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Towards a Standard Model of Diffusion in White Matter with Phase and Relaxation – A Monte-Carlo Study

Anders Dyhr Sandgaard¹, Valerij G. Kiselev², Noam Shemesh³, and Sune Nørhøj Jespersen^1,4
¹Center of Functionally Integrative Neuroscience, Department of Clinical Medicine, Aarhus University, Aarhus, Denmark, ²Division of Medical Physics, Department of Radiology, University Medical Center Freiburg, Freiburg, Germany, ³Champalimaud Research,Champalimaud Centre for the Unknown, Lisbon, Portugal, ⁴Department of Physics and Astronomy, Aarhus University, Aarhus, Denmark

Synopsis

Keywords: Microstructure, Microstructure

Motivation: Improving parameter estimation for the standard model of diffusion in white matter (SM) by modelling the subsequent decay of the spin-echo of the dMRI signal.

Goal(s): To numerically validate SMPR - an extension of SM incorporating orientation-dependent, susceptibility related relaxation rates and Larmor frequency shifts of the spin echo decay of the dMRI signal.

Approach: To perform Monte-Carlo (MC) simulations in orientationally dispersed, non-exchanging bundles of hollow magnetized cylinders, simulate a standard PGSE signal and its spin echo decay, and compare against the SMPR model.

Results: SMPR is in agreement with the MC simulations in both phase and signal magnitude.

Impact: Orientation-dependent susceptibility effects may improve parameter estimation of the Standard Model of diffusion in white matter and enable rotation-free mapping of susceptibility-related parameters.

Introduction

Estimating the parameters of the Standard Model (SM) of diffusion in white matter (WM) can be ill-posed^1-4. We have previously proposed to incorporate orientation dependent susceptibility effects to SM to improve fitting by modelling the attenuation of a PGSE signal⁵ – the SMPR model. Simulations have so far demonstrated that SMPR may improve parameter estimation and aid in estimation of susceptibility parameters without having to rotate the sample. Here we extend the investigation by examining the accuracy of SMPR against Monte-Carlo simulations of a PGSE experiment with additional sampling of its spin echo decay (figure 1). This is done using a digital phantom consisting of non-communicating bundles of magnetized, hollow cylinders.

Methods

Theory
We consider the extended SM – SMPR – with diffusion weighting $$$(b,\hat{\mathbf{g}})$$$, echo time $$$T_E$$$ and additional sampling of the spin echo decay at times $$$\Delta{T_E}$$$. Figure 1 gives an overview of the signal $$$S(b,\hat{\mathbf{g}};T_E,\Delta{T_E},{\mathbf{B}}_0)$$$ and the added compartmental transverse relaxations rates^6-9 $$$R_{2i},R_{2i}^*$$$ and Larmor frequency shifts^10-14$$$\overline\Omega_i$$$ for $$$i=e,a$$$ as defined in figure 1.

Monte-Carlo simulations
We performed MC simulations to validate SMPR (figure 2). The details are listed in figure 2A. We synthesized a digital phantom consisting of non-exchanging bundles of magnetized hollow cylinders resembling an ideal SM application. As a first step, we considered only cylinders with scalar susceptibility, and without spherical sources such that $$$\overline\Omega=\overline\Omega_e=\overline\Omega_a$$$. The bundle’s orientation distribution $$$\mathcal{P}(\hat{\mathbf{n}})$$$ (fODF) consists of 10 polar rings and 20 azimuthal points in each ring with fODF symmetry axis along $$$\hat{\mathbf{B}}_0)$$$. For this we calculated the signal kernel $$$\mathcal{K}(b,\hat{\mathbf{g}}\cdot\hat{\mathbf{n}};T_E,\Delta{T_E},\hat{\mathbf{B}}_0\cdot\hat{\mathbf{g}})$$$, as described in figure 2B including acquisition details. The SMPR signal $$$S(b,\hat{\mathbf{g}};T_E,\Delta{T_E},{\mathbf{B}}_0)$$$ for a given level of orientation dispersion was made by summing $$$\mathcal{K}(b,\hat{\mathbf{g}}\cdot\hat{\mathbf{n}};T_E,\Delta{T_E},\hat{\mathbf{B}}_0\cdot\hat{\mathbf{g}})$$$ from bundles up to a maximum polar angle $$$\theta_C$$$ (cf. figure 1A).

