Une approche stochastique à la modélisation de l’immunothérapie contre le cancer

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Une approche stochastique à la modélisation de l’immunothérapie contre le cancer

Loren Coquille -

Travail en collaboration avec

Martina Baar, Anton Bovier, Hannah Mayer (IAM Bonn) Michael Hölzel, Meri Rogava, Thomas Tüting (UniKlinik Bonn)

Institut Fourier – Grenoble

Séminaire commun IF/LJK - 17 septembre 2015

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Plan

1 Biological motivations

2 Adaptative dynamics The model State of the art

3 Only switches : Relapse caused by stochastic fluctuations Therapy with 1 type of T-cell

Biological parameters

Therapy with 2 types of T-cells

4 Only mutations : Early mutation induced by the therapy

5 Mutations and switches : Polymorphic Evolution Sequence

6 Conclusion

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Biological motivations

Plan

6 Conclusion

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Experiment on melanoma (UniKlinik Bonn)

Injection of T-cellsable to kill a specific type of melanoma.

The treatment induces an inflammation, to which the melanoma react by changing their phenotype (markers disappear on their surface, "switch").

The T-cells cannot kill them any more, the tumor continues to grow.

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Without therapy : exponential growth of the tumor.

With therapy : relapse after 140 days.

With therapy and restimulation : late relapse.

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Adaptative dynamics

Plan

6 Conclusion

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Adaptative dynamics The model

Individual-based model

Cancer cells (melanoma): each cell is characterized by a genotype and a phenotype. Each canreproduce,die,mutate(reproduction with genotypic change) orswitch(phenotypic change, without

reproduction) at prescribed rates.

Immune cells (T-cells): Each cell canreproduce,die, or kill a cancer cell of prescribed type (which produces a chemical messenger) at prescribed rates.

Chemical messenger (TNF−α): Each particle candie at a prescribed rate. Its presence influences the ability of a fixed type of cancer cell to switch.

Trait space and measure :

X =G × P t Z t W ={g₁, . . . ,g_|G|} × {p₁, . . . ,p_|P|} t {z₁, . . . ,z_|Z|} tw n= (n_(g₁_,p₁₎, . . . ,n_(g_|G|_,p_|P|₎,nz1, . . . ,nz_|Z|,nw)

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Adaptative dynamics The model

Example for 2 types of melanoma and 1 type of T-cell

The stochastic model converges, in the limit of large populations, towards the solution this dynamical system with logistic,predator-prey,switch:











n˙x =nx

b_x−d_x−c_xx·nx−c_xy·ny

+s·ny−s_w·nwnx −t_xz ·nzxnx

n˙y =ny

b_y −d_y−c_yy·ny−c_yx·nx

−s ·ny +s_w·nwnx

n˙zx = −dzx ·nzx +bzx·nzxnx

n˙w = −d_w·nw+`x·t_xz ·nxnzx

Event Rates forx Rates fory for z for w

(Re)production b_x b_y b_zxn_x

Natural death dx+cxxnx+cxyny dy +cyynx+cyxny d_zx d_w Therapy death t_xzn_z_x 0

Switch s_wn_w s

Deterministically, a number`w of TNF-αparticles are produced whenzx killsx.

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Adaptative dynamics State of the art

State of the art for the BPDL model

In general X continuous. Measure νt =PNt

i=1δxi. Markov process on the space of positive measures.

Event Rate

Clonal reproduction (1−p(x))·b(x) Reproduction with mutation m(x,dy)·p(x)·b(x)

Death d(x) +R

X c(x,y)ν(dy)

Limit of large populations and rare mutations K → ∞

µ→0

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State of the art for the BPDL model

In general X continuous. Measure νt =_K¹PNt

i=1δxi. Markov process on the space of positive measures.

