LESA: Longitudinal Elastic Shape Analysis of Brain Subcortical Structures

Introduction

Over the past 30 years, magnetic resonance imaging has become a ubiquitous tool for accurately visualizing the change and development of the brain’s subcortical structures (e.g., hippocampus). Although subcortical structures act as information hubs of the nervous system, their quantification is still in its infancy due to many challenges in shape extraction, representation, and modeling. Here, we develop a simple and efficient framework of longitudinal elastic shape analysis (LESA) for subcortical structures. Integrating ideas from elastic shape analysis of static surfaces and statistical modeling of sparse longitudinal data, LESA provides a set of tools for systematically quantifying changes of longitudinal subcortical surface shapes from raw structure MRI data. The key novelties of LESA include: (i) it can efficiently represent complex subcortical structures using a small number of basis functions and (ii) it can accurately delineate the spatiotemporal shape changes of the human subcortical structures. We applied LESA to analyze three longitudinal neuroimaging data sets and showcase its wide applications in estimating continuous shape trajectories, building life-span growth patterns, and comparing shape differences among different groups. In particular, with the Alzheimer’s Disease Neuroimaging Initiative (ADNI) data, we found that the Alzheimer’s Disease (AD) can significantly speed the shape change of ventricle and hippocampus from 60 to 75 years old compared with normal aging.

Codes and example data for implementation of LESA are available in GitHub repository.

Pipeline

pipeline

Data

We have applied LESA to study three different longitudinal brain imaging datasets: the Alzheimer’s Disease Neuroimaging Initiative (ADNI) dataset, the Human Connectome Project test-retest dataset and the OpenPain dataset. For simplicity, we mainly focus on the lateral ventricle and left hippocampus surfaces.

dataset summary
Panel (a) shows age distributions in the three datasets. The rest panels show the temporal information on scans for each subject.

PCA Results

PCA results of the ADNI dataset. (a) shows the Karcher mean of all surfaces in the ADNIGO2 dataset. Panel (b) shows the cumulative percentage of variance explained by the number of principal components (PCs). As shown here, the use of 32 and 64 PCs can represent the 95% variation of all lateral ventricle and left hippocampus surfaces seperately. Panel (c) shows the first PC direction in the shape space by reconstructing the principal geodesic as fμ+t √λ1PC1, where PC1 represents the first principal direction. We then bring the temporal labels back (the time of each observation) and plot the area trajectories for all subjects in panel (d) and PC1 score trajectories in panel (e).

  1. Lateral Ventricle
    PCA results of the ADNI’s lateral ventricle surfaces. (a) Karcher mean of all lateral ventricle surfaces. (b) Cumulative percentage of variance explained by the number of PCs. (c) First dominant PC direction reconstructed as fμ+t √λ1PC1 . The five shapes in the front view, from left to right, correspond to t = {−1; −0:5; 0; 0:5; 1}. (c) Surface area trajectories. (e) PC1 score trajectories.
    Ventricle_PCA_result

  2. Left hippocampus
    PCA results of the ADNI’s left hippocampus surfaces. (a) Karcher mean of all left hippocampus surfaces. (b) Cumulative percentage of variance explained by the number of PCs. (c) First dominant PC direction reconstructed as fμ+t √λ1PC1 . The five shapes in the front view, from left to right, correspond to t = {−1; −0:5; 0; 0:5; 1}. (c) Surface area trajectories. (e) PC1 score trajectories.
    Hippocampus_PCA_result

Shape Trajectory Fitting Results

We densely fit area trajectories and principal coefficients trajectories with two methods: PACE and MGCV, and we compare the performance under two methods.

  1. Lateral ventricle trajectories:
    (a) Trajectory fitting results of LESA from the observed sparse data. First column: sparse surface area and PC score trajectories. Second and third columns: continuous trajectories fitted by the PACE and MGCV models (black dashed lines: mean trajectories). First row: area trajectories. Second row: PC1 score trajectories.

    ventricle_PACE_MGCV

    Recovered mean surface trajectories by:
                      (b) PACE fitting;                                         (c) MGCV fitting.

    ventricle_PACE_mean_trajectory ventricle_MGCV_mean_trajectory

  2. Left hippocampus trajectories:
    (a) Trajectory fitting results of LESA from the observed sparse data. First column: sparse surface area and PC score trajectories. Second and third columns: continuous trajectories fitted by the PACE and MGCV models (black dashed lines: mean trajectories). First row: area trajectories. Second row: PC1 score trajectories.

    hippocampus_PACE_MGCV

    Recovered mean surface trajectories by:
                      (b) PACE fitting;                                         (c) MGCV fitting.

    hippocampus_PACE_mean_trajectory hippocampus_MGCV_mean_trajectory

  3. Some individual fitting examples:
    Individual surface trajectories fitted with LESA. Panels (a) and (b) show the raw and fitted trajectories for the surface area and PC1 score for three individual subjects.
    hippocampus_individual

    Reconstructed surface trajectories of:
    (c) Sub.1 with PACE;    (d) Sub.1 with MGCV;     (e) Sub.2 with PACE;    (f) Sub.2 with MGCV.
    hippocampus_indi1_PACE     hippocampus_indi1_MGCV     hippocampus_indi2_PACE     hippocampus_indi2_MGCV

