Patterns and Growth of Highly Malignant Brain Tumors

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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
2
Collaborators
E. Khain1, A.M. Stein2, C. Schneider-Mizell Physic
s 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.

• 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.
Post-op.
8 mo.

• 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 SiSj
• A model for a biologist

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Typical Invasion Models
In vitro
In vivo / In situ
cell speed 20 microns/hr
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The 3d Tumor Spheroid Assay

• 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.

3 mm
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
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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
Day 1 Day3 Day5 Day7
U87dEGFR U87WT
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PDE Model
Diffusion
Cell Shedding
Proliferation
Directed Motility
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., 92356365, 2007.
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The 4 Unknown Parameters
D Diffusion (10-4 cm2/day) 0.1 2.0
v Radial Advection (cm/day) 0 0.10
s Shed rate 106 cells/(cm2 day) 0.01 10
g Prolif. Rate (1/day) 0 0.30
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Fit Model to Different Cells
More malignant
Less malignant
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Sensitivity Analysis
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What controls shed rate?

• Cell cell adhesion is a good candidate.
• Also, it probably controls clustering.

WT
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
Phase separation and coarsening, qgtqc
50
50
100
100
150
150
200
200
1
250
250
Phase separation
300
300
0.95
350
350
0.9
100
200
300
100
200
300
(A)
(B)
q, adhesion parameter
0.85
50
50
0.8
(C)
(D)
100
100
150
150
0.75
No phase separation
200
200
0.7
250
250
0
0.1
0.2
300
300
c, average density
350
350
time
No phase separation, qltqc
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? (qltqc)
Larger cell-cell adhesion? (q gtqc)
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Experiments II
Michal O. Nowicki, E. A. Chiocca, and Sean Lawler
dEGFR
1 day
3 days
5 days
WT
3 days
1 day
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
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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Collagen-I Gel
50 µm
1.5 mg/ml, from Vader and Weitz (Harvard)
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Collagen Gel Physics
Tension Test

• Collagen is viscoelastic up to 10-15 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.

Roeder et. al., 2002
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Network Extraction
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Results on Actual Network
Image
Extended Branches
Linked Branches
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Tracking algorithm

• Microscopy data to construct network.

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Testing Algorithm withArtificial Networks

• Seed space with fiber nucleation points
• 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)

cross-links
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Testing Algorithm withArtificial 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
Black and White Image
True Network
Convolved with PSF
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
elastic beams
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
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Experimental ValidationSmall Strains
Rigid Cross-Links
Freely Rotating Cross-Links
1000-80000 Pa
Kxlink
0 Pa
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Estimation of Kxlink Small Strainfull 3d network
Kxlink (N-m)
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Collagen Networks show Nonaffine Deformations
33 µm
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
Experiment
1.5 mg/ml 1.0 mg/ml 0.5 mg/ml
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.