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Biophysical modeling of the white matter: from theory towards clinical practice
Rafael Neto Henriques1, Chantal M. W. Tax2, Noam Shemesh3, and Jelle Veraart1,4

1Champalimaud Research, Champalimaud Centre for the Unknown, Lisbon, Portugal, 2CUBRIC, School of Psychology, Cardiff University, Cardiff, United Kingdom, 3Champalimaud Centre for the Unknown, Lisbon, Portugal, 4iMinds - Vision Lab, University of Antwerp, Antwerp, Belgium

Synopsis

We address the degeneracy of the diffusion standard model and improve the precision and accuracy in parameter estimation, thus promoting clinical applicability. Acquisition of additional data is not required; instead, we introduce a more robust and accurate estimator by fitting the standard model directly to diffusion-weighted data rather than its rotational invariants. We are able to overcome the implicit assumption of data being shelled in terms of b-values. This enables the correction of gradient nonlinearities to avoid biases in model parameter estimation, whereas revising the optimal experimental design demonstrates that non-shelled encoding schemes are favorable in terms of achievable precision.

Introduction

The development of biophysical models aiming to relate diffusion MRI signal to brain tissue microstructure has seen an exponential growth in recent years. However, there is a rising awareness of the approach’s fundamental limitations1. A low precision, accuracy, and robustness of the parameters estimators of even the simplest biophysical models2 currently limit the clinical applicability of the developed techniques3,4. Factoring out the fiber orientation distribution function (fODF) using rotationally invariant metrics has reduced the complexity of the modeling problem, but in turn uncovered an intrinsic degeneracy in microstructural parameter estimation3. Moreover, such “powder averaging” approaches5,6,7 entail an implicit assumption that diffusion MRI data are perfectly shelled, i.e. different gradient directions for a finite set of $$$b$$$-values. However, this assumption is usually unmet due to gradient nonlinearities8 and poses an unnecessary constraint on experimental design9.

Theory

We here adopt the widely used two-compartment model of water inside narrow impermeable “sticks” representing axons10, embedded in an extra-cellular matrix, coined the diffusion Standard Model (SM)2,11:

$$S_{SM}(b,\hat{g})=\int d\hat{n}\mathcal{P}\left(\hat{n}\right)\mathcal{K}\left(b,\hat{g}\cdot\hat{n}\right)\;\;\;\;\left(1\right)$$

with

$$\mathcal{K}\left(b,\xi\right)=fe^{-bD_a\xi^2}+(1-f)e^{-bD_e^\perp-b\left(D_e^\parallel -D_e^\perp\right)\xi^2}\;\;\;\;\left(2\right)$$

being the signal response kernel of an individual fiber fascicle parameterized by the intra-axonal signal fraction $$$f$$$ and parallel diffusivity $$$D_a$$$, and the extra-axonal radial and axial diffusivities $$$D_e^\parallel$$$ and $$$D_e^\perp$$$, respectively. The fODF $$$\mathcal{P}\left(\hat{n}\right)$$$ is parameterized by its spherical harmonic coefficients $$$p_{lm}$$$, up to order $$$L$$$.

The $$$4+(L+1)(L+2)/2$$$ parameters can be estimated by minimizing the following non-linear object function: $$\|S(b,\hat{g})-S_{SM}(b,\hat{g})\|^2\;\;\;\;\left(3\right)$$

If data is acquired with several $$$b$$$-shells, this system can also be solved by first representing each $$$b$$$-shell by its spherical harmonic (SH) coefficients $$$S_{lm}(b)$$$:

$$\|S_{lm}(b)-p_{lm}K_l (b)\|^2\;\;\;\;\left(4\right)$$

with $$$K_l(b)$$$ the projection of $$$\mathcal{K}\left(b,\xi\right)$$$ onto Legendre polynomials3. The dimensionality of this optimization problem can be reduced by adopting the $$$l^{\mathrm{th}}$$$ rotational invariants of the signal and fODF, $$$S_l^2=\sum_{m=-l}^l|S_{lm}|^2/4\pi(2l+1)$$$ and $$$p_l^2=\sum_{m=-l}^l |p_{lm}|^2/4\pi (2l+1)$$$, respectively, thereby factoring out the full fODF3: $$\|S_{l}(b)-p_{l}K_l(b)\|^2\;\;\;\left(5\right).$$

Although all three fitting approaches, i.e. Eqs. (2), (3), and (4) are derived from the same standard model, we will demonstrate their robustness, accuracy, and precision is significantly different.

We will refer to the parameter estimators based on Eqs. (2), (3), and (4) as “SM”, “SM-SH”, and “SM-RotInv”, respectively. The maximal SH order $$$L$$$ is 6 unless stated differently.

Data

Simulations: Synthetic signals were produced by solving Eq. (1) with ground truth parameters $$$f\,=\,0.75,\,D_a\,=\,2.1\,\mathrm{\mu\,m^2/ms},\,D_e^\parallel\,=\,1.5\,\mathrm{\mu\,m^2/ms}$$$, and $$$D_e^\perp\,=\,0.5,\mathrm{\mu\,m^2/ms}$$$ and a crossing fiber geometry for $$$\mathcal{P}\left(\hat{n}\right)$$$ using the diffusion-weighted encoding scheme of the MR data. Gaussian noise was added (SNR$$$_{b=0}=50$$$) to evaluate the robustness, accuracy and precision of the different estimators.

