Anomalous diffusion of water in biological tissues

The biophysics of water in cell biology: perspectives on a keystone for both marine sciences and cancer research

The biophysics of water, has been debated over more than a century. Although its importance is still underestimated, significant breakthroughs occurred in recent years. The influence of protein condensation on water availability control was documented, new findings on water-transport proteins emerged, and the way water molecules rearrange to minimize free energy at interfaces was deciphered, influencing membrane thermodynamics. The state of knowledge continued to progress in the field of deep-sea marine biology, highlighting unknown effects of high hydrostatic pressure and/or temperature on interactions between proteins and ligands in extreme environments, and membrane structure adaptations. The role of osmolytes in protein stability control under stress is also discussed here in relation to fish egg hydration/buoyancy. The complexity of water movements within the cell is updated, all these findings leading to a better view of their impact on many cellular processes. The way water flow and osmotic gradients generated by ion transport work together to produce the driving force behind cell migration is also relevant to both marine biology and cancer research. Additional common points concern water dynamic changes during the neoplastic transformation of cells and tissues, or embryo development. This could improve imaging techniques, early cancer diagnosis, and understanding of the molecular and physiological basis of buoyancy for many marine species.

A new perspective of molecular diffusion by nuclear magnetic resonance

The diffusion-weighted NMR signal acquired using Pulse Field Gradient (PFG) techniques, allows for extrapolating microstructural information from porous materials and biological tissues. In recent years there has been a multiplication of diffusion models expressed by parametric functions to fit the experimental data. However, clear-cut criteria for the model selection are lacking. In this paper, we develop a theoretical framework for the interpretation of NMR attenuation signals in the case of Gaussian systems with stationary increments. The full expression of the Stejskal-Tanner formula for normal diffusing systems is devised, together with its extension to the domain of anomalous diffusion. The range of applicability of the relevant parametric functions to fit the PFG data can be fully determined by means of appropriate checks to ascertain the correctness of the fit. Furthermore, the exact expression for diffusion weighted NMR signals pertaining to Brownian yet non-Gaussian processes is also derived, accompanied by the proper check to establish its contextual relevance. The analysis provided is particularly useful in the context of medical MRI and clinical practise where the hardware limitations do not allow the use of narrow pulse gradients.

Disentangling the effects of restriction and exchange with diffusion exchange spectroscopy

Diffusion exchange spectroscopy (DEXSY) is a multidimensional NMR technique that can reveal how water molecules exchange between compartments within heterogeneous media, such as biological tissue. Data from DEXSY experiments is typically processed using numerical inverse Laplace transforms (ILTs) to produce a diffusion-diffusion spectrum. A tacit assumption of this ILT approach is that the signal behavior is Gaussian - i.e., the spin echo intensity decays exponentially with the degree of diffusion weighting. The assumptions that underlie Gaussian signal behavior may be violated, however, depending on the gradient strength applied and the sample under study. We argue that non-Gaussian signal behavior due to restrictions is to be expected in the study of biological tissue using diffusion NMR. Further, we argue that this signal behavior can produce confounding features in the diffusion-diffusion spectra obtained from numerical ILTs of DEXSY data - entangling the effects of restriction and exchange. Specifically, restricted signal behavior can result in broadening of peaks and in the appearance of illusory exchanging compartments with distributed diffusivities, which pearl into multiple peaks if not highly regularized. We demonstrate these effects on simulated data. That said, we suggest the use of features in the signal acquisition domain that can be used to rapidly probe exchange without employing an ILT. We also propose a means to characterize the non-Gaussian signal behavior due to restrictions within a sample using DEXSY measurements with a near zero mixing time or storage interval. We propose a combined acquisition scheme to independently characterize restriction and exchange with various DEXSY measurements, which we term Restriction and Exchange from Equally-weighted Double and Single Diffusion Encodings (REEDS-DE). We test this method on ex vivo neonatal mouse spinal cord - a sample consisting primarily of gray matter - using a low-field, static gradient NMR system. In sum, we highlight critical shortcomings of prevailing DEXSY analysis methods that conflate the effects of restriction and exchange, and suggest a viable experimental approach to disentangle them.

Statistical Evaluation of Different Mathematical Models for Diffusion Weighted Imaging of Prostate Cancer Xenografts in Mice

Purpose

To evaluate fitting quality and repeatability of four mathematical models for diffusion weighted imaging (DWI) during tumor progression in mouse xenograft model of prostate cancer.

Methods

Human prostate cancer cells (PC-3) were implanted subcutaneously in right hind limbs of 11 immunodeficient mice. Tumor growth was followed by weekly DWI examinations using a 7T MR scanner. Additional DWI examination was performed after repositioning following the fourth DWI examination to evaluate short term repeatability. DWI was performed using 15 and 12 b-values in the ranges of 0-500 and 0-2000 s/mm², respectively. Corrected Akaike information criteria and F-ratio were used to evaluate fitting quality of each model (mono-exponential, stretched exponential, kurtosis, and bi-exponential).

