The Reciprocal Linear QUEST Analysis Method Facilitates the Measurements of Chemical Exchange Rates with CEST MRI

Abstract

MRI contrast media that are detected via Chemical Exchange Saturation Transfer (CEST) often require an accurate estimation of their chemical exchange rate, k_ex. A variety of analysis methods have been proposed to estimate k_ex, including the non-linear QUEST analysis method that evaluates the CEST amplitude as a function of saturation time. We have derived a linear version of QUEST, termed the Reciprocal Linear QUEST (RL-QUEST) method. Our simulations and experimental results show that RL-QUEST performs as well as QUEST, while providing a more simplistic fitting procedure. Although CEST results should be acquired with saturation power that has a nutation rate that is faster than k_ex of the CEST agent, an exact determination of the saturation power is not required to accurately estimate k_ex with RL-QUEST.This new analysis method requires a determination of the CEST agent’s concentration, which is straightforward for the analyses of CEST agents in chemical solutions, but may be a limitation during in vivo CEST MRI studies. Based on the results of this study and previous studies, we provide recommendations for the linear analysis method that should be employed for each type of CEST MRI study.

Introduction

A variety of MRI contrast media have been developed that can be detected via Chemical Exchange Saturation Transfer (CEST) (1). During a CEST MRI experiment, radio frequency saturation is applied at the chemical shift of a labile proton on the CEST agent, eliminating the coherent net magnetic moment of this proton. Chemical exchange of the labile proton from the CEST agent to water results in transferring the effect of saturation to the water MR resonance. The resulting detection of the partially suppressed water signal using standard MRI protocols can be iterated over a range of saturation frequencies, which can be used to construct a CEST spectrum thatis used to measure CEST (Fig. 1) (2).

Figure 1.

CEST spectra of iopromide. A) A simulated CEST spectrum of iopromide at 4 μT saturation power and 0.5secsaturation time. The chemical structure of iopromide is shown in the inset. B) A simulated CEST spectrum of iopromide at 9 μT saturation power and 0.25secsaturation time (dots). Bloch fitting without a Gaussian point spread function showed a poor fit to the simulated spectrum (gray line), while Bloch fitting with a Gaussian point spread function using a sigma of 56Hz showed a good fit to the simulated spectrum (black line).

The development of a CEST agent often includes a measurement of the chemical exchange rate, k_ex, to ensure that the agent has acceptable chemical exchange properties for a specific biomedical application and/or for selecting the best CEST MRI detection protocol for the agent (3-8). A variety of methods have been created to estimate k_ex from CEST MRI results, including fitting CEST spectra with Bloch-McConnell equations modified for chemical exchange (also known as Bloch fitting), and the QUEST and QUESP methods that are dependent on saturation time and saturation power, respectively (9,10). Bloch fitting is a very accurate and precise method, but requires expertise and time to ensure that the fitting is properly performed. Although QUEST and QUESP are non-linear fitting methods, both of these methods offer a more simplistic fitting process relative to Bloch fitting. We and others have developed linear versions of QUESP that further simplify the fitting procedure (11,12).

We hypothesized that a linear QUEST analysis method may offer the same practical advantages for estimating k_ex from CEST results as shown from the linear versions of QUESP analysis. We investigated several linear versions of QUEST using simulated results that had known chemical exchange rates. We then measured CEST from iopromide (Ultravist™, Bayer Health Care, Inc.), and compared the estimates of k_ex with non-linear QUEST and a linear QUEST method. These results have led us to propose recommendations for the use of various analysis methods for estimating k_ex using CEST MR studies.

Theory

Assuming that a CEST agent instantaneously reaches steady-state saturation (but does not necessarily reach complete saturation), a previous report has shown that CEST can be described as a function of saturation time (t_sat) (Eq. [1]) (10).

$$\frac{M_0-M_S}{M_0}=\frac{\alpha k_{\mathit{ex}}\chi }{R_{1W}+k_{\mathit{ex}}\chi }[1-e^{(R_{1W}+k_{\mathit{ex}}\chi )t_{\mathit{sat}}}]$$ [1]

where:

$$\chi =\frac{n_{\mathit{CA}}\left[\mathit{CA}\right]}{n_{H_{2}O}\left[H_{2}O\right]}$$ [2]

$$\alpha =\frac{{\omega }_1^2}{{\omega }_1^2+R_1{R}_2}$$ [3]

$$R_{1,2}=R_{1A,2A}+k_{\mathit{ex}}-\frac{k_{\mathit{ex}}^2\chi }{R_{1W,2W}+k_{\mathit{ex}}\chi }$$ [4]

