Parallel Adaptive Multilevel Sampling Algorithms for the Bayesian Analysis of Mathematical Models

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, the selection of the most plausible models among these classes, given relevant observational data; third, the statistical calibration and validation of models in these classes, and finally, the prediction of key Quantities of Interest (QOIs) relevant to patient survival and the effect of various therapies. The most challenging aspects of this endeavor is that all of these issues often involve confounding uncertainties: in observational data, in model parameters, in model selection, and in the features targeted in the prediction. Our approach can be referred to as “model agnostic” in that no single model is advocated; rather, a general approach that explores powerful mixture-theory representations of tissue behavior while accounting for a range of relevant biological factors is presented, which leads to many potentially predictive models. Then representative classes are identified which provide a starting point for the implementation of OPAL, the Occam Plausibility Algorithm (OPAL) which enables the modeler to select the most plausible models (for given data) and to determine if the model is a valid tool for predicting tumor growth and morphology (in vivo). All of these approaches account for uncertainties in the model, the observational data, the model parameters, and the target QOI. We demonstrate these processes by comparing a list of models for tumor growth, including reaction–diffusion models, phase-fields models, and models with and without mechanical deformation effects, for glioma growth measured in murine experiments. Examples are provided that exhibit quite acceptable predictions of tumor growth in laboratory animals while demonstrating successful implementations of OPAL.

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“…Refs. 22, 29 and 23). The norm in (3.5) symbolizes any of several measures of distance between the PDFs d V ( θ ) and y V .…”

Section: Classes Of Parametric Models Of Tumor Growthmentioning

confidence: 99%

Selection, calibration, and validation of models of tumor growth

Lima

Oden

Hormuth

et al. 2016

Math. Models Methods Appl. Sci.

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“…The most plausible model, within the same Occam category, is selected for the validation step. The Tempered Monte Carlo algorithm (see [38]) is used for the stochastic approximation of Bayesian evidence (10). This algorithm is available in the library QUESO (Quantification of Uncertainty for Estimation, Simulation, and Optimization) [37].…”

Section: Algorithms and Numerical Methodsmentioning

confidence: 99%

“…A major challenge with MH-MCMC algorithms is that the chain of samples has a tendency to become locked in local modes of the target density being explored. One solution to this locality problem is to use Parallel Tempering which is a basis for the QUESO library [37, 38] used in this work.…”

Section: Algorithms and Numerical Methodsmentioning

confidence: 99%

Selection and validation of predictive models of radiation effects on tumor growth based on noninvasive imaging data

Lima

Oden

Wohlmuth

et al. 2017

Computer Methods in Applied Mechanics and Engineering

119

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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 systems. In this paper, large classes of parametric models of tumor growth in vascular tissue are discussed including models for radiation therapy. Observational data is obtained from MRI of a murine model of glioma and observed over a period of about three weeks, with X-ray radiation administered 14.5 days into the experimental program. Parametric models of tumor proliferation and decline are presented based on the balance laws of continuum mixture theory, particularly mass balance, and from accepted biological hypotheses on tumor growth. Among these are new model classes that include characterizations of effects of radiation and simple models of mechanical deformation of tumors. The Occam Plausibility Algorithm (OPAL) is implemented to provide a Bayesian statistical calibration of the model classes, 39 models in all, as well as the determination of the most plausible models in these classes relative to the observational data, and to assess model inadequacy through statistical validation processes. Discussions of the numerical analysis of finite element approximations of the system of stochastic, nonlinear partial differential equations characterizing the model classes, as well as the sampling algorithms for Monte Carlo and Markov chain Monte Carlo (MCMC) methods employed in solving the forward stochastic problem, and in computing posterior distributions of parameters and model plausibilities are provided. The results of the analyses described suggest that the general framework developed can provide a useful approach for predicting tumor growth and the effects of radiation.

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“…Bayes' formula can be written to make explicit the whole set of assumptions underlying the modeling and inference efforts [15,99,102]:…”

Section: Model Selection: Posterior Plausibilities Of Parametric Modementioning

confidence: 99%

Toward Predictive Multiscale Modeling of Vascular Tumor Growth

Oden

Lima

Almeida

et al. 2015

Arch Computat Methods Eng

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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 patientspecific treatments. We refer to these new approaches for studying biomedical systems as predictive medicine, a new version of medical science that involves the use of advanced computer models of biomedical phenomena, high-performance computing, new experimental methods for model data calibration, modern imaging technologies, cutting-edge numerical algorithms for treating large stochastic systems, modern methods for model selection, calibration, validation, verification, and uncertainty quantification, and new approaches for drug design and delivery, all based on predictive models. The methodologies are designed to study events at multiple scales, from genetic data, to sub-cellular signaling mechanisms, to cell interactions, to tissue physics and chemistry, to organs in living human subjects. The present document surveys work on the development and implementation of predictive models of vascular tumor growth, covering aspects of what might be called modeling-and-experimentally based computational oncology. The work described is that of a multi-institutional team, centered at ICES with strong participation by members at M. D. Anderson Cancer Center and University of Texas at San Antonio. This exposition covers topics on signaling models, cell and cell-interaction models, tissue models based on multi-species mixture theories, models of angiogenesis, and beginning work of drug effects. A number of new parallel computer codes for implementing finite-element methods, multi-level Markov Chain Monte Carlo sampling methods, data classification methods, stochastic PDE solvers, statistical inverse algorithms for model calibration and validation, models of events at different spatial and temporal scales is presented. Importantly, new methods for model selection in the presence of uncertainties fundamental to predictive medical science, are described which are based on the notion of Bayesian model plausibilities. Also, as part of this general approach, new codes for determining the sensitivity of model outputs to variations in model parameters are described that provide a basis for assessing the importance of model parameters and controlling and reducing the number of relevant model parameters. Model specific data is to be accessible through careful and model-specific platforms in the Tumor Engineering Laboratory. We describe parallel computer platforms on which large-scale calculations are run as well as specific time-marching algorithms needed to treat stiff systems encountered in some phase-field mixture models. We also cover new non-invasive imaging and data classification methods that provide in vivo data for model validation. The study concludes with a brief discussion of future work and open challenges.

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Parallel Adaptive Multilevel Sampling Algorithms for the Bayesian Analysis of Mathematical Models

Cited by 39 publications

References 36 publications

Selection, calibration, and validation of models of tumor growth

Selection, calibration, and validation of models of tumor growth

Selection and validation of predictive models of radiation effects on tumor growth based on noninvasive imaging data

Toward Predictive Multiscale Modeling of Vascular Tumor Growth

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