Development of Glioblastoma growth model.

Antonio Fernández

Francisco Guillén

Antonio Suárez

November, 21st 2018

Index

1 Introduction

2 Glioblastoma Model

3 Theoritical Results

4 Simulations

Introduction Glioblastoma Model Theoritical Results Simulations

1. Introduction Development of Glioblastoma growth model

Motivation

A. Fernández, F. Guillén, A. Suárez

Introduction Glioblastoma Model Theoritical Results Simulations

1. Introduction Development of Glioblastoma growth model

Motivation

A. Fernández, F. Guillén, A. Suárez

Introduction Glioblastoma Model Theoritical Results Simulations

1. Introduction Development of Glioblastoma growth model

Purpose

Our goal is to understand and reproduce this developmet indifferent situations. For it,

We are going to create a PDE-ODEs model with which wewill work.Later, we will start studying theoretical properties of it.Finally, we are going to show some simulations about thismodel.

A. Fernández, F. Guillén, A. Suárez

Index

1 Introduction

2 Glioblastoma Model

3 Theoritical Results

4 Simulations

Introduction Glioblastoma Model Theoritical Results Simulations

2. Glioblastoma Model Development of Glioblastoma growth model

Formulation

∂T

∂t= ∇ · (D (Φ, T )∇T ) + ρ (Φ, T ) T

(1− T +N + Φ

K

)− α f (Φ, T ) T − β N T

∂N

∂t= α f (Φ, T ) T + β N T + δ T Φ + β N Φ

∂Φ

∂t= γ f (Φ, T )

T

KΦ

(1− T +N + Φ

K

)− δ T Φ− β N Φ

A. Fernández, F. Guillén, A. Suárez

Introduction Glioblastoma Model Theoritical Results Simulations

2. Glioblastoma Model Development of Glioblastoma growth model

Formulation

∂T

∂t= ∇ · (D (Φ, T )∇T ) + ρ (Φ, T ) T

(1− T +N + Φ

K

)− α f (Φ, T ) T − β N T

∂N

∂t= α f (Φ, T ) T + β N T + δ T Φ + β N Φ

∂Φ

∂t= γ f (Φ, T )

T

KΦ

(1− T +N + Φ

K

)− δ T Φ− β N Φ

A. Fernández, F. Guillén, A. Suárez

Introduction Glioblastoma Model Theoritical Results Simulations

2. Glioblastoma Model Development of Glioblastoma growth model

Formulation

∂T

∂t= ∇ · (D (Φ, T )∇T ) + ρ (Φ, T ) T

(1− T +N + Φ

K

)− α f (Φ, T ) T − β N T

∂N

∂t= α f (Φ, T ) T + β N T + δ T Φ + β N Φ

∂Φ

∂t= γ f (Φ, T )

T

KΦ

(1− T +N + Φ

K

)− δ T Φ− β N Φ

A. Fernández, F. Guillén, A. Suárez

Introduction Glioblastoma Model Theoritical Results Simulations

2. Glioblastoma Model Development of Glioblastoma growth model

Parameters

f (Φ, T ) =

√1−

(Φ

Φ + T

)2

, D(Φ, T ) = DΦ

Φ + T, ρ (Φ, T ) = ρ

Φ

Φ + T

Variable Description ValueD(φ) Diffusion Speed cm2/dayρ (φ) Proliferation rate day−1

β Change rate to necrosis influence day−1

α Hypoxic death rate by persistent anoxia day−1

γ Vasculature proliferation rate day−1

δ Vasculature death by tumor action day−1

K Carriying capacity cel/cm3

A. Fernández, F. Guillén, A. Suárez

Index

1 Introduction

2 Glioblastoma Model

3 Theoritical Results

4 Simulations

Introduction Glioblastoma Model Theoritical Results Simulations

3. Theoritical Results Development of Glioblastoma growth model

Estimates

1 L∞ bounds0 ≤ T,Φ ≤ K

0 ≤ N ≤ N

T,N,Φ ∈ L∞ ([0, Tf ];L∞ (Ω))

