John Lowengrub

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Multiscale Models of Solid Tumor Growthand
AngiogenesisThe effect of the microenvironment

• John Lowengrub
• Dept Math and Biomed Eng., UCI
• www.math.uci.edu/lowengrb

Vittorio Cristini, Paul Macklin (UT Health Sci.
Center) Yao-Li Chuang, H. Frieboes, Fang Jin,
Steven Wise (UCI), X. Zheng (U. Mich.)
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Motivation

• Provide biophysically justified in silico virtual
  system to study
• Help experimental investigations design new
  experiments
• Therapy protocols

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Outline

• Models and analysis of invasion
• Numerical methods and results
• Models of angiogenesis
• Nonlinear coupling of angiogenesis and invasion

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The Six Basic Capabilities of Cancer (Hanahan
and Weinberg, 2000)

• Genetic-Level (Nanoscopic)
• Self-sufficiency in Growth Signals
• Insensitivity to Growth-inhibitory Signals
• Evasion of Programmed Cell Death
• Limitless Replicative Potential
• Tissue-Level (Microscopic)
• Tissue Invasion and Metastasis
• Sustained Angiogenesis

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Cartoon of solid tumor growth

• Goal Model all Phases of growth

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Cancer Multiscale Problem
Recent Reviews Araujo-McElwain (2004), Byrne et
al (2006), Roose et al (2007)
Nonlinear (continuum) simulations Cristini et al
(2003), Zheng et al. (2005), Macklin-L.
(2005,2006,2007), Hogea et al (2005,2006), Wise
et al. (in review)
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Modeling
Owen et al, 2004

• Continuum approximation super-cell macro scale
• (Collective motion)
• Role of cell adhesion and motility on tissue
  invasion and metastasis
• Idealized mechanical response of
  tissues
• Coupling between growth and angiogenesis
  (neo-vascularization) necessary for maintaining
  uncontrolled cell proliferation.
  Hybrid-continuum-discrete approach
• Genetic mutations random changes in
  microphysical parameters cell apoptosis and
  adhesion

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Key variables
Minimal set.
Tumor fraction
Will discuss refinements later.
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Equations governing tumor growth and tissue
invasion
Wise, Lowengrub, Frieboes, Cristini, Bull. Math.
Biol., in review.
-- Adhesion fluxes
S Net sources/sinks of mass
Mixture models Ambrosi-Preziosi (2002),
Byrne-Preziosi (2003)
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Adhesion
Fundamental biophysical mechanism.
Cell-cell binding through cell-surface proteins
(CAMs, cadherins)

• Cell-sorting due to cell-cell adhesion

Chick embryo Armstrong (1971)

• Cells of like kind prefer to stay together.

Cell-ECM binding through other cell-surface
proteins (integrins)
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Adhesion Energy Application to Tumor

• Assume tumor cells prefer to be together.

Different phenotypes may have different
adhesivity (can extend the model)
(expansion of nonlocal interaction potential)

• Thermodynamic consistency

Generalized Cahn- Hilliard equation
where
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Constitutive Assumptions
Simplest assumptions. Can be generalized.
(X.Li, L., Cristini, Wise)

• Water density is constant

Water decouples

• Close-packing

Excess adhesion

• Cell-velocities are matched using Darcys law

Solid
Cell mobility reflects strength of cell-cell and
cell-matrix adhesion
Oncotic (hydrostatic) solid pressure
liquid pressure
permeability
Liquid
(arise from thermodynamic considerations)
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Constitutive Assumptions Contd.
Heaviside function
Nutrient (oxygen)
Cell proliferation
Viability level of nutrient
mitosis
apoptosis
necrosis
Necrotic cells
lysing (enzymatic degradation)
Host domain
Water
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The Equations (nondimensionalized)
Total tumor cells
Dead tumor cells
Adhesive potential
Solid Pressure/Velocity
Liquid Pressure/Velocity
Sources

• Only one Cahn-Hilliard Equation to be solved for

(4th order, nonlinear advection-diffusion
equation)
time
length

• Generalizes to multiple species easily.

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Evolution of nutrient
Oxygen
uptake by viable cells
0 (quasi-steady assumption). Tumor growth time
scale (1 day) large compared to typical
diffusion time (1 min)
Source due to capillaries (angiogenesis)
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Interpretation

• D is an indirect measure of perfusion

i.e., D large
nutrient rich

is a measure of mechanical/adhesive properties of
extra-tumor tissue
i.e.,
small
tissue hard to penetrate
(less mobile)

• Although a very simplified model of these
  effects, this
• does provide insight on how the
  microenvironment influences
• tumor growth.

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Nondimensional parameters
Microenvironmental

• Diffusion ratio

• Mobility (adhesion) ratio

Cell-based

• Adhesion

• Intermixing

• Necrosis

• Viability

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Spherical Solutions

• Balance between proliferation/necrosis/lysing.
• Viable tumor cells move to center. (water moves
  outward)
• Necrotic boundary is diffuse

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Convergence to sharp interface
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Tumor Spheroids Validation in vitro
In vitro growth
No vascularization (diffusion-dominated)
Dormant (steady) states
One micron section of tumor spheroid showing
outer living shell of growing cells and inner
core of necrosis.
3-D video holography through biological tissue P.
Yu, G. Mustata, and D. D. Nolte,  Dept. of
Physics, Purdue University
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Tumor Modeling The basic model
Model validation
Growth of tumor
Viable rim
In vitro data Karim Carlsson Cancer Res.

