"Tumor Treating Fields" delivered via electromagnetic induction have varied effects across glioma cell lines and electric field amplitudes

Previous studies reported that alternating electric fields (EFs) in the intermediate frequency (100-300 kHz) and low intensity (1-3 V/cm) regime - termed "Tumor Treating Fields" (TTFields) - have a specific, anti-proliferative effect on glioblastoma multiforme (GBM) cells. However, the mechanism(s) of action remain(s) incompletely understood, hindering the clinical adoption of treatments based on TTFields. To advance the study of such treatment in vitro, we developed an inductive device to deliver EFs to cell cultures which improves thermal and osmolar regulation compared to prior devices. Using this inductive device, we applied continuous, 200 kHz electromagnetic fields (EMFs) with a radial EF amplitude profile spanning 0-6.5 V/cm to cultures of primary rat astrocytes and several human GBM cell lines - U87, U118, GSC827, and GSC923 - for a duration of 72 hours. Cell density was assessed via segmented pixel densities from GFP expression (U87, U118) or from staining (astrocytes, GSC827, GSC923). Further RNA-Seq analyses were performed on GSC827 and GSC923 cells. Treated cultures of all cell lines exhibited little to no change in proliferation at lower EF amplitudes (0-3 V/cm). At higher amplitudes (> 4 V/cm), different effects were observed. Apparent cell densities increased (U87), decreased (GSC827, GSC923), or showed little change (U118, astrocytes). RNA-Seq analyses on treated and untreated GSC827 and GSC923 cells revealed differentially expressed gene sets of interest, such as those related to cell cycle control. Up- and down-regulation, however, was not consistent across cell lines nor EF amplitudes. Our results indicate no consistent, anti-proliferative effect of 200 kHz EMFs across GBM cell lines and thus contradict previous in vitro findings. Rather, effects varied across different cell lines and EF amplitude regimes, highlighting the need to assess the effect(s) of TTFields and similar treatments on a per cell line basis.

Water exchange rates measure active transport and homeostasis in neural tissue

For its size, the brain is the most metabolically active organ in the body. Most of its energy demand is used to maintain stable homeostatic physiological conditions. Altered homeostasis and active states are hallmarks of many diseases and disorders. Yet there is currently no direct and reliable method to assess homeostasis and absolute basal activity of cells in the tissue noninvasively without exogenous tracers or contrast agents. We propose a novel low-field, high-gradient diffusion exchange NMR method capable of directly measuring cellular metabolic activity via the rate constant for water exchange across cell membranes. Exchange rates are 140 ± 16 s−1 under normal conditions in viable ex vivo neonatal mouse spinal cords. High repeatability across samples suggest that values are absolute and intrinsic to the tissue. Using temperature and drug (ouabain) perturbations, we find that the majority of water exchange is metabolically active and coupled to active transport by the sodiumpotassium pump. We show that this water exchange rate is sensitive primarily to tissue homeostasis and provides distinct functional information. In contrast, the Apparent Diffusion Coefficient (ADC) measured with sub-millisecond diffusion times is sensitive primarily to tissue microstructure but not activity. Water exchange appears independently regulated from microstructural and oxygenation changes reported by ADC and T1 relaxation measurements in an oxygen–glucose deprivation model of stroke; exchange rates remain stable for 30–40 minutes before dropping to levels similar to the effect of ouabain and never completely recovering when oxygen and glucose are restored.

"Tumor Treating Fields" delivered via electromagnetic induction have varied effects across glioma cell lines and electric field amplitudes

