Convergence analysis of a numerical scheme for a tumour growth model - PowerPoint PPT Presentation

. . . . . . . . . . . . . . Convergence analysis of a numerical scheme for a tumour growth model Gopikrishnan C. Remesan IITB - Monash Research Academy Monash Workshop on Numerical Difgerential Equations and Applications 2020 Joint work with A/Prof J. Droniou (Monash Uni.) and Prof N. Nataraj (IIT Bombay) February 11, 2020 1 / 26 . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical solutions of free boundary problems

. . . . . . . . . . . . . . . . . . Contents 2 / 26 . . . . . . . . . . . . . . . . . . . . . . Numerical solutions of free boundary problems 1 Model 2 Discretisation 3 Main Theorem 4 Compactness results 5 Convergence results 6 Numerical results

. . . . . . . . . . . . . . . . . . 2 / 26 . . . . . . . . . . . . . . . . . . . . . . Numerical solutions of free boundary problems 1 Model 2 Discretisation 3 Main Theorem 4 Compactness results 5 Convergence results 6 Numerical results

. . . . . . . . . . . . . . Model of tumour–growth cross-section of tumour spheroid Assumptions Cells and fmuid exchange matter via the processes, cell division and cell death. Mass and momentum are conserved internally. No blood vessels. Limiting nutrient - Oxygen, follows difgusion. 3 / 26 . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical solutions of free boundary problems

. . . . . . . . . . . . . . . . . Model of tumour growth tension. 4 / 26 . . . . . . . . . . . . . . . . . . Numerical solutions of free boundary problems . . . . . Domain − 0 < t < T , x ∈ ˇ Ω( t ) = (0 , ˇ ℓ ( t )) . ˇ ℓ ( t ) − tumour length, x = 0 − tumour centre. α − volume fraction of tumour cells, ˇ ˇ u − cell velocity, ˇ c − oxygen

. . . . . . . . . . . . . . . Model of tumour growth . tension. cell volume fraction (hyperbolic conservation law) Birth rate Death rate death rate. 4 / 26 . Numerical solutions of free boundary problems . . . . . . . . . . . . . . . . . . . . . . . Domain − 0 < t < T , x ∈ ˇ Ω( t ) = (0 , ˇ ℓ ( t )) . ˇ ℓ ( t ) − tumour length, x = 0 − tumour centre. α − volume fraction of tumour cells, ˇ ˇ u − cell velocity, ˇ c − oxygen ∂ ˇ ∂t + ∂ α u ) =(1+ s 1 )ˇ c ˇ α (1 − ˇ α ) − s 2 + s 3 ˇ c ∂x (ˇ α ˇ c ˇ α , 1+ s 1 ˇ c 1+ s 4 ˇ � �� � � �� � α (0 ,x ) = α 0 ( x ) . 1+(1 /s 1 ) , s 2 − maximal birth and death rates, s 3 /s 4 − minimal c ) = (1+ s 1 )(1 − ˇ α )ˇ c − s 2 + s 3 ˇ c Set f (ˇ α, ˇ 1+ s 1 ˇ c 1+ s 4 ˇ c

. . . . . . . . . . . . . . . . . tension. cell velocity (elliptic) coeffjcient. 4 / 26 . Model of tumour growth . . . . . . . . . . . Numerical solutions of free boundary problems . . . . . . . . . . . Domain − 0 < t < T , x ∈ ˇ Ω( t ) = (0 , ˇ ℓ ( t )) . ˇ ℓ ( t ) − tumour length, x = 0 − tumour centre. α − volume fraction of tumour cells, ˇ ˇ u − cell velocity, ˇ c − oxygen ( ) k ˇ α ˇ u α − µ ∂ α∂ ˇ u = − ∂ ∂x (ˇ α H (ˇ α )) , 1 − ˇ ∂x ∂x µ∂ ˇ u u ( t, 0) = 0 , ˇ ∂x ( t,ℓ ( t )) = H (ˇ α ( t,ℓ ( t ))) . µ − coeffjcient of viscosity of cell phase. k − interfacial drag α ) 2 , a + = max( a, 0) . α − α ∗ ) + / (1 − ˇ Set H (ˇ α ) = (ˇ

. . . . . . . . . . . . . . . . Model of tumour growth tension. Oxygen tension (parabolic) Ox. consumption rate 4 / 26 . . . . . . . . . . . . . Numerical solutions of free boundary problems . . . . . . . . . . . Domain − 0 < t < T , x ∈ ˇ Ω( t ) = (0 , ˇ ℓ ( t )) . ˇ ℓ ( t ) − tumour length, x = 0 − tumour centre. α − volume fraction of tumour cells, ˇ ˇ u − cell velocity, ˇ c − oxygen ∂t − ∂ 2 ˇ ∂ ˇ c c ∂x 2 = − Q ˇ α ˇ c, � �� � c (0 ,x ) = c 0 ( x ) , ∂ ˇ c ˇ ∂x ( t, 0) = 0 , ˇ c ( t,ℓ ( t )) = 1 . Q − Maximum oxygen consumption rate.

. . . . . . . . . . . . . . . . . Model of tumour growth tension. boundary evolution 4 / 26 . . . . . . . . . . . . . Numerical solutions of free boundary problems . . . . . . . . . . Domain − 0 < t < T , x ∈ ˇ Ω( t ) = (0 , ˇ ℓ ( t )) . ˇ ℓ ( t ) − tumour length, x = 0 − tumour centre. α − volume fraction of tumour cells, ˇ ˇ u − cell velocity, ˇ c − oxygen ℓ ′ ( t ) = ˇ ˇ u ( t, ˇ ℓ ( t )) , ˇ ℓ (0) = 1 .

