1 Discontinuous Hermite Collocation and Diagonally Implicit RK3 for a Brain Tumour Invasion Model John E. Athanasakis Applied Mathematics & Computers Laboratory Technical University of Crete Chania 73100, Greece gathanasakis@gmail.com
Gliomas Gliomas are primary brain tumours that originate from brain s glial cells. They usually occur in the cerebral (upper) brain s hemisphere and they are characterized by their aggressive di usive invasion of brain normal tissue.
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 cm /day) : net proliferation rate (0.01/day) g(c) = c, exponential growth c c(1 K ), logistic growth, K :carryingcapacity Zero flux BC on the anatomy boundaries : @c @ =0 Initial spatial distribution malignant cells : c(x, 0) = f (x) 3
Analytical Deterministic Models 4 Homogeneous Brain Tissue (Cruywagen et.al 1995) # D is constant # @c @t = Dr c + g(c)
Analytical Deterministic Models 5 Heterogeneous Brain Tissue (Swanson 1999) # D(x) = Dg, x in Grey Matter D w, x in White Matter, D w > D g # @c @t = r (Drc)+ g(c) continuity and flux preservation conditions at the interfaces
Exponential Growth 3-region Model in 1+1 Dimensions Dimensionless variables x q D w x, t t, c(x, t) c < D = : q D w x, t Dw N 0, apple x < w 1 1, w 1 apple x < w, = Dg D w < 1, w apple x apple b 1 1 γ γ a w 1 w b Transformation c(x, t) =e t u(x, t) Interface conditions at x = w k, k =1, 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 u t = Du xx, x R`, ` =1,, 3, t > 0 >< >: u x (, t) =0 and u x (b, t) =0 [u] =0 and [Du x ]=0 atx = w k, k =1, u(x, 0) = f (x) R 1 := (, w 1 ), R := (w 1, w ), R 3 := (w, b) t ux =0 u t = u xx u t = u xx u t = u xx ux =0 a w 1 w b 7
Domain Discretization Discretization t n+1 t h (x m,t n ) t n = n t n 1 t t =0 x = x = b x m := +(m 1)h, m =1,...,N+1, t n = n, n =0, 1,... < h 1 := (w 1 )/N 1 h = h := (w w 1 )/N, N = N 1 + N + N 3, = t : h 3 := (b w )/N 3
Hermite Collocation 9 U(x, t) = N+1 X m=1 >< m 1(x) = [ m 1 (t) m 1 (x)+ m (t) m (x)] x m x x h, x [x m 1, x m ] x m h, x [x m, x m+1 ] >: 0, otherwise h xm x h, x [x m 1, x m ] >< m(x) = h x xm h, x [x m, x m+1 ] >: 0, otherwise, (1) (s) =(1 s) (1 + s), (s) =s(1 s), s [0, 1]
Interface Conditions [DU x ]:=D + U + x D U x =0, at x = w k, k =1, # # i(x i )= i (x + i ) i(x i )= i (x + x i w 1 x i w i = N 1 +1 i = N 1 + N +1 i ) i(x) = >< >: # # h x i x h h x x i h 0 10 i(x) = >< >: h x i x h h x x i h 0
Hermite Elements at the Interfaces x i 1 x i x i+1 11 φ i 1 (x) x i 1 x i x i +1 φ i (x) i = N 1 +1 φ i 1 (x) φ i (x) i = N 1 + N +1
Boundary Collocation equations 1 U x (, t) =0! (t) =0! (t) =0 U x (b, t) =0! N+ (t) =0! N+ (t) =0
