Fast quantification of uncertainty in non-linear diffusion MRI models for artifact detection and more power in group studies

Diffusion MRI (dMRI) allows for non-invasive investigation of brain tissue microstructure. By fitting a model to the dMRI signal, various quantitative measures can be derived from the data, such as fractional anisotropy, neurite density and axonal radii maps. The uncertainty in these dMRI measures is often ignored, while previous work in functional MRI has shown that incorporating uncertainty estimates can lead to group statistics with a higher statistical power. We propose the Fisher Information Matrix (FIM) as a generally applicable method for quantifying the parameter uncertainties in non-linear diffusion MRI models. In direct comparison with Markov Chain Monte Carlo sampling, the FIM produces similar uncertainty estimates at lower computational cost. Using acquired and simulated data, we then list several characteristics that influence the parameter variances, like data complexity and signal-to-noise ratio. In individual subjects, the parameter standard deviations can help in detecting white matter artifacts as patches of relatively large standard deviations. In group statistics, we recommend using the parameter standard deviations by means of variance weighted averaging. Doing so can reduce the overall variance in group statistics and reduce the effect of data artifacts without discarding data from the analysis. Both these effects can lead to a higher statistical power in group studies.

FIM can be interpreted as an approximation to the CRLB, we follow the results in astrophysics and only interpret the FIM as a measure of uncertainty around the estimated parameters (Vallisneri, 2008).
an asymptotic estimator of the covariance matrix (Pawitan, 2013;Gelman et al., 2013). Formally, let l(x) be a log-likelihood function with maximum 117 likelihood estimatex. A second order Taylor approximation of l(x) cen-118 tered atx is then given by: ignoring the higher terms and having dropped the linear term since the 120 first derivative of a function is zero at the mode. Considering the first term, 121 l(x), a constant and the second term, 1 2 (x −x) T ∂ 2 ∂x 2 l(x)(x −x), proportional 122 to the logarithm of a normal density, we get the approximation: where I(x) is the observed Fisher Information Matrix: For the Hessian to be positive definite, this theory requiresx to lie within with J f the Jacobian matrix of f . More succinctly, the covariance matrix of y = f (θ) is given by: which holds as a generally applicable formula for linear propagation of co-143 variances (Arras, 1998). In the case of an univariate output y = f (θ), the 144 Jacobian can be formulated as a gradient vector ∇ f , leading to the follow- 145 ing expression for the variance in y: and a weighted standard deviation as: with m for the number of non-zero weights, included here to allow for non-166 normalized weights. It has been shown that the weights that minimize the 167 variance of the weighted average are the reciprocals of the variances of each 168 of the data points z i (Shahar, 2017). That is, given the variances σ 2 i for each 169 z i , the weights that minimize Var( i w i z i ) is given by: Incidentally, these weights are also the maximum likelihood estimator of 171 the weighted mean and variance under the assumption that the data points 172 z i are independent and normally distributed with the same mean (Cochran,    ing and weighted averaging over these remaining 30 subjects.

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As a comparison method between regular and weighted averaging we com-317 puted (µ weighted − µ regular )/µ regular and (σ weighted − σ regular )/σ regular as dif-318 ference measure for the mean and standard deviation estimates between 319 regular and weighted averaging.

Effect of SNR on parameter variances
Lower SNR per data point (i.e. single diffusion volume) is expected to lead 399 to higher uncertainty in fitted parameter estimates. This issue is of extra im-400 portance in brain dMRI by the fact that SNR is non-uniform over the brain,         presence of what appears to be a lower-expressed artifact in the remaining 578 subjects. Due to this mechanism, subjects no longer need to be excluded 579 from analysis, thereby improving the power of one's study.

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Near the gray-white matter border we noticed some voxels where weighted 581 averaging provides a higher variance than regular averaging. We note that although weighted averaging is shown here over subjects, The analytical gradient of this function is given by: The covariance matrix of the weights can be defined as: with σ 2 w i denoting the variance of weight w i , and σ w i w j denoting the co- The derivative of FA with respect to the first diffusivity can be written as: and similar derivatives can be derived for the second and third diffusivity The covariance matrix of the diffusivities can be defined as: with σ 2 d i denoting the variance of diffusivity d i , and σ d i d j denoting the co-