A log-logistic survival model from multimodal data for prediction of Alzheimer's Disease

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HAL Id: hal-02430943

https://hal.inria.fr/hal-02430943

Submitted on 7 Jan 2020

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A log-logistic survival model from multimodal data for prediction of Alzheimer’s Disease

Pascal Lu, Olivier Colliot

To cite this version:

Pascal Lu, Olivier Colliot. A log-logistic survival model from multimodal data for prediction of Alzheimer’s Disease. SAfJR 2019 - Survival Analysis for Junior Researchers, Apr 2019, Copenhague, Denmark. �hal-02430943�

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A log-logistic survival model from multimodal data for prediction of Alzheimer’s Disease

Pascal Lu^1,2and Olivier Colliot^1,2,

for the Alzheimer’s Disease Neuroimaging Initiative

1Sorbonne Université, Inserm, CNRS, Institut du cerveau et la moelle (ICM), AP-HP - Hôpital Pitié-Salpêtrière, Boulevard de l’hôpital, 75013, Paris, France

2INRIA Paris, ARAMIS project-team, 75013, Paris, France

The early diagnosis of Alzheimer’s disease (AD) is important for pro- viding adequate care and is currently the topic of very active research. In particular, a current challenge is to predict the future occurrence of AD in patients with mild cognitive impairment. Multimodal data (such as cognitive/clinical, imaging and genetic) can provide complementary information for the prediction. Whereas clinical data, such as cognitive scores, provide an accurate estimate of the current subject’s state, genetic variants help to identify whether a subject would develop Alzheimer’s disease faster than another. In the state of the art, most papers put clinical and genetic vari- ables on the same level in order to predict the current or future subject’s state, although they do not provide the same type of information.

In this work, we propose a new survival model based on multimodal data to estimate the conversion date to AD from genetics and clinical data.

We chose the log-logistic model which provides a parametric framework where the parameters depends on both clinical and genetic data. The haz- ard function is unimodal, which seems well-suited to our model. In our pro- posed formulation, genetic datax_Gonly influences the speedv(x_G)at which the conversion would happen, whereas clinical datax_Cinfluences the initial state of the subject p(x_C). If S denotes the survival function and t_1/2 the median survival time, we sett_1/2 = p(x_C)/v(x_G), andS⁰(t_1/2) = v(x_G). By determiningv(x_G)and p(x_C), we are able to determine the associated survival functionS.

We tested our model on the ADNI-1 dataset (from the Alzheimer’s Dis- ease Neuroimaging Initiative), using cognitive scores (such as MMSE, ADAS13, RAVLT) and genetic information (APOE, gender) and compared it to the Cox-regression model using the Kaplan-Meier estimate, and the classical parametric log-logistic model where the parameters depends on the covari- ates using a log-linear model.

E-mail: pascal.lu@inria.fr

Keywords: Log-logistic model, multimodal data, Alzheimer’s Disease References:

R. Marinescu, et al, TADPOLE Challenge: Prediction of Longitudinal Evolution in Alzheimer’s Disease,https://arxiv.org/abs/1805.03909

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1 1

1 t _1/2

S (t) = P{ T > t | }

x S (t | x) = 1

1 + λ(x)t ^α(x) λ(x) > 0 α(x) > 0

x _G x _I

v(x _G , x _I ) p(x _G , x _I )

t _1/2 S (t) = ¹ ₂ t _1/2 = ^p(x _v(x ^G ^,x ^I ⁾

G ,x _I )

t _1/2 S ^′ (t _1/2 ) = − v(x _G , x _I ) P{ T ≥ t }

t _1/2

S ^′ (t _1/2 ) = − v

A log-logistic survival model from multimodal data for prediction of Alzheimer's Disease

HAL Id: hal-02430943

https://hal.inria.fr/hal-02430943

A log-logistic survival model from multimodal data for prediction of Alzheimer’s Disease

Pascal Lu, Olivier Colliot

To cite this version:

A log-logistic survival model from multimodal data for prediction of Alzheimer’s Disease

1 1

1

t 1/2

S (t) = P{ T > t | }

x S (t | x) = 1

1 + λ(x)t α(x) λ(x) > 0 α(x) > 0

x G x I

v(x G , x I ) p(x G , x I )

t 1/2 S (t) = 1 2 t 1/2 = p(x v(x G ,x I )

G ,x I )

t 1/2 S ′ (t 1/2 ) = − v(x G , x I ) P{ T ≥ t }

t 1/2

S ′ (t 1/2 ) = − v

0 t 0.5

α = − 4t 1/2 S ′ (t 1/2 ) λ = t α 1 1/2

S (t | x G , x I ) = P{ T ≥ t | x G , x I } = 1 1 + exp !

4p(x G , x I ) !

ln t − ln ! p(x

G ,x I ) v(x G ,x I )

"""

v > 0 p > 0

0.2 0.4 0.6 0.8 1.0

1 2 3 4

P{ T ≥ t }

p ∈ { 2, 3, 4 } v = 1.3

0.2 0.4 0.6 0.8 1.0

1 2 3 4

P{ T ≥ t }

p = 4 v ∈ { 1, 2, 3 }

p v S (t) p S (t) v

p S (t)

p(x G , x I ) = β ⊤ I x I v(x G , x I ) = γ ⊤ G x G

β I γ I γ G

p(x i G , x i I ) > 0 v(x i G , x i I ) > 0 i

h(t) = h 0 (t) β ⊤ I x I +β ⊤ G x G S (t) = (S 0 (t)) β

⊤ I x I +β ⊤ G x G

t 1/2 (x I , x G ) = β ⊤ I x I + β ⊤ G x G

t 1/2 p(x G , x I ) v(x G , x I )

↗

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β γ β I , β G

↘

↗

↗

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↗

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τ i T i > τ

τ

v(x G , x I ) = γ ⊤ I x I + γ ⊤ G x G

t _1/2

1 + λ(x)t ^α(x) λ(x) > 0 α(x) > 0

x _G x _I

v(x _G , x _I ) p(x _G , x _I )

t _1/2 S (t) = ¹ ₂ t _1/2 = ^p(x _v(x ^G ^,x ^I ⁾

G ,x _I )

t _1/2 S ^′ (t _1/2 ) = − v(x _G , x _I ) P{ T ≥ t }

t _1/2

S ^′ (t _1/2 ) = − v

α = − 4t _1/2 S ^′ (t _1/2 ) λ = _t α ¹ 1/2

S (t | x _G , x _I ) = P{ T ≥ t | x _G , x _I } = 1 1 + exp !

4p(x _G , x _I ) !

ln t − ln ! _p(x

G ,x _I ) v(x _G ,x _I )

p(x _G , x _I ) = β ^⊤ _I x _I v(x _G , x _I ) = γ ^⊤ _G x _G

β _I γ _I γ _G

p(x ⁱ _G , x ⁱ _I ) > 0 v(x ⁱ _G , x ⁱ _I ) > 0 i

h(t) = h ₀ (t) ^β ^⊤ ^I ^x ^I ^+β ^⊤ ^G ^x ^G S (t) = (S ₀ (t)) ^β

t _1/2 (x _I , x _G ) = β ^⊤ _I x _I + β ^⊤ _G x _G

t _1/2 p(x _G , x _I ) v(x _G , x _I )

β γ β _I , β _G

τ i T _i > τ

v(x _G , x _I ) = γ ^⊤ _I x _I + γ ^⊤ _G x _G