# Response function estimation¶

A compulsory step in spherical deconvolution is deriving the ‘response
function (RF)’, which is used as the kernel during the deconvolution
step. For the white matter, this is the signal expected for a voxel
containing a single, coherently-oriented fibre bundle. While some groups
prefer to define this function using some *ad-hoc* template function
(e.g. a diffusion tensor with empirical diffusivities), the MRtrix
contributors are in preference of deriving this function directly from
the image data, typically by averaging the diffusion signal from a set
of empirically-determined ‘single-fibre (SF)’ voxels.

The process of estimating this function from the data is however
non-trivial; there is no single unambiguous way in which this should be
done. Earlier in the beta version of MRtrix3, we provided a command
`dwi2response`

that advertised automated determination of the response
function, based on a published
method
with a few additional enhancements. Unfortunately user testing showed
that this algorithm would not produce the desired result in a number of
circumstances, and the available command-line options for altering its
behaviour were not intuitive.

As a result, we are now instead providing dwi2response as a
*script*. This was done for a few reasons. Firstly, it means that we can
provide multiple different mechanisms / algorithms for response function
estimation, all accessible within the one script, allowing users to
experiment with different approaches. Secondly, because these Python
scripts are more accessible to most users than C++ code, the algorithms
themselves can be modified, or some may even choose to try devising
their own heiristics for response function estimation. Thirdly, it
reinforces the fact that there is unfortunately *not* a black-box,
one-size-fits-all solution to this problem.

Here I will discuss some of the technical aspects of response function estimation, and describe the mechanisms by which the currently provided algorithms work. If however you are not interested in the nitty-gritty of this process, feel free to scroll to the bottom of the page.

## Necessary steps¶

Looking at the process of response function estimation in full detail, there are four crucial steps. For each of these, I will also briefly mention the typical process used.

- Select those image voxels that are to be used when determining the
response function - the ‘single-fibre mask’.
*Typical*: Varies. - Estimate the direction of the underlying fibres in each voxel.
*Typical*: Often the diffusion tensor fit is still used for this purpose; though CSD itself can also be used as long as an initial response function estimate is available. - Rotate the signal measured in each single-fibre voxel in such a way
that the estimated fibre direction coincides with the z-axis.
*Typical*: This may be done by rotating the diffusion gradient table according to the estimated fibre direction; or if the diffusion signal is converted to spherical harmonics, then a spherical convolution can be used. - Combine these signals to produce a single response function.
*Typical*: The`m=0`

terms of the spherical harmonic series (which are rotationally symmetric about the z-axis) are simply averaged across single-fibre voxels.

Of these steps, the first is the one that has caused the greatest difficulty, and is also the principle mechanism where the provided response function estimation algorithms vary. It will therefore be the primary focus of this document, though note that the other aspects of this process may also change with ongoing development.

`dwi2response`

algorithms¶

`fa`

¶

In the previous version of MRtrix (‘0.2’), the following heuristic was suggested in the documentation for deriving the response function:

- Erode a brain mask by a few voxels, to omit any voxels near the edge of the brain;
- Select those voxels within the mask that have a Fractional Anisotropy (FA) of 0.7 or greater;
- The
`estimate_response`

command would then be used to generate a response function, which would internally perform diffusion tensor estimation to get the fibre directions as well as the gradient reorientation.

Rather than this series of commands, `dwi2response`

now provides a
similar heuristic in-built as the `fa`

algorithm. The primary
difference is that by default, it will instead select the 300 voxels
with the highest FA (though this can be modified at the command-line).

This algorithm is provided partly for nostalgic purposes, but it also
highlights the range of possibilities for single-fbre voxel selection.
One of the problems associated with this approach (over and above the
feeling of uncleanliness from using the tensor model) is that in white
matter regions close to CSF, Gibbs ringing can make the signal in *b=0*
images erroneously low, which causes an artificial increase in FA, and
therefore such voxels get included in the single-fibre mask.

