J. T. Oden

• University of Texas at Austin

A key parameter of interest recovered from hyperpolarized (HP) MRI measurements is the apparent pyruvate-to-lactate exchange rate, kPL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \beg...

Formulating mathematical models that estimate tumor growth under therapy is vital for improving patient-specific treatment plans. In this context, we present our recent work on simulating non-small-scale cell lung cancer (NSCLC) in a simple, deterministic setting for two different patients receiving an immunotherapeutic treatment. At its core, our...

This chapter develops methods and computational infrastructures that comprise a Dynamic Data-Driven Applications Systems (DDDAS) framework for real-time monitoring and prediction of the behavior of complex physical systems. The developed DDDAS framework for materials damage prediction enables harnessing imaging data acquired in real-time to train p...

This work focuses on developing methods for approximating the solution operators of a class of parametric partial differential equations via neural operators. Neural operators have several challenges, including the issue of generating appropriate training data, cost-accuracy trade-offs, and nontrivial hyperparameter tuning. The unpredictability of...

Cancer is a disease driven by random DNA mutations and the interaction of many complex phenomena. To improve the understanding and ultimately find more effective treatments, researchers leverage computer simulations mimicking the tumor growth in silico. The challenge here is to account for the many phenomena influencing the disease progression and...

Cancer is a disease driven by random DNA mutations and the interaction of many complex phenomena. To improve the understanding and ultimately find more effective treatments, researchers leverage computer simulations mimicking the tumor growth in silico. The challenge here is to account for the many phenomena influencing the disease progression and...

Identifying parameters of computational models from experimental data, or model calibration, is fundamental for assessing and improving the predictability and reliability of computer simulations. In this work, we propose a method for Bayesian calibration of models that predict morphological patterns of diblock copolymer (Di-BCP) thin film self-asse...

Formulating tumor models that predict growth under therapy is vital for improving patient-specific treatment plans. In this context, we present our recent work on simulating non-small-scale cell lung cancer (NSCLC) in a simple, deterministic setting for two different patients receiving an immunotherapeutic treatment. At its core, our model consists...

We explore using neural operators, or neural network representations of nonlinear maps between function spaces, to accelerate infinite-dimensional Bayesian inverse problems (BIPs) with models governed by nonlinear parametric partial differential equations (PDEs). Neural operators have gained significant attention in recent years for their ability t...

In this work, a Bayesian model calibration framework is presented that utilizes goal-oriented a-posterior error estimates in quantities of interest (QoIs) for classes of high-fidelity models characterized by PDEs. It is shown that for a large class of computational models, it is possible to develop a computationally inexpensive procedure for calibr...

Current clinical decision-making in oncology relies on averages of large patient populations to both assess tumour status and treatment outcomes. However, cancers exhibit an inherent evolving heterogeneity that requires an individual approach based on rigorous and precise predictions of cancer growth and treatment response. To this end, we advocate...

Directed self-assembly (DSA) of block-copolymers (BCPs) is one of the most promising developments in the cost-effective production of nanoscale devices. The process makes use of the natural tendency for BCP mixtures to form nanoscale structures upon phase separation. The phase separation can be directed through the use of chemically patterned subst...

The Ohta--Kawasaki (OK) model is a phase field model that characterizes the self-assembly of diblock copolymers with a Ginzburg--Landau type free energy. The nonlocal Cahn--Hilliard equation, i.e., the H^{-1} gradient flow of the OK energy, is a commonly used approach for solving the OK model. However, such an approach is unnecessarily slow if one...

Hyperpolarized (HP) $^{13}$C-Pyruvate MR imaging provides unique information about metabolic alterations indicative of the aggressiveness of certain cancer types. The information content of the HP-MRI data is fundamentally limited by the physics and chemistry of specific processes that take place at an atomic scale during a well-defined chemical re...

We consider the Bayesian calibration of models describing the phenomenon of block copolymer (BCP) self-assembly using image data produced by microscopy or X-ray scattering techniques. To account for the random long-range disorder in BCP equilibrium structures, we introduce auxiliary variables to represent this aleatory uncertainty. These variables,...

