Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations

Osher, Stanley; Sethian, James A.

doi:10.1016/0021-9991(88)90002-2

Cited by 12,158 publications

(8,027 citation statements)

References 27 publications

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“…Level set methods were first developed by Osher and Sethian in [44] and have been used to study the evolution of moving surfaces that experience frequent topology changes (e.g., merger of regions and fragmentation), particularly in the contexts of fluid mechanics and computer graphics. (See the books [43,48] and references [42,44,49].)…”

Section: Narrow Band/local Level Set Methodsmentioning

confidence: 99%

“…(See the books [43,48] and references [42,44,49].) In the level set method, the location of a region Ω is captured implicitly by introducing an auxilliary signed distance function ϕ that satisfies (6) In the level set approach, instead of explicitly tracking the position of interface Σ and manually handling topology changes, the level set function is updated by solving a PDE, which automatically accounts for the interface motion and all topology changes.…”

Section: Narrow Band/local Level Set Methodsmentioning

confidence: 99%

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A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

Macklin

Lowengrub

2008

J Sci Comput

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In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasisteady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics-an effect observed in real tumor growth.

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Section: Narrow Band/local Level Set Methodsmentioning

confidence: 99%

Section: Narrow Band/local Level Set Methodsmentioning

confidence: 99%

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

Macklin

Lowengrub

2008

J Sci Comput

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“…Active contour methods typically solve the contour evolution equation using the Level Set Method (Osher and Sethian, 1988;Sethian, 1999). This approach ensures numerical stability and allows the contour to change topology.…”

Section: Active Contour Evolutionmentioning

confidence: 99%

User-guided 3D active contour segmentation of anatomical structures: Significantly improved efficiency and reliability

Yushkevich¹,

Piven²,

Hazlett³

et al. 2006

NeuroImage

7,361

4,931

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Active contour segmentation and its robust implementation using level set methods are well established theoretical approaches that have been studied thoroughly in the image analysis literature. Despite the existence of these powerful segmentation methods, the needs of clinical research continue to be fulfilled, to a large extent, using slice-by-slice manual tracing. To bridge the gap between methodological advances and clinical routine, we developed an open source application called ITK-SNAP, which is intended to make level set segmentation easily accessible to a wide range of users, including those with little or no mathematical expertise. This paper describes the methods and software engineering philosophy behind this new tool and provides the results of validation experiments performed in the context of an ongoing child autism neuroimaging study. The validation establishes SNAP intra/interrater reliability and overlap error statistics for the caudate nucleus and finds that SNAP is a highly reliable and efficient alternative to manual tracing.Analogous results for lateral ventricle segmentation are provided.

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“…The above scheme for surface matching is complicated to implement due to the need to maintain information on surface triangulations and compute numerical derivatives of quantities such as the surface Laplacians of fields defined on surfaces, and the components and Christoffel symbols of the surface metric. Intriguingly, the approach can be greatly simplified if distance functions to the surfaces are computed in 3D (the socalled dlevel setT approach; Osher and Sethian, 1988). After some mathematical manipulation of the PDEs press, this volume), all computations can be performed in the 3D image, which eases numerical implementations.…”

Section: Mathematics Of Matching: Covariant Pdesmentioning

confidence: 99%

Mapping cortical change in Alzheimer's disease, brain development, and schizophrenia

et al. 2004

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This paper describes algorithms that can identify patterns of brain structure and function associated with Alzheimer's disease, schizophrenia, normal aging, and abnormal brain development based on imaging data collected in large human populations. Extraordinary information can be discovered with these techniques: dynamic brain maps reveal how the brain grows in childhood, how it changes in disease, and how it responds to medication. Genetic brain maps can reveal genetic influences on brain structure, shedding light on the nature-nurture debate, and the mechanisms underlying inherited neurobehavioral disorders. Recently, we created time-lapse movies of brain structure for a variety of diseases. These identify complex, shifting patterns of brain structural deficits, revealing where, and at what rate, the path of brain deterioration in illness deviates from normal. Statistical criteria can then identify situations in which these changes are abnormally accelerated, or when medication or other interventions slow them. In this paper, we focus on describing our approaches to map structural changes in the cortex. These methods have already been used to reveal the profile of brain anomalies in studies of dementia, epilepsy, depression, childhoodand adult-onset schizophrenia, bipolar disorder, attention-deficit/ hyperactivity disorder, fetal alcohol syndrome, Tourette syndrome, Williams syndrome, and in methamphetamine abusers. Specifically, we describe an image analysis pipeline known as cortical pattern matching that helps compare and pool cortical data over time and across subjects. Statistics are then defined to identify brain structural differences between groups, including localized alterations in cortical thickness, gray matter density (GMD), and asymmetries in cortical organization. Subtle features, not seen in individual brain scans, often emerge when population-based brain data are averaged in this way. Illustrative examples are presented to show the profound effects of development and various diseases on the human cortex. Dynamically spreading waves of gray matter loss are tracked in dementia and schizophrenia, and these sequences are related to normally occurring changes in healthy subjects of various ages. D 2004 Published by Elsevier Inc.

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Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations

Cited by 12,158 publications

References 27 publications

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

User-guided 3D active contour segmentation of anatomical structures: Significantly improved efficiency and reliability

Mapping cortical change in Alzheimer's disease, brain development, and schizophrenia

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