Closed form solution of improved Bloch NMR fluid flow equation using Elzaki transform method

This study aims to solve the improved Bloch Nuclear Magnetic Resonance (NMR), which defines the magnetization of blood-spinning protons. Specifically, to further understand the behavior of nuclear spins in arterial, venous, and capillary blood samples, the impact of transverse relaxation time on magnetization was investigated. The Elzaki transform method, complex inversion integral, and Cauchy’s residue theorem were employed to obtain an analytical solution for the modified Bloch NMR fluid flow equation. MATLAB and Origin software tools were used for data generation and simulation. The simulated results show that as the transverse relaxation times of arterial, venous, and capillary blood samples increase, the magnetization of blood-spinning nuclei forms 3D curved surfaces. This indicates that the phase coherence of spinning nuclei flowing along the blood vessel changes due to random interactions among the nuclei. Consequently, the magnetization of blood-spinning nuclei is affected by blood flow time and distance along the blood vessel. The results demonstrate the significance of transverse relaxation in modulating blood magnetization, with implications for Magnetic Resonance Imaging (MRI). The study’s findings have applications in cardiovascular research and may be utilized by spectroscopists as a means of assessing the accuracy of results produced by NMR spectrometers.