FEM Model for Tumor Growth Analysis Presenter Liu

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

Contents l l l 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 2

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

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

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

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

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

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

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

Contents l l l 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 10

Some Parameters for the Elemental Description of the Tumor Cells l 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 l 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 11

Some Parameters for the Elemental Description of the Tumor Cells l 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 12

Some Parameters for the Elemental Description of the Tumor Cells l 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; 13

Some Parameters for the Elemental Description of the Tumor Cells l Coordinates and displacements of nodes: – – – l 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. 14

Contents l l l l 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 15

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

Area Expression l Because l Define a area vector 17

Differential of the Area According to Lagrange identity 18

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

Differential and Dispalcement l In the finite element model 20

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

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

Contents l l l 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 23

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

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

Differential Volume in Form of Matrix l A volume spring vector 26

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

Contents l l l 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 28

Meshing Overview l l Homogeneous equilateral triangle Nodal Ring used i=0 Cell is divided into 2 n 7 i=1 sections equably in space interval ∈[0, ] i=2 18 19 35 i=3 i counter is for the 36 37 increment of j counter is for the Longitude L 5 increment of x 29 z Element Belt 1 ⑥ 2 8 ⑦ ① ⑧ 9 i=1 ② 4 i=2 3 12 ⑩ 11 ⑨ 10 i=3 26 25 20 21 o 22 23 24 y Longitude L 0 Longitude L 1 Longitude L 2

Meshing Algorithm l 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 30

Meshing Algorithm l Local area number z – Increasing with nodal Nodal Ring i=0 number; 7 – Increasing once with i=1 18 nodes located on a i=2 19 longitude; 35 i=3 – Increasing twice with 36 37 other nodes – The element number in Longitude L 5 each element belt is x 6*(2 i-1) before n l Detail seen the program 31 Element Belt 1 ⑥ 2 8 ⑦ ① ⑧ 9 i=1 ② 4 i=2 3 12 ⑩ 11 ⑨ 10 i=3 26 25 20 21 o 22 23 24 y Longitude L 0 Longitude L 1 Longitude L 2

Contents l l l 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 32

Aims of the algorithm for the cell division l l To choose to proper spatial surfaces to “cut” a cell C into two cells C 1, C 2 under the condition of averaging the cell’s volume; To mesh the new cut surfaces for the two cells 33

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

Meshing new interface l l l 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, …, (nabs(n-k))-1 nodes equably; All nodes connecting their neighbor nodes to consist triangle elements 35

Meshing new interface L 4 L 4 L 3 L 3 L 5 L 5 C L 0 C L 2 L 1 C L 2 L 0 L 1 36 L 0 L 2 L 1

Heritage Nodal and Elemental Number from Undivided Cell l Cell C 1 – The nodal number and element number before the k element belt will inherit from the cell C directly l Cell C 2 – 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 (2 n-k) element belt of cell C 2 can inherit from reversed cell C directly 37

Matrix assembly 38

Thank you! 39