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Magn Reson Med. Author manuscript; available in PMC 2013 July 1.

Published in final edited form as:

PMCID: PMC3295910

NIHMSID: NIHMS317763

Samuel A. Hurley, Medical Physics, University of Wisconsin, Madison, WI;

The publisher's final edited version of this article is available at Magn Reson Med

See other articles in PMC that cite the published article.

A new time-efficient and accurate technique for simultaneous mapping of *T _{1}* and

Accurate measurements of longitudinal relaxation time (*T _{1}*) are essential to many quantitative MRI techniques and clinical applications. Traditional inversion recovery (IR)

The variable flip angle (VFA) method (6,7), also known as driven-equilibrium single-pulse observation of *T _{1}* (DESPOT1) (7), uses several short

It is therefore commonly accepted that VFA *T _{1}* mapping at 3T or higher field strengths needs to be combined with an appropriate

It is important to note that a 3D *B*_{1} mapping technique with matched geometry is preferable in the context of VFA *T _{1}* mapping to simplify registration between data volumes. One 3D approach encodes the flip angle in the phase of the MR signal (18). This reduces the

The accurate combination of a *B _{1}* mapping technique with 3D VFA

$${\mathit{\text{TR}}}_{\mathit{\text{AFI}}1,2}\text{}\ll {T}_{1}$$

[1]

The calculated flip angle map can then be applied as a pre-calibration step to a traditional VFA *T _{1}* measurement. AFI is well suited for the correction of VFA measurements because of the identical readout scheme to the SPGR pulse sequence, the use of an identical RF excitation pulse, and lack of a need for additional magnetization preparation with specialized RF pulses (20,23,24).

In spite of the many benefits of AFI, recent studies have shown that large spoiler gradients are required for effective suppression (spoiling) of transverse magnetization and thus accurate FA measurements (27). Such large gradients require a considerable increase in the *TR* compared to a standard short *TR* SPGR sequence, and therefore an overall increase in scan time. While proper spoiling is essential for accurate FA quantification, increasing the *TR* of the AFI sequence may lead to a violation of the main assumption (Eq. [1]) (25), posing another significant problem with the accuracy and overall efficiency of the technique. Similarly, strong spoiler gradients are also necessary in VFA SPGR measurements for accurate *T _{1}* mapping, especially at larger flip angles (27,28). This further increases the overall time of the technique.

An independent mechanism for violation of the AFI assumption (Eq. [1]) is *T _{1}* shortening due to contrast agents, which may render post-contrast FA measurements highly inaccurate. Post-contrast

AFI and VFA are highly complementary methods; AFI suffers inaccuracies due to an assumption about *T _{1}*, while VFA suffers inaccuracies due to an assumption about FA. In this paper, we propose an efficient method to tackle the quantification of

*T _{1}* measurements based on VFA SPGR acquisitions have been previously described in (6–8,33). Given ideal spoiling of transverse magnetization, the SPGR signal can be described as follows:

$$\begin{array}{c}\hfill {S}_{\mathit{\text{SPGR}}}={M}_{0}\phantom{\rule{thinmathspace}{0ex}}\text{sin}\phantom{\rule{thinmathspace}{0ex}}(\mathrm{\alpha})\frac{1-{E}_{1}}{1-{E}_{1}\phantom{\rule{thinmathspace}{0ex}}\text{cos}(\mathrm{\alpha})}\hfill \\ \hfill {E}_{1}=\text{exp}\phantom{\rule{thinmathspace}{0ex}}\left(\frac{-\mathit{\text{TR}}}{{T}_{1}}\right)\hfill \end{array}$$

[2]

where *TR* and excitation flip angle α are control parameters prescribed from the operator console and the equilibrium magnetization *M _{0}* and

The AFI sequence is a spoiled gradient-recalled echo acquisition with two identical excitation RF pulses with flip angle α separated by two interleaved repetitions times *TR _{AFI1,2}* (25). Given complete spoiling of transverse magnetization at the end of each repetition time (25), the steady-state signal from the AFI sequence is described by the following equations:

$$\begin{array}{c}\hfill {S}_{\mathit{\text{AFI}}1,2}={M}_{0}\phantom{\rule{thinmathspace}{0ex}}\text{sin}\left({\mathrm{\alpha}}_{\mathit{\text{AFI}}}\right)\frac{1-{E}_{1}^{(\mathit{\text{AFI}}2,1)}+\left(1-{E}_{1}^{(\mathit{\text{AFI}}1,2)}\right){E}_{1}^{(\mathit{\text{AFI}}1,2)}\text{cos}\left({\mathrm{\alpha}}_{\mathit{\text{AFI}}}\right)}{1-{E}_{1}^{(\mathit{\text{AFI}}1,2)}{E}_{1}^{(\mathit{\text{AFI}}2,1)}\phantom{\rule{thinmathspace}{0ex}}{\text{cos}}^{2}\phantom{\rule{thinmathspace}{0ex}}\left({\mathrm{\alpha}}_{\mathit{\text{AFI}}}\right)}\hfill \\ \hfill {E}_{1}^{(\mathit{\text{AFI}}1,2)}=\text{exp}\left(\frac{-{\mathit{\text{TR}}}_{\mathit{\text{AFI}}1,2}}{{T}_{1}}\right)\hfill \end{array}$$

