Impact of Undersampling on Preclinical Lung ${\mathrm{T}}_2\mathrm{*}$ Mapping with 3D Radial UTE MRI at 7T

Abstract

Lung diseases are almost invariably heterogeneous and progressive, making it imperative to capture temporally and spatially explicit information to understand the disease initiation and progression. Imaging the lung with MRI—particularly in the preclinical setting—has historically been challenging because of relatively low lung tissue density, rapid cardiac and respiratory motion, and rapid transverse (${\mathrm{T}}_2\mathrm{*}$) relaxation. These limitations can largely be mitigated using ultrashort-echo-time (UTE) sequences, which are intrinsically robust to motion and avoid significant ${\mathrm{T}}_2\mathrm{*}$ decay. A significant disadvantage of common radial UTE sequences is that they require inefficient, center-out k-space sampling, resulting in long acquisition times relative to conventional Cartesian sequences. Therefore, pulmonary images acquired with radial UTE are often under sampled to reduce acquisition time. However, under sampling reduces image SNR, introduces image artifacts, and degrades true image resolution. The level of under sampling is further increased if offline gating techniques like retrospective gating are employed, because only a portion (~40–50%) of the data is used in the final image reconstruction. Here, we explore the impact of undersampling on SNR and ${\mathrm{T}}_2\mathrm{*}$ mapping in mouse lung imaging using simulation and in-vivo data. Increased scatter in both metrics was noticeable at around 50% sampling. Parenchymal apparent SNR only decreased slightly (average decrease ~1.4) with as little as 10% sampling. Apparent ${\mathrm{T}}_2\mathrm{*}$ remained similar across undersampling levels, but it became significantly increased (p<0.05) below 80% sampling. These trends suggest that undersampling can generate quantifiable, but moderate changes in the apparent value of ${\mathrm{T}}_2\mathrm{*}$. Moreover, these approaches to assess the impact of undersampling are straightforward to implement and can readily be expanded to assess the quantitative impact of other MR acquisition and reconstruction parameters.

Graphical Abstract

1. Introduction

Relative to other solid organs, the lungs have historically been difficult to assess with MRI because of relatively low tissue density (10–20% that of other organs, depending on inflation level) and rapid cardiac and respiratory motion. Moreover, strong magnetic susceptibility gradients at the air-tissue interface generate rapid apparent transverse magnetization (${\mathrm{T}}_2\mathrm{*}$) relaxation in the parenchyma, with ${\mathrm{T}}_2\mathrm{*}$ ~1 ms or less at the clinically relevant field strengths of 1.5 and 3 T [1–3]. The challenges inherent to lung MRI are usually exacerbated when performing small animal imaging, because physiological motion is ~10-fold more rapid than in humans [4, 5], and the lung volume is reduced from ~5 L to only ~1 mL in mice. Thus, >1000-fold smaller voxels are required to achieve anatomically equivalent image resolution [6]. To partially compensate for the lower signal originating from these minute voxels, most preclinical MRI scanners operate at high magnetic field strengths of 4.5 to 21T. Unfortunately, these higher fields lead to even more rapid ${\mathrm{T}}_2\mathrm{*}$ relaxation in the lung parenchyma (e.g., 0.4 ms at 7 T [7]), thus further complicating preclinical lung MRI.

To overcome rapid ${\mathrm{T}}_2\mathrm{*}$ losses, non-Cartesian ultra-short echo-time (UTE) imaging sequences have become increasingly popular for lung MRI. These sequences forgo conventional gradient and spin echoes, which typically require echo times (TEs) of many milliseconds, and encode k-space from the center out to the periphery, directly along the free induction decay (FID) [8]. Center-out FID encoding enables data collection with effective TEs as short as ~10 μs, thereby sampling transverse magnetization before it dephases. Additionally, multi-echo variants of these UTE sequences allow the ${\mathrm{T}}_2\mathrm{*}$ of lung tissue to measured, which is of intrinsic interest. That is, despite experimental challenges caused by rapid apparent transverse relaxation, ${\mathrm{T}}_2\mathrm{*}$ is determined by important tissues properties, including magnetic susceptibility, water diffusion, chemical composition, and microstructural order. As such, practical and robust ${\mathrm{T}}_2\mathrm{*}$ mapping could provide fundamental insights into the biophysical changes in lung tissue during disease initiation and progression [9, 10].

Unfortunately, UTE sequences suffer from a key disadvantage. They sample k-space inefficiently and are thus time consuming, relative to conventional Cartesian sequences. For example, 3D radial UTE sequences require approximately π-times more radial views than a Cartesian sequence to satisfy Nyquist sampling criteria [11]. To reduce scan times, lung UTE images are often undersampled, which reduces image signal-to-noise ratio (SNR), introduces artifacts, and degrades true resolution. While radial sequences are sufficiently robust to undersampling for many applications, such as contrast-enhanced angiography [12–15], the impact of radial undersampling on quantitative imaging has received little attention. In particular, the effect of undersampling on regional ${\mathrm{T}}_2\mathrm{*}$ measurements have never been investigated.

In this work, the quantitative impact of undersampling on parenchymal SNR and ${\mathrm{T}}_2\mathrm{*}$ measurements are quantified for 3D radial UTE using sequence parameters that were previously found to be robust and reliable for routine mouse lung imaging [7, 16, 17]. Undersampling was examined via simulations as a function of matrix size of the reconstructed images and sampling percentages, using lung-like 3D phantom derived from a modified Shepp-Logan phantom. Fully sampled, multi-TE radial images of the lungs were also acquired from C57BL/6J mice. These fully sampled k-space data were systematically undersampled by removing random projections prior to image reconstruction. Apparent signal-to-noise ratio (aSNR) and apparent ${\mathrm{T}}_2\mathrm{*}$ ($\text{a}{\mathrm{T}}_2\mathrm{*}$) were measured within the lung parenchyma as a function of the level of undersampling. As undersampling increased, image quality was degraded, parenchymal aSNR decreased, and parenchymal $\text{a}{\mathrm{T}}_2\mathrm{*}$ increased.

2. Materials and Methods

2.1. Phantom Lung Simulations

Multi-TE simulations of lung ${\mathrm{T}}_2\mathrm{*}$ decay were performed in MATLAB 2019b (MathWorks, Natick, MA) using modified, 3D Shepp-Logan phantoms (3D isotropic matrix sizes: 64³, 96³, and 128³, Figure 1). The signal intensity pattern of the phantom prior to ${\mathrm{T}}_2\mathrm{*}$ decay was structured to resemble SNR patterns observed from murine lungs in-vivo. Bright periphery was incorporated to represent extra-thoracic tissues that surround low SNR regions that represent lung parenchyma and comprised ~10% of the total phantom volume. Bright, ellipsoid structures were imbedded within the parenchymal volume to mimic lung vasculature and comprised ~5% of the “thoracic cavity” volume or ~0.5% of the image volume. The relative differences in signal intensity between compartments were chosen to closely match those found in-vivo (simulation input values: extra-thoracic tissue=1×10⁶, vasculature=1.05×10⁶, heart=9×10⁵, and lung parenchyma=5×10⁵).

