Rapid 2D variable flip angle method for accurate and precise T ₁ measurements over a wide range of T ₁ values

Abstract

Purpose

To perform dynamic T ₁ mapping using a 2D variable flip angle (VFA) method, a correction for the slice profile effect is needed. In this work we investigated the impact of flip angle selection and excitation RF pulse profile on the performance of slice profile correction when applied to T ₁ mapping over a range of T ₁ values.

Methods

A correction of the slice profile effect is proposed, based on Bloch simulation of steady‐state signals. With this correction, Monte Carlo simulations were performed to assess the accuracy and precision of 2D VFA T ₁ mapping in the presence of noise, for RF pulses with time‐bandwidth products of 2, 3 and 10 and with flip angle pairs in the range [1°‐90°]. To evaluate its performance over a wide range of T ₁, maximum errors were calculated for six T ₁ values between 50 ms and 1250 ms. The method was demonstrated using in vitro and in vivo experiments.

Results

Without corrections, 2D VFA severely underestimates T ₁. Slice profile errors were effectively reduced with the correction based on simulations, both in vitro and in vivo. The precision and accuracy of the method depend on the nominal T ₁ values, the FA pair, and the RF pulse shape. FA pairs leading to <5% errors in T ₁ can be identified for the common RF shapes, for T ₁ values between 50 ms and 1250 ms.

Conclusions

2D VFA T ₁ mapping with Bloch‐simulation‐based correction can deliver T ₁ estimates that are accurate and precise to within 5% over a wide T ₁ range.

2D imaging could improve the temporal resolution of T ₁ mapping, but T ₁ values need to be corrected for the errors due to the non‐uniform slice excitation. We propose an approach for dynamic 2D T ₁ mapping with a look‐up table correction, and we evaluate its performance over a wide range of T₁ values.

Abbreviations

DCE

dynamic contrast enhanced

FA

flip angle

GRE

gradient echo

IR

inversion recovery

LUT

look‐up table

MEP

maximum error projection

MRF

MR fingerprinting

PDw

proton density weighted

ROI

region of interest

SNR

signal to noise ratio

SPGR

spoiled gradient recalled

TBP

time‐bandwidth product

T _R

repetition time

VFA

variable flip angle

1. INTRODUCTION

An important parameter for MRI studies is the longitudinal relaxation time, T ₁. It is an intrinsic tissue property, the value of which depends on the structure of the tissue at a molecular level. Quantitative T ₁ mapping may help to non‐invasively characterize and discriminate biological tissues. This is of interest for various organs comprising the heart,¹, ² liver and kidney,² and brain.³ Methods based on 3D gradient echo (GRE) acquisitions at either multiple flip angles (FAs) or multiple repetition times (T _R) are used for T ₁ mapping.⁴ These techniques are usually faster than more conventional T ₁‐mapping sequences based on inversion recovery (IR).⁵, ⁶

Some applications require rapid time‐resolved T ₁ mapping. In T ₁‐based MR thermometry, it is used for temperature mapping, e.g. for monitoring of thermal therapy.⁷ In dynamic contrast‐enhanced (DCE) MRI, it provides input for pharmacokinetic analysis of contrast agent uptake and washout.⁸, ⁹ For these applications, T ₁ values may change considerably and rapidly. These changes are monitored through a series of fast repeated sequences in the same scan. For example, for T ₁‐based MR thermometry in fat during ablation procedures, T ₁ values range between 200 and 600 ms.¹⁰ For other DCE imaging applications, upper T ₁ values of approximately 1,200 ms have been reported.¹¹, ¹², ¹³ For such applications, accurate and precise T ₁ mapping with an adequate temporal resolution over a wide dynamic range of T ₁ values is desired. Here, conventional 3D GRE‐based methods are of limited use, because the temporal resolution they offer is typically insufficient. However, the methods can be sped up considerably by switching to 2D acquisition.¹⁴, ¹⁵

In 2D, the non‐uniform slice excitation profile causes errors in T ₁ estimation, as has been shown for the dual‐T _R T ₁‐mapping method,¹⁴ and for the variable flip angle (VFA) method.¹⁵ Parker et al proposed a method for simultaneous correction of slice profile effects and B ₁ inhomogeneity effects for 2D T ₁ mapping with dual‐T _R GRE, which uses T ₁‐weighted and proton‐density‐weighted (PDw) images.¹⁴ Their correction was based on the time domain simulation of the slice profile, for a range of nominal (i.e. requested by the user on the scanner interface) FAs and a realistic RF‐pulse shape. Their method, implemented for accurate multi‐slice scans, was not designed for rapid T ₁ mapping: their scan duration was relatively long because of the long repetition times (T _R = 1500 ms) needed for the PDw scans.

