The Physics in BiologyThe Physics in BiologyModeling Tumor Growth and Angiogenesis

Rui Travasso

Centro de Física Computacional

Universidade de Coimbra

Physics Today

100

10-27

10-31

10-21

electrons

atoms

DNA

10-12

1024

10301031

1040

dust

Man

Earth

Sunblack hole

galaxy

Mass

Number of Particles

?

Material PropertiesSuperconductivity

SuperfluidityTurbulence

ChaosLife

ConsciousnessSocial Relations

G. Relativity

Quantum Mech.

Classical Mech.

Physics in Biology

Physics is needed Physical processes entangled with biology

Tumor growth Embryonic development Consciousness

Interdisciplinary subject Physics Biology Mathematics Chemistry Informatics

Simple Systems

Liquid membranes Canham-Helfrish energy

Minimization of energy provided surface and volume constant

Curvature Energy Relevant

Influence of changing c0

Constant: pearling instability

Gradient: tube formation

So?

Simple models present rich behavior Biologically relevant

Mechanical effects are important in cell behaviour Red blood cells change mechanical

properties if patient has malaria Organization of endothelial cells

through mechanical adhesion

But Insight is important but not sufficient Interdisciplinary study is essential for advance of field

Cancer and Physics

Physics important in developing imaging tools for detection andfollowing tumor growth

but recently...

Physics may be important for understanding tumor growth

Physics meets Biology meets Chemistry Mechanical interactions, viscoelastic

dynamics, protein diffusion, chemicalreactions, gene regulatory networks, population dynamics, evolution

Physics World, June 2010

Crescimento de Tumores - Mutações

Fase 1: Mutações genéticas Genes que regulam processos essenciais

Ciclo celular Reprodução descontrolada Sistemas de reparação do DNA e de proteínas Perda de mecanismo de morte programada

Crescimento de Tumores - Tecido

Fase 2: Interacção com o tecido celular Células cancerígenas inibem células imunitárias Ou recrutam células imunitárias

(que recrutam vasos sanguíneos) Sobrevivem em condições adversas

(ambiente ácido e baixos níveis de oxigénio)

Célula Tumoral

Célula do sist. imunitário

Crescimento de Tumores - Caderinas

Fase 3: “Cadherin switch” Células interagem com vizinhas através

de proteínas da membrana Caderinas

Mutação deste mecanismo pode levar a altas taxas de proliferação mesmo quando densidade celular alta.

Crescimento de Tumores - Esferóides

Fase 4: Células cancerígenas ganham forma: Esferóide Difusão macroscópica de células Formação de zonas necróticas Tumor com diâmetro 1-2 mm

Zona Necrótica

Reprodução Descontrolada

Células SaudáveisNecroticasQuiescentesProliferativas

Alta Pressão

Crescimento de Tumores - Angiogénese

Tumor necessita nutrientes para crescer Busca activa de nutrientes

Fase 5: “Angiogenic switch” Segregação de proteínas

que promovem formaçãode novos vasos sanguíneos

Rede vascular aberrante

M. D. Anderson Cancer Center, Univ. of Texas

Crescimento de Tumores - Metástase

Fase 6: Metástase Células cancerígenas entram na

circulação sanguínea Invasão de regiões saudáveis

Pulmão Fígado

Alguns Tópicos sobre Tumores

Reprodução desregulada de células cancerínenas Grande diversidade de material genético das células Maior adaptabilidade

Tumor vive num ambiente que lhe é extremamente hostil A destruição do hospitaleiro é uma vitória da adaptação. Infelizmente isso significa a morte do tumor também

Vasos saguíneos frágeis O tumor sangra

Angiogénesis contínua O tumor é uma ferida que não sara

Understanding Tumors Through Modeling

Effect of pressure inside tumors in affecting circulation Vessel collapse

Tumor surface instabilities as a function of limitations in transport of nutrients May lead to phenotypic alterations Balance between cell-cell adhesion

and nutrient delivery Tumor adaptability and tumor

stem cells

Guide treatment Use of modeling as a tool for predicting patient-specific evolution

and treatment of tumors

Tumor Modeling

Many models Review article:

Nonlinearity, 23, R1 (2010) 578 references

Each paper introducesdifferent model for a specific application

Classification of models Discrete: Cellular automata, Agent based, ... Continuous: Multiphase, Interface focused, ...

