Relaxation Model and Mapping of Magnetic Field Gradients in MRI

Abstract

The purpose of the work is development of algorithms for separate mapping of T ₂ relaxation time and gradients, using gradient recalled echo (GRE) sequence. Application of three-dimensional (3D) model of gradients and their volumetric averaging within a voxel lead to analytical model of relaxation function, which is consistent with experimental data for both regular macroscopic and randomized micro- and mesoscopic gradients. The model is verified by fitting into experimental data obtained on specially made phantoms. Verification of algorithms is completed by comparing gradient maps obtained on specially made cylindrical phantoms with theoretical maps of their exact 3D electro-dynamic solutions. Analytical model of relaxation function proved to be in good agreement with experimental relaxation curves. On the basis of this model, fast and unambiguous fittingless algorithms were developed. Gradient maps measured on special cylindrical phantoms are in good qualitative agreement with theory. 3D statistical model and fittingless algorithms provide the basis for separating the GRE signal into two meaningful parameters—T ₂ and gradients, thus doubling information from magnetic resonance imaging.

Appendix

Functions R _p (x) in Eq. 12 may be obtained by averaging sinc(αt) in Eq. 6 over α not only with Gaussian distribution Eq. 9, but with any other suitable probability distributions. Any probability density function for the random variable α, specified by its characteristic spread σ and average value $\bar{\alpha }$, may be written in parametric form

$$\frac{1}{\sigma }P_{p} \left( {\frac{\alpha }{\sigma }} \right)\,{\text{d}}\alpha = P_{p} (u)\,{\text{d}}u,$$

(35)

with dimensionless argument u = α/σ and parameter $p = {{\bar{\alpha }} \mathord{\left/ {\vphantom {{\bar{\alpha }} \sigma }} \right. \kern-0pt} \sigma }$. Then average relaxation function may be written as a function of a dimensionless argument x = σt:

$$\frac{1}{\sigma }\int {P_{p} \left( {\frac{\alpha }{\sigma }} \right)} \, \cdot {\text{sinc}}(\alpha \,t)\;{\text{d}}\alpha = \int {P_{p} \left( u \right)} \, \cdot {\text{sinc}}(u\,x)\;{\text{d}}\alpha \equiv R_{p} (x).$$

(36)

For example, consider three simplest limited-range distributions together with the unlimited Gaussian one:

$$\begin{aligned} &{\rm uniform} \hfill \\ &P_{p} (u) = \frac{1}{2}\left\{ {\begin{array}{*{20}l} {1\,,\quad p - 1 < u < p + 1;} \hfill \\ {0\quad {\text{otherwise;}}} \hfill \\ \end{array} } \right.\end{aligned}$$

(37)

$$\begin{aligned} &{\rm cosine} \hfill \\ &P_{p} (u) = \frac{\alpha }{2}\left\{ {\begin{array}{*{20}l} {\cos \left[ {\alpha (u - p)} \right]\,,\quad \left| {\alpha (u - p)} \right| < \frac{\pi }{2};} \hfill \\ {0\quad {\text{otherwise;}}} \hfill \\ \end{array} } \right.\end{aligned}$$

(38)

α = arccos (1/e);

$$\begin{aligned} &{\rm cosine\; square}\hfill \\ &P_{p} (u) = \frac{2\,\beta }{\pi }\left\{ {\begin{array}{*{20}l} {\cos^{2} \left[ {\beta (u - p)} \right]\,,\quad \left| {\beta (u - p)} \right| < \frac{\pi }{2};} \hfill \\ {0\quad {\text{otherwise}} .} \hfill \\ \end{array} } \right. \end{aligned}$$

(39)

$$\beta = \arccos (1/\sqrt e );$$

$$\begin{aligned} &{\rm Gaussian} \hfill \\ &P_{p} (u) = \frac{1}{\sqrt \pi }e^{{ - (u - p)^{2} }} \end{aligned}$$

(40)

All these distributions are normalized to unity integral and to equal characteristic spread σ when the function falls to 1/e of its maximum (Fig. 13). This is a standard criterion for exponential functions.

Fig. 13

Normalized probability distributions drawn for p = 5, as an example. Characteristic spread at the level 1/e is the same for all four distributions

In the text, we calculated the integral (Eq. 36) analytically for Gaussian distribution (Eq. 40), using complex variables. Now consider results of numerical computations in comparison with limited-range distributions (Eqs. 37–39). The result is shown in Fig. 14, and is almost identical for all the four probability distributions. Thus, it is possible to conclude that actual form of a probability density of gradients does not change the result within practically expected precision of measurements, and Gaussian distribution can be used without limitations.

Fig. 14

Computed relaxation functions R _p (x): a p = 0; b p = 5