. . . . . . . . . . . . . . . . Idea of extended model . original model extended model velocity and oxygen tension extended by 0 and 1, respectively. 5 / 26 . Numerical solutions of free boundary problems . . . . . . . . . . . . . . . . . . . . . . T T D T ∂ t α + ∂ x ( uα ) = αf ( α, c ) α > 0 ˇ ) time ( t ) t α > 0 α = 0 ( (ˇ α, ˇ u, ˇ c ) e m ˇ u | D T \ D T = 0 ℓ ( t ) i t c D T \ D T = 1 D T D T \ D T ˇ ˇ 0 ℓ (0) ℓ m 0 ℓ (0) ℓ m space ( x ) space ( x ) ˇ ℓ as the interface between α > 0 and α = 0 .

. . . . . . . . . . . . . . . Idea of threshold model . extended model threshold model velocity and oxygen tension extended by 0 and 1, resp. numerically. 6 / 26 . Numerical solutions of free boundary problems . . . . . . . . . . . . . . . . . . . . . . . T T ∂ t α + ∂ x ( uα ) = αf ( α, c ) ∂ t α + ∂ x ( uα ) = αf ( α, c ) time ( t ) ) t ( α ≤ α thr α > 0 α = 0 e m u | D T \ D T = 0 u | D T \ D T = 0 i t c | D T \ D T = 1 c | D T \ D T = 1 D T D T D T \ D T D T \ D T ˇ ˇ 0 ℓ (0) ℓ m 0 ℓ (0) ℓ m space ( x ) space ( x ) ˇ ℓ as the interface between α > 0 and α < = α thr . α thr facilitates estimates on cell velocity and is required

. . . . . . . . . . . . . . . . . Threshold solution of and the following hold: 7 / 26 . . . . . . . . . . . . . . . . . . Numerical solutions of free boundary problems . . . . . A threshold solution (with threshold α thr ∈ (0 , 1) ) and domain D thr T the threshold model in D T is a 4-tuple ( α,u,c, Ω) such that: 0 < m 11 ≤ α | Ω( t ) ≤ m 12 < 1 for all t ∈ [0 ,T ] , m 11 ≤ m 01 , m 12 ≥ m 02 c ≥ 0 ,

. . . . . . . . . . . . . . . . . Threshold solution cell volume fraction 7 / 26 . . . . . . . . . . . . . . . . . . Numerical solutions of free boundary problems . . . . . The volume fraction α ∈ L ∞ ( D T ) is such that ∀ φ ∈ C ∞ c ([0 ,T ) × (0 ,ℓ m )) , ∫ ∫ ( α,uα ) ·∇ t,x φ d t d x + φ (0 ,x ) α 0 d x D T Ω(0) ∫ ( α − α thr ) + f ( α,c )d x = 0 . + D T

. . . . . . . . . . . . . . . . . Threshold solution tumour boundary is of the form 7 / 26 . . . . . . . . . . . . . . . . . . Numerical solutions of free boundary problems . . . . . The set D thr T D thr = ∪ 0 <t<T ( { t }× Ω( t )) , T where Ω( t ) = (0 ,ℓ ( t )) , and we have α ≤ α thr on D T \ D thr T .

. . . . . . . . . . . . . . . . Threshold solution cell velocity (1) (2) (3) 7 / 26 . . . . . . . . . . . . . Numerical solutions of free boundary problems . . . . . . . . . . . H 1 ,u ∂x ( D thr T ) := { v ∈ L 2 ( D thr T ) : ∂ x v ∈ L 2 ( D thr T ) and v ( t, 0) = 0 ∀ t ∈ (0 ,T ) } . u ∈ H 1 ,u T ) and ∀ v ∈ H 1 ,u ∂x ( D thr ∂x ( D thr T ) , satisfjes ∫ T ∫ T a t ( u ( t, · ) ,v ( t, · ))d t = L t ( v ( t, · ))d t, 0 0 where a t : H 1 (Ω( t )) × H 1 (Ω( t )) → R is a bilinear form and L t : H 1 (Ω( t )) → R is a linear form as follows: ( ) α a t ( u,v )= k 1 − αu,v + µ ( α∂ x u,∂ x v ) Ω( t ) and Ω( t ) L t ( v )= ( H ( α ) ,∂ x v ) Ω( t ) . Extend u to D T by setting u | D T \ D thr := 0 . T

. . . . . . . . . . . . . . . . . Threshold solution oxygen tension (1) 7 / 26 . . . . . . . . . . . . . Numerical solutions of free boundary problems . . . . . . . . . . H 1 ,c ∂x ( D thr T ) := { v ∈ L 2 ( D thr T ) : ∂ x v ∈ L 2 ( D thr T ) and v ( t,ℓ ( t )) = 0 ∀ t ∈ (0 ,T ) } . c − 1 ∈ H 1 ,c ∂x ( D thr T ) satisfjes, ∫ ∫ ∫ − c∂ t v d x d t + λ ∂ x c∂ x v d x d t − c 0 ( x ) v (0 ,x )d x D thr D thr Ω(0) T T ∫ − Q αcv d x d t = 0 , D thr T ∀ v ∈ H 1 ,c ∂x ( D thr T ) such that ∂ t v ∈ L 2 ( D thr T ) . Extend c to D T by setting c | D T \ D thr := 1 . T

. Threshold value - comments . . . . . . . . . . An unavoidable disadvantage . Residual volume fraction - creates spurious growth outside the tumour domain. Essential from numerical vantage point. Modifjed source term eliminates spurious growth. As , approaches . 8 / 26 . . . . . . . . . . . . . . . . Numerical solutions of free boundary problems . . . . . . . . . . . . To obtain a lower bound strictly greater than zero for α . Facilitates bounded variation estimates on α . To obtain supremum norm bounds on u and ∂ x u .