Simulating Solid Tumor Growth using Multigrid Algorithms

Transcription

1 Simulating Solid Tumor Growth using Multigrid Applied Mathematics, Statistics, and Program Advisor: Doron Levy, PhD Department of Mathematics/CSCAMM University of Maryland, College Park September 30, 2014

2 Overview

3 Tumor Growth Over Time [2]

4 Tumor Growth Tumor Growth Over Time [2]

5 Tumor Growth Tumor Growth Over Time [2]

6 Simulate solid tumor growth in two and, time permitting, three dimensions with a Cahn-Hilliard-type convection-reaction-diffusion mathematical model using multigrid algorithms [1].

7 Mathematical Model Model Parameters[1]

8 Mathematical Model Model Parameters[1] φ V : volume fraction of viable tissue φ D : volume fraction of dead tissue φ H : volume fraction of healthy tissue u S : tissue velocity ε: thickness of interface between healthy and tumoral tissue p: cell-to-cell (solid) pressure n: nutrient concentration

9 Mathematical Model Model Parameters[1] φ V : volume fraction of viable tissue φ D : volume fraction of dead tissue φ H : volume fraction of healthy tissue u S : tissue velocity ε: thickness of interface between healthy and tumoral tissue p: cell-to-cell (solid) pressure n: nutrient concentration

10 Mathematical Model If φ represents the tissue volume fraction, then φ T = φ V + φ D (1) φ T t = M (φ T µ) + S T (u S φ T ) (2) φ D = M (φ T µ) + S D (u S φ D ) t (3) 0 = (D(φ T ) n) + T c (φ T, n) n(φ T φ D ), (4) where M is the mobility constant related to phase separation between tumoral and healthy tissue and S T and S D are the net and dead source of tumoral cells, respectively.

11 Mathematical Model When looking strictly at equation (2), we have φ T t = M (φ T µ) + S T (u S φ T )

12 Mathematical Model When looking strictly at equation (2), we have φ T t a diffusion term = M (φ T µ) + S T (u S φ T )

13 Mathematical Model When looking strictly at equation (2), we have φ T t = M (φ T µ) + S T (u S φ T ) a diffusion term, a source term

14 Mathematical Model When looking strictly at equation (2), we have φ T t = M (φ T µ) + S T (u S φ T ) a diffusion term, a source term, and a convection term.

15 Tissue Velocity Equations u S = κ(φ T, φ D )( p γ ε φ T ) (5) with κ the tissue motility function and γ 0 the excess adhesion force at the diffuse tumor/host-tissue interface. Assuming the host tissue remains constant we get, u S = S T (6) Combined we have a Poisson equation in p, (κ(φ T, φ D ) p) = S T (κ(φ T, φ D ) γ ε φ T ) (7) Thus, p can be solved for independently then used to find u S in equation (5).

16 Source Terms

17 Source Terms Assuming that the net source of the tumor cells S T is due to cell proliferation, we have S T = ng(φ T )φ V λ L φ D, (8) where λ L is the rate at which the lysing of dead cells form water and 1 if 3ε φ 2 G(x) = φ/ε 1 if ε 2 2 < φ < 3ε (9) 2 0 if φ ε. 2

18 Source Terms Assuming that the net source of the tumor cells S T is due to cell proliferation, we have S T = ng(φ T )φ V λ L φ D, (8) where λ L is the rate at which the lysing of dead cells form water and 1 if 3ε φ 2 G(x) = φ/ε 1 if ε 2 2 < φ < 3ε (9) 2 0 if φ ε. 2 Assuming that the net source of the dead cells S D is due to apoptosis and necrosis with rates λ A and λ N, respectively, we have S D = (λ A + λ N H (n N n))(φ T φ D ) λ L φ D, (10) where H is the Heaviside function and n N is the necrotic limit, which below the tumor tissue dies due to lack of nutrients.

19 Nutrient Equation 0 = (D(φ T ) n) + T c (φ T, n) n(φ T φ D ) (11) Assuming (i) nutrient diffusion occurs on a faster time scale than cell proliferation and (ii) nutrient taken by dead cells and healthy cells is negligible in comparison to that of viable cells we have, D(φ) = D H (1 Q(φ T )) + Q(φ T ), (12) T c (φ T, n) = (v H p (1 Q(φ T )) + v T p Q(φ T ))(n c n), (13) and 1 if 1 φ Q(φ) = 3φ 2 2φ 3 orφ if 0 < φ < 1 0 if φ 0. (14)

20 Coupled System where, φ T t = M (φ T µ) + S T (u Sφ T ) (15) φ D t = M (φ T µ) + S D (u Sφ D ) (16) 0 = (D(φ T ) n) + T c(φ T, n) n(φ T φ D ) (17) µ = f (φ T ) ε 2 2 φ T (18) f (φ) = φ 2 (1 φ) 2 /2 (19) u S = κ(φ T, φ D )( p γ ε φ T ) (20) S T = ng(φ T )φ V λ L φ D, (21) S D = (λ A + λ N H (n N n))(φ T φ D ) λ L φ D. (22) Thus, the system is complete.

