DTI cone of uncertainty and HARDI equivalent

Hi there,

can someone of you tell me whether, using MRtrix, I can get any info about the uncertainty associated to the main eigenvector from tensor reconstruction? If so, is there a way to visualize it, e.g. using cones?

Besides, is there a way to get the same information when using CSD? Are there uncertainties associated with each of them main directions extracted for each voxel?

Thanks in advance for any help.



I’m afraid not. It would technically be possible to do: the tensor_prob algorithm in tckgen uses the wild bootstrap approach to obtain random samples from this distribution, it wouldn’t take a monumental amount of work to create a new app using this algorithm to derive estimates of uncertainty. But it’s not been done, and I don’t anticipate any of us will find the time to do this any time soon. Happy to assist anyone who does want to take this on, though.

No, there isn’t. Probably the simplest would be to map the uncertainty to a spherical harmonic representation, which could then be displayed trivially. Otherwise you might be able to use the vector plot tool to display a number of random sample orientations, which would also give you a representation of the uncertainty - although not as pretty as cones, admittedly…

This one is a bit more difficult to answer. If you’re talking about the uncertainty in the peak orientation, the way to do this would be to use something like the residual bootstrap approach taken in Ben Jeurissen’s tractography algorithm. But this isn’t implemented in MRtrix…

However, CSD is designed to estimate the distribution of fibre orientations, which in my opinion will almost invariably be broader than the uncertainty in the peak orientation. It’s now widely accepted that there is a fair amount of dispersion in the fibre directions that we see in vivo, and that introduces a spread in the fibre orientation distribution that is intrinsic to the data. Any attempt to sharpen that distribution (e.g. by taking the peak) implicitly assumes that there is no dispersion, which I don’t think is defensible - unless you are very clear that what you’re after is the uncertainty in the mean orientation, not the uncertainty in the individual fibres themselves.

So the way I see it, yes there is uncertainty in the fibre orientations, but the dominant source is probably the inherent dispersion in the fibres themselves - not the noise (for a decent quality dataset, that is). And the way I see it, that uncertainty is already included in the CSD output, since we are estimating the distribution of fibre orientations.

That said, the choice of lmax does impact on the angular resolution of the FOD, and hence the apparent uncertainty. Based on the results of my earlier NMR Biomed paper, I reckon lmax 8 is about right here: it looks like increasing the b-value beyond 3000 doesn’t increase the sharpness of the DW signal, so we’ve probably hit the limit of the inherent dispersion (although I appreciate this is somewhat speculative and open to debate). At this point lmax=8 is about the highest harmonic term we can realistically measure. Which is why I reckon lmax 8 is roughly the right level to capture the uncertainty in the fibre orientations. Bearing in mind of course that this is based on data acquired at relatively low resolution (3mm) on older hardware - there’s a chance newer scanners might contradict my naïve logic here…

Thanks for your clear and detailed explanation.

By reading your NMR Biomed paper I got almost all the answers I was seeking, as my aim is to find the best b-value and number of DW directions using a 7T scanner. I thought to find them by investigating on uncertainties but, now I’m wondering whether your findings could be also extended to a 7T scanner, i.e. whether I can find the answer by evaluating the SNR of my signal and using the “power vs SNR” plots you found at different b-values to see the corresponding power of each SH term (let’s say at b=3000 s/mm2) in order to select the best b-value and the minimum (plus 2 or 3) number of directions corresponding to the higher SH term is worth to be evaluated.

By the way, what do you mean by “overall effective SNR”? How did you evaluate that?

Thanks in advance and I apologize if my question is a bit out of the original subject of this topic.

What I mean is that as long as the number of measurements is larger than the number of coefficients to be estimated (and assuming directions are uniformly distributed), then there should be no bias in the estimated coefficients due to undersampling (i.e. aliasing). Beyond this point, additional measurements essentially contribute to the SNR of the estimated coefficients (this is what I refer to as the effective SNR), and that contribution should scale in the typical sqrt(N) fashion.

So in other words, what I mean by effective SNR is the SNR in the estimated SH coefficients, rather than in the raw images themselves. Does that make sense…?