Practical questions about multi-shell acquisition

This information provided on this page may be helpful to those who would like to acquire multi-shell DWI data using GE scanners.

The following questions may be of relevance to you as you plan your multi-shell DWI acquisition using GE scanners.

1. Since GE scanner only takes in a single maximum b-value, how should I scale the gradient vectors to acquire different b-values?

By convention, unit vectors are associated with b_max . If v = (g_x, g_y, g_z) is unit length and it is associated with some smaller b-value (here, smaller than b_max), say b_s, you should not place v directly into the tensor.dat file. Rather, you should create a new vector, say v_new , which is no longer unit length, and the recipe for creating this new vector is v_new = sqrt(b_s/b_max)*v. The new non-unit length vector, v_new , is the one that should be placed into the tensor.dat file.

Here is the reason why the new vector has to be scaled that way.

Recall the tensor model is a multivariate exponential function with the following dependence exp(- b_s v^T D v ). Note that v is unit length.

Due to the restriction that only a single b_max is applied in the pulse sequence, we would like to know how the gradient vector should be scaled so that we have the following condition: exp(- b_s v^T D v ) = exp(- b_max v_new^T D v_new ). By splitting the b-value is two of the same factors as follows

exp(- sqrt(b_s) v^T D sqrt(b_s) v ) = exp(- sqrt(b_max) v_new^T D sqrt(b_max) v_new ), it is then quite easy to see that the new vector should be expressed as

v_new = sqrt(b_s/b_max)*v.

2. Once you have acquired your data and if you happen to convert the DWI data into the NifTi format, you may also get a copy of the bval and bvec files. Note that the bval file will contain b_max only and the bvec file will contain v_new . This format is not wrong, it is just a different way of representing the same information. If you want to display bval and bvec with unit-length vectors in the bvec file, you have to compute the new b-values and normalize the gradient vectors. How should you scale the b-values?

The new b-values can be obtained from the above equation by taking the norm on both side and perform a few steps of algebraic manipulations. That is,

||v_new || = sqrt(b_s/b_max) and b_s = b_max ||v_new ||².