  4. Performance comparison:
    Mean squared prediction errors of PACE and MGCV.

    Lateral Ventricle Lateral Ventricle Left Hippocampus Left Hippocampus
    PACE MGCV PACE MGCV
    Area 59.7086 70.5375 17.7408 22.7865
    PC1 0.0357 0.0424 0.0238 0.0239
    PC2 0.0248 0.0256 0.0462 0.0474
    PC3 0.0448 0.0526 0.0316 0.0383
    PC4 0.0208 0.0225 0.0695 0.0852
    PC5 0.0531 0.0580 0.0272 0.0293
    .... .... .... .... ....
    Average PC MSPE 0.0383 0.0400 0.0350 0.0359

Life-span Shape Change

Life-span (22-90 years old) left ventricle and left hippocampus growth trajectories. (a-b) Observed sparse data and fitted mean trajectories (black solid line).
lifespan_trajectory

Reconstructed life-span mean surface trajectories. Color on each surface indicates the surface’s deformation size compared with the surface at age 22:
               (c) Lateral Ventricle;                               (d) Left Hippocampus.
ventricle_lifespan_trajectory hippocampus_lifespan_trajectory

Group Difference Analysis

  1. Lateral ventricle:
    Comparison of shape change patterns among AD, MCI and NC. (a) Mean surface area trajectories of the three groups (blue: AD; red: MCI; yellow: NC). (b) Changing rate of the area trajectories calculated as 100∗(α(ti+1) − α(ti))/α(ti).
    ventricle_AD_Comparison

    (c) Reconstructed mean shape trajectories. Color on the surface represents shape difference compared with the NC surface at the corresponding time:
                          AD                                       MCI                                         NC
    ventricle_AD_shape ventricle_MCI_shape ventricle_NL_shape

    (d) Reconstructed mean surface trajectories. Color on the surface represents shape difference compared with the NC surface at the corresponding time:
                          AD                                       MCI                                         NC
    ventricle_AD ventricle_MCI ventricle_NL

  2. Left hippocampus:
    Comparison of shape change patterns among AD, MCI and NC. (a) Mean surface area trajectories of the three groups (blue: AD; red: MCI; yellow: NC). (b) Changing rate of the area trajectories calculated as 100∗(α(ti+1) − α(ti))/α(ti).
    hippocampus_AD_Comparison

    (c) Reconstructed mean shape trajectories. Color on the surface represents shape difference compared with the NC surface at the corresponding time:
                          AD                                       MCI                                         NC
    hippocampus_AD_shape hippocampus_MCI_shape hippocampus_NL_shape

    (d) Reconstructed mean surface trajectories. Color on the surface represents shape difference compared with the NC surface at the corresponding time:
                          AD                                       MCI                                         NC
    hippocampus_AD hippocampus_MCI hippocampus_NL

Shape-trajectory-on-scalar Regression Analysis

  1. Evaluation:
    Evaluation of the shape-trajectory-on-scalar regression on the ADNIGO2 dataset. (a) Histogram of the percentage of improvement in prediction error when comparing the shapetrajectory-on-scalar regression with the baseline model. (b) Examples of original sparse surface, surface reconstructed by the regression’s prediction, and the global mean surface. Color indicates the small patch’s difference level. First row: lateral ventricle; second row: left hippocampus.
    ventricle_regression_goodoffit
    hippocampus_regression_goodoffit

  2. Covariates’ effect to the surface trajectory:
    Lateral ventricle:
    Exploration of the covariates’ effect to the surface trajectory on ADNIGO2 dataset. Panel (i) shows results for the left ventricle and panel (ii) shows results for the left hippocampus. In each sub-panel (a), we fixed gender, marriage status, education years and ApoE4 type and varied the diagnosis status. In each sub-panel (b), we fixed gender, marriage status, education years and diagnosis status, and varied the ApoE4 type.
    ventricle_regression_control

    (c) Each sub-panel shows the reconstructed shape trajectory by fixing gender, marriage status, education years and ApoE4 type, while varying the diagnosis status. Color on each surface represents shape difference compared with the NC surface at the same age:
                             AD                                         MCI                                       NC
    ventricle_AD_regression ventricle_AD_regression ventricle_AD_regression

    (d) Each sub-panel shows the reconstructed surface trajectory by fixing gender, marriage status, education years and ApoE4 type, while varying the diagnosis status. Color on each surface represents shape difference compared with the NC surface at the same age:
                             AD                                         MCI                                       NC
    ventricle_AD_regression_surface ventricle_AD_regression_surface ventricle_AD_regression_surface

    Left hippocampus:
    Exploration of the covariates’ effect to the surface trajectory on ADNIGO2 dataset. Panel (i) shows results for the left ventricle and panel (ii) shows results for the left hippocampus. In each sub-panel (a), we fixed gender, marriage status, education years and ApoE4 type and varied the diagnosis status. In each sub-panel (b), we fixed gender, marriage status, education years and diagnosis status, and varied the ApoE4 type.
    hippocampus_regression_control

    (c) Each sub-panel shows the reconstructed shape trajectory by fixing gender, marriage status, education years and ApoE4 type, while varying the diagnosis status. Color on each surface represents shape difference compared with the NC surface at the same age:
                             AD                                         MCI                                       NC
    hippocampus_AD_regression hippocampus_AD_regression hippocampus_AD_regression

    (d) Each sub-panel shows the reconstructed surface trajectory by fixing gender, marriage status, education years and ApoE4 type, while varying the diagnosis status. Color on each surface represents shape difference compared with the NC surface at the same age:
                             AD                                         MCI                                       NC
    hippocampus_AD_regression_surface hippocampus_AD_regression_surface hippocampus_AD_regression_surface