Non-invasive quantitative imaging of selective microstructure-sizes with magnetic resonance

The development of nanosized sensors with novel quantum technologies is aiming at nanoscale imaging of biological tissues for unveiling the biophysics of pathologies at such relevant scales (Staudacher et al., 2015; Wang et al., 2019; Barry et al., 2016; Glenn et al., 2015). Such imaging proposals are still based on invasive techniques. By contrast, Magnetic Resonance Imaging (MRI) has proven to be an excellent tool for acquiring non-invasive images, being applied on a daily basis for clinical diagnosis. However, the weak sensitivity for detecting the nuclear spins inherent to the biological tissues, typically limits the spatial resolution of in-vivo MRI to hundred of micrometers in pre-clinical scanners and to millimeters in clinical systems. This limitation imposes a challenge for existing methods to early detect diseases that produce changes at the cellular level (Padhani et al., 2009; Drago et al., 2011; White et al., 2013, 2014; Enzinger et al., 2015). Detecting these kind of pathologies in a development stage based on quantitative imaging of tissue microstructure parameters, will allow MRI to advance towards a new early diagnostic paradigm (Le Bihan, 2003; White et al., 2013, 2014; Xu et al., 2014; Enzinger et al., 2015; Grussu et al., 2017; Alexander et al., 2019).

Diffusion Weighted MRI (DWI) is a promising tool to probe microstructure information based on monitoring the dephasing of the nuclear spin precession due to Brownian molecular motion (Le Bihan, 2003; Grebenkov, 2007; Callaghan, 2011). The diffusion dynamics of molecules depends on tissue properties such as cell sizes, density, and other morphological features. A strong magnetic field gradient is applied to sense the microscopic motion of spins so that the the precession frequency depends on their instantaneous position. In addition, modulating the gradient strength as a function of time allows to probe the time dependent diffusion process of the molecules within tissues (Stepisnik, 1993; Callaghan, 2011; Álvarez et al., 2013; Shemesh et al., 2013). These dynamical control techniques are based on the Hahn spin-echo concept (Hahn, 1950) and its generalization to multiple echoes (Carr and Purcell, 1954). Within DWI they are called modulated gradient spin-echo (MGSE) sequences, where the phase accumulated by the spin’s precession is refocused at given times allowing to infer microscopic parameters in tissues and porous media (Stepisnik, 1993; Shemesh et al., 2013; Drobnjak et al., 2016; Nilsson et al., 2017; Novikov et al., 2019). However, obtaining detailed microstructure information is still challenging on in-vivo studies, as it requires tens of images demanding about an hour of acquisition time (Assaf et al., 2008; Alexander et al., 2010; Xu et al., 2014; Shemesh et al., 2015; Alexander et al., 2019; Novikov et al., 2019).

The decay of the nuclear spin signal under MGSE sequences is typically characterized by a decay-rate (Grebenkov, 2007; Callaghan, 2011). Here, we report that these decaying signals manifest a “decay-shift” that can be exploited to selectively probe microstructure-sizes. We develop size-filters based on the Non-uniform Oscillating Gradient Spin-Echo (NOGSE) concept that contrast the signal generated by two spin-echo sequences (Álvarez et al., 2013; Shemesh et al., 2013). The method probes different diffusion time scales while factoring out other relaxation mechanisms induced by gradient-modulation imperfections and $T_{2}$ effects. We exploit the NOGSE modulations to selectively probe the spin-echo decay-shift for producing quantitative images based on contrast intensities that reflect the probability of finding a specific microstructure-size. The present approach requires only two measurements, therefore significantly reducing the acquisition time compared to state-of-the-art methods that typically require tens of measurements to obtain quantitative information on microstructure-sizes based on fitting parameters (Assaf et al., 2008; Alexander et al., 2010; Ong and Wehrli, 2010; Xu et al., 2014; Shemesh et al., 2015; Alexander et al., 2019; Novikov et al., 2019). We analytically and experimentally demonstrate that these microstructure-size filters can be implemented with current technologies presenting a novel mechanism for quantitative and precision imaging diagnostic tools.

MRI of molecular-diffusion. The nuclear spins $S\!=\!\frac{1}{2}$ in molecules intrinsic to biological tissues, mainly from water’s protons, are typically observed on in-vivo MRI. They interact with an external uniform magnetic-field $B_{0}$$\hat{z}$ and a magnetic field gradient $G\hat{r}$ applied along the direction $\hat{r}$ for spatial encoding. In a frame rotating at the resonance frequency $\gamma B_{0}$, the precession frequency $\omega(t)=\gamma Gr(t)$ fluctuates reflecting the random motion of the molecular diffusion process (Grebenkov, 2007; Callaghan, 2011). Here, $r(t)$ is the instantaneous position of the nuclear spin along the gradient direction and $\gamma$ is the gyromagnetic ratio of the nucleus. The effective gradient strength may vary along time as $G(t)$, either by applying $\pi$-pulses if the gradient is constant or directly by modulating the gradient’s sign and amplitude. The spins dephasing induced by the diffusion process is refocused by these control modulations forming the so-called spin-echoes (Hahn, 1950; Carr and Purcell, 1954) (Fig. 1a,b). At the evolution time $T_{E}$ of the control sequence, the spin-echo magnetization decays depending on how nuclear spins were scrambled by the diffusion process. The resulting magnetization at $T_{E}$ is then spatially encoded with an MRI acquisition sequence.

The spins acquire a random phase $\phi(T_{E})$ that typically follows a Gaussian distribution (Stepisnik, 1999). The magnetization in a voxel of the image becomes

The quadratic phase is averaged over the spin ensemble as (Stepisnik, 1993; Grebenkov, 2007; Callaghan, 2011)

which is expressed in terms of the control applied to the system by $G(t)$ and the molecular displacement autocorrelation function $\left\langle\Delta r(0)\Delta r(t)\right\rangle=D_{0}\tau_{c}e^{-|t|/\tau_{c}}$ (Stepisnik, 1993; Álvarez et al., 2013; Shemesh et al., 2013). Here $\Delta r(t)=r(t)-\left\langle r(t)\right\rangle$ is the instantaneous displacement of the spin position from its mean value, $D_{0}$ is the free diffusion coefficient, and $\tau_{c}$ is the correlation time of the diffusion process. The restriction length $l_{c}$ of the microstructure compartment in which molecular diffusion is taking place is then determined by the Einstein-Diffusion equation $l_{c}^{2}=D_{0}\tau_{c}$ (Callaghan, 2011). Then, by monitoring the spin-echo decay by applying suitable control sequences, one can infer microstructure-sizes (Ong and Wehrli, 2010; Álvarez et al., 2013; Shemesh et al., 2013; Drobnjak et al., 2016; Nilsson et al., 2017; Novikov et al., 2019).

Spin-echo decay-shift as a probing-microstructure paradigm. The spin-echo magnetization decay is usually characterized by its decay-rate (Grebenkov, 2007; Callaghan, 2011). Under the control sequences discussed in Fig. 1a,b, the decay-rate is typically reduced as the number of refocusing periods $N$ increases (Carr and Purcell, 1954). This effect is shown in Fig. 1a-c, where the Hahn spin-echo $(N\!=\!1)$ decay is compared with the signal after a CPMG sequence ($N\!>\!1$) with multiple echoes (Carr and Purcell, 1954). However, in the restricted diffusion regime when the refocusing periods are longer than the correlation time $\tau_{c}$, the spin-echo decays as $M(T_{E})/M(0)\approx e^{-\gamma^{2}G^{2}D_{0}\tau_{c}^{2}(T_{E}-(1+2N)\tau_{c})}$ (see Methods). There, the spins have been fully scrambled within the compartment, and the dephasing cannot be refocused leading to a decay-rate $\gamma^{2}G^{2}D_{0}\tau_{c}^{2}$ independent of $N$. Yet, the spin-echo retains information of the transition from the free to the restricted diffusion regime. This information is manifested as a “decay-shift” $\gamma^{2}G^{2}D_{0}\tau_{c}^{3}(1+2N)$ independent of the evolution time on the spin-echo decay signal, as shown in Fig. 1c. We here demonstrate that this shift can be exploited to selectively probe microstructure sizes as it is $\propto l_{c}^{6}$.

As this decay-shift depends on $N$, it can be selectively probed by concatenating a Hahn with a CPMG gradient modulation and changing the ratio between the relative refocusing periods of each component of the sequence as shown in Fig. 1d. This control sequence produces an effective modulated gradient $G(t)$, that conforms the Non-uniform Oscillating-Gradient Spin-Echo (NOGSE) (Álvarez et al., 2013; Shemesh et al., 2013). This sequence also factorizes out other relaxation mechanisms allowing to probe selectively the diffusion induced decay.

We define the NOGSE contrast (NOGSEc) as the amplitude $\Delta M$ given by the difference between the CPMG and the Hahn signal (Fig. 1c). It is obtained by evaluating NOGSE at the refocusing periods $t_{C}=t_{H}$ and then at $t_{C}\rightarrow 0$ using the definitions shown in Fig. 1d. Then both measurements are subtracted (see Methods). Within the restricted diffusion regime, this contrast amplitude is

which is very sensitive to the restricted diffusion length $l_{c}$ as it has a parametric dependence $l_{c}^{6}\propto\tau_{c}^{3}$ provided by the spin-echo decay-shift (Álvarez et al., 2013; Shemesh et al., 2013).

