Validation of Single Reference Variable Flip Angle (SR-VFA) dynamic T₁ mapping with T₂* correction using a novel rotating phantom

Abstract

Purpose:

To validate single reference variable flip angle (SR-VFA) dynamic T₁ mapping with and without T₂* correction against inversion recovery (IR) T₁ measurements.

Methods:

A custom cylindrical phantom with 3 concentric compartments was filled with variably doped agar to produce a smooth spatial gradient of the T₁ relaxation rate as a function of angle across each compartment. IR T₁, VFA T₁, and B₁⁺ measurements were made on the phantom before rotation, and multi-echo stack-of-radial dynamic images were acquired during rotation via an MRI-compatible motor. B₁⁺ corrected SR-VFA and SR-VFA-T2* T₁ maps were computed from the sliding window reconstructed images and compared against rotationally-registered IR and VFA T₁ maps to determine the percentage error.

Results:

Both VFA and SR-VFA-T2* T₁ maps fell within 10% of IR T₁ measurements for a low rotational speed, with a mean accuracy of 2.3% ± 2.6% and 2.8% ± 2.6% respectively. Increasing rotational speed was found to decrease the accuracy due to increasing temporal smoothing over ranges where the T₁ change had a non-constant slope. SR-VFA T₁ mapping was found to have similar accuracy as the SR-VFA-T2* and VFA methods at low echo times (~<2 ms), whereas accuracy degraded strongly with later echo times. T₂* correction of the SR-VFA T₁ maps was found to consistently improve accuracy and precision, especially at later echo times.

Conclusion:

SR-VFA-T2* dynamic T₁ mapping was found to be accurate against reference IR T₁ measurements within 10% in an agar phantom. Further validation is needed in mixed fat-water phantoms and in vivo.

1. Introduction

The longitudinal magnetization relaxation time constant, T₁, is a fundamental time constant in magnetic resonance imaging (MRI) acquisitions that depends on the local tissue environment and allows MRI to produce excellent soft tissue image contrast. Quantitative T₁ measurements have been used for a variety of clinical indications, such as hypoxia in tumors,^1,2 liver diseases,^3,4,5, pancreatitis,^6,7 heart pathologies,^{8,9, 10,11,12} pulmonary disease,¹³ pulmonary cancer,¹⁴ and beyond.^15,16,17 Furthermore, dynamic quantitative T₁ mapping has been used to measure oxygenation in the brain¹⁸ and lungs,¹⁹ for local perfusion measurements using dynamic contrast enhanced imaging,^20,21 and has been considered for T₁-based thermometry in monitoring thermal therapies.^{22, 23, 24, 25} In these cases, high temporal resolution dynamic T₁ mapping is desired to capture the dynamic processes (contrast perfusion, temperature change) being measured.

The widely accepted gold standard method for T₁ mapping uses inversion recovery (IR) acquisitions, which are slow, even with fast spin-echo acquisitions. To obtain T₁ measurements more efficiently, multiple different acquisition strategies have been developed, including the Look-Locker method²⁶ and its derivatives such as MOLLI²⁷ and shMOLLI,²⁸ saturation recovery methods such as SAPPHIRE,²⁹ and the variable flip angle (VFA) method.³⁰ The VFA method is notable for being a fast T₁ mapping method with good noise efficiency.³¹ However, it requires a minimum of two images per T₁ map, and its accuracy is dependent on a wide range of factors, from flip angle choices and TRs^32,33,34,35 to B₁⁺ mapping^{36, 37} and adequate spoiling^36,38,39. The acquisition time of the VFA method is also increased by the time required to reach a steady state after changing between flip angles.

A speed improvement for the VFA method, called the single-reference variable flip angle (SR-VFA) technique, measures T₁ dynamically using subsequently acquired images at the high flip angle after a reference T₁ map and baseline image at a low flip angle are acquired. By eliminating the need to alternate between flip angles, this method can generate dynamic and volumetric T₁ maps at a high-temporal resolution. While the SR-VFA method has been evaluated in theory and simulation,^35,40 its experimental accuracy has not been validated against gold standard IR T₁ measurements.

The long acquisition time required for an IR T₁ measurement makes it difficult to produce ground truth T₁ measurements for any rapidly changing T₁ situation, such as during contrast injection or focal heating. IR T₁ measurements can only be obtained before and/or after dynamic T₁ changes or at intermediate points if the T₁ can be held constant. Further, it is difficult to obtain highly repeatable T₁ values during these dynamic situations in which the T₁ over time cannot be easily controlled.

To circumvent experimental challenges in evaluating dynamic T₁ mapping protocols, we have developed an innovative rotating cylindrical agar phantom. The phantom was designed to have smooth variation in T₁ values with respect to position by variably doping the agar along concentric rings. When the phantom is stationary, IR methods can be used to accurately measure the ground truth T₁ distribution. When the phantom is rotating, each imaging voxel samples a signal that changes with the spatial variation in T₁, thereby mimicking changes in T₁ that might occur during, e.g., contrast agent passage or focal heating. The SR-VFA method can be used to perform dynamic T₁ measurements during phantom rotation. By rotational registration of the IR T₁ maps to their SR-VFA T₁ counterparts, SR-VFA T₁ mapping can be more accurately evaluated for accuracy against the gold standard, and inferences can be made on the relationship between the accuracy of dynamic T₁ changes and the imaging temporal resolution.

