Introduction

Many methods crucial for diagnostic imaging to non-invasive studies of neuroscience, physiology, and biology, rely on the fundamental physical NMR phenomenon of transverse relaxation. Transverse relaxation results from variations, in time or in space, of the magnetic field and can be observed following an initial excitation of spin precession: individual spins experience varying Larmor frequency offsets due to the local magnetic environment and acquire different precession phases (dephase), such that the signal magnitude, resulting from the vector sum of the contributions from each spin, gradually decreases (FID), or correspondingly, the spectral line broadens (line broadening). While completely uniform fluids undergo some dephasing due to random molecular motion, the spatially heterogeneous environment of biological tissues with varying magnetic susceptibilities [1, 2] additionally contributes to dephasing at the molecular (nanometer), cellular (mesoscopic, micrometer), and macroscopic (MR voxel, millimeter) scales. At MRI temporal resolutions, the Brownian motion and diffusion length of spin carrying molecules is on the order of the cellular scale of structural organization, and the interplay between diffusion and transverse relaxation bestrides two idealized limiting cases: (1) the motional-, or diffusion-narrowing regime, when spins sample sufficiently large portions of a medium during their Brownian motion, their phases will have the same statistical distribution; (2) the static dephasing regime, when spins are static, phase differences between remote spins will be due to the spatially variable magnetic field [3,4,5]. The resulting transverse relaxation pattern over time holds valuable biophysical information about cellular scale microstructure.

Signal decay in measurements involving spin echoes (SE) correspond to irreversible transverse relaxation and can differ from measurements involving gradient echoes (GE), which correspond to reversible transverse relaxation. Considering effects of irreversible and reversible transverse relaxation jointly may further improve understanding of the underlying micro-structural complexity [6]. Simultaneous investigation of irreversible and reversible transverse relaxation is possible using pulse sequences such as gradient-echo sampling of the FID and echo (GESFIDE) [7] or gradient-echo sampling of the spin-echo (GESSE) sequences [8]. GESFIDE and GESSE sample signals during dephasing by both irreversible and reversible transverse relaxation, as well as during rephasing by reversible processes working against dephasing by irreversible processes. When reversible transverse relaxation is linear exponential, GESFIDE offers a clear SNR advantage over GESSE, however, if not, then the temporal behavior of the FID and left- and right-hand sides of SEs differ significantly from the expected.

Signal decay with reversible transverse relaxation often appears nearly exponential (especially at longer minimal echo times) and is typically modelled using a corresponding Lorentzian intravoxel frequency distribution (equivalent to a Cauchy distribution). However, this signal decay is not always exponential [4], and the Lorentzian model is a poor approximation since the distribution of metabolites in living tissue are typically not homogeneous, susceptibilities are spatially dependent with a distribution of resonance frequencies [9], resulting in rather non-mono-exponential signal decays [5]. The theoretical expectation that intra‐voxel precession frequency distribution is sometimes better modeled as non-Lorentzian (i.e., Gaussian) has also been recently supported experimentally, in the brain [10,11,12], lungs [13], prostate [14] and teeth [15] as well as blood [16].

Fitting spectra with inappropriate line shapes introduces systematic errors in quantification [9, 14]. The Voigt distribution is defined as the convolution of Lorentzian and Gaussian distributions, and has previously been shown to model in vivo MRS resonances better than purely Lorentzian or Gaussian distributions, with considerably smaller residual spectrums [9, 17]. The simplified pseudo-Voigt distribution is the linear combination of Lorentzian and Gaussian distributions, and also performed better than either purely Lorentzian or Gaussian distributions for chemical exchange saturation transfer (CEST) and nuclear Overhauser enhancement (NOE) quantification in mice at 11.7 T [18]. More recently, the Symmetric α-Stable (SAS) distribution was reported to better fit field distributions compared to purely Lorentzian or Gaussian distributions for many voxels in the brain [10]. Both Voigt and SAS distributions are more generalized line shapes, they have one additional degree of freedom relative to a Lorentzian or Gaussian, and they can reduce to either a Lorentzian or a Gaussian (which are simple subtypes of these more generalized distributions) for certain values of this additional parameter. In other words, both Voigt and SAS can become equivalent to a Lorentzian or a Gaussian or can provide intermediate distributions between a Lorentzian and Gaussian Such two-parameter descriptions may better quantify the underlying frequency distribution than the linewidth of a poorly fitting Lorentzian. Lorentzian, Gaussian, as well as sample Voigt and SAS distributions are depicted in Fig. 1. The Voigt distribution based on convolution was used in this work (rather than the simplified pseudo-Voigt distribution). More accurate quantification of relaxation may allow for better connections to specific biophysical models of water interactions in tissue that ultimately lead to disparate relaxation patterns, compared to oversimplified relaxation “maps” based on purely Lorentzian distributions with purely exponential decays.

