Alexey Vladimirovich Protopopov^{1} and Michael Bock^{1}

The parameter T_{2}* is often used to
describe the apparent rate of spin-spin relaxation in the presence of local magnetic
field gradients, which is commonly assumed to be mono-exponential. However, the behavior of the transverse relaxation is more
complex, since structural characteristics of biological
tissues are encoded in the shape of relaxation curve which cannot be described
by a single parameter.
Several attempts have been made to introduce more
accurate relaxation models. In this work we present a concept for the quantitative
analysis of the relaxation curve shape in gradient-recalled echo (GRE) imaging
based on physical parameters of the signal.

**Methods**

In our spin relaxation model, we define a set of numerical parameters to characterize the tissue structure and to describe the unique relaxation curve shape in each voxel. The following analytical formula for signal relaxation within a voxel has been derived from this model:

$$S(t)=S_{0}v\cdot e^{-t/T_{2}}\cdot R_{p_{x}}(\alpha_{x} t)\cdot R_{p_{y}}(\alpha_{y} t)\cdot R_{p_{z}}(\alpha_{z} t), \hspace{2mm} R_{p}(x)=\frac{1}{x}\int_{0}^{x} \cos(pu)e^{-u^{2}}du,\hspace{2mm}(1)$$

where

$$\alpha_{x,y,z}=\frac{\gamma}{4}\sigma_{x,y,z}l_{x,y,z},\hspace{2mm} p_{x,y,z}=\frac{2\overline{G}_{x,y,z}}{\sigma_{x,y,z}}.\hspace{2mm}(2)$$

Here, $$$l_{x,y,z}$$$ are the voxel dimensions, $$$v$$$ is the voxel volume, $$$\overline{G}_{x,y,z}$$$ are the average values of the magnetic field gradients with standard deviations $$$\sigma_{x,y,z}$$$. Equation (1) cannot be fitted effectively, but a simplified version, which is a 3D generalization of the Hahn relaxation formula [11] GRE signal:

$$S(t)=S_{0}\cdot\exp\left[-\frac{t}{T_{2}}-(\alpha^{2}-\beta^{2})t^{2}-\frac{kt^{3}}{3}\right].\hspace{2mm}(3)$$

Here $$$\alpha$$$ and $$$\beta$$$ account for gradients of
magnetic field and transverse relaxation time over the voxel, and $$$k$$$ determines diffusion. However,
equation (3) was derived using the Taylor expansion, so in order to approximate
the relaxation time and to compute physical parameters on the entire range
between zero and infinity, the Full-Range Approximation (FRA) was constructed:

$$S(t)\approx \exp\left[a-bt-w\cdot\left(\exp^{-\epsilon t}-1+\epsilon t\right)\right].\hspace{2mm}(4)$$

Parameters $$$a$$$ and $$$b$$$ are provided by the multi-point algorithm with clamping (MPC), while $$$w$$$ and $$$\epsilon$$$ are derived from them in order to match the function with the experimental curve. Figure 1 shows an example of the FRA being fitted over an experimental curve.

In order to maintain signal-to-noise ratio (SNR), the concept of branching averaging with statistical weighting was introduced (Fig. 2). As the signal amplitude (unlike noise) decays with TE, the loss of SNR is compensated by an increased averaging of neighboring pixels for subsequent TEs. This averaging leads to reduced spatial resolution, and to mitigate this, the number of averaged pixels is set proportional to TE.

The method presented in this article was tested on a number of sequences performed on volunteers. These scans were performed on a Siemens Symphony scanner using an unmodified GRE sequence. Twelve separate echoes were acquired with each sequence, with echo times between 2.6 ms and 70 ms and a TR of 75 ms.

**Results**

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Fig.1. Measured signal decay as a function of TE together with fit
curve.

Fig.2. Concept of branching averaging. Pixels used for the averaging
calculation are shown in black. The decrease in signal amplitude with
subsequent TEs is depicted by the opacity of each layer of the image.

Fig.3. From top to bottom: brain, abdomen, hips. In each row, the
first image is the original GRE image, followed by a colourmap of gradients
created, using RPCC, and the maps of T_{2} and T_{0.5}.