For 2D VFA T ₁ mapping, which uses spoiled gradient recalled (SPGR) images acquired at different FAs, the slice profile and B ₁ inhomogeneity effects were simultaneously corrected with a dual‐FA approach combined with a FA scaling map by Svedin and Parker¹⁵ and with a three‐FA implementation and a B ₁ map acquisition by Dieringer et al.¹⁶

However, the ability of these methods to correct the slice profile effect for dynamic applications, where T ₁ will vary, has not been investigated.

In this paper, we propose a correction of the slice profile effect for dynamic T ₁ mapping applications, based on a look‐up table (LUT) of nominal T ₁ values and simulated apparent T ₁ values. Here, we investigated the performance of 2D VFA T₁ mapping including the correction over a wide range of T ₁ values. Moreover, we will study the impact of FA selection and excitation RF profile on the performance of the method, in terms of accuracy and precision.

2. THEORY

2.1. T₁ mapping using the VFA method

The conventional 3D VFA T ₁ mapping method was proposed by Homer and Beevers.¹⁷ It is based on a 3D RF spoiled gradient echo sequence, the steady‐state signal of which is described by

$$S\propto M_0\frac{\left(1-E_1\right)\sin \theta }{1-E_1\cos \theta }{E}_2^{*}$$ (1)

where M ₀ is the equilibrium magnetization, θ the FA, E ₁ is defined as exp(−T _R/T ₁), and $E_2^{*}$ as $\exp \left(-T_{\mathrm{E}}/T_2^{*}\right).$ ¹⁸

Equation 1 can be written in a linearized form4, 19

$$\frac{S}{\sin \theta }=E_1\frac{S}{\tan \theta }+M_0\left(1-E_1\right),$$ (2)

which allows us to estimate T ₁ from the signal measurements at multiple FAs, by determining the slope k of S/sin θ versus S/tan θ to obtain E ₁.

To determine the slope, measuring the signal for at least two FAs is required. The choice of FAs will influence the performance of the method. It has been shown empirically⁴ and analytically²⁰ that an FA pair can be found that maximizes T ₁ measurement precision.

2.2. Correction of 2D VFA T ₁ mapping

Truncation and apodization of RF pulses lead to deviations from the optimal rectangular slice profile,¹⁴, ¹⁵ causing non‐uniform excitation in the slice selection direction. In 3D VFA, this problem is solved by phase encoding in the slice direction and excluding the partitions near the edges of the slab. In 2D, however, these slice profile effects will lead to errors in T ₁ values estimated using the VFA method.¹⁵

We propose an approach to estimate 2D VFA T ₁ maps corrected for the non‐uniform excitation effect, considering the full slice profile. We neglect partial volume effects by assuming that each voxel can be assigned a single T ₁ value. The response of the magnetization to an excitation pulse can be described by Bloch's equations.¹⁸ The available magnetization at an angular frequency ω′ in the rotating frame of reference, represented by M(ω′, t),is initially aligned with the B ₀ field, along the z‐axis (M ₀ = (0,0,1)^T). Then, it is rotated by the B ₁ field of the RF pulse towards the transverse plane over a certain FA θ. For a circularly polarized RF field in the transverse plane, defined as ${\mathbf{B}}_1=B_1\left(t\right)\widehat{\mathbf{y}}\prime$, the motion of the magnetization in the rotating frame is described by

$$\frac{d\mathit{M}\left({\omega }^{\prime },t\right)}{\mathit{dt}}=\gamma \mathit{M}\left({\omega }^{\prime },t\right)\times \mathbf{B}\left({\omega }^{\prime },t\right)$$ (3)

where

$$\mathbf{B}\left({\omega }^{\prime },t\right)={\left(0,B_1\left(t\right),\frac{{\omega }^{\prime }}{\gamma }\right)}^{\mathrm{T}}$$

(neglecting T ₁ and T ₂ decay during the RF pulse). Thus, for a specific RF pulse (B ₁(t)), the magnetization is obtained by integrating Equation 3 over time. From the resulting magnetization, the actual FA θ(ω′) at a frequency ω′ is calculated:

$$\theta \left(\omega \prime \right)={\tan }^{-1}\frac{\sqrt{M_x{\left({\omega }^{\prime }\right)}^2+M_y{\left({\omega }^{\prime }\right)}^2}}{M_z\left({\omega }^{\prime }\right)}$$ (4)

where M _x(ω′), M _y(ω′) and M _z(ω′) are the x, y and z components of the magnetization vector. The phase φ(ω′) of the transverse magnetization for that frequency ω′ is given by

$$\phi \left(\omega \prime \right)={\tan }^{-1}\frac{M_y\left({\omega }^{\prime }\right)}{M_x\left({\omega }^{\prime }\right)}.$$ (5)

Finally, the complex demodulated steady‐state signal is obtained by discrete summation of all magnetization vectors over a frequency range six times the bandwidth of the pulse, to include all the relevant side lobes:

$$\widehat{S}=M_0{\sum }_{n=1}^N\frac{\left(1-E_1\right)\sin \theta \left({\omega }_n^{\prime }\right)}{1-E_1\cos \mathrm{\theta }\left({\omega }_n^{\prime }\right)}{\mathrm{e}}^{\mathrm{i}\phi \left({\omega }_n^{\prime }\right)}$$ (6)

where N is the number of frequency bins.

