CMOR Publications

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A Branch Decomposition Algorithm for the p-Median Problem
(INFORMS, 2017) Fast, Caleb C.; Hicks, Illya V.
In this paper, we use a branch decomposition technique to improve approximations to the p-median problem. Starting from a support graph produced either by a combination of heuristics or by linear programming, we use dynamic programming guided by a branch decomposition of that support graph to find the best p-median solution on the support graph. Our results show that when heuristics are used to build the support graph and the support graph has branchwidth at most 7, our algorithm is able to provide a solution of lower cost than any of the heuristic solutions. When linear programming is used to build the support graph and the support graph has branchwidth at most 7, then our algorithm provides better solutions than popular heuristics and is faster than integer programming. Thus, our algorithm is a useful practical tool when support graphs have branchwidth at most 7.
A DEIM Induced CUR Factorization
(SIAM, 2016) Sorensen, D.C.; Embree, Mark
We derive a CUR approximate matrix factorization based on the discrete empirical interpolation method (DEIM). For a given matrix ${\bf A}$, such a factorization provides a low-rank approximate decomposition of the form ${\bf A} \approx \bf C \bf U \bf R$, where ${\bf C}$ and ${\bf R}$ are subsets of the columns and rows of ${\bf A}$, and ${\bf U}$ is constructed to make $\bf C\bf U \bf R $ a good approximation. Given a low-rank singular value decomposition ${\bf A} \approx \bf V \bf S \bf W^T$, the DEIM procedure uses ${\bf V}$ and ${\bf W}$ to select the columns and rows of ${\bf A}$ that form ${\bf C}$ and ${\bf R}$. Through an error analysis applicable to a general class of CUR factorizations, we show that the accuracy tracks the optimal approximation error within a factor that depends on the conditioning of submatrices of ${\bf V}$ and ${\bf W}$. For very large problems, ${\bf V}$ and ${\bf W}$ can be approximated well using an incremental QR algorithm that makes only one pass through ${\bf A}$. Numerical examples illustrate the favorable performance of the DEIM-CUR method compared to CUR approximations based on leverage scores.
A discrepancy-based penalty method for extended waveform inversion
(Society of Exploration Geophysicists, 2017) Fu, Lei; Symes, William W.; The Rice Inversion Project
Extended waveform inversion globalizes the convergence of seismic waveform inversion by adding nonphysical degrees of freedom to the model, thus permitting it to fit the data well throughout the inversion process. These extra degrees of freedom must be curtailed at the solution, for example, by penalizing them as part of an optimization formulation. For separable (partly linear) models, a natural objective function combines a mean square data residual and a quadratic regularization term penalizing the nonphysical (linear) degrees of freedom. The linear variables are eliminated in an inner optimization step, leaving a function of the outer (nonlinear) variables to be optimized. This variable projection method is convenient for computation, but it requires that the penalty weight be increased as the estimated model tends to the (physical) solution. We describe an algorithm based on discrepancy, that is, maintaining the data residual at the inner optimum within a prescribed range, to control the penalty weight during the outer optimization. We evaluate this algorithm in the context of constant density acoustic waveform inversion, by recovering background model and perturbation fitting bandlimited waveform data in the Born approximation.
A Distributed-Memory Randomized Structured Multifrontal Method for Sparse Direct Solutions
(Society for Industrial and Applied Mathematics, 2017) Xin, Zixing; Xia, Jianlin; de Hoop, Maarten V.; Cauley, Stephen; Balakrishnan, Venkataramanan
We design a distributed-memory randomized structured multifrontal solver for large sparse matrices. Two layers of hierarchical tree parallelism are used. A sequence of innovative parallel methods are developed for randomized structured frontal matrix operations, structured update matrix computation, skinny extend-add operation, selected entry extraction from structured matrices, etc. Several strategies are proposed to reuse computations and reduce communications. Unlike an earlier parallel structured multifrontal method that still involves large dense intermediate matrices, our parallel solver performs the major operations in terms of skinny matrices and fully structured forms. It thus significantly enhances the efficiency and scalability. Systematic communication cost analysis shows that the numbers of words are reduced by factors of about $O(\sqrt{n}/r)$ in two dimensions and about $O(n^{2/3}/r)$ in three dimensions, where $n$ is the matrix size and $r$ is an off-diagonal numerical rank bound of the intermediate frontal matrices. The efficiency and parallel performance are demonstrated with the solution of some large discretized PDEs in two and three dimensions. Nice scalability and significant savings in the cost and memory can be observed from the weak and strong scaling tests, especially for some 3D problems discretized on unstructured meshes.
