Patterns and Growth of Highly Malignant Brain Tumors

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Patterns and Growth of Highly Malignant Brain Tumors

Leonard M. SanderDepartment of Physics &

Michigan Center for Theoretical Physics,University of Michigan, Ann Arbor, MI

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E. Khain1, A.M. Stein2, C. Schneider-MizellPhysics Department, University of Michigan

M. O. Nowicki, E. A. Chiocca, S. Lawler Department of Neurological Surgery, The Ohio State University

Collaborators

T. Demuth, M. E. Berens The Translational Genomics Research Institute, Phoenix, Arizona

T. DeisboeckComplex 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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• 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.

Introduction to Malignant Brain Cancer




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Pre-op. Post-op. 8 mo.

Why Glioblastoma has been Untreatable

•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:–



In vitro In vivo / In situ

Typical Invasion Models

cell speed ~ 20 microns/hr

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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

The 3d Tumor Spheroid Assay3

mm

Bright Field Image

• 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.

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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 Day 1 Day3 Day5 Day7

U87

dEG

FR

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Diffusion Cell Shedding ProliferationDirected Motility

•Invasive cell motion has a random component and a directed component

•Core radius expands at a “slow”, constant velocity.

•Invasive cells are shed from the core surface

•Invasive cells proliferate

Rcore

PDE Model

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.



DDiffusion

(10-4 cm2/day)0.1 2.0

vRadial Advection

(cm/day)0 0.10

sShed rate

106 cells/(cm2 day)0.01 10

gProlif. Rate

(1/day)0 0.30

The 4 Unknown Parameters

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Fit Model to Different Cells

More malignant

Less malignant





DD

ss

vv

gg

Sensitivity Analysis

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WT

dEGFR

What controls shed rate?

Cluster, possibly due to cell-cell adhesion.

A secondary tumor?

• Cell cell adhesion is a good candidate.

• Also, it probably controls clustering.

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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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No phase separation, q<qc

Phase separation and coarsening, q>qc

0 0.1 0.20.7

0.75

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Phase separation

c, average density

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esio

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eter

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(C) (D)

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Simulations of clustering

time

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Experiments: Glioma cells on a surface

WT dEGFR

No clustering Clustering

Smaller cell-cell adhesion? (q<qc)

Larger cell-cell adhesion? (q >qc)

Michal O. Nowicki, E. A. Chiocca, and Sean Lawler

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1 day 3 days 5 days

1 day 3 days 5 days

Experiments II

dEGFR

WT

Michal O. Nowicki, E. A. Chiocca, and Sean Lawler

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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

1.5 mg/ml, from Vader and Weitz (Harvard)

50

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Collagen Gel Physics• 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

Tension Test

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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 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 NetworksBlack and White

ImageTrue Network

Convolved with PSF PSF + Noise

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Extracting Artificial Networks

True NetworkBW ImagePSFPSF + Noise

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Actual Networks

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Mechanical Modeling of Networks

Impose Displacement

MinimizeEnergy

Pinned

Node

SlidingNodes

elastic beams

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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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0

0 0.05

KKxlinkxlink

Experimental ValidationSmall Strains

Freely Rotating Cross-LinksRigid Cross-Links

0.1-50 Pa

1000-80000 Pa

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

Free and Fixed cross-links

33

μm

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/ml1.0 mg/ml0.5 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.

Patterns and Growth of Highly Malignant Brain Tumors

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