To determine the relation between nominal (ie prescribed as input) T ₁values and T ₁ estimated with 2D VFA mapping, simulations were performed for a range of nominal FAs and for three commonly used RF pulse shapes. For an input T ₁value and two nominal FAs, the steady‐state signals were calculated with Equation 6 and used to find the apparent longitudinal relaxation time $\widehat{T_1}$ using the conventional VFA T ₁ estimation applied to the 2D VFA data, from the slope in Equation 2. In this way, $\widehat{T_1}$ includes the effects of the non‐uniform slice excitation in the case of 2D acquisition (Figure 1I). Finally, by repeating the simulation for a range of T ₁values an LUT was created for $\widehat{T_1}$ versus the nominal T ₁. In the correction step of our method, this table is used to relate measured $\widehat{T_1}$ values to the nominal T ₁ values. The LUT needs to be computed for the T _R value and RF shape used in the VFA pulse sequence.

FIGURE 1.

I. Flowchart of building the T ₁ LUT. For each nominal T ₁ the steady‐state signal is simulated for two nominal FAs. From the simulated signals, the apparent $\widehat{T_1}$ is computed using the VFA method. The values of apparent $\widehat{T_1}$ are then inserted in the table for comparison with the corresponding nominal T ₁. II. LUT structure and correction procedure. The LUT is indexed by the FA pair and nominal T ₁. LUT entries are apparent T ₁ for each combination of FA₁, FA₂ and nominal T ₁. In the case of B ₁ ⁺ correction the nominal FA is scaled to actual FA in the voxel. At this FA pair (Arrow A) the apparent T ₁ is matched to entries of the LUT and the corresponding nominal T ₁ is looked up (Arrow B)

3. METHODS

We build the LUTs of T ₁ values for three commonly used RF pulse modulations (five central lobes of a SINC, asymmetric lobe of a SINC, Gaussian). To maintain a comparable slice thickness, all RF pulses had the same bandwidth and different durations. Each RF pulse had a specific time‐bandwidth product (TBP): the SINC pulse had TBP 10, the asymmetric SINC pulse TBP 3 and the Gaussian pulse TBP 2 (Figure 2).

FIGURE 2.

RF pulse shapes considered: Gaussian RF pulse, TBP 2 (A), asymmetric lobe of a SINC pulse, TBP 3 (B), and five central lobes of a SINC pulse, TBP 10 (C). In the simulations, slice thickness and RF bandwidth were maintained constant, whereas RF pulse duration was changed

The LUT is a multidimensional array containing the calculated apparent T ₁ values, ie the T ₁ values that would be measured in the presence of the slice profile effect, indexed by FA₁, FA₂ and nominal T ₁ (Figure 1II). FA values ranged between 1° and 90° with 1° step size and the nominal T ₁ between 50 and 1500 ms, with 1 ms steps, yielding a total of 1450 values. The LUT had 90 × 90 × 1450 entries, but it could be expanded to include T _R and/or TBP, if necessary. To build the LUT, simulations were performed with T _R 10 ms, slice thickness 7 mm, for every combination of two FAs, namely FA₁ and FA₂, between 1° and 90°, with 1° steps; the MR parameter settings in the simulation were as in the MR experiments (see Table 1). The RF duration, RF B ₁ amplitude and slice selection gradient strength were obtained by taking the values used by the MRI scanner for FA = 90° and subsequent rescaling for the FA considered.

TABLE 1.

Imaging protocol for 2D and 3D VFA scans. The VFA sequence consists of two SPGR scan at FAs 6°and 40°

2D VFA scan
Type	SPGR
Scan mode	2D
RF pulse	SINC Gauss, TBP 3
T R	10 ms
T E	4.6 ms
FA	6°, 40°
Water fat shift	1.6 pixels
Dynamic scan duration	2.2 s
Dummy scans	200
FOV	112 × 112 mm2
Voxel size	2 × 2 × 7 mm3
RF‐spoiling phase increment	117°

3D VFA scan
Type	SPGR
Scan mode	3D
RF pulse	SINC Gauss, TBP 10
TR	10 ms
TE	4.6 ms
FA	6°, 40°
Water fat shift	1.6 pixels
Number of slices	7
FOV	112 × 112 mm2
Voxel size	2 × 2 × 7 mm3
RF‐spoiling phase increment	117°

3.1. Applicability of the linearization in the 2D VFA method

The VFA method is based on the linearization of the steady‐state signal equation (Equation 2) to estimate T ₁. In order to investigate whether this assumption still holds with non‐uniform slice excitation, the behavior of S/sin θ versus S/tan θ as a function of the FAs used (ie between 1° and 90°) was studied for nominal T ₁ values from 50 ms to 1250 ms. The signals for three RF shapes at different nominal T ₁ values was simulated, as described in Section 2.2. No correction with T ₁ LUTs was applied at this stage.

