Yulia Shcherbakova^{1}, Cornelis A.T. van den Berg^{2}, Chrit T.W. Moonen^{1}, and Lambertus W. Bartels^{1}

The PLANET method has been recently proposed to quantify the relaxation
parameters T_{1} and T_{2}, the banding free magnitude, the
local off-resonance ∆f_{0}, and the RF phase from RF
phase-cycled balanced steady-state free precession (bSSFP) data. The PLANET
model requires a static B_{0} field over the course of the acquisition.
However, due to gradient activity, B_{0} drift can happen. In this work
we present a study of the influence of B_{0} drift on the performance
of the method and we propose a strategy for correction.

The complex phase-cycled bSSFP signal without drift can be represented as [2,3]:

$$$I = M_{eff}\frac{1-ae^{i(2πTRΔf_{0}+Δ\theta)}}{1-b\cos(2πTRΔf_{0}+Δ\theta)}e^{i(2πTEΔf_{0}+φ_{RF})} $$$ [1]

where M_{eff}, a, b depend
on T_{1}, T_{2}, TR, FA.
∆θ is the user-controlled RF phase increment, Δf_{0} is the local off-resonance, φ_{RF} is the combined RF transmit and receive phase.

With frequency drift modeled as Δf_{0} → Δf_{0} + Δf_{drift}(t), Eq. [1] becomes:

$$$ I = M_{eff}\frac{1-ae^{i(2πTRΔf_{0}+Δ\theta+2πTRΔf_{drift})}}{1-b\cos(2πTRΔf_{0}+Δ\theta+2πTRΔf_{drift})}e^{i(2πTEΔf_{0}+φ_{RF})}e^{i2πTEΔf_{drift}} $$$ [2]

The first part
of Eq.[2] multiplied by the first exponential represents the same elliptical equation
as Eq.[1] with only an offset in the RF phase increment: Δθ_{new} = Δθ +2πTRΔf_{drift}. This corresponds
to a drift-dependent displacement of all data points along the ellipse, see Figure
1(b). The second exponential corresponds to an additional drift-dependent rotation
of all data points around the origin, see
Figure 1(c,d). Ignoring the presence of drift, fitting an ellipse to the
”drifted” data points leads to a different fit compared to the fit for the
“non-drifted” case, see Figure (e).
After performing PLANET post-processing, this would result in errors in the
parameter estimates.

We propose a 3-step correction algorithm:

1. Calculation of the
spatial time-dependent B_{0} drift during PLANET acquisition Δf_{drift n} (i,j)(t), where n – is the number of dynamic acquisition, t - is the time, corresponding to n^{th}
acquisition.
Assuming temporally linear drift, the frequency drift over n^{th}
phase-cycled acquisition is estimated by:

$$$Δf_{drift,n} (i,j)(t)=n\frac{Δf_{total, drift}(i,j)}{N_{cycles}} $$$ (Hz), n = {1, ...N_{cycles}} [3]

where
total drift over PLANET acquisition Δf_{total drift} (i,j) is calculated
by subtracting two reference B_{0} maps acquired right before and after
PLANET acquisition.

2. Correction of M_{eff}, T_{1}, T_{2}
by multiplying the experimental complex data by e^{-i2πTEΔfdrift n (i,j)(t)}, the geometrical equivalent of which is the
rotation of the “drifted” data points around the origin back to the
“non-drifted” ellipse.

3. Correction of ∆f_{0} and φ_{RF} by
defining Δθ_{new} (i,j)(t) = Δθ(t)+2πTRΔf_{drift} (i,j)(t), which geometrically moves the “drifted” data points
along the ellipse back to their “non-drifted” positions. Δθ(t)– is the
user controlled RF phase increment.

To test the correction algorithm, MRI experiments on a phantom (bottle
filled with aqueous solution of MnCl_{2}·4H_{2}O) and in the brain
of healthy volunteer were performed on a clinical 1.5T MR scanner (Philips
Ingenia, Best, The Netherlands). A 16-channel head receive coil was used. Sequence parameter settings
for 3D bSSFP: FOV 160x160x159 mm^{3} (phantom), 220x220x100 mm^{3} (the
brain) , voxel size 1.1x1.1x3 mm^{3} (phantom), 0.98x0.98x4 mm^{3} (brain), TR 10 ms, FA 30˚, 10 phase cycles with Δθ = π/5. Reference B_{0} maps were calculated using a dual echo approach.
Reference B_{1} maps were acquired for FA correction. Reference T_{1}
and T_{2} maps were calculated using 2D MIXED method [4].

FIGURE 1. Schematic representation of
the influence of B_{0} drift on the data points in the complex signal
plane: a) the ellipse without drift; b) time dependent displacement of the data
points along the ellipse due to B_{0} drift. The “non drifted” data
points are blue, the displaced data points are red; c) time dependent rotation
of the data points around the origin. The rotated data points are green; d) the
ellipse without drift (blue) and the “drifted” ellipse (green); e) the vertical
conic forms of the ellipse without drift (blue) and the “drifted” ellipse
(green).

FIGURE 2. Experimental results obtained
in the phantom: a) B_{0} drift over 14-min PLANET acquisition; b) Estimated T_{1},
T_{2}, ∆f_{0}, φ_{RF} maps; c) T_{1}, T_{2}, ∆f_{0}, φ_{RF} maps after drift correction; d) The reference
T_{1}, T_{2}, and ∆f_{0} maps; e) T_{1} and T_{2}
profiles along the selected lines on estimated, corrected and reference T_{1}
and T_{2} maps; f) The magnitude image with white vertical and
horizontal lines in the middle of the slice used for T_{1} and T_{2}
profiles; g) The reference T_{1} and
T_{2} values, the average calculated T_{1} and T_{2}
values before and after drift correction for one slice of the phantom.

FIGURE 3. Experimental results
obtained in the brain: a) The reference B_{0} maps before (1) and after
(2) PLANET acquisition and the corresponding drift map (1-2) ; b) Banding free
magnitude image, the reference T_{1} and T_{2} maps; c) T_{1},
T_{2}, Δf_{0}, and φ_{RF} maps before
drift correction; d) T_{1}, T_{2}, Δf_{0}, and φ_{RF} maps after
drift correction; e) Relative errors ε (in percent) in T_{1} and T_{2}
values compared to the corrected value.