Parameter estimation in a stochastic model for immunotherapy of

Context
Modeling of tumor evolution
Parameter estimation
Results
References
Parameter estimation in a stochastic model for immunotherapy
of cancer
Modibo DIABATE
PhD Student in Applied Mathematics
Laboratoire Jean Kuntzmann - University Grenoble Alpes
Supervised by
Loren COQUILLE: UGA - IF
Adeline LECLERCQ SAMSON: UGA - LJK
Septièmes rencontres des Jeunes Statisticien-ne-s
Porquerolles, Avril 2017
1/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Outline
1
Context
2
Modeling of tumor evolution
3
Parameter estimation
4
Results
2/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Context: Immunotherapy/ACT therapy
Previous works:
stochastic model for skin cancer
immunotherapy (Baar et al., 2015)
Adoptive Cell Transfer (ACT)
therapy (Landsberg et al., 2012)
3/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Context: Immunotherapy/ACT therapy
Previous works:
stochastic model for skin cancer
immunotherapy (Baar et al., 2015)
Adoptive Cell Transfer (ACT)
therapy (Landsberg et al., 2012)
• Extinction of T-cells & Relapse
3/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Context: Immunotherapy/ACT therapy
Previous works:
stochastic model for skin cancer
immunotherapy (Baar et al., 2015)
Adoptive Cell Transfer (ACT)
therapy (Landsberg et al., 2012)
• Extinction of T-cells & Relapse
Objective: understanding the resistance of tumors
3/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Context: Immunotherapy/ACT therapy
Previous works:
stochastic model for skin cancer
immunotherapy (Baar et al., 2015)
Adoptive Cell Transfer (ACT)
therapy (Landsberg et al., 2012)
• Extinction of T-cells & Relapse
Objective: understanding the resistance of tumors
• Modeling the mechanism
• Estimating parameters using real data
3/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Experimental data
Therapy and groups of mice
CTRL mice (5): no therapy
ACT mice (7): ACT therapy (70th
day)
ACT+Re mice (7): ACT therapy
(70th, 160th day)
Tumor evolution measurement
obs
yij