3.2. Performance of 2D VFA T ₁ mapping over a range of FAs

In order to assess the dependence of accuracy and precision of 2D VFA T ₁ mapping with our proposed correction on the FA choice, white Gaussian noise was added to the simulated 2D complex steady‐state signals, with standard deviation of the noise distribution equal to 1 and the equilibrium magnetization M ₀ equal to 1000, resulting in a realistic signal to noise ratio (SNR) of 30‐150 for T ₁ = 250 ms. T ₁ values were then computed from the noisy signals varying both FAs in the range from 1° to 90° with 1° increments. The simulation was repeated 1000 times with T _R = 10 ms for six T ₁ values equal to 50 ms, 250 ms, 500 ms, 750 ms, 1000 ms and 1250 ms.

The accuracy of the 2D VFA T ₁‐mapping method with slice profile correction was evaluated for each combination of FAs as the relative difference between the noisy T ₁ estimates (T _1,estim) and the nominal T ₁ (T _{1, nom}):

$${\epsilon }_{T_1}\left({\mathrm{FA}}_1,{\mathrm{FA}}_2,T_1\right)=\frac{1}{N}{\sum }_{i=1}^N\frac{T_{1,i,\text{estim}}-T_{1,\mathrm{nom}}}{T_{1,\mathrm{nom}}}\times 100\%$$

where N is the number of simulations. To evaluate the effect of the slice profile correction, the accuracy was evaluated with and without the correction. Similarly, the precision of the VFA T ₁ mapping method was evaluated using the relative T ₁ standard deviation ( ${\sigma }_{T_1}$):

$${\sigma }_{T_1}\left({\mathrm{FA}}_1,{\mathrm{FA}}_2,T_1\right)=\frac{\sqrt{\frac{1}{N}{\sum }_{i=1}^N{\left(T_{1,i,\text{estim}}-⟨T_1⟩\right)}^2}}{⟨T_1⟩}\times 100\%$$

where ⟨T ₁⟩ is the average value of the T ₁ estimates and N is the number of simulations.

To evaluate the performance of the VFA method across a T ₁ range, maximum error projections (MEPs) were taken through ${\epsilon }_{T_1}\left({\mathrm{FA}}_1,{\mathrm{FA}}_2,T_1\right)$ and ${\sigma }_{T_1}\left({\mathrm{FA}}_1,{\mathrm{FA}}_2,T_1\right)$ volumes along the T ₁ dimension. We then identified the region of FA pairs for which the MEPs of ${\epsilon }_{T_1}$ and ${\sigma }_{T_1}$ were below 5%.

3.3. Experimental validation

To investigate the performance of the correction method, both phantom and human volunteer experiments were performed on a clinical 1.5 T MR scanner (Philips Achieva, Best, The Netherlands) using the integrated body coil as a transmitter and an eight‐channel head coil as a receiver. All image processing and all analyses were done offline using MATLAB 2018a (MathWorks, Natick, MA).

3.3.1. In vitro validation of the 2D T ₁‐mapping correction

To evaluate T ₁ mapping with 2D VFA, a validation of the method was performed in a phantom.

The phantom experiments were performed on a calibrated phantom consisting of gel tubes with T ₁ known to within ±3% accuracy (TO5, Eurospin II test system, Diagnostic Sonar, Livingston, UK). Eleven tubes with nominal T ₁ values in the range of 200‐1250 ms were placed on a polystyrene foam holder and scanned at 21 °C.

Reference T ₁ values were obtained from a 2D turbo IR spin‐echo scan, with the following acquisition parameter settings: repetition time (T _R) = 7000 ms, echo time (T _E) = 20 ms, matrix = 112 × 112, and variable inversion time (T _I) = [50, 100, 200, 400, 800, 1600, 3200] ms. T ₁ was calculated using the following non‐linear least squares three‐parameter fit function:

$$M_z\left(T_{\mathrm{I}}\right)=M_0\left[1-\left(1-\cos {\theta }_{\mathrm{inv}}\right){\mathrm{e}}^{-\frac{T_{\mathrm{I}}}{T_1}}+{\mathrm{e}}^{-\frac{T_{\mathrm{R}}}{T_1}}\right],$$

where the three parameters fitted were the equilibrium magnetization M ₀, the T ₁ and the FA of the inversion angle θ _inv. Phase errors have been corrected with the method proposed by Xiang et al.²¹

For the VFA experiments, a 2D SPGR sequence was used with the parameter settings reported in Table 1. The RF pulse was an asymmetric SINC‐Gauss pulse with TBP 3, a pulse commonly used for 2D SPGR sequences.