This work tested the validity of the SR-VFA method of T₁ mapping and its recently published T₂* correction³⁵ (hereafter referred to as the SR-VFA-T2* method) against VFA and gold standard IR T₁ maps for both static and “dynamic” acquisitions in an agar phantom. This work will enable future validation studies in vitro and in vivo towards application in fast dynamic T₁ mapping.

2. Theory

A brief overview of the theory of VFA, SR-VFA, and SR-VFA-T2* T₁ mapping is given here for the benefit of the reader, with full derivations and discussions in corresponding referenced work.

2.1. Variable flip angle (VFA) T₁ mapping

VFA T₁ mapping employs 2 or more scans at differing flip angles, and fits the resulting magnitude image intensity values to the steady state spoiled gradient echo signal equation,^30,41,42

$$S=M_0\frac{\left(1-E_1\right)\text{sin}\alpha }{1-E_1\text{cos}\alpha }{E}_2$$ [1],

where $\alpha$ is the flip angle (FA), $E_1=e^{-TR/T_1}$, $E_2=e^{-TE/T_2^{*}}$, and $M_0$ is the equilibrium magnetization. By alternating FAs, a quantitative T₁ map can be generated every 2 image sets. Adequate spoiling and accurate knowledge of FAs are required to produce accurate T₁ maps with this method.^36,43,44,45 Additionally, a priori knowledge of the approximate T₁ range of interest may be required for FA determination when SNR is limited, since the accuracy and precision of VFA T₁ maps are influenced by the choice of FAs relative to the Ernst angle for each voxel.^32,34,35

2.2. Single reference variable flip angle (SR-VFA) T₁ mapping

SR-VFA T₁ mapping extends the VFA method by applying the VFA method twice: once on a set of baseline images, and once on an image set consisting of one baseline image and one “dynamic” image, or an image taken at the new T₁ that is to be measured. By combining the incorrect estimate of the “dynamic” T₁ with the baseline T₁ with an analytic correction, the true dynamic T₁ can be obtained.⁴⁰ This method relies on the same assumptions as the VFA method, with the additional requirement that TE << T₂* and/or that no change in T₂* occurs between baseline and dynamic images.

2.3. Single reference variable flip angle T₁ mapping with T₂* correction (SR-VFA-T2*)

An in-depth simulation/mathematical analysis of the relation between T₁ accuracy and the assumption of negligible T₂* effects has been published previously.³⁵ It was found that SR-VFA T₁ maps suffer from a systematic bias, in which changes in T₁ are underestimated due to changes in T₂* weighting between the baseline and dynamic images. This bias is correctable by incorporating an estimate of T₂* obtained from multi-echo acquisitions into the original SR-VFA T₁ calculation.^{35, 40}

3. Methods

3.1. Setup

Two phantoms were custom designed in SOLIDWORKS (Dassault Systèmes, Waltham, MA) and 3D-printed with PLA filament material. Both phantoms include 3 inner concentric rings, each filled with an agar solution that was doped to give T₁ and T₂ values that varied smoothly around the phantom in each ring (see Figure 1a,c). The dimensions of the rings are given in Table S1, and the 3D model files for both phantoms can be found at https://www.github.com/mikemalm6/SRVFA_Exp_Paper. Each of the three thick inner rings in the rotating phantom were set to contain a specific range of T₁ and T₂ values by varying the dopant amounts of CuSO₄ (Copper sulfate pentahydrate, Acros Organics, Thermo Fisher Scientific, Waltham, MA) and MnCl₂ (Manganese chloride tetrahydrate, Alfa Aesar, Thermo Fisher Scientific, Waltham, MA), similar to the concept used by Gopalan et al.,⁴⁶ except the calibration of each solution’s T₁ value was not performed. In the stationary phantom, the three rings were filled to have initial concentrations that yielded T₁ values in the range of ~200–1500 ms. Due to diffusion of dopant between adjoining compartments and the geometry of the phantom, the T₁ ranges for the innermost to outermost concentric rings were, ~300–900 ms, ~200–1300 ms, and ~200–1400 ms, respectively. The three thick rings in the rotating phantom, from innermost to outermost, included the following approximate T₁ ranges: short (~150–500 ms), moderate (~350–1000 ms), and long (~750–2150 ms). Splitting the whole T₁ range into parts in this way keeps the maximum ratio between two T₁s in a ring at approximately three, which is the amount of T₁ change over which the SR-VFA method was previously analyzed theoretically,³⁵ and which provides a conservative estimate for the maximum T₁ change expected during a focused ultrasound ablative procedure. The outermost thin ring was constructed to have a uniform concentration aimed to emulate similar relaxation characteristics to adipose tissue at body temperature.⁴⁷ This region having a constant T₁ and T₂ relaxation rate could then be utilized as a reference during phantom rotation to more easily identify image artifacts. The concentrations of dopant used to create each phantom compartment is given in Tables S2 and S3.