Event Rate

Clonal reproduction (1−µp(x))·b(x) Reproduction with mutation m(x,dy)·µp(x)·b(x)

Death d(x) +R

X c(x,y)

K ν(dy)

Limit of large populations and rare mutations K → ∞

µ→0

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Scalings and time scales

K → ∞,µfixed, T <∞ :

Law of large numbers, deterministic limit [Fournier, Méléard, 2004]

K → ∞,µ→0,T <∞ :

Law of large numbers, deterministic limit without mutations.

K → ∞,µ→0,T ∼log(1/µ) : Deterministic jump process [Bovier, Wang, 2012]

(K, µ)→(∞,0)t.q. _µK¹ logK,T ∼ _µK¹ : Random jump process

[Champagnat, Méléard, 2009,2010]

Trait Substitution Sequence Polymorphic Evolution Sequence

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Scalings and time scales

K → ∞,µfixed, T <∞ :

Law of large numbers, deterministic limit [Fournier, Méléard, 2004]

limit dynamical systems (with switch) are not classified K → ∞,µ→0,T <∞ :

Law of large numbers, deterministic limit without mutations.

K → ∞,µ→0,T ∼log(1/µ) : Deterministic jump process [Bovier, Wang, 2012]

(K, µ)→(∞,0)t.q. _µK¹ logK,T ∼ _µK¹ : Random jump process

[Champagnat, Méléard, 2009,2010]

Trait Substitution Sequence Polymorphic Evolution Sequence

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Only switches : Relapse caused by stochastic fluctuations

Plan

6 Conclusion

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Only switches : Relapse caused by stochastic fluctuations Therapy with 1 type of T-cell

Solution of the determinisitic system

Legend : Melanomax,melanomay,T-cells, TNF-α

0 2 4 6 8 10

1 2 3 4

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3 fixed points

With reasonable parameters we have :

Pxy000

Pxy0zyw Pxyzxzyw

P00000 Pxyzx0w

2 4 6 8 nx

1 2 3 4 ny

Pxyz is stable.

Pxy0 is stable on the invariant sub-space{n_z_x =0}.

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Relapse towards Pxyz , (K = 200)

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Relapse towards Pxy 0 due to the death of z

_x

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Only switches : Relapse caused by stochastic fluctuations Biological parameters

Adjustment of parameters : data

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Only switches : Relapse caused by stochastic fluctuations Biological parameters

Adjustment of parameters : simulations (K = 10

⁵

)

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Only switches : Relapse caused by stochastic fluctuations Therapy with 2 types of T-cells

Therapy with 1 types of T-cells











n˙x =nx

b_x −d_x −c_xx·nx −c_xy·ny

−t_xz ·nzxnx+s ·ny −s_w ·nwnx

n˙y =ny

b_y −d_y−c_yy·ny −c_yx·nx

−t_yz·nzyny

−s ·ny +s_w·nwnx

n˙zx = −d_zx ·nzx +b_zx ·nzxnx

n˙zy = −d_zy ·nzy +b_zy ·nzyny

n˙w = −d_w ·nw +`_x·t_xz ·nxnzx

+`_y·t_yz ·n_yn_z_y

Event Rates for x Rates for y

Reproduction b_x b_y

Natural death dx+cxxnx+cxyny dy+cyynx+cyxny

Death due to therapy txznzx 0

Switch s_wn_w s

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Therapy with 2 types of T-cells











n˙x =nx

b_x −d_x −c_xx·nx −c_xy·ny

−t_xz ·nzxnx+s ·ny −s_w ·nwnx

n˙y =ny

b_y −d_y−c_yy·ny −c_yx·nx

−t_yz ·nzyny−s ·ny +s_w·nwnx

n˙zx = −d_zx ·nzx +b_zx ·nzxnx

n˙zy = −d_zy ·nzy+b_zy ·nzyny

n˙w = −d_w ·nw +`_x·t_xz ·nxnzx+`_y·t_yz ·nynzy

Event Rates for x Rates for y

Reproduction b_x b_y

Natural death dx+cxxnx+cxyny dy+cyynx+cyxny

Death due to therapy txznzx tyznzy

Switch s_wn_w s

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Solution of the deterministic limit

Legend : Melanomax,melanomay,T-cellz_x,T-cellz_y,TNF-α

0 5 10 15 20 25

0.5 1.0 1.5 2.0

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5 fixed points

Pxy000

Pxy0zyw Pxyzxzyw

P00000 Pxyzx0w

2 4 6 8 nx

1 2 3 4 n_y

Pxyz_xz_y is stable.

