We tackle the curve registration problem in which we learn time-warping functions $\left\{ h_{i}\right\} _{i=1,...,n}$ from noisy observations of registered curves $\left\{ X_{i}\circ h_{i} \right\} _{i=1,...,n}$. Still, in our case a priori knowledge regarding the unregistered curve dynamics is available under the form of a parametric ordinary differential equations (ODE)s $\dot{X}_{i}=f_{i}(X_{i},t)$. From this combination of descriptive non-parametric model and causal parametric one, we aim to locate as accurately and exhaustively as possible the effect of a given therapy on a treated population. From the causal representation, we quantify treatment effects on well identified mechanisms, specified as ODE parameter covariates. From the descriptive one, we infer global action of treatment due to other mechanisms missed by the ODE but accounted for by time-warping functions, leading to distorted dynamics for treated subjects compared to the control group. The joint estimation of time-warping functions and ODE parameters is then cast as a non-linear regression problem in a mixed effect setting to account for inter-subject variability. We then confirm on simulated data the capacity of our method to estimate treatment effects on the general evolution of some variables of interests as well as on specific mechanisms acting on the patient dynamic. We conclude this work by analyzing pre-clinical and clinical data from trials testing HIV cures.