Research article Special Issues

Calculating all multiple parameter solutions of ODE models to avoid biological misinterpretations

  • Received: 11 January 2019 Accepted: 05 July 2019 Published: 11 July 2019
  • Biological system's dynamics are increasingly studied with nonlinear ordinary differential equations, whose parameters are estimated from input/output experimental data. Structural identifiability analysis addresses the theoretical question whether the inverse problem of recovering the unknown parameters from noise-free data is uniquely solvable (global), or if there is a finite (local), or an infinite number (non identifiable) of parameter values that generate identical input/output trajectories. In contrast, practical identifiability analysis aims to assess whether the experimental data provide information on the parameter estimates in terms of precision and accuracy. A main difference between the two identifiability approaches is that the former is mostly carried out analytically and provides exact results at a cost of increased computational complexity, while the latter is usually numerically tested by calculating statistical confidence regions and relies on decision thresholds. Here we focus on local identifiability, a critical issue in biological modeling. This is the case when a model has multiple parameter solutions which equivalently describe the input/output data, but predict different behaviours of the unmeasured variables, often those of major interest. We present theoretical background and applications to locally identifiable ODE models described by rational functions. We show how structural identifiability analysis completes the practical identifiability results. In particular we propose an algorithmic approach, implemented with our software DAISY, to calculate all numerical parameter solutions and to predict the corresponding behaviour of the unmeasured variables, which otherwise would remain hidden. A case study of a locally identifiable HIV model shows that one should be aware of the presence of multiple parameter solutions to comprehensively describe the biological system and avoid biological misinterpretation of the results.

    Citation: Maria Pia Saccomani, Karl Thomaseth. Calculating all multiple parameter solutions of ODE models to avoid biological misinterpretations[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 6438-6453. doi: 10.3934/mbe.2019322

    Related Papers:

  • Biological system's dynamics are increasingly studied with nonlinear ordinary differential equations, whose parameters are estimated from input/output experimental data. Structural identifiability analysis addresses the theoretical question whether the inverse problem of recovering the unknown parameters from noise-free data is uniquely solvable (global), or if there is a finite (local), or an infinite number (non identifiable) of parameter values that generate identical input/output trajectories. In contrast, practical identifiability analysis aims to assess whether the experimental data provide information on the parameter estimates in terms of precision and accuracy. A main difference between the two identifiability approaches is that the former is mostly carried out analytically and provides exact results at a cost of increased computational complexity, while the latter is usually numerically tested by calculating statistical confidence regions and relies on decision thresholds. Here we focus on local identifiability, a critical issue in biological modeling. This is the case when a model has multiple parameter solutions which equivalently describe the input/output data, but predict different behaviours of the unmeasured variables, often those of major interest. We present theoretical background and applications to locally identifiable ODE models described by rational functions. We show how structural identifiability analysis completes the practical identifiability results. In particular we propose an algorithmic approach, implemented with our software DAISY, to calculate all numerical parameter solutions and to predict the corresponding behaviour of the unmeasured variables, which otherwise would remain hidden. A case study of a locally identifiable HIV model shows that one should be aware of the presence of multiple parameter solutions to comprehensively describe the biological system and avoid biological misinterpretation of the results.