A Matrix-Free Trust-Region SQP Method for Equality Constrained Optimization
(SIAM, 2014) Heinkenschloss, Matthias; Ridzal, Denis
We develop and analyze a trust-region sequential quadratic programming (SQP) method for the solution of smooth equality constrained optimization problems, which allows the inexact and hence iterative solution of linear systems. Iterative solution of linear systems is important in large-scale applications, such as optimization problems with partial differential equation constraints, where direct solves are either too expensive or not applicable. Our trust-region SQP algorithm is based on a composite-step approach that decouples the step into a quasi-normal and a tangential step. The algorithm includes critical modifications of substep computations needed to cope with the inexact solution of linear systems. The global convergence of our algorithm is guaranteed under rather general conditions on the substeps. We propose algorithms to compute the substeps and prove that these algorithms satisfy global convergence conditions. All components of the resulting algorithm are specified in such a way that they can be directly implemented. Numerical results indicate that our algorithm converges even for very coarse linear system solves.
A rapid, low-cost, and highly sensitive SARS-CoV-2 diagnostic based on whole-genome sequencing
(Public Library of Science, 2023) Adastra, Per A.; Durand, Neva C.; Mitra, Namita; Pulido, Saul Godinez; Mahajan, Ragini; Blackburn, Alyssa; Colaric, Zane L.; Theisen, Joshua W. M.; Weisz, David; Dudchenko, Olga; Gnirke, Andreas; Rao, Suhas S. P.; Kaur, Parwinder; Aiden, Erez Lieberman; Aiden, Aviva Presser; Center for Theoretical Biological Physics
Early detection of SARS-CoV-2 infection is key to managing the current global pandemic, as evidence shows the virus is most contagious on or before symptom onset. Here, we introduce a low-cost, high-throughput method for diagnosing and studying SARS-CoV-2 infection. Dubbed Pathogen-Oriented Low-Cost Assembly & Re-Sequencing (POLAR), this method amplifies the entirety of the SARS-CoV-2 genome. This contrasts with typical RT-PCR-based diagnostic tests, which amplify only a few loci. To achieve this goal, we combine a SARS-CoV-2 enrichment method developed by the ARTIC Network (https://artic.network/) with short-read DNA sequencing and de novo genome assembly. Using this method, we can reliably (>95% accuracy) detect SARS-CoV-2 at a concentration of 84 genome equivalents per milliliter (GE/mL). The vast majority of diagnostic methods meeting our analytical criteria that are currently authorized for use by the United States Food and Drug Administration with the Coronavirus Disease 2019 (COVID-19) Emergency Use Authorization require higher concentrations of the virus to achieve this degree of sensitivity and specificity. In addition, we can reliably assemble the SARS-CoV-2 genome in the sample, often with no gaps and perfect accuracy given sufficient viral load. The genotypic data in these genome assemblies enable the more effective analysis of disease spread than is possible with an ordinary binary diagnostic. These data can also help identify vaccine and drug targets. Finally, we show that the diagnoses obtained using POLAR of positive and negative clinical nasal mid-turbinate swab samples 100% match those obtained in a clinical diagnostic lab using the Center for Disease Control’s 2019-Novel Coronavirus test. Using POLAR, a single person can manually process 192 samples over an 8-hour experiment at the cost of ~$36 per patient (as of December 7th, 2022), enabling a 24-hour turnaround with sequencing and data analysis time. We anticipate that further testing and refinement will allow greater sensitivity using this approach.
A short note on a Bernstein-Bezier basis for the pyramid
(Society for Industrial and Applied Mathematics, 2016) Chan, Jesse; Warburton, T.
We introduce a Bernstein--Bezier basis for the pyramid, whose restriction to the face reduces to the Bernstein--Bezier basis on the triangle or quadrilateral. The basis satisfies the standard positivity and partition of unity properties common to Bernstein polynomials and spans the same space as nonpolynomial pyramid bases in the literature. Procedures for differentiation and integration of these basis functions are also discussed.
A Topological Model of the Hippocampal Cell Assembly Network
(Frontiers Media S.A., 2016) Babichev, Andrey; Ji, Daoyun; Mémoli, Facundo; Dabaghian, Yuri A.