Figure 1. Schematic Depicting Simulation of Undersampled Multi-echo UTE MRI.

Simulations used modified 3D Shepp-Logan phantoms, augmented with ellipsoids to mimic mouse lung vasculature inside the lung parenchyma. Phantoms were created at three, 3D isotropic matrix sizes: 64³, 96³, and 128³. 3D radial projections generated were generated along golden angle sampling trajectories (a). Tissue-specific ${\mathrm{T}}_2\mathrm{*}$ decay was introduced by adjusting the signal based on an exponential decay (lung parenchyma: ${\mathrm{T}}_2\mathrm{*}=0.4\text{ms}$; lung vasculature ${\mathrm{T}}_2\mathrm{*}=3.16\text{ms}$). K-space data were generated via Fast Fourier Transform (FFT) of decayed images. The corresponding k-space data were sampled to generate Free Induction Decay data (FIDs). Gaussian noise (with mean μ = 0 and standard deviation σ = 50) was added to both the real and imaginary k-space data to produce noisy FIDs to yield parenchymal SNR similar to that observed from in-vivo images (~30) (b). The total radial projections for each matrix size were: 12,868 for 64³, 28,953 for 96³, and 51,472 for 128³. Images were reconstructed at each TE using from noisy FIDs data employing Cartesian re-gridding, iterative density compensation, and FFT (c).

${\mathrm{T}}_2\mathrm{*}$ relaxation was incorporated such signal decay was modeled during both the nominal echo time (i.e., the delay between excitation and data acquisition) and the radial readout. Lung parenchyma was assigned a value of ${\mathrm{T}}_2\mathrm{*}=0.40\text{ms}$, and vasculature was assigned ${\mathrm{T}}_2\mathrm{*}=3.16\text{ms}$. Image signal was then varied at the voxel level according to:

$$S_{k,i}=S_0{e}^{-\left(T{E}_k+(i-1)t_d\right)/T_2\mathrm{*}},$$ (1)

where $S_0$ is the initial signal without decay (the initial phantom signal intensity), $T{E}_k=[0.08,0.25,0.50,1.25,\mathrm{a}\mathrm{n}\mathrm{d}2.50]\mathrm{m}\mathrm{s}$ are the nominal echo times with index $k$, $i$ is the index of the number of points along the radial projections, and ${\mathrm{T}}_d$ is the dwell time (fixed at ${\mathrm{T}}_d=0.0036\mathrm{m}\mathrm{s}$).

For each unique combination of $T{E}_k$ and $i$, k-space data were generated by performing a Fast Fourier Transform (FFT) of the signal-decayed images. The resulting k-space data were sampled along 3D, radial projections using golden angle sampling [7] to create the radial FIDs. Radial trajectories were rounded to the nearest grid point to avoid interpolation (Figure 1 a). The number of radial projections and points per FID were determined based on matrix size, and sampling percentage up to full Nyquist sampling (projections = 12,868 for 64³, 28,953 for 96³, and 51,472 radial projections for 128³). After data sampling, Gaussian noise (μ = 0, with σ = 50) was added separately to the real and the imaginary portions of the complex k-space data to generate the noisy FIDs that produce image SNR levels typically observed in fully sampled, in-vivo mouse lung images (parenchymal SNR ~30, Figure 1 b). Images were then reconstructed from noisy FIDs data and prescribed trajectories using Cartesian re-gridding, iterative density compensation and FFT [18–23] (Figure 1 c). The simulation MATLAB code is available at (https://github.com/cchmc-cpir/Multi_TE_UTE_Simulation).

2.2. In-vivo Imaging

C57BL/6J mice (Jackson Laboratory, Bar Harbor, ME) (N = 14) were housed inside a pathogen-free barrier and handled following protocols approved by the Institutional Animal Use and Care Committee of the Cincinnati Children’s Hospital Research Foundation. Imaging was performed on a 7 T horizontal, 30-cm bore, small animal scanner (Bruker BioSpec, Billerica, MA) with a home-built quadrature transmit/receive birdcage coil (inner diameter = 35 mm; length = 50 mm). Prior to experiments, the scanner frequency, shim settings, and animal position were determined using Bruker built-in routines. During experiments, free-breathing mice were anesthetized with 1–3% isoflurane (mixed with air) to maintain the breathing and imaged head-first in supine position at the magnet isocenter. Throughout imaging, a small-animal physiological monitoring system (Small Animal Instruments Inc., Stony Brook NY) continuously monitored mouse respiration via respiratory bellows and body temperature by a thermocouple placed near the body. Body temperature was maintained >34°C and respiration rate was maintained at ~100 breaths/min. After imaging, mice recovered in an isolated heated cage, set to 36 °C, and returned to the barrier room until fully recovered from anesthesia.

2.2. In-vivo Image Acquisition and Reconstruction

Fully sampled FID data were acquired using a 3D radial sequence with five interleaved TEs. Golden angle sampling was used to prescribe k-space trajectories in an approximately uniform fashion, and each trajectory was repeated for each TE [24]. RF and gradient excitations were prospectively triggered at expiration using a respiratory bellows [25, 26]. In addition, RF and gradient spoiling continued between breaths to maintain incoherent steady-state magnetization [27, 28]. To maximize steady-state signal intensity from the lung parenchyma, a flip angle of α = 6.3° was used (i.e. the Ernst angle assuming T₁ = 1452 ms [7]). Other MR acquisition parameters include: 4.3 μs RF block pulse; 51,472 radial projections per image; 128 points on FID; field-of-view (FOV) = 32 × 32 × 64 mm³; image matrix size = 128 × 128 × 128 voxels³; receiver bandwidth (BW) = 277 kHz; TEs = [0.08, 0.25, 0.50, 1.25, and 2.50] ms; and TR = 8.24 ms. Total acquisition time for each multi-TE scan was ~90 minutes.

K-space trajectories were corrected via Duyn’s method prior to image reconstruction using data acquired from a water phantom [17, 29, 30]. Non-Cartesian data were reconstructed by first using iterative density compensation and then interpolating the data onto a Cartesian grid prior to FFT [18–23]. All processes for undersampling and image reconstruction were performed in MATLAB 2019b.

2.3. Undersampling of Raw K-Space Data

Undersampled data were generated by pseduorandomly removing FIDs from the fully sampled k-space data for all echo time images. Each FID was assigned a unique index ranging from 1 to the total number of radial projections, and a specified percentage of these indices were selected for removal by a Mersenne Twister random number generator [31]. The fraction of unique FIDs–and thus k-space trajectories–removed from the fully sampled data increased in increments of 10% (i.e., 90%, 80%, 70%… to 10% sampled). Data decimation was iterated 10 times for each fully sampled image, at each echo time, and at every level of sampling. To ensure a unique set of indices was generated for every iteration and at each level of Nyquist sampling, the random number generator seeding was changed to the date and time of execution for each iteration. A total of 1,260 distinct undersampled and 14 fully sampled multi-TE sets (5 TE images per set) were generated.

