Erica M. Rutter

Prediction and Computer Model Calibration Using Outputs From Multifidelity Simulators

Computer simulators are widely used to describe and explore physical processes. In some cases, several simulators are available, each with a different degree of fidelity, for this task. In this work, we combine field observations and model runs from deterministic multifidelity computer simulators to build a predictive model for the real process. The resulting model can be used to perform sensitivity analysis for the system, solve inverse problems, and make predictions. Our approach is Bayesian and is illustrated through a simple example, as well as a real application in predictive science at the Center for Radiative Shock Hydrodynamics at the University of Michigan. The Matlab code that is used for the analyses is available from the online supplementary materials.

Calibrating a large computer experiment simulating radiative shock hydrodynamics

We consider adapting a canonical computer model calibration apparatus, involving coupled Gaussian process (GP) emulators, to a computer experiment simulating radiative shock hydrodynamics that is orders of magnitude larger than what can typically be accommodated. The conventional approach calls for thousands of large matrix inverses to evaluate the likelihood in an MCMC scheme. Our approach replaces that costly ideal with a thrifty take on essential ingredients, synergizing three modern ideas in emulation, calibration and optimization: local approximate GP regression, modularization, and mesh adaptive direct search. The new methodology is motivated both by necessity - considering our particular application - and by recent trends in the supercomputer simulation literature. A synthetic data application allows us to explore the merits of several variations in a controlled environment and, together with results on our motivating real-data experiment, lead to noteworthy insights into the dynamics of radiative shocks as well as the limitations of the calibration enterprise generally.

Spline-Based Emulators for Radiative Shock Experiments With Measurement Error

Radiation hydrodynamics and radiative shocks are of fundamental interest in the high-energy-density physics research due to their importance in understanding astrophysical phenomena such as supernovae. In the laboratory, experiments can produce shocks with fundamentally similar physics on reduced scales. However, the cost and time constraints of the experiment necessitate use of a computer algorithm to generate a reasonable number of outputs for making valid inference. We focus on modeling emulators that can efficiently assimilate these two sources of information accounting for their intrinsic differences. The goal is to learn how to predict the breakout time of the shock given the information on associated parameters such as pressure and energy. Under the framework of the Kennedy–O’Hagan model, we introduce an emulator based on adaptive splines. Depending on the preference of having an interpolator for the computer code output or a computationally fast model, a couple of different variants are proposed. Those choices are shown to perform better than the conventional Gaussian-process-based emulator and a few other choices of nonstationary models. For the shock experiment dataset, a number of features related to computer model validation such as using interpolator, necessity of discrepancy function, or accounting for experimental heterogeneity are discussed, implemented, and validated for the current dataset. In addition to the typical Gaussian measurement error for real data, we consider alternative specifications suitable to incorporate noninformativeness in error distributions, more in agreement with the current experiment. Comparative diagnostics, to highlight the effect of measurement error model on predictive uncertainty, are also presented. Supplementary materials for this article are available online.

Mathematically modeling the biological properties of gliomas: A review.

Although mathematical modeling is a mainstay for industrial and many scientific studies, such approaches have found little application in neurosurgery. However, the fusion of biological studies and applied mathematics is rapidly changing this environment, especially for cancer research. This review focuses on the exciting potential for mathematical models to provide new avenues for studying the growth of gliomas to practical use. In vitro studies are often used to simulate the effects of specific model parameters that would be difficult in a larger-scale model. With regard to glioma invasive properties, metabolic and vascular attributes can be modeled to gain insight into the infiltrative mechanisms that are attributable to the tumors aggressive behavior. Morphologically, gliomas show different characteristics that may allow their growth stage and invasive properties to be predicted, and models continue to offer insight about how these attributes are manifested visually. Recent studies have attempted to predict the efficacy of certain treatment modalities and exactly how they should be administered relative to each other. Imaging is also a crucial component in simulating clinically relevant tumors and their influence on the surrounding anatomical structures in the brain.

