Dependence Of Apparent Diffusion Coefficient Research Articles

Pulse-field gradient nuclear magnetic resonance of protein translational diffusion from native to non-native states.

Hydrodynamic radii (Rh -values) calculated from diffusion coefficients measured by pulse-field-gradient nuclear magnetic resonance are compared for folded and unfolded proteins. For native globular proteins, the Rh -values increase as a power of 0.35 with molecular size, close to the scaling factor of 0.33 predicted from polymer theory. Unfolded proteins were studied under four sets of conditions: in the absence of denaturants, in the presence of 6M urea, in 95% dimethyl sulfoxide (DMSO), and in 40% hexafluoroisopropanol (HFIP). Scaling factors under all four unfolding conditions are similar (0.49-0.53) approaching the theoretical value of 0.60 for a fully unfolded random coil. Persistence lengths are also similar, except smaller in 95% DMSO, suggesting that the polypeptides are more disordered on a local scale with this solvent. Three of the proteins in our unfolded set have an asymmetric sequence-distribution of charged residues. While these proteins behave normally in water and 6M urea, they give atypically low Rh -values in 40% HFIP and 95% DMSO suggesting they are forming electrostatic hairpins, favored by their asymmetric sequence charge distribution and the low dielectric constants of DMSO and HFIP. While diffusion-ordered NMR spectroscopy can separate small molecules, we show a number of factors combine to make protein-sized molecules much more difficult to resolve in mixtures. Finally, we look at the temperature dependence of apparent diffusion coefficients. Small molecules show a linear temperature response, while large proteins show abnormally large apparent diffusion coefficients at high temperatures due to convection, suggesting diffusion reference standards are only useful near 25°C.

Modeling an equivalent b-value in diffusion-weighted steady-state free precession.

PurposeDiffusion‐weighted steady‐state free precession (DW‐SSFP) is shown to provide a means to probe non‐Gaussian diffusion through manipulation of the flip angle. A framework is presented to define an effective b‐value in DW‐SSFP.TheoryThe DW‐SSFP signal is a summation of coherence pathways with different b‐values. The relative contribution of each pathway is dictated by the flip angle. This leads to an apparent diffusion coefficient (ADC) estimate that depends on the flip angle in non‐Gaussian diffusion regimes. By acquiring DW‐SSFP data at multiple flip angles and modeling the variation in ADC for a given form of non‐Gaussianity, the ADC can be estimated at a well‐defined effective b‐value.MethodsA gamma distribution is used to model non‐Gaussian diffusion, embedded in the Buxton signal model for DW‐SSFP. Monte‐Carlo simulations of non‐Gaussian diffusion in DW‐SSFP and diffusion‐weighted spin‐echo sequences are used to verify the proposed framework. Dependence of ADC on flip angle in DW‐SSFP is verified with experimental measurements in a whole, human postmortem brain.ResultsMonte‐Carlo simulations reveal excellent agreement between ADCs estimated with diffusion‐weighted spin‐echo and the proposed framework. Experimental ADC estimates vary as a function of flip angle over the corpus callosum of the postmortem brain, estimating the mean and standard deviation of the gamma distribution as 1.50·10-4 mm2/s and 2.10·10-4 mm2/s.ConclusionDW‐SSFP can be used to investigate non‐Gaussian diffusion by varying the flip angle. By fitting a model of non‐Gaussian diffusion, the ADC in DW‐SSFP can be estimated at an effective b‐value, comparable to more conventional diffusion sequences.

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 9.4T with three radical prostatectomy specimens, with diffusion times in the range 10-120ms and b-values 0-3000s/mm2 . ADC and FA were calculated from DTI measurements at b-values of 800 and 1600s/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 0.10-0.15μm2 /ms as diffusion time increased from 10 to 120ms. FA increased with diffusion time at b-values of 800 and 1600s/mm2 but was predicted to be independent of diffusion time at b=3000s/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.

