Steffen Goerke^{1}, Johannes Windschuh^{1}, Moritz Zaiss^{1,2}, Jan-Eric Meissner^{1}, Mark E Ladd^{1}, and Peter Bachert^{1}

A
novel concentration-independent approach is presented to determine the pH-dependence
of exchange rates employing a single CEST image of a set of model solutions at
different pH. Not only the comparatively short acquisition time, but also the
robustness against variations in relaxation parameters makes this modality an
elegant way to determine exchange rates *in
vitro*. The calibrated functions are required for accurate pH mapping *in vivo* using CEST, as well as for
design of exogenous CEST contrast agents.

The
isolated CEST signal calculated by the apparent exchange dependent relaxation
(AREX)^{4} can be described as followed^{2}:

$$AREX = c_{1}\cdot DC\cdot f\cdot k_{sw}\frac{(γB_{1})^{2}}{(γB_{1})^{2}+c_{2}^{2}\cdot k_{sw}(k_{sw}+R_{2s})}$$

with
the relative proton fraction f and the pulsed saturation parameters: mean amplitude B_{1}, duty cycle
DC, and the form factors c_{1,2} considering the pulse shape. AREX as a
function of k_{sw} reaches a maximum at $$$k_{sw,max}=\frac{γB_{1}}{c_{2}}$$$ (Fig. 2a) independent of concentration and R_{2s} (Fig.
2b).

Assuming
a base-catalyzed exchange process: $$$k_{sw}=k_{b}\cdot 10^{pH–pK_{w}}=k_{c}\cdot 10^{pH}$$$, AREX (eq. 1) can be transformed into
a function of pH (Fig. 2c,d). The exchange process and thus also the transformation is
fully characterized by the pre-exponential factor $$$k_{c}=k_{b}\cdot 10^{–pK_{w}}$$$.

CEST
image data of Rerich et al.^{5} was used for evaluation. Model solutions containing 50 mM creatine at
different pH ranging from 6.3 to 7.6 were measured at 37 °C.

$$$AREX(Δω)=\frac{1}{T_{1}}\cdot (\frac{1}{Z}–\frac{1}{Z_{ref}})$$$ at frequency offset Δω was
calculated using the other side of the Z-spectrum as the reference Z_{ref}.

CEST
imaging was performed on a 7 T whole body MR tomograph (MAGNETOM 7T, Siemens
Healthineers, Germany). Pre-saturation was achieved by
a train of 50 Gaussian-shaped RF pulses (c_{2}=0.6171, t_{pulse}=100ms, DC=50%, t_{sat}=10s) with a mean amplitude B_{1} ranging from 1.2 to 3.1 µT. AREX was
corrected for B_{0}- and B_{1}-inhomogeneities.

An
analytical form of the function AREX(pH) was derived. The expression for the position
of its maximum pH_{max} (eq. Fig. 2c) allows direct calculation of k_{c}
and hence full quantification of the exchange process. Remarkably, pH_{max}
is independent of R_{2s} (Fig. 2d) leading to a unique accuracy for the
determined exchange rates. In addition, the half-width of the symmetric
resonances AREX(pH) is nearly constant under variations of B_{1} (Fig.
2c) and R_{2s} (Fig. 2d), which facilitates a robust fitting of the
function and consequently robust determination of pH_{max}.

Experimental
AREX values of creatine as a function of pH (Fig. 3a) agree well with theoretical
expectations (Fig. 2c). For full quantification of the exchange process,
acquisition of data at one B_{1} is sufficient. However, to demonstrate
the robustness of the presented method, AREX values at several B_{1}
were evaluated. The calculated values k_{c} agree very well, with a
mean value of (70.7 ± 0.9) µHz. In a comparison to the reference value
determined by the Ω-plot method^{1,2} k_{c} = (66.5 ± 6.0) µHz
the error was reduced approximately by an order of magnitude.

The
presented method is a powerful tool to robustly quantify exchange rates as a
function of pH with a unique accuracy. It was already shown by Woessner et al.
that the maximal CEST signal yields insight into the exchange rate.^{6}
We were able to extend this insight by showing that a full characterization of
the exchange process is possible by acquisition of just one AREX image at one
specific B_{1}. This allows a high throughput quantification of samples
and therefore e.g. to investigate the exchange processes under different
molecular environments. In contrast, the concentration-independent Ω-plot
method^{1,2} requires a series of AREX images at several B_{1}.

In
this study, the method was verified under the assumption of a dominant
base-catalyzed exchange, which is correct for the CEST signals appearing *in vivo* at an intermediate B_{1}
around 1 µT. Nonetheless, the theory is also extendable to acid-catalyzed exchange
processes.

Finally,
the method was used to establish calibration functions for amide (Δω = 3.5 ppm)
and guanidinium (Δω = 2.0 ppm) protons *in
vivo*. Investigation of homogenized pig brain tissue (data not shown) led to
k_{c} = 1.54 and 85.1 µHz for amide and guanidinium protons,
respectively. Corresponding exchange rates under physiological conditions (pH 7.1 and 37 °C) are 19.4 and 1071 Hz, respectively.

1. Dixon WT, Ren J, Lubag AJM, et al. A Concentration-Independent Method to Measure Exchange Rates in PARACEST Agents. Magn Reson Med 2010;63:625-632.

2. Meissner J-E, Goerke S, Rerich E, et al. Quantitative pulsed CEST-MRI using Ω-plots. NMR Biomed 2015;28(10):1196-1208.

3. Sun PZ. Xiao G, Zhou IY, et al. A method for accurate pH mapping with chemical exchange saturation transfer (CEST) MRI. Contrast Media Mol Imaging 2016;11(3):195-202.

4. Zaiss M, Xu J, Goerke S, et al. Inverse Z-spectrum analysis for spillover-, MT-, and T1-corrected steady-state pulsed CEST-MRI – application to pH-weighted MRI of acute stroke. NMR Biomed. 2014;27(3):240-252.

5. Rerich E, Zaiss M, Korzowski A, et al. Relaxation-compensated CEST-MRI at 7 T for mapping of creatine content and pH – preliminary application in human muscle tissue in vivo. NMR Biomed. 2015;28(11):1402-1412.

6. Woessner DE, Zhang S, Merritt ME, et al. Numerical Solution of the Bloch Equations Provides Insights Into the Optimum Design of PARACEST Agents for MRI. Magn Reson Med. 2005;53:790-799.

Determination of pH_{max} in a multi-pH phantom
directly enables calculation of the pre-exponential factor k_{c}, which
fully characterizes the pH-dependence of the exchange rate k_{sw}. The
saturation amplitude B_{1} has to be chosen such that variations in the
AREX contrast due to labeling are covered sufficiently.

Simulations of the AREX signal (eq. 1) as a
function of the exchange rate k_{sw} for varying saturation amplitudes
B_{1} (a) and transversal relaxation rates R_{2s} (b). AREX
reaches a maximum, whose position k_{sw,max} is independent of concentration and R_{2s}.
Transformation of the k_{sw}-axis into pH-values leads to formation of
symmetric resonances of define half-widths (c,d).

(a) AREX signal of creatine guanidinium protons at
Δω = 1.9 ppm as a function of pH. The position of the maximum pH_{max}
was determined by a fit using equation 1. (b) For each B_{1}, one value
k_{c} can be calculated. As a reference, k_{c} was additionally
determined by the Ω-plot method^{1,2} (black lines).