FEM Model for Tumor Growth Analysis Presenter ： Liu Changyu( 刘昌余 ) Supervisor ： Prof....

FEM Model for Tumor Growth AnalysisFEM Model for Tumor Growth Analysis

Presenter ： Liu Changyu( 刘昌余 )Supervisor ： Prof. Shoubin Dong( 董守斌 )Field ： High Performance Computing

Otc. 10th, 2012

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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

CN

CCCA

AN

AA VVSJH

1

*

1

)2(

CN

jCCC

CN

CCA

AN

AA

CN

CCCA

AN

AA

CN

CCCCAA

AN

AA

VVVVSJ

VVSJVVVSSJH

1

*

1

2

1

1

2*

11

2*

1

)(d2)d(d

)()d()d(

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,

}{][}{ __ uTu CALCAL

}{)}({}){][))({(dd T

1

T

1_ uKuTKVV CV

ALN

ALAL_CAL_CV

ALN

ALCALC

7

Energy Increment in Forms of Nodal Displacement (Cont’)

Increment of a cell

Increment of a minor area

Energy increment

}{))})({}({(}{

}){)}({)(2(}){)}({(

1

TT

T

1

*T

1

uKKu

uKVVuKJH

CN

CCVCV

CV

CN

CCCAS

AN

AA

}{)}({}){][))({(dd T

1

T

1_ uKuTKVV CV

ALN

ALAL_CAL_CV

ALN

ALCALC

}{)}({}{][)}({}{)}({d T____ uKuTKuKS ASCAL

TCALSCAL

TCALSA

8

Minimum Energy Principle

For all possible surface displacements {u} of cells, the real one make the energy increment △H minimum.

Finite element equation

0}{

)(

uH

AS

AN

AACV

CN

CCC

CN

CCVCV KJKVVuKK }{

21}{)(}{))})({}({(

11

*

1

T

9

Tumor Growth Stiffness Matrix & Tumor Growth Driving Force

Tumor growth stiffness matrix

Tumor growth driving force

FEM Equation

CN

CCVCV KKK

1

T)}({}{][

AS

AN

AACV

CN

CCC KJKVVF }{

21}{)(}{

11

*

}{}]{[ FuK

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;

1ALN2ALN3ALN

1),(1 NLALCN AL

2),(2 NLALCN AL

3),(3 NLALCN AL

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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

S123 is a minor surface of a cell surface, O is the centroid of the cell

O (0, 0, 0)

1(x1, y1, z1)

2(x2, y2, z2)

y

z

3(x3, y3, z3)

a

b

c

n

1r

2r

3r

x

12 rra

23 rrb

31 rrc

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Area Expression

Because

Define a area vector

baS

21

)(21

)()(21

21

133221

1231

rrrrrr

rrrrbaS

SS

ccbbaa ,,

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Differential of the Area

||ddd

SSSSS

)ddd

ddd()(41d

131332

322121133221

rrrrrr

rrrrrrrrrrrrS

According to Lagrange identity ))(())(()()( cbdadbcadcba

33211122

213231121113231222

2331113212

2313311133123221

1331213222

2333231211323322

d])2( )()[(

d])( )2()[(

d])( )()2[(d4

rrrrrrrrrrrrrrrrrrrrrrrrrr

rrrrrrrrrrrrrrrrrrrrrrrrrr

rrrrrrrrrrrrrrrrrrrrrrrrrrS

19

Differential of the Area (Cont’)

Side’s relation within a triangle

Final expression of differential area

221

2313

223

2232

222

2121

2

2

2

crrrr

brrrr

arrrr

3321

2321

1321

d])()()[(

d])()()[(

d])()()[(d4

rraarbarba

rracrccrbc

rrabrcbrbbS

20

Differential and Dispalcement

In the finite element model

3

3

3

3

2

2

2

2

1

1

1

1 d,d,dwvu

rwvu

rwvu

r

21

Nodal Displacement Vector and Surface Spring Vector

zzz

yyy

xxx

zzz

yyy

xxx

zzz

yyy

xxx

S

raarcarbaraarcarbaraarcarbaracrccrbcracrccrbcracrccrbcrabrcbrbbrabrcbrbbrabrcbrbb

K

wvuwvuwvu

321

321

321

321

321

321

321

321

321

3

3

3

3

2

2

1

1

1

)()()()()()()()()()()()()()()()()()()()()()()()()()()(

41}{,{u}

Nodal Displacement Vector

Surface Spring Vector

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Differential Area in Form of Matrix

Differential area

Introduce a note “AL_C” to representative the minor surface “AL” in Cell “C”

}{}{d uKS TS

CALT

CALSCALA uKSS ___ }{)}({dd

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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

)(61

321 rrrV

O (0, 0, 0)

1(x1, y1, z1)

2(x2, y2, z2)

y

z

3(x3, y3, z3)

a

b

c

n

1r

2r

3r

x

25

Differential Volume

Differential to a tetrahedron volume

According to vector’s identify

Differential volume

)d()d()(dd6 321321321 rrrrrrrrrV

)()()( bacacbcba

)(d)(d)(dd6 213132321 rrrrrrrrrV

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Differential Volume in Form of Matrix

A volume spring vector

xyyx

zxxz

yzzy

xyyx

zxxz

yzzy

xyyx

zxxz

yzzy

V

rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr

K

2121

2121

2121

1313

1313

1313

3232

3232

3232

61}{

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Differential Volume in Form of Matrix (Cont’)

Differential volume in form of matrix

Similar to the area form

}{}{d uKS TS

CALAL_CVCAL uKV _T

_ }{)}{(d

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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

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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

o

i=1

i=2

i=3

Element Belt1

2 347

① ②

⑦

⑥

⑧⑨

⑩8

911

121819

10

20

Nodal Ring

i=0

i=1

i=2

i=3

21 22 2324

2526

35

3637

x

y

z

Longitude L0 Longitude L1

Longitude L2Longitude L5

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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

NLNL

NLNLNL

NLNLNL

RzRyRx

cossinsincossin

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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

o

i=1

i=2

i=3

Element Belt1

2 347

① ②

⑦

⑥

⑧⑨

⑩8

911

121819

10

20

Nodal Ring

i=0

i=1

i=2

i=3

21 22 2324

2526

35

3637

x

y

z

Longitude L0 Longitude L1

Longitude L2Longitude L5

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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 Vi

Dome volume DVi,

Rule to judge the half volume O

i=1

i=2

i=3

ij

jji VDV

1

121

21

,21

1

1

ikelseikthenVDVDVVif

DVVDVWhen

CiiC

iCi

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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

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Meshing new interface

L0 L1

L2

L3

L4

L5

C

L0 L1

L2

L3

L4

L5

C

L0 L1

L2

L3

L4

L5

C

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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

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T1T1 T2T2 T3T3

T1T1

T2T2

T3T3

T2-1T2-1 T2T2 T1T1 T3T311 NNNN

11

T2-1T2-1

T2T2

T1T1

T3T3

NNNN

Matrix assembly

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Thank you!Thank you!