# Response function estimation

A prerequisite for spherical deconvolution is obtaining the response function(s), which is/are used as the kernel(s) by the deconvolution algorithm. For the white matter, the response function models the signal expected for a voxel containing a single, coherently oriented bundle of axons [Tournier2004] [Tournier2007]. In case of multi-tissue variants of spherical deconvolution, response functions for other tissue types are introduced as well; typically to represent grey matter(-like) and/or CSF(-like) signals [Jeurissen2014] [Dhollander2016a].

In MRtrix3, the dwi2response script offers a range of algorithms
to estimate these response function(s) directly from your dataset itself.
This process of estimating response function(s) from the data is
non-trivial. No single algorithm works for *any* possible scenario,
although some have proven to be more widely applicable than others.

## General recommendations

### Choice of algorithm

While many algorithms exist, the following appear to perform well in a wide range of scenarios, based on experience and testing from both developers and the MRtrix3 community:

**Single-tissue CSD:** If you intend to perform (single-tissue)
Constrained spherical deconvolution (e.g. via `dwi2fod csd`

),
the tournier algorithm is a convenient and reliable way to estimate
the single-fibre white matter response function:

```
dwi2response tournier dwi.mif wm_response.txt
```

Other options include the fa or tax algorithms.

**Multi-tissue CSD or global tractography:** If you intend to perform a
*multi-tissue* analysis, such as Multi-shell multi-tissue constrained spherical deconvolution (e.g. via ```
dwi2fod
msmt_csd
```

) or Global tractography (e.g. via `tckglobal`

), the
dhollander algorithm is a convenient and reliable way to estimate the
single-fibre white matter response function as well as the grey matter and
CSF response functions:

```
dwi2response dhollander dwi.mif wm_response.txt gm_response.txt csf_response.txt
```

Other options include the msmt_5tt algorithm.

### Checking the results

In general, it’s always worthwhile checking your response function(s):

```
shview wm_response.txt
```

Use the left and right arrow (keyboard) keys in this viewer to switch between the different b-values (‘shells’) of the response function, if it has more than one b-value (this would for example be the case for the outputs of the dhollander algorithm).

It may also be helpful to check which voxels were selected by the
algorithm to estimate the response function(s) from. For any
dwi2response algorithm, this can be done by adding the `-voxels`

option, which outputs an image of these voxels. For example, for
the tournier algorithm:

