Modeling origin and natural evolution of low-grade gliomas

Mathilde BadoualParis Diderot University, IMNC lab

2nd HTE workshop: Mathematical & Computer Modeling to study tumors heterogeneity in its ecosystem, November 14th, 2018

Gliomas

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Grade I: the grade I tumors may be curable by surgeryGrade II diffuse astrocytomas or oligodendrogliomas: evolve 7-8 years in anaplastic tumorsGrade III anaplastic gliomas: fatal evolution in 2 to 4 years.Grade IV glioblastoma multiforme: Average survival of 6 months to 2 years (based on feasible treatment).

solid tumor only solid tumor+ isolated tumor cells isolated tumor cells only

Grade I Grade III and IV Grade II

Solid tumor tissue

Isolated tumor cells

Gliomas

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Grade I: the grade I tumors may be curable by surgeryGrade II diffuse astrocytomas or oligodendrogliomas: evolve 7-8 years in anaplastic tumorsGrade III anaplastic gliomas: fatal evolution in 2 to 4 years.Grade IV glioblastoma multiforme: Average survival of 6 months to 2 years (based on feasible treatment).

solid tumor only solid tumor+ isolated tumor cells isolated tumor cells only

Grade I Grade III and IV Grade II

Solid tumor tissue

Isolated tumor cells

heterogeneity

Gliomas are rare tumors, but grade II (and more) gliomas cannot be cured

systematic recurrence, even after treatments

Glioma cells migrate normal surrounding tissue, causing recurrence of the tumor.⇒ Invasion plays a key role in the poor outcome of patients

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Shibahara, I et al (2015) Malignant clinical features of anaplastic gliomas without IDH mutation , Neuro Oncol., 17, 136-144.

Diffuse low-grade gliomas: recurrence

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Mandonnet E et al (2003) Continuous growth of mean tumor diameter in a subset of grade II gliomas. Ann Neurol 53, 524–528

A linear growth of the tumor radius

€

C(r,t) =N0

(4πDt)3 / 2eκte−r

2 / 4DtSolution in 3D:

large

€

κD

small

⇥C(⇤r, t)

⇥t= r(D(⇤r, t)rC(⇤r, t)) + �(⇤r, t)C(⇤r, t)

⇥C(r, t)

⇥t= Dr2C(r, t) + �C(r, t)

Cook J et al. (1995) Resection of gliomas and life expectancy, J Neurooncol. 24, 131

if D and κ are uniforms and constants

€

κD

Modeling tumor growth

r(t) =

s

4Dt(�t+ ln(N0

C⇤(4⇥Dt)3/2)) r(C⇤, t ! 1) =

p4D�t

Assumption: diameter of the tumor on a MRI scan= iso cell density curve (C*)

- <v> = 2 mm/yr - Linear evolution since r = 10 mm ⟹ for the model rmin = 15 mm

detection threshold

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�7

The natural history of low grade gliomas

-Very invasive tumors but patients can live more than ten years after diagnosisPallud J et al, (2008) Les gliomes infiltrants de bas grade, REG, Neurologies 11, 94-101

no symptoms symptoms

Onset

Time

Mea

n tu

mor

dia

met

er

Epilepsy

No mass effect no contrast

enhancement

Anaplastic transformation

Mass effectEdema

Contrast enhancement

Necrosis

Clinical diagnosis Death

anaplastic transformation = trigger of angiogenesis?

Grade II ∼ 10 years Grade III and IV ∼ 1 year

➣ OPCs are the most widely distributed population of cycling cells in adult brain. ➣ In contrast, a small population of NSCs is found in the SVZ lining the lateral ventricles.

Geha S et al., (2010), NG2+/Olig2+ cells are the major cycle-related cell population of the adult human normal brain, Brain Pathol., 20, 399-411

Oligodendrocyte precursor cells (OPCs)

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➣ Cycling cells in the adult brain are mainly OPCs (NG2+ cells)

Ilkanizadeh S et al, (2014), Glial Progenitors as Targets for Transformation in Glioma, Adv Cancer Res., 121, 1–65.

OPCs at the origin of gliomas?

Zong H et al , (2012) The cellular origin for malignant glioma and prospects for clinical advancement, Expert Rev Mol Diagn., 12, 383-94

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➣ Mutated OPCs trigger gliomas in mouse.

⇒ OPCs (Oligodendrocyte Precursor cells) are strongly suspected to be the cell of

origin of some gliomas.