Results

Significant changes were observed in DWI data during the tumor growth, indicated by ADC_m, ADC_s, and ADC_k. Similar results were obtained using low as well as high b-values. No marked changes in model preference were present between the weeks 1−4. The parameters of the mono-exponential, stretched exponential, and kurtosis models had smaller confidence interval and coefficient of repeatability values than the parameters of the bi-exponential model.

Conclusion

Stretched exponential and kurtosis models showed better fit to DWI data than the mono-exponential model and presented with good repeatability.

The sensitivity of diffusion MRI to microstructural properties and experimental factors

Diffusion MRI is a non-invasive technique to study brain microstructure. Differences in the microstructural properties of tissue, including size and anisotropy, can be represented in the signal if the appropriate method of acquisition is used. However, to depict the underlying properties, special care must be taken when designing the acquisition protocol as any changes in the procedure might impact on quantitative measurements. This work reviews state-of-the-art methods for studying brain microstructure using diffusion MRI and their sensitivity to microstructural differences and various experimental factors. Microstructural properties of the tissue at a micrometer scale can be linked to the diffusion signal at a millimeter-scale using modeling. In this paper, we first give an introduction to diffusion MRI and different encoding schemes. Then, signal representation-based methods and multi-compartment models are explained briefly. The sensitivity of the diffusion MRI signal to the microstructural components and the effects of curvedness of axonal trajectories on the diffusion signal are reviewed. Factors that impact on the quality (accuracy and precision) of derived metrics are then reviewed, including the impact of random noise, and variations in the acquisition parameters (i.e., number of sampled signals, b-value and number of acquisition shells). Finally, yet importantly, typical approaches to deal with experimental factors are depicted, including unbiased measures and harmonization. We conclude the review with some future directions and recommendations on this topic.

Removing rician bias in diffusional kurtosis of the prostate using real-data reconstruction

PURPOSE

To compare prostate diffusional kurtosis imaging (DKI) metrics generated using phase-corrected real data with those generated using magnitude data with and without noise compensation (NC).

METHODS

Diffusion-weighted images were acquired at 3T in 16 prostate cancer patients, measuring 6 b-values (0-1500 s/mm2 ), each acquired with 6 signal averages along 3 diffusion directions, with noise-only images acquired to allow NC. In addition to conventional magnitude averaging, phase-corrected real data were averaged in an attempt to reduce rician noise-bias, with a range of phase-correction low-pass filter (LPF) sizes (8-128 pixels) tested. Each method was also tested using simulations. Pixelwise maps of apparent diffusion (D) and apparent kurtosis (K) were calculated for magnitude data with and without NC and phase-corrected real data. Average values were compared in tumor, normal transition zone (NTZ), and normal peripheral zone (NPZ).

RESULTS

Simulations indicated LPF size can strongly affect K metrics, where 64-pixel LPFs produced accurate metrics. Relative to metrics estimated from magnitude data without NC, median NC K were lower (P < 0.0001) by 6/11/8% in tumor/NPZ/NTZ, 64-LPF real-data K were lower (P < 0.0001) by 4/10/7%, respectively.

CONCLUSION

Compared with magnitude data with NC, phase-corrected real data can produce similar K, although the choice of phase-correction LPF should be chosen carefully.

Myelin basic protein dynamics from out-of-equilibrium functional state to degraded state in myelin

Living matter is a quasi-stationary out-of-equilibrium system; in this physical condition, structural fluctuations at nano- and meso-scales are needed to understand the physics behind its biological functionality. Myelin has a simple ultrastructure whose fluctuations show correlated disorder in its functional out-of-equilibrium state. However, there is no information on the relationship between this correlated disorder and the dynamics of the intrinsically disordered Myelin Basic Protein (MBP) which is expected to influence the membrane structure and overall functionality. In this work, we have investigated the role of this protein structural dynamics in the myelin ultrastructure fluctuations in various conditions, by using synchrotron Scanning micro X Ray Diffraction and Small Angle X ray Scattering. We have induced the crossover from out-of-equilibrium functional state to in-equilibrium degeneration changing the pH to values far from physiological condition. The observed compression of the cytosolic layer thickness probes that the intrinsic large MBP fluctuations preserve the cytosol structure also in the degraded state. Thus, the transition of myelin ultrastructure from correlated to uncorrelated disordered state, is principally affected by the deformation of the membrane and extracellular domain.

Modeling the diffusion-weighted imaging signal for breast lesions in the b = 200 to 3000 s/mm2 range: quality of fit and classification accuracy for different representations

PURPOSE

To evaluate different non-Gaussian representations for the diffusion-weighted imaging (DWI) signal in the b-value range 200 to 3000 s/mm² in benign and malignant breast lesions.