21 Discretization Discretize the mathematical model for solid tumor growth in 2D both spatially by finite difference and in time using an implicit Crank-Nicolson scheme due to 4th order diffusion term. Implement well known matrix solver to solve the final system of equations. Uniform Multigrid Due to expected complexity on boundary of tumor, fine grid multigrid schemes allow for high resolution. Thus, a uniform multigrid algorithm will solve the discretized system.

22 Spatial Discretization: Finite Difference Assuming a rectangular tissue domain with uniform grid points, the Laplacian operator and the Laplacian with non-constant diffusivity/mobility are approximated to second order (18) and (19), respectively. d φ i,j = φ i+1,j + φ i 1,j + φ i,j+1 + φ i,j 1 4φ i,j h 2 (23) d (m d φ) i,j = Axm i+ 1 2,j(φ i+1,j φ i,j ) + A xm i 1 2,j (φ i,j φ i 1,j ) + Ay m i,j+ 1 2 where A x and A y are averaging operators h 2 (f i,j+1 f i,j ) + A y m i,j 1 2 (f i,j f i,j 1 ) h 2, (24)

23 Temporal Discretization: Crank-Nicolson If we begin with a partial derivative equation in two dimensions u t u = F (t, x, y, x, u y, 2 u x 2, 2 u ), (25) y 2 the Crank-Nicolson discretization is a 2 nd order convergence in time based on the trapezoidal rule as follow: u n+1 i ui n t = 1 2 n+1 [Fi (t, x, y, u x, u y, 2 u 2 u x 2 y 2 ) +F n i (t, x, y, u x, u y, 2 u x 2, 2 u y 2 )] (26)

24 At each grid level Uniform Multigrid: V-Cycle [3] Pre-Smoothing: Gauss-Seidel Scheme When solving Au = f Decompose A = L + U where L is lower triangular and U is strictly upper triangular Solve the system L(u k+1 ) = f Ux k if coarsest level solve Au = f else Restrict Residual Correction for coarser grid Update end Post-Smoothing: Gauss-Seidel Scheme

25 Uniform vs Locally-Refined Multigrid

26 Software: Matlab Hardware: Personal Computer Databases: N/A

27 When testing the multigrid algorithm, two tests problems will be used: 1 The heat equation: 2 4th Order PDE 2 u = u t 4 u = u t (27) (28)

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29 Given initial conditions from [1], starting with a coarse grid of 64 2, the finest grid of 512 2, and a time step of 1x10 2, we anticipate the following results.

30 Given initial conditions from [1], starting with a coarse grid of 64 2, the finest grid of 512 2, and a time step of 1x10 2, we anticipate the following results. 2D Results

31 Given initial conditions from [1], starting with a coarse grid of 64 2, the finest grid of 512 2, and a time step of 1x10 2, we anticipate the following results. 2D Results Time Permitting

32 Timeline

33 Timeline November 30: Discretize the mathematical model and use an implicit MATLAB solver to get an initial solution

34 Timeline November 30: Discretize the mathematical model and use an implicit MATLAB solver to get an initial solution January 31: Have an initial uniform multigrid algorithm programmed and tested on verification problems.

35 Timeline November 30: Discretize the mathematical model and use an implicit MATLAB solver to get an initial solution January 31: Have an initial uniform multigrid algorithm programmed and tested on verification problems. March 31: Combine uniform multigrid algorithm and discretized 2D solid tumor growth mathematical model

36 Timeline November 30: Discretize the mathematical model and use an implicit MATLAB solver to get an initial solution January 31: Have an initial uniform multigrid algorithm programmed and tested on verification problems. March 31: Combine uniform multigrid algorithm and discretized 2D solid tumor growth mathematical model Time Permitting: Extend uniform multigrid algorithm to the locally refined multigrid and have it tested on 2D model. Extend the multigrid algorithms to the 3D solid tumor growth model.

37 1 Proposal Presentation and Report 2 Mid-Year Presentation and Report 3 Final Presentation and Report 4 Matlab Code for implicit solver for discretized solid tumor growth model 5 Matlab Code for uniform multigrid algorithm 6 Documentation for tumor growth model simulation

38 References S.M. Wise, J.S. Lowengrubb, and V. Cristini. An adaptive multigrid algorithm for simulating solid tumor growth using mixture models Mathematical and Computer Modelling, 2011, 53, Chaste Cell-based Chaste: a multiscale computational framework for modelling cell populations University of Oxford: Department of Computer Science, Howard C. Elman, David J. Silvester and Andrew J. Wathen Finite Elements and Fast Iterative Solvers Oxford University Press, 2014, 2.

39 Questions?