Discrete Models

Focus on individual cells Mutations Contact forces Cell division Movement and growth Gene regulatory networks

Advantage Some parameters may be obtained from single cell experiments

Limitations Challenging to simulate millions of cells Large number of parameters (which ones are controlling factors?)

Shirinifard et al, PLoS One, 4, e7190

Continuous Models

Interface focused Map tumor surface behavior to existing interface models In general do not include biological details

Multiphase modeling From mixture theory

Consider different components Conservation laws (mass, momentum) Constitutive relations specific

for each component Thermodynamic consistency

Possibility of including biological processes Fewer parameters than discrete methods

Preziosi et al, J.Math.Biol., 58, 625

Approach to moving boundary problems Phases associated with value of

Interface implies = 0 Diffuse interface

Original problem obtained when → 0

Dynamics of Can be derived from a free energy F[,]

Non-conserved order parameter: Allen-Cahn equation

Conserved order parameter: Cahn-Hilliard equation

Phase-Field Models

Phase 1

Phase 2= -1

= 1

δφδφ F

t−=

∂

∂

δφδφ F

t2∇=

∂

∂

f

1-1

Examples

Canham-Helfrisch energy

Phase separation of elastic phases

Dendritic growth

Phase-field model in tumor growth

Travasso, Castro, Oliveira, Phil. Mag. (2011)

Example of Multiphase and Phase-Field

A multiphase model Cristini et al, J.Math.Biol., 58, 723 (2009)

Mass balance for each component

Incompressibility

Momentum conservation

Constitutive Relations

Example of Multiphase and Phase-Field

Formation of ramified structures

More dramatic at low proliferation rate

Fingering occurs at zero chemotaxis

Instability driven by non-linear mobility

Cristini et al, J.Math.Biol., 58, 723 (2009)

Therefore...

Phase-Field is focused at the interface Link between phase-field and multiphase

Further reduction of parameters Variability of existing phase-field models

lead to possibility of direct applicationin tumor growth

Able to answer questions on the evolutionof tumor size

BUT...

Do not include competing populations oftumor cells or mutations Hybrid models are a possible solution

Tumor Growth - Competition - Evolution

Deregulated proliferation Mutations Darwin selection

Metabolism and migration

Anaerobic matabolism 2 ATP instead of 36 No need of Oxygen Produces acid Helps migration

Prevailing phenotype Acid resistant Gerlee, Anderson, J Theor Biol 2007

Acid

Tumor Growth - Angiogenesis Switch - Vascular Phase

The tumor promotes thedevelopment of nearbyvessels to have oxygen

Challenging simulations Many parameters Cell based Continuous Hybrid

MackLin et al, J Math Biol 2009

Chaplain et al, Annu Rev Biomed Eng 2006

Angiogenesis

Sprouting of new blood vessels from existing ones

Relevant in varied situations Morphogenesis Inflammation Wound healing Neoplasms Diabetic Retinopathy

For tumors Altered vessel network Dense, no hierarchical structure Capillaries are fragile, permeable, with variable diameter Capillary network carries both nutrients and drugs

Gerhardt et al, Cell (2003)

Lee et al, Cell (2007)

Two types of cells

Tip cells are special Have filopodia Follow gradients of VEGF Produce MMPs which degrade ECM Construct path Do not proliferate

Stalk cells Proliferation regulated by VEGF Not diggers

Follow tip cell created pathway

Gerhardt et al, Cell (2003)

Gerhardt et al, Cell (2003)