METHODS

Forty-three patients diagnosed with benign (n = 18) or malignant (n = 25) tumors of the breast underwent DWI (b-values 200, 600, 1200, 1800, 2400, and 3000 s/mm² ). Six different representations were fit to the average signal from regions of interest (ROIs) at different b-value ranges. Quality of fit was assessed by the corrected Akaike information criterion (AICc), and the Friedman test was used for assessing representation ranks. The area under the curve (AUC) of receiver operating characteristic curves were used to evaluate the power of derived parameters to differentiate between malignant and benign lesions. The lesion ROI was divided in central and peripheral parts to assess potential effect of heterogeneity. Sensitivity to noise-floor correction was also evaluated.

RESULTS

The Padé exponent was ranked as the best based on AICc, whereas 3 models (kurtosis, fractional, and biexponential) achieved the highest AUC = 0.99 for lesion differentiation. The monoexponential model at b_max = 600 s/mm² already provides AUC = 0.96, with considerably shorter acquisition time and simpler analysis. Significant differences between central and peripheral parts of lesions were found in malignant lesions. The mono- and biexponential models were most stable against varying degrees of noise-floor correction.

CONCLUSION

Non-Gaussian representations are required for fitting of the DWI curve at high b-values in breast lesions. However, the added clinical value from the high b-value data for differentiation of benign and malignant lesions is not clear.

Magnetic resonance measurements of cellular and sub-cellular membrane structures in live and fixed neural tissue

We develop magnetic resonance (MR) methods for real-time measurement of tissue microstructure and membrane permeability of live and fixed excised neonatal mouse spinal cords. Diffusion and exchange MR measurements are performed using the strong static gradient produced by a single-sided permanent magnet. Using tissue delipidation methods, we show that water diffusion is restricted solely by lipid membranes. Most of the diffusion signal can be assigned to water in tissue which is far from membranes. The remaining 25% can be assigned to water restricted on length scales of roughly a micron or less, near or within membrane structures at the cellular, organelle, and vesicle levels. Diffusion exchange spectroscopy measures water exchanging between membrane structures and free environments at 100 s^-1.

Influence of the size and curvedness of neural projections on the orientationally averaged diffusion MR signal

Neuronal and glial projections can be envisioned to be tubes of infinitesimal diameter as far as diffusion magnetic resonance (MR) measurements via clinical scanners are concerned. Recent experimental studies indicate that the decay of the orientationally-averaged signal in white-matter may be characterized by the power-law, Ē(q) ∝ q^-1, where q is the wavenumber determined by the parameters of the pulsed field gradient measurements. One particular study by McKinnon et al. [1] reports a distinctively faster decay in gray-matter. Here, we assess the role of the size and curvature of the neurites and glial arborizations in these experimental findings. To this end, we studied the signal decay for diffusion along general curves at all three temporal regimes of the traditional pulsed field gradient measurements. We show that for curvy projections, employment of longer pulse durations leads to a disappearance of the q^-1 decay, while such decay is robust when narrow gradient pulses are used. Thus, in clinical acquisitions, the lack of such a decay for a fibrous specimen can be seen as indicative of fibers that are curved. We note that the above discussion is valid for an intermediate range of q-values as the true asymptotic behavior of the signal decay is Ē(q) ∝ q^-4 for narrow pulses (through Debye-Porod law) or steeper for longer pulses. This study is expected to provide insights for interpreting the diffusion-weighted images of the central nervous system and aid in the design of acquisition strategies.

Fundamentals of diffusion MRI physics

Diffusion MRI is commonly considered the "engine" for probing the cellular structure of living biological tissues. The difficulty of this task is threefold. First, in structurally heterogeneous media, diffusion is related to structure in quite a complicated way. The challenge of finding diffusion metrics for a given structure is equivalent to other problems in physics that have been known for over a century. Second, in most cases the MRI signal is related to diffusion in an indirect way dependent on the measurement technique used. Third, finding the cellular structure given the MRI signal is an ill-posed inverse problem. This paper reviews well-established knowledge that forms the basis for responding to the first two challenges. The inverse problem is briefly discussed and the reader is warned about a number of pitfalls on the way.

Limitations and Prospects for Diffusion-Weighted MRI of the Prostate

Diffusion-weighted imaging (DWI) is the most effective component of the modern multi-parametric magnetic resonance imaging (mpMRI) scan for prostate pathology. DWI provides the strongest prediction of cancer volume, and the apparent diffusion coefficient (ADC) correlates moderately with Gleason grade. Notwithstanding the demonstrated cancer assessment value of DWI, the standard measurement and signal analysis methods are based on a model of water diffusion dynamics that is well known to be invalid in human tissue. This review describes the biophysical limitations of the DWI component of the current standard mpMRI protocol and the potential for significantly improved cancer assessment performance based on more sophisticated measurement and signal modeling techniques.