y if yij ∈]3 mm, 10 mm[

 ij
1.9 mm if yij ∈ [0 mm, 1.9 mm]
=

 3 mm if yij ∈ [2 mm, 3 mm]
10 mm if yij ≥ 10 mm
4/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Modeling of tumor evolution: Birth and death process (BDP)
Cell populations involved in the dynamic are modeled by BDP (continuous time
Markov process).
Individual level:
• → ••
• → †
with a rate b
with a rate d
waiting time before an event modeled by an exponential distribution
5/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Modeling of tumor evolution: Birth and death process (BDP)
Cell populations involved in the dynamic are modeled by BDP (continuous time
Markov process).
Individual level:
• → ••
• → †
with a rate b
with a rate d
waiting time before an event modeled by an exponential distribution
Population level:
M (t) → M (t) + 1
M (t) → M (t) − 1
with a rate b × M (t)
with a rate d × M (t)
5/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Modeling of tumor evolution: Birth and death process (BDP)
Cell populations involved in the dynamic are modeled by BDP (continuous time
Markov process).
Individual level:
• → ••
• → †
with a rate b
with a rate d
waiting time before an event modeled by an exponential distribution
Population level:
M (t) → M (t) + 1
M (t) → M (t) − 1
with a rate b × M (t)
with a rate d × M (t)
Large population approximation: M K (t) = MK(t)
1
M K (t) → M K (t) + K
with a rate b × M K (t) × K
1
M K (t) → M K (t) − K
with a rate d × M K (t) × K
5/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Modeling of tumor evolution: BDP - Large population approximation
Let M K (t) =
M (t)
,
K
lim M K (t) (in law) = n∗
K→∞
where n∗ is the solution of ṅM = (b − d)nM with initial condition n0 .
(Ethier and Kurtz, 2009)
6/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Modeling of tumor evolution: BDP - Large population approximation
Let M K (t) =
M (t)
,
K
lim M K (t) (in law) = n∗
K→∞
where n∗ is the solution of ṅM = (b − d)nM with initial condition n0 .
(Ethier and Kurtz, 2009)
6/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Modeling of tumor evolution: Stochastic model
Let:
•: T cell, T (t): population of therapy
cells
•: differentiated cell, M (t): population of
differentiated cells at time t
•: dedifferentiated cell, D(t): population
of dedifferentiated cells
•: cytokine T N Fα , A(t): population of
cytokines T N Fα
7/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Modeling of tumor evolution: Stochastic model
Let:
•: T cell, T (t): population of therapy
cells
•: differentiated cell, M (t): population of
differentiated cells at time t
•: dedifferentiated cell, D(t): population
of dedifferentiated cells
•: cytokine T N Fα , A(t): population of
cytokines T N Fα
Stochastic model = BDP + switch + predator-prey (Baar et al., 2015)
Z(t) = (M (t), D(t), T (t), A(t))
7/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Modeling of tumor evolution: Stochastic model
Let:
•: T cell, T (t): population of therapy
cells
•: differentiated cell, M (t): population of
differentiated cells at time t
•: dedifferentiated cell, D(t): population
of dedifferentiated cells
•: cytokine T N Fα , A(t): population of
cytokines T N Fα
Stochastic model = BDP + switch + predator-prey (Baar et al., 2015)
Z(t) = (M (t), D(t), T (t), A(t))
prod
µ = (bM , bD , bT , lA
, dM , dD , tT , dT , dA , sA , sM D , sDM )
7/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Modeling of tumor evolution: Deterministic model
Large population approximation: Stochastic model → Deterministic model
Let M K (t) =
M (t)
,
K
DK (t) =
D(t)
,
K
T K (t) =
T (t)
,
K
AK (t) =
A(t)
K
lim (M K (t), DK (t), T K (t), AK (t)) (in law) = (n∗M , n∗D , n∗T , n∗A )
K→∞
8/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Modeling of tumor evolution: Deterministic model
Large population approximation: Stochastic model → Deterministic model
Let M K (t) =
M (t)
,
K
DK (t) =
D(t)
,
K
T K (t) =
T (t)
,
K
AK (t) =
A(t)
K
lim (M K (t), DK (t), T K (t), AK (t)) (in law) = (n∗M , n∗D , n∗T , n∗A )
K→∞
with (n∗M , n∗D , n∗T , n∗A ) the solution of the deterministic system:

ṅM = (bM − dM )nM −tT nT nM −sM D nM + sDM nD − sA nA nM


 ṅ = (b − d )n
+sM D nM − sDM nD + sA nA nM
D
D
D
D
 ṅT = −dT nT +bT nM nT


prod
ṅA = −dA nA +lA
bT nM nT
with initial condition (nM0 , nD0 , nT0 , nA0 ) (Baar et al., 2015).
prod
µ = (bM , bD , bT , lA
, dM , dD , tT , dT , dA , sA , sM D , sDM ) (to be estimated)
8/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Parameter estimation: Mixed Effects Models
9/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Parameter estimation: Mixed Effects Model - Likelihood function
• NonLinear Mixed Effects Model (NLMEM):
yij = f (ψi , tij ) + ij , i ∼ N (0, σ 2 Ini ), ψi = µ + ri , with ri ∼ N (0, Ω),
1
∗
3 , n∗ , n∗ : deterministic system solution
f (ψi , t) = (n∗
M (ψi , t) + nD (ψi , t))
M
D
10/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Parameter estimation: Mixed Effects Model - Likelihood function
• NonLinear Mixed Effects Model (NLMEM):
yij = f (ψi , tij ) + ij , i ∼ N (0, σ 2 Ini ), ψi = µ + ri , with ri ∼ N (0, Ω),
1
∗
3 , n∗ , n∗ : deterministic system solution
f (ψi , t) = (n∗
M (ψi , t) + nD (ψi , t))
M
D
µ = (bM , bD , bT , . . . , sA , sM D , sDM ),
ri = (bMi , bDi , bTi , . . . , sAi , sM Di , sDMi ): vector of random effects,
Ω: variance matrix of the random effects
Θ = {µ, σ, Ω}
10/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Parameter estimation: Mixed Effects Model - Likelihood function
• NonLinear Mixed Effects Model (NLMEM):
yij = f (ψi , tij ) + ij , i ∼ N (0, σ 2 Ini ), ψi = µ + ri , with ri ∼ N (0, Ω),
1
∗
3 , n∗ , n∗ : deterministic system solution
f (ψi , t) = (n∗
M (ψi , t) + nD (ψi , t))
M
D
µ = (bM , bD , bT , . . . , sA , sM D , sDM ),
ri = (bMi , bDi , bTi , . . . , sAi , sM Di , sDMi ): vector of random effects,
Ω: variance matrix of the random effects
Θ = {µ, σ, Ω}
• Likelihood function:
L(y obs , δ; θ) =
N Z
Y
L(yiobs , δi , yicens , ψi ; θ)dψi dyicens
i=1
ψi : individual parameters; yicens : censored data (δij = 0); yiobs : observed data (δij = 1),
obs
L(yi
cens
, δ i , yi
, ψi ; θ) =
Y
(i,j)|δij =1
obs
p(yij |ψi ; θ)p(ψi ; θ)
Y
cens
p(yij
|ψi ; θ)p(ψi ; θ)
(i,j)|δij =0
obs
cens
p(yij
|ψi ; θ) and p(yij
|ψi ; θ) are gaussian (Samson et al., 2006)
10/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Parameter estimation: SAEM (Stochastic Approximation EM)
SAEM: Stochastic Approximation Expectation Maximization
11/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Parameter estimation: SAEM (Stochastic Approximation EM)
SAEM: Stochastic Approximation Expectation Maximization
Iteration k of the algorithm:
step E:
• Simulation step: draw (y cens(k) , ψ (k) ) with the conditional distribution
p(y cens , ψ|y, θ̂k−1 )
11/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Parameter estimation: SAEM (Stochastic Approximation EM)
SAEM: Stochastic Approximation Expectation Maximization
Iteration k of the algorithm:
step E:
• Simulation step: draw (y cens(k) , ψ (k) ) with the conditional distribution
p(y cens , ψ|y, θ̂k−1 )
• Stochastic approximation step:
h
i
Qk (θ) = Qk−1 (θ) + γk log p(y obs , δ, y cens(k) , ψ (k) ; θ) − Qk−1 (θ)
(γk ) is a decreasing sequence:
P∞
k=1
γk = ∞,
P∞
k=1
γk2 < ∞
11/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Parameter estimation: SAEM (Stochastic Approximation EM)
SAEM: Stochastic Approximation Expectation Maximization
Iteration k of the algorithm:
step E:
• Simulation step: draw (y cens(k) , ψ (k) ) with the conditional distribution
p(y cens , ψ|y, θ̂k−1 )
• Stochastic approximation step:
h
i
Qk (θ) = Qk−1 (θ) + γk log p(y obs , δ, y cens(k) , ψ (k) ; θ) − Qk−1 (θ)
(γk ) is a decreasing sequence:
P∞
k=1
γk = ∞,
P∞
k=1
γk2 < ∞
step M:
• Maximization step: update θ̂k−1 according to
θ̂k = arg max Qk (θ)
θ∈Θ
11/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Parameter estimation: SAEM - MCMC
SAEM-MCMC for regular exponential families:
p(y
obs
,y
cens
e obs , y cens , ψ), Ψ(θ)i}
, ψ; θ) = exp{−ξ(θ) + hS(y
Iteration k of the algorithm:
• Simulation step: simulation of (y cens(k) , ψ (k) ) through the simulation of a Markov Chain
having p(y cens , ψ|y, θ̂k−1 ) as stationary distribution
• Stochastic approximation: update sk according to
e obs , y cens(k) , ψ (k) ) − sk−1 )
sk = sk−1 + γk (S(y
• Maximization step: θ̂k = arg maxθ∈Θ {−ξ(θ) + hsk , Ψ(θ)i}
12/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Parameter estimation: SAEM - MCMC
SAEM-MCMC for regular exponential families:
p(y
obs
,y
cens
e obs , y cens , ψ), Ψ(θ)i}
, ψ; θ) = exp{−ξ(θ) + hS(y
Iteration k of the algorithm:
• Simulation step: simulation of (y cens(k) , ψ (k) ) through the simulation of a Markov Chain
having p(y cens , ψ|y, θ̂k−1 ) as stationary distribution
• Stochastic approximation: update sk according to
e obs , y cens(k) , ψ (k) ) − sk−1 )
sk = sk−1 + γk (S(y
• Maximization step: θ̂k = arg maxθ∈Θ {−ξ(θ) + hsk , Ψ(θ)i}
SAEM is considered to estimate correctly population parameters as well as having good theoretical
properties
Convergence of the estimate sequence of the SAEM-MCMC to a local maximum of the likelihood
proved by (Delyon et al., 1999) using:
• the assumption that p(y, ψ, θ) belongs to a regular exponential families,
e
• some continuity and differentiability assumptions on the likelihood and S(y,
ψ),
• the uniform ergodicity assumption on Markov Chains in the MCMC algorithm.
12/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Estimated parameters: Individuals fits (group ACT+Re)
13/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
T-cells extinction & Relapse: Simulation of extinction with the stochastic
model using estimated parameters
Scenario (a): T cells survive
Scenario (b): Extinction of T cells
14/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
T-cells extinction & Relapse : Rare events probability estimation
Monte Carlo, Importance Sampling
15/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
T-cells extinction & Relapse : Rare events probability estimation
Monte Carlo, Importance Sampling
Importance Splitting (Jacquemart-Tomi et al., 2013)
Tt : stochastic process
S : threshold
tS = inf{t ≥ 0, Tt ≤ S}
The rare event :
{tS ≤ tF } = {Tt ≤ S, for t ≤ tF }
15/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
T-cells extinction & Relapse : Rare events probability estimation
Monte Carlo, Importance Sampling
Importance Splitting (Jacquemart-Tomi et al., 2013)
Tt : stochastic process
S : threshold
tS = inf{t ≥ 0, Tt ≤ S}
The rare event :
{tS ≤ tF } = {Tt ≤ S, for t ≤ tF }
Extinction probability P(tS ≤ tF )
P(tS ≤ tF ) =
m
Y
pk
k=1
p1 = P(t1 ≤ tF ) = P{Tt ≤ S1 , for t ≤ tF }
pk = P(tk ≤ tF | tk−1 ≤ tF ) for k = 2, . . . , m
15/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Conclusion & Future Works
Conclusion
Simplification/adaptation of the model for parameter estimation
Parameter estimation using real biological data (censored data)
Estimation of the probability of T cell population extinction (relapse)
16/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Conclusion & Future Works
Conclusion
Simplification/adaptation of the model for parameter estimation
Parameter estimation using real biological data (censored data)
Estimation of the probability of T cell population extinction (relapse)
Future Works
Optimal selection of parameters for the relapse probability estimation
Parameter estimation (with an Expectation Propagation algorithm) from a
diffusion approximation of the stochastic model
(Barthelmé and Chopin, 2014), (Cseke et al., 2016)
Development (and parameter estimation) of a probabilistic model for lung
cancer using MRI images
16/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
Thank you for your attention !
17/18
Context
Modeling of tumor evolution
Parameter estimation
Results
References
References
Martina Baar, Loren Coquille, Hannah Mayer, Michael Hölzel, Meri Rogava, Thomas
Tüting, and Anton Bovier. A stochastic individual-based model for immunotherapy
of cancer. Scientific Reports, 2015.
Bernard Delyon, Marc Lavielle, and Eric Moulines. Convergence of a stochastic
approximation version of the EM algorithm. Annals of statistics, pages 94–128,
1999.
Stewart N Ethier and Thomas G Kurtz. Markov processes: characterization and
convergence, volume 282. John Wiley & Sons, 2009.
Damien Jacquemart-Tomi, Jérôme Morio, and François Le Gland. A combined
importance splitting and sampling algorithm for rare event estimation. Proceedings
of the 2013 Winter Simulation Conference, Washington 2013, Dec 2013,
WASHINGTON, United States., 2013.
Jennifer Landsberg, Judith Kohlmeyer, Marcel Renn, Tobias Bald, Meri Rogava, Mira
Cron, Martina Fatho, Volker Lennerz, Thomas Wölfel, Michael Hölzel, et al.
Melanomas resist T-cell therapy through inflammation-induced reversible
dedifferentiation. Nature, 490(7420):412–416, 2012.
Adeline Samson, Marc Lavielle, and France Mentré. Extension of the SAEM algorithm
to left-censored data in nonlinear mixed-effects model: Application to HIV dynamics
model. Computational Statistics and Data Analysis, 51(3):1562–1574, 2006.
18/18

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