Interior Collocation equations 13 XN+1 XN+1 [ j 1 (t) j 1 ( i )+ j (t) j ( i )] = D j 1 (t) 00 j 1( i )+ j (t) 00 j( i ) j=1 j=1 where i, i = 1,...,N are the Gauss points (two per subinterval) # C (0) j 6 4 j 1 j j+1 Elemental Collocation equations 3 3 7 5 = D h C () j 6 4 j 1 j j+1 7 5, j =1,...,N j+ j+
Elemental Collocation matrices 14 C (apple) j = 6 4 s (apple) 1 s (apple) 3 h j s (apple) s (apple) 3 h j s (apple) 4 s (apple) 1 h s(apple) j 4 h s(apple) j 3 7 5, apple =0, s (0) 1 = 9+4p 3 1, s (0) = 3+p 3 s () 1 = p 3, s () = 1 3 = 9 4p 3 1, s (0) 4 = 3 p 3 36, p () 3, s 3 = p 3 and s () 4 = 1+ p 3 36, s (0) < 1, j 6= N 1 + N +1 j = :, j = N 1 + N +1 < 1, j 6= N 1, j =. :, j = N 1
Collocation system of ODEs 15 Aȧ = Ba where ȧ = 1 3 N+1 T, a = 1 3 N+1 T Ã 1 B 1 A B A = 6 4 A N 1 B N 1 A N BN 3 7 5, B = 6 4 F 1 G 1 F G F N 1 G N 1 F N 3 7 5. GN Aj B j = C (0) j and F j G j = D h C () j. ȧ = C(a, t) where C(a, t) =A 1 Ba
Backward Euler (BE) - Crank Nicolson (CN) 16 BE: a (n+1) a (n) = C (n+1) (a, t) or (A B)a (n+1) = Aa (n) CN: a (n+1) (A a (n) = 1 (C (n+1) (a, t)+c (n) (a, t)) or B) a(n+1) =(A + B) a(n)
Diagonally Implicit RK3 (DIRK) 17 DIRK: a (n,1) = a (n) + C (n,1) (a, t) a (n,) = a (n) + (1 )C n,1 (a, t)+ C n, (a, t) a (n+1) = a (n) + h i C (n,1) (a, t)+c (n,) (a, t) or (A B) a (n,1) = A a (n) (A B) a (n,) = A a (n) + (1 )B a (n,1) A a (n+1) = A a (n) + [B a(n,1) + B a (n,) ]
Numerical Results - Model Problem 1 1 R 1 := ( 5, 1), R := ( 1, 1), R 3 := (1, 5), =0.5 and f (x) = 1 p e x /, with =0.. 1 τ=0.1 t max =4 10 Tumour growth c(x,t) 6 4 0 5 4 3 1 0 1 3 4 5 Spatial variable x
Numerical Results - Model Problem R 1 := ( 5, 1), R := (1, 1.5), R 3 := (1.5, 5), =0.5 and f (x) = 1 p (e (x+3.5) / + e (x 3) / ), with =0.. 1 τ=0.1, t max =4 16 14 Tumour growth c(x,t) 1 10 6 4 0 5 4 3 1 0 1 3 4 5 Spatial variable x 19
Numerical Results - Spatial Relative Error 0 Model Problem 1 Model Problem 10 1 g=0.5 ksi=0 w1= 1 w=1 [ 5 5] dt=0.01 10 1 10 10 3 DIRK CN BE 10 10 3 DIRK CN BE 10 4 10 4 Relative Error 10 5 10 6 Relative Error 10 5 10 6 10 7 10 7 10 10 10 9 10 9 10 10 10 10 10 Number of Elements 10 11 10 Number of Elements
Logistic Growth 3-region Model in 1+1 Dimensions 1 c t = Dc xx + c(1 c), x R`, ` =1,, 3, t 0 >< >: c x (a, t) =0 and c x (b, t) =0 [c] =0 and [Dc x ]=0 atx = w k, k =1, c(x, 0) = f (x) R 1 := (, w 1 ), R := (w 1, w ), R 3 := (w, b) <, apple x < w 1 D(x) = 1, w 1 apple x < w :, w apple x apple b
Interior Collocation equations N+ X N+ X N+ X [ j (t) j ( i )] = D [ j (t) 00 j ( i )] + [ j (t) j ( i )] j=1 j=1 j=1 N+ X [ j (t) j ( i )] j=1! C (0) j 6 4 j 1 j j+1 j+ 3 7 5 = DC () j 6 4 j 1 j j+1 j+ 3 7 5 + C (0) j 6 4 j 1 j j+1 j+ 3 7 5 0 B @ C (0) j 6 4 j 1 j j+1 j+ 31 7C 5A.