Novel Magnetic Resonance Imaging Techniques in Brain Tumors

Magnetic resonance imaging is a powerful, noninvasive imaging technique with exquisite sensitivity to soft tissue composition. Magnetic resonance imaging is primary tool for brain tumor diagnosis, evaluation of drug response assessment, and clinical monitoring of the patient during the course of their disease. The flexibility of magnetic resonance imaging pulse sequence design allows for a variety of image contrasts to be acquired, including information about magnetic resonance-specific tissue characteristics, molecular dynamics, microstructural organization, vascular composition, and biochemical status. The current review highlights recent advancements and novel approaches in MR characterization of brain tumors.

Ultra-low-field NMR relaxation and diffusion measurements using an optical magnetometer

Nuclear magnetic resonance (NMR) relaxometry and diffusometry are important tools for the characterization of heterogeneous materials and porous media, with applications including medical imaging, food characterization and oil-well logging. These methods can be extremely effective in applications where high-resolution NMR is either unnecessary, impractical, or both, as is the case in the emerging field of portable chemical characterization. Here, we present a proof-of-concept experiment demonstrating the use of high-sensitivity optical magnetometers as detectors for ultra-low-field NMR relaxation and diffusion measurements.

Optimization of b-value distribution for four mathematical models of prostate cancer diffusion-weighted imaging using b values up to 2000 s/mm(2): simulation and repeatability study

PURPOSE

To find optimal b-value distributions for monoexponential, stretched exponential, kurtosis, and biexponential models of prostate cancer (PCa) diffusion-weighted imaging (DWI) using simulations and repeated DWI examinations.

METHODS

Simulations aiming to minimize estimation accuracy error were performed. Ten PCa patients underwent in total four repeated 3-tesla DWI examinations using 12 equally spaced b values (0-2000 s/mm(2) ). Normalized mean signal intensities of regions-of-interest placed in normal tissue and PCa were fitted. In total, 210 different b-value combinations consisting of six b values, 0 and 100 s/mm(2) included in every b-value distribution, were evaluated in terms of accuracy and repeatability.

RESULTS

The simulations and in vivo DWI data suggest the optimal b-value distribution for the monoexponential model consists of four to five equally distributed b values in the range of 0 to 1200 s/mm(2) . The parameters of the stretched exponential and kurtosis models are best estimated using five to seven b values in the ranges of 300 to 700 and close to 2000 s/mm(2) , in addition to low b value. B-value distribution consisting of eight to 10 b values in the ranges of 0 to 100, 800 to 1200, and 1800 to 2000 s/mm(2) is the preferred method for estimation of the biexponential model parameters of PCa DWI.

CONCLUSION

The optimized b-value distributions demonstrated improved estimation accuracy and repeatability of DWI signal decay-derived parameters.

Anomalous diffusion and response in branched systems: a simple analysis

We revisit the diffusion properties and the mean drift induced by an external field of a random walk process in a class of branched structures, as the comb lattice and the linear chains of plaquettes. A simple treatment based on scaling arguments is able to predict the correct anomalous regime for different topologies. In addition, we show that even in the presence of anomalous diffusion, Einstein's relation still holds, implying a proportionality between the mean square displacement of the unperturbed systems and the drift induced by an external forcing.

Anomalous diffusion of single particles in cytoplasm

The crowded intracellular environment poses a formidable challenge to experimental and theoretical analyses of intracellular transport mechanisms. Our measurements of single-particle trajectories in cytoplasm and their random-walk interpretations elucidate two of these mechanisms: molecular diffusion in crowded environments and cytoskeletal transport along microtubules. We employed acousto-optic deflector microscopy to map out the three-dimensional trajectories of microspheres migrating in the cytosolic fraction of a cellular extract. Classical Brownian motion (BM), continuous time random walk, and fractional BM were alternatively used to represent these trajectories. The comparison of the experimental and numerical data demonstrates that cytoskeletal transport along microtubules and diffusion in the cytosolic fraction exhibit anomalous (nonFickian) behavior and posses statistically distinct signatures. Among the three random-walk models used, continuous time random walk provides the best representation of diffusion, whereas microtubular transport is accurately modeled with fractional BM.

Tracing Molecular Propagation in Dextran Solutions by Pulsed Field Gradient NMR

We have exploited the pulsed field gradient (PFG) technique of NMR to measure molecular diffusion in aqueous solutions of a mixture of dextran molecules. From detailed studies by fluorescence correlation spectroscopy (FCS), the lighter component of such mixtures is known to undergo subdiffusion, up to diffusion path lengths on the order of 0.5 μm. Our studies provide clear evidence of a crossover to normal diffusion for diffusion path lengths from this range up to about 1 μm.