Results

Figure 3A shows the signal magnitude for a given $$$T_E,\Delta{T_E}$$$ for varying levels of orientation dispersion, b-values and gradient directions, and figure 3B shows the overall Larmor frequency shift $$$\overline\Omega$$$. We find SMPR agrees with MC in all cases. The orientation dependence of $$$\overline\Omega$$$ depends on the amount of dispersion and angle between the fODF symmetry axis and $$$\hat{\mathbf{B}}_0$$$. $$$\overline\Omega$$$ varies little for low levels of dispersion since $$$\hat{\mathbf{B}}_0$$$ is parallel to the fODF symmetry axis, and because we only included a single weak bulk susceptibility, 0.05ppm.

Discussion

Our simulations demonstrate that SMPR describes the spin echo decay of the PGSE signal for an ideal digital SM² phantom well. Here we considered linear diffusion encoding, which unfortunately reduces the magnitude of the measured phase because the convolution $$$\mathcal{P}\otimes\mathcal{K}$$$ (cf. figure 1) is roughly taken over a plane perpendicular to the direction of the strong diffusion gradient. Planar encoding^15,16 would restore the phase towards the range of values one would measure in a normal gradient echo as $$$\mathcal{P}\otimes\mathcal{K}$$$ is then centered around one direction instead of a plane. However, planar encoding comes at the expense of lower SNR¹⁷. The optimal compromise will be investigated in the future. We also plan to include susceptibility anisotropy of myelin and spherical inclusions. However, anisotropy of myelin will not affect the functional form of the relaxation rate and Larmor frequencies^18,19, as it only adds an additional constant frequency shift in the intra-axonal space. Our phantom was fairly simple and constructed to be in agreement with the model assumptions, in order to test if the model did not leave out any important signal contributions, e.g., from 3rd and 4th order cumulants²⁰. We therefore plan to test SMPR against MC simulations in more realistic phantoms including segmented WM from 3D electron microscopy²¹.

Conclusion

We showed that our analytical SMPR model - incorporating orientation dependent susceptibility effects into the standard model of diffusion in white matter – agrees with Monte-Carlo simulations of PGSE in a phantom of dispersed cylinder bundles. We believe our results bring us one step closer to better SM parameter estimation and further enable susceptibility properties of WM to be measured without impractical sample rotations.

Acknowledgements

This study is funded by the Independent Research Fund (grant 10.46540/3103-00144B)

References

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Figures

Figure 1 - Overview of SPMR signal and its parameters. R2*, R2 and Ω are orientation dependent and described by rotation invariants, A, A*,B,B*,α,β, while γ(B) is not invariant as it contains all orientation dependent macroscopic frequency shifts. SMPR is similar to SM in terms of diffusion but includes frequency shifts and relaxation due to the susceptibility of the myelin sheath and iron-containing cells.

Figure 2 - A: Diffusion was simulated in bundles of parallel hollow cylinders (colored by orientation). Each bundle was used to compute 5 different orientations. Summing the signal from bundles allowed us to vary the orientation dispersion. B: Overview of the generated PGSE signal with sampling of the spin echo decay for a single bundle. T_E in magenta is 70 ms, and the ΔT_E in green happens at 70,71,…,80 ms. Red indicates the diffusion gradients²², and blue the RF pulses, where a 180 degree flip occurs at 35 ms.

Figure 3 - Simulated and model predicted signals and Larmor frequency shifts: A shows the signal magnitude for different dispersion levels θ_C (titles), the x-axis indicates gradient directions with increasing polar angle to symmetry axis of the cylinder distribution, and colors b-values. B shows the Larmor frequency shifts in a similar manner. Uniform dispersion leads to a fully linear dependence on (g·B₀)² as all orientation to B₀ are sampled evenly.

Proc. Intl. Soc. Mag. Reson. Med. 32 (2024)

0589

DOI: https://doi.org/10.58530/2024/0589