Umberto Villani^{1,2}, Simona Schiavi^{3}, and Alessandra Bertoldo^{1,2}

^{1}Padova Neuroscience Center, University of Padova, Padova, Italy, ^{2}Department of Information Engineering, University of Padova, Padova, Italy, ^{3}Department of Computer Science, University of Verona, Verona, Italy

Generalized sensitivity functions provide insights about which temporal observations are most informative for the estimation of biological model parameters. We formulate the same concept in the dMRI field to investigate how biophysical models/data representations react to HARDI acquisitions of different b-values. This approach handily shows how different parameters feature enhanced estimation precision at different b-values and exposes potential correlations between them, shedding light on possible a posteriori identifiability issues. Requiring only byproducts of standard optimization routines, generalized sensitivity functions can easily be integrated in standard analyses when proposing either a new model or a modification of existing ones.

$$ \mathbf{GS}(b_{J})=\sum_{j=1}^{J}{\left[\sum_{m=1}^{M}{\mathbf{I_\theta}}{(b_{m})}\right]^{-1} \mathbf{I_\theta}}({b_{j})}\hphantom{aaaaaaah}[1] $$

where $$$ \mathbf{GS} $$$ is the Generalized Sensitivity matrix, $$$ M $$$ is the length of the b-value vector $$$\mathbf{b}=[b_{1},b_{2},...,b_{M}] $$$, and $$$ \mathbf{I_\theta} $$$ is the shell-wise Fisher Information matrix, whose exact formulation depends on the biophysical model and the noise probability density function. For the most common Gaussian case, $$$ \mathbf{I_\theta} $$$ becomes

$$ (\mathbf{I_{\theta}})_{i,j}(b)=\sum_{\mathbf{g}\in\mathbf{G}}{\frac{1}{\sigma^{2}(b)}}\mathbf{\nabla}_{\theta}[\tilde{S}(b,\mathbf{g},\theta_{i})]\mathbf{\nabla}_{\theta}[\tilde{S}(b,\mathbf{g},\theta_{i})]\hphantom{aaaaaaah}[2] $$

where $$$\mathbf{G}$$$ is the set containing all the diffusion gradient directions, $$$ \nabla_{\theta} $$$ denotes the gradient operator with respect to the model parameters and $$$\tilde{S}$$$ is the studied biophysical model prediction. The generalized sensitivity functions for the model parameters are then extracted from the GS matrix in [1]:

$$ \mathbf{gsf_{\theta}}(b_{J}) =diag(\mathbf{GS}(b_{J}))\hphantom{aaaaaaah}[3] $$

As GSFs are cumulative functions, the most informative shells are associated with the steepest increase of their trajectories

We demonstrate the usefulness of GSFs by providing their visualization for the DTI

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Table 1: dMRI models and corresponding estimators used for GSFs
generation. Different estimators may imply a different formulation for the
Fisher Information matrix. LLS: Linear Least Squares NNL: Non-Linear Least
Squares ML: Maximum Likelihood.

Figure 1: Generalized sensitivity functions for the six parameters of
the DTI model. The [0-1] interval has been highlighted to visualize possible overshoots/undershoots
due to parameter correlations.

Figure 2: Generalized sensitivity functions for the twenty-one
parameters of the DKI model. The [0-1] interval has been highlighted to visualize possible overshoots/undershoots
due to parameter correlations.

Figure 3: Generalized sensitivity functions for the three scalar
parameters of the NODDI model. The [0-1] interval has been highlighted to visualize possible overshoots/undershoots
due to parameter correlations.