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. 2026 May 9;16:21267. doi: 10.1038/s41598-026-51466-2

Construction of a quantitative extracellular pH map of rat brain glioma using CEST-MRI with iobitridol as the contrast agent

Xiaolei Zhang 1,#, Haochuan Gan 1,#, Zhiwei Shen 2, Fengfeng Lin 1, Gang Xiao 3,✉, Renhua Wu 1,✉
PMCID: PMC13346539  PMID: 42106420

Abstract

This study aimed to evaluate whether chemical exchange saturation transfer (CEST) imaging, in combination with the nonionic X-ray iodinated contrast agent Iobitridol, can detect extracellular pH (pHe) in rats with gliomas and enable the construction of quantitative pHe maps. CEST pH imaging was performed both on Iobitridol phantoms and on rat models bearing brain gliomas, using a 7.0 Tesla small animal MRI scanner (Agilent Technologies) and employing varying radiofrequency (RF) powers (1.5, 3.0, and 6.0 µT) based on the ratio of apparent exchange-dependent relaxation (AREXratio) technique (specifically, 1.5/6.0 µT and 3.0/6.0 µT). The results indicated that AREXratio can more effectively eliminate the influence of magnetization transfer (MT) effects from the CEST signal, thereby enabling more accurate quantification of pH. In vivo CEST pHe imaging distinctly delineated the glioma regions, and quantitative analysis demonstrated that the mean extracellular pH values within gliomas were closely aligned and exhibited an acidic profile. These results further verify the reliability and accuracy of CEST imaging for quantitative assessment of the tumor microenvironment’s acidity in gliomas. In conclusion, this study is the first to demonstrate that non-invasive CEST imaging can accurately detect the acidic extracellular microenvironment of brain gliomas and produce quantitative pHe maps with good spatial resolution, highlighting its significant potential for clinical translation.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-51466-2.

Keywords: Iobitridol, Chemical exchange saturation transfer, Contrast agents, PH, Glioma

Subject terms: Cancer imaging, Cancer imaging

Introduction

Brain glioma represents the most common primary intracranial tumor, with malignant gliomas (grade III–IV) accounting for approximately 70% of all primary malignant brain tumors1–6. Although comprehensive treatment regimens—including maximal surgical resection7, radiotherapy8 and chemotherapy9, as well as new therapeutic modalities such as molecular targeted therapy10 and immunotherapy11 have been applied in practice, the survival rate for malignant glioblastoma (GBM) remains dismally low12.

As with many other malignant tumors, the extracellular pH (pHe) of gliomas is typically acidic, ranging from 6.2 to 6.9, primarily due to increased fermentative metabolism and poor perfusion13–16. Lowered pHe facilitates tumor growth, local invasion, and metastasis14. Experimental models have shown that administration of sodium bicarbonate increases peritumoral pH and inhibits tumor growth and invasion, thus supporting the acid-mediated invasion hypothesis13,17,18. Therefore, noninvasive detection of extracellular acidosis in gliomas could be of great value for their diagnosis and for monitoring therapeutic efficacy.

Various MRI-based techniques have been developed in recent years to measure tumor extracellular pH. Magnetic resonance spectroscopy (MRS), using IEPA (2-imidazole-1-yl-3-ethoxycarbonyl propionic acid) as an exogenous agent, has been applied to quantify pHe in rat C6 glioblastoma multiforme models19,20. Despite offering accurate and sensitive quantification, MRS is limited by long acquisition times and poor spatial resolution.

Clinically, paramagnetic gadolinium (Gd3+) chelates are widely used to enhance MRI signal intensity by shortening the longitudinal (T1) relaxation time. pH-dependent paramagnetic metal chelates can probe extracellular pH changes, but their signal intensity depends on both the agent’s concentration and pHe. To address this, a dual-injection approach using both Dy-DOTP5- (pH-insensitive) and Gd-DOTA-4AmP5- (pH-sensitive) was developed20–22. Although high spatial resolution pHe maps were obtained, differences in biodistribution and higher total dose of contrast agent remain drawbacks. Advances in dynamic nuclear polarization (DNP) now allow in vivo imaging of tumor pHe at high sensitivity using hyperpolarized 13 C-labeled molecules23,24, although major limitations include the need for specialized 13 C coils and hyperpolarized molecule generation.

