Chapter 3 - Mathematics of diffusion measurement

Abstract

To calculate the distribution of signal phases by molecular motion the first factor that needs to be determined is how a phase distribution in space is introduced by a gradient pulse with a strength of G and a duration of δ. The diffusion constant (D) of water inside the brain is about 1.0×10-3mm2/s and the diffusion time is approximately 30 ms. Using Einstein's equation it is calculated that water moves approximately 8 μm on average. This means that the phase of water signal is found to be about 90° off between two places separated by 8 μm .This is the condition after the first gradient pulse. After the phase gradient is introduced across the sample, water molecules start moving during the time period, D. Although the water molecules move in a 3D space, interest is focused only on the movement along the gradient axis, which is, the horizontal orientation (x-axis). After the time period D and a rephasing pulse, each water molecule gets a phase shift that is proportional to the amount of movement. The further the water moves, the more phase shift it gets.