In this work, a Bayesian model calibration framework is presented that utilizes goal-oriented a-posterior error estimates in quantities of interest (QoIs) for classes of high-fidelity models characterized by PDEs. It is shown that for a large class of computational models it is possible to develop a computationally inexpensive procedure for calibra...

In this work, we present mixed dimensional models for simulating blood flow and transport processes in breast tissue and the vascular tree supplying it. These processes are considered, to start from the aortic inlet to the capillaries and tissue of the breast. Large variations in biophysical properties and flow conditions exist in this system neces...

In this work, we present mixed dimensional models for simulating blood flow and transport processes in breast tissue and the vascular tree supplying it. These processes are considered, to start from the aortic inlet to the capillaries and tissue of the breast. Large variations in biophysical properties and flow conditions exist in this system neces...

Manufactured materials usually contain random imperfections due to the fabrication process, e.g., the 3D-printing, casting, etc. In this work, we present a new flexible class of digital surrogate models which imitate the manufactured material respecting the statistical features of random imperfections. The surrogate model is constructed as the leve...

In this work, we present a coupled 3D–1D model of solid tumor growth within a dynamically changing vascular network to facilitate realistic simulations of angiogenesis. Additionally, the model includes erosion of the extracellular matrix, interstitial flow, and coupled flow in blood vessels and tissue. We employ continuum mixture theory with stocha...

Tumor-associated vasculature is responsible for the delivery of nutrients, removal of waste, and allowing growth beyond 2–3 mm3. Additionally, the vascular network, which is changing in both space and time, fundamentally influences tumor response to both systemic and radiation therapy. Thus, a robust understanding of vascular dynamics is necessary...

In this work, we present and analyze a mathematical model for tumor growth incorporating ECM erosion, interstitial flow, and the effect of vascular flow and nutrient transport. The model is of phase-field or diffused-interface type in which multiple phases of cell species and other constituents are separated by smooth evolving interfaces. The model...

Current clinical decision-making in oncology relies on averages of large patient populations to both assess tumor status and treatment outcomes. However, cancers exhibit an inherent evolving heterogeneity that requires an individual approach based on rigorous and precise predictions of cancer growth and treatment response. To this end, we advocate...

In this work, we present a coupled 3D-1D model of tumor growth within a dynamically changing vascular network to facilitate realistic simulations of angiogenesis. Additionally, the model includes ECM erosion, interstitial flow, and coupled flow in vessels and tissue. We employ continuum mixture theory with stochastic Cahn--Hilliard type phase-field...

We consider a mixture-theoretic continuum model of the spread of COVID-19 in Texas. The model consists of multiple coupled partial differential reaction-diffusion equations governing the evolution of susceptible, exposed, infectious, recovered, and deceased fractions of the total population in a given region. We consider the problem of model calibr...

We propose a fast and robust scheme for the direct minimization of the Ohta-Kawasaki energy that characterizes the microphase separation of diblock copolymer melts. The scheme employs a globally convergent modified Newton method with line search which is shown to be mass-conservative, energy-descending, asymptotically quadratically convergent, and...

In this work, we present and analyze a mathematical model for tumor growth incorporating ECM erosion, interstitial flow, and the effect of vascular flow and nutrient transport. The model is of phase-field or diffused-interface type in which multiple phases of cell species and other constituents are separated by smooth evolving interfaces. The model...

We develop a general class of thermodynamically consistent, continuum models based on mixture theory with phase effects that describe the behavior of a mass of multiple interacting constituents. The constituents consist of solid species undergoing large elastic deformations and compressible viscous fluids. The fundamental building blocks framing th...

We present and analyze new multi-species phase-field mathematical models of tumor growth and ECM invasion. The local and nonlocal mathematical models describe the evolution of volume fractions of tumor cells, viable cells (proliferative and hypoxic cells), necrotic cells, and the evolution of matrix-degenerative enzyme (MDE) and extracellular matri...

A mathematical analysis of local and nonlocal phase-field models of tumor growth is presented that includes time-dependent Darcy–Forchheimer–Brinkman models of convective velocity fields and models of long-range cell interactions. A complete existence analysis is provided. In addition, a parameter-sensitivity analysis is described that quantifies t...

Random microstructures of heterogeneous materials play a crucial role in the material macroscopic behavior and in predictions of its effective properties. A common approach to modeling random multiphase materials is to develop so-called surrogate models approximating statistical features of the material. However, the surrogate models used in fatigu...

