continuum methods iiilowengrb/research/multimedia/lecture3.pdfmechanism of tumor neo-vascularization...
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Continuum Methods III
John LowengrubDept MathUC Irvine
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Outline
•Angiogenesis, neovascularization in context ofsolid tumor growth
•Systems Biology: PDE models of cell signaling.
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Mechanism of tumor neo-vascularization
•Tumor cells prefer glucose as a nutrient. •Angiogenic phenotype is the result of a net balance of endogenous stimulators and inhibitors of angiogenesis.•The expression of angiogenic growth factors is increased in the tumor regions neighboring its necrotic area.•Both hypoxia and hypoglycaemia may increase the expression of angiogenic factors.
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Present model
•Continuum approximation: super-cell macro scale
•Role of cell adhesion and motility on tissue invasion and metastasisIdealized mechanical response of tissues
•Coupling between growth and angiogenesis (neo-vascularization): necessary for maintaining uncontrolled cell proliferation
•Genetic mutations: random changes in microphysical parameters cell apoptosis and adhesion
•Limitations: poor feedback from macro scale to micro scale(Greenspan, Byrne & Chaplain, Anderson & Chaplain,Levine…)
highly-vascularizedexterior
Healthy tissue(may have vessels)
ProliferatingTumor region(may have vessels)
Necroticcore
Capturedregion
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Cell proliferation and tissue invasion
( ) in
in | ( , )
on
M A P
N N Nt
PP
λ σ λλ σ σ
τ κν µ
− Ω∇• = − Ω = ≤
= Σ
− ∆ = − ∇
ux x
u u
Cell proliferation: in the tumor is a balance of mitosisand apoptosis (mitosis is responsible for reproduction of mutated genes) and is one of the two main factors responsible for tissue invasion
Spatial distribution of the oncotic pressure
Cell mobility: reflect strength of cell adhesionto other cells and to the Extra-Cellular Matrix (ECM), the other main factor leading to tissue invasion
Assume constant tumor cell density: cell velocity
Darcy-Stokes
Cell death responsible for release of angiogenic factors: INPUT TO ANGIOGENESIS
Assume 1 diffusing nutrient of concentration σ
Cell-to-celladhesion
Greenspan, Chaplain, Byrne, …
Rate of enzymatic breakdownof necrotic cells (death due to lack of nutrient)
Viability concentrationviscosity
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Evolution of nutrient: Oxygen/GlucoseGreenspan, Chaplain, Byrne, …
( ) ( ,P , , )C B B BD P ttσ σ λ σ λ σ σ∂= ∇• ∇ − ⋅ + − −
∂x
Oncotic pressure: affects blood flow and delivery of nutrients (and chemotherapy drugs)
Blood-to-tissue nutrient transfer rate function.Spatial distribution of capillaries: OUTPUT FROM ANGIOGENESIS
Nutrient consumption by the cells
Diffusion
=0 (quasi-steady assumption). Tumor growth time scale (~1 day) large compared to typical diffusion time (~1 min)
nutrientconcentrationin blood
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More complex Biophysics
•Blood-tissue transfer of nutrient
( ) ( ), , , ( )B B B B B BP P t h P Pλ σ σ λ σ σ +− − = − ⋅ −x
•Avascular, angiogenesis and fully vascularized growth
•Simplified cell-cycling model ( )M bλ σ σ=
•Nonlinear interaction betweendeveloping vasculature and tumorgrowth
( ) ( )B B Capillaryh σ σ σ σ δ− = −
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Angiogenesis
Angiogenic factors:VEGF (Vascular Endothelial cell Growth Factor)FGF (Fibroblast Growth Factor)AngiogeninTGF (Transforming Growth Factor),….
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ECM/MMP Regulation of VEGFLee, Jilani, Nikolova, Carpizo, Iruela-Arispe JCB. 2005.
VEGF-A isoformssoluble
insoluble
•Insoluble VEGF + Matrix Metalloproteinases Soluble VEGF+ MMPs(ECM) (Endothelial cells)
•Different signaling outcomes through VEGFR2
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Effect on EC growthBeads containing cells embedded in fibrin/fibronectin gels
•VEGF 113: SheetsVEGFD108-118: ChordsVEGF 164: Both
(stain to measureproliferation)
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Effect on Vessel MorphologyWild Soluble Insoluble
•Morphology stronglydepends on type
(diameters) 15 mµ 109 mµ 16 mµNumber density perimeter
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Effect on tumor growth
•Soluble VEGF poor prognosticator of tumorprogression
•Matrix-bound VEGFyields more efficientangiogenic response
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Mathematical modelEndothelial cell concentration e: form the lining of the capillary
Chemotaxis Haptotaxis
Proliferation
Anderson, Chaplain, Macdougall, Levine, Sleeman, Sun, et al BMB 2005,…
Tumor angiogenic factor (e.g., VEGF-A): potent mitogen, drives motion
Uptake by the endothelial cellsDecay
Cell receptor ligand (e.g., Fibronectin) in the ECM. Regulates cell adhesion and motion
production degradation
•Recast in a biased random-walk model to follow the evolution of the capillaries (Anderson, Chaplain)
N1 onc = Σ
Zheng,Wise,Cristini BMB 2005
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Numerical method
Resolution of physical scales
•Level-set/Finite-element formulationZheng, Wise, Cristini,. Bull. Math. Biol. 2005 Zheng, Anderson, Cristini, J. Comp. Phys. 2005Zheng, Lowengrub, Anderson, Cristini, J. Comp. Phys. 2005
•Adaptive computational meshCristini et al. J. Comp. Phys. 2001, Zheng et al. J. Comp. Phys (2005):
(Mixed methods, LDG, EPC)
Mesh: System of springs (energy)Local Operations Minimum energy Optimal mesh
1 2min( , ,...)eql l l=
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•Vary DandcD β to mimic Soluble/Insoluble VEGF-ADay 0 Day 10 Day 20
Insoluble
Partlysoluble
Soluble
•Chemotaxis/Branching enhancedwith insoluble VEGF
•Qualitative agreementwith experiments
Parameters from literature.