T₁ maps were estimated with the VFA method and then corrected for slice profile effects. The relative error in T ₁ estimation relative to the reference values obtained with IR was defined as

$${\epsilon }_{T_1}=\frac{T_{1,\mathrm{VFA}}-T_{1,\mathrm{IR}}}{T_{1,\mathrm{IR}}}\times 100\%$$

where T _1,VFA are the T values estimated with 2D VFA mapping and T _1,IR are the reference T ₁ values from the IR experiment.

A correction for in‐plane B ₁ inhomogeneities was applied. The relative error in T ₁ estimates corrected for B ₁ inhomogeneities was compared with the relative error without this correction. In‐plane B ₁ inhomogeneities were corrected using B ₁ maps, acquired using the actual flip‐angle imaging method²²: T _R1 30 ms, T _R2 150 ms, T _E 4.40 ms, FA 60°.

The B ₁ correction was applied prior to correction for the slice profile effects, as follows: the actual FA in each voxel was calculated by rescaling the nominal FA using the B ₁ value for that voxel. Next, the LUT indexed by the actual FA pair in each voxel was used to correct for the slice profile effect (Figure 1I).

The quantitative evaluation was based on a region of interest (ROI) placed at the center of each tube and the spatial error was assessed using the spatial standard deviation of the T ₁ maps over the voxels inside the ROI.

3.3.2. In vivo demonstration of the 2D T ₁‐mapping correction

To demonstrate the method in vivo, experiments were performed on the brains of two healthy volunteers. The volunteer study was performed with the approval of the institutional review board of the University Medical Center Utrecht (NL53099.041.15), and written informed consent was obtained from the volunteer.

Experimental T ₁ maps were calculated from single‐slice 2D and 3D VFA acquisitions (Table 1). The parameter settings for the 2D VFA scans and B ₁ maps were the same as those for the phantom study. The qualitative evaluation was assessed comparing 2D T ₁ maps with and without the slice profile correction with 3D T ₁ maps. For 3D VFA scans, the analysis has been performed on the central slice of the volume.

4. RESULTS

The FA, phase and magnetization profiles across a range of frequencies for the three RF pulses considered are shown in the Supporting Information Figures S1 and S2, along with the signal versus FA curves for TBP 2, 3 and 10.

4.1. Applicability of the linearization in the 2D VFA method

Figure 3 shows the plots of S/sin θ versus S/tan θ for RF pulses with TBP 2, 3 and 10. Slice profile effects in 2D VFA cause deviations some from linearity for all TBPs. Yet, TBP 2 and 3 show more linear behavior than TBP 10, especially at long T ₁ and high FAs.

FIGURE 3.

Plot of S/sin θ versus S/tan θ for simulated steady‐state signals with RF pulses with TBP 2, 3 and 10, T _R = 10 ms and T ₁ = 50, 250, 500, 750, 1000 and 1250 ms. The high FAs lie closer to the origin of the graphs, as indicated by the arrow. The right‐hand column shows a magnified view to detail the behavior at high FA

The deviation from linearity leads to a change of the slope (k) as a function of the FA combination, as shown in Figure 4 with a fixed FA₁ for T ₁ = 300 and 1000 ms. The natural logarithm of the slope (ln(k)) is negative for all combinations of FAs with TBP 2 and 3, whereas it increases at higher FAs for TBP 10 and becomes positive especially with long T ₁. When ln(k) at high FA becomes positive, the T ₁ values estimated with the VFA method become negative, which is unrealistic, and also the slice profile correction using the LUT will fail.

FIGURE 4.

Plot of the natural logarithm of k, for TBP 2, 3 and 10 and the ideal case without slice profile effects, with fixed FA₁ = 4°, T _R = 10 ms and T ₁ = 300 ms (left) and 1000 ms (right)

4.2. Performance of 2D VFA T ₁ mapping (over a range of FAs)

Figure 5A shows the simulated T ₁ estimates calculated without slice profile correction as a percentage of the true T ₁. For nearly all FAs, the VFA method without correction for the slice profile severely underestimates T ₁. When the correction with the T ₁ LUT is applied, the error in the T ₁ estimate is effectively reduced, as Figure 5B shows. For example, for T ₁ = 500 ms and an RF shape with TBP 3, a large range of FA pairs has ${\epsilon }_{T_1}$ < 5% after correction.

FIGURE 5.

Relative error in 2D VFA T ₁ mapping as a function of FA combination, without (A) and with (B) the T ₁ correction with LUT. Data is shown for a single T ₁ of 500 ms. Simulation input: RF pulse with TBP 3, nominal T ₁ 500 ms and T _R 10 ms. Note: because of symmetry, only half of the FA combinations are presented

An overview of the relative error ${\epsilon }_{T_1}$ and standard deviation ${\sigma }_{T_1}$ is presented in Figure 6, for T ₁ values across the range examined. The number of FA pairs with an acceptable (ie <5%) ${\epsilon }_{T_1}$ and ${\sigma }_{T_1}$ decreases as the nominal T ₁ increases. The FA pairs that gave the highest accuracy (in Figure 6A) and precision (in Figure 6B), ie minimum ${\epsilon }_{T_1}$ and ${\sigma }_{T_1}$, are marked with a blue cross: as expected, they are dependent on the T ₁ value. Moreover, as shown in Figure 6B, they differ from the FAs calculated using the method proposed by Deoni et al for 3D VFA.⁴, ²⁰ Typically, the minima occur for higher FAs than those calculated for 3D VFA for a given T ₁.