We present and analyze new multi-species phase-field mathematical models of tumor growth and ECM invasion. The local and nonlocal mathematical models describe the evolution of volume fractions of tumor cells, viable cells (proliferative and hypoxic cells), necrotic cells, and the evolution of MDE and ECM, together with chemotaxis, haptotaxis, apopt...

Methods for generating sequences of surrogates approximating fine scale models of two-phase random heterogeneous media are presented that are designed to adaptively control the modeling error in key quantities of interest (QoIs). For specificity, the base models considered involve stochastic partial differential equations characterizing, for exampl...

Whether the nom de guerre is Mathematical Oncology, Computational or Systems Biology, Theoretical Biology, Evolutionary Oncology, Bioinformatics, or simply Basic Science, there is no denying that mathematics continues to play an increasingly prominent role in cancer research. Mathematical Oncology—defined here simply as the use of mathematics in ca...

A mathematical analysis of local and nonlocal phase-field models of tumor growth is presented that includes time-dependent Darcy-Forchheimer-Brinkman models of convective velocity fields and models of long-range cell interactions. A complete existence analysis is provided. In addition, a parameter-sensitivity analysis is described that quantifies t...

Abstract Two of the central challenges of using mathematical models for predicting the spatiotemporal development of tumors is the lack of appropriate data to calibrate the parameters of the model, and quantitative characterization of the uncertainties in both the experimental data and the modeling process itself. We present a sequence of experimen...

Methods for generating sequences of surrogates approximating fine-scale models of two-phase random heterogeneous media are presented that are designed to adaptively control modeling error in key quantities of interest (QoIs). For specificity, the base models considered involve stochastic partial differential equations characterizing, for example, s...

The use of computational models and simulations to predict events that take place in our physical universe, or to predict the behaviour of engineered systems, has significantly advanced the pace of scientific discovery and the creation of new technologies for the benefit of humankind over recent decades, at least up to a point. That ‘point’ in rece...

Cancer results from a complex interplay of different biological, chemical, and physical phenomena that span a wide range of time and length scales. Computational modeling may help to unfold the role of multiple evolving factors that exist and interact in the tumor microenvironment. Understanding these complex multiscale interactions is a crucial st...

This paper describes versions of OPAL, the Occam-Plausibility Algorithm (Farrell et al. in J Comput Phys 295:189–208, 2015) in which the use of Bayesian model plausibilities is replaced with information-theoretic methods, such as the Akaike information criterion and the Bayesian information criterion. Applications to complex systems of coarse-grain...

The use of mathematical and computational models for reliable predictions of tumor growth and decline in living organisms is one of the foremost challenges in modern predictive science, as it must cope with uncertainties in observational data, model selection, model parameters, and model inadequacy, all for very complex physical and biological syst...

Most biological systems encountered in living organisms involve highly complex heterogeneous multi-component structures that exhibit different physical, chemical, and biological behavior at different spatial and temporal scales. The development of predictive mathematical and computational models of multiscale events in such systems is a major chall...

Introduction: We aim to develop a multiple spatio-temporal scale (MSTS) framework capable of bridging tissue, cellular, and subcellular scales. Our overarching hypothesis is that an accurate and precise MSTS model, initialized with patient-specific information, can dramatically outperform current standard-of-care for diagnosis and selecting therapy...

As a result of the highly heterogeneous tumor microenvironment and a series of complex phenomena that occur at different scales, several modeling approaches have emerged. The tumor growth models can be categorized by the behaviors they aim to capture, the hallmarks of cancer that are being considered, and the temporal and biological scales represen...

This paper presents general approaches for addressing some of the most important issues in predictive computational oncology concerned with developing classes of predictive models of tumor growth. First, the process of developing mathematical models of vascular tumors evolving in the complex, heterogeneous, macroenvironment of living tissue; second...

This chapter addresses core issues of convergence and transdisciplinary research and academic study. It describes one approach toward creating an infrastructure for transdisciplinary research and true interdisciplinary study within a contemporary university in the United States. Solutions for overcoming traditional obstacles to this type of program...