Figure 1 –

Rotating phantom setup. (A) – 3D-printed phantom model with one outer ring and 3 thick concentric rings with spacers for the first agar filling inserted in alternating sections via slot guides in the print. Fiducial-marker is in the center. The high T₁ region for each ring (i.e., with the lowest dopant concentration) is assigned to the largest section of each ring to maintain the desired T₁ range for a longer period of time (weeks) despite diffusion. Diffusion is intended to smooth out the sharp T₁ transition between sections. (B) – Rotating platform assembly with power cord and encased MRI-compatible motor with separate power box/grounding cable. Phantom is locked in place in the center of the platform via 4 aligning screws along the platform’s edge. (C) Rotating phantom filling procedure. CuSO₄ and MnCl₂ are mixed in custom amounts to produce 5×3 distinct “pre-agar” aqueous solutions, where yellow, red, and blue in the small beakers correspond with the outer, middle, and inner rings of the phantom respectively, and solutions are in order of decreasing T₁ values (i.e., 1 = highest T₁ in ring). Each solution is combined in a 1:3 ratio with hot 1% agar, mixed, allowed to cool to 50°C, and then poured into alternating sections of the phantom as shown above. After the agar sets in the fridge the remaining spacers are carefully removed and the resulting empty sections are filled with their respective agar mixes in the same fashion.

3.2. Filling process

The goal of the rotating phantom was to provide a smooth variation in T₁ and T₂ values with time during phantom rotation. To accomplish this, eight sections were filled in each thick ring with homogeneous doped agar mixtures to create a quasi-sinusoidal T₁ pattern along each ring (see Figure 1c). Natural diffusion gradually smoothed out the sharp transitions between each section over several days. Dopant concentrations for each section were determined using relaxivities given by Thangavel et al.,⁴⁸ and were adjusted so that a 3:1 mix of hot 1% w/v agar and the doped aqueous mix for each section would produce the desired dopant concentration in the resulting 0.75% w/v agar. Alternating sections of alternating rings of the phantom were filled with these mixtures, while in-between sections were blocked using removable 3D-printed spacers (see Figure 1a,c). After the spaced agar solutions had set, the spacers were removed and remaining sections were filled to produce continuous rings of variably doped agar. To prevent temperature-induced warping of the 3D-printed plastic, the agar mixtures had to cool to approximately 50 °C before pouring into each phantom compartment. Further cooling of the thin walls was done by filling the rings that weren’t being filled with the hot agar mix with cold water. The whole phantom was allowed to set in the fridge, and then to equilibrate to room temperature for 8+ hours before scanning. Figure 1c gives a visual description of the filling process for the rotating phantom, and Figure S1 gives a similar description for the stationary phantom.

3.3. Rotating platform

A custom MRI-compatible voltage-controlled electromagnetic motor⁴⁹ was used to actuate a 3D-printed rotating platform. The platform used plastic spheres as ball-bearings to enable the rotation (see Figure 1b). The motor’s speed could be manually adjusted by changing the voltage input. In these experiments, the speed was controlled by switching between 1.25 V, 2.5 V, 3.7 V, and 7.4 V battery sources, thereby achieving rotational speeds at the MRI magnet’s isocenter of 1 rotation per ~254, 119, 83, and 41 seconds (Speeds 1, 2, 3 and 4), respectively. Thus, at the fastest speed, the time to cover the full T₁ range within each ring was ~20 seconds, which is comparable to the temperature rise time during a focused ultrasound ablative procedure. The phantom described above was secured to the center of the rotating platform using plastic aligning screws.

3.4. Experiments

All experiments were performed on a Siemens Prisma Fit 3T MRI scanner (Siemens Healthcare, Erlangen, Germany). Turbo spin echo (TSE) IR measurements with an adiabatic slice-selective inversion pulse were used to determine reference standard T₁ measurements in all cases. Detailed scan parameters are given in Table 1. Separate validation was performed of our TSE IR measurements against values from a calibrated NIST System Phantom (CaliberMRI, Boulder, CO) for the T₁ values relevant to our custom build phantom (T₁ > 120 ms, T₂ > 14 ms). Accuracy (mean ± 2 SD) of the inversion recovery measurements used as reference standard for this work was found to be −0.43 ± 9.95 %, with a maximum deviation of 13.6%.

Table 1:

MRI acquisition parameters

Scan Type	Parameter	Value – Static Exp	Value – Dynamic Exp
All	Field-of-view	256 mm	400 mm
	RF spoiling phase increment	117° (GRE scans) 50° (IR)
	Bandwidth	1184 Hz/pix	501 Hz/pix
	Slice thickness	5 mm (GRE), 10 mm (IR)	2 mm (GRE & IR)
Inversion recovery (turbo spin echo)	TR	10.0 sec	8.0 sec
	TI	[25 75 150 400 900 1500 3500] ms	[25 75 200 500 1000 2000 3000 7000] ms
	Image size	128×128	192×192
	ETL	14
AFI B1+ mapping (GRE)	TR1 / TR2	30 ms / 150 ms
	TEs	2.17 ms	3.0 ms
	Image size	128×128×10	96×96×8
	FA	60°	48°
VFA/SR-VFA (stack-of-radial GRE)	TR	20 ms
	TE	[1.24, 2.80, 4.36] ms	[1.94, 3.81, 5.68, 7.55] ms
	Image size	128×128×10	192×192×8
	FA	[7°, 25°]	[5.5°, 23°]
	# Radial views / view increment	377 / ~68.7533°
	# Repetitions	1	FA 5.5° - 2; FA 23° - 7
	# Averages	1	FA 5.5° - 4; FA 23° - 1

3.4.1. Static experiment

Static scans were acquired to verify the SR-VFA method in the absence of dynamically changing T₁ values. To do this, baseline TSE IR T₁ measurements were performed, followed by pseudo-golden angle (PGA) stack-of-radial⁵⁰ baseline measurements with FAs of 7° and 25°, and B₁⁺ mapping with the Actual Flip angle Imaging (AFI) method.^{36, 52} Following this, PGA stack-of-radial measurements were repeated at each discrete 30° rotation from 0° to 360°, and B₁⁺ maps were acquired after every 90° of rotation to ensure consistency of the B₁⁺ field at each rotational interval.

3.4.2. Dynamic rotation experiment

Additional scans were acquired during rotation of the phantom to determine how accurately SR-VFA T₁ mapping can capture true T₁ even when T₁ within a given voxel changes during the acquisition. As in the static experiment, TSE IR T₁ scans and B₁⁺ mapping with the AFI method were performed before rotation, in addition to baseline PGA stack-of-radial⁵⁰ VFA measurements at FAs of 5.5° and 23°. FAs were independently chosen for each experiment to maximize precision in the SR-VFA method according to (35) for the range of T₁ values present in the phantom and the TR used. PGA stack-of-radial measurements were chosen so that the k-space trajectory repeated itself after a fixed number of lines, thereby allowing for a trajectory-matched baseline comparison for each dynamic image.

Following this, additional radial measurements were acquired at 23° during rotation of the phantom. Rotational position was recorded before and after acquisition, as well as the elapsed rotation time, and then the rotating platform’s voltage was manually switched to acquire equivalent stack-of-radial measurements with the new rotation speed in a similar fashion. This was done for each of the 4 voltages (rotational speeds) mentioned previously. IR measurements were repeated after data acquisition for the first two rotation speeds was complete (halfway through the experiment), and again at the end to validate the phantom’s ground truth T₁ consistency over the experiment’s duration. B₁⁺ mapping was repeated before each dynamic rotation and again post-experiment to ensure B₁⁺ consistency across the various rotational positions.

3.5. Simulation

Simulation was performed to determine the effects of our dynamic measurement setup on T₁ accuracy for our measured IR T₁ data. Simulated signal data was obtained via the spoiled gradient echo steady state signal equation at the FAs used in the dynamic rotation experiment at 1440 × 3 points, corresponding with the 0.25°-spaced ROI positions in each of the 3 rings of the IR T₁ analysis (see next section). Echo times and T₂* values were chosen to approximately match those from the dynamic rotation experiment.

To simulate rotation for the “dynamic” dataset for a single reference T₁ map, the high FA signal data was filtered using a moving average over the number of points corresponding with the angle traversed by the phantom at each of the 4 measured speeds over the time it takes to acquire 55 radial views during the dynamic experiment. Since radial acquisitions cross the center of k-space with each view, each view contributes to the bulk contrast of the image, and thus to the T₁. Then, as a simple approximation, the effect of rotation can be modeled as an average over the rotational angle corresponding with the time to fill “enough” of the center of k-space. To this end, 55 views were chosen, since the data were reconstructed using a sliding window with k-space weighted image contrast (KWIC)⁵¹ that uses a Fibonacci number of views in each KWIC section, and the section with 55 views covers a moderate portion of the center of frequency space. Using this algorithm with pseudo-golden angle acquisition, the k-space measurements near the center of k-space are acquired closely spaced in time with the time footprint increasing for measurements obtained farther from the center of k-space.

SR-VFA-T2* T₁ values were calculated using these signals and compared against the reference IR dataset used in the imaging analysis. The percentage error was determined, and this simulated percentage error was compared against the actual percentage error observed in the data (see next section).

3.6. Image analysis

All images were reconstructed offline using in-house MATLAB code (MathWorks, Natick, MA) that utilized convolution gridding⁵³ and iterative density compensation.⁵⁴ All images contributing to T₁ maps were Hamming-filtered in the frequency domain during reconstruction to limit the effects of noise on the images, and zero-fill interpolated by a factor of 2 to reduce partial-volume effects.⁵⁵ IR T₁ maps were reconstructed with the Reduced Dimension Nonlinear Least Squares algorithm.⁵⁶ VFA T₁ maps were computed on a voxel-by-voxel basis by fitting signal magnitude values to Equation 1.⁴¹ SR-VFA T₁ maps were obtained via the process given by Svedin et al.,⁴⁰ using the “baseline” images acquired at 0° rotation (static experiment) or the pre-rotation images (dynamic experiment) in conjunction with the high FA “dynamic” images acquired at each rotational angle (static experiment) or during rotation (dynamic rotation experiment). Four B₁⁺ maps acquired during each experiment were averaged and applied during T₁ calculation to account for FA errors.

Once all T₁ maps were created, the average T₁ was extracted from circle ROIs of radius 4 voxels at 1440 evenly spaced intervals (every ~0.25°) along the center of each thick concentric ring. These data were then registered to the same starting rotation angle. Following this registration, the stationary T₁ maps’ ROI data (IR and VFA) underwent a simulated “rotation”, during which they were circularly shifted by the number of ROIs corresponding with the angle offset between time frames, as determined by the measured rotational speed and approximate start time of the rotation. Following this simulated rotation, the stationary and dynamic T₁ maps’ ROI data could be compared directly to determine a percentage error. This ROI-based percentage error data was then grouped based on IR T₁ values into ~20 ms bins, and the mean and standard deviation (SD) of T₁ accuracy was determined for each bin. A visual description of this process is shown in Figure 2. To calculate the error in T₁ percentage change as opposed to absolute T₁, the T₁ percentage change from baseline was calculated for each ROI, and the IR results were subtracted from the SR-VFA methods’ results.

Figure 2:

Dynamic rotation phantom experiment processing. Static phantom experiment processing was done similarly to the above, except since rotational position was not changing with time during acquisition, the 13 discrete rotation states (0 to 360° in 30° increments) replaces the time dimension on the x-axis. Additionally, in the static experiment, only IR measurements needed rotation via circular ROI shifting, as VFA measurements were acquired at every 30° increment. *Inputs to step 3 for the dynamic rotation phantom experiment were direct measurements, with the exception of rotation start time, which was estimated based on a start time within a few seconds of 2×377 stationary views.

4. Results

4.1. Accuracy and Precision

Figure 3 shows the binned T₁ accuracy and precision of the SR-VFA, SR-VFA-T2*, and VFA methods for each ring of the phantom relative to IR T₁ measurements in the static and dynamic rotation experiments. The first echo’s results are shown in both cases (TE = 1.24 ms and 1.94 ms, respectively), and Figure 3d–f’s results were taken from the lowest rotation speed acquisition (~254 sec/rev). The T₁ accuracy of all methods fell approximately within ± 10%, with the stationary VFA method achieving slightly better accuracy and precision than the SR-VFA method. The T₂* correction improved the accuracy and precision of the SR-VFA T₁ measurements to bring the time-normalized precision to be comparable to the VFA method. Notably, the T₂* correction most strongly improved the SR-VFA T₁ accuracy and precision in the inner ring, in which the lowest T₁ and T₂ values were found. Oscillations are also seen in the accuracy plots for the dynamic single-reference T₁ measurements (Figure 3d,e) that are not found in those for the stationary measurements (Figure 3a,b).

Figure 3:

Accuracy of SR-VFA, SR-VFA-T₂*, and dual flip angle VFA T₁ mapping relative to inversion recovery T₁ values for the static experiment (A-C) and dynamic rotation experiment (D-F). Data were obtained from the mean of T₁ values within ROIs, as described in the Methods section. Precision is given as ± 2 SD for the VFA method, and ± √2 SD for the single reference methods to normalize for dynamic acquisition time per T₁ map. Red horizontal lines at ± 10% are given as points of reference.

Figure 4 shows an overall comparison of the mean T₁ accuracy and precision across all phantom rings and T₁ values for each experiment, T₁ mapping method, and echo time. With the SR-VFA method, the average absolute T₁ percentage error and precision worsened with echo time and with rotation speed. The SR-VFA-T2* method improved the accuracy and precision in all cases and produced consistent T₁ maps across the echo time dimension. At the lowest rotation speed (Speed 1), the SR-VFA-T2* method was comparable in accuracy and precision to the stationary VFA data. The static experiment showed similar results, with the SR-VFA-T2* T₁ maps approaching the accuracy and precision of the VFA T₁ maps. As in the other figures in this work, the SR-VFA and SR-VFA-T2* precision is given as sqrt(2) times the SD as opposed to 2 times the SD for the VFA method to account for the dynamic acquisition speed improvement afforded by the single reference methods.

Figure 4:

Mean Absolute T₁ % error and mean precision for the SR-VFA, SR-VFA-T2*, and VFA T₁ mapping methods across the entire T₁ range and all rings for the static and dynamic experiment. Precision is given as ± 2 SD for the VFA method, and ± √2 SD for the single reference methods to normalize for dynamic acquisition time per T₁ map. (A) – Dynamic Rotation Experiment: Rotation speeds from lowest to highest are shown with increasing echo times in each group. One set of stationary VFA data was acquired before each rotation, and these data were all combined for the VFA column. (B) – Static Experiment. (A&B) – gray bar on the left of the SR-VFA-T2* groupings in each experiment indicates the combined echoes’ T₁ map, as described in (35).

