Level Set Tumor Growth Model Andrew Nordquist and Rakesh Ranjan, PhD University of Texas, San Antonio July 29, 2013 Andrew Nordquist and Rakesh Ranjan, PhD (University Level Set of Texas, TumorSan Growth Antonio) Model July 29, 2013 1 / 14
Level Set vs Mixture Theory The Level Set model is based on Mixture Theory. While Mixture Theory has the concept of Volume Fractions, κ n α = α=1 κ α=1 ρ α = 1, (1) ραr Level Set says, We have volume fractions, two solids and one liquid-dissolved nutrient. The two solids do not mix. The nutrient is consumed by one of the two solids. One solid is Tumor tissue, the other is Host (or normal) tissue. Yes, host tissue consumes nutrient, but at a much lower rate than tumor tissue. Thus, we ignore homeostasis for this model. Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, 2013 2 / 14
Level Set departure from Mixture Theory Mixture Theory concerns itself with Balance of Mass and Momentum equations of Continuum Mechanics: Balance of Mass - constituent s mass rate ˆρ α, and velocity v α n α t + (n α v α ) = ˆρ α ρ αr (2) Balance of Momentum - Cauchy stress tensor T α, body force b α ( ) T α + ρ αr n α b α dvα + ˆp α ˆρ α v α = 0 (3) dt Balance of Energy (1st Law of Thermo) is combined with Entropy Inequality (2nd Law of Thermo) to help construct constitutive equations to go with above equations. Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, 2013 3 / 14
Level Set departure from Mixture Theory, (continued) We temporarily shift to 2 dimensions, without loss of generality. Assume the two solid tissues act as fluids, with only one volume fraction (n [0, 1]), and incompressible Navier-Stokes behavior: v x x + v y y = 0 (4) Here is x-component of momentum balance equations, ( v x t + v v x x x + v v x 2 ) y y + ν v x x 2 + 2 v x y 2 = P x + ρg x + kcn (5) and y-component of momentum balance equations, ( v y t + v v y x x + v v y 2 y y + ν v y x 2 ) + 2 v y y 2 = P y + ρg y + kcn (6) Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, 2013 4 / 14
Level Set Methodology: Governing Equations for Nutrient and Interface The Interface is defined by the volume fraction of tumor tissue, n (AKA level set ). Where n = 1, we have tumor tissue, and where n = 0 we have normal host tissue. Equation to move the interface with velocity field v: n t + v n = γ [ ( ɛ n n(1 n) n )] n Evolution equation for nutrient as a dissolved species ( c t + v c x x + v c 2 y y D v x x 2 (7) ) + 2 v y y 2 = ṙ c (c, n) (8) Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, 2013 5 / 14
Level Set Methodology: Description of Symbols n if n = 1, tumor tissue; if n = 0, host tissue v x, v y x- and y- components of tumor tissue velocity field v P pressure ν the viscosity tensor ρ the (tumor) tissue density g x, g y x- and y- components of the gravitational field g (negligible) c concentration of a dissolved nutrient species, such as oxygen k rate of consumption of the nutrient D the diffusivity tensor ɛ thickness of the interface region γ amount of reinitialization or stabilization of the level set function Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, 2013 6 / 14
Level Set - COMSOL Implementation Phase initialization step - GI is reciprocal of initial interface distance Time Dependent step GI GI + σ w GI ( GI ) = (1 + 2σ w )GI 4 (9) I w = 1 GI I ret 1 (10) ρ v [ ] t + ρ(v )v = ρi + µ( v + ( v) T ) + ρg + f (11) v = 0, n t [ + v n = γ ɛ ls n n(1 n) n ] n (12) Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, 2013 7 / 14
Level Set - COMSOL Model Setup Length scale factor: 1 meter in model is 10 microns (10 5 m) in reality Time scale factor: 1 second in model is 1 day (86400 s) in reality Model is 100µm on each side capillary structure is 20µm in diameter, 10µm away from tumor tumor starts out at 20µm in diameter intent is to see tumor grow, and prefer to grow around capillary, forming a tumor cord (... Show Pictures Now...) Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, 2013 8 / 14
What needs to happen now to the Level Set model? Calculate scaling parameters for: density, viscosity, velocity, etc. Find a transform to convert viscosity to some kind of tissue hardness unit Make model more realistic by converting real parameters to scaled parameters Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, 2013 9 / 14
That s all, Folks! Thank You! Any Questions? Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, 2013 10 / 14
Thermodynamics: First Law Consider our mixture in a 3D volume Ω, and an increment of that volume, dv. Ω s surface is Ω. This is a widely accepted mathematical form for the First Law of Thermodynamics, applicable to each constituent (α) in our mixture: ρ εα t T α : L α ρ α ˆρ c + q α = 0 (13) ρ α the density of the constituent ε α internal energy of the constituent T α : L α stress power of the constituent ˆρ c reduction of energy due to consumption of c internal to dv q α heat flux through dv T α Cauchy stress tensor L α velocity gradient = D α (symmetric) + W α (skew symmetric) Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, 2013 11 / 14
First law of thermodynamics, continued Since the Cauchy stress tensor is symmetric, the stress power simplifies to T : L = T : D = tr(td) (14) So the First Law simplifies to ρ ε t tr(td) ρˆρc + q = 0 (15) Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, 2013 12 / 14
Thermodynamics: Second Law The mathematical form for the Second Law of Thermodynamics, (AKA Entropy Inequality) is: d ρη dv ρ ˆρc dt Ω Ω θ dv q ˆn da, where (16) Ω θ η the specific entropy of the constituent ˆn the unit vector normal to S. The divergence theorem changes our surface integral to a volume integral. Due to the arbitrary nature of the volumes under consideration, we may then discard the volume integrals after converting the surface integral, leaving the integrands: ρ η ρˆρ c θ + q θ q θ θ 2 0 (17) Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, 2013 13 / 14
Second law of thermodynamics, continued After multiplying Equation (17) through by θ, we subtract Equation (13) from it to get: ρ ηθ ρ ε t q θ + T : D 0 (18) θ We now introduce the Helmholtz Free Energy, defined as ψ = ε ηθ (19) Differentiate with respect to time and rearrange to get, ηθ = ε t η θ ψ t (20) Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, 2013 14 / 14
The Entropy Inequality Replace the ηθ term in Equation (18) by the right-hand side of the last Equation, and we get: ρ ε t ρη θ ρ ψ t ρ ε t q θ + T : D 0 (21) θ The terms involving ε cancel each other, and this equation simplifies to T : D ρη θ ρ ψ t q θ 0 (22) θ This is known as the modified Clausius-Duhem inequality. Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, 2013 15 / 14