The use of coarse-grained approximations of atomic systems is the most common methods of constructing reduced-order models in computational science. However, the issue of central importance in developing these models is the accuracy with which they approximate key features of the atomistic system. Many methods have been proposed to calibrate coarse...

A general adaptive modeling algorithm for selection and validation of coarse-grained models of atomistic systems is presented. A Bayesian framework is developed to address uncertainties in parameters, data, and model selection. Algorithms for computing output sensitivities to parameter variances, model evidence and posterior model plausibilities fo...

New directions in medical and biomedical sciences have gradually emerged over recent years that will change the way diseases are diagnosed and treated and are leading to the redirection of medicine toward patient-specific treatments. We refer to these new approaches for studying biomedical systems as predictive medicine, a new version of medical sc...

The development of predictive computational models of tumor initiation, growth, and decline is faced with many formidable challenges. Phenomenological models which attempt to capture the complex interactions of multiple tissue and cellular species must cope with moving interfaces of heterogeneous media and the sprouting vascular structures due to a...

Carcinogenesis, as every biological process, is not purely deterministic as all systems are subject to random perturbations from the environment. In tumor growth models, the values of the parameters are subjected to many uncertainties that can arise from experimental variations or due to patient-specific data. The present work is devoted to the dev...

In the present study, a general dynamic data-driven application system (DDDAS) is developed for real-time monitoring of damage in composite materials using methods and models that account for uncertainty in experimental data, model parameters, and in the selection of the model itself. The methodology involves (i) data data from uniaxial tensile exp...

Coarse-grained models of atomic systems, created by aggregating groups of atoms into molecules to reduce the number of degrees of freedom, have been used for decades in important scientific and technological applications. In recent years, interest in developing a more rigorous theory for coarse graining and in assessing the predictivity of coarse-g...

In this paper we describe a stochastic dynamic data-driven application system (DDDAS) for monitoring, in real-time, material damage in aerospace structures. The work involves experiments, different candidate damage models, finite element discretization, Bayesian analysis of the candidate models, Bayesian filtering with the most plausible model, par...

We propose a virtual statistical validation process as an aid to the design of experiments for the validation of phenomenological models of the behavior of material bodies, with focus on those cases in which knowledge of the fabrication process used to manufacture the body can provide information on the micro-molecular-scale properties underlying m...

We address general approaches to the rational selection and validation of mathematical and computational models of tumor growth using methods of Bayesian inference. The model classes are derived from a general diffuse-interface, continuum mixture theory and focus on mass conservation of mixtures with up to four species. Synthetic data are generated...

The idea that one can possibly develop computational models that predict the emergence, growth, or decline of tumors in living tissue is enormously intriguing as such predictions could revolutionize medicine and bring a new paradigm into the treatment and prevention of a class of the deadliest maladies affecting humankind. But at the heart of this...

In this paper, we develop a thermodynamically consistent four-species model of tumor growth on the basis of the continuum theory of mixtures. Unique to this model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial...

A modern approach to mathematical modeling, featuring unique applications from the field of mechanics An Introduction to Mathematical Modeling: A Course in Mechanics is designed to survey the mathematical models that form the foundations of modern science and incorporates examples that illustrate how the most successful models arise from basic prin...

The idea of adaptive control of modeling error is expanded to include ideas of statistical calibration, validation, and uncertainty quantification. Copyright © 2010 John Wiley & Sons, Ltd.

A posteriori estimates of errors in quantities of interest are developed for the nonlinear system of evolution equations embodied in the Cahn–Hilliard model of binary phase transition. These involve the analysis of wellposedness of dual backward-in-time problems and the calculation of residuals. Mixed finite element approximations are developed and...

A new adaptive method based on an optimal control approach is proposed for adaptive modeling of an atomic-to-continuum coupling method constructed from the Arlequin framework. The coupling method provides for an approximation of the solution to a fully atomic model in which errors may arise due to the misplacement of the overlap region. The objecti...

The main objective of this work is to identify spurious effects and propose a corrective method when coupling a particle model involving long–range interaction potentials with a first gradient homogenized model using the Arlequin framework. The most significant spurious effects are generally created by the so-called ghost forces that arise in coupl...