Fig. 1
figure 1

Lorentzian, Gaussian, Voigt, and SAS distributions. Voigt and SAS distributions are generalized line shapes and can provide intermediate distributions between a Lorentzian and Gaussian or match a Lorentzian or a Gaussian. The distributions are demonstrated for comparison, normalized, using the following parameters: Common: \({R}_{2}\) = 0.5, \({{R}_{2}}^{^{\prime}}\) = 2, σ = 2; Voigt1: \({{R}_{2}}^{^{\prime}}\) = 1.08, σ = 1.18; Voigt2: \({{R}_{2}}^{^{\prime}}\) = 1.43, σ = 0.95; SAS1: α = 1.67, SAS2: α = 1.33

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Correct signal characterization is important for establishment of accurate MRI biomarkers. A biomarker for hypoxia, or reduced oxygen tension, in tumors is desirable, and may better guide treatment of hypoxic tumors which tend to have increased metastatic potential, are more resistant to radiation treatment, and result in poor patient outcomes [19,20,21,22,23,24]. Hypoxia is prevalent among cervical cancers, with up to 48% showing this aggressive phenotype [25], which relative to non-hypoxic tumors increases mortality by two- to threefold [26]. Improved targeting of hypoxia in tumors may improve treatment response and survival [27]. To date, efforts at identifying intra-tumoral hypoxia have mainly targeted the effective relaxation of GE signal decays from reversible and irreversible relaxation effects [28,29,30,31]. In this work we compared Lorentzian, Gaussian, Voigt, or SAS distributions as models of signal attenuation with reversible transverse relaxation, in terms of information content, on data acquired using GESSE sequences, in biological phantoms and women with cervical cancers, at 3 T. The Voigt distribution has not previously been utilized in this context in MRI, and cervical cancers have not previously been investigated with GESSE or the SAS model. Our findings may help improve tumor characterization in women with cervical cancers at 3 T, with better modeling of (i.e., hypoxia related) susceptibility effects, treatment guidance at baseline, and monitoring following therapy.

Methods

Theory

GESSE sequences facilitate assessment of the effects of intravoxel Larmor frequency distribution, by following the evolution of transverse magnetization at various dephasing states during a SE experiment, both before and after complete rephasing of all components, and by correctly accounting for irreversible relaxation.

Signal magnitude in a voxel at readout time t, for a Lorentzian distribution of spins is [12]:

$${S}_{\mathrm{Lorentzian}}=\left|\rho {e}^{-\left(\uptau +\mathrm{t}\right) {\mathrm{R}}_{2}- \left|\left(\uptau -\mathrm{t}\right)\right| {\mathrm{R}}_{2}^{{^{\prime}}}}\right|,$$
(1)

where τ denotes the period between excitation and refocusing (τ = TESE/2), t denotes the time following refocusing (such that the spin echo forms at t = τ), ρ is the pseudo proton density, including the proton density and appropriate T1-weighting factors influencing longitudinal magnetization prior to excitation, \({R}_{2}\) and \({{R}_{2}}^{^{\prime}}\) denote transverse relaxation rates associated with irreversible and reversible processes, respectively. The full width at half‐maximum (FWHM) of the corresponding Lorentzian distribution is \({{2R}_{2}}^{^{\prime}}\). Complete spoiling of transverse magnetization before each excitation, and the same \({R}_{2}\) across all frequency components (as may be expected for most tissues that contain only water, and where fat does not exist or has been suppressed) is assumed.

The key difference between left- and right-hand sides of the SE in a GESSE sequence is reflected in Eq. (1) by the exponential term for \({{R}_{2}}^{^{\prime}}\): the argument has an absolute value, indicating that \({R}_{2}\) – \({{R}_{2}}^{^{\prime}}\) governs signal evolution (which could be decay or growth) on the left-hand side (irreversible processes dephase spins and reversible processes rephase spins), while \({R}_{2}\) + \({{R}_{2}}^{^{\prime}}\) governs signal evolution (decay) before the refocusing pulse and on the right-hand side (both irreversible and reversible transverse relaxation dephase spins).

For a Gaussian distribution of spins [12]:

$${S}_{\mathrm{Gaussian}}=\left|\rho {e}^{-\left(\uptau +\mathrm{t}\right) {\mathrm{R}}_{2}- ({\left(\uptau -\mathrm{t}\right)\sigma )}^{2}/2}\right|,$$
(2)

where the FWMH is \(2\sqrt{2\mathrm{log}(2)}\sigma\). All other quantities are the same as those for the Lorentzian distribution, except the reversible transverse relaxation term is now σ rather than \({{R}_{2}}^{^{\prime}}\).