2.5. Image Analysis

Simulated thoracic cavity images (parenchymal and vascular regions in the modified Shepp-Logan phantoms) were automatically segmented using the known locations. Binary masks of the thoracic cavity from the fully sampled in-vivo images were automatically segmented using a deep-learning algorithm. The architecture of the algorithm is modeled after Ronneberger’s U-Net Convolutional Network and adapted to accommodate 3-D datasets [32]. The deep-learning model had been trained on 113 manually segmented lung masks spanning the entire lung volume [17, 30]. Masks were manually edited in Amira (FEI Company, Natick MA), if needed, to improve accuracy. To minimize the impact of partial volume effects, lung masks were then subjected to morphometric erosion using a 1-voxel structuring element in MATLAB. Data analysis was confined to the parenchymal tissues by drawing a region-of-interest (ROI) within the heart in the fully sampled images. The mean signal from this ROI was then used to weight the signal values within the lung mask [17, 30]. Voxels with weighted signal falling below the threshold value of 0.8 were treated as lung parenchyma in subsequent ${\mathrm{T}}_2\mathrm{*}$ fitting and SNR measurements [17, 30].

aSNR was calculated from TE = 0.08 ms image by averaging, non-weighted, lung parenchyma signal and dividing by the standard deviation of signal within an ROI drawn outside the mouse body in a region free from obvious image artifacts. $\text{a}{\mathrm{T}}_2\mathrm{*}$ was estimated voxel-by-voxel via iterative nonlinear least squares fitting of the signal intensity as a function of TE to [30]

$$S_I=S_0{e}^{-\frac{TE}{T_2\mathrm{*}}}+c,$$ (2)

where $c$ is a constant representing the non-zero noise floor.

The MATLAB Parallel Computing Toolbox was used for all fittings. Systematic trends in average aSNR and $\text{a}{\mathrm{T}}_2\mathrm{*}$, fully sampled and undersampled images were assessed by comparing histogram distributions and by measuring the percent change between undersampled and fully sampled images.

2.6. Statistical Analysis

For both simulated and in-vivo datasets, $\text{a}{\mathrm{T}}_2\mathrm{*}$ values obtained at different sampling levels were correlated by linear regression, and the strength of the correlation was assessed with Pearson's correlation coefficient. The slope of the best fit line was also reported. To examine the significant impact of undersampling on $\text{a}{\mathrm{T}}_2\mathrm{*}$ maps, a Hoteling t-test (for simulated data) and a two-sample t-test (for in-vivo data) were conducted using the difference of $\text{a}{\mathrm{T}}_2\mathrm{*}$ maps obtained at different levels of undersampling with the $\text{a}{\mathrm{T}}_2\mathrm{*}$ maps acquired from fully sampled images. Different statistical tests were employed because intraclass correlation between iterations was higher for in-vivo than for simulated data. Significance for all comparisons was determined based on a p value < 0.05. The standard error of aSNR and $\text{a}{\mathrm{T}}_2\mathrm{*}$ was calculated as the standard deviation divided by the square root of total number of iterations.

3. Results

3.1. Impact of Undersampling on aSNR and $\text{a}{\mathrm{T}}_2\mathrm{*}$ in Phantom Simulations

Figure 2 shows representative images reconstructed from the modified Shepp-Logan phantom at the shortest echo time (TE = 0.08 ms) at sampling levels of 100%, 70%, 50%, and 10% for all three matrix sizes used in the multi-TE simulations. As expected, the image with the smallest matrix (64³, Figure 2 a–d) displayed steadily degrading image resolution as sampling was reduced from 100% to 10%. In particular, the level of signal blurring from the bright periphery and “vascular” into “lung parenchyma” increased steadily as sampling was reduced below 70%. Between 70% and 50% sampling, undersampling-induced blurring began to noticeably manifest, with significant loss of true image resolution visible at 10% sampling. Also as expected, undersampling also reduced aSNR of the parenchyma, with aSNR degrading by >20% (reduced from 27 to 20) as sampling was reduced from 100% to 10%. Similar trends in resolution and aSNR reduction were observed for the 96³ (Figure 1 e–h) and 128³ (Figure 1 i–l) matrix sizes—albeit at noticeably reduced levels.

Figure 2. Influence of Undersampling on Modified Shepp-Logan Phantoms.

Axial slices of modified Shepp-Logan phantom lung with simulated TE = 0.08 ms are shown at progressively decreasing levels of sampling (left to right) and increasing matrix size (top to bottom). Windowing and leveling were identical for all images. (a-d) 64³ phantom. Apparent ${\mathrm{T}}_2\mathrm{*}$ ($\text{a}{\mathrm{T}}_2\mathrm{*}$) maps of the 64³ matrix phantom show increased $\text{a}{\mathrm{T}}_2\mathrm{*}$ in the phantom parenchyma volume compared to the input ${\mathrm{T}}_2\mathrm{*}$ of 0.4 ms. Increased $\text{a}{\mathrm{T}}_2\mathrm{*}$ mean and standard deviation are observed with spurious “hot spots” of long $\text{a}{\mathrm{T}}_2\mathrm{*}$ voxels near the simulated vasculature regions. (e-h) 96³ matrix phantom. Increased $\text{a}{\mathrm{T}}_2\mathrm{*}$ deviation persists (±0.20 ms) but is reduced relative to the 100% sampled image. (i-l) 128³ matrix phantom. The $\text{a}{\mathrm{T}}_2\mathrm{*}$ maps display mean $\text{a}{\mathrm{T}}_2\mathrm{*}=0.4\text{ms}$ with increased homogeneity away from vessels, even when image is sampled only 10%.

The corresponding phantom $\text{a}{\mathrm{T}}_2\mathrm{*}$ maps of each matrix size are also shown in Figure 2. aT2* maps of all phantom sizes show increased mean $\text{a}{\mathrm{T}}_2\mathrm{*}$ of ~0.44 ms in the phantom parenchyma volume, relative to the uniform simulation input of ${\mathrm{T}}_2\mathrm{*}=0.4\text{ms}$. However, for sampling levels of 70% and 50%, the mean $\text{a}{\mathrm{T}}_2\mathrm{*}$ value remained relatively unchanged for all three matrix sizes. However, in $\text{a}{\mathrm{T}}_2\mathrm{*}$ maps from 10% sampled data, localized “hot spots” with long relaxation times (i.e., ${\mathrm{T}}_2\mathrm{*}>0.7\text{ms}$) were observed from voxels near the phantom vasculature. Spurious, voxel-level fits were less apparent as matrix size was increased, likely resulting from reduced partial volume effects.

