On the perils of multiexponential fitting of diffusion MR data

Quantitative diffusion MRI in prostate cancer: Image quality, what we can measure and how it improves clinical assessment

Diffusion-weighted imaging is a dependable method for detection of clinically significant prostate cancer. In prostate tissue, there are several compartments that can be distinguished from each other, based on different water diffusion decay signals observed. Alterations in cell architecture, such as a relative increase in tumor infiltration and decrease in stroma, will influence the observed diffusion signal in a voxel due to impeded random motion of water molecules. The amount of restricted diffusion can be assessed quantitatively by measuring the apparent diffusion coefficient (ADC) value. This is traditionally calculated using a monoexponential decay formula represented by the slope of a line produced between the logarithm of signal intensity decay plotted against selected b-values. However, the choice and number of b-values and their distribution, has a significant effect on the measured ADC values. There have been many models that attempt to use higher-order functions to better describe the observed diffusion signal decay, requiring an increased number and range of b-values. While ADC can probe heterogeneity on a macroscopic level, there is a need to optimize advanced diffusion techniques to better interrogate prostate tissue microstructure. This could be of benefit in clinical challenges such as identifying sparse tumors in normal prostate tissue or better defining tumor margins. This paper reviews the principles of diffusion MRI and novel higher order diffusion signal analysis techniques to improve the detection of prostate cancer.

Metabolic activity diffusion imaging (MADI): I. Metabolic, cytometric modeling and simulations

Evidence mounts that the steady-state cellular water efflux (unidirectional) first-order rate constant (k_io [s^-1 ]) magnitude reflects the ongoing, cellular metabolic rate of the cytolemmal Na⁺ , K⁺ -ATPase (NKA), ^c MR_NKA (pmol [ATP consumed by NKA]/s/cell), perhaps biology's most vital enzyme. Optimal ¹ H₂ O MR k_io determinations require paramagnetic contrast agents (CAs) in model systems. However, results suggest that the homeostatic metabolic k_io biomarker magnitude in vivo is often too large to be reached with allowable or possible CA living tissue distributions. Thus, we seek a noninvasive (CA-free) method to determine k_io in vivo. Because membrane water permeability has long been considered important in tissue water diffusion, we turn to the well-known diffusion-weighted MRI (DWI) modality. To analyze the diffusion tensor magnitude, we use a parsimoniously primitive model featuring Monte Carlo simulations of water diffusion in virtual ensembles comprising water-filled and -immersed randomly sized/shaped contracted Voronoi cells. We find this requires two additional, cytometric properties: the mean cell volume (V [pL]) and the cell number density (ρ [cells/μL]), important biomarkers in their own right. We call this approach metabolic activity diffusion imaging (MADI). We simulate water molecule displacements and transverse MR signal decays covering the entirety of b-space from pure water (ρ = V = 0; k_io undefined; diffusion coefficient, D₀ ) to zero diffusion. The MADI model confirms that, in compartmented spaces with semipermeable boundaries, diffusion cannot be described as Gaussian: the nanoscopic D (D_n ) is diffusion time-dependent, a manifestation of the "diffusion dispersion". When the "well-mixed" (steady-state) condition is reached, diffusion becomes limited, mainly by the probabilities of (1) encountering (ρ, V), and (2) permeating (k_io ) cytoplasmic membranes, and less so by D_n magnitudes. Importantly, for spaces with large area/volume (A/V; claustrophobia) ratios, this can happen in less than a millisecond. The model matches literature experimental data well, with implications for DWI interpretations.

NMR shutter-speed elucidates apparent population inversion of 1 H2 O signals due to active transmembrane water cycling

Purpose

The desire to quantitatively discriminate the extra‐ and intracellular tissue ¹H₂O MR signals has gone hand‐in‐hand with the continual, historic increase in MRI instrument magnetic field strength [B₀]. However, recent studies have indicated extremely valuable, novel metabolic information can be readily accessible at ultra–low B₀. The two signals can be distinguished, and the homeostatic activity of the cell membrane sodium/potassium pump (Na⁺,K⁺,ATPase) detected. The mechanism allowing ¹H₂O MRI to do this is the newly discovered active transmembrane water cycling (AWC) phenomenon, which we found using paramagnetic extracellular contrast agents at clinical B₀ values. AWC is important because Na⁺,K⁺,ATPase can be considered biology’s most vital enzyme, and its in vivo steady‐state activity has not before been measurable, let alone amenable to mapping with high spatial resolution. Recent reports indicate AWC correlates with neuronal firing rate, with malignant tumor metastatic potential, and inversely with cellular reducing equivalent fraction. We wish to systematize the ways AWC can be precisely measured.

Methods

We present a theoretical longitudinal relaxation analysis of considerable scope: it spans the low‐ and high–field situations.

Results

We show the NMR shutter‐speed organizing principle is pivotal in understanding how trans–membrane steady–state water exchange kinetics are manifest throughout the range. Our findings illuminate an aspect, apparent population inversion, which is crucial in understanding ultra‐low field results.

Conclusions

Without an appreciation of apparent population inversion, significant misinterpretations of future data are likely. These could have unfortunate diagnostic consequences.

Optimising sampling patterns for bi-exponentially decaying signals

A recently reported method, based on the Cramér-Rao Lower Bound theory, for optimising sampling patterns for a wide range of nuclear magnetic resonance (NMR) experiments is applied to the problem of optimising sampling patterns for bi-exponentially decaying signals. Sampling patterns are optimised by minimizing the percentage error in estimating the most difficult to estimate parameter of the bi-exponential model, termed the objective function. The predictions of the method are demonstrated in application to pulsed field gradient NMR data recorded for the two-component diffusion of a binary mixture of methane/ethane in a zeolite. It is shown that the proposed method identifies an optimal sampling pattern with the predicted objective function being within 10% of that calculated from the experiment dataset. The method is used to advise on the number of sampled points and the noise level needed to resolve two-component systems characterised by a range of ratios of populations and diffusion coefficients. It is subsequently illustrated how the method can be used to reduce the experiment acquisition time while still being able to resolve a given two-component system.

Optimising magnetic resonance sampling patterns for parametric characterisation

Sampling strategies are often central to experimental design. Choosing efficiently which data to acquire can improve the estimation of parameters and reduce the acquisition time. This work is focused on designing optimal sampling patterns for Nuclear Magnetic Resonance (NMR) applications, illustrated with respect to the best estimate of the parameters characterising a lognormal distribution. Lognormal distributions are commonly used as fitting models for distributions of spin-lattice relaxation time constants, spin-spin relaxation time constants and diffusion coefficients. A method for optimising the choice of points to be sampled is presented which is based on the Cramér-Rao Lower Bound (CRLB) theory. The method's capabilities are demonstrated experimentally by applying it to the problem of estimating the emulsion droplet size distribution from a pulsed field gradient (PFG) NMR diffusion experiment. A difference of <5% is observed between the predictions of CRLB theory and the PFG NMR experimental results. It is shown that CLRB theory is stable down to signal-to-noise ratios of ∼10. A sensitivity analysis for the CRLB theory is also performed. The method of optimizing sampling patterns is easily adapted to distributions other than lognormal and to other aspects of experimental design; case studies of optimising the sampling scheme for a fixed acquisition time and determining the potential for reduction in acquisition time for a fixed parameter estimation accuracy are presented. The experimental acquisition time is typically reduced by a factor of 3 using the proposed method compared to a constant gradient increment approach that would usually be used.

Using 1H2O MR to measure and map sodium pump activity in vivo

The cell plasma membrane Na⁺,K⁺-ATPase [NKA] is one of biology's most [if not the most] significant enzymes. By actively transporting Na⁺ out [and K⁺ in], it maintains the vital trans-membrane ion concentration gradients and the membrane potential. The forward NKA reaction is shown in the Graphical Abstract [which is elaborated in the text]. Crucially, NKA does not operate in isolation. There are other transporters that conduct K⁺ back out of [II, Graphical Abstract] and Na⁺ back into [III, Graphical Abstract] the cell. Thus, NKA must function continually. Principal routes for ATP replenishment include mitochondrial oxidative phosphorylation, glycolysis, and creatine kinase [CrK] activity. However, it has never been possible to measure, let alone map, this integrated, cellular homeostatic NKA activity in vivo. Active trans-membrane water cycling [AWC] promises a way to do this with ¹H₂O MR. Inthe Graphical Abstract, the AWC system is characterized by active contributions totheunidirectional rate constants for steady-state water efflux and influx, respectively, k_io(a) and k_oi(a). The discovery, validation, and initial exploration of active water cycling are reviewed here. Promising applications in cancer, cardiological, and neurological MRI are covered. This initial work employed paramagnetic Gd(III)chelate contrast agents [CAs]. However, the significant problems associated with in vivo CA use are also reviewed. A new analysis of water diffusion-weighted MRI [DWI] is presented. Preliminary results suggest a non-invasive way to measure the cell number density [ρ (cells/μL)], the mean cell volume [V (pL)], and the cellular NKA metabolic rate [^cMR_NKA(fmol(ATP)/s/cell)] with high spatial resolution. These crucial cell biology properties have not before been accessible invivo. Furthermore, initial findings indicate their absolute values can be determined.