For a Voigt distribution of spins [32, 33]:

$${S}_{\mathrm{Voigt}}=\left|\rho {e}^{-\left(\uptau +\mathrm{t}\right) {\mathrm{R}}_{2}- \left|\left(\uptau -\mathrm{t}\right)\right| {\mathrm{R}}_{2}^{{^{\prime}}}- ({\left(\uptau -\mathrm{t}\right)\sigma )}^{2}/2}\right|,$$
(3)

such that the Voigt distribution becomes equivalent to the Lorentzian distribution when σ = 0 and to the Gaussian distribution when \({{R}_{2}}^{^{\prime}}=0\), and where the FWHM \({{\approx R}_{2}}^{^{\prime}}+ \sqrt{{{{(R}_{2}}^{^{\prime}})}^{2}+ {8\mathrm{log}\left(2\right)\sigma }^{2}}\) [32].

For a SAS distribution of spins [10]:

$${S}_{\mathrm{SAS}}=\left|\rho {e}^{-\left(\uptau +\mathrm{t}\right) {\mathrm{R}}_{2}- {2}^{(1-{\alpha })}{\left|\sigma \left(\uptau -\mathrm{t}\right)\right|}^{{\alpha }} }\right|,$$
(4)

where σ describes the width of the distribution and α describes the shape of the distribution with 1 ≤ α ≤ 2, such that the SAS distribution becomes equivalent to the Lorentz distribution when α = 1 and to the Gaussian distribution when α = 2. When α is not 1 or 2, the underlying distribution has no analytic form [10].

Subjects

Six consecutive patients (ages 60 ± 18 years, weight 64 ± 19 kg) diagnosed with cervical squamous cell carcinoma (International Federation of Gynecology, FIGO, stage IIB-IIIB [34], were imaged in this Institutional Review Board-approved study following their written informed consent. Biological phantoms (one potato and one lemon) were used to assess potential effects of differences in microscopic structure on transverse relaxation: potato was used due to its relatively homogeneous isotropic texture (no strongly ordered anisotropic structures such as fibers or canals), and lemon was used due to its relatively anisotropic fibrous texture (with fibers oriented in multiple directions which might potentially mirror muscle fibers of various orientations in the cervix).

Data acquisition

Data were acquired at 3 T (Verio, IMRIS, Minnetonka, MN). Matrix coils, specifically the 6-element body and 24-element spine (only the subset near the pelvis) were used. Two-dimensional \({T}_{2}\)-weighted turbo-spin-echo (TSE) images were acquired in axial, coronal, and sagittal orientations, using the parameters: repetition time (TR) = 7740 ms, echo time (TE) = 102 ms, flip angle = 140º, field of view (FOV) = 22 × 22 cm2, matrix = 256 × 320, resolution = 0.9 × 0.7 mm2, 2 averages, 30 slices with a thickness of 4 mm and gap of 0.8 mm. Two-dimensional axial, GESSE images were acquired on patients with TR = 2.5 s, TE = 80 ms, gradient‐echo spacing was ΔTE = 4.52 ms, and the TE values were: 48.36, 52.88, 57.4, 61.92, 66.44, 70.96, 75.48, 80, 84.52, 89.04, 93.56, 98.08, 102.6, 107.12 and 111.64 ms, such that 8th of 15 unipolar gradient echoes coincided with the spin echo. FOV was 38 × 38 cm2, matrix = 128 × 128, resolution = 2.969 × 2.969 mm2, 19 slices were acquired with a thickness of 4 mm, gap of 2 mm, and 1 average. The SE was at the center of the train of gradient echoes, such that readouts were symmetrical with respect to the SE. All GESSE parameters were the same for biological phantoms except for the acquisition of more echoes and fewer slices: the 16th of 31 unipolar gradient echoes coincided with the spin echo, and gradient‐echo spacing was smaller at ΔTE = 2.48 ms, and the TE values were: 42.80, 45.28, 47.76, 50.24, 52.72, 55.2, 57.68, 60.16, 62.64, 65.12, 67.6, 70.08, 72.56, 75.04, 77.52, 80, 82.48, 84.96, 87.44, 89.92, 92.4, 94.88, 97.36, 99.84, 102.32, 104.8, 107.28, 109.76, 112.24, 114.72 and 117.2 ms, for 12 slices. Frequency-encoding was superior (S) to inferior (I), and phase encoding was anterior (A) to posterior (P). Higher cellularity in malignancy is thought to restrict motion of water molecules resulting in lower apparent diffusion coefficients (ADC). On patients, two-dimensional axial, diffusion-weighted images (DWI) were also acquired, and ADCs were calculated for tumor localization from readout segmented multi-shot EPI at b = 0 and 600 mm2/s, fat saturation, TE = 56 ms, TR = 5.7 s, FOV = 35 × 28 cm2, matrix = 160 × 128, resolution = 2.1875 × 2.1875 mm2, 34 slices with a thickness of 4.8 mm, gap of 0 mm, and three averages. Macroscopic magnetic field inhomogeneities were minimized by shimming in a sub-volume covering the phantom, or the tumor and surrounding tissues, in all cases.