The interaction between apparent diffusion coefficients and transverse relaxation rates of human brain metabolites and water studied by diffusion-weighted spectroscopy at 7 T

The dependence of apparent diffusion coefficients (ADCs) of molecules in biological tissues on an acquisition-specific timescale is a powerful mechanism for studying tissue microstructure. Unlike water, metabolites are confined mainly to intracellular compartments, thus providing higher specificity to tissue microstructure. Compartment-specific structural and chemical properties may also affect molecule transverse relaxation times (T₂). Here, we investigated the correlation between diffusion and relaxation for N-acetylaspartate, creatine and choline compounds in human brain white matter in vivo at 7 T, and compared them with those of water under the same experimental conditions. Data were acquired in a volume of interest in parietal white matter at two different diffusion times, Δ = 44 and 246 ms, using a matrix of three echo times (T(E)) and five diffusion weighting values (up to 4575 s/mm²). Significant differences in the dependence of the ADCs on T(E) were found between water and metabolites, as well as among the different metabolites. A significant decrease in water ADC as a function of TE was observed only at the longest diffusion time (p < 0.001), supporting the hypothesis that at least part of the restricted water pool can be associated with longer T₂, as suggested by previous studies in vitro. Metabolite data showed an increase of creatine (p < 0.05) and N-acetylaspartate (p < 0.05) ADCs with TE at Δ = 44 ms, and a decrease of creatine (p < 0.05) and N-acetylaspartate (p = 0.1) ADCs with TE at Δ = 246 ms. No dependence of choline ADC on TE was observed. The metabolite results suggest that diffusion and relaxation properties are dictated not only by metabolite distribution in different cell types, but also by other mechanisms, such as interactions with membranes, exchange between "free" and "bound" states or interactions with microsusceptibility gradients.

Hybrid multidimensional T2 and diffusion‐weighted MRI for prostate cancer detection

To study the dependence of apparent diffusion coefficient (ADC) and T2 on echo time (TE) and b-value, respectively, in normal prostate and prostate cancer, using two-dimensional MRI sampling, referred to as "hybrid multidimensional imaging." The study included 10 patients with biopsy-proven prostate cancer who underwent 3 Tesla prostate MRI. Diffusion-weighted MRI (DWI) data were acquired at b = 0, 750, and 1500 s/mm(2) . For each b-value, data were acquired at TEs of 47, 75, and 100 ms. ADC and T2 were measured as a function of b-value and TE, respectively, in 15 cancer and 10 normal regions of interest (ROIs). The Friedman test was used to test the significance of changes in ADC as a function of TE and of T2 as a function of b-value. In normal prostate ROIs, the ADC at TE of 47 ms is significantly smaller than ADC at TE of 100 ms (P = 0.0003) and T2 at b-value of 0 s/mm(2) is significantly longer than T2 at b-value of 1500 s/mm(2) (P = 0.001). In cancer ROIs, average ADC and T2 values do not change as a function of TE and b-value, respectively. However, in many cancer pixels, there are large decreases in the ADC as a function of TE and large increases in T2 as a function of b-value. Cancers are more conspicuous in ADC maps at longer TEs. Parameters derived from hybrid imaging that depend on coupled/associated values of ADC and T2 may improve the accuracy of MRI in diagnosing prostate cancer.

Meaningful Interpretation of Subdiffusive Measurements in Living Cells (Crowded Environment) by Fluorescence Fluctuation Microscopy