Temporal scaling characteristics of diffusion as a new MRI contrast: findings in rat hippocampus

Features of the diffusion-time dependence of the diffusion-weighted magnetic resonance imaging (MRI) signal provide a new contrast that could be altered by numerous biological processes and pathologies in tissue at microscopic length scales. An anomalous diffusion model, based on the theory of Brownian motion in fractal and disordered media, is used to characterize the temporal scaling (TS) characteristics of diffusion-related quantities, such as moments of the displacement and zero-displacement probabilities, in excised rat hippocampus specimens. To reduce the effect of noise in magnitude-valued MRI data, a novel numerical procedure was employed to yield accurate estimation of these quantities even when the signal falls below the noise floor. The power-law dependencies characterize the TS behavior in all regions of the rat hippocampus, providing unique information about its microscopic architecture. The relationship between the TS characteristics and diffusion anisotropy is investigated by examining the anisotropy of TS, and conversely, the TS of anisotropy. The findings suggest the robustness of the technique as well as the reproducibility of estimates. TS characteristics of the diffusion-weighted signals could be used as a new and useful marker of tissue microstructure.

Two-step anomalous diffusion tensor imaging

We extend the formalism of anomalous diffusion imaging to include directional anisotropy of fitted parameters. The resulting technique is termed anomalous diffusion tensor imaging (aDTI), and allows the directional properties of the distributed diffusion coefficient (α) and the anomalous diffusion exponent, (γ) to be analysed using the same analytical techniques as regular diffusion tensor imaging (DTI). Together, these parameters quantify the rate of diffusion (α) and the complexity of the diffusion environment (γ). We generated tensor images for the anomalous exponent tensor (Γ) and distributed diffusivity tensor (A) from in vivo human brain data and present images of eigenvalues, eigenvectors, Trace/3 (Tr), fractional anisotropy (FA) and tensor shape measures. In white matter, A is found to have a median Tr = 0.56 × 10(- 3) mm(2) s(- 1), FA = 0.58 and Γ Tr = 0.69, FA = 0.13. We observed that white matter shows a similar anisotropic geometry for the distributed diffusion tensor as for the regular diffusion tensor, whereas the anomalous exponent tensor exhibits a different shape characteristic which may be informative of tissue microstructure.

Erratum to: Hydrodynamic dispersion in β-lactoglobulin gels measured by PGSE NMR

The displacement scale dependent molecular dynamics of solvent water molecules flowing through β-lactoglobulin gels are measured by pulse gradient spin echo (PGSE) nuclear magnetic resonance (NMR). Gels formed under different p H conditions generate structures which are characterized by magnetic resonance imaging (MRI) and PGSE NMR measured dynamics as homogeneous and heterogeneous. The data presented clearly demonstrate the applicability of the theoretical framework for modeling hydrodynamic dispersion to the analysis of protein gels.

Recent advances in diffusion MRI modeling: Angular and radial reconstruction

Recent advances in diffusion magnetic resonance image (dMRI) modeling have led to the development of several state of the art methods for reconstructing the diffusion signal. These methods allow for distinct features to be computed, which in turn reflect properties of fibrous tissue in the brain and in other organs. A practical consideration is that to choose among these approaches requires very specialized knowledge. In order to bridge the gap between theory and practice in dMRI reconstruction and analysis we present a detailed review of the dMRI modeling literature. We place an emphasis on the mathematical and algorithmic underpinnings of the subject, categorizing existing methods according to how they treat the angular and radial sampling of the diffusion signal. We describe the features that can be computed with each method and discuss its advantages and limitations. We also provide a detailed bibliography to guide the reader.

Hydrodynamic dispersion in β-lactoglobulin gels measured by PGSE NMR

The displacement scale dependent molecular dynamics of solvent water molecules flowing through [Formula: see text] -lactoglobulin gels are measured by pulse gradient spin echo (PGSE) nuclear magnetic resonance (NMR). Gels formed under different p H conditions generate structures which are characterized by magnetic resonance imaging (MRI) and PGSE NMR measured dynamics as homogeneous and heterogeneous. The data presented clearly demonstrate the applicability of the theoretical framework for modeling hydrodynamic dispersion to the analysis of protein gels.

