Synopsis

Keywords: Electromagnetic Tissue Properties, Susceptibility

Typically MRI resolution is limited to millimeters in clinical scanners, but other sources of information, derived from diffusion processes of water in biological tissues, allow us to get information at micrometric scales. Here we consider extracting morphological information by probing the internal magnetic gradients induced by the heterogeneous magnetic susceptibility of tissues. From the cumulant expansion of the magnetization signal we derive an internal gradient distribution tensor (IGDT) expansion and propose modulating gradient spin-echo sequences to probe them. These IGDTs contains microstructural information of tissues. Our results provide a framework to describe IGDTs to exploit them as quantitative diagnostic tools.

Introduction

Magnetic resonance imaging (MRI) is a powerful technique for non-invasive medical diagnosis. Different methods exploit physical properties to generate contrast that can unveil tissue microstructures$^{1-8}$. The intrinsic heterogeneity of biological tissues is imitated by their magnetic susceptibility changes$^{7-10}$. In presence of an external magnetic field, internal magnetic field gradients are induced by the susceptibility discontinuities$^{7-14}$. Here we develop a cumulant expansion framework to derive internal gradient distribution tensors (IGDT)$^{11}$. The framework is then exploited to design modulated gradient spin-echo (MGSE) sequences to enhance and extract the IGDT terms. We evaluate the feasibility of measuring the IGDT-expansion with simulations using typical brain tissue parameters.

Methods

Internal gradient ensemble model
Internal gradients are probed by the spin-dephasing $$\phi\left[T,\boldsymbol{x},\boldsymbol{G}_{0}\right]=\gamma\int_{0}^{T}dt'\boldsymbol{x}(t')\cdot\left(\boldsymbol{G}f_{G}(t')+\boldsymbol{G}_{0}f_{0}(t')\right)\qquad\qquad(1)$$ induced by molecular diffusion in the presence of a combination of applied gradient $\boldsymbol{G}$ and internal gradients $\boldsymbol{G}_0$. During the diffusion time $T$, we assume the spin-phase is characterized by an effective and spatially constant internal gradient $\boldsymbol{G}_{0}$ $^{14}$. The internal gradient may be different for each particle in the sample depending on their position and diffusion pathway $\boldsymbol{x}(t)$. Here we consider the applied and internal gradients can be controlled in time through the modulations $f_{G}(t)$ and $f_{0}(t)$ respectively. Figure 1 shows a schematic representation of the model, where we consider an ensemble of spin bearing particles and an ensemble of internal gradients$^{11,12}$. We account also for anisotropic diffusion processes as illustrated in Fig. 2.

Cumulant expansion framework for internal gradient distribution tensors
We describe the spin magnetization $M(T)$ using a cumulant expansion for the spin-phase. The cumulants are then written in terms of IGDT with different ranks. The magnetization attenuation factor is $$\ln{M(T)}=\chi_{\mathrm{app}}(T)+\chi_{\mathrm{bck}}(T)+\chi_{\mathrm{odd-cross}}(T)+\chi_{\mathrm{even-cross}}(T)\qquad\qquad(2)$$ with $$\chi_{\mathrm{app}}(T)=-\frac{1}{2}G_{i}\beta_{ij}^{GG}(T)G_{j},$$ $$\chi_{\mathrm{bck}}(T)=-\frac{1}{2}\beta_{ji}^{00}(T)\left\langle\boldsymbol{G}_{0}\boldsymbol{G}_{0}\right\rangle_{ij}+\frac{1}{2}\beta_{ij}^{00}(T)\beta_{kl}^{00}(T)\left\langle\boldsymbol{G}_{0}\right\rangle_{i}\left\langle\boldsymbol{G}_{0}\right\rangle_{l}\left\langle\Delta\boldsymbol{G}_{0}\Delta\boldsymbol{G}_{0}\right\rangle_{jk}\\+\frac{1}{2}\beta_{ij}^{00}(T)\beta_{lk}^{00}(T)\left\langle\boldsymbol{G}_{0}\right\rangle_{i}\left\langle\Delta\boldsymbol{G}_{0}\Delta\boldsymbol{G}_{0}\Delta\boldsymbol{G}_{0}\right\rangle_{jkl}+\mathcal{O}\left[\boldsymbol{\beta}(T)^{3}\right],$$ $$\chi_{\mathrm{odd-cross}}(T)=-G_{i}\beta_{ij}^{0G}(T)\left\langle\boldsymbol{G}_{0}\right\rangle_{j}+G_{i}\beta_{ij}^{0G}(T)\beta_{lk}^{00}(T)\left\langle\boldsymbol{G}_{0}\right\rangle_{l}\left\langle\Delta\boldsymbol{G}_{0}\Delta\boldsymbol{G}_{0}\right\rangle_{jk}\\+\frac{1}{2}G_{i}\beta_{ij}^{0G}(T)\beta_{lk}^{00}(T)\left\langle\Delta\boldsymbol{G}_{0}\Delta\boldsymbol{G}_{0}\Delta\boldsymbol{G}_{0}\right\rangle_{jkl}+\mathcal{O}\left[\boldsymbol{\beta}(T)^{3}\right],$$ and $$\chi_{\mathrm{even-cross}}(T)=\frac{1}{2}G_{i}G_{l}\beta_{ij}^{0G}(T)\beta_{lk}^{0G}(T)\left\langle\Delta\boldsymbol{G}_{0}\Delta\boldsymbol{G}_{0}\right\rangle_{jk}+\mathcal{O}\left[\boldsymbol{\beta}(T)^{3}\right].$$ The moments $\left\langle \boldsymbol{G}_{0}\boldsymbol{G}_{0}\cdots\right\rangle_{ij\cdots}$ and the central moments $\left\langle \Delta\boldsymbol{G}_{0}\Delta\boldsymbol{G}_{0}\cdots\right\rangle_{ij\cdots}$ define the matrix elements of IGDTs. The attenuation matrices $\boldsymbol{\beta}^{ab}(T)=\gamma^{2}\int_{[0,T]^{2}}dt_{1}dt_{2}f_{a}(t_{1})f_{b}(t_{2})\left\langle\Delta\boldsymbol{x}(t_{1})\Delta\boldsymbol{x}(t_{2})\right\rangle$ include all the time-dependence of the magnetization decay, and depends on the self-correlation tensor of the spin displacement $\left\langle \Delta\boldsymbol{x}(t_{1})\Delta\boldsymbol{x}(t_{2})\right\rangle$ and the gradient modulation functions $f_G(t)$ and $f_0(t)$.

