Patterns and Growth of Highly
Malignant Brain Tumors
Leonard M. Sander
Department of Physics &
Michigan Center for Theoretical Physics,University of Michigan, Ann Arbor, MI
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Collaborators
E. Khain1, A.M. Stein2, C. Schneider-Mizell
Physics Department, University of Michigan
M. O. Nowicki, E. A. Chiocca, S. Lawler
Department of Neurological Surgery, The Ohio State University
T. Demuth, M. E. Berens
The Translational Genomics Research Institute, Phoenix, Arizona
T. Deisboeck
Complex Biosystems Modeling Laboratory, Harvard-MIT (HST);
A. A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital
NIH grant R01 CA085139-01A2.
1.
Now at Oakland University, Michigan
2.
Now at IMA, Minneapolis
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Introduction to Malignant
Brain Cancer
•
•
•
•
18,000 people/year in the US are
diagnosed with primary brain tumors.
9,000 have glioblastoma multiforme
(GBM), the most malignant form.
After diagnosis:
•
•
50% of GBM patients die within 1 year.
98% of GBM patients die within 5 years.
No significant advances in the last 30
years.
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Why Glioblastoma has been
Untreatable
Pre-op.
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Post-op.
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8 mo.
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•Surgery fails:
•Cancer is highly invasive.
•Some areas of the brain cannot be removed.
•Chemotherapy and radiation fail:
•Invasive cells proliferate slowly.
•Blood-brain barrier blocks drug delivery.
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Vocubulary: the word ‘model’
• A model for a physicist:
–
H = -ij Si•Sj
• A model for a biologist:
–
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Typical Invasion Models
In vitro
In vivo / In situ
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cell speed ~ 20 microns/hr
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The 3d Tumor Spheroid Assay
3 mm
• Put a clump of cultured
tumor cells (a tumor
spheriod) in a gel. (We
use collagen.
•Spheriod grows.
•Single cells invade.
• A reasonable model for
invasion in the brain.
Bright Field
Image
T. S. Deisboeck et. al. (2001) Pattern of self-organization in tumour
systems: complex growth dynamics in a novel brain tumour spheroid
model. Cell Prolif, 34, 115-134
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Growth and invasion in vitro
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Cell tracking
A. M. Stein, D. A. Vader, L. M. Sander, and D. A. Weitz. Mathematical Modeling of
Biological Systems, volume I. Birkhauser, 2006.
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Cell paths from confocal
microscopy
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Cells are Biased Random
Walkers
vr
vθ
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Results of short-time tracking
• Bias to move away from spheroid is
clear, and decays in time.
• Bias depends on cell line.
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Longer-time behavior
Day3
Day5
Day7
U87dEGFR
U87WT
Day 1
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PDE Model
Diffusion Directed Motility
Cell Shedding
Proliferation
•Invasive cell motion has a random
component and a directed
component
•Core radius expands at a “slow”,
constant velocity.
Rcore
•Invasive cells are shed from the
core surface
•Invasive cells proliferate
A. M. Stein, T. Demuth, D. Mobley, M. E. Berens, and L. M. Sander. A mathematical
model of glioblastoma tumor spheroid invasion in a three-dimensional in vitro
experiment. Biophys. J., 92:356–365, 2007.
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The 4 Unknown Parameters
D
v
s
g
Diffusion
0.1
-4
2
(10 cm /day)
Radial Advection
0
(cm/day)
Shed rate
0.01
6
2
10 cells/(cm day)
Prolif. Rate
0
(1/day)
2.0
0.10
10
0.30
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Fit Model to Different Cells
More
malignant
Less
malignant
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Sensitivity Analysis
D
v
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s
g
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What controls shed rate?
• Cell cell adhesion is a good
candidate.
WT
• Also, it probably controls
clustering.
dEGFR
Cluster, possibly due
to cell-cell adhesion.
A secondary tumor?
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Shed rate, clustering, and
adhesion
• Cells with large adhesion should have
difficulty detaching from spheriod.
• Clusters should result from adhesion.
– Indirect measurement of adhesion through
cell clustering.
• Possible clinical significance: shed rate
should correlate with invasiveness.
– Can we use shed rate to guide surgery/
radiation, etc?
