Ser. Anal., 3 (1982), 253-264.  doi: 10.1111/j.1467-9892.1982.tb00349.x.  Google Scholar

[23]

I.M. Sobol, Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates, Mathematics and Computers in Simulation, 55 (2001), 271-280.  doi: 10.1016/S0378-4754(00)00270-6.  Google Scholar

[24]

H. Thieme, Mathematics in Population Biology, Princeton Series in theoretical and Computational Biology., 2003  Google Scholar

[25]

E. A. Wan, R. Van der Merwe and A. T. Nelson, Dual estimation and the unscented transformation, in Advances in Neural Information Processing Systems (ed. M. I. J. et al.), MIT Press, 2000. Google Scholar

[26]

C. Wu, On the convergence properties of the EM algorithm, The Annals of Statistics, 11 (1983), 95-103.  doi: 10.1214/aos/1176346060.  Google Scholar

[27]

Z. Xi, S. Khare, A. Cheung, B. Huang, T. Pan, W. Zhang, F. Ibrahim, C. Jin and S. Gabos, Mode of action classification of chemicals using multi-concentration time-dependent cellular response profiles, Comp. Biol. Chem., 49 (2014), 23-35.  doi: 10.1016/j.compbiolchem.2013.12.004.  Google Scholar

[28]

J. Xing, L. Zhu, S. Gabos and L. Xie, Microelectronic cell sensor assay for detection of cytotoxicity and prediction of acute toxicity, Toxicology in Vitro, 20 (2006), 995-1004.  doi: 10.1016/j.tiv.2005.12.008.  Google Scholar

[29]

M. Zhang, D. Aguilera, C. Das, H. Vasquez, P. Zage, V. Gopalakrishnan and J. Wolff, Measuring cytotoxicity: A new perspective on LC50, Anticancer Res., 27 (2007), 35-38.   Google Scholar

Figure 1.  TCRCs for (a) PF431396 and (b) monastrol
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Figure 2.  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) $CE(0)>\frac{\beta\eta_1^2}{\alpha\lambda_1^2}=6.51$
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Figure 3.  The separation between persistence and extinction according to the initial values $n(0)$ and $CE(0)$, red $*$: persistence; blue $\circ$: extinction
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Figure 4.  Negative control data fitted by logistic model, dot: experimental data, line: logistic model
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Figure 5.  Smooth spline approximation, dot: experimental data, line: smooth spline
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Figure 6.  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
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Figure 7.  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
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Figure 8.  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
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Figure 10.  (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
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Figure 11.  (a) Experimental TCRCs for ABT888 for CE(0)=308uM, 103uM, 34uM (b) Expected cell index and probability of extinction for different concentrations for ABT888
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Figure 9.  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
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Figure 12.  The first order GSA indices ranking for PF431396 (higher rank means more sensitive)
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Figure 13.  The first order GSA indices ranking for ABT888 (higher rank means more sensitive)
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Figure 14.  Network graph visualizing the second order GSA indices for (a) PF431396 with CE(0)=10uM (b) ABT888 with CE(0)=400uM
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Table 1.  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
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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
Table 2.  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 $\hat{E}$
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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 $\hat{E}$
Table 3.  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
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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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