Previous studies reported that alternating electric fields (EFs) in the intermediate frequency (100 - 300 kHz) and low intensity (1 - 3 V/cm) regime - termed "Tumor Treating Fields" (TTFields) - have a specific, anti-proliferative effect on glioblastoma multiforme (GBM) cells. However, the mechanism(s) of action remain(s) incompletely understood, hindering the clinical adoption of treatments based on TTFields. To advance the study of such treatment in vitro , we developed an inductive device to deliver EFs to cell cultures which improves thermal and osmolar regulation compared to prior devices. Using this inductive device, we applied continuous, 200 kHz electromagnetic fields (EMFs) with a radial EF amplitude profile spanning 0 - 6.5 V/cm to cultures of primary rat astrocytes and several human GBM cell lines - U87, U118, GSC827, and GSC923 - for a duration of 72 hours. Cell density was assessed via segmented pixel densities from GFP expression (U87, U118) or from staining (astrocytes, GSC827, GSC923). Further RNA-Seq analyses were performed on GSC827 and GSC923 cells. Treated cultures of all cell lines exhibited little to no change in proliferation at lower EF amplitudes (0 - 3 V/cm). At higher amplitudes (> 4 V/cm), different effects were observed. Apparent cell densities increased (U87), decreased (GSC827, GSC923), or showed little change (U118, astrocytes). RNA-Seq analyses on treated and untreated GSC827 and GSC923 cells revealed differentially expressed gene sets of interest, such as those related to cell cycle control. Up- and down-regulation, however, was not consistent across cell lines nor EF amplitudes. Our results indicate no consistent, anti-proliferative effect of 200 kHz EMFs across GBM cell lines and thus contradict previous in vitro findings. Rather, effects varied across different cell lines and EF amplitude regimes, highlighting the need to assess the effect(s) of TTFields and similar treatments on a per cell line basis.

Interrogating the relationship between the microstructure of amphiphilic poly(ethylene glycol-b-caprolactone) copolymers and their colloidal assemblies using non-interfering techniques

Understanding the microstructural parameters of amphiphilic copolymers that control the formation and structure of aggregated colloids (e.g., micelles) is essential for the rational design of hierarchically structured systems for applications in nanomedicine, personal care and food formulations. Although many analytical techniques have been employed to study such systems, in this investigation we adopted an integrated approach using non-interfering techniques - diffusion nuclear magnetic resonance (NMR) spectroscopy, dynamic light scattering (DLS) and synchrotron small-angle X-ray scattering (SAXS) - to probe the relationship between the microstructure of poly(ethylene glycol-b-caprolactone) (PEG-b-PCL) copolymers [e.g., block molecular weight (MW) and the mass fraction of PCL (f_PCL)] and the structure of their aggregates. Systematic trends in the self-assembly behaviour were determined using a large family of well-defined block copolymers with variable PEG and PCL block lengths (number-average molecular weights (M_n) between 2 and 10 and 0.5-15 kDa, respectively) and narrow dispersity (Ð < 1.12). For all of the copolymers, a clear transition in the aggregate structure was observed when the hydrophobic f_PCL was increased at a constant PEG block M_n, although the nature of this transition is also dependent on the PEG block M_n. Copolymers with low M_n PEG blocks (2 kDa) were observed to transition from unimers and loosely associated unimers to metastable aggregates and finally, to cylindrical micelles as the f_PCL was increased. In comparison, copolymers with PEG block M_n of between 5 and 10 kDa transitioned from heterogenous metastable aggregates to cylindrical micelles and finally, well-defined ellipsoidal micelles (of decreasing aspect ratios) as the f_PCL was increased. In all cases, the diffusion NMR spectroscopy, DLS and synchrotron SAXS results provided complementary information and the grounds for a phase diagram relating copolymer microstructure to aggregation behaviour and structure. Importantly, the absence of commonly depicted spherical micelles has implications for applications where properties may be governed by shape, such as, cellular uptake of nanomedicine formulations.

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.

