# Implementation of a Disease Progression Model accounting for covariates

Gabrielle Casimiro, Sophie Tezenas Du Montcel, Sofia Kaisaridi

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Gabrielle Casimiro, Sophie Tezenas Du Montcel, Sofia Kaisaridi. Implementation of a Disease Progression Model accounting for covariates. ISCB 2025 - 46th Annual Conference of the International Society for Clinical Biostatistics, Aug 2025, Basel, Switzerland. hal-05296137

HAL Id: hal-05296137

https://inria.hal.science/hal-05296137v1

Submitted on 3 Oct 2025

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# Implementation of a Disease Progression Model accounting for covariates

G.CASIMIRO¹, S. TEZENAS DU MONTCEL¹, S.KAISARIDI¹

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ARAMIS
LAB
BRAIN DATA SCIENCE

1 - Inria, centre of Paris, ARAMIS team, Sorbonne Université, Institut du Cerveau / Paris Brain Institute AP-HP, INSERM, CNRS, University Hospital, Paris, France

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https://github.com/aramis-lab/leaspy

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gabrielle.casimiro@icm-institute.org

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http://www.aramislab.fr/

## Objectives

Disease progression models are promising tools for analyzing longitudinal data presenting multiple modalities. Such models can be used to estimate long-term disease progression and reconstruct individual trajectories. **Inter-patient variability** is often modeled as random perturbations around a fixed reference. However, much of this variability is driven by external factors such as genetic mutations, gender, level of education or socio-economic status.

In this work, we extend the Disease Course Mapping Model, a non-linear mixed-effects disease progression model implemented in the opensource library Leaspy, implementing a multivariate logistic framework to explicitly account for covariates.

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## Context

### Disease Course Mapping model

<figure><img src="image_7.png" /><figcaption>Figure 1: Spatio-temporal transformations - From population to individual trajectories</figcaption></figure>

Population parameters ($t_0, p_0, v_0$) and Individual parameters ($\tau_i, \alpha_i, w_i$)

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### MCMC SAEM Algorithm

1. Sample Latent Variables given posterior distribution
2. Update sufficient statistics by averaging the new samples with previous estimates
3. Update Model Parameters by maximizing the expected complete-data log-likelihood

## Methods

### Covariate-augmented model

Population Parameters ($t_0, p_0, v_0$) are now estimated as a deterministic function of the covariates:

$$ (t_0, p_0, v_0) = f_{\phi}(c) $$

Link Function $f_{\phi}(c) = \phi_{\text{mod}} \cdot c + \phi_{\text{ref}}$

New Latent Variables with the following Prior Distribution:

$$ \begin{bmatrix} \phi_{\text{mod}} \\ \phi_{\text{ref}} \end{bmatrix} \sim \mathcal{N} \left( \begin{bmatrix} \bar{\phi}_{\text{mod}} \\ \bar{\phi}_{\text{ref}} \end{bmatrix}, \begin{bmatrix} \sigma_{\text{mod}}^2 & \rho \sigma_{\text{mod}} \sigma_{\text{ref}} \\ \rho \sigma_{\text{mod}} \sigma_{\text{ref}} & \sigma_{\text{ref}}^2 \end{bmatrix} \right) $$

introducing new Model Parameters to be estimated via the MCMC SAEM Algorithm.

<figure><img src="image_9.png" /><figcaption>Figure 2: Mean trajectories varying with a covariate</figcaption></figure>

## Conclusion

This ongoing work presents a modeling approach to study how a covariate may influence the progression of multiple features over time. The approach is under implementation and will be tested on real data.

**References** : 
Jean-Baptiste Schiratti, Stéphanie Allassonnière, Olivier Colliot, and Stanley Durrleman. "A Bayesian mixed-effects model to learn trajectories of changes from repeated manifold-valued observations". In: Journal of Machine Learning Research 18.133(2017), pp. 1-33.

• Nemo Fournier and Stanley Durrleman. "A Multimodal Disease Progression Model for Genetic Associations with Disease Dynamics". In: Medical Image Computing and Computer Assisted Intervention (MICCAI). https://hal.archives-ouvertes.fr/hal-04295080

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SORBONNE
UNIVERSITÉ

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ICM

Paris Brain
Institute

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Inserm

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cnrs

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ASSISTANCE
PUBLIQUE

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HÔPITAUX
DE PARIS