```
dwi2response tournier dwi.mif wm_response.txt -voxels voxels.mif
```

The resulting `voxels.mif`

image can be overlaid on the `dwi.mif`

dataset using the mrview image viewer for further inspection.

## Available algorithms

The available algorithms differ in a few general properties, related to what they deliver (as output) and require (as input), notably

**single- versus multi-tissue**: whether they only estimate a single-fibre white matter response function (tournier, tax and fa) or also additional response functions for other tissue types (dhollander and msmt_5tt both output a single-fibre white matter response function as well as grey matter and CSF response functions)**single versus multiple b-values**: whether they only output response function(s) for a single b-value (tournier, tax and fa) or for all—or a selection of— b-values (dhollander and msmt_5tt)**input requirements**: whether they only require the DWI dataset as input (tournier, dhollander, tax and fa) or also additional input(s) (msmt_5tt requires a 5TT segmentation from a spatially aligned anatomical image)

Beyond these general categories, the algorithms differ mostly in the actual strategy used to determine the voxels that will be used to estimate the response function(s) from.

The manual choice is an exception to most of the above, in that it
allows/*requires* you to provide the voxels yourself, and even allows
you to provide single-fibre orientations manually as well. It should
only be considered in case of exceptional kinds of data, or otherwise
exceptional requirements. Caution is advised with respect to *interpretation*
of spherical deconvolution results using manually defined response
function(s).

The following sections provide more details on each algorithm specifically.

### dhollander

This algorithm is the official implementation of the strategy proposed
in [Dhollander2016b] (including improvements proposed in [Dhollander2019])
to estimate multi b-value (single-shell + b=0, or multi-shell) response functions
for single-fibre white matter (*anisotropic*), grey matter and CSF (both
*isotropic*), which can subsequently be used for multi-tissue (constrained)
spherical deconvolution algorithms. It has the distinct advantage of requiring
*only* the DWI data as input, in contrast to other multi-tissue response function
estimation methods, making it the simplest and most accessible method, and a
sensible default for applications that require multi-tissue responses.

This is a fully automated unsupervised algorithm that leverages the relative diffusion properties of the 3 tissue response functions with respect to each other, across all b-values and the angular domain, to select the most appropriate voxels from which to estimate the response functions. It has been used successfully in a wide range of conditions (overall data quality, pathology, developmental state of the subjects, animal data and ex-vivo data). Additional insights into its performance are presented in [Dhollander2018a]. Due to its ability to deal with the presence of extensive white matter (hyperintense) lesions, it was for example also successfully used in [Mito2018a]. The response functions as obtained in this particular way also form the basis of the 3-tissue framework to study the microstructure of lesions and other pathology [Dhollander2017] [Mito2018b].

The algorithm has been further improved in [Dhollander2019]. While the 2016 version identified the voxels to estimate the single-fibre white matter response function using the tournier algorithm, the new 2019 version relies on a novel strategy that optimises these voxels using properties of the signal across all b-values (and the full angular domain). It’s also faster than the original approach.

In almost all cases, the algorithm runs and performs well out of the box.
In *exceptional* cases where the anisotropy in the data is particularly *low*
(*very* early development, ex-vivo data, (with) low b-value, …), it is *sometimes*
advisable to set the `-fa`

parameter *lower* than its default value of 0.2.
See [Dhollander2018b] for a good example of a dataset where changing this
parameter was required to obtain good results. This FA threshold should be set
so as to roughly separate the bulk of WM from the rest (GM and CSF). Further
imperfections are corrected by the algorithm itself during a later stage.

As always, check the `-voxels`

option output in unusually (challenging) cases.

For more information, refer to the dhollander algorithm documentation.

### fa

This algorithm is an implementation of the strategy proposed in [Tournier2013] to estimate a single b-value (single-shell) response function of single-fibre white matter, which can subsequently be used for single-tissue (constrained) spherical deconvolution. The algorithm estimates this response function from the 300 voxels with the highest FA value in an eroded brain mask. There are also options to change this number or provide an absolute FA threshold.

Due to relying *only* on FA values, this strategy is relatively
limited in its abilities to select the best voxels. In white matter
close to CSF, for example, Gibbs ringing can affect FA values.
More advanced iterative strategies, such as the tournier and tax
algorithms have been proposed more recently.

For more information, refer to the fa algorithm documentation.

### manual

This algorithm is provided for cases where none of the available automated 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).

For more information, refer to the manual algorithm documentation.

### msmt_5tt

This algorithm is a reimplementation of the strategy proposed in
[Jeurissen2014] to estimate multi b-value response functions of single-fibre
white matter (*anisotropic*), grey matter and CSF (both *isotropic*), which can
subsequently be used for multi-tissue (constrained) spherical deconvolution.
The algorithm is primarily driven by a prior (The 5TT format) tissue segmentation,
typically obtained from a spatially aligned anatomical image. This also
requires prior correction for susceptibility-induced (EPI) distortions of the
DWI dataset. The algorithm selects voxels with a segmentation partial volume of
at least 0.95 for each tissue type. Grey matter and CSF are further
constrained by an (upper) 0.2 FA threshold. Single-fibre voxels within the WM
segment are then extracted using the tournier algorithm (in contrast
to original publication, see Replicating original publications below).

The input tissue segmentation can be estimated using the same pre-processing pipeline as required for Anatomically-Constrained Tractography (ACT), namely: correction for motion and (EPI and other) distortions present in the diffusion MR data, registration of the structural to (corrected) EPI data, and spatial segmentation of the anatomical image. This process is therefore also dependent on the accuracy of each of these steps, so that the T1 image can be reliably used to select pure-tissue voxels in the DWI volumes. Failure to achieve high accuracy for each of these individual steps may result in inappropriate voxels being used for response function estimation, with concomitant errors in tissue estimates.

The dhollander algorithm does not rely on a number of these steps. A comparison is presented in [Dhollander2018a].

For further information, refer to the msmt_5tt algorithm documentation.

### tax

This algorithm is a reimplementation of the iterative approach proposed in [Tax2014] to estimate a single b-value (single-shell) response function of single-fibre white matter, which can subsequently be used for single-tissue (constrained) spherical deconvolution. The algorithm iterates between performing CSD and estimating a response function from all voxels detected as being ‘single-fibre’ from the CSD result itself. The criterion for a voxel to be ‘single-fibre’ is based on the ratio of the amplitude of second tallest to the tallest peak. The method is initialised with a ‘fat’ response function; i.e., a response function that is safely deemed to be much less ‘sharp’ than the true response function.

This algorithm has occasionally been found to be unstable and converge towards suboptimal solutions. The tournier algorithm has been engineered with the intention to overcome some of the issues believed to be the cause of these instabilities (see some discussion on this topic here and here).

For more information, refer to the tax algorithm documentation.

### tournier

This algorithm is a reimplementation of the iterative approach proposed in [Tournier2013] to estimate a single b-value (single-shell) response function of single-fibre white matter, which can subsequently be used for single-tissue (constrained) spherical deconvolution. The algorithm iterates between performing CSD and estimating a response function from a set of the best ‘single-fibre’ voxels, as detected from the CSD result itself. Notable differences between this implementation and the algorithm described in [Tournier2013] include:

This implementation is initialised by a sharp lmax=4 response function as opposed to one estimated from the 300 brain voxels with the highest FA.

This implementation uses a more complex metric to measure how ‘single-fibre’ FODs are: √|peak1| × (1 − |peak2| / |peak1|)², as opposed to a simple ratio of the two tallest peaks. This new metric has a bias towards FODs with a larger tallest peak, to avoid favouring small, yet low SNR, FODs.

This implementation only performs CSD on the 3000 best ‘single-fibre’ voxels (of the previous iteration) at each iteration.

While the tournier algorithm has a similar iterative structure as the tax algorithm, it was adjusted with the intention to overcome some occasional instabilities and suboptimal solutions resulting from the latter. Notable differences between the tournier and tax algorithms include:

The tournier algorithm is initialised by a

*sharp*(lmax=4) response function, while the tax algorithm is initialised by a*fat*response function.This implementation of the tournier algorithm uses a more complex metric to measure how ‘single-fibre’ FODs are (see above), while the tax algorithm uses a simple ratio of the two tallest peaks.

The tournier algorithm estimates the response function at each iteration only from the 300

*best*‘single-fibre’ voxels, while the tax algorithm uses*all*‘single-fibre’ voxels.

Due to these differences, the tournier algorithm is currently believed to be more robust in a wider range of scenarios (for further information on this topic, refer to some of the discussions here and here).

For more information, refer to the tournier algorithm documentation.

## Replicating original publications

For completeness, we provide below instructions for replicating the approaches
used in previous relevant publications. Note that the implementations provided
below are not necessarily *exactly* as published, but aim to be close
approximations nonetheless.

### Spherical deconvolution and Constrained spherical deconvolution

In the original spherical deconvolution [Tournier2004] and constrained spherical deconvolution [Tournier2007] papers, the response function was estimated by extracting the 300 voxels with the highest FA values within a brain mask, eroded to avoid noisy voxels near the edge of the brain. This can be performed using the fa method directly:

```
dwi2response fa dwi.mif response.txt
```

where:

`dwi.mif`

is the input DWI data set,`response.txt`

is the estimated response function, produced as output

### MSMT-CSD and Global tractography

In the original multi-shell multi-tissue CSD [Jeurissen2014] and global
tractography [Christiaens2015] papers, response functions were estimated using
a prior tissue segmentation obtained from a coregistered structural T1 scan.
For the WM response, a further hard FA threshold was used: respectively 0.7 in
the MSMT-CSD paper and 0.75 in the global tractography paper. This pipeline can be
replicated using the 5ttgen command and msmt_5tt algorithm with
the `-sfwm_fa_threshold`

option in this fashion:

```
5ttgen fsl T1.mif 5tt.mif
dwi2response msmt_5tt dwi.mif 5tt.mif wm_response.txt gm_response.txt csf_response.txt -sfwm_fa_threshold 0.7
```

where:

`T1.mif`

is a coregistered T1 data set from the same subject (input)`5tt.mif`

is the resulting tissue type segmentation, used subsequently used in the response function estimation (output/input)`dwi.mif`

is the same dwi data set as used above (input)`<tissue>_response.txt`

is the tissue-specific response function as used above (output)

To replicate the global tractography paper, specify a value of 0.75 instead of 0.7 as shown in the command line above.