[3]

It was shown in (25) that under the condition of Eq. [1] there exists an efficient first-order exponential approximation without an explicit dependence on *M _{0}* or

$$\begin{array}{c}\hfill {\alpha}_{\mathit{\text{AFI}}}=\text{arccos}\phantom{\rule{thinmathspace}{0ex}}\left(\frac{\mathit{\text{rn}}-1}{n-r}\right)\hfill \\ \hfill n={\mathit{\text{TR}}}_{\mathit{\text{AFI}}2}/{\mathit{\text{TR}}}_{\mathit{\text{AFI}}1},\phantom{\rule{thinmathspace}{0ex}}r={S}_{\mathit{\text{AFI}}2}/{S}_{\mathit{\text{AFI}}1}\hfill \end{array}$$

[4]

In order to avoid the main AFI assumption (Eq. [1]), we propose a novel approach, VAFI, which takes as an input one AFI dataset and one or more VFA SPGR datasets. In VAFI, the dependence of the AFI flip angle on *T _{1}* is not ignored, but instead coupled in a single nonlinear optimization procedure for an unbiased simultaneous estimation of both parameters. We assume that

$$[{M}_{0}\text{}{T}_{1}\text{}\kappa ]=\underset{{M}_{0}\text{}{T}_{1}\text{}\kappa}{\text{arg min}}\phantom{\rule{thinmathspace}{0ex}}\left({\displaystyle \sum _{i=1}^{2}}{({S}_{\mathit{\text{AFI}},i}-{S}_{\mathit{\text{AFI}},i}\phantom{\rule{thinmathspace}{0ex}}({\alpha}_{\mathit{\text{AFI}}}\xb7\kappa ))}^{2}+{\displaystyle \sum _{i=1}^{{N}_{\mathit{\text{SPGR}}}}}{({s}_{\mathit{\text{SPGR}},i}-{S}_{\mathit{\text{SPGR}}}\phantom{\rule{thinmathspace}{0ex}}({\alpha}_{i}\xb7\kappa ))}^{2}\right)$$

[5]

where $\text{arg}\phantom{\rule{thinmathspace}{0ex}}\underset{x}{\text{min}}\phantom{\rule{thinmathspace}{0ex}}(f(x))$ is the set of values *x* for which *f (x)* attains its smallest value.

This procedure simultaneously yields a solution for three unknown parameters: *M _{0}*,

As the transmit field varies slowly with spatial position, an additional term may be introduced to Eq. [5] to penalize the roughness of κ (r) and improve the noise properties of reconstructed parametric maps in a way similar to regularized fat/water imaging (36), which may be computationally challenging for 3D reconstructions. To reduce computational time, we developed a simplified three-step version of the regularization procedure:

**Step 2:**Apply spatial smoothing to κ (**r**) to yield κ_{s}(**r**).**Step 3.**Solve $[{M}_{0}\text{}{T}_{1}]=\underset{{M}_{0}\text{}{T}_{1}}{\text{arg min}}\left({\displaystyle \sum _{i=1}^{2}}{({s}_{\mathit{\text{AFI}},i}-{S}_{\mathit{\text{AFI}},i}\phantom{\rule{thinmathspace}{0ex}}({\alpha}_{\mathit{\text{AFI}}}\xb7{\kappa}_{s}))}^{2}+{\displaystyle \sum _{i=1}^{{N}_{\mathit{\text{SPGR}}}}}{({s}_{\mathit{\text{SPGR}},i}-{S}_{\mathit{\text{SPGR}}}\phantom{\rule{thinmathspace}{0ex}}({\alpha}_{i}\xb7{\kappa}_{s}))}^{2}\right)$

The proposed VAFI fitting method was implemented in MATLAB (The Mathworks; Natick, MA) using a built-in optimization algorithm. As an initial guess, we used the maximum signal intensity for *M _{0}*,

An AFI pulse sequence was implemented on a clinical GE 3.0T Discovery MR750 (GE Healthcare; Waukesha, WI) based on a product 3D SPGR sequence with the addition of a second *TR* as described in (25). Experiments were performed using a 32-element phased array receive coil (*in-vivo* experiments) and an eight-element torso coil (phantom experiments). Spoiler gradients were designed such that their areas *A _{G1,2}* were related as

Monte Carlo noise simulations were implemented to compare the accuracy (bias in the estimated values) and precision (noise in the estimated values) of standard two flip angle VFA with AFI correction (henceforth referred to as VFA) and VAFI. AFI and SPGR signals were generated using Eqs. [2],[3], with simulation parameters chosen to match the *in-vivo* experiments described in the next section (SPGR: *TR* = 10 ms, α = [3 18]°; AFI: *TR _{AFI1}* = 30 ms, α

A second simulation was performed to investigate the effect of flip angle selection on the VAFI experiment. For two-angle VFA, a pair of "ideal angles" has been previously identified as a way to optimize experiment design for a specific *T _{1}*, however these angles were derived based on the linearized version of the SPGR equation (33) which is no longer valid in the VAFI technique. In order to assess the noise performance of our new method, a set of 45,000 noise realizations (standard deviation of 1e-4) were generated over a range of

A third simulation was performed to evaluate the effect of gradient and RF spoiling on the accuracy of VAFI quantification. AFI and SPGR signals were generated using a combined isochromat summation and diffusion propagator model (27), which takes into account transverse signal dephasing due to both gradient and diffusion effects. Signals were generated for prototypical tissues at 3T over a range of RF phase increments _{o} = 0 to 180° (white matter (WM): *T _{1}/T_{2}* = 1000/70 ms, diffusion coefficient D = 0.70 × 10

To test the effects of the suboptimal AFI regime (*TR ≈ T _{1}*) on both VFA and VAFI, a set of ten phantoms were created from deionized water doped with Gd-BOPTA to concentrations of C = [0.00 0.10 0.25 0.50 1.00 2.00 2.50 3.00 3.50 4.00] mM. The phantoms were placed in a heated water bath lightly doped with Gd-BOPTA and held at 36.8 ± 0.5 °C for the duration of the experiment. SPGR scans were acquired with

$${R}_{1}(C)={R}_{1,0}+{r}_{1}\xb7C$$

[6]

where *r _{1}* is the relaxivity, and

It was shown before that simultaneously increasing *TR* and acceleration factor *R* while maintaining a constant acquisition time constant leads to an improvement of the overall SNR efficiency of several steady-state gradient echo techniques (38). In the context of AFI, in addition to signal boost from longer *TR*, this increase also offers an opportunity to increase the length of the spoiler gradient and improve the accuracy of FA quantification (27). To confirm this effect for the AFI sequence, a uniform bottle phantom doped with Gd-DTPA to a *T _{1}* of 1.4 s was scanned using an 8-channel receive-only phased array knee coil. Two AFI scans were acquired (α

To demonstrate the application of VAFI to *in-vivo* brain imaging, one healthy human volunteer was scanned. Informed written consent was obtained in accordance with the local institutional policy. SPGR scans were acquired with *TR* = 10 ms and α_{SPGR} = [3 18]°, which is optimal for *T _{1}* = 1.2 s as described in (33). To demonstrate the benefits of increased

Figure 1 shows the results of simulations comparing the accuracy and precision of the methods. Over the range investigated, the standard AFI method shows a systematic underestimation of flip angle with decreasing *T _{1}* (> 5% for

Monte Carlo simulations of standard VFA (dotted line) and VAFI (dash line for two flip angle fit and solid line for one flip angle fit) over a range of *TR*_{AFI1}/*T*_{1} for flip angle accuracy (a), flip angle-to-noise ratio (b), *T*_{1} accuracy (c), and *T*_{1}-to-noise-ratio **...**

Figure 2 shows the results of VAFI *T _{1}NR* and α

Monte Carlo simulations demonstrating the impact of flip angle selection in the VAFI technique on the *T*_{1}-to-noise-ratio (a,c) and flip angle-to-noise-ratio (b,d) for a *TR*_{AFI1}/*T*_{1} = 0.150 (dotted line), 0.050 (dot-dash line), 0.030 (dash line), and 0.021 **...**

Figure 3 shows the results of simulations assuming ideal AFI spoiling and non-ideal SPGR spoiling for WM and GM tissues. For standard VFA, increasing the gradient spoiling minimizes the dependence of *T _{1}* accuracy on the RF phase increment. However, the choice of a proper phase increment (i.e. the choice of

Simulation results showing the dependence of *T*_{1} accuracy on SPGR phase increment for weak (a, b), intermediate (c, d), and strong (e, f) spoiling regimes in prototypical white matter (*T*_{1}/T_{2} = 1000/70 ms, D = 0.70 × 10^{−3} mm^{2}/s) (a, c, e) **...**

