This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1 Robust Calibration of High Dimension Nonlinear Dynamical Models for Omics Data: An Application in Cancer Systems Biology Fortunato Bianconi , Member, IEEE, Chiara Antonini, Member, IEEE, Lorenzo Tomassoni, Member, IEEE, and Paolo Valigi , Member, IEEE Abstract— In computational mathematical modeling of biological systems, most model parameters, such as initial con- ditions, kinetics, and scale factors, are usually unknown because they cannot be directly measured. Therefore, key issues in system identification of nonlinear systems are model calibration and identifiability analysis. Currently, existing methodologies for parameter estimation are divided in two classes: frequentist and Bayesian methods. The first optimize a cost function, while the second estimate the posterior distribution of parameters through different sampling techniques. However, when dealing with high-dimensional models, these methodologies suffer from an increasing computational cost due to the important volume of -omic data necessary to carry out reliable and robust solutions. Here, we present an innovative Bayesian method, called condi- tional robust calibration (CRC), for model calibration and identi- fiability analysis. The algorithm is an iterative procedure based on a uniform and joint perturbation of the parameter space. At each step the algorithm returns the probability density functions of all parameters that progressively shrink toward specific points in the parameter space. These distributions are estimated on parameter samples that guarantee a certain level of agreement between each observable and the corresponding in silico measure. We apply CRC to a nonlinear high-dimensional ordinary differential equations model representing the signaling pathway of p38MAPK in multiple myeloma. The available data set consists of time courses of proteomic cancerous data. We test CRC performances in comparison with profile-likelihood and approximate Bayesian computation sequential Monte Carlo. We obtain a more precise and robust solution with a reduced computational cost. Index Terms— Bayesian calibration, cancer systems biology, nonlinear model, omics data, parameter estimation, system identification. I. I NTRODUCTION B IOLOGICAL systems are naturally complex, since their behavior results from the interaction of numerous molec- ular components [1]. Molecules that govern the basic cell func- tions are connected in networks and pathways through nonlin- Manuscript received November 1, 2017; revised March 27, 2018; accepted May 23, 2018. Manuscript received in final form June 1, 2018. This work was supported by the Italian Association for Cancer Research under Grant 15713/2014. Recommended by Associate Editor B. Jayawardhana. (Corresponding author: Fortunato Bianconi.) The authors are with the Department of Engineering, University of Perugia, 06132 Perugia, Italy (e-mail: fortunato:bianconi@gmail:com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2018.2844362 ear reaction kinetics in which feedback loops play a key role. Moreover, cells communicate among each other to form higher organization levels, such as tissues and organs [2]. Thus, understanding the dynamic interplays in the cell and among the cells is of crucial importance in order to gain major insights about mechanisms of complicated diseases [3]. In the context of systems biology, mathematical, and computational model- ing are of central importance for reproducing the dynamic evolution of static pathways and for characterizing biologi- cal systems from a quantitative point of view [1], [2], [4]. Through control and system identification theory, mathemat- ical models allow for an abstract and formal representation of a natural system, while their computational implemen- tation permits the simulation of the temporal behavior of model variables, i.e., molecules participating to biochemical reactions [2], [4], [5]. Although a mathematical model is not an exact replica model, it can be used for making hypotheses and predictions on inter or intra-cellular processes [6]. Through in silico simulations of a model, it is possible to investigate how a biological process reacts to an external or internal signal, thus reducing the costs of expensive experiments [7]. The model building process in systems biology is a cycle of activities organized in the following way. First, it is necessary to understand the nature of the phenomenon of interest and to establish the objective and application of the model. Second, the optimal modeling technique should be chosen in order to build a first draft model, based also on the a priori biological knowledge. After that, unknown model parameters are estimated using the experimental data. Finally, the model is validated through new experiments, until it reaches the desired level of agreement with them [4]. Models that pass this iterative process are declared to be consistent with existing experimental evidence [8]. As a result, the generation of the experimental data is strictly related to the modeling process, since it represents the information that guides both model development and test [1], [2]. Cancer systems biology represents the application of sys- tems biology approach to the analysis of how the intracellular networks of normal cells are perturbed during carcinogen- esis to develop effective predictive models that can assist scientists and clinicians in the validations of new targeted 1063-6536 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.