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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