Feature Preserving Image Smoothing Using a Continuous Mixture of Tensors

Many computer vision and image processing tasks require the preservation of local discontinuities, terminations and bifurcations. Denoising with feature preservation is a challenging task and in this paper, we present a novel technique for preserving complex oriented structures such as junctions and corners present in images. This is achieved in a two stage process namely, (1) All image data are pre-processed to extract local orientation information using a steerable Gabor filter bank. The orientation distribution at each lattice point is then represented by a continuous mixture of Gaussians. The continuous mixture representation can be cast as the Laplace transform of the mixing density over the space of positive definite (covariance) matrices. This mixing density is assumed to be a parameterized distribution, namely, a mixture of Wisharts whose Laplace transform is evaluated in a closed form expression called the Rigaut type function, a scalar-valued function of the parameters of the Wishart distribution. Computation of the weights in the mixture Wisharts is formulated as a sparse deconvolution problem. (2) The feature preserving denoising is then achieved via iterative convolution of the given image data with the Rigaut type function. We present experimental results on noisy data, real 2D images and 3D MRI data acquired from plant roots depicting bifurcating roots. Superior performance of our technique is depicted via comparison to the state-of-the-art anisotropic diffusion filter.

A CONTINUOUS MIXTURE OF TENSORS MODEL FOR DIFFUSION-WEIGHTED MR SIGNAL RECONSTRUCTION

Diffusion MRI is a non-invasive imaging technique that allows the measurement of water molecular diffusion through tissue in vivo. In this paper, we present a novel statistical model which describes the diffusion-attenuated MR signal by the Laplace transform of a probability distribution over symmetric positive definite matrices. Using this new model, we analytically derive a Rigaut-type asymptotic fractal law for the MR signal decay which has been phenomenologically used before. We also develop an efficient scheme for reconstructing the multiple fiber bundles from the DW-MRI measurements. Experimental results on both synthetic and real data sets are presented to show the robustness and accuracy of the proposed algorithms.

Water diffusion heterogeneity index in the human brain is insensitive to the orientation of applied magnetic field gradients

The alpha diffusion-weighted imaging (DWI) method was developed to study heterogeneous water diffusion in the human brain using magnetic resonance imaging (MRI). An advantage of this model is that it does not require an assumption about the shape of the intravoxel distribution of apparent diffusion rates, and it has a calculable relationship to this distribution. The alpha-DWI technique is useful for detecting microstructural tissue changes associated with brain tumor invasion, and may be useful for directing therapy to invading tumor cells. In previous work, alpha-DWI was performed with magnetic field gradients applied along a single direction in order to avoid artificially introducing a source of heterogeneity to the decay. However, it is known that restricted diffusion is anisotropic in the brain, and the alpha-DWI method must take this into account to be complete. In this work the relationship between the applied magnetic field gradients and the fitted stretched-exponential model parameters was studied in the human brain. It was found the distributed diffusion coefficient (DDC) varies with the direction of applied gradients, while the heterogeneity index alpha is relatively direction-insensitive. It is proposed that in clinical use, maps of alpha can be created using diffusion-weighting gradients applied in a single direction that reflect the tissue heterogeneity.

Observation of anomalous diffusion in excised tissue by characterizing the diffusion-time dependence of the MR signal

This report introduces a novel method to characterize the diffusion-time dependence of the diffusion-weighted magnetic resonance (MR) signal in biological tissues. The approach utilizes the theory of diffusion in disordered media where two parameters, the random walk dimension and the spectral dimension, describe the evolution of the average propagators obtained from q-space MR experiments. These parameters were estimated, using several schemes, on diffusion MR spectroscopy data obtained from human red blood cell ghosts and nervous tissue autopsy samples. The experiments demonstrated that water diffusion in human tissue is anomalous, where the mean-square displacements vary slower than linearly with diffusion time. These observations are consistent with a fractal microstructure for human tissues. Differences observed between healthy human nervous tissue and glioblastoma samples suggest that the proposed methodology may provide a novel, clinically useful form of diffusion MR contrast.

Stochastic modeling of protein motions within cell membranes

A simple model in which immobilizing events are imposed onto otherwise free Brownian diffusion [R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000) and a recent adaptation due to S. Khan and A. M. Reynolds, Physica A 350, 183 (2005)] is shown to encapsulate the peculiar transport characteristics of individual cell receptors within plasma membranes observed in single-particle tracking (SPT) experiments. These characteristics include the occurrence of normal diffusion; non-Gaussian subdiffusion; confined diffusion; a superdiffusive mode of transport that is not due to flow of the membrane or molecular motor attachment; and the occurrence of transitions between these transport modes. Model predictions are shown to be in close agreement with a reanalysis of existing SPT data.

