Discontinuous Hermite Collocation and DiagonallyImplicit RK3 for a Brain Tumour Invasion Model

John E. Athanasakis

Applied Mathematics & Computers Laboratory

Technical University of Crete

Chania 73100, Greece

gathanasakis@gmail.com

1

Gliomas

Gliomas are primary brain tumours that originate from brain’s glialcells. They usually occur in the cerebral (upper) brain’s hemisphereand they are characterized by their aggressive di↵usive invasion ofbrain normal tissue.

2

Analytical Deterministic Models

@c

@t= r · (Drc) + ⇢g(c) , (Cruywagen et.al 1995)

c(x, t) : tumour cell density at location x and time t

D : di↵usion coe�cient representing the active motility(0.0013 cm2/day)

⇢ : net proliferation rate (0.012/day)

g(c) =

⇢c , exponential growth

c(1� c

K

) , logistic growth , K : carrying capacity

Zero flux BC on the anatomy boundaries : @c@⌘ = 0

Initial spatial distribution malignant cells : c(x, 0) = f (x)

3

Analytical Deterministic Models

Homogeneous Brain Tissue (Cruywagen et.al 1995)

#

D is constant

#

@c

@t= Dr2

c + ⇢g(c)

4

Analytical Deterministic Models

Heterogeneous Brain Tissue (Swanson 1999)

#

D(x) =

⇢D

g

, x in Grey MatterD

w

, x in White Matter, D

w

> D

g

#

@c

@t= r · (Drc) + ⇢g(c)

continuity and flux preservation conditions at the interfaces

5

Exponential Growth 3-region Model in 1+1 Dimensions

Dimensionless variables

x q

⇢D

w

x , t ⇢t , c(x , t) c

⇣q⇢D

w

x , ⇢t⌘

D

w

⇢N0

D =

8<

:

� , ↵ x < w

1

1 , w

1

x < w

2

, � =D

g

D

w

< 1

� , w

2

x b

ba

γ γ

11

w1

w2

Transformationc(x , t) = e

t

u(x , t)

Interface conditions at x = w

k

, k = 1, 2⇢

Continuity : [u] := u

+ � u

� = 0 ,Conservation of flux : [Du

x

] := D

+

u

+

x

� D

�u

�x

= 0 ,

6

Exponential Growth 3-region Model in 1+1 Dimensions

8>>>>>>>><

>>>>>>>>:

u

t

= Du

xx

, x 2 R` , ` = 1, 2, 3 , t > 0

u

x

(↵, t) = 0 and u

x

(b, t) = 0

[u] = 0 and [Dux

] = 0 at x = w

k

, k = 1, 2

u(x , 0) = f (x)

R1

:= (↵,w1

) , R2

:= (w1

,w2

) , R3

:= (w2

, b)

t

a w1 w2 b

ux

=0

ux

=0

ut = �uxx ut = uxx ut = �uxx

u(x, 0) = f(x)

7

Domain Discretization

Discretization

�t

x = ↵ x = bt = 0

h�

(xm, tn)

tn = n�t

tn+1

tn�1

x

m

:= ↵+(m�1)h , m = 1, . . . ,N+1 , t

n

= n⌧ , n = 0, 1, . . .

h =

8<

:

h

1

:= (w1

� ↵)/N1

h

2

:= (w2

� w

1

)/N2

h

3

:= (b � w

2

)/N3

, N = N

1

+N

2

+N

3

, ⌧ = �t

8

Hermite Collocation

U(x , t) =N+1X

m=1

[↵2m�1

(t)�2m�1

(x) + ↵2m

(t)�2m

(x)]

�2m�1

(x) =

8>>>><

>>>>:

��x

m

�x

h

�, x 2 [x

m�1

, xm

]

��x�x

m

h

�, x 2 [x

m

, xm+1

]

0 , otherwise

,

�2m

(x) =

8>>>><

>>>>:

�h �x

m

�x

h

�, x 2 [x

m�1

, xm

]

h �x�x

m

h

�, x 2 [x

m

, xm+1

]

0 , otherwise

(1)

�(s) = (1� s)2(1 + 2s) , (s) = s(1� s)2 , s 2 [0, 1]

