A monte carlo study of restricted diffusion: Implications for diffusion MRI of prostate cancer

Spatial profiling of in vivo diffusion-weighted MRI parameters in the healthy human kidney

OBJECTIVE

Diffusion-weighted MRI is a technique that can infer microstructural and microcirculatory features from biological tissue, with particular application to renal tissue. There is extensive literature on diffusion tensor imaging (DTI) of anisotropy in the renal medulla, intravoxel incoherent motion (IVIM) measurements separating microstructural from microcirculation effects, and combinations of the two. However, interpretation of these features and adaptation of more specific models remains an ongoing challenge. One input to this process is a whole organ distillation of corticomedullary contrast of diffusion metrics, as has been explored for other renal biomarkers.

MATERIALS AND METHODS

In this work, we probe the spatial dependence of diffusion MRI metrics with concentrically layered segmentation in 11 healthy kidneys at 3 T. The metrics include those from DTI, IVIM, a combined approach titled "REnal Flow and Microstructure AnisotroPy (REFMAP)", and a multiply encoded model titled "FC-IVIM" providing estimates of fluid velocity and branching length.

RESULTS

Fractional anisotropy decreased from the inner kidney to the outer kidney with the strongest layer correlation in both parenchyma (including cortex and medulla) and medulla with Spearman correlation coefficients and p-values (r, p) of (0.42, <0.001) and (0.37, <0.001), respectively. Also, dynamic parameters derived from the three models significantly decreased with a high correlation from the inner to the outer parenchyma or medulla with (r, p) ranges of (0.46-0.55, <0.001).

CONCLUSIONS

These spatial trends might find implications for indirect assessments of kidney physiology and microstructure using diffusion MRI.

Molecular Imaging with Aquaporin-Based Reporter Genes: Quantitative Considerations from Monte Carlo Diffusion Simulations

Aquaporins provide a unique approach for imaging genetic activity in deep tissues by increasing the rate of cellular water diffusion, which generates a magnetic resonance contrast. However, distinguishing aquaporin signals from the tissue background is challenging because water diffusion is influenced by structural factors, such as cell size and packing density. Here, we developed a Monte Carlo model to analyze how cell radius and intracellular volume fraction quantitatively affect aquaporin signals. We demonstrated that a differential imaging approach based on subtracting signals at two diffusion times can improve specificity by unambiguously isolating aquaporin signals from the tissue background. We further used Monte Carlo simulations to analyze the connection between diffusivity and the percentage of cells engineered to express aquaporin and established a mapping that accurately determined the volume fraction of aquaporin-expressing cells in mixed populations. The quantitative framework developed in this study will enable a broad range of applications in biomedical synthetic biology, requiring the use of aquaporins to noninvasively monitor the location and function of genetically engineered devices in live animals.

Cardiac Phase and Flow Compensation Effects on REnal Flow and Microstructure AnisotroPy MRI in Healthy Human Kidney

BACKGROUND

Renal diffusion-weighted imaging (DWI) involves microstructure and microcirculation, quantified with diffusion tensor imaging (DTI), intravoxel incoherent motion (IVIM), and hybrid models. A better understanding of their contrast may increase specificity.

PURPOSE

To measure modulation of DWI with cardiac phase and flow-compensated (FC) diffusion gradient waveforms.

STUDY TYPE

Prospective.

POPULATION

Six healthy volunteers (ages: 22-48 years, five females), water phantom.

FIELD STRENGTH/SEQUENCE

3-T, prototype DWI sequence with 2D echo-planar imaging, and bipolar (BP) or FC gradients. 2D Half-Fourier Single-shot Turbo-spin-Echo (HASTE). Multiple-phase 2D spoiled gradient-echo phase contrast (PC) MRI.