In living cell or its nucleus, the motions of molecules are complicated due to the large crowding and expected heterogeneity of the intracellular environment. Randomness in cellular systems can be either spatial (anomalous) or temporal (heterogeneous). In order to separate both processes, we introduce anomalous random walks on fractals that represented crowded environments. We report the use of numerical simulation and experimental data of single-molecule detection by fluorescence fluctuation microscopy for detecting resolution limits of different mobile fractions in crowded environment of living cells. We simulate the time scale behavior of diffusion times τD(τ) for one component, e.g. the fast mobile fraction, and a second component, e.g. the slow mobile fraction. The less the anomalous exponent α the higher the geometric crowding of the underlying structure of motion that is quantified by the ratio of the Hausdorff dimension and the walk exponent d f /dw and specific for the type of crowding generator used. The simulated diffusion time decreases for smaller values of α ≠ 1 but increases for a larger time scale τ at a given value of α ≠ 1. The effect of translational anomalous motion is substantially greater if α differs much from 1. An α value close to 1 contributes little to the time dependence of subdiffusive motions. Thus, quantitative determination of molecular weights from measured diffusion times and apparent diffusion coefficients, respectively, in temporal auto- and crosscorrelation analyses and from time-dependent fluorescence imaging data are difficult to interpret and biased in crowded environments of living cells and their cellular compartments; anomalous dynamics on different time scales τ must be coupled with the quantitative analysis of how experimental parameters change with predictions from simulated subdiffusive dynamics of molecular motions and mechanistic models. We first demonstrate that the crowding exponent α also determines the resolution of differences in diffusion times between two components in addition to photophyscial parameters well-known for normal motion in dilute solution. The resolution limit between two different kinds of single molecule species is also analyzed under translational anomalous motion with broken ergodicity. We apply our theoretical predictions of diffusion times and lower limits for the time resolution of two components to fluorescence images in human prostate cancer cells transfected with GFP-Ago2 and GFP-Ago1. In order to mimic heterogeneous behavior in crowded environments of living cells, we need to introduce so-called continuous time random walks (CTRW). CTRWs were originally performed on regular lattice. This purely stochastic molecule behavior leads to subdiffusive motion with broken ergodicity in our simulations. For the first time, we are able to quantitatively differentiate between anomalous motion without broken ergodicity and anomalous motion with broken ergodicity in time-dependent fluorescence microscopy data sets of living cells. Since the experimental conditions to measure a selfsame molecule over an extended period of time, at which biology is taken place, in living cells or even in dilute solution are very restrictive, we need to perform the time average over a subpopulation of different single molecules of the same kind. For time averages over subpopulations of single molecules, the temporal auto- and crosscorrelation functions are first found. Knowing the crowding parameter α for the cell type and cellular compartment type, respectively, the heterogeneous parameter γ can be obtained from the measurements in the presence of the interacting reaction partner, e.g. ligand, with the same α value. The product α⋅γ=γ˜ is not a simple fitting parameter in the temporal auto- and two-color crosscorrelation functions because it is related to the proper physical models of anomalous (spatial) and heterogeneous (temporal) randomness in cellular systems. We have already derived an analytical solution for γ˜ in the special case of γ = 3/2 . In the case of two-color crosscorrelation or/and two-color fluorescence imaging (co-localization experiments), the second component is also a two-color species gr, for example a different molecular complex with an additional ligand. Here, we first show that plausible biological mechanisms from FCS/ FCCS and fluorescence imaging in living cells are highly questionable without proper quantitative physical models of subdiffusive motion and temporal randomness. At best, such quantitative FCS/ FCCS and fluorescence imaging data are difficult to interpret under crowding and heterogeneous conditions. It is challenging to translate proper physical models of anomalous (spatial) and heterogeneous (temporal) randomness in living cells and their cellular compartments like the nucleus into biological models of the cell biological process under study testable by single-molecule approaches. Otherwise, quantitative FCS/FCCS and fluorescence imaging measurements in living cells are not well described and cannot be interpreted in a meaningful way.

MO‐D‐204B‐09: Characterization of ADC and T2 Responses of Tumor Tissue to Radiation with Histological Interpretation

Purpose: To characterize the temporal, dose, and oxygenation dependence of apparent diffusion coefficient (ADC) and transverse relaxation time (T2) responses of tumor tissue after treatment with ionizing radiation, in the context of histology and observed volume changes. Methods: Human glioblastoma multiforme (GBM) tumors were grown subcutaneously in mice. Multiple groups treated with 200 kVp x‐rays and two unirradiated control groups were formed (n=6 per group). Single fraction groups received doses ranging from 50 cGy to 800 cGy. Fractionated deliveries were 800 cGy in 1, 2 or 3 fractions and fractions were separated by 24 or 72 hours. Oxygenation dependence was investigated with 800 cGy in a single fraction with a ligature in place to deplete the tumor's oxygen supply. Images were acquired on a 9.4 T MR system at multiple time points before and after treatment. Quantitative ADC and T2 maps were produced from each image set. Results: A dose dependent ADC increase was observed in irradiated tumor tissue. Maximum response was observed 7 days post‐treatment. Tumor T2 was maximum 3 days after treatment and had fallen below baseline by 14 days after treatment. Fractionation did not affect amplitude of ADC response, but delayed the time at which maximum ADC occurred and the time for T2 to fall below baseline. Hypoxic tissues showed reduced ADC response after treatment. Preliminary pathology results show ADC response correlated with reduced tumor growth rate and reduced mitotic cell count. Early increases in ADC and T2 were associated with early‐stage necrosis and edema. Later responses were associated with evolving necrosis and extracellular macromolecule accumulation. Conclusion: These results further the understanding of ADC and T2 responses and their development as biomarkers of treatment response.