OPCs organize in a grid-like manner, with individual cells occupying almost non-overlapping domain

Xu G et al, (2014), Spatial organization of NG2 glial cells and astrocytes in rat hippocampal CA1 region, Hippocampus, 24, 383-95

OPCs dynamics in vivo

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Hughes EG et al, (2013), Oligodendrocyte progenitors balance growth with self-repulsion to achieve homeostasis in the adult brain, Nat Neurosci., 16, 668-76.

OPCs maintain a constant density in vivo

death

differentiation

proliferation

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Modeling OPCs dynamics

100μm

Model: a cellular automaton without lattice (continuous space)

Rules

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A cell can:1. proliferate (⇒ proliferation rule)2. migrate (⇒ migration rule)3. and disappear (differentiate or die) (differentiation rule)

The formation of a glioma: different scenarios

- Apparition of an immortal cell.

- Apparition of a cell that has lost its contact inhibition.

- Apparition of a highly proliferative cell.

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The daughter cells have the same proliferative properties than the mother cells.

The formation of a glioma: different scenarios

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First scenario: Apparition of an immortal cell

Time (days)500 1000

1600

2000

2400

2800

3200

X-Title

Y-Title

Time (days)

# c

ell/m

m3

TumorNormal

400 900

1600

2000

2400

2800

500 10000

20

40

60

X-Title

Y-Title

400 9000

20

40

60

# c

ell/m

m3 /d

ay

Cell density in a 1mm3 volume Proliferative cell density

The formation of a glioma: different scenarios

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First scenario: Apparition of an immortal cell

Time (days)500 1000

1600

2000

2400

2800

3200

X-Title

Y-Title

Time (days)

# c

ell/m

m3

TumorNormal

400 900

1600

2000

2400

2800

500 10000

20

40

60

X-Title

Y-Title

400 9000

20

40

60

# c

ell/m

m3 /d

ay

Cell density in a 1mm3 volume Proliferative cell density

The tumor cell proliferation goes to

zero !Not compatible with experimental data

The formation of a glioma: different scenarios

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100 150 200

2000

6000

10000

# c

ell/m

m3

TumorNormal

0 50 100 100 150 2000

100

200

300

400

500

Time (days)

# c

ell/m

m3 /d

ay

0 50 100Time (days)

Second scenario: Apparition of a cell without contact inhibition

Cell density in a 1mm3 volume Proliferative cell density

The formation of a glioma: different scenarios

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100 150 200

2000

6000

10000

# c

ell/m

m3

TumorNormal

0 50 100 100 150 2000

100

200

300

400

500

Time (days)

# c

ell/m

m3 /d

ay

0 50 100

Very high cell and proliferation cell density⇒ high-grade glioma

Cell density in a 1mm3 volume Proliferative cell density

Second scenario: Apparition of a cell without contact inhibition

Time (days)

High-grade vs low-grade glioma

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MIB-1

Singh SK et al (2004), Identification of human brain tumour initiating cells, Nature, 432, 396-401.

Low grade

H & E(immunostaining of proliferative cells)

High grade

Low-grade glioma

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0

5

10

15

0

200

400

600

800

1000

1200

1400

# c

ell/m

m2

1200

400

0

800

15

5

0

10

# M

IB-1

pos

itive

cel

ls/m

m2

Normal Tumor Normal Tumor

H&E staining of a tumor tissue

MIB1 immunostaining

The formation of a glioma: different scenarios

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150 350 550 7501800

2000

2200

Time (days)

# c

ell/m

m3

Normal Tumor

18000 200 400 600 150 350 550 750

0

20

40

60

Time (days)

# c

ell/m

m3 /d

ay

0 200 400 600

Cell density in a 1mm3 volume Proliferative cell density

Third scenario: Apparition of a highly proliferative cell

The formation of a glioma: different scenarios

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150 350 550 7501800

2000

2200

Time (days)

# c

ell/m

m3

Normal Tumor

18000 200 400 600 150 350 550 750

0

20

40

60

Time (days)

# c

ell/m

m3 /d

ay

0 200 400 600

Cell density in a 1mm3 volume Proliferative cell density

Third scenario: Apparition of a highly proliferative cellHigher cell and proliferation cell density inside the tumor but not too high (a new equilibrium)

⇒ low-grade glioma

A highly proliferative cell at the origin of low-grade glioma

A very proliferative cell in redNormal OPCs are in blue

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Modeling the formation of a glioma

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0

1000

2000

X-Title

Y-Title

200 400 600

Distance to the center (µm)

2000

1000

0

# c

ell/m

m3

Tumor cells are yellow to red (cell clock increasing)Normal OPCs are in blue to green (cell clock increasing) Red curves: tumor cells; blue curves: normal cells

Dufour A et al, (2018), Modeling the dynamics of oligodendrocyte precursor cells and the genesis of gliomas, PLoS Comput Biol., 14, e1005977.