Michaelis-Menten kinetics under spatially constrained conditions: application to mibefradil pharmacokinetics

Two different approaches were used to study the kinetics of the enzymatic reaction under heterogeneous conditions to interpret the unusual nonlinear pharmacokinetics of mibefradil. Firstly, a detailed model based on the kinetic differential equations is proposed to study the enzymatic reaction under spatial constraints and in vivo conditions. Secondly, Monte Carlo simulations of the enzyme reaction in a two-dimensional square lattice, placing special emphasis on the input and output of the substrate were applied to mimic in vivo conditions. Both the mathematical model and the Monte Carlo simulations for the enzymatic reaction reproduced the classical Michaelis-Menten (MM) kinetics in homogeneous media and unusual kinetics in fractal media. Based on these findings, a time-dependent version of the classic MM equation was developed for the rate of change of the substrate concentration in disordered media and was successfully used to describe the experimental plasma concentration-time data of mibefradil and derive estimates for the model parameters. The unusual nonlinear pharmacokinetics of mibefradil originates from the heterogeneous conditions in the reaction space of the enzymatic reaction. The modified MM equation can describe the pharmacokinetics of mibefradil as it is able to capture the heterogeneity of the enzymatic reaction in disordered media.

Anomalous fluid transport in porous media induced by biofilm growth

Magnetic resonance measurements of the transition from normal to anomalous hydrodynamic dispersion in porous media due to biological activity are presented. Fractional advection-diffusion equations are shown to provide models for the measured impact of biofilm growth on porous media transport dynamics.

Intravoxel distribution of DWI decay rates reveals C6 glioma invasion in rat brain

The hypothesis was tested that the intravoxel distribution of water diffusion rates, as measured with a stretched-exponential model of diffusion-weighted imaging (DWI), is a marker of brain tumor invasion. Eight rats underwent intracerebral inoculation of C6 glioma cells. In three rats, cells were labeled with a fluorescent dye for microscopy. One rat was inoculated with a saline solution, and five more rats were imaged without inoculation as controls. Five healthy uninoculated rats were also imaged. DWI was performed 14-15 days after inoculation, with diffusion-weighting factor b = 500 to 6500 sec/mm2, and the resulting signal attenuation was fitted with the stretched-exponential model. The heterogeneity index values were significantly lower (P < 0.05) in the peritumor ROI than in normal gray matter and significantly higher than in normal white matter. The distributed diffusion coefficient values were significantly lower than in normal white matter or normal gray matter. Fluorescence microscopy confirmed the presence of tumors in the peritumor region that could be histologically distinguished from the main tumor mass. There was no change in proton density or T2-weighted images in the peritumor region, making vasogenic edema unlikely as a source of contrast. It is therefore thought that the heterogeneity parameter alpha is a marker of brain tumor invasion.

Characterization of continuously distributed cortical water diffusion rates with a stretched-exponential model

Experience with diffusion-weighted imaging (DWI) shows that signal attenuation is consistent with a multicompartmental theory of water diffusion in the brain. The source of this so-called nonexponential behavior is a topic of debate, because the cerebral cortex contains considerable microscopic heterogeneity and is therefore difficult to model. To account for this heterogeneity and understand its implications for current models of diffusion, a stretched-exponential function was developed to describe diffusion-related signal decay as a continuous distribution of sources decaying at different rates, with no assumptions made about the number of participating sources. DWI experiments were performed using a spin-echo diffusion-weighted pulse sequence with b-values of 500-6500 s/mm(2) in six rats. Signal attenuation curves were fit to a stretched-exponential function, and 20% of the voxels were better fit to the stretched-exponential model than to a biexponential model, even though the latter model had one more adjustable parameter. Based on the calculated intravoxel heterogeneity measure, the cerebral cortex contains considerable heterogeneity in diffusion. The use of a distributed diffusion coefficient (DDC) is suggested to measure mean intravoxel diffusion rates in the presence of such heterogeneity.

Towards rapid and unique curve resolution of low-field NMR relaxation data: trilinear SLICING versus two-dimensional curve fitting

In this work an alternative method, named SLICING, for two-dimensional and noniterative T(2) decomposition of low-field pulsed NMR data (LF-NMR) is proposed and examined. The method is based on the Direct Exponential Curve Resolution Algorithm (DECRA) proposed by W. Windig and A. Antalek (1997, Chemom. Intell. Lab. Syst.37, 241-254) and takes advantage of the fact that exponential decay functions, when translated in time, retain their characteristic relaxation times while only their relative amounts or concentrations change. By such simple translations (slicing) it is possible to create a new "pseudo" direction in the relaxation data and thus facilitate application of trilinear (multiway) data-analytical methods. For the application on LF-NMR relaxation data, the method has two basic requirements in practice: (1) two or more samples must be analyzed simultaneously and (2) all samples must contain the same qualities (i.e., identical sets of distinct T(2) values). In return, if these requirements are fulfilled, the SLICING (trilinear decomposition) method provides very fast and unique curve-resolution of multiexponential LF-NMR relaxation curves and, as a spin-off, calibrations to reference data referring to individual proton components require only scaling of the resulting unique concentrations. In this work the performance of the SLICING method (including multiple slicing schemes) is compared to a traditional two-dimensional curve fitting algorithm named MATRIXFIT through application to simulated data in a large-scale exhaustive experimental design and the results validated by application to two small real data sets. Finally a new algorithm, Principal Phase Correction (PPC) based on principal component analysis, is proposed for phase rotation of CPMG quadrature data, an important prerequisite to optimal SLICING analysis.

