Analyzing the Impact of Vascular Compartments Blood Relaxation Times on Resultant Spectrum Area Using Improved Bloch NMR Fluid Flow Equations

Nuclear Magnetic Resonance (NMR) spectrum, an invaluable tool in the frequency domain, is generated from the Fourier transformation of time-domain NMR signal. Previous research has utilized this method, applying the Fourier transform to data that represents the transient state solution of the improved Bloch NMR fluid flow equation. The current study developed a new resultant spectrum equation designed to enhance the accuracy of the NMR spectrum. Additionally, this research explores how the relaxation times of arterial, venous, and capillary blood affect the area beneath the resultant spectrum (A_r) produced by spinning blood protons. This study employed the Fourier transform to theoretically construct the frequency domain spectrum, while Laplace transform method and Heaviside expansion theorem were used to generate the NMR signals, which are the time-dependent solutions of the improved Bloch NMR fluid flow equations. For the purpose of data processing and visualization, MATLAB and Origin Pro software were employed. The arterial blood simulation results show corresponding values of A_r = 263.48746 Arads^-1m^-1, 261.59038 Arads^-1m^-1, 259.62945 Arads^-1m^-1, and 257.62471 Arads^-1m^-1 respectively at specific input values of arterial blood flow parameter T₁ = 1387 ms, 1419 ms, 1451 ms, and 1483 ms respectively. The simulation results produce values of A_r = 263.54293 Arads^-1m^-1, 281.53685 Arads^-1m^-1 , 299.23162 Arads^-1m^-1 , and 318.59282 Arads^-1m^-1 respectively for arterial blood when the input arterial blood flow parameter is set to T₂ =228 ms, 245 ms, 262 ms, 279 ms respectively. The simulation results produce values of A_r = 185.78128 Arads^-1m^-1, 185.81027 Arads^-1m^-1, 186.25025 Arads^-1m^-1, and 184.71834 Arads^-1m^-1 respectively for venous blood when the input venous blood flow parameter is set to T₁ = 1381 ms, 1461 ms, 1451 ms, and 1486 ms respectively. The simulation results produce values of A_r = 189.30669 Arads^-1m^-1 ,189.25119 Arads^-1m^-1, 220.72525 Arads^-1m^-1, and 236.86291 Arads^-1m^-1 respectively for venous blood when the input venous blood flow parameter is set to T₂ = 158 ms, 173 ms, 188 ms, and 203 ms. The results of simulation for capillary blood show corresponding values of A_r = 10.78129 Arads^-1m^-1, 6.453 Arads^-1m^-1, 5.85084 Arads^-1m^-1, and 5.63595 Arads^-1m^-1 at specific capillary blood flow parameter values of T₁ = 300 ms, 1200 ms, 2100 ms, and 3000 ms respectively. The simulation results produce values of A_r = 10.77763 Arads^-1m^-1, 102.02507 Arads^-1m^-1, 377.54099 Arads^-1m^-1 and 14345.86911 Arads^-1m^-1 respectively for capillary blood when the input capillary blood flow parameter is set to T₂ = 10 ms, 73 ms, 136 ms and 199 ms respectively. These findings indicate that the number of vascular compartments blood protons undergoing spin varies as a result of differences in relaxation times of arterial, venous, and capillary blood during circulation through blood vessels.