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What is DBM?
DBM usually refers to looking at volume changes using the determinant of the Jacobian matrix field, referred to as “the Jacobian” in SPM. This is calculated during spatial normalization steps, and saved with the “j_*.nii” prefix.
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Thus, the Jacobian as termed in SPM has at each voxel a scale (ratio) relative to the template volume. Thus, a value of 1.1 would be a voxel with 10% higher volume (native > template), whereas a value of 0.95 would be 5% lower volume (native < template).
What is the Jacobian, and how does it relate to VBM and regional gray matter volume or density?
More about jacobian
In neuroimaging, we use the term “Jacobian” to mean the determinant of the Jacobian matrix, which is a term from vector calculus. The Jacobian matrix is calculated at each voxel as a vector representation of the deformation field (“field” for short) represented by the a warping - usually spatial normalization. Think of the “field” as a vector (x,y,z) that describes the direction and size of warping that voxel has to undergo to be moved into normalized space.
There is a nasty little confound in that the deformation field can be forward or inverse (native to template or template to native), so the Jacobian and be forward or inverse…. (“j_*.nii or ij_*.nii in SPM). In SPM, the Jacobian calculated during normalization steps is such that numbers > 1.0 in the Jacobian reflect larger native space (99% sure on this, need to check).
OK, so what about VBM, gay matter…?
Gray matter density is based on the intensity of a voxel relative to other voxel intensities, and it’s closeness to the gray matter “tissue probability map” (tpm) after normalization. For example, let’s say GM is around average 60 intensity (arbitrary), WM is 100, and CSF is 10; this could be a T1 where WM is bright, GM is gray, and CSF is dark-black. The unified segmentation, DARTEL and SHOOT methods in SPM all determine GM density based on how close 60 a voxel value is, AND the degree to which it overlaps a prior “tpm” GM map. This “c1” is termed gray matter probability, and analyzing this with VBM is termed a gray matter density analysis.
To get what is termed “regional gray matter volume” (the original meaning of VBM), this c1 is multiplied by the Jacobian (called “modulated”). Thus, in the normalized images (prior to smoothing for VBM analysis), a voxel in a large part of a brain with a probably of 0.95 being GM (c1) and a Jacobian of 1.1 would be 0.95 * 1.1 = 1.045, whereas a voxel in a small part of a brain with the same “density” but and Jacobian of 0.9 would have a value of 0.05 * 0.9 = 0.855. Thus, the “modulation” means gray matter accounting for volume will be analyzed. More volume, larger number.
So, when doing “VBM” of “regional gray matter volume”, we are combining the Jacobian and the GM density.
References
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant
DBM using field
More sophisticated DBM would use the actual field:
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