Chemical exchange saturation transfer (CEST) is an emerging MRI technique based on magnetization transfer (MT), capable of detecting low-concentration solutes25–30. Through selective RF irradiation, exchangeable protons on CEST agents (endogenous or exogenous) are selectively saturated and subsequently exchange with bulk water protons, leading to a reduction in the water signal—allowing indirect detection of the CEST signal. As the chemical exchange process is sensitive to factors such as pH, temperature, and mobile proton concentration, several exogenous pH-sensitive agents have been developed for quantitative pHe mapping using ratiometric approaches31–33.

Iodinated contrast agents are extensively utilized in clinical practice as CT contrast agents and serve as excellent CEST pH-sensitive probes due to their incorporation of one or more mobile protons34. The conventional ratiometric method requires agents with multiple magnetically nonequivalent protons (e.g., iopamidol) to be irradiated with the same RF power; however, at low field strengths, the chemical shift difference between such protons narrows, posing challenges35. To address this, Longo et al. proposed an RF power-based ratiometric pH MRI method employing the nonionic X-ray contrast agent iobitridol36. This method has been further validated in renal and xenografted breast tumor models, though only two RF power levels (1.5 µT and 6.0 µT in vivo) were utilized. In addition, the technique is significantly influenced by magnetization transfer (MT) effects and other asymmetric effects.

This study aimed to evaluate whether CEST-pH mapping employing the ratio of apparent exchange-dependent relaxation (AREXratio)37 technique, with iobitridol as the contrast agent, can achieve quantitative and high spatial resolution measurement of extracellular pH (pHe) in gliomas. Since the AREXratio technique can more effectively eliminate the influence of MT effects on the CEST signal, it is expected to enhance the accuracy and reliability of pHe quantification.

Materials and methods

Phantom preparation

Phantoms were prepared consisting of six microcentrifuge tubes, each containing 30 mM iobitridol (Guerbet, Aulnay-sous-Bois, France) in phosphate-buffered solution. The pH levels of the iobitridol solutions were individually titrated to 5.7, 6.0, 6.3, 6.7, 7.0, and 7.3 using HCl or NaOH. All phantoms were embedded in agarose solution and solidified at 40 °C, the actual scans were performed at 37 °C.

Cell culture

The C6 glioma cell line was purchased from the Shanghai branch of the Chinese Academy of Sciences. Cells were cultured in Dulbecco’s modified Eagle’s medium (DMEM/F12) supplemented with 10% fetal bovine serum and 1% amphotericin at 37 °C in a humidified incubator with 5% CO₂. The medium was refreshed every 1 to 2 days.

Glioma animal model

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.

Eleven adult male Sprague-Dawley rats (weighing 250–300 g), obtained from the Animal Center at Shantou University Medical College, were used to establish the glioma model. The animals were anesthetized with 4% isoflurane delivered in a mixture of oxygen and air, and their body temperature was maintained throughout the procedure. During the MRI experiments, the isoflurane concentration was reduced to 1.4–1.5%. Body temperature (maintained at 37 °C) and respiratory rate were continuously monitored using an MRI-compatible small animal monitoring system (SAII Technologies, USA). The rats’ heads were fixed in a stereotaxic apparatus. Using a microsyringe, approximately 1 × 106 C6 glioma cells in exponential phase (suspended in 10 µL) were stereotactically injected into the right basal ganglia. After recovery, rats were housed in the animal facility and MRI scans were performed 14 to 21 days post-implantation.