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Simulations of clustering
q, adhesion parameter
Phase separation and coarsening, q>qc
1
Phase separation
0.95
0.9
50
50
100
100
150
150
200
200
250
250
300
300
350
350
100
(A)
200
300
100
200
300
(B)
0.85
0.8
(C)
0.75
0.7
(D)
No phase separation
0
0.1
0.2
c, average density
time
50
50
100
100
150
150
200
200
250
250
300
300
350
350
No phase separation, q<qc
200
300
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Experiments: Glioma cells on a surface
Michal O. Nowicki, E. A. Chiocca, and Sean Lawler
WT
dEGFR
No clustering
Clustering
Smaller cell-cell
adhesion? (q<qc)
Larger cell-cell
adhesion? (q >qc)
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Experiments II
Michal O. Nowicki, E. A. Chiocca, and Sean Lawler
dEGFR
1 day
3 days
5 days
WT
1 day
3 days
5 days
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Shed rate
• We can measure the shed rate directly.
• But, adhesion might also be important
for secondary tumor formation.
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Cause of Velocity Bias is
Unknown
• Chemotaxis
• Nutrient gradients (glucose, O2)
• Waste product gradients
• Cell matrix interactions
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Cell-Gel Interactions
Two spheroids, 5mm apart
D. Vader
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A good model for cell-gel
interactions requires a mechanical
model for collagen
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Single Cell in Collagen
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Vader and Weitz (Harvard)
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Collagen is the primary animal structural protein. It is
found in bone, cartilage, tendons, ECM, and jello.
1 nm
~100 nm
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50 μm
Collagen-I Gel
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1.5 mg/ml, from Vader and Weitz
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Collagen Gel Physics
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Collagen is viscoelastic up to 1015% strains.
Significant strain stiffening and
plastic deformation occur at
larger strains.
Many other biological gel
networks have these properites,
e.g. actin.
A micromechanical model is
needed to understand strain
stiffening and plasticity.
Tension Test
Roeder et. al., 2002
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Network Extraction
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Results on Actual Network
Image
Extended BranchesLinked Branches
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Tracking algorithm
• Microscopy data to construct network.
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Testing Algorithm with
Artificial Networks
•Seed space with fiber
nucleation points
cross-links
•Chose random direction
•Extend fibers along a
persistent (lp) random walk
•Create cross-link when two
fibers are less than a fiber
diameter (d) apart.
•Stop extending fibers when
the reach max length (L)
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Testing Algorithm with
Artificial Networks
•All pixels within a radius (r)
from the fiber backbone are
set to one
•To mimic confocal
microscope, images are
convolved with a gaussian
point spread function,
elongated in z
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Extracting Artificial Networks
True Network
Convolved with PSF
Black and White
Image
PSF + Noise
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Extracting Artificial Networks
True Network
BW Image
PSF
PSF + Noise
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Actual Networks
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Mechanical Modeling of
Networks
Impose
Displacement
Sliding
Nodes
elastic beams
Pinne
d
Minimize
Energy
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Mechanical Model
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Mechanical Modeling of Fibers
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Mechanical Modeling of
Cross-links
Minimize Total Energy
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Experimental Validation
Small Strains
Rigid Cross-Links
Freely Rotating Cross-Links
1000-80000 Pa
Kxlink
0.1-50 Pa
0 Pa
0
0
0.05
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Estimation of Kxlink: Small Strain
full 3d network
Kxlink (N-m)
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33 μm
Collagen Networks show
Nonaffine Deformations
Free and Fixed cross-links
More than 99% of energy in network is in
bending
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Strain Stiffening
Model - 3d network
projected to 2d
1.5 mg/ml
1.0 mg/ml
0.5 mg/ml
Experiment
2 mg/ml
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We seem to have forgotten
about the cells
Work in progress:
•
•
•
•
•
Treat cells as a force monopole or
force dipole.
Look for characteristic length for
deformation decay for single cell.
Model individual cell motility.
Look at fiber orientation decay for a
spheroid.
Consider plastic deformations.
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Summary
• Lots of physics in glioma invasion.
• Two processes:
– Shedding of cells from tumor spheroids.
• Depends on cell phenotype probably through
cell-cell adhesion.
– Motility.
• Seems to depend on cell-environment
interactions, at least in vitro.
• First step in understanding cell-ECM
interactions.
– Mechanics of a collagen network.
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