## An Eulerian PDE approach for computing tissue thickness (2003)

Venue: | IEEE Trans. Med. Imag |

Citations: | 33 - 5 self |

### BibTeX

@ARTICLE{Yezzi03aneulerian,

author = {Anthony J. Yezzi and Jerry L. Prince and Senior Member},

title = {An Eulerian PDE approach for computing tissue thickness},

journal = {IEEE Trans. Med. Imag},

year = {2003},

volume = {22},

pages = {1332--1339}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract—We outline an Eulerian framework for computing the thickness of tissues between two simply connected boundaries that does not require landmark points or parameterizations of either boundary. Thickness is defined as the length of correspondence trajectories, which run from one tissue boundary to the other, and which follow a smooth vector field constructed in the region between the boundaries. A pair of partial differential equations (PDEs) that are guided by this vector field are then solved over this region, and the sum of their solutions yields the thickness of the tissue region. Unlike other approaches, this approach does not require explicit construction of any correspondence trajectories. An efficient, stable, and computationally fast solution to these PDEs is found by careful selection of finite differences according to an upwinding condition. The behavior and performance of our method is demonstrated on two simulations and two magnetic resonance imaging data sets in two and three dimensions. These experiments reveal very good performance and show strong potential for application in tissue thickness visualization and quantification. Index Terms—Correspondence trajectory, numerical methods, partial differential equations (PDEs), thickness. I.

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Citation Context ...icitly constructing a correspondence trajectory. We now describe an efficient numerical solution of (3) and (4). IV. NUMERICAL IMPLEMENTATION There are many standard numerical methods for solving (1) =-=[15]-=-–[17], any one of which can be used to obtain , and follows immediately from (2). There are also many alternative ways to define and compute the tangent field without using the Laplace equation. There... |

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Citation Context ...ply a special case of this case. The notation , , and denotes the components of at the grid point ( , , ), and backward and forward differences are given by the following standard notation [7], [18], =-=[19]-=-: (5)sYEZZI AND PRINCE: EULERIAN PDE APPROACH FOR COMPUTING TISSUE THICKNESS 1335 A. Upwind Differencing We start by considering various combinations of the above first-order differences to approximat... |

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Citation Context ...pondence trajectories. There are many possible choices, and the framework we will outline applies equally well to any particular choice. One possible choice is a normalized gradient vector flow field =-=[14]-=-, where the boundaries play the role of edge maps. Another choice is the normalized gradient of the unique harmonic function over that interpolates between 0 along and 1 along . This is the function u... |

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Citation Context ...case is simply a special case of this case. The notation , , and denotes the components of at the grid point ( , , ), and backward and forward differences are given by the following standard notation =-=[7]-=-, [18], [19]: (5)sYEZZI AND PRINCE: EULERIAN PDE APPROACH FOR COMPUTING TISSUE THICKNESS 1335 A. Upwind Differencing We start by considering various combinations of the above first-order differences t... |

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Citation Context ...es, which are selected according to an upwinding condition. Very fast solutions are obtained by using heap-based algorithms similar to the “fast marching method” used for solving the Eikonal equation =-=[6]-=-–[8]. In Section II, we motivate the definition of thickness we have adopted here by discussing some drawbacks associated with a variety of alternative definitions. In Section III, we outline our Eule... |

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Citation Context ...ness can also provide an indication of disease. For example, thinning of the gray matter in the brain cortex is thought to be associated with Alzheimer’s disease and other neurodegenerative disorders =-=[2]-=-. Thickness might also prove to be the basis for image segmentation. For example, it is well known that the anterior and posterior banks of the central sulcus in the human brain cortex can be distingu... |

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Citation Context ...N PDE APPROACH FOR COMPUTING TISSUE THICKNESS 1333 Fig. 1. Problems related to thickness definitions. Brain cortex thickness has been defined in several ways. Coupled-surface methods, such as that in =-=[10]-=-, define thickness as the distance between point pairs uniquely associated between the two surfaces. One problem with this approach is that the thickness measures will be artificially high if the two ... |

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Citation Context ...is simply a special case of this case. The notation , , and denotes the components of at the grid point ( , , ), and backward and forward differences are given by the following standard notation [7], =-=[18]-=-, [19]: (5)sYEZZI AND PRINCE: EULERIAN PDE APPROACH FOR COMPUTING TISSUE THICKNESS 1335 A. Upwind Differencing We start by considering various combinations of the above first-order differences to appr... |

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Citation Context ... of a human brain. Representative MR cross sections are shown in the left-hand-side column of Fig. 8. The inner and outer surfaces of the cortical gray matter, segmented from the MR images (cf. [10], =-=[21]-=-, and [22]), are shown on the upper right-hand side. Thickness was computed in the region between these two boundaries, and the results for the three MR cross sections shown are shown in the middle co... |

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Citation Context ...he definition of thickness as the length of correspondence trajectories (curved, in general), which run from one surface to the other. While conceptually analogous to the thickness definition used in =-=[5]-=-, our definition is more general and our computational approach is fast and stable. Furthermore, the computational procedure outlined in this paper is purely Eulerian in nature, using only the structu... |

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Citation Context ...hen there is a pronounced bulge in the opposing surface. Both of these problems are demonstrated in Fig. 1(c). It is possible to create point associations between the surfaces by shape matching [12], =-=[13]-=-. However, the standard definitions of thickness for coupled surfaces would now apply, and these suffer from the problems outlined above. Another class of methods define thickness relative to a centra... |

35 |
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Citation Context ...oach when there is a pronounced bulge in the opposing surface. Both of these problems are demonstrated in Fig. 1(c). It is possible to create point associations between the surfaces by shape matching =-=[12]-=-, [13]. However, the standard definitions of thickness for coupled surfaces would now apply, and these suffer from the problems outlined above. Another class of methods define thickness relative to a ... |

29 |
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Citation Context ...iori point associations between the two surfaces. A simple measure of thickness in this case is to measure the distance from each point on a given surface to the closest point on the opposing surface =-=[11]-=-. The most obvious problem with this definition is the lack of symmetry—the thickness is not the same when the surfaces are interchanged. Also, the thickness can be dramatically underestimated using t... |

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Citation Context ...sis for image segmentation. For example, it is well known that the anterior and posterior banks of the central sulcus in the human brain cortex can be distinguished by a difference in thickness alone =-=[3]-=-. Finally, thickness can be used as a basis for efficient characterization of anatomical shape when coupled with a central axis representation [4]. Manuscript received May 24, 2003; revised January 30... |

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Citation Context ...l thickness is most often defined within a cross-sectional image, and is assumed to be the distance1 between the endocardium and epicardium along a line passing through the long axis of the ventricle =-=[9]-=- (thought of as the origin), as shown in Fig. 1(a). This definition does not capture the three-dimensional (3-D) aspect of the heart wall, requires that the positions of the endocardium and epicardium... |

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Citation Context ... thickness of a particular structure might provide an indication of its functional performance. For example, myocardial thickening during systole is an important indicator of healthy cardiac function =-=[1]-=-. Thickness can also provide an indication of disease. For example, thinning of the gray matter in the brain cortex is thought to be associated with Alzheimer’s disease and other neurodegenerative dis... |