MRI scanning

All phantom and in vivo scans were conducted using a 7 T MRI scanner (Agilent, VNMRS, USA) at room temperature, equipped with a standard body coil (internal diameter 63 mm) for both transmission and reception. T2-weighted images were acquired using a fast spin echo sequence with the following parameters: TR = 2500 ms, TE = 4.1 ms, NEX = 2, FOV = 40 × 40 mm, slice thickness = 2 mm, and matrix = 128 × 128. CEST images were acquired using an echo planar imaging (EPI) sequence with continuous wave RF saturation applied for 5 s and RF powers of 1.5, 3.0 and 6 µT to obtain CEST Z-spectra of both phantoms and animals. Additional imaging parameters included: TR = 5.0 s, TE = 4.1 ms, NEX = 4, FOV = 30 × 30 mm for animals and 35 × 35 mm for phantoms, slice thickness = 4 mm, and matrix = 64 × 64. Z-spectra were collected from − 8 to 8 ppm (− 2400 to 2400 Hz) relative to the water resonance, at 0.4 ppm intervals. The main magnetic field (B0) was shimmed and the RF field (B1) calibrated prior to scanning. For Z-spectra normalization, a non-irradiation control image S0 was acquired in both the phantom and in vivo experiments. The saturation pulse offset was set to 10,000 Hz, corresponding to approximately 33.3 ppm at 7 T, which is far off-resonance from water and therefore is not expected to produce substantial saturation of endogenous or exogenous exchanging pools. The signal at each frequency offset was normalized to this reference image.

Animal studies

During MR imaging, rats were anesthetized by inhalation of 1.5% isoflurane in oxygen, and respiratory and heart rates were continuously monitored using a respiratory gating device. Respiratory gating triggered the pulse sequence immediately after exhalation, maintaining a rate of 40–50 breaths per minute. CEST images at three RF powers (1.5, 3.0 and 6.0 µT) were acquired before and after rapid intravenous injection of a total of 4 mL of iobitridol solution via the tail vein. Following MRI acquisition, rats were sacrificed and tumor tissues excised for histological HE staining.

Data analysis

All CEST data were processed using software implemented in MATLAB (The Mathworks, Inc., Natick, MA, USA). Anatomical and Z-spectra images were initially segmented using an intensity-threshold filter. For each voxel, Z-spectra were interpolated using smoothing splines to accurately identify the bulk water resonance, thereby minimizing B₀ inhomogeneity artefacts. The interpolated Z-spectra was shifted so that the bulk water resonance aligned with zero frequency, and corrected intravoxel saturation transfer effects were subsequently calculated.

This study used the apparent exchange-dependent relaxation (AREX) or the inverse Z-values (MTRRex)38 to specifically quantify CEST effects. The exchange dependent relaxation rate (Rex) for a general exchanging pool i can be approximated by

graphic file with name d33e386.gif 1

withInline graphic. In Eq. 1,Inline graphicis the amplitude of the RF field,Inline graphicis a fraction of the total proton for the ith pool,Inline graphicis its exchange rate with water in Hz,Inline graphicis its transversal relaxation rate,Inline graphicis the frequency offset with respect to the CEST pool i. The AREX or MTRRex at a particular off-resonant frequencyInline graphiccan be simplified as

graphic file with name d33e434.gif 2

whereInline graphicdenotes the longitudinal relaxation rate of water, andInline graphic.

Following Eq. 1 and Eq. 2, a new ratiometric method for the pH assessment has been adopted based on different RF irradiation powers named AREXratio37. For the phantoms, the ratiometric value is calculated according to the following equation:

graphic file with name d33e461.gif 3

where AREXRF1 and AREXRF2 denote the AREX values at different RF power levels (e.g., 1.5 µT and 6 µT), respectively. For in vivo, the ratiometric value is defined as

graphic file with name d33e471.gif 4

where ΔAREXRF1 and ΔAREXRF2 represent the difference between post- and pre-injection AREX values at different RF power levels, respectively.

Results

Phantom experiments

To establish the quantitative relationship between pH and AREXratio, two independent sets of phantom experiments were performed, and the corresponding fitting equations were obtained separately. These calibration equations derived from the phantom studies were then applied to two independent in vivo experiments for the generation and quantitative analysis of pHe maps. The use of two groups also aimed to assess the stability and reproducibility of the method.

Figure 1 presents the Z-spectra of the first group of phantoms containing 30 mM iobitridol in PBS, acquired under different B1 field strengths (1.5 µT, 3.0 µT, and 6.0 µT), demonstrating a pronounced CEST effect at 5.6 ppm downfield from the water signal. This effect exhibited strong dependence on both RF power and pH. The AREX imaging of iobitridol PBS phantoms was measured at B1 field strengths of 1.5 µT, 3.0 µT and 6.0 µT, as shown in Fig. 2 (a, b, c). Compared to 1.5 µT, the AREX was significantly higher at 3.0 µT and 6.0 µT under identical pH conditions. For each B1 field strength, AREX increased with pH until reaching an optimal value—approximately pH 6.4 at 6.0 µT, pH 6.7 at 3.0 µT and pH 7.0 at 1.5 µT—after which it declined as pH continued to rise.