However, the systematic nature of these undersampling effects is better appreciated in the voxel-wise correlation between $\text{a}{\mathrm{T}}_2\mathrm{*}$ values obtained from fully sampled images and $\text{a}{\mathrm{T}}_2\mathrm{*}$ values calculated at images with different sampling levels (Figure 3). Within each subplot, a strong (R²>0.5) and statistically significant (p < 0.001) positive correlation is observed between the $\text{a}{\mathrm{T}}_2\mathrm{*}$ measured with full sampling in a given voxel with the values measured from the same voxel with undersampling. However, the slope of the correlation decreases as the sampling rate decreases. This indicates a systematic shift towards higher $\text{a}{\mathrm{T}}_2\mathrm{*}$ values in the maps acquired with lower sampling rates compared to the fully sampled ones.

Figure 3. Voxel-level Correlations between Undersampled and Fullly Sampled $\text{a}{\mathrm{T}}_2\mathrm{*}$.

(a-d) For phantom size 64³ (a-d), $\text{a}{\mathrm{T}}_2\mathrm{*}$ maps at sampling rates of 90%, 70%, 50%, and 10% exhibit strong and significant correlations with fully sampled $\text{a}{\mathrm{T}}_2\mathrm{*}$ maps, respectively. However, as the sampling level decreases, the correlation intercept also decreases, suggesting a shift towards higher $\text{a}{\mathrm{T}}_2\mathrm{*}$ values in the different sampling rates compared to fully sampled data. Similar trends are observed for phantom sizes 96³ (e-h) and 128³ (i-l).

The undersampling-induced trends in mean aSNR and mean $\text{a}{\mathrm{T}}_2\mathrm{*}$ of the simulated phantoms are summarized in Figure 4, which shows the percent change in mean aSNR and mean $\text{a}{\mathrm{T}}_2\mathrm{*}$ for each matrix size across all iterations of sampling (Figure 4 a–c). Smaller matrices showed a sharper decline in aSNR as the sampling level decreased. This trend held true for all matrix sizes, but the effect was more pronounced for smaller matrices. Conversely, the $\text{a}{\mathrm{T}}_2\mathrm{*}$ relaxation time showed a more systematic increase in its percentage change as the sampling level was reduced (Figure 4 d–f). However, the percent change in $\text{a}{\mathrm{T}}_2\mathrm{*}$ varied with sampling iteration, and the highest level of scatter was observed for the 64³ phantom, which varied by ±0.01 ms (±2.4%) across sampling levels. The scatter was reduced to ±0.003 ms (±1.1%), and to ±0.001 ms (±0.48%) for the 96³ and 128³ phantoms, respectively. Moreover, the influence of sampling on $\text{a}{\mathrm{T}}_2\mathrm{*}$ demonstrated notable variability depending on the size of the phantom. In the case of the 64³ phantom, $\text{a}{\mathrm{T}}_2\mathrm{*}$ values exhibited significant divergence (p < 0.05) from the $\text{a}{\mathrm{T}}_2\mathrm{*}$ values derived from the fully sampled image when the degree of sampling was reduced to 20%. Conversely, the critical thresholds for significant divergence were ≤60% and ≤80% sampling for the 96³ and 128³ phantoms, respectively.

Figure 4. Sampling-dependent aSNR and $\text{a}{\mathrm{T}}_2\mathrm{*}$ Changes across Phantom Matrix Sizes.

Sampling level vs percent difference in apparent signal-to-noise ratio (aSNR) (a) and $\text{a}{\mathrm{T}}_2\mathrm{*}$ (d) in 64³ phantom. Maximum percent difference for both metrics occurred at 10% sampling, where ΔaSNR decreased by 23.9% and $\Delta \text{a}{\mathrm{T}}_2\mathrm{*}$ increased by 3.2%. The 64³ phantom showed significant divergence (p < 0.05, vertical dashed line) in $\text{a}{\mathrm{T}}_2\mathrm{*}$ values from fully sampled images below 20% sampling. Sampling level vs percent difference in aSNR (b) and $\text{a}{\mathrm{T}}_2\mathrm{*}$ (e) in 96³ phantom. The maximum percent difference was also at 10% sampling, where ΔaSNR decreased by 17.0% and $\Delta \text{a}{\mathrm{T}}_2\mathrm{*}$ increased by 4.7%. The 96³ phantom showed significant divergence (p < 0.05, vertical dashed line) in $\text{a}{\mathrm{T}}_2\mathrm{*}$ values from fully sampled images below 60% sampling. Lastly, sampling level vs percent difference in aSNR (c) and $\text{a}{\mathrm{T}}_2\mathrm{*}$ (f) in 128³ phantom. The maximum percent difference was again at 10% sampling where ΔaSNR decreased by 13.5% and $\Delta \text{a}{\mathrm{T}}_2\mathrm{*}$ increased by 4.6%. The 128³ phantom showed significant divergence (p < 0.05, vertical dashed line) in $\text{a}{\mathrm{T}}_2\mathrm{*}$ values from fully sampled images below 80% sampling.

3.2. Impact of Undersampling on aSNR and $\text{a}{\mathrm{T}}_2\mathrm{*}$ in In-Vivo

Images of the entire lung volume were successfully reconstructed for all mice at all levels of sampling, and motion artifacts (e.g., blurring and coherent streaking) were largely mitigated with prospective gating at expiration. At the shortest echo time (TE = 0.08 ms, Figure 5 a), the lung volume qualitatively shows low aSNR in the lung parenchyma and high aSNR in the pulmonary vasculature (i.e., dark to bright voxels respectively). As expected from previous work in mice at 7 T, lung vasculature displayed minimal signal decay as TE was increased. In contrast, parenchymal signal decays quickly as a function of TE (Figure 5 b–e). Any detectable parenchymal signal is absent by TE = 2.50 ms, reflecting the expected rapid ${\mathrm{T}}_2\mathrm{*}$ decay within the gas exchange tissues. By fitting the signal intensity of individual voxels at each TE to Equation 2, an $\text{a}{\mathrm{T}}_2\mathrm{*}$ map (Figure 5 f) can be extracted from the parenchyma volume. In this multi-TE, the signal intensity of the vasculature decayed by ~53% (by TE = 2.50 ms) with an average $\text{a}{\mathrm{T}}_2\mathrm{*}$ of 3.16 ± 0.02 ms. In contrast, parenchymal signal was fully decayed by 2.50 ms, with an average $\text{a}{\mathrm{T}}_2\mathrm{*}$ of 0.44 ± 0.12 ms and only 5.0% of parenchymal voxels yielding greater $\text{a}{\mathrm{T}}_2\mathrm{*}>0.6\text{ms}$. For all fully sampled mouse images, the mean ± standard deviation of the parenchymal aSNR and $\text{a}{\mathrm{T}}_2\mathrm{*}$ were 26.1 ± 2.3 and 0.44 ± 0.12 ms, respectively.