Pxyz_x0 is stable in the invariant subspace {n_z_y =0} Pxy0z_y is stable in the invariant subspace {n_z_x =0}

Pxy00 is stable in the invariant subspace {n_z_x =0} ∩ {n_z_y =0}

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Metastable transitions between several possible relapses

P_xyz_x_z_y_w

P_xy0z_y_w

Pxyzx0w

Pxy000

zy= 0 zx= 0

invariant space

P00000

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Metastable transitions between several possible relapses

P_xyz_x_z_y_w

P_xy0z_y_w

Pxyzx0w

Pxy000

zy= 0 zx= 0

initial condition invariant space

invariant space deterministic system

P00000

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Metastable transitions between several possible relapses

P_xyz_x_z_y_w

P_xy0z_y_w

Pxyzx0w

Pxy000

zy= 0 zx= 0

initial condition invariant space

invariant space deterministic system stochastic system

P00000

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Stochastic system close to the deterministic system

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Relapse towards Pxyz

_x

0 caused by the death of z

_y

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Relapse towards Pxy 0z

_y

caused by the death of z

_x

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Relapse towards Pxy 00 caused by the death of z

_x

and z

_y

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Cure ! (P 0000)

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Only mutations : Early mutation induced by the therapy

Plan

6 Conclusion

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Example for 2 types of melanoma and 1 type of T-cell

BPDL + therapy with usual competition

Event Rates for x

Clonal reproduction (1−µ)bx

Mutation towards y µbx

Natural death d_x+c_xxn_x+c_xyn_y Death due to therapy t_zxn_z

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Example for 2 types of melanoma and 1 type of T-cell

BPDL + therapy with birth-reducing competition

Event Rates for x

Clonal reproduction (1−µ)bb_x−c_xxn_x−c_xyn_yc₊ Mutation towards y µbb_x−c_xxn_x−c_xyn_yc₊ Natural death d_x +bb_x−c_xxn_x −c_xyn_yc−

Death due to therapy t_zxn_z

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Example for 2 types of melanoma and 1 type of T-cell

BPDL + therapy with birth-reducing competition

Event Rates for y

Clonal reproduction bb_y−c_yyn_y −c_yxn_xc₊ Mutation towards x 0

Natural death d_y +bb_y−c_yyn_y −c_yxn_xc₋ Death due to therapy 0

Event Rates for z Reproduction b_zxn_x

Death d_z

No switch⇒ The chemical messenger (TNF-α) has a trivial role : n˙w =0

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Limiting determinisitic system

When (K, µ)→(∞,0)such that

µ·K →α >0

thenµ disappears from the deterministic system on the time scaleT <∞ (mutant appear after a time O(1/µK) =O(1) but needO(log(K))→ ∞ to become macroscopic).











n˙x =nx

b_x−d_x−c_xx·nx−c_xy·ny

−t_xz·nzxnx

n˙_y =n_y

b_y −d_y−c_yy·n_y−c_yx·n_x n˙zx = −d_zx ·nzx +b_zx·nzxnx

The deterministic system doesn’t "see" the difference between death enhancing and birth-reducing competiton.

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With birth-reducing competition

Let n(0) = (n_x(0),0,0),

then the initial mutation rateis quadratic in the populationn_x :

m(nx) :=µbb_x−c_xxn_xc+n_x

bx 2cxx

bx cxx

0 m(nx)

nx

¯ n_x

dx cxx

µ_4c^b^x_xx²

¯nx := ^b^x_c^−d^x

xx is the equilibrium of the initial x population.