While this work is primarily focused on absolute T₁ accuracy, the accuracy of percentage changes in T₁ was also analyzed for the SR-VFA-T2* method relative to IR T₁ values. It was found that for the range in which T₁ changed by less than a factor of 3 (T_1,dynamic / T_1,BL ∈ [0.33 3]), the error in percentage T₁ change (mean ± 2 SD of SR-VFA-T2* T₁ percentage change minus IR T₁ percentage change) was 0.88 ± 5.28% for the stationary experiment and 0.01 ± 6.38% for the dynamic rotation experiment at the lowest rotational speed. Additional details can be found in Figure S2.

4.2. SR-VFA vs. SR-VFA-T2*

Figures 5 and 6 gives the mean binned T₁ accuracy of the SR-VFA method versus the SR-VFA-T2* method for each echo time in the static and dynamic rotation experiments respectively, grouped by phantom ring and by the relationship between the baseline (pre-rotation) T₁ and the “dynamic” (post-rotation for static exp., during-rotation for dynamic exp.) T₁. Each marker represents a different range of the ratio $\frac{T_{1,\mathit{\text{dynamic}} }}{T_{1,\mathit{\text{baseline}}}}$.

Figure 5:

SR-VFA mean T₁ measurement accuracy and precision before (top row) and after T₂* correction (bottom row) for the static experiment. Each column is a different echo time, (TE = [1.24, 2.8, 4.36] ms). The overall accuracy is improved after application of the T₂* correction. Different marker types indicate the relationship between the dynamic mean T₁ (T_1,dyn) and the baseline mean T₁ (T_1,BL) for a given ROI.

Figure 6:

SR-VFA mean T₁ measurement accuracy and precision before (top row) and after T₂* correction (bottom row) for the dynamic rotation experiment. Each column is a different echo time, (TE = [1.94, 3.81, 5.68, 7.55] ms). In all cases, the accuracy is improved after application of the T₂* correction. Different marker types indicate the relationship between the dynamic mean T₁ (T_1,dyn) and the baseline mean T₁ (T_1,BL) for a given ROI.

Figure 6 also indicates that the SR-VFA method underestimates changes in T₁ relative to the T₁ at the baseline acquisition. For points at which $\frac{T_{1,\mathit{\text{dynamic}} }}{T_{1,\mathit{\text{baseline}}}}$ < 75%, as shown by the dot markers in Figure 6, the SR-VFA method overestimated the true T₁ values by up to ~30%. When $\frac{T_{1,\mathit{\text{dynamic}} }}{T_{1,\mathit{\text{baseline}}}}$ > 133%, as shown by the open circle markers in Figure 6, the SR-VFA method underestimated the true T₁ values by up to ~10%. Echo time had only a minor effect on the T₁ accuracy when $\frac{T_{1,\mathit{\text{dynamic}} }}{T_{1,\mathit{\text{baseline}}}}$ was closer to 1. In both experiments accuracy decreased with increasing echo time. Additionally, the bias increased with echo time most strongly in the inner ring, where both T₁ and T₂ values are the smallest. Similarly, when the T₂* correction was applied, the inner ring experienced the largest change, though accuracy improved to varying degrees across the entire T₁ range. As shown in Figures 4–6, even just two later echoes can be sufficient to provide a T₂* estimate sufficient for correction, though the T₂* correction improves with a better estimate of T₂* (e.g., by using more echoes).

4.3. Averaging effect – acquisition speed vs accuracy

A comparison between the simulation and measured data for the SR-VFA-T2* method is shown in Figure 7. The moving average effect of radial acquisitions produces an alternating over- and under-estimation at locations in the phantom where the concentration of dopants has not diffused sufficiently for a smooth T₁ transition. This produces the oscillations shown in Figures 3 and 6, as proven by the matching T₁ positions of the oscillations in Figures 7b and 7c. Increasing rotation speed increases the T₁ smoothing effect, and thus increases the amplitude of the oscillations in Figures 7b and 7c. After taking the average difference between the measured and simulated SR-VFA-T₂* T₁ percentage error measurements, a residual trace remains in Figure 7d that bears strong resemblance to the VFA mean T₁ accuracy traces found in Figure 3f.

Figure 7:

Simulation of dynamic rotation effects on SR-VFA-T2* T₁ accuracy. Simulated signals were generated by inputting measured IR T₁ data (solid black traces, plot A) into the steady state spoiled gradient echo signal equation and applying moving averages to simulate phantom rotation (see Methods section 3.5). SR-VFA-T2* T₁ values were calculated from these signals. Colors denote different phantom rings, and line styles in plots A-C correspond to different rotational speeds. (A) - Mean T₁ trace from ROIs around each phantom ring from inversion recovery data and SR-VFA-T2* simulated data at each rotational speed. (B & C) – Binned mean % error of SR-VFA-T₂* T₁ values at each rotational speed vs inversion recovery T₁ values from the simulated data (B) and measured data from the dynamic rotation experiment (C). (D) – Difference plot between lines in B and C averaged over all speeds. The moving average effect of radial acquisitions combined with the nonlinear T₁ gradient in the phantom is shown to produce the oscillations seen in single reference methods’ T₁ accuracy.

