Upload
magdalen-bruce
View
243
Download
0
Embed Size (px)
DESCRIPTION
Three phases for tumor growth 3 1. Avascular phase 2. Angiogenesis 3. Vascular tumor growth
Citation preview
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
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
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
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
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
17
Area Expression
Because
Define a area vector
baS
21
)(21
)()(21
21
133221
1231
rrrrrr
rrrrbaS
SS
ccbbaa ,,
18
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
22
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
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
)(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
26
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}{
27
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
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
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
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
NLNL
NLNLNL
NLNLNL
RzRyRx
cossinsincossin
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
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
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 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
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
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
T1T1 T2T2 T3T3
T1T1
T2T2
T3T3
T2-1T2-1 T2T2 T1T1 T3T311 NNNN
11
T2-1T2-1
T2T2
T1T1
T3T3
NNNN
Matrix assembly
39
Thank you!Thank you!