Fig. 1.

Fig. 1

Z-spectra of the first group of phantoms containing 30 mM iobitridol in PBS at pH values of 5.3, 6.0, 6.4, 6.7, 7.0 and 7.3, acquired under B1 powers of 1.5 µT (a), 3.0 µT (b), and 6.0 µT (c).

Fig. 2.

Fig. 2

AREX imaging of the first group of iobitridol PBS phantoms were measured at B1 field strengths of 1.5 µT (a), 3.0 µT (b) and 6.0 µT (c). AREXratio imaging for the 1.5µT/6.0µT (d) and 3.0µT/6.0µT (e) conditions were calculated using Eq. 3. The relationships between AREXratio and pH were fitted using model (log (1 + x) + a)/b for the 1.5µT/6.0µT (f) and 3.0µT/6.0µT (g) conditions.

Based on the AREXratio, AREXratio imaging for the 1.5 µT/6.0 µT and 3.0 µT/6.0 µT conditions were calculated. Within the pH range of 5.7–7.3, theInline graphicdemonstrated significantly greater sensitivity to pH changes compared toInline graphic, as depicted in Fig. 2 (d, e). The relationships between AREXratio and pH were fitted, as illustrated in Fig. 2 (f, g). Specifically, the fitted equations are as follows: pH = (log (1 +Inline graphic) + 1.708)/0.2872, and pH = (log(1 +Inline graphic) + 1.184)/0.2271.

The second group of phantoms containing 30 mM iobitridol in PBS, measured under different B1 field strengths (1.5 µT, 3.0 µT, and 6.0 µT), yielded similar results, as demonstrated in Figs. 3 and 4. The corresponding fitted equations are as follows: pH = (log (1 +Inline graphic) + 1.373)/0.2281, and pH = (log(1 +Inline graphic) + 1.023)/0.1975.

Fig. 3.

Fig. 3

Z-spectra of the second group of phantoms containing 30 mM iobitridol in PBS at pH values of 5.3, 6.0, 6.4, 6.7, 7.0 and 7.3, acquired under B1 powers of 1.5 µT (a), 3.0 µT (b), and 6.0 µT (c).

Fig. 4.

Fig. 4

AREX imaging of the second group of iobitridol PBS phantoms were measured at B1 field strengths of 1.5 µT (a), 3.0 µT (b) and 6.0 µT (c). AREXratio imaging for the 1.5µT/6.0µT (d) and 3.0µT/6.0µT (e) conditions were calculated using Eq. 3. The relationships between AREXratio and pH were fitted using model (log (1 + x) + a)/b for the 1.5µT/6.0µT (f) and 3.0µT/6.0µT (g) conditions.

In Vivo experiments

Herein, we explored the feasibility of applying the AREXratio method for CEST-based pHe imaging in rat gliomas, a novel approach in the literature. Following rapid tail vein infusion of Iobitridol at 1 ml/min, the first group of ΔAREX images of rat brain gliomas were obtained at saturation RF powers of 1.5, 3.0, and 6.0 µT, as shown in Fig. 5 (a, b, c). Figures 5 (d) and 5(e) displayed theInline graphicandInline graphicimages generated using Eq. 4, respectively.

Fig. 5.

Fig. 5

Experimental results from the first group of rat brain gliomas. The ΔAREX images under three RF irradiation levels of 1.5 µT (a), 3.0 µT (b), and 6 µT (c). The correspondingInline graphicmap (d) andInline graphicmap (e) were generated using Eq. 4. The corresponding pH1.5µT/6.0µT map (f) and pH3.0µT/6.0µT map (g) were calculated by fitted curve that obtained from the first group of phantoms. The pH1.5µT/6.0µT map (h) and pH3.0µT/6.0µT map (i) were calculated by fitted curve that obtained from the second group of phantoms. Finally, (h) shows the Z spectra at the three RF irradiation levels of 1.5 µT, 3.0 µT, and 6 µT (pre- and post-Iobitridol injection).