M_S, M₀: water signal with saturation (S) and without saturation (0) [CA], [H₂O]: concentration of the contrast agent and water n_CA, n_H2O: number of magnetically equivalent exchangeable protons on the agent and water R_1A, R_2A, R_1w, R_2w: R₁ and R₂ relaxation rates of the agent (A) and water (w)

The QUEST method fits the exponential term of Eq. [1] to CEST measurements determined with a range of saturation times. Eq. [1] can be simplified for CEST measured with a very long saturation time (Eq. [5]). The ratio of CEST measurements at short saturation times relative to a very long saturation time is only dependent on an exponential term (Eq. [6]). This non-linear relationship can be rearranged to create linear relationships (Eq. [7]).

$${\left(\frac{M_0-M_S}{M_0}\right)}_{t=\infty }=\frac{\alpha k_{\mathit{ex}}\chi }{R_{1W}+k_{\mathit{ex}}\chi }$$ [5]

$$\frac{{\left(\frac{M_0-M_S}{M_0}\right)}_{t<\infty }}{{\left(\frac{M_0-M_S}{M_0}\right)}_{t=\infty }}=1-e^{-(R_{1W}+k_{\mathit{ex}}\chi )t_{\mathit{sat}}}$$ [6]

$$-\ln \left[1-\frac{{\left(\frac{M_0-M_S}{M_0}\right)}_{t<\infty }}{{\left(\frac{M_0-M_S}{M_0}\right)}_{t=\infty }}\right]=(R_{1W}+k_{\mathit{ex}}\chi )t$$ [7a]

$$\frac{1}{-\ln \left[1-\frac{{\left(\frac{M_0-M_S}{M_0}\right)}_{t<\infty }}{{\left(\frac{M_0-M_S}{M_0}\right)}_{t=\infty }}\right]}=\frac{1}{(R_{1W}+k_{\mathit{ex}}\chi )t}$$ [7b]

$$1=\frac{-\ln \left[1-\frac{{\left(\frac{M_0-M_S}{M_0}\right)}_{t<\infty }}{{\left(\frac{M_0-M_S}{M_0}\right)}_{t=\infty }}\right]}{(R_{1W}+k_{\mathit{ex}}\chi )t}$$ [7c]

Equation 7a is known as the Linear QUEST (L-QUEST) method and places greater weight on CEST measured with longer saturation times. Equation 7b is known as the Reciprocal Linear QUEST (RL-QUEST) method and places greater weight on CEST measured with shorter saturation times. The chemical exchange rate, k_ex, can then be determined from the slope of these linear equations, assuming that the concentration of the agent is known and is sufficiently high that k_exχ>> R_1w. Equation 7c is known as the Unitary Linear QUEST (UL-QUEST) method and places equal weight on CEST measured at all saturation times. A value of k_ex can be determined at each time point, and the average value of k_ex can be used as the estimate of the chemical exchange rate.

Results

Simulations

The Bloch-McConnell equations were used to generate CEST spectra of iopromide using a range of chemical exchange rates and saturation times (Fig. 1). Each CEST spectrum was fit with a function of Lorentzian line shapes to measure the CEST effects of both amide protons, to simulate our protocol used to analyze experimental CEST spectra. Previous reports have shown that the Lorentzian line shape fitting can measure CEST with excellent accuracy and can account for the effects of direct saturation of water (13).

The CEST measurements were used to estimate exchange rates using the QUEST method and the three linear QUEST methods (Fig. 2). The QUEST method showed an outstanding fit to the simulated results throughout the entire range of saturation times. Of the three linear QUEST methods, RL-QUEST places greater weight on CEST measured with shorter saturation times, which results in relationship that has the largest dynamic range. Based on these results, only QUEST and RL-QUEST were compared for the remainder of this study.

Figure 2.

