Biexponential diffusion attenuation in various states of brain tissue: implications for diffusion-weighted imaging

Non-Gaussian diffusion imaging for enhanced contrast of brain tissue affected by ischemic stroke

Recent diffusion MRI studies of stroke in humans and animals have shown that the quantitative parameters characterising the degree of non-Gaussianity of the diffusion process are much more sensitive to ischemic changes than the apparent diffusion coefficient (ADC) considered so far as the "gold standard". The observed changes exceeded that of the ADC by a remarkable factor of 2 to 3. These studies were based on the novel non-Gaussian methods, such as diffusion kurtosis imaging (DKI) and log-normal distribution function imaging (LNDFI). As shown in our previous work investigating the animal stroke model, a combined analysis using two methods, DKI and LNDFI provides valuable complimentary information. In the present work, we report the application of three non-Gaussian diffusion models to quantify the deviations from the Gaussian behaviour in stroke induced by transient middle cerebral artery occlusion in rat brains: the gamma-distribution function (GDF), the stretched exponential model (SEM), and the biexponential model. The main goal was to compare the sensitivity of various non-Gaussian metrics to ischemic changes and to investigate if a combined application of several models will provide added value in the assessment of stroke. We have shown that two models, GDF and SEM, exhibit a better performance than the conventional method and allow for a significantly enhanced visualization of lesions. Furthermore, we showed that valuable information regarding spatial properties of stroke lesions can be obtained. In particular, we observed a stratified cortex structure in the lesions that were well visible in the maps of the GDF and SEM metrics, but poorly distinguishable in the ADC-maps. Our results provided evidence that cortical layers tend to be differently affected by ischemic processes.

Differentiating white matter lesions in multiple sclerosis and migraine using monoexponential and biexponential diffusion measurements

PURPOSE

To compare the white matter lesions seen in multiple sclerosis and migraine using monoexponential and high b-value biexponential diffusion measurements.

MATERIALS AND METHODS

Diffusion-weighted images were acquired on a 3.0-Tesla magnetic resonance imaging system. Diffusion parameters were estimated using monoexponential (0-1000 s/mm(2) ) and biexponential (0-5000 s/mm(2) ) approaches from 15 multiple sclerosis patients, 15 patients with migraine and 15 healthy control subjects. The study was performed in accordance with the approval of the Regional Research Ethics Committee. The apparent diffusion coefficient (ADC) values were measured in the lesions and the normal-appearing white matter of patients and in the white matter of controls.

RESULTS

High lesional ADCmono values were detected in both patient groups without significant differences between the groups (10.72 and 9.86 × 10(-4) mm(2) /s for MS and migraine respectively, P = 0.2134). The biexponential measurements showed significantly higher ADCfast , ADCslow , and Pslow values in the migraine lesions than in the multiple sclerosis lesions (16.47 versus 14.29, 1.41 versus 0.76, and 20.34 versus 12.01 all values in 10(-4) mm(2) /s; P = 0.0344, P = 0.0019, P = 0.0021, respectively).

CONCLUSION

Biexponential diffusion analysis may help to differentiate multiple sclerosis-related white matter lesions from migraine-related ones.

Non-Gaussian water diffusion kurtosis imaging of prostate cancer

PURPOSE

To evaluate the non-Gaussian water diffusion properties of prostate cancer (PCa) and determine the diagnostic performance of diffusion kurtosis (DK) imaging for distinguishing PCa from benign tissues within the peripheral zone (PZ), and assessing tumor lesions with different Gleason scores.

MATERIALS AND METHODS

Nineteen patients who underwent diffusion weighted (DW) magnetic resonance imaging using multiple b-values and were pathologically confirmed with PCa were enrolled in this study. Apparent diffusion coefficient (ADC) was derived using a monoexponential model, while diffusion coefficient (D) and kurtosis (K) were determined using a DK model. Differences between the ADC, D and K values of benign PZ and PCa, as well as those of tumor lesions with Gleason scores of 6, 7 and ≥8 were assessed. Correlations between parameters D and K in PCa were analyzed using Pearson's correlation coefficient. ADC, D and K values were correlated with Gleason scores of 6, 7 and ≥8, respectively.

RESULTS

ADC and D values were significantly (p<0.001) lower in PCa (0.79±0.14μm(2)/ms and 1.56±0.23μm(2)/ms, respectively) compared to benign PZ (1.23±0.19μm(2)/ms and 2.54±0.24μm(2)/ms, respectively). K values were significantly (p<0.001) greater in PCa (0.96±0.20) compared to benign PZ (0.59±0.08). D and K showed fewer overlapping values between benign PZ and PCa compared to ADC. There was a strong negative correlation between D and K values in PCa (Pearson correlation coefficient r=-0.729; p<0.001). ADC and K values differed significantly in tumor lesions with Gleason scores of 6, 7 and ≥8 (p<0.001 and p=0.001, respectively), although no significant difference was detected for D values (p=0.325). Significant correlations were found between the ADC value and Gleason score (r=-0.828; p<0.001), as well as the K value and Gleason score (r=0.729; p<0.001).

CONCLUSION

DK model may add value in PCa detection and diagnosis. K potentially offers a new metric for assessment of PCa.

Modeling of the diffusion MR signal in calibrated model systems and nerves

Diffusion NMR is a powerful tool for gleaning microstructural information on opaque systems. In this work, the signal decay in single-pulsed-field gradient diffusion NMR experiments performed on a series of phantoms of increasing complexity, where the ground truth is known a priori, was modeled and used to identify microstructural features of these complex phantoms. We were able to demonstrate that, without assuming the number of components or compartments, the modeling can identify the number of restricted components, detect their sizes with an accuracy of a fraction of a micrometer, determine their relative populations, and identify and characterize free diffusion when present in addition to the components exhibiting restricted diffusion. After the accuracy of the modeling had been demonstrated, this same approach was used to study fixed nerves under different experimental conditions. It seems that this approach is able to characterize both the averaged axon diameter and the relative population of the different diffusing components in the neuronal tissues examined.

q-Space diffusion MRI (QSI) of the disease progression in the spinal cords of the Long Evans shaker: diffusion time and apparent anisotropy

q-Space diffusion MRI (QSI) was used to study the spinal cords of Long Evans shaker (les) rats, a model of dysmyelination, and their age-matched controls at different maturation stages. Diffusion was measured parallel and perpendicular to the fibers of the spinal cords of the two groups and at different diffusion times. The results showed that QSI is able to detect the dysmyelination process that occurs in this model in the different stages of the disease. The differences in the diffusion characteristics of the spinal cords of the two groups were found to be larger when the diffusion time was increased from 22 to 100 ms. We found that the radial mean displacement is a much better parameter than the QSI fractional anisotropy (FA) to document the differences between the two groups. We observed that the degree of myelination affects the diffusion characteristics of the tissues, but has a smaller effect on FA. All of the extracted diffusion parameters that are affected by the degree of myelination are affected in a diffusion time-dependent fashion, suggesting that the terms apparent anisotropy, apparent fractional anisotropy and even apparent root-mean-square displacement (rmsD) are more appropriate.

Diffusion kurtosis imaging study of prostate cancer: preliminary findings

PURPOSE

To evaluate the differences in parameters of diffusion kurtosis imaging (DKI) between prostate cancer, benign prostatic hyperplasia (BPH), and benign peripheral zone (PZ).

MATERIALS AND METHODS

Twenty-four foci of prostate cancer, 41 BPH nodules (14 stromal and 27 nonstromal hyperplasia), and 20 benign PZ from 20 patients who underwent radical prostatectomy were investigated. Diffusion-weighted imaging (DWI) was performed using 11 b-values (0-1500 s/mm(2) ). DKI model relates DWI signal decay to parameters that reflect non-Gaussian diffusion coefficient (D) and deviations from normal distribution (K). A mixed model analysis of variance and receiver operating characteristic (ROC) analyses were performed to assess the statistical significance of the metrics of DKI and apparent diffusion coefficient (ADC).

RESULTS

K was significantly higher in prostate cancer and stromal BPH than in benign PZ (1.19 ± 0.24 and 0.99 ± 0.28 versus 0.63 ± 0.23, P < 0.001 and P < 0.001, respectively). K showed a trend toward higher levels in prostate cancer than in stromal BPH (1.19 ± 0.24 versus 0.99 ± 0.28, P = 0.051). On the ROC analyses, a significant difference in area under the curve was not observed between K and ADC, however, K showed the highest sensitivity among three parameters.

CONCLUSION

DKI may contribute to the imaging diagnosis of prostate cancer, especially in the differential diagnosis of prostate cancer and BPH.

Differentiation of true progression from pseudoprogression in glioblastoma treated with radiation therapy and concomitant temozolomide: comparison study of standard and high-b-value diffusion-weighted imaging

PURPOSE

To explore the role of histogram analysis of apparent diffusion coefficient (ADC) maps obtained at standard- and high-b-value (1000 and 3000 sec/mm(2), respectively) diffusion-weighted (DW) imaging in the differentiation of true progression from pseudoprogression in glioblastoma treated with radiation therapy and concomitant temozolomide.

MATERIALS AND METHODS

This retrospective study was approved by the institutional review board of Seoul National University Hospital, and informed consent requirement was waived. Thirty patients with histopathologically proved glioblastoma who had undergone concurrent chemotherapy and radiation therapy (CCRT) with temozolomide underwent diffusion-weighted MR imaging with b values of 1000 and 3000 sec/mm(2), and corresponding ADC maps were calculated from entire newly developed or enlarged enhancing lesions after completion of CCRT. Histogram parameters of each ADC map between true progression (n = 15) and pseudoprogression (n = 15) groups were compared by using the unpaired Student t test. Receiver operating characteristic analysis was used to determine the best cutoff values for predictors in the differentiation of true progression from pseudoprogression. Results were validated in an independent test set of nine patients by using the best cutoff value to predict differentiation of true progression from pseudoprogression. The accuracy of the selected best cutoff value in the independent test set was then calculated.

RESULTS

In terms of cumulative histograms, the fifth percentile of both ADC at b value of 1000 sec/mm(2) (ADC1000) and the ADC at b value of 3000 sec/mm(2) (ADC3000) were significantly lower in the true progression group than in the pseudoprogression group (P = .049 and P < .001, respectively). In contrast, neither the mean ADC1000 nor the mean ADC3000 was significantly different between the two groups. The diagnostic values of the parameters derived from ADC1000 and ADC3000 were compared, and a significant difference (0.224, P = .016) was found between the area under the receiver operating characteristic curve of the fifth percentile for ADC1000 and that for ADC3000. The accuracies were 66.7% (six of nine patients) and 88.9% (eight of nine patients) based on the fifth percentile of both ADC1000 and ADC3000 in the independent test set, respectively.

CONCLUSION

The fifth percentile of the cumulative ADC histogram obtained at a high b value was the most promising parameter in the differentiation of true progression from pseudoprogression of the newly developed or enlarged enhancing lesions after CCRT with temozolomide for glioblastoma treatment. Online supplemental material is available for this article.

Avian egg latebra as brain tissue water diffusion model

PURPOSE

Simplified models of non-monoexponential diffusion signal decay are of great interest to study the basic constituents of complex diffusion behavior in tissues. The latebra, a unique structure uniformly present in the yolk of avian eggs, exhibits a non-monoexponential diffusion signal decay. This model is more complex than simple phantoms based on differences between water and lipid diffusion, but is also devoid of microscopic structures with preferential orientation or perfusion effects.

METHODS

Diffusion scans with multiple b-values were performed on a clinical 3 Tesla system in raw and boiled chicken eggs equilibrated to room temperature. Diffusion encoding was applied over the ranges 5-5,000 and 5-50,000 s/mm(2). A low read-out bandwidth and chemical shift was used for reliable lipid/water separation. Signal decays were fitted with exponential functions.

RESULTS

The latebra, when measured over the 5-5,000 s/mm(2) range, exhibited independent of preparation clearly biexponential diffusion, with diffusion parameters similar to those typically observed in in vivo human brain. For the range 5-50,000 s/mm(2), there was evidence of a small third, very slow diffusing water component.

CONCLUSION

The latebra of the avian egg contains membrane structures, which may explain a deviation from a simple monoexponential diffusion signal decay, which is remarkably similar to the deviation observed in brain tissue.

Multiparameter MR imaging in the 6-OPRI variant of inherited prion disease

BACKGROUND AND PURPOSE

Inherited prion diseases represent over 15% of human prion cases and are a frequent cause of early onset dementia. The purpose of this study was to define the distribution of changes in cerebral volumetric and microstructural parenchymal tissues in a specific inherited human prion disease mutation combining VBM with VBA of cerebral MTR and MD.

MATERIALS AND METHODS

VBM and VBA of cerebral MTR and MD were performed in 16 healthy control participants and 9 patients with the 6-OPRI mutation. An analysis of covariance consisting of diagnostic grouping with age and total intracranial volume as covariates was performed.

RESULTS

On VBM, there was a significant reduction in gray matter volume in patients compared with control participants in the basal ganglia, perisylvian cortex, lingual gyrus, and precuneus. Significant MTR reduction and MD increases were more anatomically extensive than volume differences on VBM in the same cortical areas, but MTR and MD changes were not seen in the basal ganglia.

CONCLUSIONS

Gray matter and WM changes were seen in brain areas associated with motor and cognitive functions known to be impaired in patients with the 6-OPRI mutation. There were some differences in the anatomic distribution of MTR-VBA and MD-VBA changes compared with VBM, likely to reflect regional variations in the type and degree of the respective pathophysiologic substrates. Combined analysis of complementary multiparameter MR imaging data furthers our understanding of prion disease pathophysiology.

Hypothalamic metabolic compartmentation during appetite regulation as revealed by magnetic resonance imaging and spectroscopy methods

We review the role of neuroglial compartmentation and transcellular neurotransmitter cycling during hypothalamic appetite regulation as detected by Magnetic Resonance Imaging (MRI) and Spectroscopy (MRS) methods. We address first the neurochemical basis of neuroendocrine regulation in the hypothalamus and the orexigenic and anorexigenic feed-back loops that control appetite. Then we examine the main MRI and MRS strategies that have been used to investigate appetite regulation. Manganese-enhanced magnetic resonance imaging (MEMRI), Blood oxygenation level-dependent contrast (BOLD), and Diffusion-weighted magnetic resonance imaging (DWI) have revealed Mn²⁺ accumulations, augmented oxygen consumptions, and astrocytic swelling in the hypothalamus under fasting conditions, respectively. High field ¹H magnetic resonance in vivo, showed increased hypothalamic myo-inositol concentrations as compared to other cerebral structures. ¹H and ¹³C high resolution magic angle spinning (HRMAS) revealed increased neuroglial oxidative and glycolytic metabolism, as well as increased hypothalamic glutamatergic and GABAergic neurotransmissions under orexigenic stimulation. We propose here an integrative interpretation of all these findings suggesting that the neuroendocrine regulation of appetite is supported by important ionic and metabolic transcellular fluxes which begin at the tripartite orexigenic clefts and become extended spatially in the hypothalamus through astrocytic networks becoming eventually MRI and MRS detectable.

White matter organization in relation to upper limb motor control in healthy subjects: exploring the added value of diffusion kurtosis imaging

Diffusion tensor imaging (DTI) characterizes white matter (WM) microstructure. In many brain regions, however, the assumption that the diffusion probability distribution is Gaussian may be invalid, even at low b values. Recently, diffusion kurtosis imaging (DKI) was suggested to more accurately estimate this distribution. We explored the added value of DKI in studying the relation between WM microstructure and upper limb coordination in healthy controls (N = 24). Performance on a complex bimanual tracking task was studied with respect to the conventional DTI measures (DKI or DTI derived) and kurtosis metrics of WM tracts/regions carrying efferent (motor) output from the brain, corpus callosum (CC) substructures and whole brain WM. For both estimation models, motor performance was associated with fractional anisotropy (FA) of the CC-genu, CC-body, the anterior limb of the internal capsule, and whole brain WM (r s range 0.42-0.63). Although DKI revealed higher mean, radial and axial diffusivity and lower FA than DTI (p < 0.001), the correlation coefficients were comparable. Finally, better motor performance was associated with increased mean and radial kurtosis and kurtosis anisotropy (r s range 0.43-0.55). In conclusion, DKI provided additional information, but did not show increased sensitivity to detect relations between WM microstructure and bimanual performance in healthy controls.

Diffusion tensor imaging of normal-appearing white matter in unilateral cerebral arterial occlusive disease

PURPOSE

To use MR with diffusion tensor imaging (DTI) and conventional and high b value to assess diffusion changes in normal-appearing white matter (NAWM) in patients with unilateral, severe stenosis, or occlusion of the middle cerebral artery (MCA).

MATERIALS AND METHODS

In total, 28 patients with NAWM and unilateral, severe stenosis, or occlusion of the MCA underwent DTI with b values 1000 and 2200 s/mm(2) at 3.0T MR. Fractional anisotropy (FA), apparent diffusion coefficient (ADC), radial diffusivity (eigenvalues λ1 , λ2 ), and axial diffusivity (eigenvalue λ3 ) were measured for the ipsilateral and contralateral corona radiata.

RESULTS

Mean FA was significantly lower for the ipsilateral than contralateral corona radiata with high b value, 2200 s/mm(2) , and ipsilateral corona radiata with conventional low b value, 1000 s/mm(2) (all P < 0.01). Mean ADC, λ1 , λ2 , and λ3 were significantly higher for the ipsilateral than contralateral corona radiata with high b value (all P < 0.05) but not for ipsilateral than contralateral corona radiata with low b value (P > 0.05).

CONCLUSION

DTI with a high b value detects diffusion changes in NAWM in patients with unilateral, severe stenosis, or occlusion of the MCA not seen with conventional b value or conventional MRI contrasts.

White matter maturation in the brains of Long Evans shaker myelin mutant rats by ex-vivo QSI and DTI

The brains of Long Evans shaker (les) rats, a model of dysmyelination, and their age- matched controls were studied by ex-vivo q-space diffusion imaging (QSI) and diffusion tensor imaging (DTI). The QSI and DTI indices were computed from the same acquisition. The les and the control brains were studied at different stages of maturation and disease progression. The mean displacement, the probability for zero displacement and kurtosis were computed from QSI data while the fractional anisotropy (FA) and the eigenvalues were computed from DTI. It was found that all QSI indices detect the les pathology, at all stages of maturation, while only some of the DTI indices could detect the les pathology. The QSI mean displacement was larger in the les group as compared with their age-matched controls while the probability for zero displacement and the kurtosis were both lower all indicating higher degree of restriction in the control brains. Since all the DTI eigenvalues were higher in the les brains as compared to controls, the less efficient DTI measure for discerning the les pathology was found to be the FA. Clearly, the most sensitive DTI parameter to the les pathology is λ3, i.e., the minimal diffusivity. Since the QSI and DTI data were obtained from the same acquisition, despite the somewhat higher SNR of the QSI data compared to the DTI data, it seems that the higher diagnostic capacity of the QSI data in this experimental model of dysmyelination, originates mainly from the higher diffusing weighting of the QSI data.

Sensitivities of statistical distribution model and diffusion kurtosis model in varying microstructural environments: a Monte Carlo study

The aim of this study was to investigate the microstructural sensitivity of the statistical distribution and diffusion kurtosis (DKI) models of non-monoexponential signal attenuation in the brain using diffusion-weighted MRI (DWI). We first developed a simulation of 2-D water diffusion inside simulated tissue consisting of semi-permeable cells and a variable cell size. We simulated a DWI acquisition of the signal in a volume using a pulsed gradient spin echo (PGSE) pulse sequence, and fitted the models to the simulated DWI signals using b-values up to 2500 s/mm(2). For comparison, we calculated the apparent diffusion coefficient (ADC) of the monoexponential model (b-value=1000 s/mm(2)). In separate experiments, we varied the cell size (5-10-15 μm), cell volume fraction (0.50-0.65-0.80), and membrane permeability (0.001-0.01-0.1mm/s) to study how the fitted parameters tracked simulated microstructural changes. The ADC was sensitive to all the simulated microstructural changes except the decrease in membrane permeability. The ADC increased with larger cell size, smaller cell volume fraction, and larger membrane permeability. The σstat of the statistical distribution model increased exclusively with a decrease in cell volume fraction. The Kapp of the DKI model was exclusively increased with decreased cell size and decreased with increasing membrane permeability. These results suggest that the non-monoexponential models of water diffusion have different, specific microstructural sensitivity, and a combination of the models may give insights into the microstructural underpinning of tissue pathology.

Brain connectivity plasticity in the motor network after ischemic stroke

The motor function is controlled by the motor system that comprises a series of cortical and subcortical areas interacting via anatomical connections. The motor function will be disturbed when the stroke lesion impairs either any of these areas or their connections. More and more evidence indicates that the reorganization of the motor network including both areas and their anatomical and functional connectivity might contribute to the motor recovery after stroke. Here, we review recent studies employing models of anatomical, functional, and effective connectivity on neuroimaging data to investigate how ischemic stroke influences the connectivity of motor areas and how changes in connectivity relate to impaired function and functional recovery. We suggest that connectivity changes constitute an important pathophysiological aspect of motor impairment after stroke and important mechanisms of motor recovery. We also demonstrate that therapeutic interventions may facilitate motor recovery after stroke by modulating the connectivity among the motor areas. In conclusion, connectivity analyses improved our understanding of the mechanisms of motor recovery after stroke and may help to design hypothesis-driven treatment strategies and sensitive measures for outcome prediction in stroke patients.

High b-value diffusion tensor imaging of the remote white matter and white matter of obstructive unilateral cerebral arterial regions

AIM

To assess diffusion changes in the remote white matter and areas of white matter with cerebral artery obstruction without magnetic resonance imaging (MRI) evidence of brain parenchymal abnormalities using high b-value diffusion tensor imaging (DTI).

MATERIALS AND METHODS

A total of 34 patients with severe unilateral stenosis (≥75%) or occlusion of the middle cerebral artery (MCA) without abnormal brain parenchymal signals at MRI underwent DTI with a b value of 2200 s/mm(2) at 3 T. Fractional anisotropy (FA), apparent diffusion coefficient (ADC), axial diffusivity (eigenvalue λ₁) and radial diffusivity (eigenvalue λ₂₃) were measured at the bilateral corona radiata, anterior and posterior limbs of the internal capsule, cerebral peduncle, and pons.

RESULTS

The mean FA was significantly lower at the ipsilateral corona radiata and anterior and posterior limbs of the internal capsule than at the contralateral corona radiata and anterior and posterior limbs of the internal capsule (p < 0.05). The mean ADC, λ₁ and λ₂₃ were significantly higher at the ipsilateral corona radiata than at the contralateral corona radiata (p < 0.01). The mean λ₂₃ were significantly higher at the ipsilateral anterior and posterior limb of the internal capsule than at the contralateral anterior and posterior limb of the internal capsule (p < 0.05). The mean ADC, λ₁ and λ₂ ₃were not significantly different between the ipsilateral cerebral peduncle and pons.

CONCLUSIONS

High b-value DTI could sensitively reveal diffusion changes in white matter in regions of cerebral artery obstruction without abnormal anisotropy and diffusivity of the remote white matter of patients with severe MCA stenosis or occlusion without MRI evidence of brain parenchymal abnormalities.

Associations among q-space MRI, diffusion-weighted MRI and histopathological parameters in meningiomas

OBJECTIVES

The purposes of this MR-based study were to calculate q-space imaging (QSI)-derived mean displacement (MDP) in meningiomas, to evaluate the correlation of MDP values with apparent diffusion coefficient (ADC) and to investigate the relationships among these diffusion parameters, tumour cell count (TCC) and MIB-1 labelling index (LI).

METHODS

MRI, including QSI and conventional diffusion-weighted imaging (DWI), was performed in 44 meningioma patients (52 lesions). ADC and MDP maps were acquired from post-processing of the data. Quantitative analyses of these maps were performed by applying regions of interest. Pearson correlation coefficients were calculated for ADC and MDP in all lesions and for ADC and TCC, MDP and TCC, ADC and MIB-1 LI, and MDP and MIB-1 LI in 17 patients who underwent subsequent surgery.

RESULTS

ADC and MDP values were found to have a strong correlation: r = 0.78 (P = <0.0001). Both ADC and MDP values had a significant negative association with TCC: r = -0.53 (p = 0.02) and -0.48 (P = 0.04), respectively. MIB-1 LI was not, however, found to have a significant association with these diffusion parameters.

CONCLUSION

In meningiomas, both ADC and MDP may be representative of cell density.

KEY POINTS

• Diffusion-weighted MRI offers possibilities to assess the aggressiveness of meningiomas. • The q-space imaging-derived mean displacement correlates strongly with apparent diffusion coefficients. • Both diffusion parameters showed a strong negative association with tumour cell counts. • Derived mean displacement may help assess the aggressiveness of meningiomas preoperatively.

The role of tissue microstructure and water exchange in biophysical modelling of diffusion in white matter

Biophysical models that describe the outcome of white matter diffusion MRI experiments have various degrees of complexity. While the simplest models assume equal-sized and parallel axons, more elaborate ones may include distributions of axon diameters and axonal orientation dispersions. These microstructural features can be inferred from diffusion-weighted signal attenuation curves by solving an inverse problem, validated in several Monte Carlo simulation studies. Model development has been paralleled by microscopy studies of the microstructure of excised and fixed nerves, confirming that axon diameter estimates from diffusion measurements agree with those from microscopy. However, results obtained in vivo are less conclusive. For example, the amount of slowly diffusing water is lower than expected, and the diffusion-encoded signal is apparently insensitive to diffusion time variations, contrary to what may be expected. Recent understandings of the resolution limit in diffusion MRI, the rate of water exchange, and the presence of microscopic axonal undulation and axonal orientation dispersions may, however, explain such apparent contradictions. Knowledge of the effects of biophysical mechanisms on water diffusion in tissue can be used to predict the outcome of diffusion tensor imaging (DTI) and of diffusion kurtosis imaging (DKI) studies. Alterations of DTI or DKI parameters found in studies of pathologies such as ischemic stroke can thus be compared with those predicted by modelling. Observations in agreement with the predictions strengthen the credibility of biophysical models; those in disagreement could provide clues of how to improve them. DKI is particularly suited for this purpose; it is performed using higher b-values than DTI, and thus carries more information about the tissue microstructure. The purpose of this review is to provide an update on the current understanding of how various properties of the tissue microstructure and the rate of water exchange between microenvironments are reflected in diffusion MRI measurements. We focus on the use of biophysical models for extracting tissue-specific parameters from data obtained with single PGSE sequences on clinical MRI scanners, but results obtained with animal MRI scanners are also considered. While modelling of white matter is the central theme, experiments on model systems that highlight important aspects of the biophysical models are also reviewed.

Biexponential apparent diffusion coefficients values in the prostate: comparison among normal tissue, prostate cancer, benign prostatic hyperplasia and prostatitis

Objective

To investigate the biexponential apparent diffusion parameters of diverse prostate tissues and compare them with monoexponential apparent diffusion coefficient (ADC) value in the efficacy to discriminate prostate cancer from benign lesions.

Materials and Methods

Eleven healthy volunteers and 61 patients underwent a conventional (b-factors 0, 1000 s/mm²) and a 10 b-factor (0 to 3000 s/mm²) diffusion-weighted imaging (DWI). The monoexponential ADC value and biexponential parameters of fast ADC (ADCf), fraction of ADCf (f), slow ADC (ADCs) value for 29 prostate cancer, 28 benign prostatic hyperplasia (BPH), 24 prostatitis lesions and normal tissue were calculated and compared. Receiver operating characteristic analysis was performed to determine the sensitivity, specificity and optimal cut-off points.

Results

Prostate cancer had lower ADC, ADCf, f, and ADCs than all other tissues (p < 0.01). Prostatitis exhibited a lower ADC, ADCf, ADCs and f than the peripheral zone tissue (p < 0.01), and BPH showed a lower ADC and ADCf than the central gland tissue (p < 0.01). The ADCf demonstrated a comparable accuracy with ADC in differentiating cancer from BPH [area under the curve (AUC) 0.93 vs. 0.92] and prostatitis AUC 0.98 vs. 0.99) (both p > 0.05), but the AUC of f and ADCs in differentiating cancer from BPH (0.73 and 0.81) and prostatitis (0.88 and 0.91) were significantly lower than ADC (all p < 0.05).

Conclusion

The biexponential DWI appears to provide additional parameters for tissue characterization in prostate, and ADCf helps to yield comparable accuracy with ADC in differentiating cancer from benign lesions.

Serial MR analysis of early permanent and transient ischemia in rats: diffusion tensor imaging and high b value diffusion weighted imaging

Objective

To evaluate the temporal evolution and diagnostic values of the diffusion tensor imaging (DTI) and the high b value diffusion weighted imaging (DWI) in the early permanent and transient cerebral ischemia.

Materials and Methods

For permanent or 30-minute transient-ischemia induced 30 rats, DTI and DWIs at both high b (b = 3000 s/mm²) and standard b value (b = 1000 s/mm²) were obtained at the following conditions: at 15, 30, 45, 60 minutes after the occlusion of what for hyperacute permanent ischemia; at 1, 3, 5, 7, 9 hours after the occlusion for acute permanent ischemia; and at 15 minutes before reperfusion, 0.5, 2.5, and 24 hours after reperfusion for transient ischemia. The diffusion parameters and their ratios were obtained and compared between different b values, and among different time points and groups, respectively.

Results

For both b values, the apparent diffusion coefficient (ADC) ratio decreased for first three hours, and then slightly increased until 9 hours after the occlusion during a gradual continuous increase of DWI signal intensity (SI) ratio, with excellent correlation between ADC ratios and DWI SI ratios. The DWI showed a higher contrast ratio, but the ADC map showed a lower contrast ratio for permanent ischemia at high b value than at standard b value. Fractional anisotropy (FA) increased for 1 hour, then gradually decreased until 9 hours after the occlusion in permanent ischemia and showed transient normalization and secondary decay along with change in ADC in transient ischemia.

Conclusion

This study presents characteristic initial elevation and secondary decay of FA, higher contrast ratio of DWI, and lower contrast ratio of ADC map at high b value, in addition to the time evolutions of diffusion parameters in early permanent and transient ischemia.

Probing tissue microstructure with restriction spectrum imaging: Histological and theoretical validation

Water diffusion magnetic resonance imaging (dMRI) is a powerful tool for studying biological tissue microarchitectures in vivo. Recently, there has been increased effort to develop quantitative dMRI methods to probe both length scale and orientation information in diffusion media. Diffusion spectrum imaging (DSI) is one such approach that aims to resolve such information based on the three-dimensional diffusion propagator at each voxel. However, in practice, only the orientation component of the propagator function is preserved when deriving the orientation distribution function. Here, we demonstrate how a straightforward extension of the linear spherical deconvolution (SD) model can be used to probe tissue orientation structures over a range (or "spectrum") of length scales with minimal assumptions on the underlying microarchitecture. Using high b-value Cartesian q-space data on a rat brain tissue sample, we demonstrate how this "restriction spectrum imaging" (RSI) model allows for separating the volume fraction and orientation distribution of hindered and restricted diffusion, which we argue stems primarily from diffusion in the extraneurite and intraneurite water compartment, respectively. Moreover, we demonstrate how empirical RSI estimates of the neurite orientation distribution and volume fraction capture important additional structure not afforded by traditional DSI or fixed-scale SD-like reconstructions, particularly in gray matter. We conclude that incorporating length scale information in geometric models of diffusion offers promise for advancing state-of-the-art dMRI methods beyond white matter into gray matter structures while allowing more detailed quantitative characterization of water compartmentalization and histoarchitecture of healthy and diseased tissue.

Intravoxel incoherent motion MR imaging for prostate cancer: an evaluation of perfusion fraction and diffusion coefficient derived from different b-value combinations

There has been a resurgent interest in intravoxel incoherent motion (IVIM) MR imaging to obtain perfusion as well as diffusion information on lesions, in which the diffusion was modeled as Gaussian diffusion. However, it was observed that this diffusion deviated from expected monoexponential decay at high b-values and the reported perfusion in prostate is contrary to the findings in dynamic contrast-enhanced (DCE) MRI studies and angiogenesis. Thus, this work is to evaluate the effect of different b-values on IVIM perfusion fractions (f) and diffusion coefficients (D) for prostate cancer detection. The results show that both parameters depended heavily on the b-values, and those derived without the highest b-value correlated best with the results from DCE-MRI studies; specifically, f was significantly elevated (7.2% vs. 3.7%) in tumors when compared with normal tissues, in accordance with the volume transfer constant (K(trans); 0.39 vs. 0.18 min(-1)) and plasma fractional volume (v(p) ; 8.4% vs. 3.4%). In conclusion, it is critical to choose an appropriate range of b-values in studies or include the non-Gaussian diffusion contribution to obtain unbiased IVIM measurements. These measurements could eliminate the need for DCE-MRI, which is especially relevant in patients who cannot receive intravenous gadolinium-based contrast media.

Preoperative histological grading of meningiomas using apparent diffusion coefficient at 3T MRI

PURPOSE

We assessed whether a high b-value DWI at b=4000s/mm(2) would discriminate the histopathological differentiation of the tumor grade of meningiomas, and also focused on the relationship between radiologic features and the tumor grade.

MATERIALS AND METHODS

We acquired DWI at 3T with b=1000 and b=4000s/mm(2) in 77 patients (42, 31 and 4 patients were WHO grades I (G1), II (G2), and III (G3), respectively). The apparent diffusion coefficient (ADC) was measured by placing multiple regions of interest (ROIs) on ADC maps. The ADC values of each tumor were determined preoperatively from several ROIs, and expressed as the minimum (ADCMIN), mean (ADCMEAN), and maximum absolute values (ADCMAX). We evaluated the relationship between ADCs and histological findings, and assessed the radiologic features such as tumor location, tumor size, presence/absence of peritumoral edema, shape of the tumor, presence/absence of bone destruction or hyperplasia, status of contrast enhancement, presence/absence of calcification and cyst.

RESULTS

ADCs of the meningiomas were inversely correlated with the histological grade of meningiomas. According to results of the discriminant analysis, the apparent log likelihood value was greatest for ADCMIN at b=4000. Furthermore, only the ADCMIN value at b=4000 was significantly correlated with the histological grade of meningiomas when we performed a multiple logistic regression analysis to identify the significant independent factors such as shape of tumor, presence/absence of bone destruction, status of contrast enhancement, presence/absence of cyst and ADCMIN at b=4000.

CONCLUSION

A meningioma with a low ADCMIN at a high b-value might imply a high-grade meningioma.

The physical and biological basis of quantitative parameters derived from diffusion MRI

Diffusion magnetic resonance imaging is a quantitative imaging technique that measures the underlying molecular diffusion of protons. Diffusion-weighted imaging (DWI) quantifies the apparent diffusion coefficient (ADC) which was first used to detect early ischemic stroke. However this does not take account of the directional dependence of diffusion seen in biological systems (anisotropy).Diffusion tensor imaging (DTI) provides a mathematical model of diffusion anisotropy and is widely used. Parameters, including fractional anisotropy (FA), mean diffusivity (MD), parallel and perpendicular diffusivity can be derived to provide sensitive, but non-specific, measures of altered tissue structure. They are typically assessed in clinical studies by voxel-based or region-of-interest based analyses.The increasing recognition of the limitations of the diffusion tensor model has led to more complex multi-compartment models such as CHARMED, AxCaliber or NODDI being developed to estimate microstructural parameters including axonal diameter, axonal density and fiber orientations. However these are not yet in routine clinical use due to lengthy acquisition times.In this review, I discuss how molecular diffusion may be measured using diffusion MRI, the biological and physical bases for the parameters derived from DWI and DTI, how these are used in clinical studies and the prospect of more complex tissue models providing helpful micro-structural information.

Cramér-Rao bound for Intravoxel Incoherent Motion Diffusion Weighted Imaging fitting

The precision of parameter estimation for Intravoxel Incoherent Motion Diffusion Weighted Imaging (IVIM-DWI) was investigated by examining their Cramér-Rao bounds (CRBs) under the presence of Rician noise. Monte Carlo (MC) simulation was also conducted to validate the CRB results. The estimation uncertainties of true diffusion coefficient (D) and perfusion fraction (f0) could reach 3.89% and 11.65% respectively with typical parameter values at a moderate signal-to-noise ratio (SNR) of 40. However, to estimate pseudo diffusion coefficient (D*) within 10% uncertainty requires SNR>122. The results also showed that the estimation precision of each parameter is not only dependent on SNR but also their true values, while this mutual dependency is complicated. Under some particular cases, estimation uncertainty for certain parameters might be smaller than 5% at a moderate SNR of 40. However, the simultaneous precise estimation for all three parameters is theoretically difficult and highly SNR demanding.

Diffusion Tensor Metric Measurements as a Function of Diffusion Time in the Rat Central Nervous System

MRI and Monte Carlo simulated data of pulsed gradient spin echo experiments were used to study the effects of diffusion time, gradient strength and b-value on diffusion tensor (DT) metrics using real and simulated fixed rat spines. Radial (λ⊥) in grey matter and simulation data, axial (λ||) in both grey and white matter in fixed rat spinal cords and mean diffusivity in all tissues showed a significant decrease with diffusion time at b = 1 μm2/ms. All diffusivities significantly decreased with b-value at g = 116 mT/m and at Δeff = 23 ms. The fractional anisotropy (FA) significantly increased with diffusion time at b = 1 μm2/ms in the simulation data and grey matter. FA significantly increased in white matter and simulation data and significantly decreased in grey matter with b-value at g = 116 mT/m and at Δeff = 23 ms. These data suggest that DTI metrics are highly dependent on pulse sequence parameters.

Parkinson's disease patients show reduced cortical-subcortical sensorimotor connectivity

Reduced dopamine input to cortical and subcortical brain structures, particularly those in the sensorimotor network, is a hallmark of Parkinson's disease (PD). The extent to which dopamine dysfunction affects connectivity within this and other brain networks remains to be investigated. The purpose of this study was to measure anatomical and functional connectivity in groups of PD patients and controls to determine whether connectivity deficits within the cortico-basal ganglia thalamocortical system could be attributed to PD, particularly in sensorimotor connections. A neuroimaging paradigm involving diffusion-weighted magnetic resonance imaging (MRI) and resting-state functional MRI was implemented in a large cohort of PD patients and control subjects. Probabilistic tractography and functional correlation analyses were performed to map connections between brain structures and to derive indices of connectivity that were then used to compare groups. Anatomical connectivity deficits were demonstrated in PD patients, specifically for sensorimotor connections. Functional deficits were also found in some of the same connections. In addition, functional connectivity was found to increase in associative and limbic connections in PD patients compared with controls. This study lends support to findings regarding the dysfunction of the sensorimotor circuit in PD. As deficits in anatomical and functional connectivity within this circuit were in some cases concordant in PD patients, a possible link between brain structure and function is suggested. Increases in functional connectivity in other cortico-basal ganglia thalamocortical circuits may be indicative of compensatory effects in response to system deficits elsewhere.

Methodology of diffusion-weighted, diffusion tensor and magnetisation transfer imaging

MRI offers a number of opportunities to examine characteristics of tissue well below the spatial resolution of the imaging technique. The best known of these is diffusion imaging, which allows the production of images whose contrast reflects the ability of water molecules to move through the extravascular extracellular space. Less well-known, but increasingly important, is magnetisation transfer imaging, which produces contrast based on the ability of protons to move between the free water pool and local macromolecules. Both of these techniques offer unique information about the microscopic and molecular structure of tumour tissue. This article will briefly review the underlying theory and technical aspects associated with these imaging techniques.

Histogram analysis of apparent diffusion coefficient map of standard and high B-value diffusion MR imaging in head and neck squamous cell carcinoma: a correlation study with histological grade

RATIONALE AND OBJECTIVES

A histologic grade in head and neck squamous cell carcinoma (HNSCC) is clinically important because of its association with prognosis. The purpose of this study was to investigate the efficacy of histographic analysis of apparent diffusion coefficient (ADC) maps on the basis of the entire tumor volume in differentiating histologic grades in HNSCC at standard (b = 1000 s/mm(2)) and high (b = 2000 s/mm(2)) b values.

MATERIALS AND METHODS

Fifty-four patients with HNSCC, including well-differentiated (WD; n = 35), moderately differentiated (MD; n = 13) and poorly differentiated (PD; n = 6) carcinomas, were retrospectively evaluated. ADC maps were obtained at two different b values (1000 and 2000 s/mm(2)) in each patient. Tumors were delineated on each slice of ADC maps, and data were collected to obtain a histogram for the entire tumor volume. Histographic parameters were calculated, including mean, standard deviation, kurtosis, skewness, and the ratio of the kurtosis measured at b values of 1000 and 2000 s/mm(2). These parameters were correlated with histologic grades.

RESULTS

There was no significant correlation between tumor grades and histographic parameters obtained from ADC maps at b = 1000 s/mm(2). However, mean ADC at b = 2000 s/mm(2) was significantly higher in WD HNSCC (881 ± 131 × 10(-6) mm(2)/s) than in MD and PD HNSCC (770 ± 163 and 780 ± 158 × 10(-6) mm(2)/s, respectively) (P < .05). Kurtosis ratio was significantly higher in PD HNSCC (115 ± 10%) compared to WD and MD HNSCC (91 ± 21% and 86 ± 26%, respectively) (P < .05). Diagnostic accuracy was 100%, 76.9%, and 65.8% for PD, MD, and WD HNSCC, respectively.

CONCLUSIONS

Histographic analysis of ADC maps on the basis of the entire tumor volume can be useful in differentiating histologic grades of HNSCC using mean ADC at b = 2000 s/mm(2) and kurtosis ratio.

Imaging hypothalamic activity using diffusion weighted magnetic resonance imaging in the mouse and human brain

Hypothalamic appetite regulation is a vital homeostatic process underlying global energy balance in animals and humans, its disturbances resulting in feeding disorders with high morbidity and mortality. The objective evaluation of appetite remains difficult, very often restricted to indirect measurements of food intake and body weight. We report here, the direct, non-invasive visualization of hypothalamic activation by fasting using diffusion weighted magnetic resonance imaging, in the mouse brain as well as in a preliminary study in the human brain. The brain of fed or fasted mice or humans were imaged at 7 or 1.5 Tesla, respectively, by diffusion weighted magnetic resonance imaging using a complete range of b values (10<b<2000s.mm(-2)). The diffusion weighted image data sets were registered and analyzed pixel by pixel using a biexponential model of diffusion, or a model-free Linear Discriminant Analysis approach. Biexponential fittings revealed statistically significant increases in the slow diffusion parameters of the model, consistent with a neurocellular swelling response in the fasted hypothalamus. Increased resolution approaches allowed the detection of increases in the diffusion parameters within the Arcuate Nucleus, Ventromedial Nucleus and Dorsomedial Nucleus. Independently, Linear Discriminant Analysis was able to classify successfully the diffusion data sets from mice and humans between fed and fasted states. Present results are consistent with increased glutamatergic neurotransmission during orexigenic firing, a process resulting in increased ionic accumulation and concomitant osmotic neurocellular swelling. This swelling response is spatially extendable through surrounding astrocytic networks until it becomes MRI detectable. Present findings open new avenues for the direct, non-invasive, evaluation of appetite disorders and other hypothalamic pathologies helping potentially in the development of the corresponding therapies.