ASSESSMENT

BP and FC water signal decays were qualitatively compared. Renal arteries and velocities were visualized on PC-MRI. Systolic (peak velocity), diastolic (end stable velocity), and pre-systolic (before peak velocity) phases were identified. Following mutual information-based retrospective self-registration of DWI within each kidney, and Marchenko-Pastur Principal Component Analysis (MPPCA) denoising, combined IVIM-DTI analysis estimated mean diffusivity (MD), fractional anisotropy (FA), and eigenvalues (λi) from tissue diffusivity (Dt ), perfusion fraction (fp ), and pseudodiffusivity (Dp , Dp,axial , Dp,radial ), for each tissue (cortex/medulla, segmented on b0/FA respectively), phase, and waveform (BP, FC). Monte Carlo water diffusion simulations aided data interpretation.

STATISTICAL TESTS

Mixed model regression probed differences between tissue types and pulse sequences. Univariate general linear model analysis probed variations among cardiac phases. Spearman correlations were measured between diffusion metrics and renal artery velocities. Statistical significance level was set at P < 0.05.

RESULTS

Water BP and FC signal decays showed no differences. Significant pulse sequence dependence occurred for λ1 , λ3 , FA, Dp , fp , Dp,axial , Dp,radial in cortex and medulla, and medullary λ2 . Significant cortex/medulla differences occurred with BP for all metrics except MD (systole [P = 0.224]; diastole [P = 0.556]). Significant phase dependence occurred for Dp , Dp,axial , Dp,radial for BP and medullary λ1 , λ2 , λ3 , MD for FC. FA correlated significantly with velocity. Monte Carlo simulations indicated medullary measurements were consistent with a 34 μm tubule diameter.

DATA CONCLUSION

Cardiac gating and flow compensation modulate of measurements of renal diffusion.

EVIDENCE LEVEL

2 TECHNICAL EFFICACY STAGE: 2.

Molecular imaging with aquaporin-based reporter genes: quantitative considerations from Monte Carlo diffusion simulations

Aquaporins provide a new class of genetic tools for imaging molecular activity in deep tissues by increasing the rate of cellular water diffusion, which generates magnetic resonance contrast. However, distinguishing aquaporin contrast from the tissue background is challenging because water diffusion is also influenced by structural factors such as cell size and packing density. Here, we developed and experimentally validated a Monte Carlo model to analyze how cell radius and intracellular volume fraction quantitatively affect aquaporin signals. We demonstrated that a differential imaging approach based on time-dependent changes in diffusivity can improve specificity by unambiguously isolating aquaporin-driven contrast from the tissue background. Finally, we used Monte Carlo simulations to analyze the connection between diffusivity and the percentage of cells engineered to express aquaporin, and established a simple mapping that accurately determined the volume fraction of aquaporin-expressing cells in mixed populations. This study creates a framework for broad applications of aquaporins, particularly in biomedicine and in vivo synthetic biology, where quantitative methods to measure the location and performance of genetic devices in whole vertebrates are necessary.

Relationship between kurtosis and bi-exponential characterization of high b-value diffusion-weighted imaging: application to prostate cancer

BACKGROUND

High b-value diffusion-weighted imaging has application in the detection of cancerous tissue across multiple body sites. Diffusional kurtosis and bi-exponential modeling are two popular model-based techniques, whose performance in relation to each other has yet to be fully explored.

PURPOSE

To determine the relationship between excess kurtosis and signal fractions derived from bi-exponential modeling in the detection of suspicious prostate lesions.

MATERIAL AND METHODS

This retrospective study analyzed patients with normal prostate tissue (n = 12) or suspicious lesions (n = 13, one lesion per patient), as determined by a radiologist whose clinical care included a high b-value diffusion series. The observed signal intensity was modeled using a bi-exponential decay, from which the signal fraction of the slow-moving component was derived ( SFs). In addition, the excess kurtosis was calculated using the signal fractions and ADCs of the two exponentials ( KCOMP). As a comparison, the kurtosis was also calculated using the cumulant expansion for the diffusion signal ( KCE).