Analyses of chemical exchange rates using A) the QUEST method, B) the L-QUEST method, C) the RL-QUEST method, and D) the UL-QUEST method. The slope of the L-QUEST plot or the RL-QUEST plot can be used to measure the chemical exchange rate, k_ex Each data point of the UL-QUEST plot can be used to determine k_ex, and the average of these determinations of k_ex can be used as the estimate of the chemical exchange rate.

We compared chemical exchange rates estimated with QUEST and RL-QUEST using different saturation powers (Fig. 3). The nearly identical estimates from both methods showed that the simplistic fitting with LR-QUEST can substitute for the more complicated non-linear fitting with QUEST. This result also shows that both methods can accurately estimate slow chemical exchange rates, but faster chemical exchange rates are underestimated. This underestimation can be mitigated by using high saturation powers, as shown previously for QUEST analyses (10). These results support the guideline that the nutation rate of the saturation pulse must be greater than the chemical exchange rate to accurately measure exchange rates with CEST MR methods (9).

Figure 3.

The dependence of QUEST and RL-QUEST on saturation power. The chemical exchange rate, k_ex, estimated with QUEST (dashed lines) and RL-QUEST (solid lines) from simulated CEST spectra showed accurate estimations of slow exchange rates at all saturation powers, but underestimated fast exchange rates especially at low saturation powers. The underestimations were comparable for QUEST and RL-QUEST. A dotted line with slope=1 is shown to aid visualization of the results.

Because obtaining a CEST effect with an infinite saturation time is impractical, we analyzed the effect of truncating the CEST_t=∞ saturation time on the estimate of the chemical exchange rate (Fig. 4). These results showed that chemical exchange rate was overestimated with a truncated saturation time used for CEST_t=∞. However, this overestimation was negligible for long saturation times of 1 to 4 seconds that are practical to implement. Furthermore, a 50% reduction in CEST_t=∞ with a saturation time of 0.25 sec (Fig. 4B) only produced a 5% overestimation in the chemical exchange rate (Fig. 4C), which further demonstrated the insensitivity of estimating chemical exchange rates with finite saturation times used to measure CEST_t=∞.

Figure 4.

The dependence of estimated k_ex on CEST_tsat=∞ when using RL-QUEST. A) Short saturation times for CEST_tsat=∞ caused lower CEST. B) An underdetermined CEST_tsat=∞ causedk_ex to be overestimated. C) k_ex was only overestimated by 4% when CEST_tsat=∞ was underdetermined by 50%, demonstrating that estimates of k_ex are insensitive to underdetermined CEST_tsat=∞ values when using RL-QUEST.

Experimental Results

CEST spectra were acquired at various saturation times and saturation powers. The saturation powers were calibrated using a 360° excitation pulse to mitigate the effects of radiation damping. Bloch fitting was used to simultaneously fit to a series of 36 CEST spectra of iopromide that were acquired with different saturation times and saturation powers (Fig. 5) (14). Our past results have shown that simultaneously fitting a series of CEST spectra produces more precise estimates of k_ex relative to fitting a single spectrum (15). The chemical exchange rates of the amide proton resonating at 4.2 ppm and 5.6 ppm were estimated to be 133.1 and 1130.8 Hz, respectively (Table 1).

Figure 5.

The estimation of experimental k_ex. A) A CEST spectrum of iopromide was acquired with 4.45 uT saturation power and 4 second saturation time. The kex of the proton resonating at 4.2 ppm was estimated with B) The QUEST method and C) the RL-QUEST method. The k_ex of the proton resonating at 5.6 ppm was also estimated with D) The QUEST method and E) the RL-QUEST method. Experimental data points are shown as circles, the fitted result is shown as a black line, and the 95% confidence intervals are shown as gray lines.

Table 1.

Estimates of Chemical Exchange Rates (Hz)

Method	4.2 ppm	5.6 ppm
Bloch Fitting	133.1 (117.5, 150.7)    [−11.7%, +13.2%]	1130.8 (1074.3, 1190.2)    [−5.00%,+5.25%]
QUEST	80.0 (74.4, 86.1)    [−7.0%, +7.6%]	815.0 (726.3, 914.4)    [−10.9%, +12.2%]
RL-QUEST	112.8 (99.6, 130.0)    [−11.7%, +15.2%]	824.6 (792.3, 859.6)    [−3.9%, +4.2%]

The 95% confidence intervals expressed in Hz are shown in parentheses.

