Quantitative separation of CEST effect by R_ex-line-fit analysis of Z-spectra

Abstract

The process of chemical exchange saturation transfer (CEST) is quantified by evaluating a Z-spectra, where CEST signal quantification and Z-spectra fitting have been widely used to distinguish the contributions from multiple origins. Based on the exchange-dependent relaxation rate in the rotating frame (R_ex), this paper introduces an additional pathway to quantitative separation of CEST effect. The proposed R_ex-line-fit method is solved by a multi-pool model and presents the advantage of only being dependent of the specific parameters (solute concentration, solute‐water exchange rate, solute transverse relaxation, and irradiation power). Herein we show that both solute‐water exchange rate and solute concentration monotonously vary with R_ex for Amide, Guanidino, NOE and MT, which has the potential to assist in solving quantitative separation of CEST effect. Furthermore, we achieve R_ex imaging of Amide, Guanidino, NOE and MT, which may provide direct insight into the dependency of measurable CEST effects on underlying parameters such as the exchange rate and solute concentration, as well as the solute transverse relaxation.

Introduction

Chemical exchange saturation transfer (CEST)-magnetic resonance imaging (MRI) is a new contrast enhancement technique that indirectly measures molecules with exchangeable protons and exchange-related properties, providing high detection sensitivity^1–4. In practice, the saturation transfer effects of CEST-MRI are often assessed and quantified using a Z-spectra where the water signal is plotted as a function of the applied saturation frequency. For in vivo CEST-MRI, proper parameter quantifications demand careful measurement of the CEST effects (uncontaminated and with sufficient SNR) such as solute concentration and solute‐water exchange rate, thus rendering quantitative CEST a challenging task⁵.

Theoretically, the CEST parameter quantification through Z-spectra fitting demonstrated by Bloch-McConnell (BM) equations could provide a feasible approach, and yet there are the common problems of slow operation speed and converging to local optimal solution. Nevertheless, scholars spent their efforts and carried on studies of CEST quantification in another way. The symmetric magnetization transfer ratio (MTR_asym) that calculated from a Z-spectra is the most used CEST quantification method^6–8. However, MTR_asym is confounded by several types of contamination, including direct saturation (DS), semisolid macromolecular magnetization transfer (MT) and nuclear Overhauser effect (NOE)^9,10.

To further boost CEST specificity, Z-spectra fitting has been successfully applied to differentiate the contributions from multiple origins¹¹, such as multiple‐pool Lorentzian fit^12–14, the Lorentzian difference (LD) analysis^15,16, and three‐point method^11,12. For a specific solute along with overlapping signals from nearby pools, the LD analysis that employs a single Lorentzian line may not provide an accurate reference signal^10,16. The same problem would still occur for a three‐point method that relies on two nearby signals as a reference. The multiple‐pool Lorentzian fit strongly relies on assumption that each CEST signal can be approximated as a Lorentzian lineshape¹⁷.

Recent advancements in the quantification methods of CEST and NOE techniques have significantly improved their application in biomedical imaging, particularly in the context of brain tumors detection^18–22. For example, Glang et al. proposed a deep neural network with uncertainty quantification that can efficiently and accurately predict Lorentzian parameters from CEST MRI spectra, providing fast and reliable CEST contrast image reconstruction while indicating prediction trustworthiness¹⁸. Cui et al. proposed a new method termed as 2π-CEST to reduce the contribution from APT in detecting NOE, offering a more accurate strategy than the conventional asymmetric analysis²⁰. The study concludes that NOE (− 3.5 ppm) serves as a highly sensitive MRI contrast for imaging membrane lipids in the brain, with lipids being the primary contributor to NOE (− 3.5 ppm) signals, rather than proteins, explaining variations in signals between tumors, gray matter, and white matter²¹.

Theoretically, the CEST specificity through Z-spectra rely on the pool size, exchange rate and relaxation time, as described by BM equations. Particularly, the exchange-dependent relaxation rate in the rotating frame (R_ex) that solved from the BM equations by an eigenspace approach, operates independently of non‐specific tissue parameters and depends on specific parameters (solute concentration, solute‐water exchange rate, solute transverse relaxation and irradiation power), therefore it is able to make the CEST more specific^2,4,16.