    加载中


    [1] J. DiStefano, Dynamic System Biology Modeling and Simulation, 1st ed., Academic Press, Elsevier, USA, 2014.
    [2] L. Ljung, System Identification - Theory For the User, 2nd ed., PTR, Prentice Hall, Upper Saddle River, N.J., USA, 1999.
    [3] E.D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd ed., Springer, New York, 1998.
    [4] L. Ljung and S. T. Glad, On global identifiability for arbitrary model parameterizations, Automatica, 30(1994), 265–276.
    [5] A. Raue, J. Karlsson, M. P. Saccomani, et al., Comparison of approaches for parameter identifiability analysis of biological systems, Bioinformatics, 30(2014), 1440–1448.
    [6] M. P. Saccomani, S. Audoly and L. D'Angiò, Parameter identifiability of nonlinear systems: the role of initial conditions, Automatica, 39(2003), 619–632.
    [7] J. D. Stigter and J. Molenaar, A fast algorithm to assess local structural identifiability, Automatica, 58(2015), 118–124.
    [8] H. Hong, A. Ovchinnikov, G. Pogudin, et al., SIAN: software for structural identifiability analysis of ODE models, Bioinformatics, (2019), bty1069.
    [9] T. S. Ligon, F. Fröhlich, O. T. Chis, et al., GenSSI 2.0: multi-experiment structural identifiability analysis of SBML models, Bioinformatics, 34(2018), 1421–1423.
    [10] A. F. Villaverde, A. Barreiro and A. Papachristodoulou, Structural identifiability of dynamic systems biology models, PLoS Comput. Biol., 12(2016), e1005153.
    [11] M. Anguelova, J. Karlsson and M. Jirstrand, Minimal output sets for identifiability, Math. Biosci., 239(2012), 139–153.
    [12] A. Raue, C. Kreutz, T. Maiwald, et al., Addressing Parameter Identifiability by Model-based Experimentation, IET Syst. Biol., 5(2011), 120–130.
    [13] M. Rodriguez-Fernandez, M. Rehberg, A. Kremling, et al., Simultaneous model discrimination and parameter estimation in dynamic models of cellular systems, BMC Syst. Biol., 7(2013), 1–14.
    [14] T. R. B. Grandjean, M. J. Chappell, J. W. T. Yates, et al., Structural identifiability analysis of candidate models for in vitro Pitavastatin hepatic uptake, European J. Pharmac. Sci., 46(2013), 259–271.
    [15] D. L. I. Janzén, L. Bergenholm, M. Jirstrand, et al., Parameter identifiability of fundamental pharmacodynamic models, Front. Physiol., 7(2016), 590.
    [16] H. Klett, M. Rodriguez-Fernandez, S. Dineen, et al., Modeling the inflammatory response in the hypothalamus ensuing heat stroke: Iterative cycle of model calibration, identifiability analysis, experimental design and data collection, Math. Biosci., 260(2015), 35–46.
    [17] M. P. Saccomani, An effective automatic procedure for testing parameter identifiability of HIV/AIDS models, B. Math. Biol., 73(2011), 1734–1753.
    [18] K. Thomaseth, J. J. Batzel, M. Bachar, et al., Parameter estimation of a model for baroreflex control of unstressed volume, in Mathematical Modeling and Validation in Physiology (eds. J.J. Batzel, M. Bachar, F. Kappel), Springer-Verlag, Berlin, (2013), 215–246.
    [19] J. A. Egea, D. Henriques, T. Cokelaer, et al., MEIGO: an open-source software suite based on metaheuristics for global optimization in systems biology and bioinformatics, BMC Bioinformatics, 15(2014), 136.
    [20] T. Maiwald and J. Timmer, Dynamical Modeling and Multi-Experiment Fitting with PottersWheel, Bioinformatics, 24(2008), 2037–2043.
    [21] G. Bellu, M. P. Saccomani, S. Audoly, et al., DAISY: A new software tool to test global identifiability of biological and physiological systems, Comp. Meth. Prog. Biom., 88(2007), 52–61.
    [22] J. F. Ritt, Differential Algebra, Providence, RI: American Mathematical Society, (1950).
    [23] S. Audoly, G. Bellu, L. D'Angiò, et al., Global identifiability of nonlinear models of biological systems, IEEE Transact. Biomed. Eng., 48(2001), 55–65.
    [24] K. Thomaseth and M. P. Saccomani, Local identifiability analysis of nonlinear ODE models: how to determine all candidate solutions, IFAC PapersOnLine, 51(2018), 529–534.
    [25] P. Stapor, D. Weindl, B. Ballnus, et al., PESTO Parameter EStimation TOolbox, Bioinformatics, 34(2008), 705–707.
    [26] M. P. Saccomani and K. Thomaseth, Structural vs Practical Identifiability of Nonlinear Differential Equation Models in Systems Biology, in Dynamics of Mathematical Models in Biology (eds. A. Rogato, V. Zazzu, M.R. Guarracino), Springer International Publishing, Switzerland, (2016), 31–42.
    [27] J. P. Norton, An Investigation of the Sources of Nonuniqueness in Deterministic Identifiability, Math. Biosci., 60(1982), 89–108.
    [28] A. S. Perelson, D. E. Kirschner and R. DeBoer, Dynamics of HIV Infection of CD4+ T cells, Math. Biosci., 114(1993), 81–125.
    [29] R. Ferrentino and C. Boniello, On the Well-Posedness for Optimization Problems: A Theoretical Investigation, Appl. Math., 10(2019), 19–38.
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4927) PDF downloads(797) Cited by(1)

Article outline

Figures and Tables

Figures(4)  /  Tables(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog