Multiexponential Signal Research Articles

Toward Optimal Fitting Parameters for Multi-Exponential DWI Image Analysis of the Human Kidney: A Simulation Study Comparing Different Fitting Algorithms

In DWI, multi-exponential signal analysis can be used to determine signal underlying diffusion components. However, the approach is very complex due to the inherent low SNR, the limited number of signal decay data points, and the absence of appropriate acquisition parameters and standardized analysis methods. Within the scope of this work, different methods for multi-exponential analysis of the diffusion signal in the kidney were compared. To assess the impact of fitting parameters, a simulation was conducted comparing the free non-negative (NNLS) and rigid non-linear least square (NLLS) fitting methods. The simulation demonstrated improved accuracy for NNLS in combination with area-under-curve estimation. Furthermore, the accuracy and stability of the results were further enhanced utilizing optimized parameters, namely 350 logarithmically spaced diffusion coefficients within [0.7, 300] × 10−3 mm2/s and a minimal SNR of 100. The NNLS approach shows an improvement over the rigid NLLS method. This becomes apparent not only in terms of accuracy and omitting prior knowledge, but also in better representation of renal tissue physiology. By employing the determined fitting parameters, it is expected that more stable and reliable results for diffusion imaging in the kidney can be achieved. This might enable more accurate DWI results for clinical utilization.

Study of the collagen tissue nanostructure by analyzing the echo decay obtained using the MRI technique.

The multicomponent relaxation observed in nuclear magnetic resonance experiments in biological tissues makes it difficult to establish a correlation between specific relaxation times and tissue structural parameters. The analysis of a nanostructure (the characteristic size of 10-1000 nm) is usually based on formulas for relaxation times which depend on structural parameters at the atomic or molecular levels in the size range of 0.1-5 nm. We have recently developed an analysis method in which relaxation times' anisotropy in a sample is explicitly related to its structure of nanocavities containing a liquid or gas. However, the method is based on the analysis of experimental data on the anisotropy of relaxation times obtained by using the standard NMR technique and rotating the sample relative to a magnetic field and requires a series of experiments. In the present study, to address this challenge, we develop a new method of analysis of a multi-exponential magnetic resonance signal that does not require determining relaxation times and eliminates the sample rotation and the necessity of a series of experiments. Using the magnetic resonance imaging (MRI) technique, the total signal from the whole sample was obtained as a sum of the signals (echo decays) from all voxels. In contrast to previous research, the volumes of nanocavities and their angular distribution can be obtained by analyzing a single total signal for the entire cartilage. In addition, within the framework of this approach, it is possible to identify the reason for the multicomponent nature of relaxation. The proposed method for analyzing a single multi-exponential signal (transverse relaxation) was implemented on cartilage. Using the signal, three anatomical zones of cartilage were studied, revealing significant structural differences between them. The proposed method not only avoids the need for sample rotation but also enables repeated application of layer-by-layer magnetic resonance imaging with micron resolution. The study results allow us to suggest that water molecules contributing to the echo decay are more likely located in nanocavities formed by the fibrillar structure rather than inside the fibrils.

Low-field, high-gradient NMR shows diffusion contrast consistent with localization or motional averaging of water near surfaces

Nuclear magnetic resonance (NMR) measurements of water diffusion have been extensively used to probe microstructure in porous materials, such as biological tissue, however primarily using pulsed gradient spin echo (PGSE) methods. Low-field single-sided NMR systems have built-in static gradients (SG) much stronger than typical PGSE maximum gradient strengths, which allows for the signal attenuation at extremely high b-values to be explored. Here, we perform SG spin echo (SGSE) and SG stimulated echo (SGSTE) diffusion measurements on biological cells, tissues, and gels. Measurements on fixed and live neonatal mouse spinal cord, lobster ventral nerve cord, and starved yeast cells all show multiexponential signal attenuation on a scale of b with significant signal fractions observed at b × D0 ≫ 1 with b as high as 400 ms/μm2. These persistent signal fractions trend with surface-to-volume ratios for these systems, as expected from porous media theory. An exception found for the case of fixed vs. live spinal cords was attributed to faster exchange or permeability in live spinal cords than in fixed spinal cords on the millisecond timescale. Data suggests the existence of multiple exchange processes in neural tissue, which may be relevant to the modeling of time-dependent diffusion in gray matter. The observed multi-exponential attenuation is from protons on water and not macromolecules because it remains proportional to the normalized signal when a specimen is washed with D2O. The signal that persists to b × D0 ≫ 1 is also drastically reduced after delipidation, indicating that it originates from lipid membranes that restrict water diffusion. The multi-exponential or stretched exponential character of the signal attenuation at b × D0 ≫ 1 appears mono-exponential when viewed on a scale of (b×D0)1/3, suggesting it may originate from localization or motional averaging of water near membranes on sub-micron length scales. To try to disambiguate these two contributions, signal attenuation curves were compared at varying temperatures. While the curves align when normalizing them using the localization length scale, they separate on a motional averaging length scale. This supports localization as the source of non-Gaussian displacements, but this interpretation is still provisional due to the possible confounds of heterogeneity, exchange, and relaxation. Measurements on two types of gel phantoms designed to mimic extracellular matrix, one with charged functional groups synthesized from polyacrylic acid (PAC) and another with uncharged functional groups synthesized from polyacrylamide (PAM), both exhibit signal at b × D0 ≫ 1, potentially due to water interacting with macromolecules. These preliminary finding motivate future research into contrast and attenuation mechanisms in tissue with low-field, high-gradient NMR.

Distortion Characterization of Exponential Signal Reconstructed by Low-Chirp Signal

This paper is an extension of the procedure pointed out to evaluate the distortion of the exponential signal, modeled as multiexponential signal with different amplitudes and time constants. This procedure evaluates the distortion by applying the Prony-like method to the signal reconstructed by zero-crossing time evaluated on the difference between the exponential original signal and the reference sinusoidal one. The extension proposed in this paper concerns with the use of the low-chirp signal as reference instead of the sinusoidal signal. The low-chirp signal permits to increase the number of the zero-crossing time in the section of exponential signal with the higher slope and decreasing in the section with the lower slope. This setting permits to estimate the distortion of the exponential signal with higher accuracy and precision. The parameters’ amplitude, phase, and start frequency of the reference low-chirp signal are used to reconstruct the exponential signal in the zero-crossing time in order to overcome the problem of the high-resolution sampling. Their estimation is carried out by the time-varying sine-fitting algorithm that processes the resulting signal that is the difference between the exponential original signal and the low-chirp signal. Advantages of the proposed extended procedure regard the direct estimation of the reference low-chirp signal parameters from the resulting signal over-sampled at low resolution, and the use of only one input acquisition channel that permits to overcome the synchronization problem between the reference and the resulting signal. The numerical and experimental tests validate the extended procedure.

Faint and clustered components in exponential analysis

An important hurdle in multi-exponential analysis is the correct detection of the number of components in a multi-exponential signal and their subsequent identification. This is especially difficult if one or more of these terms are faint and/or covered by noise. We present an approach to tackle this problem and illustrate its usefulness in motor current signature analysis (MCSA), relaxometry (in FLIM and MRI) and magnetic resonance spectroscopy (MRS).The approach is based on viewing the exponential analysis as a Padé approximation problem and makes use of some well-known theorems from Padé approximation theory. We show how to achieve a clear separation of signal and noise by computing sufficiently high order Padé approximants, thus modeling both the signal and the noise, rather than filtering out the noise at an earlier stage and return a low order approximant.We illustrate the usefulness of the approach in different practical situations, where some exponential components are difficult to detect and retrieve because they are either faint compared to the other signal elements or contained in a cluster of similar exponential components.

Separation of extra- and intracellular metabolites using hyperpolarized 13C diffusion weighted MR

This work demonstrates the separation of extra- and intracellular components of glycolytic metabolites with diffusion weighted hyperpolarized 13C magnetic resonance spectroscopy. Using b-values of up to 15,000smm−2, a multi-exponential signal response was measured for hyperpolarized [1-13C] pyruvate and lactate. By fitting the fast and slow asymptotes of these curves, their extra- and intracellular weighted diffusion coefficients were determined in cells perfused in a MR compatible bioreactor. In addition to measuring intracellular weighted diffusion, extra- and intracellular weighted hyperpolarized 13C metabolites pools are assessed in real-time, including their modulation with inhibition of monocarboxylate transporters. These studies demonstrate the ability to simultaneously assess membrane transport in addition to enzymatic activity with the use of diffusion weighted hyperpolarized 13C MR. This technique could be an indispensible tool to evaluate the impact of microenvironment on the presence, aggressiveness and metastatic potential of a variety of cancers.

Transient multiexponential signals analysis using Bayesian deconvolution

A new method based on Bayesian deconvolution is proposed for multiexponential transient signal analysis. The multiexponential signal is initially converted to a convolution model using logarithmic and differential transformation after which the Bayesian iteration is used to deconvolve the data. The numerical simulation is applied on four different multiexponential signals with different levels of noise. Thermal transient experiment data of the high power light emitting diodes are also analyzed using the proposed method. Simulation and experimental results indicate that the present method performs efficiently in accurately estimating the decay rates except at low SNR case.

Simple and reliable determination of intravoxel incoherent motion parameters for the differential diagnosis of head and neck tumors.

Intravoxel incoherent motion (IVIM) imaging can characterize diffusion and perfusion of normal and diseased tissues, and IVIM parameters are authentically determined by using cumbersome least-squares method. We evaluated a simple technique for the determination of IVIM parameters using geometric analysis of the multiexponential signal decay curve as an alternative to the least-squares method for the diagnosis of head and neck tumors. Pure diffusion coefficients (D), microvascular volume fraction (f), perfusion-related incoherent microcirculation (D*), and perfusion parameter that is heavily weighted towards extravascular space (P) were determined geometrically (Geo D, Geo f, and Geo P) or by least-squares method (Fit D, Fit f, and Fit D*) in normal structures and 105 head and neck tumors. The IVIM parameters were compared for their levels and diagnostic abilities between the 2 techniques. The IVIM parameters were not able to determine in 14 tumors with the least-squares method alone and in 4 tumors with the geometric and least-squares methods. The geometric IVIM values were significantly different (p<0.001) from Fit values (+2±4% and −7±24% for D and f values, respectively). Geo D and Fit D differentiated between lymphomas and SCCs with similar efficacy (78% and 80% accuracy, respectively). Stepwise approaches using combinations of Geo D and Geo P, Geo D and Geo f, or Fit D and Fit D* differentiated between pleomorphic adenomas, Warthin tumors, and malignant salivary gland tumors with the same efficacy (91% accuracy = 21/23). However, a stepwise differentiation using Fit D and Fit f was less effective (83% accuracy = 19/23). Considering cumbersome procedures with the least squares method compared with the geometric method, we concluded that the geometric determination of IVIM parameters can be an alternative to least-squares method in the diagnosis of head and neck tumors.

Measurement of distorted exponential signal components using maximum likelihood estimation

Exponential signal is a suitable stimulus for dynamic ADC testing because of the simplicity of the generating RC circuit. A potential distortion source of the ideal exponential shape is the dielectric absorption of the capacitor, whose effect can be represented by additional superimposed exponential components with longer time constant and smaller peak value. Measurement of the distortion of the exponential signal by using a reference waveform recorder with known nonlinearity is the initial step in the calibration of an ADC testing stand with exponential stimulus, along with the assessment of its uncertainty. Lack of the orthogonality of stimulus signal components makes classical analysis methods difficult to apply.This paper presents a method for measurement of multiexponential signal components as an example of the more general task of signal decomposition where signal components are non-orthogonal. The proper optimization procedure based on the ML method will be presented, which usually reaches the global minimum of the cost function. Effectiveness will be shown by simulation, and by application to measurement of a multiexponential signal acquired by a reference waveform recorder with known error parameters.

Fast hydrogen exchange affects 15N relaxation measurements in intrinsically disordered proteins

Unprotected amide protons can undergo fast hydrogen exchange (HX) with protons from the solvent. Generally, NMR experiments using the out-and-back coherence transfer with amide proton detection are affected by fast HX and result in reduced signal intensity. When one of these experiments, (1)H-(15)N HSQC, is used to measure the (15)N transverse relaxation rate (R2), the measured R2 rate is convoluted with the HX rate (kHX) and has higher apparent R2 values. Since the (15)N R2 measurement is important for analyzing protein backbone dynamics, the HX effect on the R2 measurement is investigated and described here by multi-exponential signal decay. We demonstrate these effects by performing (15)N R 2 (CPMG) experiments on α-synuclein, an intrinsically disordered protein, in which the amide protons are exposed to solvent. We show that the HX effect on R 2 (CPMG) can be extracted by the derived equation. In conclusion, the HX effect may be pulse sequence specific and results from various sources including the J coupling evolution, the change of steady state water proton magnetization, and the D2O content in the sample. To avoid the HX effect on the analysis of relaxation data of unprotected amides, it is suggested that NMR experimental conditions insensitive to the HX should be considered or that intrinsic R 2 (CPMG) values be obtained by methods described herein.

