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And Varifold-Based Registration of Lung Vessel and Airway Trees Current- and Varifold-Based Registration of Lung Vessel and Airway Trees Yue Pan, Gary E. Christensen, Oguz C. Durumeric, Sarah E. Gerard, Joseph M. Reinhardt The University of Iowa, IA 52242 {yue-pan-1, gary-christensen, oguz-durumeric, sarah-gerard, joe-reinhardt}@uiowa.edu Geoffrey D. Hugo Virginia Commonwealth University, Richmond, VA 23298 [email protected] Abstract 1. Introduction Registration of lung CT images is important for many radiation oncology applications including assessing and Registering lung CT images is an important problem adapting to anatomical changes, accumulating radiation for many applications including tracking lung motion over dose for planning or assessment, and managing respiratory the breathing cycle, tracking anatomical and function motion. For example, variation in the anatomy during radio- changes over time, and detecting abnormal mechanical therapy introduces uncertainty between the planned and de- properties of the lung. This paper compares and con- livered radiation dose and may impact the appropriateness trasts current- and varifold-based diffeomorphic image of the originally-designed treatment plan. Frequent imaging registration approaches for registering tree-like structures during radiotherapy accompanied by accurate longitudinal of the lung. In these approaches, curve-like structures image registration facilitates measurement of such variation in the lung—for example, the skeletons of vessels and and its effect on the treatment plan. The cumulative dose airways segmentation—are represented by currents or to the target and normal tissue can be assessed by mapping varifolds in the dual space of a Reproducing Kernel Hilbert delivered dose to a common reference anatomy and com- Space (RKHS). Current and varifold representations are paring to the prescribed dose. The treatment plan can then discretized and are parameterized via of a collection of be adapted periodically during therapy to help mitigate the momenta. A momenta corresponds to a line segment via the impact of these changes by ensuring the cumulative deliv- coordinates of the center of the line segment and the tangent ered dose is concordant with the prescribed dose[13, 9, 15]. direction of the line segment at the center. A varifold-based Furthermore, image registration can also help measure how registration approach is similar to currents except that the tumor changes during or after treatment, which can two varifold representations are aligned independent of potentially assist in predicting early response to therapy. the tangent vector orientation. An advantage of varifolds These applications all rely on accurate tracking of lung mo- over currents is that the orientation of the tangent vectors tion over the breathing cycle and anatomical and functional can be difficult to determine especially when the vessel changes over time. and airway trees are not connected. In this paper, we Accurate image registration is critical for clinical imple- examine the image registration sensitivity and accuracy of mentation of these strategies. However, the ability of an current- and varifold-based registration as a function of the algorithm to match anatomy throughout the lung may be number and location of momentum used to represent tree limited by the complex variations in the anatomy and lim- like-structures in the lung. The registrations presented in ited image contrast. One approach to improving registration this paper were generated using the Deformetrica software accuracy is to highlight and extract known anatomy such package ([Durrleman et al. 2014]). as pulmonary airways or blood vessels[2] to improve the matching at these tissue locations. Image registration correspondence can be defined ei- Keywords: Diffeomorphic image registration, currents, ther through intensity-based or feature-based approaches. varifolds, momenta, Reproducing Kernel Hilbert Space Intensity-based approaches register images by minimizing (RKHS) differences in intensities between the moving (deformed template) and target images. In general, intensity-based 126 registration approaches have the advantage of not needing 2.1. Current Representation user intervention but often do a poor job of aligning fea- The current representation of a curve L is defined by the tures such as points, lines and surfaces contained in the im- path/line integral along the curve through a test vector field ages. Feature-based approaches are attractive because they ω via directly match features, but they often require a priori point- t to-point correspondence, which can be challenging in radio- L(ω)= ω(x) τ(x)dλ(x) (1) ZL therapy applications where this is often not known. Current- and varifold-based image registration are feature-based reg- where, τ is the tangent of the curve at point x and dλ is the istration approaches with the advantage that no point cor- Lebesgue measure on the curve. The test vector field ω is an respondence is assumed between the objects being regis- element of a space of possible vector fields W , where W is tered ([14],[5],[6]). The currents framework has been suc- a Reproducing Kernel Hilbert Space (RKHS). In this work, W is a space of square integrable vector fields convolved cessfully applied to perform registrations of MR images W ([14],[4],[5]) and also lung CT images ([8]). A varifold- with a smoothing Gaussian kernel: ω(x) = K (x,.)