Subsequently, two types of pHe maps were generated based on the correspondingInline graphicandInline graphicmaps, using pH = (log (1 +Inline graphic) + 1.708)/0.2872 and pH = (log(1 +Inline graphic) + 1.184)/0.2271 that obtained from the first group of phantoms, as depicted in Fig. 5 (f, g). Additionally, Fig. 5 (h, i) present two more pHe maps, calculated using the equations pH = (log (1 +Inline graphic) + 1.373)/0.2281 and pH = (log(1 +Inline graphic) + 1.023)/0.1975 that obtained from the second group of phantoms, respectively. The mean pHe values using fitted equations of the first group of phantoms were pH1.5µT/6.0µT = 6.9494 and pH3.0µT/6.0µT = 7.1153; using fitted equations of the second group of phantoms, they were pH1.5µT/6.0µT = 6.9331 and pH3.0µT/6.0µT = 6.9300. The resulting pHe maps indicated an acidic microenvironment within the gliomas. Compared to the 31P MRS results (pH = 7.084)19, the estimated pH value was similar. Figure 5 (j) presents the Z-spectra of the first group of rat brain gliomas, acquired under different B1 field strengths (1.5 µT, 3.0 µT, and 6.0 µT).

The second group of rat brain gliomas, measured at different B1 field strengths (1.5 µT, 3.0 µT, and 6.0 µT), yielded results consistent with previous experiments, as shown in Fig. 6. The corresponding pH values are as follows: using fitted equations of the first group of phantoms, they were pH1.5µT/6.0µT = 6.9425 and pH3.0µT/6.0µT = 7.0594; using fitted equations of the second group of phantoms, they were pH1.5µT/6.0µT = 7.0747 and pH3.0µT/6.0µT = 6.9861.

Fig. 6.

Fig. 6

Experimental results from the second group of rat brain gliomas. The ΔAREX images under three RF irradiation levels of 1.5 µT (a), 3.0 µT (b), and 6 µT (c). The correspondingInline graphicmap (d) andInline graphicmap (e) were generated using Eq. 4. The corresponding pH1.5µT/6.0µT map (f) and pH3.0µT/6.0µT map (g) were calculated by fitted curve that obtained from the first group of phantoms. The pH1.5µT/6.0µT map (h) and pH3.0µT/6.0µT map (i) were calculated by fitted curve that obtained from the second group of phantoms. Finally, (h) shows the Z spectra at the three RF irradiation levels of 1.5 µT, 3.0 µT, and 6 µT (pre- and post-Iobitridol injection).

Discussion

Quantitative pH detection of gliomas

In this study, we employed 7 Tesla CEST imaging using the iodinated contrast agent iobitridol to noninvasively detect the extracellular pH (pHe) of brain gliomas, achieving high spatial resolution quantitative pHe mapping in vivo.

Following tail vein administration, iobitridol disperses into the extravascular and extracellular spaces of the tumor. Due to the rapid growth and invasiveness of gliomas, the severely compromised blood-brain barrier allows contrast agent to accumulate extensively in the extracellular microenvironment, enabling accurate pHe imaging. Our results indicate that the overall pHe mapping of gliomas was distinctly acidic, aligning closely with previously reported 31P MRS data19.

Since tumor cell viability within the microenvironment depends on pH regulation, disruption of pH regulatory systems is regarded as a potential therapeutic target39. Clinical studies support that an acidic tumor microenvironment promotes metastasis and increases resistance to chemotherapy and radiotherapy14. Therefore, the non-invasive and high-resolution iobitridol CEST-pH approach provides significant advantages for the quantitative detection of glioma pHe. After iobitridol administration (4 g I/kg, i.v.), we obtained Z-spectra at three RF powers (1.5, 3.0, 6.0 µT), revealing a pronounced nuclear Overhauser effect (NOE) in the downfield region at 1.5 µT compared to 3.0 µT and 6.0 µT40,41.