9

Interface Conditions

[DUx

] := D

+

U

+

x

� D

�U

�x

= 0 , at x = w

k

, k = 1, 2

# #

� �2i

(x�i

) = �2i

(x+i

) �2i

(x�i

) = ��2i

(x+i

)x

i

⌘ w

1

x

i

⌘ w

2

i = N

1

+ 1 i = N

1

+ N

2

+ 1

# #

�2i

(x) =

8>>>><

>>>>:

� h

� �x

i

�x

h

�

h �x�x

i

h

�

0

�2i

(x) =

8>>>><

>>>>:

�h �x

i

�x

h

�

h

� �x�x

i

h

�

0

10

Hermite Elements at the Interfaces

φ2i−1

(x)

xi−1 x

ix

i+1

φ2i

(x)

i = N

1

+ 1

φ2i−1

(x)

φ2i

(x)

xi−1

xi

xi+1

i = N

1

+ N

2

+ 1

11

Boundary Collocation equations

U

x

(↵, t) = 0 �! ↵2

(t) = 0 �! ↵2

(t) = 0

U

x

(b, t) = 0 �! ↵2N+2

(t) = 0 �! ↵2N+2

(t) = 0

12

Interior Collocation equations

N+1X

j=1

[↵2j�1

(t)�2j�1

(�i

) + ↵2j

(t)�2j

(�i

)] = DN+1X

j=1

⇥↵2j�1

(t)�002j�1

(�i

) + ↵2j

(t)�002j

(�i

)⇤

where �i

, i = 1, . . . , 2N are the Gauss points (two per subinterval)

#

Elemental Collocation equations

C

(0)

j

2

666666664

↵2j�1

↵2j

↵2j+1

↵2j+2

3

777777775

=D

h

2

C

(2)

j

2

666666664

↵2j�1

↵2j

↵2j+1

↵2j+2

3

777777775

, j = 1, . . . ,N

13

Elemental Collocation matrices

C

()j

=

2

64s

()1

h

⇣j

s

()2

s

()3

� h

�j

s

()4

s

()3

h

⇣j

s

()4

s

()1

� h

�j

s

()2

3

75 , = 0, 2

s

(0)

1

= 9+4

p3

18

, s

(0)

2

= 3+

p3

36

, s

(0)

3

= 9�4

p3

18

, s

(0)

4

= 3�p3

36

,

s

(2)

1

= �2p3 , s

(2)

2

= �1�p3 , s

(2)

3

= 2p3 and s

(2)

4

= �1 +p3

⇣j

=

8<

:

1 , j 6= N

1

+ N

2

+ 1

� , j = N

1

+ N

2

+ 1, �

j

=

8<

:

1 , j 6= N

1

� , j = N

1

.

14

Collocation system of ODEs

Aa = Ba

where a =⇥↵1

↵3

· · · ↵2N+1

⇤T

, a =⇥↵1

↵3

· · ·↵2N+1

⇤T

A =

2

66664

A

1

B

1

A

2

B

2

�A

N�1

B

N�1

A

N

B

N

3

77775, B =

2

66664

F

1

G

1

F

2

G

2

�F

N�1

G

N�1

F

N

G

N

3

77775.

⇥A

j

B

j

⇤= C

(0)

j

and⇥F

j

G

j

⇤=

D

h

2

C

(2)

j

.

a = C (a, t)

where C (a, t) = A

�1

Ba

15

Backward Euler (BE) - Crank Nicolson (CN)

BE:a

(n+1) � a

(n)

⌧= C

(n+1)(a, t)

or

(A� ⌧B)a(n+1) = Aa

(n)

CN:a

(n+1) � a

(n)

⌧=

1

2(C (n+1)(a, t) + C

(n)(a, t))

or

(A� ⌧

2B) a(n+1) = (A+

⌧

2B) a(n)

16

Diagonally Implicit RK3 (DIRK)

DIRK:

a

(n,1) = a

(n) + ⌧ � C

(n,1)(a, t)

a

(n,2) = a

(n) + ⌧⇥(1� 2�)Cn,1(a, t) + �Cn,2(a, t)

⇤

a

(n+1) = a

(n) +⌧

2

hC

(n,1)(a, t) + C

(n,2)(a, t)i

or

(A� ⌧�B) a(n,1) = A a

(n)

(A� ⌧�B) a(n,2) = A a

(n) + ⌧(1� 2�)B a

(n,1)

A a

(n+1) = A a

(n) +⌧

2[B a

(n,1) + B a

(n,2)]

17

Numerical Results - Model Problem 1

R1

:= (�5,�1) , R2

:= (�1, 1) , R3

:= (1, 5), � = 0.5

and f (x) = 1

⌘p

⇡e

�x

2/⌘2

, with ⌘ = 0.2.