α, based registration approach [3] is similar to current-based where the pair (x,α) is called a momentum. The tangent registration approach except that two varifold representa- vector along the curve gives a natural action of the curve on tions can be aligned in an orientation invariant manner vector fields. The norm of the current is defined in the dual ∗ which will be discussed in the next section. space (currents space) W of W , as the maximum action of the current among all possible test vector fields. W is a Current- and varifold-based image registration is built W using the Large Deformation Diffeomorphic Metric Map- closed span of the vector fields ω(x) = K (x,.)α. The dual space of W denoted as W ∗ is a closed span of Dirac ping (LDDMM) framework ([11],[12]). This framework α delta currents δx , where a Dirac delta current is the dual produces correspondence maps (transformations) between W images that are guaranteed to be diffeomorphisms.1 In this representation of the basis vector field K (x,.)α. Based work, the velocity field of the LDDMM framework is rep- on the Riesz representation theorem, there is a linear map- ∗ L ∗ resented by currents ([4]) and the control points of the de- ping between W and its dual space W , W : W → W formation field are not necessarily dense in order to get de- such that L ′ ′ sired registration results ([5],[7]). The results presented in W (ω)(ω )=< ω,ω >W (2) this paper were generated using the Defometrica software α L W Therefore, δx = W (K (x,.)α) ([Durrleman et al. 2014]), which is publicly available at In a discrete setting, curves may be represented as polyg- www.deformetrica.org. The contribution of this paper is a onal lines where the direction of the tangent is constant over sensitivity analysis of the current- and varifold-based im- each line segment. In this case, the current representation of age registration methods to the number and location of mo- a polygonal curve is given by menta representing tree-like structures in the lung such as t the centerline of the pulmonary vessel and airway trees. L(ω)= ω(xk) τ(xk) (3) Xk 2. Methods where xk is the center of each line segment and τ(xk) is Currents and varifolds are mathematical objects that can the tangent vector at xk. The magnitude of τ(xk) is propor- be used to model general geometrical objects. A current is a tional to the length of the line segment centered at xk. linear functional on a smooth manifold which is continuous in the sense of distributions. Any set of curves or surfaces 2.2. Varifold Representation can be represented in terms of currents. The advantage of A varifold is a generalization of a current in the sense using currents to register images is that the similarity mea- that the tangent vector of its representative momenta are not sure is defined in the space of currents, which does not as- oriented. A varifold representation provides an advantage sume any kind of point-correspondence between structures. over a current representation for representing structures in A varifold can be considered a generalization of the idea which the orientation of the constitute line segments are un- of a current in the sense that the tangent vector of its repre- known or difficult to discern. This is true for the current sentative momenta are not oriented. Theoretically, varifolds application in which the vessel trees may consist of discon- are weaker objects than currents due to the lack of orienta- nected line segments due to, for example, where a tumor or tion of the tangent vector of the momenta used to represent other pathology interrupts the vessel trees. a shape. However, this “weaker” side of varifolds is a desir- The varifold representation of a curve L is defined by the able property when matching line segments with uncertain path/line integral along the curve through a test vector field tangent orientation. ω via 2 1A diffeomorphism is a bijective, differentiable map between two man- t τ(x) ifolds such that its inverse is also differentiable. L(ω)= ω(x) dλ(x) (4) ZL ||τ(x)|| 127 α β where, τ is the tangent of the curve at point x and dλ is the an abuse of notation, let T1 = n δx and T2 = m δy be Lebesgue measure on the curve. The test vector field ω is the Dirac delta varifold representationP of L1 and LP2, respec- an element of a space of possible vector fields W , where tively, using Eq.
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