Unbiased template

A wide range of different unbiased templates from the control group images were obtained using different values of the registration parameters {

*α*,

*σ*} in

Eq. (3) that define the amount of smoothness and the balance between intensity matching and regularization respectively. illustrates a sagittal view of the Nor group template estimated using the following values of the registration parameters

*α* = [0.5, 1, 2, 5, 10] and

*σ* = [0.2, 0.5, 1, 2, 5].

Large values of the regularization parameter, i.e., *σ* = 5, produce an important blur in the templates for all values of the smoothing parameter *α*. On the other hand, unrealistic structures can be seen in most of the templates using *α*≤2 (see corpus callosum–lateral ventricle boundary). A possible reason can be that small values of the smoothing parameter *α* yield many local minima in the energy function to be minimized by the registration algorithm.

The values of the registration parameters in the interval {*α* = [5, 10], *σ* = [1, 2]} provide a good trade-off between regularization and smoothing. We visually checked that these templates preserve most of the anatomical details of the normal brain anatomy. For the rest of the study, the template was chosen as the one obtained with the values {*α* = 5, *σ* = 1}.

Student’s t-statistic STV plots

In order to compute non-rigid registration from the template to all brain images, the range of values of the registration parameters {*α*, *σ*} were slightly adjusted according to the results shown in . The value of *σ* = 5 was disregarded because the corresponding template did not show enough anatomical detail due to poor image matching; additionally a larger value of the smoothness parameter was considered. The new set of values of the registration parameters were *α* = [0.5, 1, 2, 5, 10, 20] and *σ* = [0.2, 0.5, 1, 2].

The STV curves of the Student’s *t*-statistic in illustrate the sensitivity to detect significant brain volume changes between AD–Nor and MCI–Nor groups when using different values of the registration parameters {*α*, *σ*}. The STV curves corresponding to the null distribution were also computed comparing the two independent normal groups (Nor–Nor2). As only large values of the *t*-statistic are of interest, either positive for atrophy or negative for expansion, the horizontal axis shows values |*t*|≥3.

An important asymmetry between atrophy and expansion can be observed in . For large enough values of the smoothing parameter *α*, the number of voxels with significant atrophy is larger than for expansion with the same significance level.

Most of the STV curves for AD–Nor group comparison show an increasing sensitivity to detect brain volume changes when increasing the value of the smoothing parameter *α*. The values of the registration parameters yielding voxels with larger *t*-statistic are {*α* = [5, 10], *σ* = 2}.

For each curve, a random permutation test with 10,000 permutations was performed to estimate the *t*_{p}-threshold that controls FWE with significance level *p*. The values of *t*_{p} are illustrated in for *p* = [0.05, 0.01, 0.005]. All STV curves of the AD–Nor group comparison showed FWE-corrected significant voxels at level *p* = 0.05.

The optimal pattern for a STV curve would be the one that maximizes the number of voxels with higher significance, i.e. larger values of |*t*|. As the regularization is an extra penalty term to ensure smoothness of the mapping, a reasonable criterion could be to select the lowest value of *α* among the values that achieve a similar pattern of the STV curve. Accordingly, the values of the registration parameters {*α* = [5, 10], *σ* = 2} would be a good choice.

Brain atrophy statistical maps

In order to illustrate the effect of using different values of the registration parameters in the spatial distribution of the brain atrophy, three sets of values were selected to represent different conditions: low-level smoothing with small regularization {*α* = 0.5, *σ* = 0.5}, large smoothing with large regularization {*α* = 20, *σ* = 2} and a point with intermediate smoothing {*α* = 5, *σ* = 2}. These working conditions are a representative sample of the different performance of STV curves illustrated in . Student’s *t*-statistic maps are shown in .

Assessment of statistical significance corrected for multiple comparisons is required in order to compare and to give an interpretation to Student’s *t*-maps. For each value of the registration parameters {*α*, *σ*}, 10,000 random permutations were used to correct for multiple comparisons with FWE- and FDR-based methods. illustrates the corrected *p*-values for the three set of values of the registration parameters shown in . As statistical maps are typically shown with either *t*- or uncorrected *p*-value maps, two panels were used to illustrate the dependence of the corrected *p*-values on both measures. This information is redundant due to the known mapping between *t*-statistic and uncorrected *p*-values, but it may be helpful for comparison purposes. Note that while a *t*-threshold is used to control FWE, uncorrected *p*-value thresholds are used to estimate the overall significance.