Figure 4 shows the results of simulations assuming non-ideal AFI spoiling and ideal SPGR spoiling for WM and GM. The behavior of spoiling-related errors is similar for the VFA and VAFI. Unlike the case of SPGR spoiling, the choice of a proper AFI phase increment is essential for accurate FA and *T _{1}* quantification for all techniques. The proper phase increment (i.e. the choice of

Figure 5 compares FA maps in Gd phantoms estimated using standard AFI and VAFI. A significant discontinuity between the short *T _{1}* vials (phantoms 6–10) and the background media (longer

Plot of *R*_{1} vs. gadolinium concentration for the vial phantoms, using standard VFA and VAFI. The standard method shows a significant underestimation of *R*_{1} at short *T*_{1} times (due to errors in the AFI FA map propagating into the *T*_{1} estimate) and the relaxivity **...**

Figure 7 illustrates the effect of increasing the SENSE acceleration factor on the accuracy and precision of AFI maps for the strong spoiling regime and a comparison with an AFI map in the short *TR* (weak spoiling) regime. Simulated AFI steady-state signals grow with increasing *TR* faster than the SNR loss due to *R*-factor, giving rise to an overall increase in SNR efficiency for realistic reduction factors (*R* < 8) and constant scan time (Fig. 7a). Indeed, long *TR* (strong spoiling) AFI (Fig. 7b) demonstrates improved precision over a short *TR* (weak spoiling) AFI (Fig. 7c) obtained in an equivalent scan time (*R*=3). The weak spoiling regime in the short *TR* AFI also leads to significant artifacts in the estimated FA map (Fig. 7c).

Figure 8 shows a representative axial *T _{1}* map from the

The variable flip angle SPGR *T _{1}* mapping continues to be an accurate and efficient method for mapping

Similar to VAFI, DESPOT1-HIFI (24) is a variation on the SPGR sequence, which also utilizes simultaneous fitting to map FA and *T _{1}*. To generate unique information to fit for an additional FA parameter, DESPOT1-HIFI utilizes an inversion pulse prior to SPGR readout. Both VAFI and DESPOT1-HIFI require some assumptions about RF pulse properties. In the case of DESPOT1-HIFI, it is assumed that the adiabatic inversion pulse is completely insensitive to any parameters affecting the spin system (

We demonstrated that, with properly tuned parameters (proper choice of AFI and SPGR flip angle, phase increment of RF spoiling, and spoiler gradient area for the tissue of interest, the accuracy and precision of VAFI fit can surpass that of the standard VFA fit (when used with AFI for flip angle calibration) (Fig. 1). This is true even when one SPGR measurement is eliminated, further increasing the overall time efficiency of the technique. This improvement comes from the explicit utilization of least squares estimation for solving Eq. [9], a benefit which is not available with linearized calculation of the flip angle and *T _{1}* in the standard AFI and VFA methods. However, this improvement is at the cost of longer processing times than those required for original AFI and VFA approaches. An alternative and potentially faster minimization approach is to solve VAFI Eq. [5] through interleaved estimations of flip angle by AFI and

A major advantage of the traditional VFA technique is the ease with which acquisition settings can be optimized, either for a single *T _{1}* time using a pair of “ideal” flip angles (33) or using sets of combined ideal angles to image over a wide range of

A major limitation of both AFI and VFA is the need for proper spoiling parameters to ensure accurate quantification (27). While strong AFI spoiling is still a necessity for VAFI, simulations and in *vivo* studies have shown that the need for strong SPGR spoiling for the case of VAFI may be greatly reduced if a single flip angle SPGR measurement is used. This may allow reduction of spoiling gradients for SPGR part of VAFI and, as a result, more time-efficient overall mapping of the flip angle and *T _{1}* than that is available with AFI and VFA alone. Notably, the dependence of

One must consider the time efficiency of VAFI compared to the traditional VFA and AFI approaches when used separately. In the context of *T _{1}*-insensitive flip angle mapping, VAFI requires a short

The variable flip angle method is a rapid and accurate method for measuring *T _{1}*, as long as it can be paired with an equally rapid and accurate method for measuring and correcting flip angle. In this study, we presented a method that combines VFA and AFI into a single procedure, named VAFI. This allows the

This work was supported by NIH NINDS R01NS065034 and in part by NIBIB R21EB009908. We would like to thank to Dr. Alex Frydrychowicz and Dr. Ian Rowland for assistance with phantom experiments.

Samuel A. Hurley, Medical Physics, University of Wisconsin, Madison, WI.

Vasily L. Yarnykh, Radiology, University of Washington, Seattle, WA.

Kevin M. Johnson, Medical Physics, University of Wisconsin, Madison, WI.

Aaron S. Field, Radiology, University of Wisconsin, Madison, WI.

Andrew L. Alexander, Medical Physics and Psychiatry, University of Wisconsin, Madison, WI.

Alexey A. Samsonov, Radiology, University of Wisconsin, Madison, WI.

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