The 95% confidence intervals expressed as a percentage of the estimated exchange rate are shown in brackets.

Lorentzian line shapes were fit to each experimental CEST spectrum to measure CEST (Fig. 5A). These results were then used to estimate the chemical exchange rates. The amide proton resonating at 4.2 ppm was estimated to have a k_ex value of 80.0 and 112.8 Hz using QUEST and RL-QUEST, respectively (Fig. 5B,C; Table 1). This result indicated that RL-QUEST provided a more accurate estimate than QUEST, using results from Bloch fitting as the “gold standard”. These estimates used experimental results acquired with 2.25 μT saturation power. Lower saturation powers caused estimates of k_ex to be underestimated, because low saturation powers and short saturation times caused weak CEST effects relative to CEST spectral noise, which reduced the accuracy of Lorentzian line fitting (Fig. 6A). Higher saturation powers caused estimates of k_ex to be overestimated, because higher saturation powers caused wider, overlapping CEST peaks at 4.2 and 5.6 ppm, which also reduced the accuracy of Lorentzian line fittings.

Figure 6.

The estimated experimental chemical exchange rates depend on saturation power. A) The estimated k_ex of the slowly-exchanging proton resonating at 4.2 ppm showed similar estimates with RL-QUEST (squares) and QUEST (circles). Low saturation powers improved the Lorentzian line shape fitting of experimental CEST spectra, which led to accurate estimations of k_ex relative to the k_ex estimated with Bloch fitting (dashed line). However, very low saturation powers led to inaccurate estimations of k_ex due to low CEST effects. B) The estimated k_ex of the rapidly-exchanging proton resonating at 5.6 ppm showed similar estimates with RL-QUEST (circles) and QUEST (squares), but these estimates were underestimated relative to the k_ex estimated with Bloch fitting (dashed line), especially with low saturation powers.

The amide proton resonating at 5.6 ppm was estimated to have a k_ex value of 815.0 and 824.6 Hz using QUEST and RL-QUEST, respectively (Fig. 5D,E; Table 1). Again, this result indicated that RL-QUEST provided a more accurate estimate than QUEST, using results from Bloch fitting as the “gold standard”. These estimates were obtained with a saturation power of 8 μT, which has the highest nutation rate of the saturation powers used in this study. The nutation rate of the saturation at this power level was below the proton’s rapid chemical exchange rate, causing QUEST and RL-QUEST to underestimate the rapid chemical exchange rate. Lower saturation powers caused greater underestimations of k_ex for this rapidly exchanging amide proton (Fig. 6B). This experimental underestimation matched the results from the simulations (Fig. 3).

Discussion

Our simulations and experimental results demonstrated that RL-QUEST performs as well as QUEST when estimating chemical exchange rates. RL-QUEST has practical advantages relative to QUEST, because linear fitting methods are faster than non-linear fitting methods. Furthermore, RL-QUEST has y-intercept value that is zero, so that RL-QUEST analysis may be performed with only two CEST measurements (at t <∞ and t ≈∞), which further accelerates the analysis relative to non-linear QUEST.

Both QUEST and RL-QUEST determine a more accurate k_ex when a high saturation power is used or when k_ex is slow. In addition, these methods require that the concentration of the agent must be known to determine k_ex. For comparison, spin-lock MRI experiments can determine k_ex by measuring the R_1ρ relaxation rate in the rotating frame (15,16). R_1ρcan be converted to k_ex under strong saturation conditions, slow R₁ and R₂ relaxation rates, slow k_ex rates, and when the concentration of the agent is known. These similarities between QUEST, RL-QUEST and spin-lock experiments have been noted previously (16,17).

We have recently completed a similar evaluation of the non-linear QUESP analysis method and linear versions of QUESP, which evaluate the change in CEST as a function of power (18). Similar to this study, our evaluation of linear QUESP methods showed the advantages of using a linear version known as LB-QUESP when estimating slow k_ex values, and another linear version known as HW-QUESP when estimating fast k_ex values. However, these linear versions of QUESP depend on accurately determining the values of saturation powers (for QUESP and LB-QUESP) or accurately determining the increments of saturation powers (for HW-QUESP). The calibration of saturation power can be challenging during NMR studies due to radiation damping. More importantly, saturation power may be variable during in vivo MRI studies due to B₁ inhomogeneity. For comparison, the accuracy of estimating k_ex with RL-QUEST does not depend on the accuracy of determining the saturation power. RL-QUEST only requires a sufficient saturation power so that the nutation rate exceeds the chemical exchange rate.