2 L2 boundsT ∈ L∞

([0, Tf ];L2 (Ω)

)∩ L2

([0, Tf ];H1 (Ω)

)

A. Fernández, F. Guillén, A. Suárez

Introduction Glioblastoma Model Theoritical Results Simulations

3. Theoritical Results Development of Glioblastoma growth model

Existence and uniqueness for linear diffusion model

TheoremGiven T0, N0,Φ0 ∈ Ω such that 0 < T0 +N0 + Φ0 ≤ K, exits aunique solution (T,N,Φ) for the problem with linear diffusionthat satisfies

N,Φ ∈ C1([0, Tf ] ; C0

(Ω))

T ∈ C0([0, Tf ] ; C0

(Ω))

Way: Leray-Schauder fixed-point therorem with

R : C0([

0, Tf

]; C0

(Ω)) R1→

(C1

([0, Tf

]; C0

(Ω)))2 R2→ C0

([0, Tf

]; C0

(Ω))

T (N,Φ) T

A. Fernández, F. Guillén, A. Suárez

Introduction Glioblastoma Model Theoritical Results Simulations

3. Theoritical Results Development of Glioblastoma growth model

Stability

Model without diffusion has 3 types of equilibrium solutions

1 (0, 0, 0)2 (0, (N0)∗ , 0), (N0)∗ > 03 (0, 0, (Φ0)∗), (Φ0)∗ > 0

PDE-ODE system has the same equilibriums

A. Fernández, F. Guillén, A. Suárez

Index

1 Introduction

2 Glioblastoma Model

3 Theoritical Results

4 Simulations

Introduction Glioblastoma Model Theoritical Results Simulations

4. Simulations Development of Glioblastoma growth model

Vasculature fixed. Results

(a) Necrosis (b) Tumor (c) Vasculature

A. Fernández, F. Guillén, A. Suárez

Introduction Glioblastoma Model Theoritical Results Simulations

4. Simulations Development of Glioblastoma growth model

Variable vasculature. Results

(a) Necrosis (b) Tumor (c) Vasculature

A. Fernández, F. Guillén, A. Suárez

Introduction Glioblastoma Model Theoritical Results Simulations

4. Simulations Development of Glioblastoma growth model

Survival by Rim width groups

0 20 40 60 80Survival [months]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1C

umul

ativ

e su

rviv

alP = 0.014 (b)

CE rimwidth ↑(n=92)

CE rimwidth ↓(n=73)

7.69 mo

Survival byRim widthgroups

A. Fernández, F. Guillén, A. Suárez

Introduction Glioblastoma Model Theoritical Results Simulations

4. Simulations Development of Glioblastoma growth model

Model

∂T

∂t= ∇ · (D (Φ, T )∇T ) + ρ (Φ, T ) T

(1− T +N + Φ

K

)−α f (Φ, T ) T − β N T

∂N

∂t= α f (Φ, T ) T + β N T + δ T Φ + β N Φ

∂Φ

∂t= γ f (Φ, T )

T

KΦ

(1− T +N + Φ

K

)− δ T Φ− β N Φ

A. Fernández, F. Guillén, A. Suárez

Introduction Glioblastoma Model Theoritical Results Simulations

4. Simulations Development of Glioblastoma growth model

Rim width-Volume

(a) α = 1 (b) α = 500

A. Fernández, F. Guillén, A. Suárez

Introduction Glioblastoma Model Theoritical Results Simulations

4. Simulations Development of Glioblastoma growth model

Tumor and Necrosis growth

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T+N=0.2

T+N=0.85

T+N=0.95

(a) α = 1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T+N=0.2

T+N=0.65

T+N=0.85

T+N=0.95

(b) α = 500

A. Fernández, F. Guillén, A. Suárez

Introduction Glioblastoma Model Theoritical Results Simulations

5. Development of Glioblastoma growth model

Thanks !

A. Fernández, F. Guillén, A. Suárez