It is widely accepted that the hippocampal place cells' spiking activity produces a cognitive map of space. However, many details of this representation's physiological mechanism remain unknown. For example, it is believed that the place cells exhibiting frequent coactivity form functionally interconnected groups—place cell assemblies—that drive readout neurons in the downstream networks. However, the sheer number of coactive combinations is extremely large, which implies that only a small fraction of them actually gives rise to cell assemblies. The physiological processes responsible for selecting the winning combinations are highly complex and are usually modeled via detailed synaptic and structural plasticity mechanisms. Here we propose an alternative approach that allows modeling the cell assembly network directly, based on a small number of phenomenological selection rules. We then demonstrate that the selected population of place cell assemblies correctly encodes the topology of the environment in biologically plausible time, and may serve as a schematic model of the hippocampal network.
A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty
(SIAM, 2013) Kouri, D.P.; Heinkenschloss, M.; Ridzal, D.; van Bloemen Waanders, B.G.
The numerical solution of optimization problems governed by partial differential equations (PDEs) with random coefficients is computationally challenging because of the large number of deterministic PDE solves required at each optimization iteration. This paper introduces an efficient algorithm for solving such problems based on a combination of adaptive sparse-grid collocation for the discretization of the PDE in the stochastic space and a trust-region framework for optimization and fidelity management of the stochastic discretization. The overall algorithm adapts the collocation points based on the progress of the optimization algorithm and the impact of the random variables on the solution of the optimization problem. It frequently uses few collocation points initially and increases the number of collocation points only as necessary, thereby keeping the number of deterministic PDE solves low while guaranteeing convergence. Currently an error indicator is used to estimate gradient errors due to adaptive stochastic collocation. The algorithm is applied to three examples, and the numerical results demonstrate a significant reduction in the total number of PDE solves required to obtain an optimal solution when compared with a Newton conjugate gradient algorithm applied to a fixed high-fidelity discretization of the optimization problem.
An Algebraic Exploration of Dominating Sets and Vizing's Conjecture
(The Electronic Journal of Combinatorics, 2012) Margulies, S.; Hicks, I.V.
Systems of polynomial equations are commonly used to model combinatorial problems such as independent set, graph coloring, Hamiltonian path, and others. We formulate the dominating set problem as a system of polynomial equations in two di erent ways: rst, as a single, high-degree polynomial, and second as a collection of polynomials based on the complements of domination-critical graphs. We then provide a su cient criterion for demonstrating that a particular ideal representation is already the universal Gr obner bases of an ideal, and show that the second representation of the dominating set ideal in terms of domination-critical graphs is the universal Gr obner basis for that ideal. We also present the rst algebraic formulation of Vizing's conjecture, and discuss the theoretical and computational rami cations to this conjecture when using either of the two dominating set representations described above.
An accelerated Poisson solver based on multidomain spectral discretization
(Springer, 2018) Babb, Tracy; Gillman, Adrianna; Hao, Sijia; Martinsson, Per-Gunnar
This paper presents a numerical method for variable coefficient elliptic PDEs with mostly smooth solutions on two dimensional domains. The method works best for domains that can readily be mapped onto a rectangle, or a collection of nonoverlapping rectangles. The PDE is discretized via a multi-domain spectral collocation method of high local order (order 30 and higher have been tested and work well). Local mesh refinement results in highly accurate solutions even in the presence of local irregular behavior due to corner singularities, localized loads, etc. The system of linear equations attained upon discretization is solved using a direct (as opposed to iterative) solver with O(N1.5)O(N1.5) complexity for the factorization stage and O(NlogN)O(Nlog⁡N) complexity for the solve. The scheme is ideally suited for executing the elliptic solve required when parabolic problems are discretized via time-implicit techniques. In situations where the geometry remains unchanged between time-steps, very fast execution speeds are obtained since the solution operator for each implicit solve can be pre-computed.
An adaptive multiscale algorithm for efficient extended waveform inversion
(Society of Exploration Geophysicists, 2017) Fu, Lei; Symes, William W.; The Rice Inversion Project
Subsurface-offset extended full-waveform inversion (FWI) may converge to kinematically accurate velocity models without the low-frequency data accuracy required for standard data-domain FWI. However, this robust alternative approach to waveform inversion suffers from a very high computational cost resulting from its use of nonlocal wave physics: The computation of strain from stress involves an integral over the subsurface offset axis, which must be performed at every space-time grid point. We found that a combination of data-fit driven offset limits, grid coarsening, and low-pass data filtering can reduce the cost of extended inversion by one to two orders of magnitude.