Collocation System of ODEs 3 where M = 6 4 Aȧ = Ma + P 1 Q 1 P Q C (0) j a. P N 1 Q N 1 P N 3 7 5, Q N () Pj Q j = C j + C (0) j where ȧ = K(a, t) K(a, t) =A 1 h Ma + C (0) a.i
Time Discretization 4 BE: CN: (A M)a (n+1) C (0) a (n+1). = Aa (n) (A M) a(n+1) C (0) a (n+1). =(A+ M) a(n) + C (0) a (n). DIRK: (A M) a (n,1) (A M) a (n,) C (0) a (n,1). = A a (n) C (0) a (n,). = A a (n) + (1 ) M a (n,1) + C (0) a (n,1). A a (n+1) = A a (n) + [M a(n,1) + C (0) a (n,1). + M a (n,) + C (0) a (n,). ]
Numerical Results 5 Model Problem 1 Model Problem 3 τ=0.1 t max =4 3 τ=0.1 t max =4.5.5 Tumour growth c(x,t) 1.5 1 Tumour growth c(x,t) 1.5 1 0.5 0.5 0 5 4 3 1 0 1 3 4 5 Spatial variable x 0 5 4 3 1 0 1 3 4 5 Spatial variable x
Numerical Results - Spatial Relative Error 6 Model Problem 1 Model Problem 10 1 10 1 10 DIRK CN 10 DIRK CN 10 3 BE 10 3 BE 10 4 10 4 Relative Error 10 5 10 6 Relative Error 10 5 10 6 10 7 10 7 10 10 10 9 10 9 10 10 10 10 10 Number of Elements 10 11 10 Number of Elemets
Generalization in m+1 Regions R 1 =(, w 1 ), R =(w 1, w ),..., R m+1 =(w m, b) D(x) =, x [, w 1 ) [ [w, w 3 ) [...[ [w m, b] 1, x else 1 1 1 1 w m w m 1 w m w 1 w w 3 b # [c] =0 and [Dc x ]=0 atx = w k, k =1,...,m 7
Interface Conditions [DU x ]:=D + U + x D U x =0, at x = w k, k =1,...,m # # i(x) = i(x i )= i (x + i ) i(x i )= i (x + x i w k, kodd x i w k, keven P i = k P N j +1 i = k N j +1 >< >: j=1 j=1 # # h x i x h h x x i h 0 i(x) = >< >: i ) h x i x h h x x i h 0
Numerical Results 10 9 7 6 5 4 3 1 9 15 10 5 0 10 6 4 0 4 6 10 Model Problem 1 Model Problem τ=0.1 t max =4 τ=0.1 t max =4 Tumour growth c(x,t) Tumour growth c(x,t) 0 5 4 3 1 0 1 3 4 5 Spatial variable x Spatial variable x
Numerical Results - Spatial Relative Error 10 11 30 10 11 Model Problem 1 Model Problem 10 1 10 1 10 DIRK C.N. 10 DIRK C.N. 10 3 B.E. 10 3 B.E. 10 4 10 4 Relative Error 10 5 10 6 10 7 Relative Error 10 5 10 6 10 7 10 10 10 9 10 9 10 10 10 10 10 1 10 Number of Elements 10 Number of Elements
Exponential Growth 3-region Model in +1 Dimensions 31 <, (x, y) [, w 1 ) [, b] D = 1, (x, y) [w 1, w ) [, b] :, (x, y) [w, b] [, b], >< u t = Du xx + Du yy, x R`, ` =1,, 3, t > 0 @u @ =0 [u] =0 and [D x u x ]=0 atx = w k, k =1, andy (, b) >: u(x, y, 0) = f (x, y) R 1 := (, w 1 ) (, b), R := (w 1, w ) (, b), R 3 := (w, b) (, b)
Exponential Growth 3-region Model in +1 Dimensions 1.5 1 0.5 0 0.5 4 3 1 0 y 1 3 4 4 3 1 x 0 1 3 4 Di usion coe 3 cient D
Exponential Growth 3-region Model in +1 Dimensions 33 U(x, y, t) = NX x +1 m=1 N y +1 X n=1 [ m 1 (t) m 1 (x)+ m (t) m (x)] [ n 1 (t) n 1 (y)+ n (t) n (y)] # [DU x ]:=D + U + x D U x =0, at x = w k, k =1, # # i(x i )= i (x + i ) i(x i )= i (x + x i w 1 x i w i = N x1 +1 i = N x1 + N x +1 i )
-d Screenshots 34 1 t= 4 0.9 0. 0.7 0.6 0.5 0.4 0.3 0. 0.1 0 4 3 1 0 1 3 4 3 1 0 1 3 4 4 3 1 0 1 3 4 0. 0.1 0.16 0.14 0.1 0.1 0.0 0.06 0.04 0.0
-d Screenshots 35 0.9 1 t= 4 4 0.9 0. 3 0. 0.7 0.7 0.6 0.5 0.4 1 0 0.6 0.5 0.3 0. 1 0.4 0.1 0 04 3 4 4 3 1 0 1 3 4 4 3 1 0 1 3 4 0.3 0.
-D Model Problem - Solution 36 (Loading movie.mpg)