Efficient Optimization of Stimuli for Model-Based Design of Experiments to Resolve Dynamical Uncertainty

RESEARCH ARTICLE Efficient Optimization of Stimuli for Model-Based Design of Experiments to Resolve Dynamical Uncertainty Thembi Mdluli1*, Gregery T. Buzzard2, Ann E. Rundell1* 1 Weldon School of Biomedical Engineering, Purdue University, West Lafayette, Indiana, United States of America, 2 Mathematics Department, Purdue University, West Lafayette, Indiana, United States of America * (TM); (AR) Abstract a11111 OPEN ACCESS Citation: Mdluli T, Buzzard GT, Rundell AE (2015) Efficient Optimization of Stimuli for Model-Based Design of Experiments to Resolve Dynamical Uncertainty. PLoS Comput Biol 11(9): e1004488. doi:10.1371/journal.pcbi.1004488 Editor: Sergei L. Kosakovsky Pond, University of California San Diego, UNITED STATES Received: October 24, 2014 Accepted: August 5, 2015 Published: September 17, 2015 Copyright: © 2015 Mdluli et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. This model-based design of experiments (MBDOE) method determines the input magnitudes of an experimental stimuli to apply and the associated measurements that should be taken to optimally constrain the uncertain dynamics of a biological system under study. The ideal global solution for this experiment design problem is generally computationally intractable because of parametric uncertainties in the mathematical model of the biological system. Others have addressed this issue by limiting the solution to a local estimate of the model parameters. Here we present an approach that is independent of the local parameter constraint. This approach is made computationally efficient and tractable by the use of: (1) sparse grid interpolation that approximates the biological system dynamics, (2) representative parameters that uniformly represent the data-consistent dynamical space, and (3) probability weights of the represented experimentally distinguishable dynamics. Our approach identifies data-consistent representative parameters using sparse grid interpolants, constructs the optimal input sequence from a greedy search, and defines the associated optimal measurements using a scenario tree. We explore the optimality of this MBDOE algorithm using a 3-dimensional Hes1 model and a 19-dimensional T-cell receptor model. The 19-dimensional T-cell model also demonstrates the MBDOE algorithm’s scalability to higher dimensions. In both cases, the dynamical uncertainty region that bounds the trajectories of the target system states were reduced by as much as 86% and 99% respectively after completing the designed experiments in silico. Our results suggest that for resolving dynamical uncertainty, the ability to design an input sequence paired with its associated measurements is particularly important when limited by the number of measurements. Data Availability Statement: All relevant data are within the paper and its Supporting Information files. Funding: This research was supported in part by the NSF grant DMS-0900277. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. Author Summary Many mathematical models that have been developed for biological systems are limited because the complex systems are not well understood, the parameters are not known, and available data is limited and noisy. On the other hand, experiments to support model development are limited in terms of costs and time, feasible inputs and feasible PLOS Computational Biology | DOI:10.1371/journal.pcbi.1004488 September 17, 2015 1 / 23 Stimuli Optimization MBDOE measurements. MBDOE combines the mathematical models with experiment design to strategically design optimal experiments to obtain data that will contribute to the understanding of the systems. Our approach extends current capabilities of existing MBDOE techniques to make them more useful for scientists to resolve the trajectories of the system under study. It identifies the optimal conditions for stimuli and measurements that yield the most information about the system given the practical limitations. Exploration of the input space is not a trivial extension to MBDOE methods used for determining optimal measurements due to the nonlinear nature of many biological system models. The exploration of the system dynamics elicited by different inputs requires a computationally efficient and tractable approach. Our approach plans optimal experiments to reduce dynamical uncertainty in the output of selected target states of the biological system. Introduction Since experiments can be expensive and time consuming, it is important that they are planned to generate useful data. Traditional design of experiments is a well established field and has led to many advances in biology and medicine. The data obtained from strategically designed experiments has facilitated the creation of mathematical models that relate experimental stimuli to measurable outcomes. These models typically describe the system’s input-output relationship but fail to capture or encode knowledge of the system’s internal mechanisms and processes. Mechanistic and semi-mechanistic mathematical models encode the current understanding of the internal processes of the biological system even though many of these internal states or species are not directly measurable. These mechanistic models can be used to support optimal experiment design that considers the current knowledge of the system interactions and practical experimental constraints. In recent literature this type of experiment design has been referred to as model-based design of experiments (MBDOE). MBDOE produces experiments meant to reduce some measure of uncertainty in the associated model while respecting cost, time and resource constraints. Most MBDOE strategies can be categorized by three types of objectives: (1) reducing model parameter uncertainty [1–7], (2) discriminating among possible models [8–13], and (3) reducing dynamical uncertainty [14–17]. This work advances current abilities to design experiments to resolve the trajectories of target states of a biological system model, thereby reducing its dynamical uncertainty. Many of the MBDOE strategies that support reduction of parameter uncertainty and model discrimination rely on linear approximations that are locally optimal to design an experiment by optimizing a criterion of the Fisher Information Matrix (FIM) [15, 18–21]. Such techniques use the local sensitivities of parameters to design an optimal experiment which requires an initial estimate of the unknown parameters. Most biological system models are not well characterized, as data is limited and noisy, so initial estimates of the model parameters are inaccurate. Furthermore, biological models are typically (...truncated)