4.4. Measurement Stability

Figures 8 and 9 show the precision of the B₁⁺ mapping and signal across rotational positions respectively. The B₁⁺ precision over the voxels covering the center of each ring was found to be (mean ± 2 SD of the histograms’ data in Figure 8) 0.014 ± 0.011 (max 0.031) in the stationary experiment and 0.012 ± 0.010 (max 0.028) in the dynamic rotation experiment. The worse precision found in the middle ring of the static experiment can also be seen in the slightly worse T₁ precision for the VFA method’s middle ring (Figure 3c). In addition, the bright spots (higher SD) in the inner ring for the dynamic rotation experiment coincide with the positions of the low T₁ portion of the ring across the four B₁⁺ maps acquired.

Figure 8:

2x SD of B₁⁺ across the 4 B₁⁺ maps acquired in the static (A) and dynamic rotation (B) phantom experiment. B₁⁺ maps were acquired at [0°, 90°, 180°, 270°] in the static experiment (A) and at [60°, 149.5°, 237°, 303.3°] in the dynamic rotation experiment (B). Each histogram in (A) and (B) is normalized as a probability density function. Note the overall higher SD in the middle ring of the static experiment and the inner ring of the dynamic experiment.

Figure 9:

Signal variation across successive rotations of the phantom during the high flip angle scan for Speed 3. ~4 full rotations are included, with signal data taken from a single circular ROI at the top of each ring. (A) – Precision of signal vs rotational position; (B) – Coefficient of variation box-and-whisker plot for each ring. Signal from one rotation to another is highly reproducible, with higher coefficient of variation in the outer rings due to higher T₁ values in the outer rings, which produces lower signal values in the GRE acquisition.

5. Discussion

This study is the first, to the authors’ knowledge, to validate a T₁ measurement method by direct comparison of dynamic quantitative T₁ maps to gold standard IR T₁ maps. This comparison was made possible by our design of a variably-doped cylindrical phantom combined with motorized rotation during image acquisition. In this way, accuracy could be compared simultaneously for a broad range of T₁ values experiencing a broad range of T₁ changes from their baseline values.

The rotating cylinder concept used in this work provides a repeatable model to validate dynamic T₁ measurements against the gold standard. T₁ values measured on a run-to-run basis are less repeatable during a contrast injection or focal heating due to changing tissue properties or difficulty of precise control of contrast injection or heating versus time at every point. However, the design concept used in this work allowed for repeatable highs and lows in signal and T₁ over acquisition time and allowed for these to be captured at speeds varying between ~0.24 and 1.5 rotations per minute.

It was shown that SR-VFA T₁ accuracy degrades with increasing echo time, such that changes in T₁ from the baseline T₁ value are underestimated. However, these inaccuracies were found to be correctable through the application of the T₂* correction published previously, as shown in Figures 5 and 6.³⁵ Additionally, it was shown that a substantial correction can be performed even with two later echoes, such that the resulting T₁ is nearly as accurate as the T_s’s obtained without correction at the early echo times. Thus, the SR-VFA-T2* T₁ mapping method, with only a single dynamic image per T₁ map, could produce T₁ values within 10% of gold standard IR measurements. Similar accuracy was obtained for changes in T₁ (see Figure S2). Further, this study found that with the SR-VFA-T2* method, the combined echoes’ estimate of T₁ had approximately equivalent accuracy and precision to that of the earliest echo used in the correction. The lack of improvement after combining echoes could be due to the correlation present between different echo times’ corrected T₁ maps,³⁵ and decreasing SNR with increasing echo time.

In the rotation experiment, we found that temporal averaging caused a predictable over- and under-estimation of true T₁ at positions where the T₁ changed with a non-constant slope. The oscillations in the T₁ error curves in Figure 3 appear to be highly repeatable and appear to be caused by the fact that there is a rapidly changing slope of T₁ values around each ring in the cylinder, as indicated by the T₁ variation shown in Figure 7. In implementing a radial sliding window KWIC reconstruction, the measurements at the center of k-space are effectively averaged over a short time interval. When the T₁ value changes with a non-constant slope during that interval, the calculated T₁ from the resulting images may not match the true T₁ value, the T₁ at the center of the averaging window. This effect manifested as oscillations in the T₁ error curves in Figures 3d–f, 6, and 7, and was visible across all rotation speeds, as shown in Figure 7. This temporal averaging oscillatory effect was simulated and removed in Figure 7b–d, and the strong similarity between Figure 7d and the stationary VFA mean T₁ accuracy traces in Figure 3f possibly indicates that any other bulk-motion-induced artifacts in our experiments were small in comparison with the effects of temporal blurring. As expected, increasing the speed of rotation had a negative impact on the T₁ accuracy and precision, as shown in Figure 4 and Figure 7. We note that this oscillatory effect is intrinsic to our method of phantom preparation. However, this importantly indicates that for situations where T₁ may change with non-constant slope over the measurement window, especially the time window for the acquisition of the center of k-space, accurate dynamic T₁ mapping requires that either 1) the scan trajectory and acquisition parameters be chosen to shrink the measurement window until the T₁ changes over that time can be reasonably approximated as linear, else 2) other compensatory techniques that incorporate prior knowledge about the dynamics of the T₁ changes are used. Condition 1 was approximately met for the lowest rotation speed (see Figure 7a, solid-colored lines), and thus the SR-VFA-T2* method’s T₁ accuracy and time-normalized precision were comparable to the stationary VFA T₁ measurement for this case. With the use of a faster KWIC-compatible trajectory than the stack-of-radial measurements used in this work, accuracy in the higher rotation speeds would likely have been improved.