Mathematical Analysis of Glioma Growth in a Murine Model

Five immunocompetent C57BL/6-cBrd/cBrd/Cr (albino C57BL/6) mice were injected with GL261-luc2 cells, a cell line sharing characteristics of human glioblastoma multiforme (GBM). The mice were imaged using magnetic resonance (MR) at five separate time points to characterize growth and development of the tumor. After 25 days, the final tumor volumes of the mice varied from 12 mm3 to 62 mm3, even though mice were inoculated from the same tumor cell line under carefully controlled conditions. We generated hypotheses to explore large variances in final tumor size and tested them with our simple reaction-diffusion model in both a 3-dimensional (3D) finite difference method and a 2-dimensional (2D) level set method. The parameters obtained from a best-fit procedure, designed to yield simulated tumors as close as possible to the observed ones, vary by an order of magnitude between the three mice analyzed in detail. These differences may reflect morphological and biological variability in tumor growth, as well as errors in the mathematical model, perhaps from an oversimplification of the tumor dynamics or nonidentifiability of parameters. Our results generate parameters that match other experimental in vitro and in vivo measurements. Additionally, we calculate wave speed, which matches with other rat and human measurements.

Continuous structured population models for Daphnia magna

We continue our efforts in modeling Daphnia magna, a species of water flea, by proposing a continuously structured population model incorporating density-dependent and density-independent fecundity and mortality rates. We collected new individual-level data to parameterize the individual demographics relating food availability and individual daphnid growth. Our model is fit to experimental data using the generalized least-squares framework, and we use cross-validation and Akaike Information Criteria to select hyper-parameters. We present our confidence intervals on parameter estimates.

Initial conditions of radiative shock experimentsa)

We performed experiments at the Omega Laser Facility to characterize the initial, laser-driven state of a radiative shock experiment. These experiments aimed to measure the shock breakout time from a thin, laser-irradiated Be disk. The data are then used to inform a range of valid model parameters, such as electron flux limiter and polytropic γ, used when simulating radiative shock experiments using radiation hydrodynamics codes. The characterization experiment and the radiative shock experiment use a laser irradiance of ∼7 × 1014 W cm−2 to launch a shock in the Be disk. A velocity interferometer and a streaked optical pyrometer were used to infer the amount of time for the shock to move through the Be disk. The experimental results were compared with simulation results from the Hyades code, which can be used to model the initial conditions of a radiative shock system using the CRASH code.

Automated Object Tracing for Biomedical Image Segmentation Using a Deep Convolutional Neural Network

Convolutional neural networks (CNNs) have been used for fast and accurate segmentation of medical images. In this paper, we present a novel methodology that uses CNNs for segmentation by mimicking the human task of tracing object boundaries. The architecture takes as input a patch of an image with an overlay of previously traced pixels and the output predicts the coordinates of the next m pixels to be traced. We also consider a CNN architecture that leverages the output from another semantic segmentation CNN, e.g., U-net, as an auxiliary image channel. To initialize the trace path in an image, we use either locations identified as object boundaries with high confidence from a semantic segmentation CNN or a short manually traced path. By iterating the CNN output, our method continues the trace until it intersects with the beginning of the path. We show that our network is more accurate than the state-of-the-art semantic segmentation CNN on microscopy images from the ISBI cell tracking challenge. Moreover, our methodology provides a natural platform for performing human-in-the-loop segmentation that is more accurate than CNNs alone and orders of magnitude faster than manual segmentation.

Traveling Waves of a Go-or-Grow Model of Glioma Growth

Glioblastoma multiforme is a deadly brain cancer in which tumor cells excessively proliferate and migrate. The first mathematical models of the spread of gliomas featured reaction-diffusion equatio...

Estimating intratumoral heterogeneity from spatiotemporal data

Glioblastoma multiforme (GBM) is a malignant brain cancer with a tendency to both migrate and proliferate. We propose modeling GBM with heterogeneity in cell phenotypes using a random differential equation version of the reaction–diffusion equation, where the parameters describing diffusion (D) and proliferation (