Models and fitting

Matlab was used to fit GESSE signals to all models (R2017b, Mathworks, Natick, MA). Before model fitting, magnitude data was corrected for bias (from a positive noise floor), as described in [35]:

$${N}_{\mathrm{SDcorr}}=\mathrm{std}\left({R}_{\mathrm{uncorr}}\right)/\sqrt{2-\pi /2},$$
(5)
$${S}_{\mathrm{corr}}= \sqrt{\left|\mathrm{mean}{({S}_{\mathrm{uncorr}})}^{2}- {{N}_{\mathrm{SDcorr}}}^{2}\right|},$$
(6)

where a region-of-interest (ROI) in the background is manually drawn to determine Runcorr, the level of uncorrected noise, NSDcorr denotes the corrected noise standard deviation, Suncorr the uncorrected signal, and Scorr the corrected signal. Voxels with SNR higher than three for at least more than the “number of maximum fitting parameters” (i.e., SNR > 3 at a minimum of 5 TEs) number of echo times were analyzed further. For each voxel, data averaging to further reduce noise was performed over a neighborhood of 3 × 3 pixels, in-plane. For all models, the lsqnonlin function was used to perform fitting. For all four models, ρ and \({R}_{2}\), additionally, for the Lorentzian model, \({R}_{2}{^{\prime}}\); for the Gaussian model, σ; for the Voigt model, \({R}_{2}{^{\prime}}\) and σ; and finally for the SAS model, σ and α were obtained from fits of S vs TE. Constraints were that \(\rho \ge 0\), \({R}_{2}\ge 0\), \({{R}_{2}}^{^{\prime}}\ge 0\), \(\sigma \ge 0\) and \(1\le \alpha \le 2\). For subsequent model comparison, sums of the squared residual errors (SSEs) between the data and each fit were also calculated.

Model comparison

In modeling, smaller SSEs can generally be achieved by increasing the number of parameters, however, parameter variance may be high when more parameters result in overfitting. Akaike’s Information Criterion (AIC) is based on information theory and for problems involving model selection, it offers an objective solution: AIC assesses models based on how much information they can retain, weighing improvements in SSEs by the additional number of parameters used to achieve these improvements. When the number of parameters in a model is on the order of the number of available measurements, Akaike’s information criterion is corrected using a (second-order) small-sample bias adjustment [36].

The corrected Akaike’s information criterion, AICc, was used for model comparison. AICc was calculated for each model as described in [37]:

$$\mathrm{AICc}=\frac{\mathrm{SSE}}{{N}_{\mathrm{SDcorr}}}+2p+\frac{2p(p+1)}{(n-p-1)},$$
(7)

where NSDcorr is the standard deviation of noise, p denotes the number of model parameters, and n denotes the number of available data points. In our case, p = 3 for Lorentzian and Gaussian models, p = 4 for Voigt and SAS models, and n = 15. The model with higher estimated information content has lower AICc values and is considered the better model.

AICc differences relative to the model with the minimum AICc were calculated for all models. The likelihood of each model was estimated based on AICc differences and Akaike weights were calculated for each model as follows:

$${\mathrm{AICc}}_{\mathrm{difference}}\left(m\right)=\mathrm{AICc}\left(m\right)-{\mathrm{AICc}}_{\mathrm{minimum}},$$
(8)
$${\mathrm{AICc}}_{\mathrm{likelihood}}\left(m\right)={e}^{-{\mathrm{AICc}}_{\mathrm{difference}}\left(m\right)/2},$$
(9)
$${\mathrm{AICc}}_{\mathrm{weight}}\left(m\right)=\frac{{\mathrm{AICc}}_{\mathrm{likelihood}}\left(m\right)}{\sum_{i=1}^{4}{\mathrm{AICc}}_{\mathrm{likelihood}}\left(i\right)},$$
(10)

for each model, m = 1 (Lorentzian), 2 (Gaussian), 3 (Voigt), 4 (SAS). The best performing model has the minimum AICc value, an AICc difference of 0, likelihood of 1, and weight of 1. AICc maps were generated, indicating the model with the minimum AICc, at each voxel.