A single-shot measurement of time-dependent diffusion over sub-millisecond timescales using static field gradient NMR

Time-dependent diffusion behavior is probed over sub-millisecond timescales in a single shot using a nuclear magnetic resonance static gradient time-incremented echo train acquisition (SG-TIETA) framework. The method extends the Carr-Purcell-Meiboom-Gill cycle under a static field gradient by discretely incrementing the π-pulse spacings to simultaneously avoid off-resonance effects and probe a range of timescales (50-500 µs). Pulse spacings are optimized based on a derived ruleset. The remaining effects of pulse inaccuracy are examined and found to be consistent across pure liquids of different diffusivities: water, decane, and octanol-1. A pulse accuracy correction is developed. Instantaneous diffusivity, Dinst(t), curves (i.e., half of the time derivative of the mean-squared displacement in the gradient direction) are recovered from pulse accuracy-corrected SG-TIETA decays using a model-free log-linear least squares inversion method validated by Monte Carlo simulations. A signal-averaged 1-min experiment is described. A flat Dinst(t) is measured on pure dodecamethylcyclohexasiloxane, whereas decreasing Dinst(t) is measured on yeast suspensions, consistent with the expected short-time Dinst(t) behavior for confining microstructural barriers on the order of micrometers.

A Novel In Vitro Device to Deliver Induced Electromagnetic Fields to Cell and Tissue Cultures

We have developed a novel, to our knowledge, in vitro instrument that can deliver intermediate-frequency (100-400 kHz), moderate-intensity (up to and exceeding 6.5 V/cm pk-pk) electric fields (EFs) to cell and tissue cultures generated using induced electromagnetic fields (EMFs) in an air-core solenoid coil. A major application of these EFs is as an emerging cancer treatment modality. In vitro studies by Novocure reported that intermediate-frequency (100-300 kHz), low-amplitude (1-3 V/cm) EFs, which they called "tumor-treating fields (TTFields)," had an antimitotic effect on glioblastoma multiforme (GBM) cells. The effect was found to increase with increasing EF amplitude. Despite continued theoretical, preclinical, and clinical study, the mechanism of action remains incompletely understood. All previous in vitro studies of "TTFields" have used attached, capacitively coupled electrodes to deliver alternating EFs to cell and tissue cultures. This contacting delivery method suffers from a poorly characterized EF profile and conductive heating that limits the duration and amplitude of the applied EFs. In contrast, our device delivers EFs with a well-characterized radial profile in a noncontacting manner, eliminating conductive heating and enabling thermally regulated EF delivery. To test and demonstrate our system, we generated continuous, 200-kHz EMF with an EF amplitude profile spanning 0-6.5 V/cm pk-pk and applied them to exemplar human thyroid cell cultures for 72 h. We observed moderate reduction in cell density (<10%) at low EF amplitudes (<4 V/cm) and a greater reduction in cell density of up to 25% at higher amplitudes (4-6.5 V/cm). Our device can be readily extended to other EF frequency and amplitude regimes. Future studies with this device should contribute to the ongoing debate about the efficacy and mechanism(s) of action of "TTFields" by better isolating the effects of EFs and providing access to previously inaccessible EF regimes.

Real-time measurement of diffusion exchange rate in biological tissue

Diffusion exchange spectroscopy (DEXSY) provides a means to isolate the signal attenuation associated with exchange from other sources of signal loss. With the total diffusion weighting b₁+b₂=b_s held constant, DEXSY signals acquired with b₁=0 or b₂=0 have no exchange weighting, while a DEXSY signal acquired with b₁=b₂ has maximal exchange weighting. The exchange rate can be estimated by fitting a diffusion exchange model to signals acquired with variable mixing times. Conventionally, acquired signals are normalized by a signal with b₁=0 and b₂=0 to remove the decay due to spin-lattice relaxation. Instead, division by a signal with equal b_s but b₁=0 or b₂=0 reduces spin-lattice relaxation weighting of the apparent exchange rate (AXR). Furthermore, apparent diffusion-weighted R₁ relaxation rates can be estimated from non-exchange-weighted DEXSY signals. Estimated R₁ values are utilized to remove signal decay due to spin-lattice relaxation from exchange-weighted signals, permitting a more precise estimate of AXR with less data. Data reduction methods are proposed and tested with regards to statistical accuracy and precision of AXR estimates on simulated and experimental data. Simulations show that the methods are capable of accurately measuring the ground-truth exchange rate. The methods remain accurate even when the assumption that DEXSY signals attenuate with b is violated, as occurs for restricted diffusion. Experimental data was collected from fixed neonatal mouse spinal cord samples at 25 and 7°C using the strong static magnetic field gradient produced by a single-sided permanent magnet (i.e., an NMR MOUSE). The most rapid method for exchange measurements requires only five data points (an 80 s experiment as implemented) and achieves a similar level of accuracy and precision to the baseline method using 44 data points. This represents a significant improvement in acquisition speed, overcoming a barrier which has limited the use of DEXSY on living specimen.