RESULTS

Both K and KCE were found to increase with SFs within the range of SFs commonly found within the prostate. Voxel-wise receiver operating characteristic performance of SFs, KCE, and KCOMP in discriminating between suspicious lesions and normal prostate tissue was 0.86 (95% confidence interval [CI] = 0.85 - 0.87), 0.69 (95% CI = 0.68-0.70), and 0.86 (95% CI = 0.86-0.87), respectively.

CONCLUSION

In a two-component diffusion environment, KCOMP is a scaled value of SFs and is thus able to discriminate suspicious lesions with equal precision . KCE provides a computationally inexpensive approximation of kurtosis but does not provide the same discriminatory abilities as SFs and KCOMP.

Characterization of prostate microstructure using water diffusion and NMR relaxation

For many pathologies, early structural tissue changes occur at the cellular level, on the scale of micrometers or tens of micrometers. Magnetic resonance imaging (MRI) is a powerful non-invasive imaging tool used for medical diagnosis, but its clinical hardware is incapable of reaching the cellular length scale directly. In spite of this limitation, microscopic tissue changes in pathology can potentially be captured indirectly, from macroscopic imaging characteristics, by studying water diffusion. Here we focus on water diffusion and NMR relaxation in the human prostate, a highly heterogeneous organ at the cellular level. We present a physical picture of water diffusion and NMR relaxation in the prostate tissue, that is comprised of a densely-packed cellular compartment (composed of stroma and epithelium), and a luminal compartment with almost unrestricted water diffusion. Transverse NMR relaxation is used to identify fast and slow T ₂ components, corresponding to these tissue compartments, and to disentangle the luminal and cellular compartment contributions to the temporal evolution of the overall water diffusion coefficient. Diffusion in the luminal compartment falls into the short-time surface-to-volume (S/V) limit, indicating that only a small fraction of water molecules has time to encounter the luminal walls of healthy tissue; from the S/V ratio, the average lumen diameter averaged over three young healthy subjects is measured to be 217.7±188.7 μm. Conversely, the diffusion in the cellular compartment is highly restricted and anisotropic, consistent with the fibrous character of the stromal tissue. Diffusion transverse to these fibers is well described by the random permeable barrier model (RPBM), as confirmed by the dynamical exponent ϑ = 1/2 for approaching the long-time limit of diffusion, and the corresponding structural exponent p = -1 in histology. The RPBM-derived fiber diameter and membrane permeability were 19.8±8.1 μm and 0.044±0.045 μm/ms, respectively, in agreement with known values from tissue histology and membrane biophysics. Lastly, we revisited 38 prostate cancer cases from a recently published study, and found the same dynamical exponent ϑ = 1/2 of diffusion in tumors and benign regions. Our results suggest that a multi-parametric MRI acquisition combined with biophysical modeling may be a powerful non-invasive complement to prostate cancer grading, potentially foregoing biopsies.

An improved model for prostate diffusion incorporating the results of Monte Carlo simulations of diffusion in the cellular compartment

The purpose of this work was to refine a previously published model of prostate diffusion by incorporating improved estimates of cellular diffusivity obtained by Monte Carlo simulation. Stromal and epithelial cell size and intracellular volume fraction in different grades of cancer were determined from histological images. Diffusion in different mixtures of cells, corresponding to different tumor grades, was simulated and cellular apparent diffusion coefficient and kurtosis values determined. These values were incorporated into the previously published model of prostate diffusion and model predictions compared with values found in the literature. Stromal cell radius and intracellular volume fraction were 3.74 ± 0.96 μm and 13 ± 3% respectively in normal peripheral zone (PZ), and were similar in all grades of cancer. Epithelial cell radius and intracellular volume fraction were 3.40 ± 0.15 μm and 45 ± 5% respectively in normal PZ, rising to 4.75 ± 0.20 μm and 70 ± 8% in high grade cancer. Cellular apparent diffusion coefficient and kurtosis were 1.02 μm² ms^-1 and 0.58 respectively in normal PZ, and 0.61 μm² ms^-1 and 1.15 in high grade cancer (variation in simulation values are less than 0.1%). Agreement between model predictions and measurements were good, with a mean square error of 0.22 μm² ms^-1 . Incorporation of cellular diffusion coefficient and kurtosis values obtained by Monte Carlo simulation into a model of prostate diffusion gives good agreement with published results.