Figure 5. Multi-TE Images of a Representative C57BL/6J Mouse.

Axial slice of mouse lung acquired at TE = (a) 0.08 ms, (b) 0.25 ms, (c) 0.50 ms, (d) 1.25 ms, and (e) 2.50 ms. Windowing and leveling were identical for all images. Lung regions include the airways (blue arrows), parenchyma (white arrow), and vasculature tissue (red arrows). (f) Resulting $\text{a}{\mathrm{T}}_2\mathrm{*}$ map of the lung parenchyma fit from multi-TE images to an exponential decay function (Eq. 2).

Representative radial images (TE = 0.08 ms) and corresponding $\text{a}{\mathrm{T}}_2\mathrm{*}$ maps are shown in Figure 6 a–d at sampling levels of 100%, 70%, 50%, and 10%. In agreement with the simulations, images sampled 50% or more, showed no visibly obvious edge blurring in the lung volume or substantially reduced SNR (parenchymal aSNR at 100%, 70%, and 50% was 26.1 ± 2.3, 25.9 ± 2.5 and 25.7 ± 2.6, respectively). Below 50% sampling, undersampling artifacts began to manifest more obviously, with parenchymal aSNR decreasing by 3% between 100% and 30% sampled. Severe undersampling artifacts were observed at 20% and 10% sampling, and visibly obvious loss of true image resolution became apparent. The magnitude of the noise in the 10% sampled image increased by ~10%, leading to a reduced parenchymal aSNR of 24.7 ± 2.7. Consistent undersampling-dependent trends were observed for all iterations and across all mice (see Table 1).

Figure 6. Influence of Undersampling on In-vivo $\text{a}{\mathrm{T}}_2\mathrm{*}$ Mapping.

Panels (a-d) display axial lung slices (TE = 0.080 ms) sampled at 100%, 70%, 50%, and 10%. At 70% and 50% sampling rates, images closely resemble the fully sampled image, while significantly degraded resolution is observed in the 10% sampled image, particularly near lung parenchyma edges and pulmonary vasculature. Representative parenchymal $\text{a}{\mathrm{T}}_2\mathrm{*}$ maps in panels (e-h) demonstrate regions affected by increasing undersampling. Panels (e-h) exhibit consistent $\text{a}{\mathrm{T}}_2\mathrm{*}$ mean values, while panel (h) displays areas with both increased and reduced $\text{a}{\mathrm{T}}_2\mathrm{*}$ regions, indicated by red arrows and a red-dashed arrow, respectively. (i-l) Histograms of parenchymal $\text{a}{\mathrm{T}}_2\mathrm{*}$ across 10 iterations of undersampling depict distributions at sampling levels of 100%, 70%, 50%, and 10%. Scatter plots (panels m-o) show voxel-level correlations between $\text{a}{\mathrm{T}}_2\mathrm{*}$ values calculated from full sampled data with values calculated at various sampling levels. As sampling level decreases, the correlation slope diminishes, indicating a shift towards higher $\text{a}{\mathrm{T}}_2\mathrm{*}$ values compared to fully sampled data.

Table 1.

Impact of repeated undersampling on lung parenchyma aSNR.

Mouse	100 %	90%	80%	70%	60%	50%	40%	30%	20%	10%
	Sampled	Sampled	Sampled	Sampled	Sampled	Sampled	Sampled	Sampled	Sampled	Sampled
1	29.2 ± 2.8	29.1 ± 2.9	29.0 ± 2.9	28.9 ± 3.0	28.7 ± 3.0	28.5 ± 3.0	28.3 ± 3.1	28.2 ± 3.1	27.9 ± 3.1	27.8 ± 3.2
2	29.0 ± 2.7	28.9 ± 2.8	28.8 ± 2.8	28.8 ± 2.8	28.7 ± 2.8	28.6 ± 2.9	28.5 ± 2.9	28.5 ± 2.9	28.5 ± 2.9	28.3 ± 2.9
3	27.2 ± 2.2	27.1 ± 2.3	27.0 ± 2.3	26.9 ± 2.4	26.7 ± 2.4	26.6 ± 2.4	26.5 ± 2.5	26.4 ± 2.5	26.4 ± 2.5	26.3 ± 2.6
4	25.9 ± 2.7	25.9 ± 2.7	25.9 ± 2.7	25.7 ± 2.7	25.7 ± 2.7	25.6 ± 2.7	25.5 ± 2.7	25.5 ± 2.8	25.5 ± 2.8	25.4 ± 2.8
5	26.3 ± 2.4	26.3 ± 2.6	26.3 ± 2.7	26.2 ± 2.7	26.1 ± 2.7	26.0 ± 2.8	25.9 ± 2.8	25.8 ± 2.9	25.7 ± 2.9	25.2 ± 2.9
6	26.6 ± 2.5	26.5 ± 2.6	26.5 ± 2.7	26.4 ± 2.7	26.3 ± 2.8	26.2 ± 2.8	26.0 ± 2.9	25.8 ± 2.9	25.5 ± 2.9	25.2 ± 3.0
7	25.4 ± 2.3	25.4 ± 2.3	25.4 ± 2.4	25.3 ± 2.4	25.3 ± 2.4	25.1 ± 2.4	25.0 ± 2.5	24.9 ± 2.5	24.9 ± 2.5	24.8 ± 2.5
8	27.0 ± 2.5	26.9 ± 2.6	26.8 ± 2.6	26.7 ± 2.7	26.7 ± 2.7	26.5 ± 2.8	26.3 ± 2.8	26.2 ± 2.8	26.0 ± 2.9	25.6 ± 2.9
9	28.5 ± 2.5	28.5 ± 2.7	28.4 ± 2.8	28.4 ± 2.8	28.3 ± 2.8	28.2 ± 2.8	27.9 ± 2.8	27.9 ± 2.8	27.6 ± 2.8	27.0 ± 2.9
10	26.1 ± 2.4	26.0 ± 2.5	25.9 ± 2.6	25.8 ± 2.7	25.6 ± 2.7	25.4 ± 2.8	25.1 ± 2.8	24.7 ± 2.9	24.3 ± 3.0	23.7 ± 3.0
11	22.0 ± 1.9	21.9 ± 2.0	21.8 ± 2.1	21.8 ± 2.2	21.7 ±2.2	21.5 ± 2.3	21.3 ± 2.4	21.1 ±2.4	20.7 ± 2.4	20.3 ± 2.5
12	24.4 ± 1.8	24.3 ± 1.9	24.2 ± 2.0	24.1 ± 2.0	23.9 ± 2.1	23.7 ±2.1	23.4 ± 2.1	22.9 ± 2.1	22.3 ± 2.1	20.8 ± 2.2
13	26.4 ± 1.6	26.4 ± 1.7	26.4 ± 1.7	26.4 ± 1.7	26.3 ± 1.7	26.2 ± 1.7	26.1 ± 1.7	25.8 ± 1.7	25.6 ± 1.8	24.3 ± 1.8
14	21.7 ± 2.1	21.7 ± 2.1	21.7 ± 2.1	21.7 ± 2.2	21.6 ± 2.2	21.5 ± 2.2	21.5 ± 2.2	21.3 ± 2.2	21.1 ± 2.2	20.7 ± 2.2
Mean ± SD	26.1 ± 2.3	26.1 ± 2.4	26.0 ± 2.5	25.9 ± 2.5	25.8 ± 2.5	25.7 ± 2.6	25.5 ± 2.6	25.4 ± 2.6	25.2 ± 2.7	24.7 ± 2.7