Limits to flow detection in phase contrast MRI

Pulsed gradient spin echo (PGSE) complex signal behavior becomes dominated by attenuation rather than oscillation when displacements due to flow are similar or less than diffusive displacements. In this "slow-flow" regime, the optimal displacement encoding parameter q for phase contrast velocimetry depends on the diffusive length scale q s l o w = 1 / l D = 1 / 2 D Δ rather than the velocity encoding parameter v _enc = π/(qΔ). The minimum detectable mean velocity using the difference between the phase at +q _slow and -q _slow is 〈 v m i n 〉 = 1 / SNR D / Δ . These theories are then validated and applied to MRI by performing PGSE echo planar imaging experiments on water flowing through a column with a bulk region and a beadpack region at controlled flow rates. Velocities as slow as 6 μm/s are detected with velocimetry. Theories, MRI experimental protocols, and validation on a controlled phantom help to bridge the gap between porous media NMR and pre-clinical phase contrast and diffusion MRI.

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.

Cross-Correlative Single-Cell Analysis Reveals Biological Mechanisms of Nanoparticle Radiosensitization

Nanoparticle radiosensitization has been demonstrated well to enhance the effects of radiotherapy, motivate the improvement of therapeutic ratios, and decrease morbidity in cancer treatment. A significant challenge exists in optimizing formulations and translation due to insufficient knowledge of the associated mechanisms, which have historically been limited to physical concepts. Here, we investigated a concept for the role of biological mechanisms. The mere presence of gold nanoparticles led to a down-regulation of thymidylate synthase, important for DNA damage repair in the radioresistant S-phase cells. By developing a cross-correlative methodology to reveal probabilistic gold nanoparticle uptake by cell sub-populations and the associated sensitization as a function of the uptake, a number of revealing observations have been achieved. Surprisingly, for low numbers of nanoparticles, a desensitization action was observed. Sensitization was discovered to preferentially impact S-phase cells, in which impairment of the DNA damage response by the homologous recombination pathway dominates. This small but radioresistant cell population correlates with much greater proliferative ability. Thus, a paradigm is presented whereby enhanced DNA damage is not necessarily due to an increase in the number of DNA double-strand breaks (DSBs) created but can be from a nanoparticle-induced impairment of the damage response by down-regulating repair proteins such as thymidylate synthase.

Glass Dynamics and Domain Size in a Solvent-Polymer Weak Gel Measured by Multidimensional Magnetic Resonance Relaxometry and Diffusometry

Nuclear magnetic resonance measurements of rotational and translational molecular dynamics are applied to characterize the nanoscale dynamic heterogeneity of a physically cross-linked solvent-polymer system above and below the glass transition temperature. Measured rotational dynamics identify domains associated with regions of solidlike and liquidlike dynamics. Translational dynamics provide quantitative length and timescales of nanoscale heterogeneity due to polymer network cross-link density. Mean squared displacement measurements of the solvent provide microrheological characterization of the system and indicate glasslike caging dynamics both above and below the glass transition temperature.

