Synopsis

In this study we propose a theoretical model to estimate different microstructure indices from diffusion MRI and multi-echo T2 (MET2) data. The proposed estimation framework takes into account the common and complementary information provided by both modalities. While the MET2 data allow us to model the myelin compartment, the diffusion data enable us to better characterize the intra-axonal and extra-axonal compartments. Results from numerical experiments support the hypothesis that the new unified estimation is more accurate than the alternative approach based on the individual sequential fitting of both image modalities. The performance was stable for noise levels commonly found in clinical protocols.

INTRODUCTION

Different diffusion models have been proposed to estimate the axon volume fraction (AVF) in white matter using clinical protocols^1,2,3. However, due to the long echo time (TE) employed in diffusion experiments, the signal from myelin is very low and only the relative but not the absolute AVF can be estimated⁴. Recently, a new approach to combine the myelin volume fraction (MVF) obtained from magnetization transfer and the relative AVF computed from NODDI¹ was proposed⁵. In that methodology, the absolute AVF and the myelin g-ratio are estimated by fitting each dataset to its corresponding signal model⁵. Despite its advantages, there are some remaining methodological limitations that should be addressed. For instance, the nonlinear fitting of multi-compartmental diffusion models, such as NODDI, suffers from heavy bias and poor precision⁶. Besides, the current diffusion models for datasets acquired with a single TE do not include the differential T₂ relaxation times for all compartments. This can bias the volume fractions estimation in voxels with partial volume effects, such as those located in brain structures that border cerebrospinal fluid (CSF). In order to overcome these limitations, we propose a unified multi-modal model using diffusion MRI and multi-echo T₂ (MET2) data. It allows to estimate, throughout a joint fitting approach, the volume fractions, diffusivities and T₂ times of the intra-axonal, extra-axonal and myelin compartments.

METHODS

MET2 model: the multi-echo signal is modeled as a function of TE as:

$$ M(TE)=k_{1}N_{t}\left(f_{int}e^{-\frac{TE}{T_{2}^{int}}}+f_{ext}e^{-\frac{TE}{T_{2}^{ext}}}+f_{m}e^{-\frac{TE}{T_{2}^{m}}}+f_{csf}e^{-\frac{TE}{T_{2}^{csf}}}\right), \qquad [1] $$

where $$$k_{1}$$$ is a constant, $$$ N_{t} $$$ is the number of spins within the voxel, $$$ f_{int},f_{ext},f_{m},f_{csf} $$$ and $$$ T_{2}^{int},T_{2}^{ext},T_{2}^{m},T_{2}^{csf} $$$ are the absolute water fractions and T₂ relaxation times of the intra-axonal, extra-axonal, myelin and CSF compartments, respectively.

Diffusion MRI model: the diffusion signal is modeled using the Spherical Mean Technique³ (SMT). However, we extended the SMT framework in two aspects. First, we included a free water compartment to model separately the contributions from extra-axonal and CSF compartments. Second, we included the signal decay due to different T₂ relaxation times in different compartments:

$$ e(b)=k_{2}\acute{N_{t}}\left(\acute{f_{int}}e^{-\frac{\overline{TE}}{T_{2}^{int}}}e_{int}(b)+\acute{f_{ext}}e^{-\frac{\overline{TE}}{T_{2}^{ext}}}e_{ext}(b)+\acute{f_{csf}}e^{-\frac{\overline{TE}}{T_{2}^{csf}}}e_{csf}(b)\right), \qquad [2] $$

where $$$ e(b) $$$ is the spherical mean of the diffusion signal $$$ S(b) $$$ for a given b-value and $$$ e_{int},e_{ext},e_{csf} $$$ are the spherical means of the normalized signals coming from the intra-axonal, extra-axonal and CSF compartments, $$$k_{2}$$$ is a constant, $$$ \acute{N_{t}} $$$ is the number of spins within the voxel without taking into account the myelin compartment, $$$ \overline{TE} $$$ denotes the diffusion echo-time and $$$ \acute{f_{int}},\acute{f_{ext}}, \acute{f_{csf}} $$$ are the relative water fractions.

Cross-modal relationships: assuming both modalities have the same voxel-size, Eq.[2] can be rewritten in terms of absolute water fractions:

$$ e(b)=k_{2}N_{t}\left(f_{int}e^{-\frac{\overline{TE}}{T_{2}^{int}}}e_{int}(b)+f_{ext}e^{-\frac{\overline{TE}}{T_{2}^{ext}}}e_{ext}(b)+f_{csf}e^{-\frac{\overline{TE}}{T_{2}^{csf}}}e_{csf}(b)\right). \qquad [3] $$

The initial value for the constant term $$$ k_{1}N_{t} $$$ was computed using a linear inversion approach⁷, and term $$$ k_{2}N_{t} $$$ was derived from the relationship:

$$ \frac{M(\overline{TE})}{S(\overline{TE},b=0)}\thickapprox\frac{k_{1}N_{t}}{k_{2}N_{t}}. \qquad [4] $$

Global fitting: the joint estimation for all parameters is performed by minimizing the following composite cost-function, which depends on Eq.[1] and Eq.[3]:

$$ min\{c_{1}\sum_{j}(M(TE_{j})-\widetilde{M}(TE_{j}))^{2}+c_{2}\sum_{i}(e(b_{i})-\widetilde{e}(b_{i}))^{2}\}, \qquad [5] $$

where $$$ \widetilde{M} $$$ is the measured MET2 signal and $$$ \widetilde{e} $$$ is the spherical mean computed from the diffusion data; $$$ c_{1} $$$ and $$$ c_{2} $$$ are constants that depend on the signal-to-noise ratio (SNR) and the number of data points for each modality.

Synthetic data: the MET2 imaging protocol consisted in 32 TE-values, $$$TE=(10,20,...,320)*10^{-3} s$$$. The diffusion protocol used three b-values $$$(b_{1}=625s/mm^{2}, b_{2}=1250s/mm^{2}, b_{3}=2500s/mm^{2})$$$ with 64 measurements per shell and $$$ \overline{TE}=85*10^{-3}s$$$. Simulated signals for different microstructural parameters where generated. Rician noise was added with $$$SNR=30,25,20,10$$$.

In the evaluation, the following assumptions were made: $$$ T_{2}^{int}=T_{2}^{ext},T_{2}^{csf} = 2s,D_{csf}=3.0 mm^{2}/s $$$.

RESULTS

Figure 1 and 2 depicts results from numerical simulations. In figure 1 the unified estimation framework is compared with the alternative approach based on the individual fitting of both modalities. The unified method was more accurate and stable. Figure 2 shows the performance of the new method for different noise levels commonly found in clinical acquisition protocols.

DISCUSSION AND CONCLUSION

A theoretical model for the joint estimation of different microstructure indices from diffusion MRI and MET2 data was proposed. The developed framework improves the reconstruction process by using the common and complementary information provided by both modalities. While the MET2 data allow us to model the myelin compartment, the diffusion data enable us to better characterize the intra-axonal and extra-axonal compartments. In order to discriminate microscopic tissue features from fiber dispersion, the diffusion data was modeled using the SMT method, which was extended to also include T₂ relaxation. Shortly the proposed technique will be validated in real data.

Acknowledgements

References

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