Bessel Fourier Orientation Reconstruction (BFOR): an analytical diffusion propagator reconstruction for hybrid diffusion imaging and computation of q-space indices

The ensemble average propagator (EAP) describes the 3D average diffusion process of water molecules, capturing both its radial and angular contents. The EAP can thus provide richer information about complex tissue microstructure properties than the orientation distribution function (ODF), an angular feature of the EAP. Recently, several analytical EAP reconstruction schemes for multiple q-shell acquisitions have been proposed, such as diffusion propagator imaging (DPI) and spherical polar Fourier imaging (SPFI). In this study, a new analytical EAP reconstruction method is proposed, called Bessel Fourier Orientation Reconstruction (BFOR), whose solution is based on heat equation estimation of the diffusion signal for each shell acquisition, and is validated on both synthetic and real datasets. A significant portion of the paper is dedicated to comparing BFOR, SPFI, and DPI using hybrid, non-Cartesian sampling for multiple b-value acquisitions. Ways to mitigate the effects of Gibbs ringing on EAP reconstruction are also explored. In addition to analytical EAP reconstruction, the aforementioned modeling bases can be used to obtain rotationally invariant q-space indices of potential clinical value, an avenue which has not yet been thoroughly explored. Three such measures are computed: zero-displacement probability (Po), mean squared displacement (MSD), and generalized fractional anisotropy (GFA).

Noninvasive mapping of water diffusional exchange in the human brain using filter-exchange imaging

We present the first in vivo application of the filter-exchange imaging protocol for diffusion MRI. The protocol allows noninvasive mapping of the rate of water exchange between microenvironments with different self-diffusivities, such as the intracellular and extracellular spaces in tissue. Since diffusional water exchange across the cell membrane is a fundamental process in human physiology and pathophysiology, clinically feasible and noninvasive imaging of the water exchange rate would offer new means to diagnose disease and monitor treatment response in conditions such as cancer and edema. The in vivo use of filter-exchange imaging was demonstrated by studying the brain of five healthy volunteers and one intracranial tumor (meningioma). Apparent exchange rates in white matter range from 0.8±0.08 s(-1) in the internal capsule, to 1.6±0.11 s(-1) for frontal white matter, indicating that low values are associated with high myelination. Solid tumor displayed values of up to 2.9±0.8 s(-1). In white matter, the apparent exchange rate values suggest intra-axonal exchange times in the order of seconds, confirming the slow exchange assumption in the analysis of diffusion MRI data. We propose that filter-exchange imaging could be used clinically to map the water exchange rate in pathologies. Filter-exchange imaging may also be valuable for evaluating novel therapies targeting the function of aquaporins.

White matter integrity, fiber count, and other fallacies: the do's and don'ts of diffusion MRI

Diffusion-weighted MRI (DW-MRI) has been increasingly used in imaging neuroscience over the last decade. An early form of this technique, diffusion tensor imaging (DTI) was rapidly implemented by major MRI scanner companies as a scanner selling point. Due to the ease of use of such implementations, and the plausibility of some of their results, DTI was leapt on by imaging neuroscientists who saw it as a powerful and unique new tool for exploring the structural connectivity of human brain. However, DTI is a rather approximate technique, and its results have frequently been given implausible interpretations that have escaped proper critique and have appeared misleadingly in journals of high reputation. In order to encourage the use of improved DW-MRI methods, which have a better chance of characterizing the actual fiber structure of white matter, and to warn against the misuse and misinterpretation of DTI, we review the physics of DW-MRI, indicate currently preferred methodology, and explain the limits of interpretation of its results. We conclude with a list of 'Do's and Don'ts' which define good practice in this expanding area of imaging neuroscience.

WITHDRAWN: First observation of diffusion-diffraction pattern in neuronal tissue by double-pulsed-field-gradient NMR

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Head and neck squamous cell carcinoma: differentiation of histologic grade with standard- and high-b-value diffusion-weighted MRI

BACKGROUND

The correlation between the histologic grades and the apparent diffusion coefficient (ADC) in head and neck squamous cell carcinomas (HNSCCs) remains unclear. This study aimed to evaluate the potential of diffusion-weighted MRI (DW-MRI) at both standard and high b values to differentiate the histologic grades of HNSCC.

METHODS

In all, 54 consecutive patients with HNSCCs (34 well-differentiated, 10 moderately differentiated, and 10 poorly differentiated) performed DW-MRIs at both b = 1000 and 2000 s/mm(2) prior to biopsy or surgery. The ADC values were compared among the different histologic grades.

RESULTS

The ADC values of well-differentiated and poorly differentiated HNSCC were significantly different at both b values (p < .001 in both). However, significant difference between moderately differentiated and poorly differentiated HNSCC was revealed at only b = 2000 s/mm(2) (p = .014).

CONCLUSIONS

DW-MRIs at standard and high b values are helpful for differentiating histologic grades in HNSCC with better differentiation at a high b-value.

Accelerated diffusion spectrum imaging in the human brain using compressed sensing

We developed a novel method to accelerate diffusion spectrum imaging using compressed sensing. The method can be applied to either reduce acquisition time of diffusion spectrum imaging acquisition without losing critical information or to improve the resolution in diffusion space without increasing scan time. Unlike parallel imaging, compressed sensing can be applied to reconstruct a sub-Nyquist sampled dataset in domains other than the spatial one. Simulations of fiber crossings in 2D and 3D were performed to systematically evaluate the effect of compressed sensing reconstruction with different types of undersampling patterns (random, gaussian, Poisson disk) and different acceleration factors on radial and axial diffusion information. Experiments in brains of healthy volunteers were performed, where diffusion space was undersampled with different sampling patterns and reconstructed using compressed sensing. Essential information on diffusion properties, such as orientation distribution function, diffusion coefficient, and kurtosis is preserved up to an acceleration factor of R = 4.

Model-based analysis of multishell diffusion MR data for tractography: how to get over fitting problems

In this article, we highlight an issue that arises when using multiple b-values in a model-based analysis of diffusion MR data for tractography. The non-monoexponential decay, commonly observed in experimental data, is shown to induce overfitting in the distribution of fiber orientations when not considered in the model. Extra fiber orientations perpendicular to the main orientation arise to compensate for the slower apparent signal decay at higher b-values. We propose a simple extension to the ball and stick model based on a continuous gamma distribution of diffusivities, which significantly improves the fitting and reduces the overfitting. Using in vivo experimental data, we show that this model outperforms a simpler, noise floor model, especially at the interfaces between brain tissues, suggesting that partial volume effects are a major cause of the observed non-monoexponential decay. This model may be helpful for future data acquisition strategies that may attempt to combine multiple shells to improve estimates of fiber orientations in white matter and near the cortex.