An alternating direction and projection algorithm for structure-enforced matrix factorization
(Springer, 2017) Xu, Lijun; Yu, Bo; Zhang, Yin
Structure-enforced matrix factorization (SeMF) represents a large class of mathematical models appearing in various forms of principal component analysis, sparse coding, dictionary learning and other machine learning techniques useful in many applications including neuroscience and signal processing. In this paper, we present a unified algorithm framework, based on the classic alternating direction method of multipliers (ADMM), for solving a wide range of SeMF problems whose constraint sets permit low-complexity projections. We propose a strategy to adaptively adjust the penalty parameters which is the key to achieving good performance for ADMM. We conduct extensive numerical experiments to compare the proposed algorithm with a number of state-of-the-art special-purpose algorithms on test problems including dictionary learning for sparse representation and sparse nonnegative matrix factorization. Results show that our unified SeMF algorithm can solve different types of factorization problems as reliably and as efficiently as special-purpose algorithms. In particular, our SeMF algorithm provides the ability to explicitly enforce various combinatorial sparsity patterns that, to our knowledge, has not been considered in existing approaches.
An approximate inverse to the extended Born modeling operator
(Society of Exploration Geophysicists, 2015) Hou, Jie; Symes, William W.; The Rice Inversion Project
Given a correct (data-consistent) velocity model, reverse time migration (RTM) correctly positions reflectors but generally with incorrect amplitudes and wavelets. Iterative least-squares migration (LSM) corrects the amplitude and wavelet by fitting data in the sense of Born modeling, that is, replacing migration by Born inversion. However, LSM also requires a correct velocity model, and it may require many migration/demigration cycles. We modified RTM in the subsurface offset domain to create an asymptotic (high-frequency) approximation to extended LSM. This extended Born inversion operator outputs extended reflectors (depending on the subsurface offset and position in the earth) with correct amplitude and phase, in the sense that similarly extended Born modeling reproduces the data to good accuracy. Although the theoretical justification of the inversion property relies on ray tracing and stationary phase, application of the weight operators does not require any computational ray tracing. The computational expense of the extended Born inversion operator is roughly the same as that of extended RTM, and the inversion (data-fit) property holds even when the velocity is substantially incorrect. The approximate inverse operator differes from extended RTM only in the application of data- and model-domain weight operators, and takes the form of an adjoint in the sense of weighted inner products in data and model space. Because the Born modeling operator is approximately unitary with respect to the weighted inner products, a weighted version of conjugate gradient iteration dramatically accelerates the convergence of extended LSM. An approximate LSM may be extracted from the approximate extended LSM by averaging over subsurface offset.
An exact redatuming procedure for the inverse boundary value problem for the wave equation
(Society for Industrial and Applied Mathematics, 2018) de Hoop, Maarten V.; Kepley, Paul; Oksanen, Lauri
Redatuming is a data processing technique to transform measurements recorded in one acquisition geometry to an analogous data set corresponding to another acquisition geometry, for which there are no recorded measurements. We consider a redatuming problem for a wave equation on a bounded domain, or on a manifold with boundary, and model data acquisition by a restriction of the associated Neumann-to-Dirichlet map. This map models measurements with sources and receivers on an open subset $\Gamma$ contained in the boundary of the manifold. We model the wavespeed by a Riemannian metric and suppose that the metric is known in some coordinates in a neighborhood of $\Gamma$. Our goal is to move sources and receivers into this known near boundary region. We formulate redatuming as a collection of unique continuation problems and provide a two-step procedure to solve the redatuming problem. We investigate the stability of the first step in this procedure, showing that it enjoys conditional Hölder stability under suitable geometric hypotheses. In addition, we provide computational experiments that demonstrate our redatuming procedure.