Using different values of the registration parameters {*α*, *σ*} provide atrophy maps with different amount of spatial correlation, and therefore the severity of the correction for multiple comparison will change. However, the values of the *t*-threshold *t*_{p} controlling for FWE at level *p* for all values of {*α*, *σ*} differ in less than 0.5 units (see and ). This difference is difficult to appreciate in the Student’s *t*-statistic maps in .

Due to the fact that several values of the registration parameters were explored, an additional correction for multiple comparisons can be performed. For strong control of FWE, the distribution of the maximum statistic under the null hypothesis must be estimated. Accordingly, the maximum is computed not only across the voxels but also across the whole set of parameters {*α*, *σ*}. The mapping between *t*-threshold and this corrected *p*-value which takes into account the whole set of comparisons is also shown in the left panel of .

Brain atrophy statistical maps are strongly influenced by the values of the registration parameters {*α*, *σ*} used during the estimation of the warping between each subject and the template. In general, larger regions with significant differences between groups are obtained for larger values of the smoothing parameter *α*. However, too large values of *α* may produce smoothed statistical maps. For example, the statistical maps of the intermediate point {*α* = 5, *σ* = 2} in show regions with sharp boundaries in agreement with anatomical structures affected by dementia, while the corresponding maps when using {*α* = 20, *σ* = 2} are blurred. See for example the boundaries of the parahippocampal gyrus in the AD–Nor comparison. Other structures with significant atrophy, such as the frontal part of the insula, are better represented when using {*α* = 5, *σ* = 2} than {*α* = 20, *σ* = 2}. When comparing AD–Nor versus MCI–Nor patient groups, AD group showed larger areas with stronger significance affected by brain atrophy.

shows in more detail the AD–Nor brain atrophy map for the intermediate point, i.e., the values of the registration parameters are {*α* = 5, *σ* = 2}. The following brain structures showed atrophy with a strong significance: left (see slice 1) and right (slices 3–4) superior temporal sulcus; bilateral posterior part of the cingulate gyrus (precuneus region) at slices 1–5; bilateral temporo-occipital sulcus at slices 1–2, with larger significance at the left side; bilateral hippocampus at slices 2–6, mainly affecting subiculum and CA1 regions; bilateral entorhinal cortex and parahippocampal gyrus at slices 4–7; bilateral amygdala at slice 7; temporal pole, more pronounced at right side (slice 9); anterior part of the right insula at slice 11 and axial slice, with a lower significance at the left insula (slice 10).

Regression analysis maps

Voxel-wise linear regression analysis was performed with the following clinical variables: *MMSE*_{baseline}, *MMSE*_{12month} and *age*. The interest here is not to discuss deeply the clinical interpretation of the relationship between brain atrophy and clinical measurements, but to illustrate the performance of the regression maps obtained with SVF diffeomorphic registration. shows the spatial distribution of some regression features, such as the coefficient of determination *r*^{2}, regression significance (uncorrected *p*-value) and regression coefficient. These statistical maps were obtained with registration parameters {*α* = 5, *σ* = 2}. It can be seen that Jacobian determinants at the hippocampus and amygdala showed a positive (right panel in ) and significant (left panel in ) relation with *MMSE*_{baseline}, because smaller values of the Jacobian determinants were related to lower MMSE scores. Note that the *p*-value regression map with *MMSE*_{baseline} is similar to the AD–Nor atrophy statistical map in . This result was expected because the clinical variable *MMSE*_{baseline} is closely related to the diagnostic label that defines patient groups. It can be noted that the atrophy-age regression maps have a completely different pattern: the most significant correlation was found in the lateral ventricles, which was positive, i.e. an increase in age was linearly related to expansion of the ventricles. In contrast, the regions showing a stronger linear relation between brain atrophy and cognitive status, either *MMSE*_{baseline} or *MMSE*_{12month}, were located at structures known to be affected by dementia, such as hippocampus and amygdala.

Region of interest analysis

In order to assess statistical differences in the volume of subcortical regions across patient groups, univariate hypothesis testing was performed on the ROI-average Jacobian determinant of the mappings. Among the analyzed structures, only amygdalae and hippocampi presented significant volume differences, both in AD–Nor and MCI–Nor group comparisons. shows the values of the Student’s *t*-statistic for the whole set of values of the registration parameters {*α*, *σ*}. It can be noted that the magnitude of the *t*-statistic in the ROI is smaller than the voxel-wise brain statistical maps due to the spatial averaging performed in the ROI analysis, especially at those structures with a heterogeneous atrophy. In our case, while the atrophy distribution at the amygdala was roughly homogeneous, an important heterogeneity was found in the hippocampus. Again, a good candidate of the registration parameter values is {*α* = 5, *σ* = 2} because it yields large differences between patient groups.