Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

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

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.

  Article Metrics

Keywords Mathematical model; cytotoxicity; parameter estimation; persistence

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


  • [1] C. Biernacki,G. Celeux,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.
  • [2] F. Cannavó, Sensitivity analysis for volcanic source modeling quality assessment and model selection, Computers & Geosciences,, 44 (2012): 52-59.
  • [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.
  • [4] T. Hallam,C. Clark,G. Jordan, Effects of toxicants on populations: A qualitative approach Ⅱ. First order kinetics, J. Math. Biology, 18 (1983): 25-37.
  • [5] J. He,K. Wang, The survival analysis for a population in a polluted environment, Nonlinear Analysis: Real World Applications, 10 (2009): 1555-1571.
  • [6] B. Huang,J. Xing, Dynamic modeling and prediction of cytotoxicity on microelectronic cell sensor array, The Canadian Journal of Chemical Engineering, 84 (2006): 393-405.
  • [7] Q. Huang,L. Parshotam,H. Wang,C. Bampfylde,M. Lewis, A model for the impact of contaminants on fish population dynamics, Journal of Theoretical Biology, 334 (2013): 71-79.
  • [8] F. Ibrahim,B. Huang,J. Xing,S. Gabos, Early determination of toxicant concentration in water supply using MHE, Water Research, 44 (2010): 3252-3260.
  • [9] A.M. Jarrett,Y. Liu,N. Cogan,M.Y. Hussaini, Global sensitivity analysis used to interpret biological experimental results, Journal of Mathematical Biology, 71 (2015): 151-170.
  • [10] J. Jiao,W. Long,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.
  • [11] S. Julier,J. Uhlmann,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.
  • [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.
  • [13] A. Kiparissides,S. Kucherenko,A. Mantalaris,E.N. Pistikopoulos, Global sensitivity analysis challenges in biological systems modeling, Industrial & Engineering Chemistry Research, 48 (2009): 7168-7180.
  • [14] K. Kothawad,A. Pathan,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.
  • [15] M. Liu,K. Wang, Survival analysis of stochastic single-species population models in polluted environments, Ecological Modeling, 220 (2009): 1347-1357.
  • [16] M. Liu,K. Wang,X. Liu, Long term behaviors of stochastic single-species growth models in a polluted environment, Applied Mathematical Modelling, 35 (2011): 752-762.
  • [17] X. Meng,D. Van Dyk, The EM algorithm -an old folk-song to a fast new tune, J.R. Statist. Soc.B, 59 (1997): 511-567.
  • [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.
  • [19] T. Pan,B. Huang,W. Zhang,S. Gabos,D. Huang,V. Devendran, Cytotoxicity assessment based on the AUC50 using multi-concentration time-dependent cellular response curves, Anal. Chim. Acta, 764 (2013): 44-52.
  • [20] T. Pan,S. Khare,F. Ackah,B. Huang,W. Zhang,S. Gabos,C. Jin,M. Stampfl, In vitro cytotoxicity assessment based on KC50 with real-time cell analyzer (RTCA) assay, Comp. Biol. Chem., 47 (2013): 113-120.
  • [21] L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 2001.
  • [22] R. Shumway,D. Stoffer, An approach to time series smoothing and forecasting using the EM algorithm, J. Time Ser. Anal., 3 (1982): 253-264.
  • [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.
  • [24] H. Thieme, Mathematics in Population Biology, Princeton Series in theoretical and Computational Biology., 2003
  • [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.
  • [26] C. Wu, On the convergence properties of the EM algorithm, The Annals of Statistics, 11 (1983): 95-103.
  • [27] Z. Xi,S. Khare,A. Cheung,B. Huang,T. Pan,W. Zhang,F. Ibrahim,C. Jin,S. Gabos, Mode of action classification of chemicals using multi-concentration time-dependent cellular response profiles, Comp. Biol. Chem., 49 (2014): 23-35.
  • [28] J. Xing,L. Zhu,S. Gabos,L. Xie, Microelectronic cell sensor assay for detection of cytotoxicity and prediction of acute toxicity, Toxicology in Vitro, 20 (2006): 995-1004.
  • [29] M. Zhang,D. Aguilera,C. Das,H. Vasquez,P. Zage,V. Gopalakrishnan,J. Wolff, Measuring cytotoxicity: A new perspective on LC50, Anticancer Res., 27 (2007): 35-38.


This article has been cited by

  • 1. Anna Poma, Sabrina Colafarina, Eleonora Aruffo, Osvaldo Zarivi, Antonella Bonfigli, Sebastiano Di Bucchianico, Piero Di Carlo, Maria Rosaria Scarfi, Effects of ozone exposure on human epithelial adenocarcinoma and normal fibroblasts cells, PLOS ONE, 2017, 12, 9, e0184519, 10.1371/journal.pone.0184519
  • 2. Giuseppe Chichiriccò, Claudio Ferrante, Luigi Menghini, Lucia Recinella, Sheila Leone, Annalisa Chiavaroli, Luigi Brunetti, Simonetta Di Simone, Maurizio Ronci, Pierpaolo Piccone, Barbara Lanza, Stefania Cesa, Annamaria Poma, Giulia Vecchiotti, Giustino Orlando, Crocus sativus by-products as sources of bioactive extracts: Pharmacological and toxicological focus on anthers, Food and Chemical Toxicology, 2019, 126, 7, 10.1016/j.fct.2019.01.040
  • 3. Vavrinevych O.P., Antonenko A.M., Korshun M.M., Omelchuk S.T., Hygienic substantiation of calculating models for fungicides of different classes toxicity depend on their physical and chemical properties prognosis, Environment & Health, 2017, 4 (84), 52, 10.32402/dovkil2017.04.052
  • 4. A. M. Antonenko, O. P. Vavrinevych, S. T. Omelchuk, B. I. Shpak, Hygienic substantiation of calculating models for predicting toxicity of different classes insecticides (first part)., Medicni perspektivi (Medical perspectives), 2019, 24, 3, 106, 10.26641/2307-0404.2019.3.181892

Reader Comments

your name: *   your email: *  

Copyright Info: 2017, Cristina Anton, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved