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Neurite Exchange Imaging (NEXI): A minimal model of diffusion in gray matter with inter-compartment water exchange
Ileana O. Jelescu, Alexandre de Skowronski, Françoise Geffroy, Marco Palombo, Dmitry S. Novikov
TL;DR
Gray-matter diffusion MRI requires models beyond the non-exchanging Gaussian Standard Model because exchange, structural disorder, and soma contribute to the signal. The paper introduces NEXI, an anisotropic Kärger exchange model, and uses multi-b multi-t rat-brain data to show that exchange best explains the observed time-dependent signal signatures. It concludes that exchange should be included in gray-matter microstructure interpretation, while noting important model limitations.
Problem
Gray matter requires extending the Standard Model to account for membrane exchange, structural-disorder-induced non-Gaussian diffusion, and soma contributions.
Method
NEXI models neurite–extracellular water exchange as two exchanging compartments using the anisotropic Kärger model and evaluates it with multi-b multi-t rat-brain data.
Results
Exchange best explains gray-matter signal time-dependence in both low-b and high-b regimes, with kurtosis decreasing over time while diffusivity shows little time-dependence.
Takeaways & Limitations
Multi-b multi-t acquisitions are best suited to estimate NEXI parameters reliably, supporting exchange as a minimal component of gray-matter microstructure mapping.
Takeaways & Limitations
NEXI omits soma as a third compartment and assumes Gaussian compartments, while the neurite Gaussian assumption can break at higher b-values.
Abstract
from arXiv · showhide
Biophysical models of diffusion in white matter are based on what is now commonly referred to as the "Standard Model" (SM) of non-exchanging anisotropic Gaussian compartments. In this work, we focus on diffusion MRI in gray matter, which requires rethinking basic microstructure modeling blocks. In particular, at least three contributions beyond the SM need to be considered: water exchange across the cell membrane - between neurites and the extracellular space; non-Gaussian diffusion along neuronal and glial processes - resulting from structural disorder; and signal contribution from soma. For the first contribution, we propose Neurite Exchange Imaging (NEXI) as an extension of the SM of diffusion, which builds on the anisotropic Kärger model of two exchanging compartments. Using datasets acquired at multiple diffusion weightings (b) and diffusion times (t) in the rat brain in vivo, we show that for the investigated diffusion time window (~10-45 ms) there is minimal diffusivity time-dependence and more pronounced kurtosis decay with time in gray matter, which is well fit by the exchange model. Moreover, we observe lower signal for longer diffusion times at high b. In light of these observations, we identify exchange as the mechanism that best explains these signal signatures in both low-b and high-b regime, and thereby propose NEXI as the minimal model for gray matter microstructure mapping. We finally highlight multi-b multi-t acquisitions protocols as being best suited to estimate NEXI model parameters reliably. Using this approach, we estimate the inter-compartment water exchange time to be 15 - 60 ms in the rat cortex and hippocampus in vivo, which is of the same order or shorter than the diffusion time in typical diffusion MRI acquisitions. This suggests water exchange as an essential component for interpreting diffusion MRI measurements in gray matter.
1. Introduction
Gray-matter diffusion MRI requires extending the white-matter Standard Model to account for exchange, structural disorder, and soma. This work evaluates these effects in rat brain data and proposes NEXI as an exchange-based minimal model.
- Tissue compartments: Cellular processes, soma, and extracellular space form the principal compartments relevant to interpreting brain diffusion MRI.Processes include axons, dendrites, and glial processes; soma are roughly 15 μm in diameter.
- Motivation: Gray matter requires modeling three contributions beyond the Standard Model: membrane exchange, structural-disorder-induced non-Gaussian diffusion, and soma signal.These contributions reflect gray matter’s distinct microgeometry compared with white matter.
- Motivation: Water exchange across neurite membranes may be non-negligible at typical clinical diffusion times, with reported in vivo exchange times of 10–30 ms in human cortex.Related exchange-time ranges have also been reported in perfused neonatal mouse spinal cords.
- Competing mechanisms: Structural disorder can produce kurtosis time-dependence proportional to t^-1/2, but exchange can generate a faster t^-1 decay that cannot always be ruled out.Prior human-cortex observations did not resolve these mechanisms because exchange remained possible.
- Competing models: Soma occupy approximately 10–20% of gray-matter volume and motivate models such as SANDI, which represents them as impermeable spheres.SANDI uses short diffusion times to reduce the potential impact of inter-compartment exchange.
- Experimental design: The study examines diffusivity and kurtosis time-dependence together with high-b powder-average signal to assess exchange, structural disorder, and soma effects.The reported trends include negligible diffusivity time-dependence and time-dependent kurtosis consistent with exchange.
- NEXI: NEXI extends the Standard Model with exchange between neurites and extracellular space using the anisotropic Kärger model of two exchanging compartments.The study proposes NEXI as a minimal gray-matter model and finds multi-b multi-t data important for reliable parameter estimation.
- Experimental design: Rat-brain experiments probe diffusion times of 10–45 ms and strong diffusion weightings up to b = 10 ms/μm2 across gray- and white-matter regions.Cortex and hippocampus serve as gray-matter regions, with corpus callosum, internal capsule, and cingulum as white-matter references.
2. Methods
The methods develop NEXI by extending the Standard Model with exchange between neurite and extracellular compartments, while simplifying gray-matter orientation and extracellular diffusion. They derive exchange-dependent kurtosis and high-b behavior, compare alternative mechanisms, and fit NEXI and SANDI signals to multi-time data.
- NEXI: adding exchange to SM: NEXI extends the Standard Model by mixing anisotropic neurite and extracellular Gaussian compartments through barrier-limited exchange.The exchange rates obey detailed balance, and the characteristic exchange time is t_ex = 1/r.
- NEXI: adding exchange to SM: The powder-averaged NEXI model assumes isotropic extracellular diffusion and uses parameters [f, D_i,∥, D_e, t_ex].The orientational average removes dependence on the neurite orientation distribution function under the model’s gray-matter approximation.
- NEXI kurtosis: NEXI kurtosis combines inter-compartment heterogeneity that decays as 1/t through exchange with a residual offset K_∞.At long times, the decaying contribution vanishes while K_∞ represents residual kurtosis sources.
- NEXI kurtosis: The model’s residual kurtosis reflects the assumption that exchange occurs only between one neurite and its proximal extracellular space.A connected extracellular space could instead permit exchange across multiple neurites and yield K_∞ = 0 at long diffusion times.
- High-b scaling: The high-b expansion distinguishes impermeable-stick behavior, with a b^-1/2 term, from exchange-related b^-3/2 and higher-order terms.The narrow-pulse approximation requires δ ≪ t_ex; the methods compare experimental pulse durations with estimated exchange times.
- Alternative mechanisms: The study compares structural-disorder and soma alternatives with NEXI, including SANDI fits to shell-averaged signals and joint NEXI fits across diffusion times.Structural disorder can produce t^-1/2 or (ln t)/t kurtosis behavior, while SANDI models three non-exchanging compartments including soma.
3. Results
Across rat gray matter, diffusivity was largely stable over 10–45 ms while kurtosis decayed markedly, supporting exchange as a key signal mechanism. NEXI fit multi-time data and simulations showed that multi-b, multi-t acquisitions improve parameter estimation, although model applicability and precision have important boundaries.
- Exchange versus structural disorder: No significant diffusivity time-dependence was measured over 10–45 ms, whereas mean kurtosis decayed markedly and approximately as t^-1 in gray matter.The differing long-time behaviors of MD and MK were interpreted as compatible with exchange, with structural-disorder effects largely coarse-grained beyond 20 ms.
- Exchange versus structural disorder: 21 ± 4 ms in cortex and 16 ± 3 ms in hippocampus were estimated for exchange time when fitting kurtosis with K∞=0.Allowing nonzero K∞ produced exchange-time and K0 uncertainties up to 1800%, whereas fixing K∞=0 enabled more robust fits.
- Exchange versus soma: NEXI explained signal decay across multiple diffusion times better than SANDI, which predicted higher rather than experimentally observed lower signals at longer diffusion times.At a single diffusion time, both SANDI and NEXI fit the signal well, but the multi-time prediction favored NEXI.
- Simulations: In finite-SNR simulations, De was estimated precisely and f acceptably, but Di,∥ and tex could not be estimated when each diffusion time was fit separately.Joint fitting across all diffusion times improved f and tex precision and restored some sensitivity to Di,∥.
- Simulations: b-values above 2.5 ms/μm2 were needed for accuracy, while bmax=10 ms/μm2 outperformed bmax=2.5 ms/μm2 in accuracy and precision.The authors identify the available b-value range, rather than shell count alone, as critical for parameter sensitivity.
- Experimental validation: With b-values up to 10 ms/μm2 and three to four diffusion times, all NEXI parameters could be estimated at ROI and single-voxel levels in rat gray matter.Across cortex and hippocampus, average estimates were 2.5 μm2/ms for intra-neurite diffusivity, 0.75 μm2/ms for extra-neurite diffusivity, and approximately 0.3 for neurite fraction.
- Experimental validation: NEXI exchange times were estimated at 15–60 ms in rat cortex and hippocampus, comparable to diffusion times used in the study.The analysis therefore supports accounting for inter-compartment exchange in gray matter at relatively long diffusion times, especially above 20 ms.
- Experimental validation: Applying NEXI to white matter is limited because its extra-neurite compartment cannot generally be assumed isotropic.Consequently, larger estimated intra-neurite fractions in white matter should not be interpreted as direct quantitative tissue comparisons under the NEXI model.
4. Discussion
The discussion identifies inter-compartment exchange as the mechanism best explaining gray-matter diffusion-time signatures and supports NEXI as a minimal model, while defining acquisition and interpretation limits.
- NEXI and exchange: NEXI identifies exchange as the mechanism best explaining diffusion-time-dependent signal in gray matter across low- and high-b regimes.The model extends the Standard Model with exchange between neurites and extracellular space.
- Model interpretation: Gray-matter Standard Model parameters become ill-defined with exchange, making their microstructural interpretation challenging.The apparent intra-neurite fraction decreases with diffusion time as exchanged water acquires a more hindered-diffusion-like signature.
- Exchange versus disorder: Negligible diffusivity time-dependence from 20–45 ms, alongside marked kurtosis decay, supports exchange over structural disorder in rat gray matter.The observed kurtosis decrease is attributed to inter-compartment exchange rather than structural disorder, with kurtosis expected to vanish at very long times.
- Regional exchange: NEXI estimates relatively short exchange times in cortex and hippocampus, intermediate times in cingulum, and long times in heavily myelinated white-matter bundles.Reported values are approximately 15–20 ms in cortex and hippocampus, about 40 ms in cingulum, and 80–130 ms in the corpus callosum and internal capsule.
- Acquisition requirements: Multi-b multi-t data substantially improve NEXI parameter accuracy and precision, whereas a single diffusion time cannot reliably estimate exchange time and intra-neurite diffusivity.The benefit of extending b-values is critical up to 6 ms/μm2 but marginal beyond that range.
- Scope and limitations: Exchange-time estimates plateau for ground truths t_ex≥80 ms, so tissues with slower exchange require a longer diffusion-time range.The limitation follows from simulations probing only 12–40 ms diffusion times.
Limitations
NEXI is limited by its two-compartment Gaussian assumptions and by acquisition and parameter-estimation constraints. Its treatment of soma and exchange geometry may not fully represent gray-matter microanatomy.
- Acquisition and estimation: Multi-shell multi-t protocols may be difficult to implement when scan time is limited, and t_ex and D_i,|| estimates currently have large uncertainty.Protocol optimization is needed to balance scan time against parameter accuracy and precision.
- Model assumptions: The imposed constraint D_i,|| > D_e may require relaxation in pathological tissue where cytoplasmic tortuosity could reduce D_i,|| below D_e.The paper specifically identifies diseases involving intracellular tangles or protein accumulation as relevant cases.
- Model scope: NEXI does not model soma as a third compartment, instead absorbing soma into the extra-neurite space.A three-compartment extension or exchange-aware SANDI model is proposed as future work.
- Gaussian-compartment assumption: Higher b-values reveal non-Gaussian neurite effects that can bias NEXI estimates, while low b-values reduce estimation accuracy and precision.These effects may reflect finite dendritic length, branching, and related structural features.
- Exchange geometry: The NEXI exchange geometry assumes each neurite exchanges only with its immediate extracellular space, excluding sequential exchange among differently oriented neurites.This assumption may differ from gray-matter microanatomy when multiple orientations occupy one diffusion volume.
Value
The findings support accounting for inter-compartment exchange in gray-matter diffusion models at typical clinical diffusion times. NEXI also offers exchange time as a potential proxy for membrane permeability and tissue condition.
- Modeling implication: Inter-compartment exchange is not negligible in gray matter at typical PGSE or clinical diffusion times above 20 ms.The authors suggest accounting for exchange in gray-matter models and potentially in thinner or demyelinating white-matter tracts.