To evaluate how different IGDT terms contribute to the signal decay, we focus only on a diffusion process along one of the principal axes of the correlation-time tensor $\boldsymbol{\tau}_{c}$ of Fig. 2. The IGDT-expansion thus reduces to $$\ln{M(T)}=-\frac{1}{2}G^{2}\beta_{GG}(T)-G\beta_{0G}(T)\left\langle G_{0}\right\rangle-\frac{1}{2}\beta_{00}(T)\left\langle{G_{0}^{2}}\right\rangle\\+\frac{1}{2}G^{2}\beta_{0G}^{2}(T)\left\langle\Delta{G}_{0}^{2}\right\rangle+G\beta_{0G}(T)\beta_{00}(T)\left\langle{G_{0}}\right\rangle\left\langle{\Delta{G}_{0}^{2}}\right\rangle\qquad\qquad(3)\\+\frac{1}{2}\beta_{00}^{2}(T)\left\langle{G_{0}}\right\rangle^{2}\left\langle{\Delta{G}_{0}^{2}}\right\rangle+\mathcal{O}\left[\boldsymbol{\beta}(T)^{3}\right].$$

Results

Sequence design to probe IGDTs
The four groups of terms in the IGDT-expansion (2) can be probed selectively by suitable design of symmetric and asymmetric gradient modulation sequences following the protocol introduced in Ref.$^{11}$. Figure 3 shows the considered sequences.

To enhance the cross-term effects in Eqs. (2) and (3), the background gradient modulation $f_0(t)$ has to be an even function with respect to $T/2$ with a minimal number of pulses (two pulses). The applied gradient modulation has to be a MGSE sequence with smooth modulations, i.e. $|f_{G}(t)|\leq1$, with a single refocusing echo synchronizing the zero crossing time with a $\pi$-pulse to modulate the background gradient $f_0(t)$ as in Fig. 3 (b). The modulation function $f_G(t)$ has to be an odd function with respect to its middle time and should vanish at every $\pi$-pulse that modulates $f_0(t)$.

The IGDT-expansion using typical brain tissue parameters
We calculate the IGDT-expansion of Eq. (3) for typical parameter values that characterize brain tissues. We consider a free diffusion regime and a restricted diffusion regime in Fig. 4. We compare the expected magnetization signal resulting from applying the symmetric and asymmetric sequences with the IGDT expansion of Eq. (3) for both diffusion regimes. We assume a Gaussian distribution for the internal gradients for the expected signal.

Discussion

Figure 4 shows restricted and free diffusion regimens using realistic values that represent brain tissue. The IGDT terms can be extracted from the magnetization decay with a combination of mathematical operations with the symmetric and asymmetric sequence’s signals following the protocol of Ref.$^{11}$. The symmetric sequence’s signal is well described by the pure applied and pure background gradient terms of the IGDT expansion (3). The asymmetric sequence gives a decay reproduced well by including the cross-terms of the expansion (3). The signal variations provided by different IGDT terms can be of the order of 10% for diffusion times lower or comparable to typical $T_2$ $^{15}$, thus showing the feasibility of their experimental validation.

Conclusions

Brain physiology is regulated by a sort of molecules and structures as the myelin sheath of axons with significant magnetic susceptibility in comparison with the surrounding medium. The degree of axon myelination significantly affects the internal gradient distributions in white matter$^{12,15-17}$. Thus internal gradient distributions show correlations with the amount of myelin in tissues as potential biomarkers for many degenerative diseases.

Our results thus contribute to estimate IGDTs that may be especially useful for unveiling structures and fiber orientation based on susceptibility induced changes. The IGDTs are complementary to DTI as they may estimate anisotropies when diffusion is free or isotropic$^{11-13}$. In the restricted diffusion regimen, internal local gradients might be averaged by molecular motion, thus we expect this tool to be useful to characterize extra-axonal diffusion$^{12}$.

Acknowledgements

This work was supported by CNEA; CONICET; ANPCyT-FONCyT PICT-2017-3156, PICT-2017-3699, PICT-2018-4333; PIP-CONICET (11220170100486CO); UNCUYO SIIP Tipo I 2019-C028, 2022-C002, 2022-C030; Instituto Balseiro; A collaboration program from MINCyT (Argentina) and MAECI (Italy) and Erasmus+ Higher Education program from the European Commission between the CIMEC (University of Trento) and the Instituto Balseiro (Universidad Nacional de Cuyo).