Generalized Mean Apparent Propagator MRI to Measure and Image Advective and Dispersive Flows in Medicine and Biology

Water transport in biological systems spans different regimes with distinct physical behaviors: diffusion, advection, and dispersion. Identifying these regimes is of paramount importance in many in vivo applications, among them, measuring microcirculation of blood in capillary networks and cerebrospinal fluid transport in the glymphatic system. Diffusion magnetic resonance imaging (dMRI) can be used to encode water displacements, and a Fourier transform of the acquired signal furnishes a displacement probability density function known as the propagator. This transformation normally requires the use of a fast Fourier transform (FFT), which presents major feasibility challenges when scanning in vivo, mainly because of dense signal sampling, resulting in long acquisition times. A second approach to reconstruct the propagator is by using analytical representation of the signal, overcoming many of the FFT's limitations. In all analytical implementations of dMRI to date, the translational motion of water has been assumed to be exclusively diffusive, which is the case only in the absence of flow. However, retaining the phase information from the diffusion signal provides the ability to measure both mean coherent velocity and random diffusion from a single experiment. We implement and extend an analytical framework, mean apparent propagator (MAP), which can account for non-zero flow conditions. We call this method generalized MAP or GMAP. We describe a numerical optimization scheme and implement it on data from an MRI flow phantom constructed from a pack of 10- [Formula: see text] beads. The advantages of GMAP over the FFT-based method in the context of sampling density and low-flow detection were demonstrated, and analytically derived propagator moments were shown to agree with theoretical values even after data subsampling. GMAP would enable the detection of microflow in vivo that could help elucidate many important biological processes.

Rapid detection of the presence of diffusion exchange

Diffusion exchange spectroscopy (DEXSY) provides a detailed picture of how fluids in different microenvironments communicate with one another but requires a large amount of data. For DEXSY MRI, a simple measure of apparent exchanging fractions may suffice to characterize and differentiate materials and tissues. Reparameterizing signal intensity from a PGSE-storage-PGSE experiment as a function of the sum, b_s=b₁+b₂, and difference b_d=b₂-b₁ of the diffusion encodings separates diffusion weighting from exchange weighting. Exchange leads to upward curvature along a slice of constant b_s. Exchanging fractions can be measured rapidly by a finite difference approximation of the curvature using four data points. The method is generalized for non-steady-state and multi-site exchange. We apply the method to image exchanging fractions and calculate exchange rates of water diffusing across the bulk water interface of a glass capillary array.

Novel Gd-Loaded Silicon Nanohybrid: A Potential Epidermal Growth Factor Receptor Expressing Cancer Cell Targeting Magnetic Resonance Imaging Contrast Agent

Continuing our research efforts in developing mesoporous silicon nanoparticle-based biomaterials for cancer therapy, we employed here porous silicon nanoparticles as a nanocarrier to deliver contrast agents to diseased cells. Nanoconfinement of small molecule Gd-chelates (L1-Gd) enhanced the T₁ contrast dramatically compared to distinct Gd-chelate (L1-Gd) by virtue of its slow tumbling rate, increased number of bound water molecules, and their occupancy time. The newly synthesized Gd-chelate (L1-Gd) was covalently grafted on silicon nanostructures and conjugated to an antibody specific for epidermal growth factor receptor (EGFR) via a hydrazone linkage. The salient feature of this nanosized contrast agent is the capability of EGFR targeted delivery to cancer cells. Mesoporous silicon nanoparticles were chosen as the nanocarrier because of their high porosity, high surface area, and excellent biodegradability. This type of nanosized contrast agent also performs well in high magnetic fields.

NMR investigation of water diffusion in different biofilm structures

Mass transfer in biofilms is determined by diffusion. Different mostly invasive approaches have been used to measure diffusion coefficients in biofilms, however, data on heterogeneous biomass under realistic conditions is still missing. To non-invasively elucidate fluid-structure interactions in complex multispecies biofilms pulsed field gradient-nuclear magnetic resonance (PFG-NMR) was applied to measure the water diffusion in five different types of biomass aggregates: one type of sludge flocs, two types of biofilm, and two types of granules. Data analysis is an important issue when measuring heterogeneous systems and is shown to significantly influence the interpretation and understanding of water diffusion. With respect to numerical reproducibility and physico-chemical interpretation, different data processing methods were explored: (bi)-exponential data analysis and the Γ distribution model. Furthermore, the diffusion coefficient distribution in relation to relaxation was studied by D-T₂ maps obtained by 2D inverse Laplace transform (2D ILT). The results show that the effective diffusion coefficients for all biofilm samples ranged from 0.36 to 0.96 relative to that of water. NMR diffusion was linked to biofilm structure (e.g., biomass density, organic and inorganic matter) as observed by magnetic resonance imaging and to traditional biofilm parameters: diffusion was most restricted in granules with compact structures, and fast diffusion was found in heterotrophic biofilms with fluffy structures. The effective diffusion coefficients in the biomass were found to be broadly distributed because of internal biomass heterogeneities, such as gas bubbles, precipitates, and locally changing biofilm densities. Thus, estimations based on biofilm bulk properties in multispecies systems can be overestimated and mean diffusion coefficients might not be sufficiently informative to describe mass transport in biofilms and the near bulk.