RL-QUEST is complimentary to QUESTRA, which is another linear version of QUEST analysis (19). QUESTRA evaluates the ratio of water signals in a CEST spectrum at opposite chemical shift values about the direct saturation of water at 0 ppm (which a version of asymmetry analysis of a CEST spectrum). The simulations of QUESTRA showed that k_ex could be accurately estimated as long as the nutation rate of the saturation power exceeds k_ex. Only one CEST spectrum is required for QUESTRA analysis, while two CEST spectra are required for RL-QUEST analysis (at t <∞ and t ≈∞), so that QUESTRA may be faster than RL-QUEST. However, a linear analysis based on one experimental data point and a zero y-intercept is highly dependent on the variability of the single data point, so that CEST spectra with multiple saturation times are recommended for both methods to improve the quality of the fitting process.

QUESTRA automatically accounts for overlap between the peaks in a CEST spectrum that represent CEST from an amide proton and the direct saturation of water. However, QUESTRA does not account for overlapping CEST effects such as the effects observed with iopromide. In addition, QUESTRA does not account for asymmetric effects in CEST spectra such as the endogenous MT effect observed during CEST studies with tissues (20). QUESTRA is also dependent on B₀ homogeneity. As shown in our previous studies and in our current study with RL-QUEST, Lorentzian line fittings of CEST spectra avoids complications caused by the direct saturation of water, overlapping CEST effects, other asymmetric features of CEST spectra including the MT effect, and B₀ inhomogeneities. Therefore, we recommend that QUESTRA analyses should also employ Lorentzian line fitting analyses, which should provide similar benefits shown in our RL-QUEST analyses.

To estimate k_ex, RL-QUEST and QUESTRA inherently require an understanding of the concentration of the CEST agent relative to the concentration of water. Measuring the concentration of a CEST agent is straightforward for NMR studies of chemical solutions, but can be challenging during in vivo MRI studies (21). Conversely, LB-QUESP and HW-QUESP can estimate k_ex without also requiring a measurement of concentration, which provides advantages for in vivo studies. Therefore, we offer the following recommendations:

• If the concentration is known,and the saturation power is allowed to be sufficiently high so that the nutation rate of the saturation exceeds k_ex, then use RL-QUEST or QUESTRA to determine k_ex.

• If the concentration is unknown, and saturation powers are allowed to be sufficiently high so that the nutation rate of the saturation exceeds k_ex, and the saturation powers can be accurately determined, then use LB-QUESPto determine k_ex.

• If the concentration is unknown, and if saturation powers are not allowed to be sufficiently high so that the nutation rate of the saturation exceeds k_ex, and the increments between saturation powers can be accurately determined, then use HW-QUESPto determine k_ex.

• If the concentration is unknown, and the increments between the saturation powers cannot be accurately determined, then use Bloch fitting to determine k_ex. Simultaneous Bloch fitting of a series of CEST spectra is recommended to improve the accuracy of estimating k_ex.

Conclusions

Simulations and experimental results demonstrate that RL-QUEST performs as well as QUEST when estimating chemical exchange rates. RL-QUEST is a linear fitting method with a zero y-intercept, which has practical advantages relative to the non-linear QUEST fitting method. RL-QUEST requires a sufficient saturation power, but does not depend on measurements of saturation powers, and therefore has advantages relative to other analysis methods that estimate chemical exchange rates from CEST measurements based on saturation powers. Yet RL-QUEST requires an understanding of the CEST agent’s concentration, which may limit the utility of this method.

Experimental

Simulations

The Bloch-McConnell equations modified for chemical exchange were used to generate CEST spectra of iopromide (9). All simulations used chemical shifts of 5.6 ppm and 4.2 ppm for each amide resonance, 0.0 ppm for the water resonance, and 0.87 ppm for the hydroxyl group resonance that was set to an exchange rate of 400 Hz. Each simulation tested saturation times of 0.25 to 10 seconds, chemical exchange rates from 10 to 1200 Hz, and T₁ relaxation times from 0.1 to 10 seconds. The saturation power was set to 4 μT, and the concentration was set to 200 mM.Lorentzian line shape fitting was used to measure the CEST effect (M_S) for each amide from each simulated spectrum, using spectra normalized from 0% to 100% water signal (which set M₀ equal to 100%) (13). The fittings were performed with Matlab v2011a (Mathworks, Inc. Natick MA) using the trust-region-reflective algorithm to reach a convergence criterion of 10⁻¹⁶_.