• Agreement w/ observed growth
• Determine microphysical parameters

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Microphysical parameters

• A0,

(approximately 7 cells)
G is not determined
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Morphological stability
Perturbation
Underlying Growth
d2,3
(balance between proliferation, necrosis and
apoptosis)
Shape evolution
Self-similar evolution
If N0, then can also get
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• Qualitatively similar for 2D/3D

• Necrosis enhances instability

(A,Ggt0)

• Low vascularization
• (diffusion-dominated)

Stable/Shape-preserving/Unstable

• Moderate vascularization

(Alt0, Ggt0)
Stable
Experimental evidence (Polverini et al., Cancer
Res. 2001)
3. High vascularization
(Glt0)
Shape instability with high vascularization
Vascular/mechanical anisotropy
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Diffusional Instability--Avascular
2D Cristini, Lowengrub and Nie, J. Math. Biol.
46, 191-224, 2003 3D, Li, Cristini, Nie and
Lowengrub, DCDS-B, 2007.
Avascular (tumor spheroid) (low cell-to-cell
adhesion)
2D
3D

• Growth-by-bumps
• topology change

ejection of cells from bulk
Highly vascularized

• Stable evolution

(isotropic vasculature)
Boundary integral method
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Diffusional Instability

• Perturbed tumor spheroids/Complex Morphology

glioblastoma
Velocity field (simulation)
Swirling ejection from bulk
Frieboes, et al.

• Theory
• Possible mechanism for invasion into soft
  tissue

Cristini, Lowengrub, Nie J. Math. Biol
(2003) Cristini, Gatenby, et. al., Clin. Cancer
Res. 11 (2003) 6772. Macklin, Lowengrub, J.
Theor. Biol. (2007)
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Nonlinear Simulations
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Numerical Scheme

• Implicit time discretization (Gradient Stable)
• fully implicit treatment of system
• Second order accurate, centered difference
  scheme.
• Conservative form. Adaptive spatial
  discretization.
• Nonlinear, Multilevel,
• multigrid method

Chombo, Mitran
Kim, Kang, Lowengrub, J. Comp. Phys. (2004) Wise,
Lowengrub, Kim, Thornton, Voorhees, Johnson,
Appl. Phys. Lett. (2005) Wise, Kim, Lowengrub J.
Comp. Phys., in review
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Advantages of Multigrid

• Complexity is O(N)
• Optimal convergence rate
• Handles large inhomogeneity/ nonlinearity
  seamlessly (no additional cost)
• Flexible implementation of b.c.s (compare with
  pseudo-spectral, spectral methods)
• Seamlessly made adaptive
• Hard to analyze quantify smoothing properties of
  the nonlinear relaxation scheme

• Smoothing is performed by, for example, the
  nonlinear Gauss-Seidel method.

• Local linearization. No global linearization, for
  example via Newtons Method, is needed.

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Nutrient-rich host domain

• Small nutrient
• gradients in host

Wise, Lowengrub, Frieboes, Cristini, BMB in
review.
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• Tumor develops folds to increase access to
  nutrient

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Large nutrient gradients

• Large nutrient
• gradients in host

• Tumor breaks up in its search for nutrient

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Morphology diagram
Macklin, Lowengrub JTB, 2007.
A0, G20,
N0.35
(decreased mobility)

• 3 distinct regimes

• Fragmented (nutrient-poor).
• Fingered (high tissue resistance)
• Hollowed (low tissue resistance, nutrient-rich)

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Invasion Summary

• Microenvironment is a primary determinant for
• tumor growth and morphology

(fragmented, invasive fingering, hollow/necrotic)

• Internal structure (e.g. size of necrotic,
  proliferating regions)
• determined by cell-based parameters

• Link between morphology,
• microenvironment, phenotype

• Implications for therapy
• Experimental evidence for this
• behavior?

G55 human glioblastoma tumors in vivo becoming
invasive after anti-angiogenic therapy Rubinstein
et al. Neoplasia (2000)
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Comparison with experiment
Exp Frieboes et al., Cancer Res. (2006).
fetal bovine serum (FBS)
increasing
glucose
increasing

• Model is qualitatively consistent with
  experimental results

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Angiogenesis
More realistic description of in vivo tumor
microenvironment
Angiogenic factors VEGF (Vascular
Endothelial cell Growth Factor) FGF
(Fibroblast Growth Factor) Angiogenin
TGF (Transforming Growth Factor),.
(fibronectin/collagen)
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Gradient-based, biased circular random walk
Ref Plank-Sleeman, Bull. Math. Biol (2004)
Othmer-Stevens, SIAM J. App. Math. (1997)
Hill-Häder, J. Theor. Biol. (1997)
Idea track the capillary tip. Use the trace to
describe the vessel. Not lattice-based.