To assess the impact of undersampling on $\text{a}{\mathrm{T}}_2\mathrm{*}$ quantitatively, $\text{a}{\mathrm{T}}_2\mathrm{*}$ maps corresponding to the data in Figure 6 a–d are shown in Figure 6 e–h. For the 100% sampled $\text{a}{\mathrm{T}}_2\mathrm{*}$ map, mean parenchymal $\text{a}{\mathrm{T}}_2\mathrm{*}$ was 0.44 ± 0.13 ms. As sampling decreased to 70% and 50% (Figure 6 f & g), mean $\text{a}{\mathrm{T}}_2\mathrm{*}$ value remained relatively unchanged, increasing only by 0.2% and 0.5%, respectively. However, in the 10% sampled $\text{a}{\mathrm{T}}_2\mathrm{*}$ map (Figure 6 h), there are regions in the lung parenchymal volume that exhibit both increased (red arrows) and decreased $\text{a}{\mathrm{T}}_2\mathrm{*}$ (red-dashed arrow). Further, the voxel-level impact of undersampling on $\text{a}{\mathrm{T}}_2\mathrm{*}$ is shown in Figure 6 i–k, which shows histograms of parenchymal $\text{a}{\mathrm{T}}_2\mathrm{*}$ across all iterations at sampling levels of 100% (black), 70% (blue), 50% (green), and 10% (red), with the 100% sampled $\text{a}{\mathrm{T}}_2\mathrm{*}$ distribution superimposed at each sampling level (black line). Much like the apparent undersampling trends in the $\text{a}{\mathrm{T}}_2\mathrm{*}$ maps, the overall parenchymal $\text{a}{\mathrm{T}}_2\mathrm{*}$ distributions for 70% and 50% sampled images remained nearly identical to the $\text{a}{\mathrm{T}}_2\mathrm{*}$ distribution from the fully sampled data whereas, the $\text{a}{\mathrm{T}}_2\mathrm{*}$ distribution obtained from the 10% sampled data, skewing subtly, but noticeably, to the right. As the induced undersampling became more severe, this resulted in equivalent edge blurring in the lung volume which led to erroneous $\text{a}{\mathrm{T}}_2\mathrm{*}$ fittings.

Additionally, a strong positive correlation (R²=0.36 – 0.89, p < 0.001) is evident between the aT2* values obtained with full sampling and those acquired with a reduced sampling rate (Figure 6 m–o). Furthermore, the slope of the regression lines is decreasing with deceasing sampling rates. This indicates a systematic bias towards higher $\text{a}{\mathrm{T}}_2\mathrm{*}$ values in the maps acquired with a lower sampling rate compared to the fully sampled ones. Importantly, these undersampling-dependent trends in parenchymal $\text{a}{\mathrm{T}}_2\mathrm{*}$ are consistent across all mice (see Table 2).

Table 2.

Impact of undersampling on Lung Parenchyma aT₂* (ms). P value is < 0.001 for all correlations.