Field inhomogeneity effects

We estimated levels of background macroscopic field inhomogeneity from GESSE acquisitions by saving both magnitude and phase data. The phase accumulation between the first two gradient echoes was determined and unwrapped using the Statistical Parametric Mapping software package SPM12, and function pm_unwrap [33] (www.fil.ion.ucl.ac.uk/spm). Field maps (mT) were calculated by dividing the unwrapped phase difference by the TE difference:

$${B}_{0}\;\mathrm{Fieldmap}=\frac{\mathrm{pm}\_\mathrm{unwrap}\left(\left|{\mathrm{S}}_{2}\right| {e}^{i\mathrm{ \angle }\left({\mathrm{S}}_{2}\right)}/{e}^{i\mathrm{ \angle }\left({\mathrm{S}}_{1}\right)}\right)}{2\uppi \gamma \left({\mathrm{TE}}_{2}-{\mathrm{TE}}_{1}\right)},$$
(11)

where \({S}_{1}\) denotes the complex signal at \({\mathrm{TE}}_{1}\) and \({S}_{2}\) denotes the complex signal at \({\mathrm{TE}}_{2}\). Phase unwrapping failed in areas of low signal where there was extensive phase accumulation. Such voxels (typically seen in the rectum due to air, or at the edges of the bladder due to motion) were excluded from further analysis since their field distributions were unknown. Field gradient magnitudes along in-plane and through-plane directions (\({G}_{x}\), \({G}_{y}\), and \({G}_{z}\), mT/m) and their root mean square (RMS) were estimated from central differences of the \({B}_{0}\) fieldmap as follows:

$${G}_{i}=\frac{\left|{\mathrm{Fieldmap}}_{\mathrm{Next\; voxel\; along\; }i}- {\mathrm{Fieldmap}}_{\mathrm{Previous\; voxel \;along\; }i}\right|}{2\times \left(\mathrm{Inter \;voxel \;distance \;along \; }i\right)}\mathrm{\; for \; }i=x, y,z,$$
(12)
$${G}_{\mathrm{RMS}}=\sqrt{{{G}_{x}}^{2}+{{G}_{y}}^{2}+{{G}_{z}}^{2}}.$$
(13)

Results

Field distribution in biological phantom: potato (isotropic)

Magnetic field distribution could be reliably assessed in most areas of the potato. Figure 2 shows a mid-tuber transverse slice with the maximum extension of the potato along the R-L direction. The inner and outer medulla, medullary ray and vascular ring are clearly visible and labeled on the \({T}_{2}\)-weighted image (Fig. 2a). The \({B}_{0}\) field map (Fig. 2b) and \({G}_{z}\) field map (Fig. 2c) were quite uniform, remaining well under 0.02 mT and 0.002 mT/mm in the center, with slightly higher values around the periphery. The width parameter σ nearly doubled near the periphery relative to the center (Fig. 2d). The shape parameter α indicated a nearly Gaussian field distribution with values close to 2 in most of the outer medulla, with lower values of 1.5–1.8 only near the center of the inner medulla, and around the vascular ring (Fig. 2e).

Fig. 2
figure 2

Sample potato mid-tuber transverse slice. A \({T}_{2}\) weighted TSE image (arbitrary units, au.). B \({B}_{0}\) field map (μT). C \({G}_{\mathrm{RMS}}\) field gradient map (μT/m). D σ, SAS width parameter (Hz). E α, SAS shape parameter (no unit). F Best performing distribution according to AICc (no unit)

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In terms of information content, the Gaussian distribution sufficiently represented most of the outer medulla (Fig. 2f, green), and the SAS distribution performed better only near the center of the inner medulla and around the vascular ring (Fig. 2f, maroon). Lorentzian or Voigt distributions were not representative of signal decay except for a few voxels at the stem and bud ends of the tuber (Fig. 2f, blue and orange, respectively). Sample GESSE signal and best fits to each distribution are shown in Fig. 3b–e, for four voxel positions depicted in Fig. 3a. Voxel b (Fig. 3b) and voxel e (Fig. 3e) show purely Lorentzian and purely Gaussian distributions, respectively, while voxels c and d (Fig. 3c and d) are intermediate scenarios where Voigt and SAS distributions improved fits.