Analyte flux through chronically implanted subcutaneous polyamide membranes differs in humans and rats

The rat is commonly used to evaluate physiological responses of subcutaneous tissue to implanted devices. In vivo longevity of various devices and the biocompatibility of biomaterials depend on how adjacent tissue interacts. How closely the rat model predicts the human response has not been well characterized. The objective of this study was to compare rat and human subcutaneous foreign body responses by monitoring the biochemical environment at a polymer-tissue interface over 8 days using microdialysis. Polyamide microdialysis probes were implanted subcutaneously in humans and rats (n = 12). Daily microdialysis samples were analyzed for glucose, lactate, pyruvate, glycerol, and urea. Blood glucose was also monitored. Analyte concentrations differed significantly between rats and humans at the implant-tissue interface. There were also qualitative differences in the 8-day trends. For example, over 8 days, microdialysate glucose increased two- to fourfold in humans but decreased in rats (P < 0.001). This study reveals profound physiological differences at material-tissue interfaces in rats and humans and highlights the need for caution when extrapolating subcutaneous rat biocompatibility data to humans.

Non-Debye dielectric relaxation in biological structures arises from their fractal nature

What differentiates biological tissues from one another, thereby allowing their accomplishment of a physiological function, is their organization at supracellular and cellular levels. We developed general dielectric models for Cantorian (or treelike) fractal networks of transmission lines that mimic supracellular organization in numerous biological tissues and tissue surfaces, and which are compatible with both in vitro and in vivo measuring techniques. By varying a set of adjustable physical and geometrical parameters pertaining to the structure, we could numerically reproduce a variety of dielectric dispersion curves-most of them of a composite type-that suitably described experimental data from relatively organized biological tissues. We therefore conclude that the well-documented non-Debye dielectric behavior of biological structures reflects their self-similar architecture.

Analysis of fluorophore diffusion by continuous distributions of diffusion coefficients: application to photobleaching measurements of multicomponent and anomalous diffusion

Fluorescence recovery after photobleaching (FRAP) is widely used to measure fluorophore diffusion in artificial solutions and cellular compartments. Two new strategies to analyze FRAP data were investigated theoretically and applied to complex systems with anomalous diffusion or multiple diffusing species: 1) continuous distributions of diffusion coefficients, alpha(D), and 2) time-dependent diffusion coefficients, D(t). A regression procedure utilizing the maximum entropy method was developed to resolve alpha(D) from fluorescence recovery curves, F(t). The recovery of multi-component alpha(D) from simulated F(t) with random noise was demonstrated and limitations of the method were defined. Single narrow Gaussian alpha(D) were recovered for FRAP measurements of thin films of fluorescein and size-fractionated FITC-dextrans and Ficolls, and multi-component alpha(D) were recovered for defined fluorophore mixtures. Single Gaussian alpha(D) were also recovered for solute diffusion in viscous media containing high dextran concentrations. To identify anomalous diffusion from FRAP data, a theory was developed to compute F(t) and alpha(D) for anomalous diffusion models defined by arbitrary nonlinear mean-squared displacement <x2> versus time relations. Several characteristic alpha(D) profiles for anomalous diffusion were found, including broad alpha(D) for subdiffusion, and alpha(D) with negative amplitudes for superdiffusion. A method to deduce apparent D(t) from F(t) was also developed and shown to provide useful complementary information to alpha(D). alpha(D) and D(t) were determined from photobleaching measurements of systems with apparent anomalous subdiffusion (nonuniform solution layer) and superdiffusion (moving fluid layer). The results establish a practical strategy to characterize complex diffusive phenomena from photobleaching recovery measurements.

Characterisation of erythrocyte shapes and sizes by NMR diffusion-diffraction of water: correlations with electron micrographs

The utility of 1H nuclear magnetic resonance (NMR) diffusion-diffraction of water as a tool for characterising red cell shape was investigated. Experiments were conducted on various cell suspensions which contained different shapes/forms of erythrocytes prepared by manipulating the conditions of the suspension medium, such as osmolality, and altering metabolism to affect the adenosine triphosphate concentration. Abnormal red cells from patients with hereditary stomatocytosis and megaloblastic anemia were also studied in order to assess the practical application of this "new" technique. The results clearly show that NMR diffusion-diffraction is sensitive to very small changes in mean cell dimensions and that a "characteristic" q-space plot/profile can be ascribed to each erythrocyte form. It was also found that the homogeneity of the cell shape and/or size is an important factor that affects the intensity of the diffusion-diffraction peaks. This study demonstrates the potential of the NMR diffusion-diffraction technique as a diagnostic tool in hematology.