Shu Zhang^{1}, Robert E Lenkinski^{1,2}, and Elena Vinogradov^{1,2}

bSSFP was shown to be sensitive to exchange and is explored as
an alternative way for CEST/T_{1ρ} experiments (the bSSFPX method). In
this abstract, an analytical solution is derived for the magnetization behavior
in the bSSFPX. The solution describes the transient signal fluctuations for
short saturation times. The solution is verified by comparing it to full,
step-wise computation of Bloch-McConnell Equations. Overall, this solution is
in good agreement with Bloch-McConnell Equations and follows the transient signal
oscillations well. Work is in progress to use this solution to quantify
exchange rates experimentally.

**Purpose**

**Theory**

Consider only 1 pool first. In a bSSFP sequence,
^{5 }if the relaxation is ignored, $$M_{n+1}=R_z(\theta)R_x(\alpha)M_n=R_{\overrightarrow{e}}(\Phi)M_n=e^{H_{eff}TR}M_n~~~~(1)$$where $$$M_{n+1}$$$ and $$$M_{n}$$$ are
the magnetization vectors at the end of $$$n+1$$$ and $$$n$$$ cycle. $$$R_z(\theta)$$$ and $$$R_x(\alpha) $$$ are
the standard z- and x-rotation matrices respectively, where $$$\theta=2\pi\Delta{TR}$$$ is
the precession angle during TR and $$$\Delta$$$ is
the off-resonance shift. According to the Euler theorem, $$$R_z(\theta)R_x(\alpha)$$$ can
be represented by a single rotation $$$R_{\overrightarrow{e}}(\Phi)$$$, where $$$\overrightarrow{e}$$$ is the
directionality vector and $$$\Phi$$$ is
the rotation
angle about $$$\overrightarrow{e}$$$. The
effective
field (Fig.1) is
in the same direction as $$$\overrightarrow{e}$$$ and
with the
magnitude $$$\omega_{eff}^{bSSFP}$$$: $$\omega_{eff}^{bSSFP}=\Phi/TR~~~~(2)$$The $$$e^{H_{eff}TR}$$$ is
the matrix exponential form of $$$R_{\overrightarrow{e}}(\Phi)$$$.

When relaxation is included, Eq.1 becomes: $$ M_{n+1}=e^{(H_{eff}-\widetilde{R})TR}M_n-[I-e^{(H_{eff}-\widetilde{R})TR}](H_{eff}-\widetilde{R})^{-1}R_1M_0~~~(3)$$where $$$\widetilde{R}=diag(R_2,R_2,R_1)$$$ is the standard relaxation matrix and $$$M_0=[0,0,M_{0}]^T$$$. From Eq.3, the steady-state magnetization $$$M_{ss}$$$ can be obtained: $$M_{ss}=-(H_{eff}-\widetilde{R})^{-1}R_1M_0~~~(4)$$ $$$H_{eff}$$$ can be expressed in its eigenbasis as $$$H_{eff}=D\Lambda_{eff}D^{-1}$$$ where $$$D$$$ is the diagonalization matrix and $$$\Lambda_{eff}$$$ is the eigenvalue diagonal matrix. $$$\widetilde{R}$$$ can also be diagonalized in the eigenbasis of $$$H_{eff}$$$, which yields $$$\widetilde{R}=D\Lambda_{R}D^{-1}$$$ where $$$\Lambda_{R}=diag(R_{1\rho},R_{2\rho},R_{2\rho})$$$, $$$R_{1\rho}=cos^2(\Theta)R_1+sin^2(\Theta)R_2$$$, $$$R_{2\rho}=sin^2(\Theta)R_1+cos^2(\Theta)R_2$$$, and $$$\Theta$$$ is the angle between the effective field and the Z-axis (Fig.1). Using the eigensystem of $$$H_{eff}$$$, the $$$M_{ss}$$$ becomes: $$M_{ss}=-D\Lambda^{-1}D^{-1}R_1M_0~~~(5)$$where $$$\Lambda=\Lambda_{eff}-\Lambda_{R}$$$.