0 50 100 150 200 250 3000

100

200

300

400

500

600

700

800

0 100 200 300

200

400

600

0

Time (days)

Mea

n ra

dius

(µm

)

800

Modeling the formation of a glioma

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With reasonable parameters, v ≃ 1 mm/yr

0 50 100 150 200 250 3000

100

200

300

400

500

600

700

800

0 100 200 300

200

400

600

0

Time (days)

Mea

n ra

dius

(µm

)

800

Modeling the formation of a glioma

With reasonable parameters, v ≃ 1 mm/yr ⇒ consistent with clinical dataMandonnet E et al (2003) Continuous growth of mean tumor diameter in a subset of grade II gliomas. Ann Neurol, 53, 524–528

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First step of formation of a glioma

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Dufour A et al, (2018), Modeling the dynamics of oligodendrocyte precursor cells and the genesis of gliomas, PLoS Comput Biol., 14, e1005977.

OPC: oligodendrocyte precursor cell.

The appearance of a highly proliferative OPC among normal OPCs leads to the formation of a glioma-like tumor:

- invasive- slow linear increase of the radius, compatible with clinical data

⇒ first step of heterogeneity: mixture and competition between normal and cancer cells

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Increasing heterogeneity

-Very invasive tumors but patients can live more than ten years after diagnosisPallud J et al, (2008) Les gliomes infiltrants de bas grade, REG, Neurologies, 11, 94-101

no symptoms symptoms

Onset

Time

Mea

n tu

mor

dia

met

er

Epilepsy

No mass effect no contrast

enhancement

Anaplastic transformation

Mass effectEdema

Contrast enhancement

Necrosis

Clinical diagnosis Death

anaplastic transformation = trigger of angiogenesis

Grade II ∼ 10 years Grade III and IV ∼ 1 year

x 28

P1

P3

P4

16

0

-6

-16

outs

ide

the

tum

orin

side

the

tum

or

: area fraction of edema

Quantification of edema

-40 -20 0 20

100

200

Distance (mm)

Grey

leve

l

P4 P3 P2 P1

P1P2P3P4

-16 -6 5 160

(mm)

⇠ = 0.92(1.01� 10�2(Re �Ge))

⇠

Tumor tissue: normal cells + tumor cells + edema + ECM +….

0 20 40 60 800

2

4

6

8

10

Edema fraction at x=0

Patie

nt n

umbe

r

−20 −10 0 10 20 300

20

40

60

80

100oedema

0

x (mm)

Edem

a fra

ctio

n

123

54

76

89

Edema fraction

Patie

nt n

umbe

r

29

inside the tumor outside the tumor

Edema/border of the tumor

Gerin C, et al (2013) Quantitative characterization of the imaging limits of diffuse low-grade oligodendrogliomas, Neuro-Oncology, 15, 1379.

d

⇤⇥

⇤t= Dr2⇥+ �⇥(1� ⇥)

⌅⇥

⌅t= �⇤(1� ⇥)� µ⇥⌫

A model with edema

ρ: tumor cell densityξ: edema fraction

κ: proliferation D: diffusion λ: edema production μ: edema clearance

Equation for the cell density evolution:

Equation for the edema fraction evolution:

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At the center, when ρ=1, reaches its maximum value that verifies: 1 � ⇠e =�

µ⇠e

⌫⇠

Low-grade gliomas and radiotherapy

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Delay between the end of the radiotherapy and the regrowth of the tumor: Why?

Fit of clinical data

32Badoual M, et al (2014) An oedema-based model for diffuse low-grade gliomas: application to clinical cases under radiotherapy, Cell Prolif, 47, 369

clinical data

model

Conclusion

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➣ When the tumor grows the heterogeneity increases.

➣ In low-grade gliomas, the heterogeneity is still low: easier for modeling. Two models, with increased heterogeneity, corresponding to different stages of evolution of a glioma.

➣ Next step: study of the apparition of heterogeneity between tumor cells (hypoxia)

Acknowledgments

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Emilie Gontran, PhD studentAloys Dufour, undergraduate studentBasile Grammaticos Christophe Deroulers

Johan Pallud, neurosurgeonPascale Varlet, pathologist

Catherine Oppenheimer, radiologist

IMNC laboratory, Orsay, France Collaborators: Sainte-Anne hospital, Paris

Thank you for your attention !

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