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
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Earth
1024
Man
G. Relativity
?
Classical Mech.
Material Properties
Superconductivity
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Chaos
Life
Consciousness
Social Relations
100
Number of Particles
dust
10-12
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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
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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
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