−5 −4 −3 −2 −1 0 1 2 3 4 50

2

4

6

8

10

12

τ=0.1 tmax

=4

Spatial variable x

Tum

our

gro

wth

c(x

,t)

18

Numerical Results - Model Problem 2

R1

:= (�5, 1) , R2

:= (1, 1.5) , R3

:= (1.5, 5), � = 0.5

and f (x) = 1

⌘p

⇡(e�(x+3.5)2/⌘2

+ e

�(x�3)

2/⌘2

) , with ⌘ = 0.2.

−5 −4 −3 −2 −1 0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

τ=0.1, tmax

=4

Spatial variable x

Tum

our

gro

wth

c(x

,t)

19

Numerical Results - Spatial Relative Error

Model Problem 1 Model Problem 2

102

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

g=0.5 ksi=0 w1=−1 w2=1 [−5 5] dt=0.01

Number of Elements

Rela

tive E

rror

DIRK

CN

BE

102

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Number of ElementsR

ela

tive E

rror

DIRK

CN

BE

20

Logistic Growth 3-region Model in 1+1 Dimensions

8>>>>>>>><

>>>>>>>>:

c

t

= Dc

xx

+ c(1� c) , x 2 R` , ` = 1, 2, 3 , t � 0

c

x

(a, t) = 0 and c

x

(b, t) = 0

[c] = 0 and [Dcx

] = 0 at x = w

k

, k = 1, 2

c(x , 0) = f (x)

R1

:= (↵,w1

) , R2

:= (w1

,w2

) , R3

:= (w2

, b)

D(x) =

8<

:

� , ↵ x < w

1

1 , w

1

x < w

2

� , w

2

x b

21

Interior Collocation equations

2N+2X

j=1

[↵j

(t)�j

(�i

)] = D2N+2X

j=1

[↵j

(t)�00j

(�i

)] +2N+2X

j=1

[↵j

(t)�j

(�i

)]�

2N+2X

j=1

[↵j

(t)�j

(�i

)]

!2

C

(0)

j

2

664

↵2j�1

↵2j

↵2j+1

↵2j+2

3

775 = DC

(2)

j

2

664

↵2j�1

↵2j

↵2j+1

↵2j+2

3

775+ C

(0)

j

2

664

↵2j�1

↵2j

↵2j+1

↵2j+2

3

775�

0

BB@C

(0)

j

2

664

↵2j�1

↵2j

↵2j+1

↵2j+2

3

775

1

CCA

.2

22

Collocation System of ODEs

Aa = Ma+⇣C

(0)

j

a

⌘.2

where

M =

2

66664

P

1

Q

1

P

2

Q

2

�P

N�1

Q

N�1

P

N

Q

N

3

77775,

⇥P

j

Q

j

⇤= C

(2)

j

+ C

(0)

j

a = K (a, t)

where K (a, t) = A

�1

hMa+

�C

(0)

a

�.2i

23

Time Discretization

BE:

(A� ⌧M)a(n+1) � ⌧⇣C

(0)

a

(n+1)

⌘.2= Aa

(n)

CN:

(A�⌧2M) a(n+1)�⌧

2

⇣C

(0)

a

(n+1)

⌘.2= (A+

⌧

2M) a(n)+

⌧

2

⇣C

(0)

a

(n)

⌘.2

DIRK:

(A� ⌧�M) a(n,1) � ⌧�⇣C (0)

a

(n,1)⌘.2

= A a

(n)

(A� ⌧�M) a(n,2) � ⌧�⇣C (0)

a

(n,2)⌘.2

= A a

(n) + ⌧(1� 2�)

✓M a

(n,1) +⇣C (0)

a

(n,1)⌘.2◆

A a

(n+1) = A a

(n) +⌧2

[M a

(n,1) +⇣C (0)

a

(n,1)⌘.2

+M a

(n,2) +⇣C (0)

a

(n,2)⌘.2

]

24

Numerical Results

Model Problem 1 Model Problem 2

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

τ=0.1 tmax

=4

Spatial variable x

Tum

our

gro

wth

c(x

,t)