Angola Cameia Development Casing-Settlement Calculations
(Society of Petroleum Engineers, 2017) Akin, J. Ed; Dove, N. Roland; Ruddy, Ken
The amount of axial settlement of casings supported by regions of axial elastic foundations is computed. The differential equation of axial equilibrium, including the foundation stiffnesses, is solved by use of cubic axial finite elements. The analysis is applied to 101 m of a vertical 914-mm (36-in.) casing supporting a 559-mm (22-in.) casing running from 3 m above the mudline to near the bottom of the 1201-m hole. The upper casing is supported by a region of layered clay sediments. The lower casing has a long openhole region, followed by a bottom region encased in cement. A series of loading increments is applied, with the connection of the tree/blowout preventer (BOP) as the last. The settlement at the mudline was calculated to be less than 0.1 m. This study shows that for weak upper foundations, a good cement job is needed to support the tree/BOP without large mudline settlements.
Benchmarking lung cancer screening programmes with adaptive screening frequency against the optimal screening schedules derived from the ENGAGE framework: a comparative microsimulation study
(Elsevier, 2024) Hemmati, Mehdi; Ishizawa, Sayaka; Meza, Rafael; Ostrin, Edwin; Hanash, Samir M.; Antonoff, Mara; Schaefer, Andrew J.; Tammemägi, Martin C.; Toumazis, Iakovos
Background Lung cancer screening recommendations employ annual frequency for eligible individuals, despite evidence that it may not be universally optimal. The impact of imposing a structure on the screening frequency remains unknown. The ENGAGE framework, a validated framework that offers fully dynamic, analytically optimal, personalised lung cancer screening recommendations, could be used to assess the impact of screening structure on the effectiveness and efficiency of lung cancer screening. Methods In this comparative microsimulation study, we benchmarked alternative clinically relevant structured lung cancer screening programmes employing a fixed (annual or biennial) or adaptive (start with annual/biennial screening and then switch to biennial/annual at ages 60- or 65-years) screening frequency, against the ENGAGE framework. Individuals were eligible for screening according to the 2021 US Preventive Services Task Force recommendation on lung cancer screening. We assessed programmes' efficiency based on the number of screenings per death avoided (LDCT/DA) and the number of screenings per ever-screened individual (LDCT/ESI), and programmes’ effectiveness using quality-adjusted life years (QALY) gained from screening, lung cancer-specific mortality reduction (MR), and number of screen-detected lung cancer cases. We used validated natural history, smoking history generator, and risk prediction models to inform our analysis. Sensitivity analysis of key inputs was conducted. Findings ENGAGE was the best performing strategy. Among the structured policies, adaptive biennial-to-annual at age 65 was the best strategy requiring 24% less LDCT/DA and 60% less LDCT/ESI compared to TF2021, but yielded 105 more deaths per 100,000 screen-eligible individuals (10.2% vs. 11.8% MR for TF2021, p = 0.28). Fixed annual screening was the most effective strategy but the least efficient and was ranked as the fifth best strategy. All strategies yielded similar QALYs gained. Adherence levels did not affect the rankings. Interpretation Adaptive lung cancer screening strategies that start with biennial and switch to annual screening at a prespecified age perform well and warrant further consideration, especially in settings with limited availability of CT scanners and radiologists. Funding National Cancer Institute.
Chromatin architecture transitions from zebrafish sperm through early embryogenesis
(Cold Spring Harbor Laboratory Press, 2021) Wike, Candice L.; Guo, Yixuan; Tan, Mengyao; Nakamura, Ryohei; Shaw, Dana Klatt; Díaz, Noelia; Whittaker-Tademy, Aneasha F.; Durand, Neva C.; Aiden, Erez Lieberman; Vaquerizas, Juan M.; Grunwald, David; Takeda, Hiroyuki; Cairns, Bradley R.; Center for Theoretical Biological Physics
Chromatin architecture mapping in 3D formats has increased our understanding of how regulatory sequences and gene expression are connected and regulated in a genome. The 3D chromatin genome shows extensive remodeling during embryonic development, and although the cleavage-stage embryos of most species lack structure before zygotic genome activation (pre-ZGA), zebrafish has been reported to have structure. Here, we aimed to determine the chromosomal architecture in paternal/sperm zebrafish gamete cells to discern whether it either resembles or informs early pre-ZGA zebrafish embryo chromatin architecture. First, we assessed the higher-order architecture through advanced low-cell in situ Hi-C. The structure of zebrafish sperm, packaged by histones, lacks topological associated domains and instead displays “hinge-like” domains of ∼150 kb that repeat every 1–2 Mbs, suggesting a condensed repeating structure resembling mitotic chromosomes. The pre-ZGA embryos lacked chromosomal structure, in contrast to prior work, and only developed structure post-ZGA. During post-ZGA, we find chromatin architecture beginning to form at small contact domains of a median length of ∼90 kb. These small contact domains are established at enhancers, including super-enhancers, and chemical inhibition of Ep300a (p300) and Crebbpa (CBP) activity, lowering histone H3K27ac, but not transcription inhibition, diminishes these contacts. Together, this study reveals hinge-like domains in histone-packaged zebrafish sperm chromatin and determines that the initial formation of high-order chromatin architecture in zebrafish embryos occurs after ZGA primarily at enhancers bearing high H3K27ac.
