mri-signal-density-relaxation-cross-scan - Biophysical Instrumentation VII: NMR Relaxation and MRI Contrast Step by Step

Voxel signal changes with density and with cited relaxation fractions. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

MRI contrast signal has density and relaxation factors

The scan separates density changes from recovery-fraction changes while the transverse fraction remains cited.

S=\rho F_R F_T

Density and recovery both move the signal product

The first group changes density; the second group changes the cited recovery fraction.

\begin{array}{c|c|c|c|c}\text{scan}&\rho&F_R&F_T&S\\\rho\ \text{changes}&12&\frac{1}{2}&\frac{1}{2}&3\\\rho\ \text{changes}&16&\frac{1}{2}&\frac{1}{2}&4\\\rho\ \text{changes}&20&\frac{1}{2}&\frac{1}{2}&5\\F_R\ \text{changes}&12&\frac{1}{4}&\frac{1}{2}&\frac{3}{2}\\F_R\ \text{changes}&12&\frac{1}{2}&\frac{1}{2}&3\\F_R\ \text{changes}&12&\frac{3}{4}&\frac{1}{2}&\frac{9}{2}\\\end{array}

The final product cites both relaxation sources

The displayed signal is not density alone; it multiplies density by the checked recovery and transverse rows.

16\cdot\frac{1}{2}\cdot\frac{1}{2}=4