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

τ=0.1 tmax

=4

Spatial variable xT

um

our

gro

wth

c(x

,t)

25

Numerical Results - Spatial Relative Error

Model Problem 1 Model Problem 2

102

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Number of Elements

Rela

tive E

rror

DIRK

CN

BE

102

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Number of ElemetsR

ela

tive E

rror

DIRK

CN

BE

26

Generalization in m+1 Regions

R1

= (↵,w1

) , R2

= (w1

,w2

) , . . . , Rm+1

= (wm

, b)

D(x) =

⇢� , x 2 [↵,w

1

) [ [w2

,w3

) [ . . . [ [wm

, b]1 , x else

� �

1 1 1 1

↵ w1 w2 w3 bwmwm�1wm�2

#

[c] = 0 and [Dcx

] = 0 at x = w

k

, k = 1, . . . ,m

27

Interface Conditions

[DUx

] := D

+

U

+

x

� D

�U

�x

= 0 , at x = w

k

, k = 1, . . . ,m

# #

� �2i

(x�i

) = �2i

(x+i

) �2i

(x�i

) = ��2i

(x+i

)x

i

⌘ w

k

, k odd x

i

⌘ w

k

, k even

i =kP

j=1

N

j

+ 1 i =kP

j=1

N

j

+ 1

# #

�2i

(x) =

8>>>><

>>>>:

� h

� �x

i

�x

h

�

h �x�x

i

h

�

0

�2i

(x) =

8>>>><

>>>>:

�h �x

i

�x

h

�

h

� �x�x

i

h

�

0

28

Numerical Results

Model Problem 1 Model Problem 2

−5 −4 −3 −2 −1 0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

τ=0.1 tmax

=4

Spatial variable x

Tu

mo

ur

gro

wth

c(x

,t)

−10 −8 −6 −4 −2 0 2 4 6 8 100

5

10

15

τ=0.1 tmax

=4

Spatial variable x

Tu

mo

ur

gro

wth

c(x

,t)

29

Numerical Results - Spatial Relative Error

Model Problem 1 Model Problem 2

101

102

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Number of Elements

Re

lativ

e E

rro

r

DIRK

C.N.

B.E.

102

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Number of Elements

Re

lativ

e E

rro

r

DIRK

C.N.

B.E.

30

Exponential Growth 3-region Model in 2+1 Dimensions

D =

8<

:

� , (x , y) 2 [↵,w1

)⇥ [↵, b]1 , (x , y) 2 [w

1

,w2

)⇥ [↵, b]� , (x , y) 2 [w

2

, b]⇥ [↵, b],

8>>>>>>>>>><

>>>>>>>>>>:

u

t

= Du

xx

+ Du

yy

, x 2 R` , ` = 1, 2, 3 , t > 0

@u

@⌘= 0

[u] = 0 and [Dx

u

x

] = 0 at x = w

k

, k = 1, 2 and y 2 (↵, b)

u(x , y , 0) = f (x , y)

R1

:= (↵,w1

)⇥(↵, b) , R2

:= (w1

,w2

)⇥(↵, b) , R3

:= (w2

, b)⇥(↵, b)

31

Exponential Growth 3-region Model in 2+1 Dimensions

−4−3

−2−1

01

23

4

−4

−3

−2

−1

0

1

2

3

4

−0.5

0

0.5

1

1.5

xy

Di↵usion coe�cient D32

Exponential Growth 3-region Model in 2+1 Dimensions

U(x , y , t) =N

x

+1X

m=1

N

y

+1X

n=1

[↵2m�1

(t)�2m�1

(x) + ↵2m

(t)�2m

(x)]

[�2n�1

(t)�2n�1

(y) + �2n

(t)�2n

(y)]

#

[DUx

] := D

+

U

+

x

� D

�U

�x

= 0 , at x = w

k

, k = 1, 2

# #

� �2i

(x�i

) = �2i

(x+i

) �2i

(x�i

) = ��2i

(x+i

)x

i

⌘ w

1

x

i

⌘ w

2

i = N

x

1

+ 1 i = N

x

1

+ N

x

2

+ 1

33

2-d Screenshots

−4 −3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1t= 2

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

34

2-d Screenshots

−4 −3 −2 −1 0 1 2 3 4−4−2024

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 t= 4

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

35

2-D Model Problem - Solution

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36