Geometrical Modeling of the Heart

Transcription

1 Geometrical Modeling of the Heart Olivier Rousseau University of Ottawa

2 The Project Goal: Creation of a precise geometrical model of the heart Applications: Numerical calculations Dynamic of the blood flow Mechanical deformations of the heart Electro physiology Methodology: Extract heart characteristics from medical images Boundaries of muscles and cavities Fine anatomical details (valves, fibers,... ) Create a mesh Techniques: Variational methods in BV space Mumford Shah functional Level Set

3 Overview 1 What already exists Available data 3 The segmentation problem 4 Mesh generation

4 Hunter's heart (Hunter & al.) Cubic mesh From anatomical measures on canine heart.

5 Hunter's heart (Hunter & al.) Cubic mesh From anatomical measures on canine heart. Fibers

6 Deformable models (Pages & al.) Tetrahedral mesh Approximate heart is specified by user and is deformed to better fit the image. Surface is meshed, then the inside. Problems: May contain holes Lack of precision

7 . Available Data MRI (Magnetic Resonance Imaging) courtesy of Sunnybrook Health Sciences Center University of Toronto Visible Human data (Photographies) courtesy of the United States National Library of Medicine CT scan (Computed Tomography) courtesy of the Pet Center Heart Institute, University of Ottawa MRI diffusion courtesy of the Center for Cardiovascular Bio informatics and Modeling Johns Hopkins University

8 MRI Dimensions: 56 x 56 x 14 Spacing: 1. x 1. x 6 # Pixels:

9 Visible Human (Female) Dimensions: 500 x 500 x 400 Spacing: 0.33 x 0.33 x 0.33 # Pixels:

10 CT Dimensions: 51 x 51 x 199 Spacing: 0.5 x 0.5 x 1.5 # Pixels:

11 MRI diffusion Dimensions: 56 x 56 x 134 Spacing: 0.4 x 0.4 x 1 # Pixels: Fiber Orientation!

12 3. The Segmentation Problem The problem Active Contours Level Set Method Mumford Shah Functional Active Contours without Edges

13 3. The Segmentation problem Identify significant regions and boundaries

14 3. The Segmentation problem Identify significant regions and boundaries

15 Active contours (Terzopoulos & al.) Let g : ℝ be an image. 1 Look for a curve v : S that approximates well the boundaries, ie. minimizes the snake energy. E snake v = E i n t E e x t Ei nt = S 1 Elasticity term dv ds d v ds ds Rigidity term

16 Active contours (Terzopoulos & al.) Let g : ℝ be an image. 1 Look for a curve v : S that approximates well the boundaries, ie. minimizes the snake energy. E snake v = E i n t E e x t Ei nt = S 1 dv ds d v ds ds E e x t = g v s S 1

17 Active contours To solve: Start with initial contour push towards the real contour (in the direction of the negative gradient of the snake energy) 1 Discretize Advance the front 3 Re-parametrize

18 Active contours Advantages: The method works well if the initial contour is close to the real contour Disadvantages: Works bad if the initial contour is far Cannot handle topology changes

19 Active contours Advantages: The method works well if the initial contour is close to the real contour Disadvantages: Works bad if the initial contour is far Cannot handle topology changes

20 Active contours Advantages: The method works well if the initial contour is close to the real contour Disadvantages: Works bad if the initial contour is far Cannot handle topology changes

21 Level Set Method (Osher, Sethian) Describe the curve evolution via : ℝ ℝ C t = { x : x, t =0 } The problem becomes an initial condition problem d =F dt x, 0 = 0 Handles topology changes. F do not have to come from a minimization problem. This is called the Level Set Method

22 Level Set Method

23 Level Set Method The evolution equation: d dt Motion in the normal direction: =F F= V Examples: V =g, g is an edge stopper, eg. g x = V =g k, k =d i v 1 1 x is the mean curvature

24 Edge stopping Level Set Advantages: Handles topology changes Disadvantages: It is local: it depends on the region where the initial curve is. It won't detect contours that are far away from it. Hard to locate unclear edges: 1 Between regions of close intensities That are not sharp

25 Mumford Shah functional The idea of Mumford and Shah is to find an image u that minimizes N 1 u g u H J u ℝ H N 1 N N-1 Hausdorff measure J u jump set of the function u

26 The BV space The Mumford-Shah problem has a solution in the space of Special Functions of Bounded Variation BV = u L 1 : [ D ' ]N, { u d i v = d } is a finite Radon measure representing the derivative in the sense of distributions of u. In fact: N = u L u u u H Sobolev part Jump part N 1 C J u D u Cantor part

27 The BV space The Mumford-Shah problem has a solution in the space of Special Functions of Bounded Variation N 1 BV = u L : [ D ' ], { u d i v = d } is a finite Radon measure representing the derivative in the sense of distributions of u. In fact: N = u L u u u H N 1 C J u D u C SBV = {u BV : D u=0 }

28 Active Contours Without Edges (Chan & al.) Minimize the Mumford-Shah functional over binary functions u. Interface: C={ =0} u=c 1 H c 1 H =0 0 0

29 Euler Lagrange Equation The Mumford-Shah functional F u = u g u H N 1 J u becomes F u = c 1 u H c u 1 H The Euler-Lagrange equation: d dt [ = d i v g c 1 g c ]

30 Solving the problem d dt [ = d i v Explicit forward Euler = g c 1 g c Centered differences x / ]

31 Solving the problem d dt [ = d i v At each time step: 1 Compute c 1 and c from Compute n 1 explicitly n g c 1 g c ]

32 Stability A common choice for initial condition is to take 0 as the signed distance function to a given curve. In this case = 1 Then d i v simply becomes CFL condition: We can't hope to do better t x

33 Example An active contour model without edges Chan & Vese

34 More regions To obtain more than regions, there are possible strategies: 1 Evolve several curves at the same time Problem: stability Iterative segmentation

35 Results D CT pixels

36 Results D CT pixels

37 Results D CT pixels

38 Results D CT pixels

39 Results D CT pixels

40 Results D CT pixels

41 Results 3D CT pixels

42 Results 3D CT pixels

43 Results 3D CT pixels

44 Results 3D CT pixels

45 Results 3D CT pixels

46 Results 3D CT pixels

47 4. Mesh generation Regular mesh Adapted mesh Adapting tools: GIREF (Groupe Interdisciplinaire de Recherche en Elements Finis) u: function defined on regular mesh H: Hessian matrix of u (metric) Repeat the Adaptation sequence until the length of all edges is a constant c Adaptation sequence: Swap edges & move nodes Refine Swap edges & move nodes Un-refine Swap edges & move nodes

48 D Meshes Function: in the heart: 1000 x y outside: 0.0 x y elements

49 D Meshes Function: in the heart: 1000 x y outside: 0.0 x y elements

50 D Meshes Function: solution of u= f Where f has value 10 inside the heart and 0.0 outside elements

51 D Meshes Function: solution of u= f Where f has value 10 inside the heart and 0.0 outside elements

52 D Meshes Function: solution of u= f Where f has value 100 inside the heart and 0.01 outside elements

53 D Meshes Function: solution of u= f Where f has value 100 inside the heart and 0.01 outside elements