Analytical Solution of the Bloch-McConnell Equations for Steady-state CEST Z-spectra

Overview

Journal Magn Reson Imaging

Publisher Elsevier

Specialty Radiology

Date 2024 Mar 2

PMID 38430977

Authors

Mehran Shaghaghi

Kejia Cai

Affiliations

Soon will be listed here.

Abstract

Purpose: To derive an analytic expression for the steady-state Chemical Exchange Saturation Transfer (CEST) Z-spectra of a two-pool proton-exchanging system, facilitating simulations and expedited fitting of steady-state Z-spectra.

Method: The analytical expression is derived by directly solving the set of Bloch-McConnell differential equations in matrix form for a two-pool exchanging system, determining water magnetization under steady-state saturation across the entire Z-spectrum. The analytic solution is compared and validated against the numerical solution of Bloch-McConnell equations under prolonged saturation. The study also explores the line shape of a CEST peak, interpolating under-sampled Z-spectra, and Z-spectral fitting in the presence of noise.

Results: The derived analytic solution accurately reproduces spectra obtained through numerical solutions. Direct fitting of simulated CEST spectra with the analytical solution yields the physical parameters of the exchanging system. The study shows that the analytical solution enables the reproduction of fully sampled spectra from sparsely sampled Z-spectra. Additionally, it confirms the approximation of the CEST spectrum of a single exchanging proton species with a Lorentzian function. Monte Carlo simulations reveal that the accuracy and precision of Z-spectral fittings for physical parameters are significantly influenced by data noise. The study also derives and discusses the analytical solution for three-pool Z-spectra.

Conclusion: The derived analytic solution for steady state Z-spectra can be utilized for simulations and Z-spectrum fitting, significantly reducing fitting times compared to numerical methods employed for fitting CEST Z-spectra.

References

Zaiss M, Zu Z, Xu J, Schuenke P, Gochberg D, Gore J . A combined analytical solution for chemical exchange saturation transfer and semi-solid magnetization transfer. NMR Biomed. 2014; 28(2):217-30. PMC: 4297271. DOI: 10.1002/nbm.3237. View

Sun P, Zhou J, Sun W, Huang J, van Zijl P . Detection of the ischemic penumbra using pH-weighted MRI. J Cereb Blood Flow Metab. 2006; 27(6):1129-36. DOI: 10.1038/sj.jcbfm.9600424. View

Chen L, van Zijl P, Wei Z, Lu H, Duan W, Wong P . Early detection of Alzheimer's disease using creatine chemical exchange saturation transfer magnetic resonance imaging. Neuroimage. 2021; 236:118071. PMC: 8321389. DOI: 10.1016/j.neuroimage.2021.118071. View

Harston G, Tee Y, Blockley N, Okell T, Thandeswaran S, Shaya G . Identifying the ischaemic penumbra using pH-weighted magnetic resonance imaging. Brain. 2015; 138(Pt 1):36-42. PMC: 4285197. DOI: 10.1093/brain/awu374. View

Sun P . Simplified and scalable numerical solution for describing multi-pool chemical exchange saturation transfer (CEST) MRI contrast. J Magn Reson. 2010; 205(2):235-41. PMC: 2902598. DOI: 10.1016/j.jmr.2010.05.004. View

Murase K, Tanki N . Numerical solutions to the time-dependent Bloch equations revisited. Magn Reson Imaging. 2010; 29(1):126-31. DOI: 10.1016/j.mri.2010.07.003. View

Shaghaghi M, Chen W, Scotti A, Ye H, Zhang Y, Zhu W . quantification of proton exchange rate in healthy human brains with omega plot. Quant Imaging Med Surg. 2019; 9(10):1686-1696. PMC: 6828586. DOI: 10.21037/qims.2019.08.06. View

Woessner D, Zhang S, Merritt M, Sherry A . Numerical solution of the Bloch equations provides insights into the optimum design of PARACEST agents for MRI. Magn Reson Med. 2005; 53(4):790-9. DOI: 10.1002/mrm.20408. View

Snoussi K, Bulte J, Gueron M, van Zijl P . Sensitive CEST agents based on nucleic acid imino proton exchange: detection of poly(rU) and of a dendrimer-poly(rU) model for nucleic acid delivery and pharmacology. Magn Reson Med. 2003; 49(6):998-1005. DOI: 10.1002/mrm.10463. View

10.