FIGURE 6.

Relative error ${\epsilon }_{T_1}$ (A) and relative standard deviation ${\sigma }_{T_1}$ (B) in 2D VFA T ₁ mapping, as a function of the FA combination chosen and applied to different nominal T ₁ values. Results are for one pulse shape, and after correction using the lookup table. Simulation input: TBP 3, nominal T ₁ 50, 250, 500, 750, 1000 and 1250 ms and T _R 10 ms. The blue crosses show the minimum for each nominal T ₁. For B, the FA pair calculated using the method of Deoni et al⁴ is indicated with a red cross. Note that the color scale is different from that of Figure 5

In Figure 7, the MEPs of ${\epsilon }_{T_1}$ and ${\sigma }_{T_1}$ along the T ₁ axes are shown for the three RF pulses. The precision and accuracy of 2D VFA T ₁ mapping depend also on the RF pulses applied. Areas of FA pairs accurate within 5% and with precision better than 5% can be identified for all RF shapes (Figure 7C): for some combinations of FAs, the two areas overlap. Moreover, errors in the T ₁ estimations increase with the TBP of the RF pulse. The RF pulses with TBP 2 and 3 have the largest FA area where ${\epsilon }_{T_{1,\max }}$ and ${\sigma }_{T_{1,\max }}$ are below 5%. This is somewhat surprising, since these pulses have the slice profiles that deviate most from the boxcar shape.

FIGURE 7.

A, B, Performance of 2D VFA T ₁ mapping for a range of T ₁, using RF pulses with different TBP. Maps of maximum relative error ${\epsilon }_{T_{1,\max }}$ (A) and relative standard deviation ${\sigma }_{T_{1,\max }}$ (B) over six nominal T ₁ values: the value in each location is calculated as the maximum projection over six nominal T ₁ values. The blue crosses show the minima identified for the whole T ₁ range. C, Contour plot identifying combinations of FA for which ${\epsilon }_{T_{1,\max }}$ and ${\sigma }_{T_{1,\max }}$ are 5% on the contour and below 5% inside; the FA pair, calculated using the method of Deoni et al,⁴ are also reported in black for the single nominal T ₁ values. Note that the scale for FA₁ in C is different

4.3. Experimental validation

4.3.1. In vitro validation of the 2D T ₁‐mapping correction

Figure 8 shows the T ₁ value for each tube estimated with 2D VFA with and without corrections for slice profile effects and B ₁ inhomogeneities. T ₁ maps with and without slice profile correction and B ₁ correction are shown in Figure S3 and the values estimated in each tube are summarized in Table S1.

FIGURE 8.

Relative error in T ₁ estimation as a function of T ₁. The relative errors were evaluated in 30‐pixel ROIs at the center of each tube (images acquired with TBP 3; nominal FA 6°, 40°). The 2D VFA estimates are reported without corrections, with slice profile correction and with both slice profile and B ₁ correction. The grey band represents the standard deviation of the IR T ₁ estimation

Without corrections, the VFA method underestimates T ₁ values with 40‐60% error (average error ‐51.4%). The T ₁ underestimation is reduced to 5‐12% with the slice profile correction (average error ‐8%).

The B ₁ scaling factors in the tubes were between 0.91 and 0.95 (average 0.93). The application of B ₁ correction leads to a slight T ₁ overestimation: range 2% to 12% (average error 7%).

Data available in Supporting Information.

4.3.2. In vivo demonstration of the 2D T ₁‐mapping correction

Figure 9 shows the results of a comparison of T ₁ maps calculated with the VFA method for the human brain at 1.5 T. For both volunteers, 2D T ₁ maps without slice profile correction show T ₁ underestimation in all tissues when compared with the 3D T ₁ maps. This underestimation was reduced by applying the slice profile correction. Slice profile‐corrected 2D T ₁ maps are generally in agreement with the 3D T ₁ maps.

FIGURE 9.

Estimated T ₁ maps of a transverse slice in the brain of two volunteers (images acquired with TBP 3; nominal FA 6°, 40°). The T ₁ maps have been estimated with the 2D VFA method without (A) and with (B) slice profile correction, and with the 3D VFA method (C). All the maps have been corrected for B ₁ inhomogeneities

Data available in Supporting Information.

