Problem Background Difficulties Multi-Domain Spectral Method What Now? Solving the 3-D ‘Sphoisson’ Equation David Gross MATH 164 May 1st, 2008 MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? An Overview Multi-Domain Spectral Method Problem Background Basis Functions And Temporal Modeling Projections Spatio-Temporal Modeling Multiple Domains Difficulties PDE Types and Methods What Now? Coordinates Results, Issues, Future Work MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? Temporal Modeling Motivation ◮ Research from Summers ’05, ‘06 and ‘07 ◮ Supported by the National Science Foundation under 3-year grant NSF-DMS-041-4011 ◮ Optimal control theory published in EJDE , Vol. 2007, No. 171 ◮ Paper pending minor revisions in Journal of Computation and Mathematical Methods in Medicine ◮ Looking for paper to discuss spatial modeling attempts and techniques, for both spherically symmetric and spherically asymmetric case MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? Temporal Modeling ODE Variables M B : Chemotherapy drug concentration in the blood, [IU/L] D B : Dendritic cell concentration in the blood [cells/L] L B : Antigen-specific activated CD8+ T lymphocytes, [cells/L] MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? Temporal Modeling ODE Equations dM B = − ω M B M B dt dD B � T � − ω D B D B − K D B (1 − e − δ DB M B ) D B = α D B dt � T � + k D B dL B D B L B L B L B − u L B e ǫ LB D B L 2 = α L B B ζ L B + D B k L B + L B k L B + L B dt − K L B (1 − e − δ LB M B ) L B − ω L B L B where � R � π � 2 π � T � = T ( r , φ, θ ) d θ d φ dr 0 0 0 is the integral of tumor density, i.e. the volume of the tumor MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? Spatio-Temporal Modeling PDE Variables T : Tumor cell density, [cells/mm 3 ] � v : Tumor cell velocity, [mm/s] N : Nutrient density, [mol/mm 3 ] M : Chemotherapy drug density, [mg/mm 3 ] S : Chemical signal responsible for inducing CD8+ T cells to move towards the center of the tumor by chemotaxis, [mol/mm 3 ] L : Antigen specific CD8+ T lymphocytes, [cells/mm 3 ] MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? Spatio-Temporal Modeling PDE Equations N T t − ∇ · ( � vT ) = D T ∆ T + α T N + ζ T − ω T T ( L / T ) σ s + ( L / T ) σ − K T (1 − e − δ T M ) T − N = D N ∆ N − Γ N t N + ζ T M t = D M ∆ M − (¯ ω M + γ T ) M = D S ∆ S + α S − ω S S S t − µ ∆ S + ∆ L − ω L TL − K L (1 − e − δ L M ) L L t = MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? Spatio-Temporal Modeling Intratumoral Pressure via Darcy’s Law We assume that there exists an intratumoral pressure p ( r , θ, φ, t ) following Darcy’s Law. That is, for some proportionality constant ν , � v = − ν ∇ p Furthermore, we stipulate that the tumor has constant density, i.e. T = 1. This reduces equation the T PDE to L σ N s + L σ − K T (1 − e − δ T M ) ν ∆ p = α T N + ζ − ω T − and we solve for p instead of both T and � v . MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? Spatio-Temporal Modeling Tumor Boundary Evolution We let the boundary be given by R = R ( θ, φ, t ). To define the time evolution of R , we introduce the moving boundary condition as follows: � ∂ R � � � ∂ t � u · � n R = � n R · − ν ∇ p � � r=R where novel terms in this equation are defined by u ( θ, φ ) � = (cos θ sin φ, sin θ sin φ, cos φ ) � = R � F u F θ × � � F φ � n R = � � F θ × � F φ � MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? Spatio-Temporal Modeling Boundary Conditions N ( R ) = N B L ( R ) = L B ( t ) M ( R ) = M B ( t ) S ( R ) = 0 (1) � 1 R − L ( R ) � p ( R ) = α κ κ = α κ (2) 2 R 2 Note here N B is constant, but