Eliminating the MT impact using AREXratio

In this study, the AREXratio method was employed to minimize the MT impact on pHe quantification. In practice, MT may be a primary factor contributing to the discrepancy in pHe values observed between the 1.5/6 µT and 3/6 µT power pairs. AREXratio can effectively eliminate the influence of MT effects, thereby enhancing the accuracy and reliability of pHe quantification. To further investigate this, we conducted simulation studies comparing scenarios with and without inclusion of an MT pool, quantitatively evaluating the influence of MT (Supplementary Information Figure S1). The results confirmed that MT substantially affects saturation transfer (ST) signals under different saturation conditions. These findings suggest that PBS-based phantom calibrations may not be directly applicable to in vivo measurements without accounting for MT-related confounds. Note that AREXratio increases with B1, but is roughly independent of MT.

To better address the interference of MT effects, we chose fetal bovine serum (FBS) and conducted numerical simulations with B1 values set at 3.5 µT, 4.0 µT, 4.5 µT and 5.0 µT, each with a saturation time of 3 s (Supplementary Information Figure S2). With increasing B1 field strength, both FBS and MT effects were enhanced. It should be noted that FBS was used in our numerical simulations to provide a more complex protein background than PBS, in order to preliminarily evaluate the influence of biological background components on the CEST signal and AREXratio analysis, rather than to serve as a rigorous model of classical MT effects. Because native FBS mainly contains mobile proteins, it should not be considered equivalent to the classical MT effect arising from semisolid macromolecules, and the related results should therefore be interpreted with caution. In addition, this study did not employ a standard phantom capable of directly representing semisolid macromolecular MT effects; therefore, the FBS-related results should be regarded as a preliminary assessment of the influence of a complex biological background, which also represents a limitation of the present study.

For in vivo studies, we also considered minimizing MT asymmetry by subtracting pre- and post-injection images (Eq. 4). This approach may further optimize the accuracy and stability of AREXratio for in vivo pHe mapping, especially under different saturation power conditions. It should be noted that the in vivo Z-spectra did not show substantial signal decay around ± 8 ppm (Figs. 5 and 6), suggesting that the apparent MT contribution within this offset range was weaker than typically expected. Because this observation may be influenced by multiple factors, including the normalization procedure, saturation conditions, tissue microenvironment, and model assumptions, we no longer attribute it simply to high-power imaging alone and instead interpret the related results with caution.

We acknowledge that the spin-lock technique42 enables pH-sensitive and agent concentration-independent imaging, with high robustness to confounding effects. However, spin-lock typically requires very high RF power, which leads to practical hardware and physiological limitations. In contrast, our method achieves good pH sensitivity and reduced concentration dependence with lower RF power requirements, offering better clinical feasibility and broader applicability. Therefore, both techniques offer distinct advantages in terms of imaging pH sensitivity and practical applicability, and the choice of technique should depend on the specific experimental and clinical needs.

Conclusion

Our study has expanded the in vivo application of ratiometric CEST, demonstrating that the CEST-pH imaging technique can evaluate the extracellular acidic microenvironment of glioma with high spatial resolution. Furthermore, constructing a quantitative extracellular pH map of rat brain glioma using the pH ratio at RF irradiation levels of 3.0 µT and 6.0 µT shows potential for clinical application.

Supplementary Information

Below is the link to the electronic supplementary material.

41598_2026_51466_MOESM1_ESM.pdf (2.4MB, pdf)

Supplementary material 1 (PDF 2411.2 kb)

Author contributions

X.-L.Z: Methodology, Formal analysis, Investigation, Validation, Visualization, Writing – original draft, Writing – review & editing; H.-C.G: Conceptualization, Methodology, Software; Z.-W.S: Formal analysis, Data curation; F.-F.L: Formal analysis, Data curation; G.X: 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: 82471974, 82411540241, 82020108016), Medical Scientific Research Foundation of Guangdong Province (Grant/Award Numbers: B2025236), and the Medical Health Science and Technology Project of Shantou (Grant/Award Number: 240428166497960).

Data availability

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

Declarations

Competing interests

The authors declare no competing interests.

Footnotes

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Xiaolei Zhang and Haochuan Gan contributed equally to this work.

Contributor Information

Gang Xiao, Email: xiao.math@foxmail.com.

Renhua Wu, Email: rhwu@stu.edu.cn.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

41598_2026_51466_MOESM1_ESM.pdf (2.4MB, pdf)

Supplementary material 1 (PDF 2411.2 kb)

Data Availability Statement

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


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