Mouse	100%	90%	80%	70%	60%	50%	40%	30%	20%	10%
Mouse	Sampled mean ±SD	Sampled mean ± SD R2, Slope	Sampled mean ± SD R2, Slope	Sampled mean ± SD R2, Slope	Sampled mean ± SD R2, Slope	Sampled mean ± SD R2, Slope	Sampled mean ± SD R2, Slope	Sampled mean ± SD R2, Slope	Sampled mean ± SD R2, Slope	Sampled mean ± SD R2, Slope
1	0.51 ± 0.16	0.51 ± 0.16	0.51 ± 0.16	0.51 ± 0.16	0.51 ± 0.16	0.51 ± 0.16	0.51 ± 0.16	0.52 ± 0.16	0.52 ± 0.16	0.53 ± 0.17
		0.96, 0.968	0.923, 0.944	0.914, 0.934	0.87, 0.911	0.841, 0.874	0.761, 0.851	0.683, 0.816	0.574, 0.754	0.554, 0.747
2	0.49 ± 0.13	0.49 ± 0.13	0.49 ± 0.13	0.49 ± 0.13	0.49 ± 0.13	0.50 ± 0.14	0.50 ± 0.14	0.51 ± 0.14	0.52 ± 0.14	0.53 ± 0.15
		0.986, 0.986	0.977, 0.971	0.963, 0.96	0.931, 0.924	0.903, 0.902	0.851, 0.888	0.773, 0.846	0.664, 0.802	0.524, 0.736
3	0.40 ± 0.10	0.40 ± 0.1	0.40 ± 0.10	0.40 ± 0.10	0.40 ± 0.10	0.40 ± 0.10	0.40 ± 0.10	0.41 ± 0.10	0.41 ± 0.10	0.41 ± 0.10
		0.858, 0.96	0.912, 0.949	0.87, 0.931	0.901, 0.914	0.884, 0.884	0.825, 0.859	0.745, 0.801	0.64, 0.747	0.483, 0.682
4	0.38 ± 0.09	0.38 ± 0.09	0.38 ± 0.09	0.38 ± 0.0	0.38 ± 0.10	0.38 ± 0.10	0.38 ± 0.10	0.39 ± 0.11	0.39 ± 0.11	0.39 ± 0.11
		0.242, 0.644	0.228, 0.636	0.231, 0.607	0.214, 0.594	0.214, 0.575	0.325, 0.603	0.299, 0.579	0.278, 0.546	0.224, 0.467
5	0.40 ± 0.10	0.40 ± 0.10	0.40 ± 0.10	0.40 ± 0.11	0.40 ± 0.11	0.40 ± 0.11	0.40 ± 0.11	0.41 ± 0.11	0.41 ± 0.12	0.41 ± 0.12
		0.923, 0.975	0.927, 0.95	0.906, 0.922	0.884, 0.886	0.855, 0.849	0.805, 0.806	0.697, 0.744	0.617, 0.662	0.427, 0.544
6	0.44 ± 0.13	0.44 ± 0.13	0.44 ± 0.13	0.44 ± 0.13	0.45 ± 0.13	0.45 ± 0.13	0.45 ± 0.13	0.45 ± 0.13	0.46 ± 0.13	0.47 ± 0.14
		0.987, 0.988	0.982, 0.975	0.967, 0.959	0.953, 0.944	0.921, 0.91	0.866, 0.885	0.816, 0.84	0.712, 0.799	0.588, 0.72
7	0.40 ± 0.10	0.44 ± 0.13	0.44 ± 0.10	0.44 ± 0.13	0.43 ± 0.10	0.44 ± 0.14	0.45 ± 0.11	0.45 ± 0.11	0.45 ± 0.11	0.45 ± 0.14
		0.928, 0.953	0.91, 0.928	0.894, 0.903	0.858, 0.869	0.823, 0.831	0.754, 0.86	0.669, 0.731	0.555, 0.66	0.366, 0.549
8	0.44 ± 0.12	0.44 ± 0.12	0.44 ± 0.12	0.44 ± 0.12	0.44 ± 0.12	0.44 ± 0.12	0.44 ± 0.13	0.44 ± 0.13	0.45 ± 0.13	0.45 ± 0.13
		0.879, 0.945	0.871, 0.92	0.859, 0.894	0.839, 0.866	0.806, 0.835	0.752, 0.792	0.672, 0.746	0.563, 0.675	0.393, 0.567
9	0.42 ± 0.11	0.42 ± 0.11	0.42 ± 0.11	0.42 ± 0.11	0.42 ± 0.12	0.42 ± 0.12	0.42 ± 0.12	0.42 ± 0.12	0.43 ± 0.12	0.43 ± 0.13
		0.816, 0.94	0.809, 0.926	0.813, 0.897	0.792, 0.87	0.756, 0.82	0.703, 0.789	0.598, 0.714	0.565, 0.647	0.365, 0.536
10	0.44 ± 0.13	0.44 ± 0.13	0.44 ± 0.13	0.44 ± 0.13	0.44 ± 0.13	0.44 ± 0.14	0.45 ± 0.14	0.45 ± 0.14	0.45 ± 0.14	0.45 ± 0.14
		0.745, 0.906	0.742, 0.889	0.737, 0.868	0.692, 0.834	0.666, 0.802	0.611, 0.766	0.593, 0.727	0.512, 0.661	0.381, 0.572
11	0.46 ± 0.15	0.46 ± 0.15	0.46 ± 0.15	0.47 ± 0.15	0.47 ± 0.15	0.47 ± 0.15	0.47 ± 0.15	0.47 ± 0.15	0.48 ± 0.16	0.49 ± 0.16
		0.959, 0.985	0.95, 0.975	0.95, 0.971	0.928, 0.95	0.906, 0.933	0.871, 0.915	0.809, 0.876	0.746, 0.86	0.621, 0.792
12	0.44 ± 0.12	0.44 ± 0.12	0.44 ± 0.12	0.45 ± 0.12	0.45 ± 0.13	0.45 ± 0.13	0.45 ± 0.13	0.45 ± 0.13	0.46 ± 0.13	0.47 ± 0.13
		0.924, 0.977	0.903, 0.961	0.891, 0.935	0.874, 0.916	0.856, 0.891	0.803, 0.863	0.749, 0.814	0.661, 0.751	0.493, 0.666
13	0.53 ± 0.15	0.53 ± 0.15	0.53 ± 0.15	0.53 ± 0.15	0.53 ± 0.15	0.53 ± 0.15	0.53 ± 0.15	0.53 ± 0.15	0.54 ± 0.15	0.55 ±0.16
		0.97, 0.978	0.96, 0.964	0.922, 0.943	0.926, 0.915	0.887, 0.882	0.84, 0.842	0.758, 0.809	0.645, 0.745	0.469, 0.642
14	0.44 ±0.12	0.47 ± 0.13	0.47 ± 0.13	0.47 ± 0.13	0.47 ± 0.13	0.47 ± 0.13	0.47 ± 0.13	0.48 ± 0.14	0.48 ± 0.14	0.48 ± 0.14
		0.951, 0.983	0.945, 0.965	0.926, 0.945	0.91, 0.907	0.879, 0.887	0.826, 0.853	0.742, 0.789	0.65, 0.752	0.486, 0.653
Mean ± SD	26.1 ± 2.3	0.44 ± 0.12	0.44 ± 0.12	0.45 ± 0.13	0.45 ± 0.13	0.45 ± 0.13	0.45 ± 0.13	0.45 ± 0.13	0.45 ± 0.13	0.46 ± 0.14

The in-vivo, sampling-dependent changes in parenchymal aSNR and $\text{a}{\mathrm{T}}_2\mathrm{*}$ are shown in Figure 7. For an individual mouse across all iterations of reduced sampling (Figure 7 a), the percent change in aSNR displayed a low variation (i.e., less than ±0.01%) when sampled 50% or more, with the highest variation (±2.5%) measured for the 10% sampled images. Similar changes in mean aSNR across the entire population (Figure 7 b) also systematically decreased with reduced sampling in images. For all mice, mean aSNR decreased by 0.7% in 70% sampled images, 1.6% at 50% sampled, and at most by 5.7% in 10% sampled images.

Figure 7. Sampling Dependent aSNR and $\text{a}{\mathrm{T}}_2\mathrm{*}$ Changes across Individuals.

(a) Sampling level vs percent difference in aSNR (a) and $\text{a}{\mathrm{T}}_2\mathrm{*}$ (c) in an individual mouse. Maximum percent difference for both metrics occurred at 10% sampling where aSNR decreased by 5.2% and $\text{a}{\mathrm{T}}_2\mathrm{*}$ increased by 2.5%. Sampling level vs percent difference in aSNR (b) and $\text{a}{\mathrm{T}}_2\mathrm{*}$ (d) for full population. Again, the maximum percent difference was at 10% sampling, where population aSNR decreased by 5.7% and $\text{a}{\mathrm{T}}_2\mathrm{*}$ increased by 4.0%. The statistically significant divergence (p < 0.05) in $\text{a}{\mathrm{T}}_2\mathrm{*}$ values from fully sampled images is below 80% sampling (red dashed vertical line).

The percent change in $\text{a}{\mathrm{T}}_2\mathrm{*}$ for an individual mouse across all iterations (Figure 7 c) also displayed a low variation (±0.3%) for images sampled 50% or more, with the highest variation (±1.8%) again observed from the 10% sampled. As was seen in the simulations, change in $\text{a}{\mathrm{T}}_2\mathrm{*}$ for the whole population (Figure 7 d), systematically increased as sampling decreased. Across mice, mean $\text{a}{\mathrm{T}}_2\mathrm{*}$ increased by 0.4% in 70% sampled images, 0.9% at 50% sampled, and at most by 4.0% in 10% sampled images. In addition, the scattering of the data across all mice was minimal in images sampled greater than 50%, but as sampling decreased, the spread increased between animal-to-animal became more apparent. Ultimately, images sampled 80% or more yield $\text{a}{\mathrm{T}}_2\mathrm{*}$ measurements that are not significantly different (p > 0.05) than those made using fully sampled images (red dashed vertical line in Figure 7 d).

