(411f) Recent Advances in Bounding Transient PDE Models with Parametric Uncertainty

Many systems engineering designs are deemed safety-critical and require strict guarantees of safety and performance. Therefore, it is necessary to identify the worst-case performance of such safety-critical systems for mitigating the impacts of uncertainty at the design stage [1, 2, 3, 4]. For example, in precision cancer therapy, physicians must guarantee that a drug's concentration in the bloodstream is never higher than a maximum safe level while ensuring that a minimum required dose is delivered to a tumor [5]. Approaching this robustness verification problem from a model-based perspective amounts to determining the realization(s) of uncertainty that result in a simulated (worst-case) violation of safety/performance constraints by a system model. For transient systems models, this exercise is equivalent to verifying feasibility of the reachable set. This problem is formulated mathematically as a global dynamic optimization problem with a performance/safety constraint as the objective and uncertain parameters as (bounded) decision variables.

Accounting for both dynamics and spatially-dependent behavior or phenomena in the systems models poses significant computational challenges as the corresponding robustness verification problem results in a nonconvex partial differential equation (PDE)-constrained optimization problem that must be solved to guaranteed global optimality. In turn, this requires the efficient calculation of rigorous bounds enclosing the reachable set; which remains an open area of research. The focus of our work in this paper is on solving robustness verification problems for systems with transient PDE models. Currently, several state-of-the-art algorithms have been developed for bounding systems of ordinary differential equation (ODE) initial value problems (IVPs). The common methods for bounding the reachable set include differential inequalities (DI) [6, 7], finite difference approximations [8, 9], and Taylor series expansion with remainder [8, 10]. However, so far these methods have only been aimed at ODEs and have limited applications to transient PDE systems [11]. Constructing valid enclosures of the parametric solutions of PDE systems with sufficient speed and accuracy remains a major challenge which must be overcome to guarantee safety and performance of safety-critical systems at the design stage. Thus, there
is a critical need to develop new bounding methods for PDE systems.

In this work, a new algorithm is developed for efficiently computing rigorous bounds on solutions of transient PDE models for all realizations of parametric uncertainty [12]. In this approach, we combine the finite difference method (FDM), interval arithmetic (IA) [13], affine arithmetic (AA) [14], and differential inequalities (DI) to construct rigorous and tight bounds on the reachable set of transient PDEs. Specifically, we reformulate a parametric PDE as a large coupled system of ODE-IVPs using the method of lines with the FDM applied to the spatial derivatives. We employ either IA or AA for bounding the spatial derivative terms. Then, we implement both continuous-time and discrete-time DI methods to construct bounds on the parametric solutions of the resulting ODE-IVP system. We demonstrate this approach on several examples and verify that this approach may greatly reduce the conservatism of bounds with the desired computational efficiency.

Finally, we implemented the new bounding procedures within the spatial branch-and-bound algorithm for deterministic global optimization of systems modeled as transient PDEs. An example of a mechanistic transport model for drug delivery in a tumor is introduced [15,16]. This example problem represents a current focus of precision neoadjuvant therapy for improving metastatic breast cancer patient outcomes [16]. Physiological parameters representing the mass transfer resistances across vascular walls and within solid tumors are considered uncertain within specified error bounds. A global optimization problem is formulated to determine if a minimum dose of nanoparticle-encapsulated chemotherapeutic is accumulated within the tumor under the worst-case mass transport conditions. The solution of this problem represents the critical next step towards our future goal of determining a patient-specific safe neoadjuvant therapy for optimal patient outcomes. The results of this work are broadly applicable across a spectrum of domains utilizing PDE models in the design of safety-critical systems under uncertainty.

References

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[2] M. D. Stuber and P. I. Barton, “Robust simulation and design using semi-infinite programs with implicit functions,” International Journal of Reliability and Safety, vol. 5, no. 3-4, pp. 378-397, 2011.

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[12] C. Wang and M. D. Stuber, “Robust simulation of transient PDE models under uncertainty,” in AIChE Annual Meeting 2019, Orlando, 2019.

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[16] J. D. Martin, M. Panagi, C. Wang, T. T. Khan, M. R. Martin, C. Voutouri, K. Toh, Papageorgis, F. Mpekris, C. Polydorou, et al., “Dexamethasone increases cisplatin-loaded nanocarrier delivery and efficacy in metastatic breast cancer by normalizing the tumor microenvironment," ACS nano, 2019.