Zhou J, Tryggestad E, Wen Z, Lal B, Zhou T, Grossman R . Differentiation between glioma and radiation necrosis using molecular magnetic resonance imaging of endogenous proteins and peptides. Nat Med. 2010; 17(1):130-4. PMC: 3058561. DOI: 10.1038/nm.2268. View

11.

Ma B, Blakeley J, Hong X, Zhang H, Jiang S, Blair L . Applying amide proton transfer-weighted MRI to distinguish pseudoprogression from true progression in malignant gliomas. J Magn Reson Imaging. 2016; 44(2):456-62. PMC: 4946988. DOI: 10.1002/jmri.25159. View

12.

Singh A, Debnath A, Cai K, Bagga P, Haris M, Hariharan H . Evaluating the feasibility of creatine-weighted CEST MRI in human brain at 7 T using a Z-spectral fitting approach. NMR Biomed. 2019; 32(12):e4176. PMC: 11463199. DOI: 10.1002/nbm.4176. View

13.

Heo H, Jones C, Hua J, Yadav N, Agarwal S, Zhou J . Whole-brain amide proton transfer (APT) and nuclear overhauser enhancement (NOE) imaging in glioma patients using low-power steady-state pulsed chemical exchange saturation transfer (CEST) imaging at 7T. J Magn Reson Imaging. 2015; 44(1):41-50. PMC: 4902781. DOI: 10.1002/jmri.25108. View

14.

van Zijl P, Jones C, Ren J, Malloy C, Sherry A . MRI detection of glycogen in vivo by using chemical exchange saturation transfer imaging (glycoCEST). Proc Natl Acad Sci U S A. 2007; 104(11):4359-64. PMC: 1838607. DOI: 10.1073/pnas.0700281104. View

15.

Zaiss M, Bachert P . Exchange-dependent relaxation in the rotating frame for slow and intermediate exchange -- modeling off-resonant spin-lock and chemical exchange saturation transfer. NMR Biomed. 2013; 26(5):507-18. DOI: 10.1002/nbm.2887. View

16.

Zhang S, Malloy C, Sherry A . MRI thermometry based on PARACEST agents. J Am Chem Soc. 2005; 127(50):17572-3. PMC: 2570850. DOI: 10.1021/ja053799t. View

17.

Schure J, Pilatus U, Deichmann R, Hattingen E, Shrestha M . A fast and novel method for amide proton transfer-chemical exchange saturation transfer multislice imaging. NMR Biomed. 2021; 34(7):e4524. DOI: 10.1002/nbm.4524. View

18.

Cai Z, Tao Q, Scotti A, Yi P, Feng Y, Cai K . Early detection of increased marrow adiposity with age in rats using Z-spectral MRI at ultra-high field (7 T). NMR Biomed. 2021; 35(2):e4633. DOI: 10.1002/nbm.4633. View

19.

Li A, Hudson R, Barrett J, Jones C, Pasternak S, Bartha R . Four-pool modeling of proton exchange processes in biological systems in the presence of MRI-paramagnetic chemical exchange saturation transfer (PARACEST) agents. Magn Reson Med. 2008; 60(5):1197-206. DOI: 10.1002/mrm.21752. View

20.

Mougin O, Coxon R, Pitiot A, Gowland P . Magnetization transfer phenomenon in the human brain at 7 T. Neuroimage. 2009; 49(1):272-81. DOI: 10.1016/j.neuroimage.2009.08.022. View