To demonstrate the use of new models of the double-echo in steady-state (DESS) signals to quantify T₂ and reduce motion artifacts.

The DESS sequence produces two echoes S₁ and S₂^1,2,3, before and after a spoiler gradient respectively, which provide morphological images and have also been used for quantification of T₂ and ADC^4,5,6 in cartilage. The signals have very complicated contrast due to the contribution of multiple echo paths to each signal⁷. The simple relationship $$$ \frac{S_2}{S_1} = e^{-\frac{2(\mathrm{TR}-\mathrm{TE})}{T2}}$$$ [1] has previously been used for T₂ estimation⁴, but underestimates T₂.

We propose a more accurate model based on Extended Phase Graphs⁷, which can be derived by tracing the echo paths between the signals S₁ and S₂, and neglecting all paths that have been dephased by two or more spoiler gradients, as shown in Figure 1:

$$ \frac{S_2}{S_1} = e^{-\frac{2(\mathrm{TR}-\mathrm{TE})}{T2} - \left( \mathrm{TR}- \frac{\tau}{3} \right) \Delta k^2 D} \sin^2{\left( \frac{\alpha}{2} \right)} \left(\frac{1 + e^{-\frac{\mathrm{TR}}{T1} - \mathrm{TR} \Delta k^2 D}}{1 - \cos{\alpha} e^{-\frac{\mathrm{TR}}{T1} - \mathrm{TR} \Delta k^2 D}} \right) = e^{-\frac{2(\mathrm{TR}-\mathrm{TE})}{T2}} \sin^2{\left( \frac{\alpha}{2} \right)} \left(\frac{1 + e^{-\frac{\mathrm{TR}}{T1}}}{1 - \cos{\alpha} e^{-\frac{\mathrm{TR}}{T1}}} \right) \hspace{2em} [2] $$

where the gradient has magnitude G and duration τ, inducing a phase of Δk = γG τ, and D is the diffusivity. The last step applies for small spoilers. Figure 2 shows a comparison between Eqs. 1 and 2, and a simulation of the true signals, as well as T₂ estimate error. Using Eq. 2 enables better T₂ estimation in cartilage by assuming the prescribed flip angle of the scan and a typical T₁ value of cartilage. Note the low sensitivity to T₁, so small errors in the T₁ assumption should not cause large errors in estimated T₂. Furthermore, for example in knee cartilage T₁ does not change much where T₂ is sensitive to degeneration.

Alternatively, different echo paths between S₁ and S₂ can be traced exactly to form a simple relationship of the form:

$$ S_1 = a S_2 + S_{SP} \hspace{2em} [3]$$

where $$$ a = \frac{e^{-\frac{\mathrm{TR}}{T1}} - \cos{\alpha}}{1 - \cos{\alpha} e^{-\frac{\mathrm{TR}}{T1}}} e^{-\frac{2\mathrm{TE}}{T2}} $$$ and $$$S_{SP}$$$ is an RF-spoiled signal. Since Eq. 3 has no dependency on diffusion or the gradient, changing the spoiler area will move a DESS data point (S₁,S₂) along a line between a minimally spoiled DESS scan and an RF spoiled scan, as shown in Fig. 3a. DESS scans with large spoiling, often run for diffusion weighting, can deviate from this line due to motion, as shown in Fig. 3b. By projecting the motion-corrupted points to the closest point on the line, motion ghost artifacts can be reduced.

A DESS scan was run in the knee of a healthy volunteer with flip angle α = 18° and a minimal spoiler, giving very little diffusion weighting. Spin echo (SE) scans were run for comparison with multiple echo times. T₂ maps were generated using Eqs. 1 and 2 assuming Δk = 0, a typical T₁ of 1.2s, and from a monoexponential fit of the SE scans.

To demonstrate the artifact reduction from Eq. 3 in ADC estimation, two DESS scans were performed in the knee of a volunteer using spoiler areas of 156.60 and 15.66 mT/m ms and α = 35°, with fat suppression. An RF-spoiled gradient-echo scan was also run with the same parameters. The strongly spoiled scan was projected to the line between the weakly spoiled DESS scan and the RF-spoiled scan. EPI DWI scans were also run for comparison.

The T₂ maps are shown in Figure 4. Equation 1 gives an average T₂ estimate of 23.3ms in the cartilage. Equation 2 gives an average of 32.7ms, agreeing better with the SE average of 36.8ms. The newer model clearly depicts regional variation in the cartilage more closely to SE.

The effects on the ADC quantification are shown in Figure 5. Using Equation 3 changes the mean ADC estimate from 1.02μm²/ms to 1.43μm²/ms, which is closer to the (distorted) EPI DWI estimate of 1.54μm²/ms.

In this work, the prescribed flip angle was used in Equation 2, but this could be made even more accurate by collecting a flip angle map, giving a better estimate of the true flip angle at each pixel. The models could allow for many other types of analysis, such as estimation of flip angle or T₁, and could be solved together for an expression of the individual signals.