5. DISCUSSION

We have shown that 2D VFA T ₁ mapping with correction for slice profile effects can deliver accurate and precise T ₁ estimates within 5% error, over a large range of T ₁ values (50‐1250 ms). We observed that both the accuracy and the precision of such 2D VFA T ₁ measurements are dependent on the two FAs used. Moreover, also the RF pulse shape was found to have an influence on T ₁ accuracy and precision. Simulations showed that for an RF pulse shape with lower TBP (ie the FA profile differs more from a boxcar function), a larger range of FA combinations can be used to arrive at accurate and precise T ₁ estimates. For RF pulses with a high TBP, a bias in T ₁ estimation compromises the performance of the method for high FAs.

In previous studies, it was shown that in order to maximize T ₁ precision the best choice for FAs depends on the value of T ₁.⁴, ²⁰ Since for dynamic T ₁ applications good performance over a range of T ₁ values is needed, we investigated the accuracy and precision of 2D VFA over a T ₁ range for all (relevant) FA combinations. In this study, we demonstrate that it is possible to choose FA pairs resulting in acceptable errors of less than 5% for the full range of T ₁ values, from 50 ms to 1250 ms. This means that objects containing tissues with large T ₁ differences or for which T ₁ changes over time can be measured using a fixed FA pair. Moreover, we identified the limits of applicability of this 2D VFA method: it was found that as T ₁ increases the VFA method provides realistic T ₁ estimates only in the low FA regime.

Conceptually, this approach to T ₁ quantification and the correction of the 2D slice profile effects presents some similarities with MR fingerprinting (MRF).²³ In MRF, a signal dictionary is generated by modelling the spin behavior during the acquisition for a variety of acquisition strategies, such as balanced steady‐state free precession, fast imaging with steady‐state precession, RF‐spoiled GRE, GRE sampling of the FID and spin‐echo sequence.²³ Although the MRF framework has shown potential for multi‐parameter estimations, it is limited in terms of temporal resolution when only a specific single property (in this case, T ₁) is required, as the matching procedure depends on the dictionary size and the number of voxels. In our 2D VFA method, the LUT serves as a strongly conditioned dictionary that is calculated once prior to the experiment. For performance reasons, it is possible to further reduce its size and only include FAs close to the nominal FA, to accommodate for B ₁ inhomogeneities.

Another similar application to our approach is T ₂ mapping with multi‐echo spin‐echo sequences: this approach relies on a database of simulated echo modulation curves for a range of T ₂ values in a multi‐spin‐echo experiment, to correct for the signal contamination from stimulated and indirect echoes present with these schemes.²⁴ Similarly, in our case, we implemented an algorithm using Bloch simulations in order to generate the steady‐state signals for the 2D VFA method.

Previously, a method to correct 2D VFA T ₁ mapping for both the slice profile effect and the variations in the RF transmit field has been proposed.¹⁵, ²⁵ The correction was based on an average FA assumption and required an initial T ₁ estimate. This approach was tested for MR thermometry studies²⁵ and its properties were studied through simulation and experiments for a few set T ₁ values.¹⁵ The FA and the RF shape were found to have an influence on the performance of this method. In their paper, Svedin and Parker concluded that the area of accurate FA combinations widens with TBP of the RF pulse.¹⁵

In the present study, we proposed and investigated a correction for the 2D slice profile effect, based on simulation only, including the full slice profile for the FAs chosen. Similarly to Svedin, we found that the FAs and TBP of the RF pulses influence the accuracy and precision of our correction. However, in contrast to Svedin, our simulations showed that for more ideal RF pulse shapes a smaller range of FA pairs can be used to compute acceptable T ₁ estimates. This can be explained by studying the behavior of S/sin θ versus S/tan θ as the FA increases. The VFA method is based on the linearization of the steady‐state equation (Equation 2) and estimates T ₁ from the slope of this line. However, with higher TBP, deviations from linearity can be identified at high FAs. Since the curve is no longer a straight line, the slope is not constant and T ₁ estimation using the VFA method will be compromised. Moreover, for high TBP pulses the signal is low at high FAs (see Figure S2), which in turn results in a low SNR. This has an effect on the precision of T ₁ measurements. To overcome this problem, the signal could be acquired at low FAs, aiming for a compromise between SNR and VFA method accuracy.

Moreover, we compared the FA pair calculated from previous studies for 3D VFA⁴, ²⁰ with those that minimized the relative standard deviation in T ₁ for the Gaussian and the SINC pulses. We found that, when an RF shape with highly uniform slice profile is considered, the optimal FA pair and the pair calculated with methods from the literature⁴, ²⁰ show good agreement. This is understandable, since these previous studies derived and investigated FA pairs for the 3D VFA method, for which the non‐uniform slice excitation is resolved through phase encoding in the slice direction.

For the 3D case, it is possible to identify a pair of FAs which maximizes T ₁ precision for a specific T ₁.⁴ In 2D, the relatively uniform slice profile offered by high TBP pulses approaches this ideal case, which could be advantageous when measuring an approximately known T ₁. For dynamic T ₁ applications, however, rapid changes should be detected over a substantial T ₁ range. In this case, we found that the low TBP pulses, which deviate more from an ideal boxcar slice profile, are more forgiving and therefore more suitable for applications where T ₁ is unknown.