Fig. 3
figure 3

Sample potato GESSE signal (au.) and best fits to each distribution. A Four sample voxels are shown on a GESSE image (averaged over all TEs). B Lorentzian distribution performed best (α = 1). C SAS distribution performed best (α = 1.61). D Voigt distribution performed best (α = 1.92). E Gaussian distribution performed best (α = 2)

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Field distribution in biological phantom: lemon (tubular)

Magnetic field distribution could also be reliably assessed in most areas of the lemon, excluding the exocarp (peel) and seeds. Figure 4 shows a mid-berry transverse slice. The exocarp and endocarp, the individual sections (carpels) and location of some seeds are visible and labeled on the \({T}_{2}\)-weighted image (Fig. 4a). The \({B}_{0}\) and \({G}_{z}\) field maps (Fig. 4b and c, respectively) are also quite uniform for the lemon, with slightly higher \({G}_{z}\) values only seen near the exocarp and seeds. The width parameter σ increases in a few areas near the exocarp and seeds but remains low throughout most of the endocarp (Fig. 4d). The shape parameter α indicates a nearly Lorentzian field distribution with values close to 1 only near the exocarp and seeds, with higher values of 1.3–2 observed throughout the endocarp (Fig. 4e).

Fig. 4
figure 4

Sample lemon mid-berry transverse slice. A \({T}_{2}\) weighted TSE image (au.). B \({B}_{0}\) field map (μT). C \({G}_{\mathrm{RMS}}\) field gradient map (μT/m). D σ, SAS width parameter (Hz). E α, SAS shape parameter. F Best performing distribution according to AICc

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In terms of information content, the Lorentzian distribution performed best only near the exocarp and seeds (Fig. 4f, blue), and the SAS distribution performed best at the peripheral and central edges of the endocarp (Fig. 4f, maroon). Gaussian and Voigt distributions performed best throughout most of the central endocarp (Fig. 4f, green and orange, respectively). Sample GESSE signal and best fits to each distribution are shown in Fig. 5b–e, for four voxel positions depicted in Fig. 5a. Voxel b (Fig. 5b) and voxel e (Fig. 5e) show signals best represented by Lorentzian and Gaussian distributions, respectively, while voxels c and d (Fig. 5c and d) show intermediate scenarios where Voigt and SAS distributions performed best.

Fig. 5
figure 5

Sample lemon GESSE signal (au.) and best fits to each distribution. A Four sample voxels are shown on a GESSE image (averaged over all TEs). B Lorentzian distribution performed best (α = 1.02). C SAS distribution performed best (α = 1.29). D Voigt distribution performed best (α = 1.80). E Gaussian distribution performed best (α = 1.98)

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Field distribution in human pelvis, cervix, and cancer

Figure 6 presents data from Patient X, with Stage IIIB cervical squamous cell carcinoma, over three consecutive transverse slices. The MRI report indicates a nearly circumferential mass within the cervix, showing restricted diffusion, invading through the cervical stroma into parametrial tissues on the right, with A/P, S/I, and Right/Left (R/L) dimensions up to 3.6 cm × 3.3 cm × 4.5 cm, respectively, and no involvement of the bladder or rectum. \({T}_{2}\)-weighted images are shown, with R/L and A/P coordinates labeled (Fig. 6a). ADC maps are shown, and the bladder, cervix (tumor and normal), and rectum are labeled in Fig. 6b. The remaining parts of Figs. 6 and 7 are magnified displays (as indicated by the yellow rectangle, Fig. 6c). Pink and red ROIs are provided to help visualization of field maps and fitting parameters, where reference structures are not clearly visible. \({B}_{0}\) and \({G}_{z}\) field maps were calculated everywhere except where phase-unwrapping was not possible (mostly within the rectum, periphery of the bladder etc.), and were also quite uniform for the patient (Fig. 6d and e, respectively). Fitting results for the same patient are presented in Fig. 7. The width parameter σ was within the 120–160 Hz range in the PL quadrant, the 0–30 Hz range in small areas in the mid-R and AL, and the 80–90 Hz range in the remainder of the ROI (Fig. 7a). The shape parameter α indicates an intermediate field distribution in the PL quadrant and mid-A to RA, a nearly Gaussian distribution is found in the mid-R to PR and AL, and a nearly Lorentzian distribution in two small regions closer to the center of the ROI (Fig. 7b).

Fig. 6
figure 6

Sample data from Patient X with Stage IIIB cervical squamous cell carcinoma. A \({T}_{2}\) weighted TSE images (au.). B Apparent Diffusion Coefficient, ADC, images (μm2/ms). C GESSE images (averaged over all TEs, au.). D \({B}_{0}\) Field maps (μT). E \({G}_{\mathrm{RMS}}\) Field gradient maps (μT/m), in three adjacent slices

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Fig. 7
figure 7

Patient X. A σ, SAS width parameter (Hz). B α, SAS shape parameter, C Best performing distribution according to AICc