Measurement and modeling of diffusion time dependence of apparent diffusion coefficient and fractional anisotropy in prostate tissue ex vivo

The purpose of this study was to measure and model the diffusion time dependence of apparent diffusion coefficient (ADC) and fractional anisotropy (FA) derived from conventional prostate diffusion-weighted imaging methods as used in recommended multiparametric MRI protocols. Diffusion tensor imaging (DTI) was performed at 9.4 T with three radical prostatectomy specimens, with diffusion times in the range 10-120 ms and b-values 0-3000 s/mm² . ADC and FA were calculated from DTI measurements at b-values of 800 and 1600 s/mm² . Independently, a two-component model (restricted isotropic plus Gaussian anisotropic) was used to synthesize DTI data, from which ADC and FA were predicted and compared with the measured values. Measured ADC and FA exhibited a diffusion time dependence, which was closely predicted by the two-component model. ADC decreased by about 0.10-0.15 μm² /ms as diffusion time increased from 10 to 120 ms. FA increased with diffusion time at b-values of 800 and 1600 s/mm² but was predicted to be independent of diffusion time at b = 3000 s/mm² . Both ADC and FA exhibited diffusion time dependence that could be modeled as two unmixed water pools - one having isotropic restricted dynamics, and the other unrestricted anisotropic dynamics. These results highlight the importance of considering and reporting diffusion times in conventional ADC and FA calculations and protocol recommendations, and inform the development of improved diffusion methods for prostate cancer imaging.

Physical characterization using diffusion NMR spectroscopy

NMR diffusion measurements (or dNMR) provide a powerful tool for analysis of solution organization and microgeometry of the environment by probing random molecular motion. Being a very versatile method, dNMR can be applied to a large variety of samples and systems. Here, a brief introduction into dNMR and a summary of recent advances in the field are presented. The research topics include restricted diffusion, anisotropic diffusion, polymer dynamics, solution structuring and dNMR method development. The dNMR studied systems include plants, cells (cell models), liquid crystals, polymer solutions, ionic liquids, supercooled solutions, untreated water, amino acid solutions and more. It is demonstrated how a variety of dNMR methods can be applied to a system to extract the data on particular structures present among, formed by or surrounding the diffusing particles. It is also demonstrated how dNMR methods can be developed to allow probing larger geometries, low sample concentrations and faster processes. Copyright © 2016 John Wiley & Sons, Ltd.

A model describing diffusion in prostate cancer

PURPOSE

Quantitative diffusion MRI has frequently been studied as a means of grading prostate cancer. Interpretation of results is complicated by the nature of prostate tissue, which consists of four distinct compartments: vascular, ductal lumen, epithelium, and stroma. Current diffusion measurements are an ill-defined weighted average of these compartments. In this study, prostate diffusion is analyzed in terms of a model that takes explicit account of tissue compartmentalization, exchange effects, and the non-Gaussian behavior of tissue diffusion.

METHOD

The model assumes that exchange between the cellular (ie, stromal plus epithelial) and the vascular and ductal compartments is slow. Ductal and cellular diffusion characteristics are estimated by Monte Carlo simulation and a two-compartment exchange model, respectively. Vascular pseudodiffusion is represented by an additional signal at b = 0. Most model parameters are obtained either from published data or by comparing model predictions with the published results from 41 studies. Model prediction error is estimated using 10-fold cross-validation.

RESULTS

Agreement between model predictions and published results is good. The model satisfactorily explains the variability of ADC estimates found in the literature.

CONCLUSION

A reliable model that predicts the diffusion behavior of benign and cancerous prostate tissue of different Gleason scores has been developed. Magn Reson Med 78:316-326, 2017. © 2016 International Society for Magnetic Resonance in Medicine.