A Physics-Driven Approach using Single-Pore-Modes (SPM) for Estimating an Average Pore Radius and Surface Relaxativity from NMR Data

The method of Nuclear Magnetic Resonance (NMR) has found a wide range of application in geophysics. At the largest scale SNMR as a surface based method covers volumes up to thousands of cubic meters, at laboratory scale NMR can handle samples as small as cuttings from drilling. Here we focus on unconsolidated material with the purpose to derive hydraulic properties from NMR relaxation curves at laboratory scale. Until now several methods for fitting NMR data are used. All approaches utilize a mono-, multi- or stretched-exponential relationship to explain NMR decay curves. Brownstein and Tarr (1979) showed that even for single pores a decay curve composes of several mono-exponential modes which superimposes to a multi-exponential signal. Thus, common multi-exponential approaches can lead to a misinterpretation of the decay time spectrum, when purely seen as a pore radius distribution. The mono-exponential modes introduced by Brownstein and Tarr (1979) occur by solving the general diffusion equation, which can be calculated for different pore geometries. In this study we used a cylindrical geometry to describe the pore space in analogy to Kozeny (1927). Thus, the solution is given for decay times T, extended by a term for the bulk relaxation and intensities I by equation 1 and 2.

An Appraisal of Gardner Transform-Based Methods of Transient Multiexponential Signal Analysis

There are several problems in applied science in which experimental observations can be accurately represented by a sum of exponential decay functions in which the amplitudes, decay rates and number of components have different physical interpretations and need to be estimated. A parameter estimation technique of multicomponent exponential functions that has undergone many modifications is the Gardner transform in which a nonlinear transformation is used to convert the data signal into a convolution model containing the parameters of interest. Modifications of this early technique include modification of the original transform or deconvolution procedure and additional processing of the deconvolved data to obtain better estimates of the desired parameters. This paper presents an appraisal of Gardner transform and its variants. It discusses major modifications and their implications to the overall results of analysis.

Analysis of transient multiexponential signals using cepstral deconvolution

We propose and test a new method of multiexponential transient signal analysis. The method based on cepstral deconvolution is fast and computationally inexpensive. The multiexponential signal is initially converted to a deconvolution model using Gardners' transformation after which the proposed method is used to deconvolve the data. Simulation and experimental results indicate that this method is good for determining the number of components but performs poorly in accurately estimating the decay rates. Influence of noise is not considered in this paper.

Multi-exponential signal decay from diffusion in a single compartment

Multi-exponential decays in diffusion experiments are typically fitted to sums of exponentially decaying components; often this is taken as evidence for spins in multiple distinct compartments. Here we examine the signal decay due to diffusion in a single cylinder, for short diffusion times (lightly restricted). The signals are well-modeled by a sum of two exponentials, despite the single compartment housing the spins. The results agree with a previous theoretical examination of the problem. The implication for biological systems is that multiple decay signal components may not correspond to multiple physical compartments.

Quantitative mapping of cerebral deoxyhemoglobin content using MR imaging

In this study, we present a magnetic resonance imaging (MRI) method for quantifying cerebral deoxyhemoglobin content that is based on a measurement of the reversible contribution ( R 2) of the transverse relaxation rate under the assumption of a quantitative relationship between R 2 and deoxyhemoglobin content. A numerical simulation was performed assuming a specific pulse sequence conventionally employed for R 2 measurement. The results showed a near linear relationship between R 2 and deoxyhemoglobin content in the physiological range of oxygen extraction for almost all sized vessels, although a large diffusion effect for capillary sized vessels compromised the relationship. Concerning the methodology for R 2 measurement, a theoretical analysis showed that multiexponential transverse signal decay of brain parenchyma may result in a considerable underestimation or overestimation of R 2, if a conventional method is employed for R 2 quantification. A modified method for correcting the effect of multiexponential signal decay was employed for R 2 measurement of the brain in normal volunteers. The results showed a high level of agreement between the R 2 values measured in our study and those estimated from physiological parameters obtained using other modalities such as PET. In the context of BOLD contrast-based functional MRI, the method proposed provides a quantitative mapping of baseline cerebral deoxyhemoglobin content, which is the essential physiological parameter for calibrating the stimulation-induced BOLD signal change and mapping neuronal activity in a quantitative manner by functional MRI.