FEM Model for Tumor Growth Analysis Presenter ： Liu Changyu( 刘昌余 ) Supervisor ： Prof. Shoubin Dong( 董守斌 ) Field ： High Performance Computing Otc. 10 th, 2012

2 Contents Basic Model Definitions Differential of a triangle area to its three vertex ’ s coordinates Differential of a tetrahedron volume to its three vertex ’ s coordinates Algorithm for meshing the initial cell Algorithm for cell division

Three phases for tumor growth 3 1. Avascular phase 2. Angiogenesis 3. Vascular tumor growth

Avascular tumor growth phases 4 1.PRE ： Day 2 2.MID ： Day 10 3.LAST ： Day 18

5 Tumor total energy and Its Increment Tumor total energy After cells have growth, the energy may changed as

6 Energy Increment in Forms of Nodal Displacement Transmission of local nodal displacement vector {u} AL_C to global nodal displacement vector {u} [T] is a 3NLN×3NN transform matrix, each element in it is 1 or 0 Then,

7 Energy Increment in Forms of Nodal Displacement (Cont’) Increment of a cell Increment of a minor area Energy increment

8 Minimum Energy Principle For all possible surface displacements {u} of cells, the real one make the energy increment △ H minimum. Finite element equation

9 Tumor Growth Stiffness Matrix & Tumor Growth Driving Force Tumor growth stiffness matrix Tumor growth driving force FEM Equation

10 Contents Basic model Definitions Differential of a triangle area to its three vertex ’ s coordinates Differential of a tetrahedron volume to its three vertex ’ s coordinates Algorithm for meshing the initial cell Algorithm for cell division

11 Some Parameters for the Elemental Description of the Tumor Cells Cell: –CN: Current total cell number, at beginning CN=1, then CN=CN+1 in the case of a cell splitting –C: Serial number of each cell, range of C is 1~CN Minor Areas: –AN: Global total minor surface number –A: Global serial number of a minor surface –ALN: Local total minor surface number is a cell –AL: Local serial number of a minor surface in a cell –LA: It is a two dimensional array which links a local surface number to its global surface number. LA(C, AL)=A

12 Some Parameters for the Elemental Description of the Tumor Cells Nodes: –NN: Total nodal number –N: Global serial nodal number –NLN: Total nodal number in a minor surface, now NLN=3 –NL: Local serial nodal number in a surface, now NL=1,2,3 –LN: It a three dimensional array, which links a local node number to its global node number. LN(C, NL)=N

13 Some Parameters for the Elemental Description of the Tumor Cells Minor Area and its Nodes Relation Arrays – : The first node of a minor area; – : The second node of a minor area; – : The third node of a minor area;

14 Some Parameters for the Elemental Description of the Tumor Cells Coordinates and displacements of nodes: –x: coordinates of nodes in x axis, x(N) –y: coordinates of nodes in y axis, y(N) –z: coordinates of nodes in z axis, z(N) –u: x directional displacement of nodes, u(N) –v: y directional displacement of nodes, v(N) –w: z directional displacement of nodes, w(N) Surface property: –J: the surface energy in a unit area, which will have different value correspondent to the surface contact property.

15 Contents Basic ideas Definitions Differential of a triangle area to its three vertex ’ s coordinates Differential of a tetrahedron volume to its three vertex ’ s coordinates Finite Element Equations Algorithm for meshing the initial cell Algorithm for cell division

16 Geometry S 123 is a minor surface of a cell surface, O is the centroid of the cell O (0, 0, 0) 1(x 1, y 1, z 1 ) 2(x 2, y 2, z 2 ) y z 3(x 3, y 3, z 3 ) x

17 Area Expression Because Define a area vector

18 Differential of the Area According to Lagrange identity

19 Differential of the Area (Cont’) Side ’ s relation within a triangle Final expression of differential area

20 Differential and Dispalcement In the finite element model

21 Nodal Displacement Vector and Surface Spring Vector Nodal Displacement Vector Surface Spring Vector

22 Differential Area in Form of Matrix Differential area Introduce a note “ AL_C ” to representative the minor surface “ AL ” in Cell “ C ”

23 Contents Basic model Definitions Differential of a triangle area to its three vertex ’ s coordinates Differential of a tetrahedron volume to its three vertex ’ s coordinates Algorithm for meshing the initial cell Algorithm for cell division

24 Volume of a Tetrahedron Volume of a tetrahedron can be expressed as O (0, 0, 0) 1(x 1, y 1, z 1 ) 2(x 2, y 2, z 2 ) y z 3(x 3, y 3, z 3 ) x

25 Differential Volume Differential to a tetrahedron volume According to vector ’ s identify Differential volume

26 Differential Volume in Form of Matrix A volume spring vector

27 Differential Volume in Form of Matrix (Cont’) Differential volume in form of matrix Similar to the area form

28 Contents Basic model Definitions Differential of a triangle area to its three vertex ’ s coordinates Differential of a tetrahedron volume to its three vertex ’ s coordinates Algorithm for meshing the initial cell Algorithm for cell division

29 Meshing Overview Homogeneous equilateral triangle used Cell is divided into 2n sections equably in space interval  ∈ [0,  ] i counter is for the increment of  j counter is for the increment of 

30 Meshing Algorithm Local node number –Increase the nodal number with the increment of i, j; –From top pole to the equatorial nodal ring, the increment of the nodal number is 6, –After equatorial nodal ring, the nodal number inversely reduces in each nodal ring –Nodal coordinates

31 Meshing Algorithm Local area number –Increasing with nodal number; –Increasing once with nodes located on a longitude; –Increasing twice with other nodes –The element number in each element belt is 6*(2i-1) before n Detail seen the program

32 Contents Basic model Definitions Differential of a triangle area to its three vertex ’ s coordinates Differential of a tetrahedron volume to its three vertex ’ s coordinates Algorithm for meshing the initial cell Algorithm for cell division

33 Aims of the algorithm for the cell division To choose to proper spatial surfaces to “ cut ” a cell C into two cells C1, C2 under the condition of averaging the cell ’ s volume; To mesh the new cut surfaces for the two cells

34 Calculating the Half Volume Cone shell –Area belt connected to the centroid –Volume of a cone shell V i Dome volume DV i, Rule to judge the half volume O i=1 i=2 i=3

35 Meshing new interface Connecting C to the 6 nodes located on the longitudes get 6 radial lines; Inserting (n-abs(n-k))-1 nodes equably in each radial lines; Connecting new nodes in same radial layer sequentially from inner to outer; From inner to outer radial layer, each new circumferential line section is inserted 0, 1, …, (n- abs(n-k))-1 nodes equably; All nodes connecting their neighbor nodes to consist triangle elements

36 Meshing new interface L0 L1 L2 L3 L4 L5 C L0 L1 L2 L3 L4 L5 C

37 Heritage Nodal and Elemental Number from Un- divided Cell Cell C1 –The nodal number and element number before the k element belt will inherit from the cell C directly Cell C2 –Renumber both the element number and nodal number inversely in cell C –Change the nodal number in each nodal ring to match the nodal –Nodal number and element number before the (2n-k) element belt of cell C2 can inherit from reversed cell C directly

38 Matrix assembly

39 Thank you!