Aggregation of Blends of Poly(N-vinyl pyrrolidone) and End-functional Polystyrene in Aqueous Solution

AbstractAggregation behavior of blend of poly(N-vinyl pyrrolidone) (PVP) and polystyrene having a sulfonic acid group at one end (PSS) in aqueous solution was investigated by dynamic and static light scattering (DLS, SLS) measurements and scanning electron microscopy (SEM). The angular dependence of apparent diffusion coefficients suggested that the PVP-PSS aggregate shows an anisotropic structure. Comparison of an experimental SLS profile with theoretical calculations of particle scattering functions indicated that the PVP-PSS aggregate formed rodlike structure with 1200 nm height and 20 nm cross-sectional radius. Corresponding results were obtained by SEM observations.

Investigation on the early and late stage phase-separation dynamics of poly(methyl methacrylate)/poly(α-methyl styrene- co-acrylonitrile) blends through rheological and scattering functions

The phase-separation behavior of poly(methyl methacrylate)/poly(α-methyl styrene- co-acrylonitrile) (PMMA/α-MSAN) blends with two different compositions was studied by time-resolved small angle light scattering (SALS) in the spinodal decomposition (SD) regime from 160 to 210 °C. The rheological function (WLF-like equation) was introduced into the processing of light scattering data. It was found that the WLF-like equation was applicable to describe the temperature dependence of apparent diffusion coefficient D app and the relaxation time τ of normalized scattering intensity ( I( t)− I(0))/( I m− I(0)) at the early stage of SD, as well as the relaxation time τ of maximum scattering intensity I m and characteristic scattering vector q m with I m at the late stage of SD for PMMA/α-MSAN blends with two different compositions. This is in consistence with the phase-separation behavior of PMMA/SAN reported in our previous paper.

Spatial scale-dependent tracer diffusion in bulk polycarbonate studied by holographic relaxation

The diffusion of a photochromic dye tracer in polycarbonate was studied by a holographic relaxation technique (forced Rayleigh scattering) at temperatures close to the glass transition temperature. By varying the holographic grating period the results could be interpreted via the spatial scale dependence of apparent diffusion coefficients within a two-state diffusion model. This indicates inhomogeneities on the scale of a few micrometers in the polymer glass.

Kinetics of Intermolecular Interaction during Protein Folding of Reduced Cytochrome c

Kinetics of intermolecular interaction between reduced cytochrome c (Cyt c) protein and solvent during the protein-refolding process is studied by monitoring the time dependence of apparent diffusion coefficient ( D) using the pulsed-laser-induced transient grating technique. The refolding was triggered by photoinduced reduction of unfolded Fe(III) Cyt c in 3.5 M guanidine hydrochloride (GdnHCl) solution and the change in the diffusion coefficient was monitored in time domain. The relationship between D and the protein conformations under equilibrium condition were investigated at various GdnHCl concentrations using a photolabeling reagent. The time dependence of the observed transient grating signal was analyzed using these data and two models: a continuous change model of the intermolecular interaction and a two-state model. It was found that the TG signals in various time ranges can be consistently reproduced well by the two-state model. The dynamics of D is expressed well by a single exponential function with a rate constant of 22 ± 7 s −1 in a whole time range. The folding process of Cyt c is discussed based on these observations.

Residue-specific real-time NMR diffusion experiments define the association states of proteins during folding.

Characterizing the association states of proteins during folding is critical for understanding the nature of protein-folding intermediates and protein-folding pathways, protein aggregation, and disease-related aggregation. To study the association states of unfolded, folded, and intermediate species during protein folding, we have introduced a novel residue-specific real-time NMR diffusion experiment. This experiment, a combination of NMR real-time folding experiments and 3D heteronuclear pulsed field gradient NMR diffusion experiments (LED-HSQC), measures hydrodynamic properties, or molecular sizes, of kinetic species directly during the folding process. Application of the residue-specific real-time NMR diffusion experiments to characterize the folding of the collagen triple helix motif shows that this experiment can be used to determine the association states of unfolded, folded, and kinetic intermediates with transient lifetimes simultaneously. The ratio of the apparent translational diffusion coefficients of the unfolded to the folded form of the triple helix is 0.59, which correlates very well with a theoretical ratio for monomer to linear trimer. The apparent diffusion coefficients of the kinetic intermediates formed during triple helix folding indicate the formation of trimer-like associates which is consistent with previously published kinetic and relaxation data. The residue-specific time dependence of apparent diffusion coefficients of monomer and trimer peaks also illustrates the ability to use diffusion data to probe the directionality of triple helix formation. NMR diffusion experiments provide a new strategy for the investigation of protein-folding mechanisms, both to understand the role of kinetic intermediates and to determine the time-dependent aggregation processes in human diseases.