The QUEST method was performed by fitting Eq. [1] to the measured CEST effects of the simulated results to estimate the chemical exchange rate of the amide at 5.6 ppm (10). Fittings were performed by allowing R_1w to be a fitted parameter, which has been shown to produce accurate estimates of k_ex. R_2w was set to R_1w for all of these fittings. The three versions of the linear QUEST method were also used to estimate chemical exchange rates from the simulated CEST results. All fittings were performed with Matlab v2011a using the trust-region-reflective algorithm to reach a convergence criterion of 10⁻¹⁶_.

Experimental Studies

Iopromide (Ultravist™, Bayer Health Care, Inc.) was provided by the Department of Medical Imaging of the University of Arizona. Iopromide was diluted to 100 mM and adjusted to pH 6.8 using 1 M NaOH. After preparation, 10% D₂O was added to each solution to maintain a lock signal during NMR studies. The addition of D₂O caused deuteration of 10% of the labile protons of iopromide, which effectively reduced the concentration of protonated iopromide relative to protonated water. This dilution was taken into consideration when determining the value of χ.

All CEST experiments were performed using a 600 MHz Varian Inova NMR spectrometer with an inverse cryoprobe. Samples were analyzed at 37.3°C. The temperature was calibrated by measuring the separation of resonances of neat methanol and ethylene glycol samples between 25°C and 40°C (22). The probe was manually tuned to each sample, and the 360° pulse time was measured before acquiring each CEST spectrum to estimate the saturation power. The 360° pulse time was also measured after acquiring each CEST spectrum to ensure that the saturation power was constant when acquiring the CEST spectrum. The CEST spectra were acquired with a continuous-wave saturation pulse and saturation frequencies set at +8 to −8 ppm in 0.1 ppm increments. Each scan was averaged four times. Saturation times of 0.25, 0.5, 1.0, 2.0, 4.0, and 8.0 sec were tested with a relaxation delay from 7.75 to 0 seconds to maintain a total time of 8 seconds for the pre-acquisition period. These studies were conducted with a saturation power of 1.0, 1.5, 2, 3, 4, 6, and 8 μT.

Bloch fitting was used to simultaneously fit all experimental CEST spectra to obtain accurate estimates of the chemical exchange rate of the protons resonating at 5.6 and 4.2 ppm (14). Short pulse times of a continuous-wave saturation pulse can create a sinc-shaped saturation frequency profile, which can broaden the bandwidth of the saturation. To compensate for this broadening, each point of the experimental CEST spectrum was modeled as a Gaussian-shaped point spread function, which improved the fitting of the CEST spectrum acquired with a short 0.25 saturation time. The integrating factor method was used during the Bloch fitting. The lsqcurvefit routine with the trust-region-reflective algorithm was used to initially fit the CEST spectra, which resulted in rough parameter estimates. The nlinfit routine with the Levenburg-Marquardt algorithm was then used to fit the CEST spectra to obtain the final parameter estimates. The Jacobian from the nlinfit routine was used to determine confidence intervals for fitted parameters.

Lorentzian line shape fitting was used to measure the CEST effect (M_S) for each amide from each experimental spectrum as previously described, using spectra normalized from 0% to 100% water signal (which set M₀ equal to 100%) (13). The fittings were performed with Matlab v2011a using the trust-region-reflective algorithm to reach a convergence criterion of 10⁻¹⁶_.

The QUEST and RL-QUEST methods were used to estimate k_ex rates from the experimental CEST measurements determined from the Lorentzian line shape fittings (10). For the QUEST method, Eq. [1] was fit to the experimental results by varying k_ex and R_1w, while R_2w, R_1A, and R_2A were set to the value of R_1w, which has been previously shown to produce accurate estimates of k_ex. All fittings were performed with Matlab v2011a using the trust-region-reflective algorithm to reach a convergence criterion of 10⁻¹⁶_.