Since a low rotation speed is analogous to slow contrast injection or heating, both of which may be unacceptable, it is clear that fast, efficient trajectories will be needed in addition to the speed improvements afforded by SR-VFA T₁ mapping. In the case of focused ultrasound ablation, where T₁ changes in fat over the course of a ~20 second sonication could be used in combined PRF/T₁ thermometry, a faster sequence than that used in this work would be required. However, this work indicates that given sufficient time resolution, SR-VFA-T2* T₁ mapping can produce accurate, precise T₁ values from a single dynamic image per dynamic T₁ map.

Besides the effects of rotation just described, B₁⁺ inaccuracies could account for most of the remaining T₁ inaccuracy, due to our assumption of constant B₁⁺ despite phantom rotation and due to the use of the AFI method with TR₂ close to the minimum T₁ being measured. For evaluation of the first assumption, Figure 8 shows the SD of the measured B₁⁺ field throughout each experiment. It indicates that the B₁⁺ maps remained approximately constant, with the precision between the four B₁⁺ maps acquired at different positions being (mean ± 2 SD of the histograms’ data in Figure 8) 0.014 ± 0.011 for the static experiment and 0.012 ± 0.010 for the rotation experiment over the center of each phantom ring. A 1% error in B₁⁺ produces approximately a 2% error in T₁,⁴³ so thus the error associated with noise in our B₁⁺ mapping and/or slight B₁⁺ changes with phantom rotation could account for errors in T₁ accuracy as large as ~3–5% in the static experiment and ~2–5% in the dynamic rotation experiment. In regards to the accuracy of our AFI B₁⁺ mapping implementation, we found a notable decrease in T₁ accuracy at the lowest of T₁ values (~150 ms) in the dynamic experiment, and 150 ms was used as the TR₂ in our AFI B₁⁺ mapping implementation. The AFI method is known to begin to underestimate B₁⁺ when T₁ is equal to or shorter than TR₂,⁵² and thus it is possible that B₁⁺ inaccuracy was systematically greater in this region, leading to a larger than normal overestimation of T₁. This could be the reason for the bright spots on the inner ring for the dynamic rotation experiment, given that the low T₁ values’ positions during each B₁⁺ map acquisition line up well with the bright spots’ locations. However, even in these regions, T₁ errors remained within ± ~10% of inversion recovery measurements. while the B₁⁺ field was approximately constant over the course of these experiments, respiratory and other motion during in-vivo experiments could cause there to be a different B₁⁺ field between the baseline and dynamic acquisitions. The effects of this kind of B₁⁺ variation have not yet been studied and warrant further investigation in in vivo studies.

During the rotational registration of the ROIs for the SR-VFA to VFA and IR T₁ comparisons for the dynamic rotation experiment, the time of beginning rotation was adjusted to minimize the error between IR T₁ and SR-VFA T₁ values. This was done due to the difficulty of determining the precise start time of rotation relative to the acquired k-space lines. Because of this, it is possible there is a systematic measurement error in opposite directions for increasing versus decreasing T₁ values within the phantom’s rotation in this study’s favor. However, the similarly high level of agreement between the VFA, SR-VFA-T2*, and IR T₁ measurements in the static experiment implies that this adjustment of rotation time was close to accurate.

Though this study validated SR-VFA-T2* T₁ mapping in an agar phantom against IR T₁ mapping directly, the in vivo case will likely not have such a direct comparison, and will have the added difficulty of mixed fat and water voxels. The phantom in this work did include a large portion of the physiological range of T₁ in body tissues, but since VFA T₁ mapping is sensitive to partial volume effects when fat and water are both present in imaged voxels, the derived SR-VFA and SR-VFA-T2* T₁ mapping methods are also sensitive to these. Future studies and in vivo work will require fat-water separation techniques such as Dixon,⁵⁷ IDEAL,⁵⁸ or others⁵⁹ to achieve optimal results.

In summary, this work represents the beginning of validation for the SR-VFA method in aqueous materials. Further study is needed for validation of the SR-VFA methods in fat and other non-aqueous tissues such as bone.