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In terms of information content, the SAS distribution was representative of most of the ROI in the PL quadrant and mid-A to RA (Fig. 7c, maroon, near where the shape parameter indicated an intermediate distribution). The Lorentzian distribution performed best near areas where the shape parameter also indicated a Lorentzian distribution (Fig. 7c, blue). The Gaussian distribution performed best near areas where the shape parameter also indicated a Gaussian distribution (Fig. 7c, green). The Voigt distribution performed best in fewer voxels compared to the other distributions (Fig. 7c, orange, also near where the shape parameter indicated an intermediate distribution). Sample best fits to each distribution are shown in Fig. 8b–e, for four voxel positions depicted in Fig. 8a. Voxel b (Fig. 8b) and voxel e (Fig. 8e) show signals best represented by Lorentzian and Gaussian distributions, respectively, while voxels c and d (Fig. 8c and d) show intermediate scenarios where Voigt and SAS distributions were the most representative distributions, respectively.

Fig. 8
figure 8

Patient X, sample GESSE signal (au.) and best fits to each distribution. A Four sample voxels are shown on a GESSE image (averaged over all TEs). B Lorentzian distribution performed best (α = 1.01). C SAS distribution performed best (α = 1.76). D Voigt distribution performed best (α = 1.49). E Gaussian distribution performed best (α = 2). F Muscle (\({R}_{2}\)= 33 ms > \({{R}_{2}}^{^{\prime}}\)  = 12 ms, note corresponding decay on the left side of the SE)

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Figure 9 presents data from Patient Y, with Stage IIB cervical squamous cell carcinoma, on a single transverse slice. The MRI report indicates a cervical mass measuring approximately 4.4 cm × 6.6 cm × 6 cm (A/P, S/I, and R/L, respectively), involving the entire endocervical canal, lower uterine segment, and superior half of the vagina, with almost certain involvement of adjacent parametrial tissues, and possible involvement of the posterior bladder wall and anterior rectal walls. The \({T}_{2}\)-weighted image and ADC map are shown in Fig. 9a and b, respectively. The rest of Fig. 9 is magnified (as indicated by the yellow rectangle), along with ROIs (in pink) to help visualization of field maps and fitting parameters (where reference structures may not be clearly visible). \({B}_{0}\) and \({G}_{z}\) field maps were also very uniform for this patient (Fig. 9c and d, respectively). The width parameter σ was within 0–60 Hz across most of the ROI, with few values approaching 80 Hz near the center of the ROI (Fig. 9e and f, show comparable results without and with averaging for noise reduction, respectively). The shape parameter α indicates a nearly Gaussian field distribution on the R and PL, and a nearly Lorentzian distribution almost everywhere else, with very few voxels near the center of the ROI indicating intermediate distributions (Fig. 9g).

Fig. 9
figure 9

Sample data from Patient Y with Stage IIB cervical squamous cell carcinoma. A \({T}_{2}\) weighted TSE images (au.). B Apparent Diffusion Coefficient, ADC, images (μm2/ms). C \({B}_{0}\) Field maps (μT). D \({G}_{\mathrm{RMS}}\) Field gradient maps (μT/m). E GESSE images (averaged over all TEs, au.). F σ, SAS width parameter (Hz). G α, SAS shape parameter. H Best performing distribution according to AICc

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In terms of information content, the Lorentzian and Gaussian distributions were representative of nearly all the relevant anatomy (Fig. 9h, blue and green, respectively, also in agreement with the shape parameter). The SAS and Voigt distributions performed best in very few voxels near the center (Fig. 9h, maroon and orange, respectively). Sample best fits to each distribution are shown in Fig. 10b–e, for four voxel positions depicted in Fig. 10a. Voxel b (Fig. 10b) and voxel e (Fig. 10e) show signals best represented by Lorentzian and Gaussian distributions, respectively, while voxels c and d (Fig. 10c and d) show intermediate scenarios where SAS and Voigt distributions were the most representative distributions, respectively.

Fig. 10
figure 10

Patient Y, sample GESSE signal (au.) and best fits to each distribution. A Four sample voxels are shown on a GESSE image (averaged over all TEs). B Lorentzian distribution performed best (α = 1.04). C SAS distribution performed best (α = 1.51). D Voigt distribution performed best (α = 1.52). E Gaussian distribution performed best (α = 1.95). F Muscle (\({R}_{2}\)= 30 ms > \({{R}_{2}}^{^{\prime}}\)  = 8 ms, note corresponding decay on the left side of the SE)

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Muscle data is also provided in Figs. 8f and 10f. In both cases, \({R}_{2}\)> \({{R}_{2}}^{^{\prime}}\) and the left side of the SE, where rephasing by reversible processes works against dephasing by irreversible processes, decays.