In this paper, a voxel-wise R_ex-line-fit method is developed to improve the reliability of Z-spectra fitting and investigate the potential of quantitative separation (Fig. 1), in which the simulation of a 5-pool model is used to complement the program capabilities. Our study first elucidates the relationship between R_ex and parameters such as solute concentration, solute‐water exchange rate and T_2,s. Then the R_ex imaging of Amide, NOE (− 3.5 ppm), Guanidino and MT is achieved by our method. Finally, we apply the R_ex-line-fit to study CEST effect in a brain tumor model, and the performance of this method in fitting quality is evaluated.

Fig. 1.

Flow chart of data processing steps of R_ex based approach.

Materials and methods

Exchange-dependent relaxation rate in the rotating frame (R_ex)

The resulting solution for the Z-spectra can be described by the monoexponential decay of the z-magnetization as a function of time with the rate R_1ρ²

$$Z(\mathrm{\Delta }\omega ,t_{\mathit{sat}})=\left(P_{\text{zeff}}{P}_z{Z}_i-Z^{\text{ss}}(\mathrm{\Delta }\omega )\right)e^{-R_{1\rho }(\mathrm{\Delta }\omega )t_{\mathit{sat}}}+Z^{\text{ss}}(\mathrm{\Delta }\omega )$$ 1

where P_zeff is the projection factor on z-axis of the effective frame, t_sat is saturation time.

In the case of steady-state, the resulting solution for the Z-spectra at each offset $\mathrm{\Delta }\omega$ simplifies to^23,24

$$\frac{1}{Z^{\mathit{ss}}(\mathrm{\Delta }\omega )}=\frac{1}{{cos}^2\theta R_{1w}}{R}_{1\rho }(\mathrm{\Delta }\omega )=\frac{1}{{cos}^2\theta R_{1w}}(R_{\mathit{eff}}(\mathrm{\Delta }\omega )+R_{ex.1}(\mathrm{\Delta }\omega )+R_{ex.2}(\mathrm{\Delta }\omega )+\cdots +R_{ex.n}(\mathrm{\Delta }\omega ))$$ 2

where $Z^{\text{ss}}$ is the steady-state condition,$R_{1w}$ denotes the longitudinal relaxation rate of water, and $\theta ={tan}^{-1}\left({\omega }_1\left./\phantom{{\omega }_1\mathrm{\Delta }\omega }\right)\mathrm{\Delta }\omega \right)$ where ${\omega }_1=\gamma B_1$ is the amplitude of the RF field. The $R_{\text{eff}}$ which describes the relaxation of free water in the rotating frame can be approximated by

$$R_{\text{eff}}(\mathrm{\Delta }\omega )=R_{\text{1,w}}{cos}^2(\theta )+R_{\text{2,w}}{sin}^2(\theta )$$ 3

Further, the $R_{\text{ex}}$ at a particular off-resonant frequency $\mathrm{\Delta }\omega$ for a general exchanging pool i is²

$$R_{\text{ex}}(f_i,k_i,R_{\text{2,i}})=\underset{k_i\text{- term}}{\underbrace{f_i{k}_i{\underbrace{\frac{\delta {\omega }_i^2}{{\omega }_1^2+\mathrm{\Delta }{\omega }^2}}}_{\text{a - peak}}{\underbrace{\frac{{\omega }_1^2}{{\mathrm{\Gamma }}_i^2\left./\phantom{{\mathrm{\Gamma }}_i^{2}4}\right)4+\mathrm{\Delta }{\omega }_i^2}}}_{\text{b - peak}}}}+\underset{R_{2,i}\text{- term}}{\underbrace{f_i{R}_{\text{2,i}}\frac{{\omega }_1^2}{{\mathrm{\Gamma }}_i^2\left./\phantom{{\mathrm{\Gamma }}_i^{2}4}\right)4+\mathrm{\Delta }{\omega }_i^2}}}+\underset{\text{corss - term}}{\underbrace{f_i{k}_i{sin}^2\theta \frac{R_{\text{2,i}}(R_{\text{2,i}}+k_i)}{{\mathrm{\Gamma }}_i^2\left./\phantom{{\mathrm{\Gamma }}_i^{2}4}\right)4+\mathrm{\Delta }{\omega }_i^2}}}$$ 4

where $f_i$ is a fraction of the total proton for the i^th pool,$k_i$ is its exchange rate with water in Hz,$R_{\text{2,i}}$ is its transversal relaxation rate,$\delta {\omega }_i$ is its frequency offset in Hz,$\mathrm{\Delta }{\omega }_i$ is the difference in Larmor frequency between pool i and water, and the full width half maximum ${\mathrm{\Gamma }}_i$ is

$${\mathrm{\Gamma }}_i=2\sqrt{\frac{R_{\text{2,i}}+k_i}{k_i}{\omega }_1^2+{(R_{\text{2,i}}+k_i)}^2}$$ 5

Multiple-pool Lorentzian-line-fit

To estimate CEST effects from individual components, we performed the multiple-pool Lorentzian fitting of Z spectra using a non‐linear optimization algorithm²⁵:

$$S(\mathrm{\Delta }\omega )\left./\phantom{S(\mathrm{\Delta }\omega )S_0}\right)S_0=1-\sum _{i=1}^N{L}_i(\mathrm{\Delta }\omega )$$ 6

where

$$L_i(\mathrm{\Delta }\omega )=\frac{A_i}{1+\frac{{(\mathrm{\Delta }\omega -{\mathrm{\Delta }}_i)}^2}{{(0.5W_i)}^2}}$$ 7

Equation (7) represents a Lorentzian line with central frequency offset from water (${\mathrm{\Delta }}_i$), peak FWHM ($W_i$), and peak amplitude ($A_i$). The value of N is the number of fitted pools; S is the measured signal on the Z-spectra; and S₀ is the non-irradiation control signal.

In this study, a five-pool model of Lorentzian-line-fit including Amide at 3.5 ppm (L₁), Guanidyl/Amine at 2.0 ppm (L₂), Water at 0 ppm (L₃), MT at − 2.4 ppm (L₄), and NOE at − 3.5 ppm (L₅) was performed to estimate CEST effects from individual components.

In vivo MR imaging

All animal care and experimental procedures comply with the National Research Council Guide for the Care and Use of Laboratory Animals. All animal experiments were approved by the Ethics Committee of Shantou University Medical College (Approval ID: SUMC2022-204) and conducted in accordance with the ARRIVE guidelines.

For our study, we used 8-week-old male SD rats (Beijing Vital River Laboratory Animal Technology Co., Ltd.) weighing approximately 250 g to establish a tumor-bearing model. In this study, three rats were prepared, where two rats were excluded from the present study due to tumor modeling failure that could not be used during data analysis. To implant the rat glioma C6 cells, a 10 µL suspension containing approximately 2 × 10⁶ cells was injected into the right basal ganglia of the rats using a Hamilton syringe and a 30-gauge needle. After two weeks of tumor cell implantation, the rats underwent MRI. During the MRI procedure, the rats were anesthetized with a mixture of isoflurane and O₂ at a rate of 1 L/min. Anesthesia induction was achieved using 4.0% isoflurane, followed by maintenance anesthesia with 2.0–3.0% isoflurane. To monitor the breath rate, a respiratory probe was utilized throughout the MRI experiments. The rats' respiration rate and body temperature during the 7 T scan were maintained at approximately 60–70 breaths per minute and 38.5–39.5 °C, respectively.

MRI was conducted using a 7T horizontal bore small animal MRI scanner (Agilent Technologies, Santa Clara, CA, USA) equipped with a surface coil (Timemedical Technologies, China) for both transmission and reception. The positioning of the rat was carefully done to ensure that the tumor was accurately centered within the magnetic field. Imaging parameters were as follows: B₁ = 1 µT, repetition time (TR) = 6000 ms, echo time (TE) = 40 ms, array = frequency offsets, slice thickness = 2 mm, field of view (FOV) = 64 × 64 mm, matrix size = 64 × 64, spatial resolution = 1 × 1 mm, averages = 1. An echo planar imaging (EPI) readout sequence was employed to acquire CEST images, utilizing continuous wave (CW) RF irradiation on the scanners. The saturation time was set to 5.0 s, with 49 frequency offsets evenly distributed from − 6 to 6 ppm relative to the resonance frequency of water.