Asmaller population can have ahigher mutation rate.

Notem(n_x) =O(µK) =O(1).

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Without treatment and n

_x

(0) ' ¯ n

_x

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Without treatment and n

_x

(0) small

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With treatment

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Mutations and switches : Polymorphic Evolution Sequence

Plan

6 Conclusion

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Two time scales

Rares mutationsin G : (K, µ)→(∞,0) s.t. µ1/KlogK Fast switches inP : ∀p,p⁰ ∈ P s_(g,p),(g_,p⁰₎=O(1)

{g⁰⁰} × P {g⁰} × P

{g} × P

mutation mutation

fixed point

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No therapy (only melanoma)

Consider an initial population of genotype g (associated with` different phenotypesp₁, . . . ,p_`) which is able to mutate at rate µto another genotypeg⁰, associated withk different phenotypes p₁⁰, . . . ,p⁰_k.

Consider as initial conditionn(0) = (n_(g,p₁₎(0), . . . ,n_(g_,p_`₎(0))a stable fixed point,n, of the following system:¯

n˙_(g_,p_i₎=n_(g_,p_i₎



b_i −d_i −

`

X

j=1

c_ijn_(g,p_j₎−

`

X

j=1

s_ij



+

`

X

j=1

s_jin_(g_,p_j₎.

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Study of one step (example with ` = 1, k = 2)

<—1—–><—2—><——3————>

Phase 1 : approximation with supercritical multi-type branching, O(log(K)) Phase 2 : approximation with the deterministic system, O(1)

Phase 3 : approximation with subcritical multi-type branching,O(log(K))

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Invasion fitness ?

For the BPDL model (without switches) : f(x,M) =b_x−d_x−X

y∈M

c_xy¯n_y.

is the growth rate of a single individual with trait x 6∈M in the presence of the equilibrium populationn¯on M.

f(x,M)>0: positive probability for the mutant (uniformly in K) to grow to a population of sizeO(K);

f(x,M)<0: the mutant populationdies out with probability tending to one (as K → ∞) before this happens.

We need to generalize this notion to the case when fast phenotypic switches are present.

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Phase 1

As long as

n_(g,p_i₎>¯n_(g_,p_i₎−ε ∀i =1, . . . , ` Pk

i=1n_(g0,p⁰_i)6εK

the mutant population (g⁰,p₁⁰), . . . ,(g⁰,p_k⁰) is well approximated by a k-type branching process with rates:

p_i⁰ →p_i⁰p_i⁰ with rate b_i⁰ p_i⁰ →∅ with rate d_i⁰+P`

l=1c_iln¯l

p_i⁰ →p_j⁰ with rate s_ij⁰







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Multi-type branching processes have been analysed by Kesten/Stigum and Atreya/Ney. Their behavior are classified in terms of the matrix A, given by

A=







f₁ s₁₂⁰ . . . s_1k⁰ s₂₁⁰ f₂ ...

... . ..

s_k⁰₁ . . . f_k







where

fi :=b⁰_i−d_i⁰−

`

X

l=1

c_il ·¯n_l−

k

X

j=1

s_ij⁰.

The multi-type process is super-critical, if and only if the largest

eigenvalue, λ1 =λ1(A)>0. It is thus the appropriate generalization of the invasion fitness:

F(g⁰,g) :=λ1(A).

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Example : Resonance

s₁₂=s₂₁=2 f₁ =f₂=−1 F(g⁰,g) =λ₁ =1

F˜(g,g⁰) =−1

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Conclusion

Plan

6 Conclusion

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Conclusion

Still a lot to understand...

Biologically :

measure precise parameters appearing in the model check predictions (e.g. therapy with 2 types of T-cells) etc.

Mathematically :

How do the transition probabilities between different relapses scale with K ?

What happens if the deterministic system has limit cycles ? How does the birth-reducing competition affect the mutation probability in presence of treatment?

etc.

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Conclusion

Merci !