NOGSE as a selective microstructure-size filter. NOGSEc $\Delta M$ has a maximum as a function of the normalized echo-time $T_{E}/\tau_{c}$ as shown in Fig. 1e. We exploit this maximum contrast to enhance the relative contribution to the signal from specific restriction lengths $l_{c}$ from a size-distribution.

In order to perform a general analysis, Eq. (3) can be expressed in terms of dimensionless lengths $L_{D}=l_{D}/l_{G},L_{c}=l_{c}/l_{G}$ (see Methods). Here, $l_{c}=\sqrt{D_{0}\tau_{c}},\>l_{D}=\sqrt{D_{0}T_{E}},\>l_{G}=\sqrt[3]{D_{0}/\gamma G}$ are the restriction length, the diffusion length that the spin can diffuse freely during $T_{E}$ and the dephasing diffusion length that provides a phase shift of $2\pi$, respectively (Callaghan, 2011) (Fig. 1c). Then, $\Delta M$ as a function of $L_{c}$ can be approximated by a Gaussian function when $L_{D}/L_{c}\gg 1$, $L_{c}^{6}\ll 1$ and $L_{D}\gg 1$ (see Methods),

NOGSEc therefore acts as a microstructure-size “bandpass-filter” with $L_{c}^{f}$ as the filter-center size (see Fig. 2a)

The filter-band selectivity is defined by the ratio between the full-width-at-half-maximum (FWHM) and $L_{c}^{f}$, $FWHM/L_{c}^{f}\approx\sqrt{\frac{\ln 2}{3}}\approx 0.5$.

The maximum of $\Delta M$ at the size $L_{c}^{f}$ can be tuned to highlight a given restriction length $l_{c}$ based on choosing properly the sequence control parameters, i.e. the gradient strength $G$ and the evolution time $T_{E}$ (see Fig. 2a).

The filtered size decreases with increasing the control parameter $L_{D}$ according to Eq. (5). At the same time, increasing $L_{D}$ decreases the contrast amplitude which is $\propto\left(L_{c}^{f}\right)^{6}\propto 1/L_{D}^{3}$. Therefore, the minimum size that can be filtered in practice is limited by the Signal-to-Noise Ratio (SNR) and the maximum achievable gradient strength as $L_{D}\propto\sqrt[3]{G}$. This decrease of $\Delta M$ can be compensated linearly with increasing the number of refocusing periods $N$, as long as $\frac{L_{D}^{2}}{L_{c}^{2}N}=\frac{T_{E}}{\tau_{c}N}\gg 1$ to reach the restricted regime (see Eq. (4) and Fig. 2b).

A practical limitation is also the unavoidable transversal $T_{2}$-relaxation due to the intrinsic dephasing of the nuclear spins. NOGSEc decays then as

where we have defined $L_{2}=\frac{l_{2}}{l_{G}}$ as the $T_{2}$ diffusion dimensionless length with $l_{2}=(D_{0}T_{2})^{1/2}$. The $T_{2}$-relaxation effect is showed in Fig. 2c for different values of $L_{2}^{2}$. Remarkably, the filter shape and center remains the same as the case of $T_{2}\rightarrow\infty$.

Selective size-filtering in typical microstructure-size distributions. Our results show that one can produce quantitative images based on a signal contrast generated by specific microstructure-sizes from a size-distribution. This method avoids extracting the microstructure-size by fitting a curve, which is time consuming as it typically requires several measurements (Assaf et al., 2008; Alexander et al., 2010; Ong and Wehrli, 2010; Xu et al., 2014; Shemesh et al., 2015; Novikov et al., 2019). The filter amplitude $\Delta M$ in Eq. (4) remains constant by varying the gradient strength and the evolution time keeping fixed the parameter $L_{D}$, as shown in Fig. 2d-e. We exploit this property for a proof-of-principle evaluation of the performance of the NOGSEc filter applied to microstructural size-distribution inherent to heterogeneous biological tissues as shown in Fig. 3. Typical size-distributions $P(l_{c})$ are log-normal and bi-modal functions (Assaf et al., 2008; Pajevic and Basser, 2013; White et al., 2013; Liewald et al., 2014; Shemesh et al., 2015) as shown in Figs. 3a-b. NOGSEc for a size-distribution is given by