Obtaining T1-T2 distribution functions from 1-dimensional T1 and T2 measurements: The pseudo 2-D relaxation model

We present the pseudo 2-D relaxation model (P2DRM), a method to estimate multidimensional probability distributions of material parameters from independent 1-D measurements. We illustrate its use on 1-D T1 and T2 relaxation measurements of saturated rock and evaluate it on both simulated and experimental T1-T2 correlation measurement data sets. Results were in excellent agreement with the actual, known 2-D distribution in the case of the simulated data set. In both the simulated and experimental case, the functional relationships between T1 and T2 were in good agreement with the T1-T2 correlation maps from the 2-D inverse Laplace transform of the full 2-D data sets. When a 1-D CPMG experiment is combined with a rapid T1 measurement, the P2DRM provides a double-shot method for obtaining a T1-T2 relationship, with significantly decreased experimental time in comparison to the full T1-T2 correlation measurement.

The lognormal and gamma distribution models for estimating molecular weight distributions of polymers using PGSE NMR

We present comprehensive derivations for the statistical models and methods for the use of pulsed gradient spin echo (PGSE) NMR to characterize the molecular weight distribution of polymers via the well-known scaling law relating diffusion coefficients and molecular weights. We cover the lognormal and gamma distribution models and linear combinations of these distributions. Although the focus is on methodology, we illustrate the use experimentally with three polystyrene samples, comparing the NMR results to gel permeation chromatography (GPC) measurements, test the accuracy and noise-sensitivity on simulated data, and provide code for implementation.

The Power of Heterogeneity: Parameter Relationships from Distributions

Complex scientific data is becoming the norm, many disciplines are growing immensely data-rich, and higher-dimensional measurements are performed to resolve complex relationships between parameters. Inherently multi-dimensional measurements can directly provide information on both the distributions of individual parameters and the relationships between them, such as in nuclear magnetic resonance and optical spectroscopy. However, when data originates from different measurements and comes in different forms, resolving parameter relationships is a matter of data analysis rather than experiment. We present a method for resolving relationships between parameters that are distributed individually and also correlated. In two case studies, we model the relationships between diameter and luminescence properties of quantum dots and the relationship between molecular weight and diffusion coefficient for polymers. Although it is expected that resolving complicated correlated relationships require inherently multi-dimensional measurements, our method constitutes a useful contribution to the modelling of quantitative relationships between correlated parameters and measurements. We emphasise the general applicability of the method in fields where heterogeneity and complex distributions of parameters are obstacles to scientific insight.

Gamma convolution models for self-diffusion coefficient distributions in PGSE NMR

We introduce a closed-form signal attenuation model for pulsed-field gradient spin echo (PGSE) NMR based on self-diffusion coefficient distributions that are convolutions of n gamma distributions, n⩾1. Gamma convolutions provide a general class of uni-modal distributions that includes the gamma distribution as a special case for n=1 and the lognormal distribution among others as limit cases when n approaches infinity. We demonstrate the usefulness of the gamma convolution model by simulations and experimental data from samples of poly(vinyl alcohol) and polystyrene, showing that this model provides goodness of fit superior to both the gamma and lognormal distributions and comparable to the common inverse Laplace transform.