- Biological value: NEXI exchange-time estimates can serve as a proxy for membrane permeability, which may provide a biomarker of tissue integrity, metabolism, and function.The proposed relevance includes injury and neurodegeneration, where membrane permeability is known to increase.
5. Conclusions
The paper identifies exchange as the dominant explanation for gray-matter diffusion-time effects beyond the Standard Model. For t > 20 ms, NEXI is favored over a soma-based three-compartment model, while future models should ideally include both exchange and soma when data permit.
- 5. Conclusions: In rat gray matter, exchange dominates over structural disorder and explains diffusion-time signal decay better than adding a soma compartment.The authors describe diffusion time as filtering out contributions from unmyelinated neurites with stronger dispersion.
- 5. Conclusions: For t > 20 ms, the two-compartment exchanging NEXI model is better suited than a three-compartment soma model for cortical microstructure characterization.NEXI also yields an exchange-time estimate that can proxy membrane permeability.
- 5. Conclusions: Both soma and exchange should ideally be modeled when the data support estimating the larger number of parameters.
Appendix: DKI(t) for the orientationally-averaged anisotropic KM
The appendix extends the Kärger-model kurtosis formulation to orientationally averaged anisotropic compartments. It defines the orientation-dependent compartment diffusivities, averages the signal through a b^2 expansion, and notes residual long-time kurtosis and limitations from stick mixing.
- Model formulation: The anisotropic Kärger model is introduced through a DKI representation in which kurtosis follows K(τ) = K0F(τ).The function F(τ) captures the exchange-time dependence of the kurtosis decay.
- Orientation dependence: For a fascicle at angle θ, D1 is set to D_i,∥x^2 and D2 to D_e,⊥ + Δ_ex^2, with x = cos θ.The isotropic extra-cellular assumption gives Δ_e = 0 and D_e,⊥ = D_e.
- Orientational averaging: The orientationally averaged signal is obtained by expanding S_KM(b,x) through b^2, integrating term-by-term over orientations, and applying algebraic simplification.The resulting expressions include the anisotropic diffusivity terms shown in the appendix.
- Long-time behavior: The model retains residual kurtosis K∞ from the isotropic mixture of diffusion tensors, although the main text neglects K∞ under isotropic extra-cellular assumptions.The residual term arises from local exchange between a stick and its accompanying extracellular space.
- Model limitation: The model does not adequately describe mixing between sticks with different orientations at sufficiently long diffusion times.
Supplementary Material for “Neurite Exchange Imaging (NEXI): A minimal model of diffusion in gray matter with inter-compartment water exchange”
Supplementary analyses evaluate parameter stability, model-fitting accuracy, and signal-model comparisons across diffusion times, b-values, and noise conditions. They support multi-time, multi-shell acquisition for more reliable NEXI estimation and show that NEXI better describes the examined high-time signal than SANDI.
- Parameter estimation: A three-layer fully connected neural network estimates NEXI parameters from log-transformed signals, using mean-squared error for noiseless and noisy training.Parameters are scaled to [-1, 1], and training uses 10^5 simulated signals with corresponding ground-truth parameters.
- Diffusion-time dependence: Diffusivity shows no significant time-dependence over 10–45 ms, whereas kurtosis changes significantly in all ROIs, with the strongest slope in gray matter.The diffusivity confidence bounds are too large to establish significant time-dependence, while MK time-dependence is significant across ROIs.
- Diffusion-time dependence: SANDI parameter estimates are evaluated against diffusion time using percentage-difference plots, statistical comparisons, and weighted linear regressions.The analyses use a 12 ms reference point, ANOVA with Bonferroni correction, and F-tests for regression trends.
- Parameter estimation: Multi-shell, multi-diffusion-time fitting improves NEXI parameter accuracy and precision relative to single-time fits, including at SNR=100.Some sensitivity to Di,∥ and high tex values remains lost at SNR=100, but joint fitting performs significantly better than single-time fitting.
- Parameter estimation: Increasing both the number of shells and maximum b-value improves the accuracy and precision of NEXI estimates in simulations.The simulations compare bmax values of 2.5, 5.5, and 10 ms/μm^2 across four diffusion times.
- Model comparison: At t=40 ms, NEXI explains the cortical signal better than SANDI across the examined b-value range, while Callaghan’s impermeable-stick model does not describe the signal decay.The comparison includes SANDI, NEXI, Callaghan’s model, and low- and high-b NEXI approximations.