Compartmental study of diffusion and relaxation measured in vivo in normal and ischemic rat brain and trigeminal nerve

The correlation between the apparent diffusion coefficient (ADC) and T(2) of water in rat brain and trigeminal nerve was investigated using a hybrid diffusion-weighted-CPMG imaging sequence. Little dependence of ADC on T(2) was found in brain regions of interest, which is postulated to be due to rapid exchange between intra- and extracellular water. Conversely, the ADC of water in trigeminal nerve was found to change significantly with echo time (TE). Parallel to the nerve and with a constant diffusion time (t(diff) = 10.8 ms), the ADC increased by approximately 30% between TEs of 25 ms and 185 ms; perpendicular to the nerve, the ADC decreased by a similar amount over the same range of TE. Measurements made following the onset of global ischemia yielded lower ADCs, with similar dependence on TE. Observations that transverse relaxation of water in nerves is multiexponential have previously been interpreted in terms of microanatomical compartments in slow exchange. In the context of this interpretation, our data suggest that diffusional anisotropy is greater outside than within the myelinated axons. Further, data following the onset of global ischemia suggest that the mechanism(s) by which ADC is reduced affect most or all microanatomical environments of nerve, at least insofar as they are represented over the TE domain investigated. Magn Reson Med 43:837-844, 2000.

Dependence of apparent diffusion coefficients on axonal spacing, membrane permeability, and diffusion time in spinal cord white matter.

We used a numerical simulation of water self-diffusion among permeable cylinders to predict the dependence of MR-based apparent diffusion coefficients in white matter on axonal separation, barrier permeability, and diffusion time (T). The transverse apparent diffusion coefficient (tADC), calculated with simulated diffusion-sensitizing gradients perpendicular to the axon fibers, remains a function of T down to diffusion times as short as .1 microsec for a range of diffusion barrier permeability. As the diffusion time lengthens, the response of tADC depends on axon diameter, with decreases in tADC occurring earliest, and most dramatically, for the smallest fiber diameter simulated (2 microm). For a given axonal separation, asymptotic values of ADC are determined by permeability alone and are the same for 2-microm and 11-microm fibers of equal membrane permeability. The effect of increased relative intracellular volume is manifested primarily in a decrease in tADC at short T. Increases in interaxonal spacing increase the tADC at asymptotically long diffusion times and reduce the dependence on permeability. However, at the widest plausible axonal separations, permeability remains an important determinant of tADC. These simulations may enhance interpretation of measured tADC in the context of the underlying physiologic and structural changes at the cellular level that accompany white-matter disease.

Dynamics of electron hopping in redox polymer electrodes using kinetic Ising model

The analysis of diffusion-migration equations pertaining to electron hopping and physical displacement in redox polymer electrodes is carried out using kinetic Ising model formalism. It is shown that, by the appropriate choice of transition probabilities obeying detailed balancing conditions, a hierarchy of transport equations can be derived. Earlier transport equations due to Nernst-Planck and Saveant are derived as special cases. The dependence of apparent diffusion coefficient on number density of redox centres, polymer morphology etc. are pointed out. Several new insights concerning the microscopic basis underlying the hitherto known phenomenological equations are demonstrated.

Diffusion/adsorption model of cellulose dyeing. II. Ordinary cellulose–direct dye system

AbstractThe diffusion and adsorption of C.I. Direct Yellow 12 and Blue 15 in water‐swollen ordinary cellophane sheets were examined at various ionic strengths. The concentration dependence of apparent diffusion coefficients, Dc, for these dyes was obtained from the diffusion profiles in the substrate, which were measured by the use of the cylindrical film roll method. The decrease of apparent porosity with an increase in the amounts of adsorption was observed. To explain the diffusion/adsorption behaviors of these systems, a variable porosity model was proposed and was applied to analyze the concentration dependence of Dc's. The diffusion/adsorption behaviors of these dyes could be quantitatively described by this model at relatively low ionic strengths. At higher ionic strengths and/or lower values of C, i.e., at the large values of Cim/Cm, where the C's are the concentrations of immobilized (suffix im) and mobile (suffix m) species, it needed to introduce the concept of dynamic equilibria which occurred simultaneously with diffusion but deviated from the true equiliblia measured by the adsorption experiments.