Chromosome size affects sequence divergence between species through the interplay of recombination and selection
(Wiley, 2022) Tigano, Anna; Khan, Ruqayya; Omer, Arina D.; Weisz, David; Dudchenko, Olga; Multani, Asha S.; Pathak, Sen; Behringer, Richard R.; Aiden, Erez L.; Fisher, Heidi; MacManes, Matthew D.; Center for Theoretical and Biological Physics
The structure of the genome shapes the distribution of genetic diversity and sequence divergence. To investigate how the relationship between chromosome size and recombination rate affects sequence divergence between species, we combined empirical analyses and evolutionary simulations. We estimated pairwise sequence divergence among 15 species from three different mammalian clades—Peromyscus rodents, Mus mice, and great apes—from chromosome-level genome assemblies. We found a strong significant negative correlation between chromosome size and sequence divergence in all species comparisons within the Peromyscus and great apes clades but not the Mus clade, suggesting that the dramatic chromosomal rearrangements among Mus species may have masked the ancestral genomic landscape of divergence in many comparisons. Our evolutionary simulations showed that the main factor determining differences in divergence among chromosomes of different sizes is the interplay of recombination rate and selection, with greater variation in larger populations than in smaller ones. In ancestral populations, shorter chromosomes harbor greater nucleotide diversity. As ancestral populations diverge, diversity present at the onset of the split contributes to greater sequence divergence in shorter chromosomes among daughter species. The combination of empirical data and evolutionary simulations revealed that chromosomal rearrangements, demography, and divergence times may also affect the relationship between chromosome size and divergence, thus deepening our understanding of the role of genome structure in the evolution of species divergence.
Cluster-Based Toxicity Estimation of Osteoradionecrosis Via Unsupervised Machine Learning: Moving Beyond Single Dose-Parameter Normal Tissue Complication Probability by Using Whole Dose-Volume Histograms for Cohort Risk Stratification
(Elsevier, 2024) Hosseinian, Seyedmohammadhossein; Hemmati, Mehdi; Dede, Cem; Salzillo, Travis C.; van Dijk, Lisanne V.; Mohamed, Abdallah S. R.; Lai, Stephen Y.; Schaefer, Andrew J.; Fuller, Clifton D.
Purpose Given the limitations of extant models for normal tissue complication probability estimation for osteoradionecrosis (ORN) of the mandible, the purpose of this study was to enrich statistical inference by exploiting structural properties of data and provide a clinically reliable model for ORN risk evaluation through an unsupervised-learning analysis that incorporates the whole radiation dose distribution on the mandible. Methods and Materials The analysis was conducted on retrospective data of 1259 patients with head and neck cancer treated at The University of Texas MD Anderson Cancer Center between 2005 and 2015. During a minimum 12-month posttherapy follow-up period, 173 patients in this cohort (13.7%) developed ORN (grades I to IV). The (structural) clusters of mandibular dose-volume histograms (DVHs) for these patients were identified using the K-means clustering method. A soft-margin support vector machine was used to determine the cluster borders and partition the dose-volume space. The risk of ORN for each dose-volume region was calculated based on incidence rates and other clinical risk factors. Results The K-means clustering method identified 6 clusters among the DVHs. Based on the first 5 clusters, the dose-volume space was partitioned by the soft-margin support vector machine into distinct regions with different risk indices. The sixth cluster entirely overlapped with the others; the region of this cluster was determined by its envelopes. For each region, the ORN incidence rate per preradiation dental extraction status (a statistically significant, nondose related risk factor for ORN) was reported as the corresponding risk index. Conclusions This study presents an unsupervised-learning analysis of a large-scale data set to evaluate the risk of mandibular ORN among patients with head and neck cancer. The results provide a visual risk-assessment tool for ORN (based on the whole DVH and preradiation dental extraction status) as well as a range of constraints for dose optimization under different risk levels.

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