Estimating parameters of chaotic systems: chaos synchronization combined with Nelder-Mead search
Nikhil Pillai, Robert Bies, Sorell L Schwartz, Aris Dokoumetzidis, Morgan Craig, Immanuel Freedman
University at Buffalo, University at Buffalo, Georgetown University, University of Athens, Harvard University, Freedman Patent
Objectives: Deterministic chaos is a prominent feature of many biological systems. We compare an adaptive chaos synchronization (ACS)  method with an Extended Least Squares (ELS) method. We highlight challenges for the ELS method when experimental data are sparse and noisy and demonstrate advantages of ACS using the well-known Kirschner-Panetta (KP) tumor model of IL-2 immunotherapy, for which we explore steady states without treatment terms. The governing differential equations of the KP model are provided below.
dE/dT = c*T - mu2*E + p1*E*IL/(g1+IL) + s1 (1)
dT/dt = r2*(1-b*T)*T - a*E*T/(g2+T) (2)
dIL/dt = p2*E*T/(g3+T) -mu3*IL +s2 (3)
with initial conditions E(0)=E0, T(0)=T0 and IL(0)=IL0, where E, T and IL denote the concentration of effector cells, tumor cells and IL-2 cells respectively. Here, c denotes the antigenicity of the tumor, the term p1*E*IL/(g1+IL) models the stimulation of effector cells by IL-2, s1 represents the external source of effector cells (treatment term), r2*(1-b*T)*T represents the (logistic) growth of tumor cells, a denotes the loss of tumor cells by interactions with the immune system, p2*E*T/(g3+T) models the stimulation of IL-2 by the effector cells, mu3 denotes the IL-2 degradation rate, and finally s2 denotes an additional treatment term for the external input of IL-2 . As in , we explore the steady states without the treatment terms thus, in what follows, we set s1=s2=0.
Methods: We analyzed the structural identifiability of the K-P model using the GenSSI software . We applied ACS to track the system and estimate parameters that enter in a linear fashion, while using Nelder-Mead search to estimate parameters that enter in a nonlinear fashion, the objective function as Root Mean Square Error (RMSE) between the observed and predicted concentrations. We compared the performance of the ACS to the ELS method using Nelder-Mead search alone for sparse and noisy simulated data. The noisy data were simulated by adding 20% proportional error to the true simulated observations.
Results: All parameters of the KP model are structurally locally identifiable. We estimated the tumor antigenicity c and growth parameters (r2, b) with all other parameters held fixed since we contend there is rationale for obtaining the other parameters from literature or via in vitro studies [2,4]. We found that the ELS method was unable to accurately estimate parameters, resulting in highly discordant predictions versus the observations. The ACS method yielded estimates very close to nominal and the predictions closely matched the simulated observations with low percent bias. The parameters r2 and c had percent bias of 7.21% and 6.8% for sparse noiseless data and 1.05% and 1.4% for noisy data with 20% proportional error respectively. The growth parameter b had percent bias of 80% for noiseless data and 120% for sparse noisy data respectively. A sensitivity analysis for parameter b was performed and it was observed that the predicted concentrations are insensitive to fluctuations in parameter b. By comparison, the percent biases of parameter estimates was very large for the ELS method, in the order of 1000.
Conclusions: Our analysis supports the suggestion that deterministic chaotic systems are well estimated using a deterministic approach and demonstrates that the ACS method combined with Nelder-Mead search is a highly effective and robust method for estimating the parameters of an exemplary chaotic system of noisy and sparsely sampled Ordinary Differential Equations.
 Huang, D., Synchronization-based estimation of all parameters of chaotic systems from time series. Physical Review E, 2004. 69(6): p. 067201.
 Kirschner, D., Panetta, J.C., Modelling immunotherapy of the tumor-immune interaction, journal of Mathematical Biology, 1998, 37(3), 235-252.
 Chis, O., Banga, J.R., Canto, E.B., GenSSI: a software toolbox for structural identifiability analysis of biological models, Bioinformatics, 2011 27(18): 2610–2611.
 Rosenberg, S., II-2: the first effective immunotherapy for human cancer, The Journal of Immunology, 192 (12)(2014)5451-5458.