MRI experiments: Four volunteers were scanned on a Connectom 3T MR scanner. Diffusion-weighting was applied along 30 gradient directions for $$$b = 1$$$ and $$$2\,\mathrm{ms/\mu\,m^2}$$$, and 60 gradient directions for $$$b\,=\,3,\,5,\,7,\,9,\,11,\,12.1,\,13.5$$$ and $$$15\,\mathrm{ms/\mu\,m^2}$$$. Following parameters were kept constant: $$$\Delta/\delta\,=\,30/13\,\mathrm{ms}$$$, $$$\mathrm{TR/TE}=3500/62\,\mathrm{ms}$$$ and resolution $$$3\,\times\,3\times\,3\,\,\mathrm{mm}^3$$$. Data was denoised12 and Gibbs-13, eddy current-14, and Rician bias-15 corrected prior to analyses. Moreover, the $$$b$$$$-values were corrected for spatially varying gradient nonlinearities16.

Results

Degeneracy: Simulations demonstrate that, unlike “SM-SH" and “SM-RotInv", “SM” does not show the notorious cluster of biophysically plausible, yet spurious solutions, which occur even at high $$$b$$$-values (Fig.$$$\,1$$$). The results are independent from the starting point (Fig. 2). Although the degeneracy is technically not resolved, the basin of attraction is strongly reduced, leaving an apparent lack of degeneracy. The findings are confirmed in the MR data (Fig. 3). Although fits were performed from random starting points, “SM” shows to produce smooth parametric maps, even for lower $$$b$$$-values, whereas ``SM-RotInv"-derived maps show speckled noise, indicating the degeneracy.

Accuracy: Uncorrected gradient-nonlinearities bias the shell-based “SM-RotInv” and “SM-SH”- estimators (Fig.$$$\,4a$$$). Some parameters, e.g. $$$D_e^\perp$$$, show errors up to $$$15\%$$$. These biases were also observed in the MR data by comparing “SM-RotInv” and “SM” estimates (Fig.$$$\,4b$$$).

Precision: Cramer-Rao lower bound (CRLB) analysis shows that non-shelled encoding schemes improve the precision of the parameter estimators, especially for more densely sampled low $$$b$$$-values (Fig. 5).

Discussion

We here show that the full “SM” fitting provides parameter estimates with higher:

  1. Robustness: no observed degeneracy in our simulated and clinical data, even if only relatively low $$$b$$$-values were selected;

  2. Accuracy: Correcting for gradient nonlinearities is required to avoid bias in the estimated model parameters, especially for the extra-axonal space, but prevents the use of estimated that rely on shelled data;

  3. Precision: Non-shelled gradient encodings might have a higher predicted precision, especially with a denser sampling of low b-values as shown in a CRLB analysis, 
 thereby promoting the clinical applicability of the standard model. The simultaneous estimation of the fODF and the microstructural kernel does not complicate the estimation problem, nor does it lower the precision of the parameter estimator. Instead, it enables fiber tractography without relying on arbitrarily chosen –often global– signal kernels.

Acknowledgements

JV is a Postdoctoral Fellow of the Research Foundation - Flanders (FWO; grant number 12S1615N). The data was acquired at the UK National Facility for In Vivo MR Imaging of Human Tissue Microstructure funded by the EPSRC (grant EP/M029778/1), and The Wolfson Foundation.

References

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Figures

Simulation results: (a) scatter plots of the model parameters estimates obtained by the three different fitting approaches; In each scatter plot, the ground truth values are marked by black stars. Simulations are corrupted with Gaussian noise (SNR$$$_{b=0}=50$$$) to avoid unrelated Rician effects and a random starting point initialized the fitting procedures. Biophysically implausible solutions are not shown. Different ground truth values had similar results. (b) The fODF’s were often estimated accurately, even if the model parameters were spurious. This highlights that the degeneracy is intrinsic to the kernel and that the fODF cannot distinguish between the correct and spurious solution.

Prevalence of biophysically implausible solutions (blue), correct solution (red), and spurious biophysically plausible solution (yellow) for a single noise realization of the simulated data when estimated 250 times, each time with a different random starting point. Unlike ``SM-SH" and ``SM-RotInv", ``SM" did not show the spurious solutions.

Parametric maps obtained from different standard model fitting approaches “SM-RotInv (left) and “SM” (middle). Fitting approaches were initialized by a random starting point and $$$b$$$-values were corrected for gradient nonlinearity for ``SM”. “SM-RotInv” cannot account for the variability in $$$b$$$-value after the correction. We also show the parametric maps for ``SM” with $$$b$$$-values limited to $$$b=7\,\mathrm{ms/\mu\,m^2}$$$ (right).

The effects of gradient non-linearities for both a) noise-free synthetic and b) in-vivo data. Gradient nonlinearities were simulated by introducing effective gradient directions $$$g^{eff} = [g_x (1+ \Delta G), g_y, g_x (1- \Delta G)]$$$. Fitting approaches were initialized by the ground truth values for the synthetic data and initialized to a random starting point for the MR data. We observed $$$\Delta G$$$’s in the order of $$$10\%$$$ for the Connectom scanner.

Precision of the “SM” estimates for different gradient encoding schemes: (a) schematic representation of four evaluated gradient encoding schemes. The exemplary schemes have equal number of 512 gradient directions, but different sampled $$$b$$$-values (blue=shelled, red=uniform, yellow= denser sampling of large $$$b$$$-values, and purple=denser sampling of low $$$b$$$-values); (b) CRLB of the “SM” parameter estimator for different gradient encoding schemes assuming Rician distributed data and (c) Standard deviations of the “SM” parameter estimates for different gradient encoding schemes obtained from simulations show that a shelled sampling is suboptimal in terms of achievable precision.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)
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