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MechanismDistribution of VEGF:
Insoluble
Partlysoluble
Soluble
•Uptake of ECM-boundVEGF-A by EC produces large gradient in insoluble case
Day 0 Day 10 Day 20
•Gradients enhance chemotaxis
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Later times
PartlySoluble
Soluble
•Brush-border effect•penetration
•Irregular vasculardevelopment
•No penetration
•Qualitative agreement with experiment
•Experimental results consistent with increased cβcD and/or decreased
Day 45 Day 70
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Movies
SolubleInsoluble
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More sophisticated modelInsoluble
SolubleCleaving
•Variable diffusion for insoluble TAF
•Test coupling with full tumor model-tumor and vessel development nonlinearly coupled
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Fully coupled modelInsoluble Soluble
•Brush-border effect•Penetration•Growth of tumor
•Irregular vasculardevelopment
•little penetration•Less growth
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Stills
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VEGF Insoluble
Soluble
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Growth of Glioblastoma MultiformeSimulated growth time: ca. 8 years
(parameters from experiments and clinical data)
Partly soluble Tumor Angiogenesis Factor (e.g. VEGF)
•Tumor and blood vessel morphology develop together•Significant growth of both
Zheng, Wise, Cristini, BMB 2005.
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Conclusions• Developed a framework to model tumors through all
phases of growth
• Qualitative agreement with experiments by Iruela-Arispe for neovascular morphology
morphology controlled by diffusion/degradation
of VEGF-A
•Nonlinear coupling of neovascular development andtissue/tumor growth
•Needs further work: MMPs, identification ofbiophysical mechanisms
•Vascular remodeling/flow, etc.
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Ongoing and Future work
• 3D
•Direct modeling of VEGF-A/ECM/MMPinteraction on neovascular morphology.
•Realistic mechanical/diffusional description of tissue
•Cell-signaling– macro/micro nonlinear coupling
•Finite, complex domains
•Stochastic models
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Future work contd.
Cell-to-cell adhesion•Multiscale Mixture Models
•Genetic mutations, cell-differentiation and spatial structure
Non-random RandomKomarova,Macklin, L.
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Modeling growth in real organs
Breast cancer model
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Example from Systems Biology
Morphogen gradients
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Extra-Cellular Signaling—Morphogen gradients
• Arthur Lander Dept. of Cell and Developmental Biology, UCI• Fred Wan Dept. of Mathematics, UCI
• Larry Marsh Dept of Cell and Developmental Biology, UCI
• Qing Nie Dept. of Math., UCI
• Yong-Tao Zhang Dept. of Math., UCI ---- (U. of Norte Dame, next year)
• Rui Zhao Dept. of Math., UCI ---- (MBI, next year)
Supported by
• NIH R01GM67247
• NIH P20 GM66051
Key collaborators:
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asm
de
ve
me
sog
dpp
LR
mesoderm
ventral ectoderm
dorsal ectoderm
amniosersaasm
de
ve
me
D
V
D
V
A B C D
Ventral-Dorsal Patterning of a Drosophila Embyro
Sog
Dpp
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Secreted modulators of Dpp
Tld
DPP
Punt(Type II)
Tkv, Sax(Type I)
*
Signal to Nucleus
*
SCW
Tsg
SogSog
Tsg
Sog
Tsg
+
LigandsDpp, Scw
Receptors
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A Mathematical Description
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Model and Equations
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Dorsal-Ventral Patterning of Drosophila
A two-dimensional simulation
C. Mizutant, Q. Nie, et al. Developmental Cell 8(6), 2005
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New Graduate Programs at UCI
• A new gateway interdisciplinary Ph. D. program on Mathematical and Computational Biology starts in Fall, 2006
• UCI was awarded $1M from HHMI in 2006 to create an interdisciplinary Ph.D. program on Mathematical, Computational and Systems Biology
3-year development period and subsequent NIH training grant support
• Continuum and PDE modeling is one of the focus areas in the training program