A goal-oriented analysis of linear, stochastic advection–diffusion models is presented which provides both a method for solution verification as well as a basis for improving results through adaptation of both the mesh and the way random variables are approximated. A class of model problems with random coefficients and source terms is cast in a var...

The treatment times of laser induced thermal therapies (LITT) guided by computational prediction are determined by the convergence behavior of partial differential equation (PDE)-constrained optimization problems. In this paper, we investigate the convergence behavior of a bioheat transfer constrained calibration problem to assess the feasibility o...

While a large and growing literature exists on mathematical and computational models of tumor growth, to date tumor growth models are largely qualitative in nature, and fall far short of being able to provide predictive results important in life-and-death decisions. This is largely due to the enormous complexity of evolving biological and chemical...

This chapter describes a class of computational methods designed to handle multiscale modeling of large atomistic or molecular systems. The key to this approach is the estimation of relative modeling error in which errors in quantities of interest produced by averaging or homogenization are computed using a posteriori error estimates. The error is...

Minimally invasive treatments of cancer are key to improving posttreatment quality of life. ermal therapies delivered under various treatment modalities are a form of minimally invasive cancer treatment that has the potential to become an eective option to eradicate the disease, maintain the functionality of infected organs, and minimize complicati...

In this paper, we present a model-based predictive control system that is capable of capturing physical and biological variations of laser-tissue interaction as well as heterogeneity in real-time during laser induced thermal therapy (LITT). Using a three-dimensional predictive bioheat transfer model, which is built based on regular magnetic resonan...

An adaptive feedback control system is presented which employs a computational model of bioheat transfer in living tissue to guide, in real-time, laser treatments of prostate cancer monitored by magnetic resonance thermal imaging. The system is built on what can be referred to as cyberinfrastructure-a complex structure of high-speed network, large-...

Este paper descreve técnica para uma formulacáo de deslocamento para análise limite e de "shakedown" via E.F. de vasos de pressáo axi-simétricos. Assume-se que o material seja elastico-perfeitamente plástico. A técnica foi desenvolvida baseada numa formulacáo cinemática (upper bound) para condicóes de escomento linearizadas para cascas. Uma relacáo...

Laser surgery, or laser-induced thermal therapy, is a minimally invasive alternative or adjuvant to surgical resection in treating tumors embedded in vital organs with poorly defined boundaries. Its use, however, is limited due to the lack of precise control of heating and slow rate of thermal diffusion in the tissue. Nanoparticles, such as nanoshe...

a b s t r a c t In this work, the notion of a posteriori estimation and control of modeling error is extended to large-scale problems in molecular statics. The approaches developed here involve systematic methods for multiscale modeling in which sequences of hybrid particle–continuum models are generated using an adaptive goal-oriented algorithm de...

In this work, we propose to extend the Arlequin framework to couple particle and continuum models. Three different coupling strategies are investigated based on the L 2 norm, H 1 seminorm, and H 1 norm. The mathematical properties of the method are studied for a one-dimensional model of harmonic springs, with varying coefficients, coupled with a li...

The ultimate goal of cancer treatment utilizing thermotherapy is to eradicate tumors and minimize damage to surrounding host tissues. To achieve this goal, it is important to develop an accurate cell damage model to characterize the population of cell death under various thermal conditions. The traditional Arrhenius model is often used to character...

For cancerous tumors in vital internal organs, minimally invasive laser surgery may be a desirable choice for cancer treatment due to its precise control and compatibility with most of the imaging modalities such as MRI (magnetic resonance imaging). However, the complexity of tumor composition and tissue response to a thermal dose demands real time...

a b s t r a c t In this paper, we propose a multiscale coupling approach to perform Monte-Carlo simulations on systems described at the atomic scale and subjected to random phenomena. The method is based on the Arlequin framework, developed to date for deterministic models involving coupling a region of interest described at a particle scale with a...

The reliability of computer simulations of physical events or of engineering systems has emerged as the single most critical issue facing advancements in computational engineering and science. Without concrete and quantifiable measures of reliability, the confidence and usefulness of computer predictions are severely limited and the value of comput...

A new coupling term for blending particle and continuum models with the Arlequin framework is investigated in this work. The coupling term is based on an integral operator defined on the overlap region that matches the continuum and particle solutions in an average sense. The present exposition is essentially the continuation of a previous work (Ba...