Results

To assess the performance of the proposed R_ex-line-fit, simulated Z-spectra are created using 5-pool system. The R_ex fitting is conducted by using a non‐linear least square constrained optimization algorithm and referencing the pool parameters^1,26,27 in Table 1. Pseudo-code of our method for R_ex imaging and Z-spectra fitting is shown Table 2. The proposed R_ex fitting is compared experimentally to AREX²⁸ and multiple-pool Lorentzian fit²⁵. The AREX is a reduced form of R_ex and follows a Lorentzian function²⁸, so the same parameters listed in Table 2 is used. Table 3 lists the boundaries of the multiple-pool Lorentzian fit²⁵.

Table 1.

Summary of the parameters for a general exchanging pool i when we conduct R_ex and AREX fitting: solute‐water exchange rate (k_i), solute concentration (f_i), transverse relaxation time (T_2,s), and solute resonance frequency offset (Δ).

		Amide	Guanidino	Free water	MT	NOE (− 3.5)
ks (s-1)	LB–UB	1–250	1–1000	–	1–100	1–30
fs (10−3)		0.1–10	1–50	–	10–400	10–100
T2,s (10−3 s)		0.1–10	0.1–10	20–100	0.01–0.3	0.1–10
Δ (ppm)		3.5	2	0	− 2.4	− 3.5

Table 2.

Pseudo-code of the R_ex based method for Z-spectra fitting and decomposition.

Define a set of LB and UB from Table 1, and start at the point x0 = LB + (UB − LB)/2

The pixel-wise decompositions of Reff and Rex for CEST are solved by nonlinear fitting function (lsqnonlin) in MATLAB

Loop for Z-spectra of each pixel

{

compute Reff and Rex fitting parameters from Eqs. (2) and (3), respectively

compute Z-spectra of Rex-line-fit from Eq. (1)

}

Obtain fitting parameters and reconstruct the Rex imaging from Z-spectra of each pixel

Table 3.

Summary of the parameters used for Lorentzian fitting: amplitude (A), width (W), and solute resonance frequency offset (Δ).

		Amide	Guanidyl/Amine	Amino	MT	NOE (− 1.6)	NOE (− 3.5)
A	LB–UB	0–10	0–10	0–10	0–40	0–10	0–20
W		0–3	0–3	0–3	0–35	0–3	0–10
Δ (ppm)		3.6	2	3	− 1.5	− 1.6	− 3.5

Parameter separation

It is worthwhile to evaluate the correlation between R_ex and parameters (solute‐water exchange rate k_s, solute concentration f_s and solute transverse relaxation T_2,s), which may assist in elucidating the R_ex specificity and separating CEST parameters. For Amide, NOE (− 3.5 ppm), Guanidino and MT, Fig. 2 illustrates the correlations between parameters (k_s, f_s) and R_ex. The surface plots demonstrate the dependence of R_ex on k_s and f_s, where R_ex is linear monotonically increasing with parameters (k_s, f_s) for the Amide, NOE (− 3.5 ppm) and Guanidino. For MT, its R_ex is linear monotonically increasing with solute concentration f_s, while a nonlinear monotonically increasing relationship between R_ex and solute‐water exchange rate k_s is observed. It should be noted that the R_ex of Guanidino depicts a monotonically increasing trend first and then decrease corresponding to k_s, while its R_ex follows a monotonically increasing pattern in respect to f_s. In addition, R_ex is nonlinear monotonically increasing with T_2,s for the Amide, NOE (-3.5 ppm) and Guanidino, while R_ex shows a trend of slight monotonic decrease corresponding to T_2,s for MT, as illustrated in Fig. 3. In addition, Fig. 4 illustrates an example of R_ex changing with k_s and R_2,s (1/T_2,s) for Amide, where R_ex follows a increasing pattern in respect to R_2,s at different k_s and R_ex shows a trend of decrease corresponding k_s at different R_2,s.

Fig. 2.