Agent Based Component

Phase-field Component

Angiogenesis in a Nutshell

Capillaries are constituted by Endothelial cells Pericites, muscle cells

Endothelial cells

Pericites, smooth muscle cells…

VEGF

VEGF weakens capillary wall

Endothelial cells may divide

Cells follow VEGF gradient

The first cell is activated and opens way in ECM

Cells organize to form lumen

Blood flows when capillaries form loops

Blood reorganizes network

Meyer et al, A

m.J.P

ath. (1997)

The Model

The penetration length of T inside the capillary

is given by D

€

∂tφ =∇ 2μ + α φTφΘ(φ) = 1 inside capillary

= -1 outside capillary

€

v t = Dφ∇T T

Two equations Diffusion: concentration of VEGF, T Phase-Field: order parameter dynamics

Tip cell Characteristic radius Rc

Perfect Notch signaling Introduced when T > Tc

Velocity:

regulates the proliferation and D the chemotaxis

Ginzburg-Landau free energy

Chemical potential

Cahn-Hilliard dynamics

Surface tension driven, bulk material conservation

€

F = −φ2

2+

φ4

4+

ε 2

2∇φ( )

2 ⎛

⎝ ⎜

⎞

⎠ ⎟∫ dr r

€

∂φ∂t

= −∇ ⋅ −∇μ( )

€

μ =δF

δφ= −φ + φ3 −ε 2∇ 2φ

Simulation

Starting configuration Capillary close to tissue

in hypoxia Concentration of VEGF at

hypoxic cells constant

CapillaryCells in hypoxia

Blood vessel network emerge

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Proliferation

Higher proliferation rate leads to thicker and ramified vessels

Low Proliferation High Proliferation

Chemotaxis Response

Higher tip cell velocity leads to thinner and more ramified vessels

Low Chemotaxis High Chemotaxis

VEGF Prodution

Higher production of VEGF leads to more vessels but not thicker vessels

Gerhardt et al., Develop. Biol. (2003)

Low VEGF High VEGF

Matrix Metalloproteinase

MMPs implementation: Heavy VEGF isoforms get

bound to matrix if cMMP high

cMMP high in a radius RMMP

of tumor cell Diffusion in function of Th

Formation of thick vessels Thin vessel merging

Rodriguez-Manzaneque et al, PNAS (2001)

MM

P-9

In

hib

itio

nM

MP

-9 O

vere

xpre

ssed

Th

Dhigh cMMP

low cMMP

Insight is important but not sufficient

Taxa de proliferação Dependente do meio (VEGF, Ang-2)? Como?

Propriedades dos tecidos Tecido como meio viscoelástico Permeabilidade e elasticidade dos vasos

Metabolismo das células Possibilidade de respiração anaeróbia? Em que circunstâncias? Influencia do meio ácido na viabilidade das células Transporte de proteínas Reacções químicas

As células tumorais são de diferentes tipos Dinâmica de populações Evolução

Interdisciplinaridade

Simulação

• Morfogénese• Tumores• Pólipos• Retinopatia

Lab in vitro Lab in vivo

Dados Clínicos

medição exp. de parâmetros

novas hipótesese experiências

previsões decrescimento

vascular

termos relevantes in vivo

acompanhamentoclínico individualizado

observaçõesclínicas

A Física poderá ajudar, mas como um elemento de um esforço interdisciplinar Integração de técnicas e métodos de diferentes disciplinas

Physics required to tackle problems in Biology New insights New therapies Interdisciplinary context

Modeling tumor growth Variety of modeling techniques

Hybrid models are able to integrate in a continuous description cell based processes essential in tumor growth and angiogenesis

Hybrid model for angiogenesis with phase-field component Proliferation rate and matrix dependent tip cell velocity regulate

capillary network morphology High production VEGF levels lead to increased vessel density Bio-avaibility of VEGF determines network

Conclusion

Gerhardt et al, Cell (2003)

High Pressure

A Pretty One

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