  Introduction: Heterogeneity of the progression of neurodegenerative diseases
is one of the main challenges faced in developing effective therapies. With the
increasing number of large clinical databases, disease progression models have
led to a better understanding of this heterogeneity. Nevertheless, these
diseases may have no clear onset and biological underlying processes may start
before the first symptoms. Such an ill-defined disease reference time is an
issue for current joint models, which have proven their effectiveness by
combining longitudinal and survival data. Objective In this work, we propose a
joint non-linear mixed effect model with a latent disease age, to overcome this
need for a precise reference time.
  Method: To do so, we utilized an existing longitudinal model with a latent
disease age as a longitudinal sub-model and associated it with a survival
sub-model that estimates a Weibull distribution from the latent disease age. We
then validated our model on different simulated scenarios. Finally, we
benchmarked our model with a state-of-the-art joint model and reference
survival and longitudinal models on simulated and real data in the context of
Amyotrophic Lateral Sclerosis (ALS).
  Results: On real data, our model got significantly better results than the
state-of-the-art joint model for absolute bias (4.21(4.41) versus
4.24(4.14)(p-value=1.4e-17)), and mean cumulative AUC for right censored events
(0.67(0.07) versus 0.61(0.09)(p-value=1.7e-03)).
  Conclusion: We showed that our approach is better suited than the
state-of-the-art in the context where the reference time is not reliable. This
work opens up the perspective to design predictive and personalized therapeutic
strategies.