where $\Delta M(l_{c})$ is the contribution for a given restriction length $l_{c}$. Figures 3c-d show $\Delta M$ as a function of $l_{c}^{f}=L_{c}^{f}l_{G}$, where $G$ and $T_{E}$ are changed simultaneously keeping $L_{D}^{2}$ constant. The center of the filter at $l_{c}^{f}$ is therefore swept while the filter amplitude is kept constant. The resulting filtered signal as a function of the filter center $l_{c}^{f}$ therefore resembles the original size-distributions, where the log-normal peak and both Gaussian peaks are clearly identified. The effects of $T_{2}$-relaxation are shown in Fig. 3c for typical values of white-matter tissue. To further demonstrate the filter selectivity, Figure 3d shows that if one chooses $l_{c}^{f}$ equal to the center of one of the two Gaussian distributions, the other component is filtered-out. These simulations demonstrate the feasibility of performing quantitative images of a selective microstructure-size $l_{c}^{f}$ based on the NOGSEc amplitude as a “bandpass” filter.

The transversal $T_{2}$-relaxation limits the largest microstructure-size that can be filtered, as the restricted diffusion regime has to be achieved. A good SNR for $\Delta M$ is only obtained for $L_{c}<L_{D}<L_{2}$. For short $T_{2}$, the strategy is therefore using the lowest possible value of $L_{D}$ and reducing the number of refocusing periods $N$ so as to remain in the restricted regime. We consider this scenario in proof-of-principle experiments and perform a size-filter sweep varying only the gradient amplitude while keeping $T_{E}\lesssim T_{2}$ constant. We implement the microstructure-size filter method on an ex-vivo mouse brain focusing on the Corpus Callosum (CC) region (see experimental details in Methods). The CC contains aligned axons and is a paradigmatic model for log-normal size-distributions (Assaf et al., 2008; Pajevic and Basser, 2013; White et al., 2013; Liewald et al., 2014; Shemesh et al., 2015). Figure 4a shows images of $\Delta M$ for two gradient strengths. The largest gradient acts as a “bandpass” Gaussian-filter of the lower microscopic sizes of the distribution, compared to the weaker gradient that acts as a “high-pass” filter of the larger sizes (Fig. 4b). Therefore Fig. 4a clearly highlights zones of the CC with complementary colors depending of the microstructure-size. This is demonstrated quantitatively in Fig. 4b-c for three regions-of-interest (ROI). The average $\Delta M$ as a function of the gradient for the ROIs is shown in Fig. 4c together with fitted curves derived from our theoretical model following Eq. (7) (see Methods). The inferred microstructure-size distributions and the $\Delta M$ filter shapes are shown in Fig. 4b. The excellent agreement between the model and the experimental data fully demonstrates the reliability of the quantitative images shown in panel a based on the NOGSE microstructure-size filter and the assumed log-normal model.

Conclusions. The presented results introduce a method for performing non-invasive quantitative images of selective microstructure-sizes based on probing nuclear-spin dephasing induced by molecular diffusion with magnetic resonance. Conversely to standard diffusion-weighted imaging approaches that are based on observing the decay-rate of the spin signal, we exploit dynamical control with oscillating gradients to selectively probe a decay-shift on spin-echo decays. This decay-shift contains quantitative information of microstructure-sizes that restrict the molecular diffusion. We generate a contrast amplitude that behaves as a microstructure-size filter to selectively probe a restriction length determined by the control parameters. We show the usefulness and performance of the method with proof-of-principle simulations and experiments on typical size-distributions of white-matter tracts in a mouse brain. A quantitative image of specific diffusion restriction lengths is performed extracted from only two images, allowing to significantly reduce the tens of images that typically demand the inference of microstructure-sizes from data fittings (Assaf et al., 2008; Alexander et al., 2010; Ong and Wehrli, 2010; Xu et al., 2014; Shemesh et al., 2015; Novikov et al., 2019). Even though intrinsic $T_{2}$-relaxation may represent a limitation, we show excellent performance probing quantitative information of microstructure details between $\sim 0.1-10\,\mu\mathrm{m}$ on biological tissue as in the mouse white matter, being able to filter sizes much lower than the present image resolution. This work lays the foundations of a novel conceptual tool with low overhead for designing quantitative methods for non-invasive imaging of tissue microstructure. This diagnostic tool opens up a new avenue to explore for in-vivo imaging. In addition, these results can also be applied for characterizing material microstructures, such as rocks which are of particular interest for oil extraction, and for nanoscale-imaging of biological tissues with novel quantum sensors based on noise spectroscopy (Staudacher et al., 2015; Barry et al., 2016; Wang et al., 2019).