Equivalence of EPG and Isochromat-based simulation of MR signals

Shaihan J Malik^{1}, Alessandro Sbrizzi^{2}, Hans Hoogduin^{2}, and Joseph V Hajnal^{1}

Consider a voxel with uniform magnetization density and
relaxation properties. Rotation due to RF pulses is constant within this voxel
and is characterized by flip angle θ and phase angle φ. In order to simulate
generation of echoes the voxel is further subdivided into a distribution of
‘isochromats’ at positions **r** each characterised by gradient dephasing angle ψ, where

$$\psi=\gamma\int_0^\tau\mathbf{G}(t')\cdot\mathbf{r}dt'=\mathbf{\Delta\,k}\cdot\mathbf{r}.\qquad\text{[1]}$$

Bloch simulations are performed, and the predicted signal
$$$\tilde{S}$$$ is computed by averaging transverse magnetisation M_{+} over all isochromats:

$$M_+(\psi)=M_x(\psi)+iM_y(\psi)$$

$$\tilde{S}=\int_0^{2\pi}M_+(\psi^\prime)d\psi^\prime\approx\frac{1}{N}\sum_{m=0}^{N-1}M_+(\psi_m)\qquad\text{[2]}$$

The integral is approximated as a sum over a set of angles $$$\psi_m$$$. Choice of sampling scheme is the key issue for isochromat-summation simulations. The EPG method approaches the same problem in the reciprocal (Fourier) domain, by representing magnetization in terms of configuration states:

$$\tilde{F}_+(\mathbf{k})=\int_V{M_+\,e^{-i\mathbf{k}\cdot\mathbf{r}}}d^3\mathbf{r}\qquad$$

$$\tilde{F}_-(\mathbf{k})=\int_V{M_-\,e^{-i\mathbf{k}\cdot\mathbf{r}}}d^3\mathbf{r}\qquad$$

$$\tilde{Z}(\mathbf{k})=\int_V{M_z\,e^{-i\mathbf{k}\cdot\mathbf{r}}}d^3\mathbf{r}\qquad\text{[3]}$$

For a sequence with regular timing (fixed **Δk**), the possible **k** values generated form a set of
discrete values:

$$\mathbf{k}=n\mathbf{\Delta k}\qquad n\in\mathbb{Z}\qquad{[4]}$$

such that the magnetization distribution can be written as a sum over integer-indexed states:

$$M_+(\psi)=\sum_{n=-\infty}^{\infty}{\tilde{F}_n e^{in\psi}}\qquad$$

$$M_-(\psi)=\sum_{n=-\infty}^{\infty}{\tilde{F}^*_{-n} e^{in\psi}}\qquad$$

$$M_z(\psi)=Re\left\lbrace\sum_{n=0}^{\infty}{\tilde{Z}_n e^{in\psi}}\right\rbrace\qquad\text{[5]}$$

Numerical predictions are obtained by following a series of
algorithmic rules^{2.} Once numerical values for the coefficients $$$\tilde{F_n}$$$, $$$\tilde{F_{-n}}$$$ and $$$\tilde{Z_n}$$$ are known, we may determine M_{+}(ψ) for any ψ. A
key general assumption is that states with k≠0
are fully dephased: only the k=0
state contributes measurable signal. By comparison with Eq.2 we identify $$$\tilde{S}=\tilde{F}_0$$$.

The DFT may be represented as follows:

$$\tilde{X}_n=\sum_{m=0}^{N-1}{x_m e^{-2\pi\,i\,n\frac{m}{N}}}\quad\,n,m=0,\dots,N-1\qquad\text{[6]}$$

where N is the number of isochromats in the ensemble. If the isochromat angles are selected according to

$$\psi_m=\frac{2\pi\,m}{N}\quad\,m=0,\dots,N-1\qquad\text{[7]}$$

then:

$$\text{DFT}(M_+)=\sum_{m=0}^{N-1}{M_+(\psi_m)e^{-2\pi\,i\,n\frac{m}{N}}}$$

$$\qquad=\sum_{m=0}^{N-1}{M_+(\psi_m) e^{-i\,n\psi_m}}$$

$$\qquad\qquad\qquad=\tilde{F}_n\quad\quad\,n=-\left\lfloor\frac{N}{2}\right\rfloor,\dots,\left\lfloor\frac{N-1}{2}\right\rfloor\text{[8]}$$

Hence, the DFT may be used to recover the EPG representation directly from an isochromat simulation, with the resulting Fourier coefficients corresponding to the configuration state coefficients. There are two restrictions: i)$$$\psi_m$$$ must be selected according to Eq.7; ii) N≥K where K is the number of non-zero transverse configuration states– if N<K then aliasing will occur (i.e. Nyquist sampling condition).

**Maximum configuration
order of selected sequences**

Figure 1 depicts phase diagrams for SPGR (Fig.1a) and FSE
(Fig.1b) sequences. For SPGR we are interested in F_{0} directly after
each RF pulse; after p pulses** K _{SPGR}=2p-1**.
For FSE sequences we are interested in F

**1. SPGR with differing
numbers of Isochromats**

SPGR was simulated for p=100 RF pulses. Figure 2 demonstrates exact agreement between
EPG and isochromat simulations for N=199(=K_{SPGR}). Figure 3 shows the
same results computed with smaller N: for N<199, FFT(M+) shows
aliasing effects. These manifest themselves as errors in $$$\tilde{S}$$$
for N<100, when aliasing corrupts the
prediction at n=0.

**2. FSE with differing
numbers of Isochromats**

FSE was simulated for p=100 refocusing pulses. Figure 4 shows that exact agreement is obtained for N=399(=K_{FSE}). Aliasing
results when N<K_{FSE} however errors in $$$\tilde{S}$$$ only result when N≤200
and aliases corrupt the n=0 prediction. There is a marked difference in error
for odd and even N.

**3. SPGR
with diffusion
**

SPGR was simulated for p=1500 pulses with N=100 for a range
of Φ, with and without diffusion implemented by circular convolution^{3} (Figure 5). Without diffusion, N=100 results in large
errors in the isochromat calculation; inclusion of diffusion drastically reduces
errors since it acts as a low-pass filter on F_{n} – i.e. reduced
numbers of isochromats can still reproduce the isochromat signal distributions
without aliasing.

1. Hennig J. Multiecho imaging sequences with low refocusing flip angles. J Magn Reson 1988;78:397–407.

2. Weigel, M. Extended phase graphs: Dephasing, RF pulses, and echoes - pure and simple. J. Magn.Reson.Imaging 41, 266–295 (2015).

3. Gudbjartsson, H. & Patz, S. Simultaneous Calculation of Flow and Diffusion Sensitivity in Steady-State Free Precession Imaging. Magn.Reson.Med. 34, 567–579 (1995).

Figure 1: Phase diagrams for (a) SPGR and (b) FSE sequences.
Black lines denote transverse states Fn, blue dotted lines
longitudinal states Zn. After p pulses, the number of non-zero Fn
is 2p-1 for SPGR and 4p-1 for FSE (e.g. p=6). Increasing the number of pulses
linearly increases the maximum configuration order.

Figure 2: Spoiled Gradient echo (SPGR) simulation using
EPG and isochromat-summation with TR=5ms/θ=10°/T1=1500ms/T2=500ms,
and RF spoiling phase increment Φ=120°. Predicted signals are identical. The
EPG prediction Fn can be exactly reproduced from fft(M+),
likewise the isochromat distribution M+ can be exactly obtained from
ifft(Fn).

Figure 3: SPGR simulations using same parameters as
Fig.2 with different numbers of isochromats. Reduction of N leads to aliasing
for N<2p-1. However errors in predicted signal only occur for N<p: at
this level the aliasing affects the average of the isochromat ensemble – i.e.
the alias crosses the n=0 line in fft(M+).

Figure 4: FSE isochromat-summation simulations using p=100
50° refocusing pulses and T1=1500ms/T2=500ms/echo-spacing=5ms.
Aliasing is visible for N<4p-1, however errors in signal only occur for N<2p,
when aliases cross the n=0 axis. Different error levels occur for odd and even
numbers of isochromats because the odd numbered configuration states have low
populations.

Figure
5: SPGR simulation for different RF spoiling phase with and without diffusion
effects. Simulations ran for 5*T1/TR=1500 pulses but for only 100 isochromats
–i.e. N<<p. (a) Without diffusion severe aliasing occurs; (b) with
diffusion the simulations agree well. (c)&(d) Show EPG representations:
un-shaded areas indicate the FOV for N=100.
Diffusion severely restricts the ‘bandwidth’ of
Fn, hence aliasing is avoided even with restricted N.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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