The Physics in Biology
Modeling Tumor Growth and Angiogenesis
Rui Travasso
Centro de Física Computacional
Universidade de Coimbra
Physics Today
Mass
galaxy
1040
black hole
Sun
1031
1030
Earth
1024
Man
G. Relativity
?
Classical Mech.
Material Properties
Superconductivity
Superfluidity
Turbulence
Chaos
Life
Consciousness
Social Relations
100
Number of Particles
dust
10-12
DNA
10-21
atoms
10-27
electrons
10-31
Quantum 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 and
following tumor growth
but recently...
 Physics may be important for
understanding tumor growth
 Physics meets Biology meets Chemistry

Mechanical interactions, viscoelastic
dynamics, protein diffusion, chemical
reactions, 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
Necroticas
Quiescentes
Proliferativas
Alta Pressão
Zona Necrótica
Reprodução
Descontrolada
Células Saudáveis
Crescimento de Tumores - Angiogénese
 Tumor necessita nutrientes para crescer
 Busca activa de nutrientes
 Fase 5: “Angiogenic switch”
 Segregação de proteínas
que promovem formação
de 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 introduces
different 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
Shirinifard et al, PLoS One, 4, e7190
 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?)
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
Phase-Field Models
 Approach to moving boundary problems
 Phases associated with value of
Interface implies f = 0
 Diffuse interface
 Original problem obtained
when e → 0
f
f= 1

 Dynamics of
Phase 1
e
f= -1
f
Phase 2
f
 Can be derived from a free energy F[f,e]
f 1
f
F


 Non-conserved order parameter: Allen-Cahn equation
t
f
f
F
 Conserved order parameter: Cahn-Hilliard equation
 2
t
f
-1
Examples

Canham-Helfrisch energy


Dendritic growth
Phase separation of elastic phases

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
Momentum conservation
Constitutive
Relations
Incompressibility
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 application
in tumor growth
 Able to answer questions on the evolution
of tumor size
BUT...
 Do not include competing populations of
tumor cells or mutations
 Hybrid models are a possible solution
Tumor Growth - Competition - Evolution
 Deregulated proliferation
 Mutations
 Darwin selection

Acid
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
Tumor Growth - Angiogenesis Switch - Vascular Phase
 The tumor promotes the
development of nearby
vessels to have oxygen
 Challenging simulations
Chaplain et al, Annu Rev Biomed Eng 2006
 Many parameters
 Cell based
 Continuous
 Hybrid
MackLin et al, J Math Biol 2009
Angiogenesis
 Sprouting of new blood vessels from existing ones
 Relevant in varied situations
 Morphogenesis
Gerhardt et al, Cell (2003)
 Inflammation
 Wound healing
 Neoplasms
 Diabetic Retinopathy
 For tumors
 Altered vessel network
Lee et al, Cell (2007)
 Dense, no hierarchical structure
 Capillaries are fragile, permeable, with variable diameter
 Capillary network carries both nutrients and drugs
Two types of cells
 Tip cells are special
 Have filopodia
 Follow gradients of VEGF
 Produce MMPs which degrade ECM
 Construct path
 Do not proliferate
Gerhardt et al, Cell (2003)
 Stalk cells
 Proliferation regulated by VEGF
 Not diggers

Follow tip cell created pathway
Gerhardt et al, Cell (2003)
Angiogenesis in a Nutshell
 Capillaries are constituted by
 Endothelial cells
Endothelial cells
Pericites, smooth muscle cells…
 Pericites, muscle cells
 VEGF weakens capillary wall
 Endothelial cells may divide
VEGF
 Cells follow VEGF gradient
 The first cell is activated and opens way in ECM
 Cells organize to form lumen
Meyer et al, Am.J.Path. (1997)
 Blood flows when capillaries form loops
 Blood reorganizes network
The Model
 Two equations
 Diffusion: concentration of VEGF, T
 Phase-Field: order parameter dynamics
The penetration length  of
T inside the capillary
is given by D
t f     f Tf(f)
2
 Tip cell
 f 2 f 4 e 2
2 
F    
Ginzburg-Landau
free energyradius
 Characteristic
Rc 2 4  2 f dr
 Perfect Notchsignaling
F

 f  f 3  e 2 2f
Chemical
potential
 Introduced when
f T > Tc
D T
 Velocity: v t 
f f
  
Cahn-Hilliard dynamics


f = 1 inside capillary
f = -1 outside capillary
T
t
 tension
f regulates
thematerial
proliferation
and
Surface
driven, bulk
conservation
Df the chemotaxis
Simulation
 Starting configuration
Capillary
Cells in hypoxia
 Capillary close to tissue
in hypoxia
 Concentration of VEGF at
hypoxic cells constant

Blood vessel network emerge
QuickTime™ and a
Video decompressor
are needed to see this picture.
QuickTime™ and a
H.264 decompressor
are needed to see this picture.
Proliferation
Low Proliferation
High Proliferation
 Higher proliferation rate leads to thicker and ramified vessels
Chemotaxis Response
Low Chemotaxis
High Chemotaxis
 Higher tip cell velocity leads to thinner and more ramified vessels
VEGF Prodution
Gerhardt et al.,
Develop. Biol. (2003)
Low VEGF
High VEGF
 Higher production of VEGF leads to more vessels but not thicker
vessels
Matrix Metalloproteinase
 MMPs implementation:
bound to matrix if cMMP high
 cMMP high in a radius RMMP
of tumor cell
 Diffusion in function of Th
low cMMP
D
high cMMP
Th
 Formation of thick vessels
 Thin vessel merging
MMP-9 Overexpressed MMP-9 Inhibition
 Heavy VEGF isoforms get
Rodriguez-Manzaneque et al, PNAS (2001)
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
 A Física poderá ajudar, mas como um elemento de um esforço
interdisciplinar
 Integração de técnicas e métodos de diferentes disciplinas
Simulação
medição exp.
de parâmetros
Lab in vitro
novas hipóteses
e experiências
• Morfogénese
• Tumores
• Pólipos
• Retinopatia
observações
clínicas
termos relevantes
in vivo
Lab in vivo
previsões de
crescimento
vascular
acompanhamento
clínico individualizado
Dados Clínicos
Conclusion
High Pressure
 Physics required to tackle problems in Biology
 New insights
 New therapies
 Interdisciplinary context
 Modeling tumor growth
 Variety of modeling techniques
Gerhardt et al, Cell (2003)
 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
A Pretty One
QuickTime™ and a
Video decompressor
are needed to see this picture.