The correlation between R_ex and parameters (k_s, f_s) for Amide, NOE (− 3.5 ppm), Guanidino and MT. For each subplot, the red line denotes the contour.

Fig. 3.

The correlation between R_ex and T_2,s for Amide, NOE (− 3.5 ppm), Guanidino and MT.

Fig. 4.

An example of R_ex changing with k_s and R_2,s (1/T_2,s) for Amide.

R_ex, Lorentzian, and AREX imaging

Herein, we conduct an experiment of R_ex, Lorentzian and AREX imaging for Amide at 3.5 ppm, Guanidino at 2.0 ppm, MT at − 2.4 ppm and NOE at − 3.5 ppm, where each pixel of imaging is obtained by computing the peaks of R_ex and Lorentzian decompositions. Figure 5 illustrates the R_ex, Lorentzian and AREX imaging of these CEST effects. The region of pseudo color image overlaid on anatomy image is the region of interest (ROI), where the region of tumor region is marked with solid red contour and the solid red contour denotes the contralateral region. Visually, the R_ex shows different structure distributions on the R_ex imaging for Amide, Guanidino, MT and NOE, this is different from Lorentzian and AREX.

Fig. 5.

The R_ex, Lorentzian and AREX imaging of Amide at 3.5 ppm, Guanidino at 2.0 ppm, MT at − 2.4 ppm and NOE at − 3.5 ppm. The solid red contour denotes the tumor region and the dashed red contour is the contralateral region.

Figure 6 illustrate the Z-spectra fitting from the tumor region and the contralateral region using the R_ex, Lorentzian and AREX approach, respectively. The results show that the satisfied accuracy and consistency are obtained by the proposed R_ex-line-fit and it displays great agreement and follows the same tendency as the actual measurements. Table 4 lists the mean value and standard deviation of residual between the considered fitting approach and the experimental Z-spectra for the tumor region and its contralateral region.

Fig. 6.

Z-spectra fitting and decomposition from the tumor region and its contralateral region using the R_ex, Lorentzian and AREX approaches.

Table 4.

The mean value and standard deviation of residual between the considered fitting approach and the experimental Z-spectra for the tumor region and its contralateral region.

	Residual between Rex fitting and the experimental Z-spectra		Residual between Lorentzian fitting and the experimental Z-spectra		Residual between AREX fitting and the experimental Z-spectra
Mean value	1.8914 × 10−4	3.8786 × 10−4	− 1.3783 × 10−5	1.7531 × 10−5	2.6252 × 10−5	1.2 × 10−3
Standard deviation	0.0054	0.0147	0.0043	0.0070	0.0053	0.0183
	Tumor region	Contralateral region	Tumor region	Contralateral region	Tumor region	Contralateral region

We further applied the linear regression analysis²⁹ to assess the general performance of the R_ex, Lorentzian and AREX approach using the whole ROI data of CEST images at 49 frequency offsets. Figure 7 displays the R_ex, Lorentzian and AREX for fitting CEST signal by plotting the linear regression lines between the experimentally acquired data and the fitting. The excellent performance of our R_ex method is confirmed by the scatter and linear regression lines, resulting in a very high coefficient of determination (R² = 0.9937).

Fig. 7.

Linear regression analysis of the R_ex, Lorentzian and AREX fitting when they compared with the experimentally acquired data within the ROI.

Discussion

The exchange-dependent relaxation R_ex is an important parameter for CEST effects and can be used to determine the exchange rate k_s of the exchangeable protons with concentration fraction f_s and transverse relaxation T_2,s. In this study, we proposed a method that can support reliable quantitative separation of CEST effect by R_ex. This is important because specificity of in vivo CEST effect is challenging due to careful measurement of the CEST effects. Nevertheless, some discussions should be made as follows.

