Defective pelvic organ support due to injuries of the levator ani is a common problem in women and its intervention requires a thorough understanding of the structure. Three-dimensional surfaces of the levator ani have proved to be a promising method of studying this. In this paper, we propose to build a statistical shape model (SSM) of the levator ani and describe a segmentation technique based on a limited number of control points with the SSM. The SSM was achieved by the use of harmonic shape embedding with the MDL objective function to optimise parameterisation while segmentation was performed by fitting the model to a user defined set of control points. The value of the technique was demonstrated with data acquired from a group of 11 asymptomatic subjects.Accepted versio

Darzi, A

Horkaew, P

Lee, S

Yang, G

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Statistical Shape Modelling of the Levator Ani Su-Lin Lee1,2, Paramate Horkaew1, Ara Darzi2, Guang-Zhong Yang1,2 1 Royal Society/Wolfson Foundation MIC Laboratory, Department of Computing 2 Department of Surgical Oncology and Technology Imperial College, London, UK Abstract. Defective pelvic organ support due to injuries of the levator ani is a common problem in women and its intervention requires a thorough understanding of the structure. Three-dimensional surfaces of the levator ani have proved to be a promising method of studying this. In this paper, we propose to build a statistical shape model (SSM) of the levator ani and describe a segmentation technique based on a limited number of control points with the SSM. The SSM was achieved by the use of harmonic shape embedding with the MDL objective function to optimise parameterisation while segmentation was performed by fitting the model to a user defined set of control points. The value of the technique was demonstrated with data acquired from a group of 11 asymptomatic subjects.  1 Introduction Pain, urinary or faecal incontinence or constipation can be the results of injuries to the levator ani due to childbirth [1]. Locating the injuries is of prime importance to the prescription of suitable treatment (pelvic floor exercises, surgery). Due to its excellent tissue contrast, conventional 2D MR imaging techniques have been used for the assessment of the levator ani [2], with diagnosis made on the position of the organs (rectum, bladder) with respect to structural landmarks. 3D representation of the levator ani is a new idea [3, 4] that has yielded findings that differed between symptomatic and asymptomatic patients. Visual comparison also showed a continuum in levator volume degradation, loss of sling integrity and laxity in the order of asymptomatic, genuine stress incontinence and prolapse. It has also been found that the volume of the levator ani was a sign of pelvic floor dysfunction [5, 6]. In both studies, the levator ani was manually segmented from a set of image slices, a time consuming process.  The purpose of this paper is to investigate a method of segmenting a levator surface using a user defined set of control points and a statistical shape model (SSM). Cootes et al [7] investigated the use of shape models and their use in automatic segmentation of images. Segmentation using a model requires the entire set of control points to be deformed as well as a set of heuristics describing the shape in the images. With a smaller set of points, a user can quickly determine which points define the primary features of the surface. As the surface of the levator is topologically homeomorphic to a compact 2D manifold with boundary (sheet topology), the statistical shape model will be built using a novel method by Horkaew and Yang [8].  2 Methodology Triangulated surfaces were generated for the eleven levators and each was parameterised onto a unit quadrilateral base domain. Each vertex is uniquely defined in the internal mapping by the minimisation of metric dispersion – a measure of the extent to which regions of small diameter are distorted when mapped. The harmonic map [9], the minimisation of the total energy of the configuration of the points over the base domain, was solved by computing its piecewise linear approximation [10],  [ ] ( ) ( ){ }∈−=Mjiijharm jiE,221 φφκφ  (1) where the spring constants computed for each edge {i, j}are, ( ) ( )( ) ( )( ) ( )( ) ( )−×−−⋅−+−×−−⋅−=2222111121,kjkikjkikjkikjkiji vvvvvvvvvvvvvvvvκ  (2) A sparse linear system is solved for the values ( )iφ at the critical point to find the unique minimum of equation (1). A B-spline surface patch was constructed from each mesh by reparameterising the harmonic embedding over uniform knots. The approximate tensor product B-spline was calculated from a set of distinct points in the parameterised base domain. Given the minimal distortion map, the least squares approximation by B-spline with a thin-plane spline energy term bears well defined smooth surfaces. Correspondences in the training set of B-spline surfaces were found by reparameterising the surfaces over the unit base domain. This is defined by a Piecewise Bilinear Map (PBM), to which multi-resolution decomposition can be applied, which results in a hierarchy representation of the parameterisation spaces. Higher dimensions can model finer detail of the distortion.  The Minimum Description Length (MDL) was used to select the parameterisation that is used to build the statistical shape model, similar to the work by Davies et al [11]. The MDL principle proposes choosing the model that provides the shortest description of the data: both the data and model parameters. At each level of iteration in the algorithm, the parameterisations were refined and the PBM parameters optimised according to the MDL objective function. The sampling rate on each B-spline surface was also increased, creating the concurrent hierarchy on both the parameterisation domain and the shapes, thus providing reliable convergence. Polak-Ribiere’s conjugate gradient optimisation [12] was employed.  All but one of the levator ani surfaces were used in each training set (for a leave-one-out error analysis). Twelve control points were selected on the surface of each levator ani and each model was fitted to the set of points, minimising the distance from the model surface to the points. The error was calculated as the mean distance between corresponding control points in the model. The control points of the model are automatically manipulated until the error between the surface and the image control points is minimised.  