Furthermore, the time-dependent magnetization can be derived from Eqs.3 and 5: $$M_n=M_{ss}+De^{\Lambda{\cdot}nTR}D^{-1}(M_0-M_{ss})~~~(6)$$

Now consider 2 pools, water and solute, coupled
by exchange. When exchange exists, an additional term $$$R_{ex}$$$ is
introduced to the water via $$$R_{1\rho}$$$ and $$$R_{2\rho}$$$: $$$R_{1\rho,w}=cos^2(\Theta_w)R_{1w}+sin^2(\Theta_w)(R_{2w}+R_{ex})$$$ and $$$R_{2\rho,w}=sin^2(\Theta_w)R_{1w}+cos^2(\Theta_w)(R_{2w}+R_{ex})$$$. Here we follow Trott and Palmer’s work ^{6}
for $$$R_{ex}$$$
(asymmetric population limit): $$R_{ex}=\frac{p_wp_s\Delta_{ws}^2k}{(w_{eff,s}^{bSSFP}/2\pi)^2+k^2}=\frac{p_wp_s\Delta_{ws}^2k}{\Phi_s^2/(2\pi{TR})^2+k^2}~~~(7)$$where $$$p_w=M_{0w}/(M_{0w}+M_{0s})$$$, $$$p_s=M_{0s}/(M_{0w}+M_{0s})$$$, $$$\Delta_{ws}$$$ is the chemical shift between water and solute, and $$$k=k_{sw}+k_{ws}$$$, $$$k_{sw}$$$
and $$$k_{ws}$$$ are
exchange rates from solute to the water pool and vice-versa.

**Methods**

Fig.2 compares the steady-state spectra obtained using the analytical solution with the exact BME simulations. Fig.3 shows the time evolution of the Z- and XY-spectra. The analytical solution is in excellent agreement with BME for both Z- and XY-spectrum for the 1-pool model.

For the exchanging 2-pools, the solution is in
good qualitative agreement with BME. At the steady-state, the solution follows BME
for the most part but deviates slightly downward at the star labeled frequencies
close to water (Fig.2b,d). In Fig.3, the overall behavior and the agreement at both
short and long t_{sat} is excellent, indicating that the solution
correctly depicts transient oscillations with T_{2ρ} and T_{1ρ}
relaxations. However, large deviations were observed close to the saturation
bands for short t_{sat} (t_{sat}<3T_{1w}). These
results indicate that the exchange contribution (Eq.7) needs to be further
improved.

**Conclusion**

1. Zhang S, Zheng L, Lenkinski RE, et al. in: Proc Intl Soc Mag Reson Med. 2015;23:0785.

2. Zhang S, Lenkinski RE, Vinogradov E. in: Proc Intl Soc Mag Reson Med. 2016;0304.

3. McMahon MT, Gilad AA, Zhou J, et al. Quantifying exchange rates in chemical exchange saturation transfer agents using the saturation time and saturation power dependencies of the magnetization transfer effect on the magnetic resonance imaging signal (QUEST and QUESP): pH calibration for poly-L-lysine and a starburst dendrimer. Magn Reson Med. 2006;55(4):836-847.

4. Zaiss M, Bachert P. Chemical exchange saturation transfer (CEST) and MR Z-spectroscopy in vivo: a review of theoretical approaches and methods. Phys Med Biol. 2013;58:R221-R269.

5. Freeman R, Hill H. Phase and intensity anomalies in Fourier transform NMR. J Magn Reson (1969). 1971;4(3):366-383.

6. Trott O, Palmer III AG. R1ρ Relaxation outside of the Fast-Exchange Limit. J Magn Reson. 2002;154:157-160.

Figure 1. The effective RF field of bSSFP. $$$\Theta$$$ is
the angle between the effective field and the Z-axis. $$$\theta=2\pi\Delta{TR}$$$ is
the precession angle during TR, where $$$\Delta$$$ is
the off-resonance shift.

Figure 2. The simulated steady-state Z- (a,b) and XY-spectra
(c,d) for 1-pool model (a,c) and 2-pool model (b,d). Circle: Bloch-McConnell
equations. Solid line: the derived analytical solution. *: solution deviates
slightly downward from the Bloch-McConnell equations.

Figure 3. The simulated time-dependent Z- (a,b) and XY-spectra
(c,d) for 1-pool model (a,c) and 2-pool model (b,d). Circle: Bloch-McConnell
equations. Solid line: the derived analytical solution. Different colors from
light to dark correspond to different saturation times: 0.15, 0.25, 0.5, 1, and
6s.