In the Eq. (4) of R_ex, the first term named ‘k_i-term’ is the dominant factor, which comprises the product of peaks for water pool (‘a-peak’) and CEST pool (‘b-peak’), respectively. The ‘R_2,i-term’, together with the ‘$\text{b - peak}$’, denotes the exchange dependent relaxation that yields an off-resonant peak in CEST. So the R_ex turns into two peaks, but unlike Lorentzian and AREX lineshape that gives only one peak. In fact, the effect of water T1 and T2 relaxation time on Lorentzian shape is described by formula in Ref.¹⁷. In contrast, R_ex excludes water T1 and T2 contributions, which serves as a tool for calculating the CEST signal, offering a more representative depiction of chemical exchange processes than traditional CEST analysis methods^30,31. As a reduced form of R_ex, AREX is a Lorentzian function²⁸, excluding the water pool (‘a-peak’), unlike the complete Rex expression^2,28. It is interesting to study the ‘a-peak’ imaging, which will be presented in our next work.

In the R_ex imaging, the tumor regions marked with solid red contour show reduced values in correspondence with the contralateral regions (Fig. 5). In practice, the exchange rate k_i can be determined by analyzing the CEST signal as a function of pH: k_amide = 5.57 × 10^pH–6.4, k_guanidyl = 5.57 × 10^pH–6.432. An intuitive explanation is that the reduced exchange rate with lower pH in and around the tumor region causes the lowering of the R_ex peak values (Fig. 6). In fact, the mechanism behind tumors is more complex compared with the clear process of stroke. Particularly, the R_ex mechanism is considered that many factors (the exchange rate k_s, the concentration fraction f_s and transverse relaxation T_2,s) participated in this process (Eq. 4). To some extent, the R_ex imaging shows the specificity for different chemical groups, because different structure distributions on the R_ex imaging for Amide, Guanidino, MT and NOE are obtained, this is different from multiple-pool Lorentzian, AREX and T₁ map (Fig. 5). Nevertheless, some multi-pool models have been applied to tumor research, such as Refs.^33,34.

In this study, we have made some meaningful explorations and obtained promising research results. We first determined whether parameters (solute‐water exchange rate and solute concentration) and R_ex have a monotonous relationship for Amide at 3.5 ppm, Guanidino at 2.0 ppm and NOE at − 3.5 ppm (Figs. 2, 3, 4). With this knowledge in mind, this makes it possible to isolate some part parameters by extending previous approaches, where numerical simulations of R_ex can be used to obtain saturation parameters for CEST effect.

We further implemented R_ex as a novel model to provide high-accuracy CEST fitting and decomposition where multiple CEST saturation transfer pools are present. The proposed R_ex-line-fit avoids specific selection of tissue parameters and minimizes operator bias, enabling adaptive fitting and decomposition for reliable estimation of CEST effects. The accuracy of R_ex-line-fit was first validated by the test of in vivo mouse, which revealed that R_ex method provided a near-perfect approximation to the experimentally acquired Z-spectra (Table 4, Figs. 6 and 7).

Conclusion

As an improvement method that only is dependent of the specific parameters (solute concentration, solute‐water exchange rate, solute transverse relaxation, and irradiation power), our R_ex-line-fit can provide a simple, robust and more accurate approach for approximating CEST and further serve for quantitative separation of CEST effect. More in vivo validations and at the clinical field strength will be performed in the future.

Author contributions

GX: Conceptualization, Methodology, Software; X.-L.Z: Methodology, Formal analysis, Investigation, Validation, Visualization, Writing – original draft; S.-Q.W: Formal analysis, Data curation; S.-X.L: Formal analysis, Data curation; T.-T.N: Resources, Data curation; Y.-W.C: Investigation, Supervision; C.-Y.Z: Resources, Data curation; GY: Investigation, Supervision, Writing – review & editing; R.-H.W: Funding acquisition, Project administration, Writing – review & editing.

Funding

The study was supported by the National Natural Science Foundation of China (Grant/Award Number: 82020108016), the Grant for Key Disciplinary Project of Clinical Medicine under the Guangdong High-Level University Development Program (Grant/Award Number: 002-18120302), the Functional Substances in Medicinal Edible Resources and Healthcare Products (Grant/Award Number: 2021B1212040015), the 2021 Medical Research Foundation of Guangdong (Grant/Award Number: 202011102275838), the 2021 Grant for Key Science Technology and Innovation Project under the Guangdong Jieyang Development Program (Grant/Award Number: 210517084612609), and the Medical Health Science and Technology Project of Shantou (Grant/Award Number: 2022-88-16).

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.