Nagesh Adluru^{1}, Hyunwoo J Kim^{1}, Molly Prigge^{2}, Nicholas T Lange^{3}, Erin D Bigler^{4}, Janet E Lainhart^{1}, Andrew L Alexander^{1}, and Vikas Singh^{1}

Mixed effects models that include fixed and factor-specific (also known as random) effects offer a natural framework for studying longitudinal MRI data. This work extends mixed effects models to the setting where the responses lie on curved spaces such as the manifold of symmetric positive definite matrices. By treating the subject-wise diffeomorphic deformations between consecutive time points as a field of Cauchy deformation tensors, our framework can facilitate longitudinal analysis that respects the geometry of such data. While the existing body of work dealing with regression models on manifold-valued data is inherently restricted to cross-sectional studies, the proposed mixed effects formulation significantly expands the operating range of longitudinal analyses.

Representative results from estimating the following mixed models are presented in **Figs.3,4,5,** $$\mathtt{CDT}=\beta_0+\beta_1\left(\alpha_i(\textrm{age}-\tau_i-t_0)-t_0\right),$$ $$\mathtt{CDT}=\beta_0+\beta_1\left(\alpha_i(\textrm{verbal_iq}-\tau_i-t_0)-t_0\right),~\textrm{and}$$ $$\mathtt{CDT}=\beta_0+\beta_1\left(\alpha_i(\textrm{social_responsiveness_scale}-\tau_i-t_0)-t_0\right).$$

**Fig.3** visualizes the estimated $$$\alpha$$$s,$$$\tau$$$s. **Fig.4** shows contours of the cumulative link model, $$$\textrm{logit}\left(p\left(\textrm{ADOS}<\textrm{discrete-level}\right)\right)=\beta_0+\beta_1\alpha+\beta_2\tau$$$. **Fig.5** shows the scatter of subjects on the $$$\alpha,\tau$$$ axes.

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4. J.-B. Schiratti, J.-B, Allassonniere, S, et al. Learning spatiotem-poral trajectories from manifold-valued longitudinal data. Neural Information Processing Systems (NIPS). 2015; 2404-2412.

5. Lorenzi M, Pennec X. Efficient parallel transport of deformations in time series of images: from Schild's to pole ladder. Journal of Mathematical Imaging and Vision. 2014; 50(1-2):5-17.

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The key effects for performing analysis of longitudinal changes in MRI. **Left:** Each subject
could have a different progression rate (acceleration effect) of brain atrophy
and a different onset for atrophy (time shift effect). A general linear model
(GLM) with only fixed effects is insufficient to capture such effects. A mixed effects model including subject-specific (or factor-specific
more generally) slope and intercept captures the effects. **Right:** A simulated
data setting demonstrating that our core computational algorithms can be used
in estimating these effects when using the entire longitudinal deformation
tensor rather than just its determinant. High
resolution image available online.

Conceptually core components needed in estimating the acceleration and time shift effects (c.f. Fig. 1) on the deformation manifolds. The exponential ($$$\mathtt{Exp}$$$) and logarithm ($$$\mathtt{Log}$$$) operators (**left**) and parallel transport of tangent vectors along geodesics (**right**) on a unit sphere (2-dimensional manifold). These three operators provide the framework for estimating distances and perform gradient descent on a manifold. Utilizing this framework, three computationally practical algorithms were developed for robustly estimating the mixed effects models from longitudinal MRI data at each voxel in the brain. Please note that the parallel transport can be visualized as a movie in a browser.

Relationships between longitudinally changing covariates ($$$x$$$-axis) and the primary longitudinal brain change ($$$y$$$-axis). Data from one of the regions of interest, pallidum (implicated to be enlarged in autism^{6}), are presented. These relationships were estimated using our algorithms on the data with atleast three time points (since that gives us the needed minimum of two change values). For comparison, fixed effects models also were estimated and as expected (c.f., **Fig. 1-right**) the mixed effects models reveal interesting correlations. Notice that these non-zero slopes represent the rate of brain change per unit change in the clinical covariate. High resolution image available online.

In addition, relationships between the subject-specific longitudinal effects (acceleration $$$\alpha$$$ and $$$time-shift$$$) and cross-sectionally available clinical covariates can also be examined. Data from the cumulative link models (CLMs) relating acceleration and time-shifts in temporal pole and a discrete subscale of autism-diagnostic-observation-schedule (ADOS) are presented. These 'onion plots' reveal cumulative likelihoods for different discrete levels of ADOS as a function of $$$\alpha$$$ and $$$\tau$$$. Effect of $$$\alpha$$$ (acceleration) (**left**) is in opposite direction to that of the onset of brain changes ($$$\tau$$$) (**right**). **Top** panel displays the model while the **bottom** panel displays data from temporal pole. High resolution image available online.

The mixed model parameters i.e. subject specific "trait" parameters ($$$\alpha$$$ and $$$tau$$$) can also offer a sensitive basis for classifying clinical diagnoses ("state" variables). Data from several different regions are shown where the samples are projected to the $$$\alpha$$$, $$$\tau$$$ space. It can be noticed in these plots that the longitudinal acceleration of brain change with age ($$$\alpha=\frac{\partial_t CDT}{\partial_t age}$$$) is mostly positive for controls while it is more spread and also negative in the autism group. This pattern was observed in many different regions of interest defined on the automated anatomical labeling (AAL) atlas. High resolution image available online.