Discussion and conclusions

At 3 T, in much of the female pelvis, transverse relaxation as measured by GESSE, did not generally follow a simple mono-exponential decay corresponding to a Lorentzian intra-voxel frequency distribution, and non-Lorentzian distributions performed substantially better. Clear and systematic deviations from mono-exponential decays were observed in both biological phantoms and the human cervix. The potato in this study, oriented with its medullary ray orthogonal to the main magnetic field \({B}_{0}\), had larger width parameters σ (Fig. 2) compared to prior results on one oriented parallel to \({B}_{0}\), in agreement with observations on a pineapple [10] and might also suggest some anisotropy in the potato. The lemon produced noisier maps and its \({T}_{2}\) was much longer (over 200 ms) compared to potato (~ 30–50 ms) and biological tissue \({T}_{2}\) values [38], which might potentially underlie reduced fitting performance related to limited irreversible relaxation (when imaging parameters relevant to the cervix are used). Our findings not only have theoretical support [4, 5, 9], but also agree well with previous observations in the brain, lungs, prostate, teeth, and blood, using either multi-echo GE or GESSE imaging [10,11,12, 39].

Deviations from mono-exponential decay have been suggested to potentially result from the interaction between macroscopic gradients (which, especially at higher field strengths, can lead to higher susceptibility effects from interfaces between air and tissue, or bone and tissue) and non-ideal slice profiles [40] (which can deviate from the assumed rectangular profiles and appear more Gaussian in reality [41]). Calculated field maps and field gradients show that the background fields are quite uniform. Additionally, while some larger width parameters coincide with larger shape parameters (closer to 2, indicating a more Gaussian distribution), in many cases smaller width parameters, and smaller gradient \({G}_{\mathrm{RMS}}\) values often also coincide with larger shape parameters, with no apparent overlap. While this mechanism could indeed contribute, it fails to completely explain the observed deviation from mono-exponential decays. The apparent lack of correlation between shape and width parameters and the lack of influence of averaging on background gradient effects, provide further support that our observations might indeed be reflections of underlying biological circumstances.

Voigt and SAS distributions have one additional parameter compared to Lorentzian and Gaussian distributions. When the shape parameter α was near 1 or 2, additional parameters were not justified, and the best performing distributions were Lorentzian and Gaussian, respectively. When the shape parameter had intermediate values, the additional parameters of Voigt and SAS distributions were justified, and the best performing distributions were the Voigt and SAS as expected. However, the Voigt distribution was needed much less frequently than the SAS distribution overall. Even in cases when an extra parameter was justified and the Voigt distribution performed best in terms of information content, the difference between the best performing Voigt and next-best performing SAS distributions were miniscule, especially in the human cases. The central peak of a Lorentzian distribution is narrower than the central peak of a Gaussian distribution, for similar reversible transverse relaxation terms \({{R}_{2}}^{^{\prime}}\) and \(\sigma\) (Fig. 1, inset A), while the tail of a Lorentzian distribution includes a wider range of frequencies than the tail of a Gaussian distribution (Fig. 1, inset B), as expected from the differences in their FWHMs. Since both two-parameter distributions SAS and Voigt can represent Lorentzian and Gaussian distributions as well as intermediate cases, they might be expected to perform more similarly, however, the ability to place upper and lower bounds on one of the SAS parameters (\(1\le {\alpha }\le 2,\mathrm{ while\; }{R}_{2}\ge 0\), \({{R}_{2}}^{{^{\prime}}}\ge 0\), \(\sigma \ge 0\)) may potentially have rendered fitting to the SAS model less sensitive to initial values and local minima. Given the same assumptions underlying validity of SAS and Voigt distributions (i.e., no considerable fat content or efficient fat suppression in order to maintain symmetry [10]), a separate Voigt implementation may not be necessary.

Improved modeling, beyond Lorentzian fitting, is needed in the female pelvis; Gaussian, Voigt, and SAS models minimize information loss, and could improve the accuracy of parameter estimation. However, our study is limited to a small number of cases. To determine the degree to which these more accurately characterized MR parameters will correlate with biomarkers of interest, i.e., hypoxia or improve clinically important quantitative assessments requires further studies, including more cases, and ideally, using a gold standard for hypoxia (e.g., immunohistochemistry, pO2 microelectrodes, and/or PET and ROC analysis.

In conclusion, modeling irreversible transverse relaxation with Gaussian and intermediate (SAS and Voigt) distributions, as opposed to a Lorentzian distribution, improves characterization of GESSE signals in cervical cancers at 3 T; existence of through plane gradients may contribute to deviations from Lorentzian but are not the only reason for the observed deviations. Improved characterization of underlying signals leads to more accurate determination of fit parameters, increasing the potential for identifying biomarkers of relevant physiological variables.