L B and M B are functions of time. We assume that pressure is proportional to mean curvature κ = κ ( r , θ, φ ) of the boundary. Here L is used to denote the angular component of the spherical Laplacian. MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? PDE Types and Methods PDE Types Hyperbolic Boundary � � � t � � � Parabolic from Boundary to Interior � t � � � � f � � � � � � g � � � Elliptic Inside MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? PDE Types and Methods Order of Solution ◮ Update ODE (Body) ◮ Update Parabolic Equations (Chemicals) ◮ Solve Elliptic Equation (Pressure) ◮ Update Hyperbolic Equations (Boundary) MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? PDE Types and Methods How to Solve ◮ Body - Finite Difference ◮ Chemicals - ? ◮ Pressure - ? ◮ Boundary - ? MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? Coordinates Naive (and Not Naive) Methods ◮ Finite differences on the sphere - Parousia ◮ Finite Element Methods ◮ Finite Volume Methods MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? Coordinates Truncated Spherical Coordinates 0.8 0.6 0.4 0.2 0 z − 0.2 − 0.4 − 0.6 − 0.8 − 0.5 0 0.5 − 0.8 − 0.6 − 0.4 − 0.2 0 0.2 0.4 0.6 0.8 1 y x MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? Basis Functions And Projections Spherical Harmonics ◮ Atomic electron configurations ◮ Representation of the gravitational field, ◮ Magnetic field of planetary bodies ◮ Cosmic microwave background radiation ◮ Indirect lighting Y m ℓ ( θ, φ ) = P m ℓ (cos( θ ))e − im φ MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? Basis Functions And Projections Chebyshev Polynomials 1.0 0.5 ◮ Your friendly neighborhood family— � 1.0 � 0.5 0.5 1.0 of orthogonal polynomials � 0.5 ◮ Solutions to a DE � 1.0 ◮ Solutions of a recurrence relation 1.0 ◮ Family friends with Cosine. 0.5 T n (x) = cos(n cos − 1 (x)) � 3.0 � 2.5 � 2.0 � 1.5 � 1.0 � 0.5 � 0.5 � 1.0 MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? Basis Functions And Projections Chebyshev Polynomials 1.0 0.5 ◮ Your friendly neighborhood family— � 1.0 � 0.5 0.5 1.0 of orthogonal polynomials � 0.5 ◮ Solutions to a DE � 1.0 ◮ Solutions of a recurrence relation 1.0 ◮ Family friends with Cosine. 0.5 T n (x) = cos(n cos − 1 (x)) � 3.0 � 2.5 � 2.0 � 1.5 � 1.0 � 0.5 � 0.5 � 1.0 MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? Basis Functions And Projections Chebyshev Polynomials 1.0 0.5 ◮ Your friendly neighborhood family— � 1.0 � 0.5 0.5 1.0 of orthogonal polynomials � 0.5 ◮ Solutions to a DE � 1.0 ◮ Solutions of a recurrence relation 1.0 ◮ Family friends with Cosine. 0.5 T n (x) = cos(n cos − 1 (x)) � 3.0 � 2.5 � 2.0 � 1.5 � 1.0 � 0.5 � 0.5 � 1.0 MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? Basis Functions And Projections Chebyshev Polynomials 1.0 0.5 ◮ Your friendly neighborhood family— � 1.0 � 0.5 0.5 1.0 of orthogonal polynomials � 0.5 ◮ Solutions to a DE � 1.0 ◮ Solutions of a recurrence relation 1.0 ◮ Family friends with Cosine. 0.5 T n (x) = cos(n cos − 1 (x)) � 3.0 � 2.5 � 2.0 � 1.5 � 1.0 � 0.5 � 0.5 � 1.0 MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
Problem Background Difficulties Multi-Domain Spectral Method What Now? Basis Functions And Projections Chebyshev Polynomials 1.0 0.5 ◮ Your friendly neighborhood family— � 1.0 � 0.5 0.5 1.0 of orthogonal polynomials � 0.5 ◮ Solutions to a DE � 1.0 ◮ Solutions of a recurrence relation 1.0 ◮ Family friends with Cosine. 0.5 T n (x) = cos(n cos − 1 (x)) � 3.0 � 2.5 � 2.0 � 1.5 � 1.0 � 0.5 � 0.5 � 1.0 MATH 164 Sci. Comp. MATH 164 Harvey Mudd College
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