`manual`

¶

This algorithm is provided for cases where none of the available algorithms give adequate results, for deriving multi-shell multi-tissue response functions in cases where the voxel mask for each tissue must be defined manually, or for anyone who may find it useful if trying to devise their own mechanism for response function estimation. It requires manual definition of both the single-fibre voxel mask (or just a voxel mask for isotropic tissues); the fibre directions can also be provided manually if necessary (otherwise a tensor fit will be used).

`msmt_5tt`

¶

This algorithm is intended for deriving multi-shell, multi-tissue
response functions that are compatible with the new Multi-Shell
Multi-Tissue (MSMT) CSD algorithm. The response function estimation
algorithm is identical to that described in the
manuscript:
As long as EPI inhomogeneity field correction has been performed, and a
tissue-segmented anatomical image (prepared in the 5TT format for
ACT) is provided with good
prior rigid-body alignment to the diffusion images, then these
high-resolution tissue segmentations can be used to identify
single-tissue voxels in the diffusion images. This algorithm is
hard-wired to provide response functions for the most typical use case
for MSMT CSD: An isotropic grey matter response, an anisotropic white
matter response, and an isotropic CSF response; the output response
functions are provided in the format expected by the dwi2fod
command. Those wishing to experiment with different multi-tissue
response function configurations will need to use the `manual`

algorithm (which will provide a multi-shell response function if the
input DWI contains such data).

For reference, this algorithm operates as follows:

- Resample the 5TT segmented image to diffusion image space.
- For each of the three tissues (WM, GM, CSF), select those voxels that obey the following criteria:

- The tissue partial volume fraction must be at least 0.95.
- For GM and CSF, the FA must be no larger than 0.2.

- For WM, use the mask derived from step 2 as the initialisation to the
`tournier`

algorithm, to select single-fibre voxels. - Derive a multi-shell response for each tissue for each of these three
tissues. For GM and CSF, use
*lmax=0*for all shells.

`tax`

¶

This algorithm is a fairly accurate reimplementation of the approach proposed by Tax et al.. The operation of the algorithm can be summarized as follows:

- Initialise the response function using a relatively ‘fat’ profile, and the single-fibre mask using all brain voxels.
- Perform CSD in all single-fibre voxels.
- Exclude from the single-fibre voxel mask those voxels where the resulting FOD detects more than one discrete fibre population, e.g. using the ratio of the amplitudes of the first and second tallest peaks.
- Re-calculate the response function using the updated single-fibre voxel mask.
- Return to step 2, repeating until some termination criterion is achieved.

The following are the differences between the implementation in
`dwi2response`

and this manuscript:

- Deriving the initial response function. In the manuscript, this is
done using a tensor model with a low FA. I wasn’t fussed on this
approach myself, in part because it’s difficult to get the correct
intensity sscaling. Instead, the script examines the mean and
standard deviation of the raw DWI volumes, and derives an initial
*lmax=4*response function based on these. - The mechanism used to identify the peaks of the FOD. In
`dwi2response`

, the FOD segmentation algorithm described in the SIFT paper (Appendix 2) is used to locate the FOD peaks. The alternative is to use the sh2peaks command, which uses a Newton search from 60 pre-defined directions to locate these peaks. In my experience, the latter is slower, and may fail to identify some FOD peaks because the seeding directions are not sufficiently dense.

For the sake of completeness, the following are further modifications
that were made to the algorithm as part of the earlier `dwi2response`

*binary*, but have been removed from the script as it is now provided:

- Rather than using the ratio of amplitudes between the tallest and second-tallest peaks, this command instead looked at the ratio of the AFD of the largest FOD lobe, and the sum of the AFD of all other (positive) lobes in the voxel. Although this in some way makes more sense from a physical perspective (comparing the volume occupied by the primary fibre bundle to the volume of ‘everything else’), it’s possible that due to the noisy nature of the FODs at small amplitudes, this may have only introduced variance into the single-fibre voxel identification process. Therefore the script has reverted to the original & simpler peak amplitude ratio calculation.
- A second, more stringent pass of SF voxel exclusion was performed, which introduced two more criteria that single-fibre voxels had to satisfy:
- Dispersion: A measure of dispersion of an FOD lobe can be derived as
the ratio between the integral (fibre volume) and the peak amplitude.
As fibre dispersion increases, the FOD peak amplitude decreases, but
the fibre volume is unaffected; therefore this ratio increases. The
goal here was to explicitly exclude voxels from the single-fibre mask
if significant orientation dispersion was observed; this can be taken
into account somewhat by using the FOD peak amplitudes (as
orientation dispersion will decrease the amplitude of the tallest
peak), but from my initial experimentation I wanted something more
stringent. However as before, given the difficulties that many users
experienced with the
`dwi2response`

command, this algorithm in the new script errs on the side of simplicity, so this test is not performed. - Integral: By testing only the ratio of the tallest to second-tallest
FOD peak amplitude, the absolute value of the peak amplitude is
effectively ignored. This may or may not be considered problematic,
for either small or large FOD amplitudes. If the peak amplitude / AFD
is smaller than that of other voxels, it’s possible that this voxel
experiences partial volume with CSF: this may satisfy the peak ratio
requirement, but using such a voxel is not ideal in response function
estimation as its noise level will be higher and the Rician noise
bias will be different. Conversely, both in certain regions of the
brain and in some pathologies, some voxels can appear where the AFD
is much higher due to T2 shine-through; it may seem appealing to use
such voxels in response function estimation as the SNR is higher, but
as for the low-signal case, the Rician noise bias will be different
to that in the rest of the brain. The previous
`dwi2response`

binary attempted to exclude such voxels by looking at the mean and standard deviation of AFD within the single-fibre mask, and excluding voxels above or below a certain threshold. As before, while this heuristic may or may not seem appropriate depending on your point of view, it has been excluded from the new`dwi2response`

script to keep things as simple as possible.

`tournier`

¶

Independently and in parallel, Donald also developed a newer method for
response function estimation based on CSD itself; it was used in this
manuscript. It bears some
resemblance to the `tax`

algorithm, but relies on a threshold on the
number of voxels in the single-fibre mask, rather than the ratio between
tallest and second-tallest peaks. The operation is as follows:

- Define an initial response function that is as sharp as possible
(ideally a flat disk, but will be fatter due to spherical harmonic
truncation). Limit this initial function to
*lmax=4*, as this makes the FODs less noisy in the first iteration. - Run CSD for all voxels within the mask (initially, this is the whole brain).
- Select the 300 ‘best’ single-fibre voxels. This is not precisely the
ratio between tallest and second-tallest peaks; instead, the
following equation is used, which also biases toward selection of
voxels where the tallest FOD peak is larger:
`sqrt(|peak1|) * (1 - |peak2| / |peak1|)^2`

. Use these voxels to generate a new response fuction. - Test to see if the selection of single-fibre voxels has changed; if not, the script is completed.
- Derive a mask of voxels to test in the next iteration. This is the top 3,000 voxels according to the equation above, and dilated by one voxel.
- Go back to step 2.

This approach appears to be giving reasonable results for the datasets on which it has been tested. However if you are involved in the processing of non-human brain images in particular, you may need to experiment with the number of single-fibre voxels as the white matter is typically smaller.

## TL;DR¶

If this document appears far too long for your liking, or you’re not particularly interested in the details and just want to know what option to use so that you can continue with your processing, the following are our ‘cautious’ recommendations. However we emphasize that this script may not work flawlessly for all data; if it did, we wouldn’t be providing a script with so many options! Furthermore, these recommendations may change over time, so check in every now and then, and make sure to sign up to the new community forum.

- If you’re processing single-shell data, the
`tournier`

algorithm appears to be fairly robust. - If you’re processing multi-shell data, and are able to perform EPI
inhomogeneity distortion correction,
`msmt_5tt`

is currently the only fully-automated method for getting multi-shell multi-tissue response functions.