Simulated annealing was used for the fitting of the control points as well as for updating the pose parameters of the model.  The data were acquired with a Siemens Sonata 1.5T scanner with the subject assuming the supine position. A turbo spin echo not-zone selective sequence (TR=1500ms, TE=130ms, slice thickness=3mm) was used to acquire 32-36 T2-weighted, transverse images for each of eleven nulliparous, female subjects. The levator ani was manually segmented from each set of images using an in-house developed 3D Slicer that allows for visualisation in an arbitrary plane. The control points (also selected in the 3D Slicer) used for shape restoration were the anterior most two points and posterior most point on the levator surface in 4 image planes. 3 Results The statistical shape model was first built with all 11 levator ani surfaces. Figure 1(a) demonstrates the shape changes corresponding to the first three principal modes of variation. The first mode varies the height of the levator ani. The second mode corresponds mostly to the variation of the “hump”, caused by the presence of the anal canal/rectum.  In the optimal model, the first three modes of variation provide 84.8% of the total variance whilst the equivalent value in the uniform model is 82.0%. The first four modes of variation capture 88.7% and 87.7% of the total variance respectively. A graph of this quantitative comparison is shown in Figure 1(b).  Figure 2(a) shows the position of the selected control points, used to reconstruct the 3D surface from the SSM, overlaid onto the original surface. Figure 2(b) is the 3D representation of the two surfaces, one derived from the complete 3D data (green) and the other from the user define control points (gold). It is evident that most of the error is at the edges and at the extremes of the original shape, where the control points were not located. For assessment of the errors, Figure 2(c) is a scatter plot of all the B-spline surface control points in 3D space for one shape during a leave-one-out exercise (average regression ratio 0.86893). 4 Discussion and Conclusion With this study, the user defined control points for segmentation were limited to twelve. This can be increased if time permits and the user may place more points where a feature is prominent, thus heavily weighting that feature. There are no limitations to the technique with regard to number of control points and their positions on the surface. Our future work will be applying the modelling technique to subjects with pelvic floor injuries as well as investigating the modelling of the muscle dynamics.                                  00.0010.0020.0030.0040.0050.0060 2 4 6 8 10 12modeeigenvalueUniformOptimised(b)  Figure 2. (a) Two views of the control points on the original surface. (b) Two views of the model segmented shape (gold) overlaid on the original shape (green). (c) Scatter plots for the control points on the original and model segmented shapes. Figure 1. (a) The first three modes of variation captured by the optimal statistical shape model. For each mode, the shape parameters have been varied by ±2σ. (b) A comparison of the compactness of each model. The results are not normalised. -0.3-0.2-0.100.10.20.3-0.3 -0.2 -0.1 0 0.1 0.2 0.3-0.3-0.2-0.100.10.20.3-0.3 -0.2 -0.1 0 0.1 0.2 0.3-0.3-0.2-0.100.10.20.3-0.3 -0.2 -0.1 0 0.1 0.2 0.3(a) (c) original shape control points control points of surface reconstructed X Z Y (b) (a)   In summary, we have proposed a method of segmentation based on a user defined set of control points on a set of images and a statistical shape model. The statistical shape model was created with the use of harmonic shape embedding and the MDL objective function was used to find the optimal parameterisation for construction. The 3D surface reconstruction was based on fitting the model control points to the user defined ones. Quantitative results from the 11 subjects demonstrate the potential of this method. We believe that statistical shape modelling is the way forward for studying the levator ani and that the proposed segmenting technique is a fast means of delineating the muscle from this anatomically complex area.  Acknowledgements The authors would like to thank S Masood, L Crowe and DN Firmin for their assistance in MR data acquisition for this study.  References 1. JC Healy, S Halligan, RH Reznek, et al. Patterns of Prolapse in Women with Symptoms of Pelvic Floor Weakness: Assessment with MR Imaging. Radiology, 203, 1997. 77-81. 2. JR Fielding, L Hoyte and L Schierlitz. MR Imaging of Pelvic Floor Relaxation. Journal of Women’s Imaging, 2, 2000. 82-87.  3. L Hoyte, L Schierlitz, K Zou, et al. Two- and 3-dimensional MRI comparison of levator ani structure, volume, and integrity in women with stress incontinence and prolapse. Am J Obstet Gynecol, 185(1), 2001. 4. K Duchicela, et al. MRI of the Aging Female Pelvic Floor; A 3D Perspective of its Principal Organization. International Continence Society Congress, Seoul, 2001.  5. L Hoyte, JR Fielding, E Versi et al. MR-based three-dimensional modelling of levator ani: First studies of muscle volume and geometry in living women with normal GU function, prolapse and genuine stress incontinence. Archivos Espanoles de Urologia, 54, 2001.  6. JR Fielding, H Dumanli, A Schreyer, et al. MR–based three dimensional modelling of the normal pelvic floor in women: quantification of muscle mass. Am J Radiol, 174, 2000. 657-660. 7. TF Cootes, CJ Taylor, DH Cooper, et al. Active Shape Models – Their Training and Application. Computer Vision and Image Understanding, 61(1), 1995. 38-59. 8. P Horkaew and G-Z Yang. Optimal Deformable Surface Models for 3D Medical Image Analysis. IPMI, 2003. (in press) 9. AP Fordy and JC Wood. Harmonic Maps and Integrable Systems. Aspects of Mathematics, volE23, Vieweg, Braunschweig/Wiesbaden, 1994.  10. M Eck, T DeRose, T Dutchamp, et al. Multiresolution Analysis of Arbitrary Meshes. Computer Graphics Proceedings SIGGRAPH, 1995. 173-182.  11. RH Davies, CJ Twining, TF Cootes, et al. A Minimum Description Length Approach to Statistical Shape Modelling. TMI, 2002.  12. WH Press, SA Teukolsky, WT Vetterling and BP Flannery. Numerical Recipes in C, 2nd edn. Cambridge University Press, 1996. ISBN 0-521-43108-5. 

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