4. Discussion

Despite reductions in SNR and edge fidelity, radial images are often acquired intentionally undersampled to reduce the relatively long time needed to acquire images. This is particularly true when quantitative image parameters must be extracted (e.g., apparent diffusion coefficients or apparent ${\mathrm{T}}_2\mathrm{*}$ values), because multiple FIDs must be acquired along each radial trajectory. In other cases, undersampled images are generated as a means to discard undesired radial views. For example, in retrospective gating, motion-free images can be generated from free-breathing animals by discarding data acquired during motion-filled portions of the respiratory cycle [33–36]. As such, it is important to understand the exact impact that undersampling will have on quantitative metrics for a range of practical pulmonary imaging applications and sequence optimizations.

The phantom data demonstrate that both aSNR and $\text{a}{\mathrm{T}}_2\mathrm{*}$ are impacted by undersampling. At smaller matrix sizes, undersampling-induced image blurring became noticeable at around 50% sampling. In phantoms, aSNR was similarly reduced (13.5–23.9%), depending on matrix size. Similar patterns were observed in-vivo. Across all animals, mean parenchymal aSNR decreased modestly. However, scatter in the in-vivo quantitative data also increased with the level of undersampling, with variations as high as 7.6% for aSNR.

It is important to distinguish between actual SNR and aSNR (i.e., combining noise and image artifacts) when evaluating the impact of undersampling. That is, the inherent SNR level of the acquired data should not be significantly affected by sampling level. However, the aSNR calculated from reconstructed images often decreases with decreasing sampling. This decrease arises from reconstruction artifacts obscuring the underlying signal, not a true contribution from noise within the raw data. Importantly, the extent of aSNR degradation can depend on the specific reconstruction technique employed, including the use of interpolation, sampling strategy, and k-space symmetry exploitation.

Phantom $\text{a}{\mathrm{T}}_2\mathrm{*}$ was progressively overestimated as the levels sampling was reduced, and the impact of undersampling was more severe at smaller matrix sizes. Mean parenchymal $\text{a}{\mathrm{T}}_2\mathrm{*}$ systematically increased from 0.44 ms at 100% sampled to 0.46 ms at 10% sampled. Note, while our simulated a ${\mathrm{T}}_2\mathrm{*}$ values were slightly higher than the ground truth (0.44 ms vs 0.4 ms) due to noise addition, directly comparing with ground truth would not isolate the effects of factors like sampling and noise on the results. Therefore, the analysis focused on comparing images and $\text{a}{\mathrm{T}}_2\mathrm{*}$ values from undersampled data to fully sampled data, ensuring they underwent the same processing and noise influence.

The level of mean $\text{a}{\mathrm{T}}_2\mathrm{*}$ scatter across simulation iterations was reduced in the higher resolution phantoms (e.g., 128³) compared to the low-resolution phantom (e.g., 64³). Several factors contribute to this reduction. Higher resolution phantoms provide finer spatial detail, leading to more accurate representation of structure, reduced partial volume effects, and thus, more precise $\text{a}{\mathrm{T}}_2\mathrm{*}$ measurements. Higher resolution images also decrease noise influence per voxel, improving the accuracy of the signal decay curve fit and thereby reducing the standard deviation of the measured $\text{a}{\mathrm{T}}_2\mathrm{*}$. The denser sampling that occurs at higher resolution also reduces aliasing and artifacts for a given level of undersampling, leading to more consistent $\text{a}{\mathrm{T}}_2\mathrm{*}$ estimates.

Scatter in the in-vivo quantitative data also increased with the level of undersampling, with variations as high as 4.2% for $\text{a}{\mathrm{T}}_2\mathrm{*}$ in some specific animals at the highest levels of undersampling. Furthermore, a robust positive correlation was noted at the voxel level between $\text{a}{\mathrm{T}}_2\mathrm{*}$ values acquired with full and reduced sampling. However, a decrease in the slope of the regression line was also observed with decreasing sampling, suggesting a potential bias towards higher $\text{a}{\mathrm{T}}_2\mathrm{*}$ values at lower sampling rates. This finding underscores the importance of cautious interpretation when analyzing quotative results obtained under specific sampling and reconstruction conditions. Consequently, the quantitative accuracy of individual voxels becomes progressively uncertain as undersampling increases. Notably, these variations do not reach statistical significance until the sampling percentage drops below 80%.

Study Limitations

This work demonstrates a clear correlation between systematic undersampling and image quality, aSNR, and $\text{a}{\mathrm{T}}_2\mathrm{*}$ systematically, radial increasing as sampling decreases, particularly for smaller matrix sizes. However, additional considerations could arise if other forms of data exclusions are used. For example, if retrospective gating is used to mitigate breathing motion during a free breathing acquisition, variable undersampling levels and patterns will result. Such variability could impact regions of k-space differently and introduce artifacts not present when data are randomly discarded. Future investigations should carefully address the impact of variable undersampling density and mitigation strategies (e.g., compressed sensing) on SNR and ${\mathrm{T}}_2\mathrm{*}$.

This study established relationships between undersampling rates and quantitative ${\mathrm{T}}_2\mathrm{*}$ mapping under specific conditions that had previously be used to generate robust in-vivo data from mouse lungs. However, the generalizability of these findings to images obtained using difference acquisition strategies (e.g., bandwidth and projection order) and reconstruction algorithms was not demonstrated. To facilities broader, systematically analysis of potentially relevant acquisition and reconstruction parameters, the simulation and undersampling code used in this work is made publicly available.

5. Conclusion

We have rigorously investigated the effect of undersampling on radial imaging, specifically its influence on lung parenchyma apparent SNR and apparent ${\mathrm{T}}_2\mathrm{*}$ for both phantom and in-vivo data. This was accomplished by repeatedly undersampling prospectively gated, fully sampled 3D radial images of the murine lung and with modified Shepp-Logan phantoms. The scatter in the data increased noticeably at about 50% undersampling and significantly impacting apparent ${\mathrm{T}}_2\mathrm{*}$ at ≤80% sampling. Thus, for quantitative analysis, we recommend images not exceed these undersampling levels.

• Average parenchymal SNR only slightly decreased by 1.4 even at 90% undersampling.

• Parenchymal ${\mathrm{T}}_2\mathrm{*}$ significantly increased (p>0.05) at less than 80% sampling compared to full sampling.

• Increased scatter in both SNR and ${\mathrm{T}}_2\mathrm{*}$ metrics noticeable at around 50% undersampling.

• Study provides valuable insights into the acceptable degree of undersampling for preclinical lung images with 3D UTE.