For 2D VFA, the FAs that minimize errors are both higher for lower TBP. The shift in the higher FA (FA₂) is greater. A possible explanation is an increased contribution from parts of the signal profile with lower effective FA for these low TBP pulses.

We observed from Figure 7C that, with the FA pair estimated with the method of Deoni et al⁴ for T ₁ = 250 ms (approximately the T ₁ of fat at 20 °C at 1.5 T), errors in T ₁ estimation for a wide range of T ₁ values are less than 5% with all the RF pulses considered. This could be relevant for other applications with dynamic changes in T ₁ in adipose tissues, such as MR thermometry in fat. For example, with thermal therapy, eg MR‐guided high intensity focused ultrasound, fat can be heated up to 70 °C, and its T ₁ value may reach ~600 ms.¹⁰ Another application involving rapid T ₁ changes is DCE imaging of the breast.¹²

Although we have studied the accuracy and precision of T ₁ estimation using the 2D VFA method for different RF shapes and over a range of T ₁ values, some factors limit the generalizability of this approach. First, we assumed single T ₁ values per voxel. It would be interesting to study the influence of partial volume effect in mixed voxels. Moreover, the assumption of a single relaxation component does not hold for a variety of biological tissues, such as human white matter and grey matter. It has been shown that the SPGR signal in the brain is more completely characterized by a summation of two or more relaxation components, arising from a fast relaxing (due to myelin) and a slow relaxing species.²⁶, ²⁷

In our phantom study we observed differences between the VFA T ₁ estimates with slice profile and B ₁ corrections and the reference T ₁ estimates from IR. The VFA method is known to overestimate T ₁ values.²⁸, ²⁹ A number of causes for this have been reported, including incomplete spoiling of the transverse magnetization in SPGR sequences³⁰ and improper accounting for noise in the VFA fitting,³¹ and the variation of the FA within the body, caused by inhomogeneities in the RF transmit field.²⁸ Moreover, it has been shown that even small B ₁ variations (ie less than 10%, as our case), can result in considerable bias in accuracy in T ₁ values, estimated with the VFA method.²⁸

In our experiments, performed at 1.5 T, we found that slice profile correction addresses most of the bias in T ₁ estimation, whereas the impact of B ₁ error was less significant. In applications where time‐resolved scanning is used to measure dynamic changes in T ₁, such as with T ₁‐based MR thermometry, B ₁ correction may be optional. However, B ₁ inhomogeneities worsen at higher field strengths.

In addition, the VFA method assumes that magnetization is in the steady state (ie the signal behaves as described in Equation 1). This is typically accomplished by setting a proper RF‐spoiling phase increment and including a sufficient number of dummy pulses to account for the time lost by transitioning between the steady states belonging to FA₁ and FA₂.

Finally, as mentioned, it is known that incomplete elimination of the transverse magnetization in SPGR sequences can be a source of T ₁ measurement errors.³⁰ At this stage, our simulations assumed perfect spoiling, which may not always be achieved in practice.

6. CONCLUSIONS

We proposed a correction for non‐uniform 2D slice excitation in 2D VFA T ₁ mapping. The correction is based on Bloch simulation of the magnetization over the full slice profile for common RF waveforms. We have shown that accurate and precise 2D VFA measurements over a large range of T ₁ values can be obtained with this correction. We studied the influence of FAs and RF shape on the 2D VFA method and we found that a larger area of FA pair with acceptable (<5%) errors in T ₁ accuracy and precision is available for RF pulses with low TBP.

Supporting information

Figure S1. Flip Angle (S1b.) and Phase profile (S1c.) at 40° flip angle, T1 = 250 ms, for Gaussian RF pulse, Asymmetric SINC pulse and SINC pulse, reported in S1a.

Figure S2. (S2a). Signal as a function of flip angle. Comparison of the simulated steady state signals for TBP 2,3 and 10, with the theoretical behavior of the steady state signal, as it would be generated by an RF pulse with ideal slice profile (i.e. rectangular slice profile, Equation 1) for T1 = 250 ms (S2b) Real component of magnetization Mx as a function of frequency at 40° flip angle.

Figure S3. T1 maps from the in vitro experiment with and without the slice profile and the B1 correction (Images acquired with TBP 3; nominal FA 6°, 40°).

Table S1. T1s in tubes estimated with 2D VFA method compared to reference values from IR experiment, with and without correction for the slice profile effect and B1 mapping correction. All the estimates are reported in ms.

Lena B, Bos C, Ferrer CJ, Moonen CTW, Viergever MA, Bartels LW. Rapid 2D variable flip angle method for accurate and precise T ₁ measurements over a wide range of T ₁ values. NMR in Biomedicine. 2021;34:e4542. 10.1002/nbm.4542