• Endothelial cell travels with speed s with
  direction given by
• the polar and azimuthal angles (f, ?)

Hybrid continuum-discrete model
(1 EC per 50-100 Tissue/Tumor cells)
Endothelial cells/vessels treated as discrete
quantities Tumor cells/substrate species treated
as continuum
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Angiogenesis contd.
Cell receptor ligand f (e.g., Fibronectin) in the
ECM. Regulates cell adhesion and motion
Matrix degradation by vascular endothelial cells
degradation
production
Matrix degradative enzymes (MDEs) m
Endothelial cells tend to move up the gradients
of c and f
(chemotaxis, haptotaxis). Speed s proportional to
their magnitudes.
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Gradient-based, biased circular random walk
Ref Plank-Sleeman, Bull. Math. Biol (2004)
Othmer-Stevens, SIAM J. App. Math. (1997)
--- Probability Density function
Hill-Häder, J. Theor. Biol. (1997)
--- mean turning rate,
--- rotational diffusivity
Fokker-Planck Eq
Prob. Mobility (Mean turning rate)
Gradient Model Transition rate Transition
probability
VEGF gradient direction
FIB gradient direction
VEGF sensitivity (turning coefficient)
FIB sensitivity
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Gradient-based, biased circular random walk
Ref Plank-Sleeman, Bull. Math. Biol (2004)
Othmer-Stevens, SIAM J. App. Math. (1997)
Hill-Häder, J. Theor. Biol. (1997)

• Branching Tip is allowed to split with a certain
  probability

• Anastomosis If vessels are close, they may merge
  with a
• certain probability.

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Tumor-Capillary Interactions
Anderson, Chaplain, McDougall, Levine, Sleeman,
Zheng,Wise,Cristini, Frieboes et al, Macklin, et
al.
(in reality is much more complicated but this is
a start)
Tumor angiogenic factor c
Tumor angiogenic factor (e.g., VEGF-A) potent
mitogen, drives motion
Uptake by the endothelial cells
Nutrient equation n
Decay
Nutrient transfer
production
Endothelial Cell (localized) density
Nutrient inside capillary
Transfer rate
Capillary pressure
Heaviside function
Capillary density
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Simulation of Tumor-Induced Angiogenesis
Parameters appropriate for glioblastoma
Wise, Lowengrub, Frieboes, Zheng, Cristini, Bull.
Math. Biol, in review
Frieboes, Lowengrub, Wise, Zheng, , Cristini,
Neuroimage (in press)
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Vascular cooption

• Initial capillaries present
• Growing tumor surrounds
• vessels
• Uses up available vasculature
• Secondary angiogenesis
• Observe bursts of growth as
• the nutrient supply increases
• (cyclical bouts of angiogenesis)

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Progression
200um

• Note nutrient supply localized near red
  (nutrient-releasing) vessels
• Observe corresponding viable tumor cells around
  vessels.

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Simulation
Human Glioma

• Tumor architecture is determined by cellular
  metabolism and intra-tumoral diffusion
• nutrient gradients of required nutrients
  provided by the vasculature,
• Confirms the presence of substrate gradients and
  the parameter estimates for
• diffusion length used in the simulations Bar,
  200 um.

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Human glioma
simulation

• Model predicts that tumor tissue
• invasion is driven by diffusion gradients.
  Bar 100um

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Implications for therapy
Anti-invasive therapy
Anti-invasive therapy
Rubinstein et al (2000)
increase adhesion
increase adhesion
Anti-angiogenic therapy
vessel disruption
Vascular normalization
2D Cristini, et al., Cancer Res. (2006)
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Next Steps

• More complex/realistic biophysics

• Mechanical response
• Angiogenesis (Flow)
• Mutations/multiple cell-types
• Complex domains

• Even biophysically simplified modeling can
  provide
• insight though

• from morphology, infer microenvironment,
  phenotype
• and hence invasive potential

• Move beyond phenomenological to predictive
  modeling

Integrative models match molecular-scale
features of experiments. Gatenby (Invasion
and Morphologic instability)
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Tumor-Induced Angiogenesis
Macklin, McDougall, Anderson, Chaplain, Cristini,
L. In preparation.

• Flow/Wall Shear Stress enhances branching.
• Pressure cuts off vessels. Complex Nonlinear
  interaction between tumor growth and angiogenesis

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Mutations

• Include a 3rd tumor species
• (random times/random positions)
• Increase proliferation/nutrient uptake

t0
t10
t60
t212
t120
t180

• Mutation enhances/creates instability (nutrient
  gradient)

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Improved modeling

• Complex domains

• Introduction of quiescent cells
• Better treatment of necrotic core

• Mutations/heterogeneous populations

• MDE
• Haptotaxis

Macklin, Lowengrub
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Hypoxia-induced invasion

• Branched tubular structures

Hypoxia-induced matrix invasion in vitro by
MLP-29 mouse embryo liver cell spheroids.
Hypoxia-induced Upregulated HGF/Met tyrosine
kinase Pennachietti et al, Cancer Cell (2003).
Normoxic
Hypoxic
HypoxicHGF
Simulation Low proliferation. Chemotaxis.
Decreased adhesion