Diffusion-relaxation correlation spectroscopic imaging: A multidimensional approach for probing microstructure

PURPOSE

To propose and evaluate a novel multidimensional approach for imaging subvoxel tissue compartments called Diffusion-Relaxation Correlation Spectroscopic Imaging.

THEORY AND METHODS

Multiexponential modeling of MR diffusion or relaxation data is commonly used to infer the many different microscopic tissue compartments that contribute signal to macroscopic MR imaging voxels. However, multiexponential estimation is known to be difficult and ill-posed. Observing that this ill-posedness is theoretically reduced in higher dimensions, diffusion-relaxation correlation spectroscopic imaging uses a novel multidimensional imaging experiment that jointly encodes diffusion and relaxation information, and then uses a novel constrained reconstruction technique to generate a multidimensional diffusion-relaxation correlation spectrum for every voxel. The peaks of the multidimensional spectrum are expected to correspond to the distinct tissue microenvironments that are present within each macroscopic imaging voxel.

RESULTS

Using numerical simulations, experiment data from a custom-built phantom, and experiment data from a mouse model of traumatic spinal cord injury, diffusion-relaxation correlation spectroscopic imaging is demonstrated to provide substantially better multicompartment resolving power compared to conventional diffusion- and relaxation-based methods.

CONCLUSION

The diffusion-relaxation correlation spectroscopic imaging approach provides powerful new capabilities for resolving the different components of multicompartment tissue models, and can be leveraged to significantly expand the insights provided by MRI in studies of tissue microstructure. Magn Reson Med 78:2236-2249, 2017. © 2017 International Society for Magnetic Resonance in Medicine.

Evaluation of fitting models for prostate tissue characterization using extended-range b-factor diffusion-weighted imaging

PURPOSE

To compare the fitting and tissue discrimination performance of biexponential, kurtosis, stretched exponential, and gamma distribution models for high b-factor diffusion-weighted images in prostate cancer.

METHODS

Diffusion-weighted images with 15 b-factors ranging from b = 0 to 3500 s/mm² were obtained in 62 prostate cancer patients. Pixel-wise signal decay fits for each model were evaluated with the Akaike Information Criterion (AIC). Parameter values for each model were determined within normal prostate and the index lesion. Their potential to differentiate normal from cancerous tissue was investigated through receiver operating characteristic analysis and comparison with Gleason score.

RESULTS

The biexponential slow diffusion fraction f_slow , the apparent kurtosis diffusion coefficient ADC_K , and the excess kurtosis factor K differ significantly among normal peripheral zone (PZ), normal transition zone (TZ), tumor PZ, and tumor TZ. Biexponential and gamma distribution models result in the lowest AIC, indicating a superior fit. Maximum areas under the curve (AUCs) of all models ranged from 0.93 to 0.96 for the PZ and from 0.95 to 0.97 for the TZ. Similar AUCs also result from the apparent diffusion coefficient (ADC) of a monoexponential fit to a b-factor sub-range up to 1250 s/mm² . For kurtosis and stretched exponential models, single parameters yield the highest AUCs, whereas for the biexponential and gamma distribution models, linear combinations of parameters produce the highest AUCs. Parameters with high AUC show a trend in differentiating low from high Gleason score, whereas parameters with low AUC show no such ability.

CONCLUSION

All models, including a monoexponential fit to a lower-b sub-range, achieve similar AUCs for discrimination of normal and cancer tissue. The biexponential model, which is favored statistically, also appears to provide insight into disease-related microstructural changes. Magn Reson Med 79:2346-2358, 2018. © 2017 International Society for Magnetic Resonance in Medicine.