Linear signal combination T₂ spectroscopy

Abstract

A technique is presented for performing T2 spectroscopy in magnetic resonance imaging (MRI). It is based on a weighted linear combination of T2 decay data. The data is combined in a manner that acts like a filter on the T2 spectrum. The choice of weighting coefficients determines the filter specifications (e.g. passband/stopband locations, stopband suppression factors). To perform spectroscopy, a series of filters are designed with narrow passbands centered about consecutive regions of the T2 spectrum. This provides an estimate of every region of the spectrum. Taken together, an initial estimate of the full T2 spectrum is thus obtained. However, the filtering process causes a distortion of the estimate relative to the true spectrum. To reduce this distortion, deconvolution is performed. The characteristics of the technique are first evaluated through simulation. The technique is then applied to experimental MRI data to demonstrate practical feasibility. T2 spectroscopy falls into a class of problems requiring inverse transformation with a set of exponential basis functions (i.e. the Laplace Transform). It is demonstrated how the present technique may be applied to problems involving non-exponential basis functions as well.

Introduction

In pure water, transverse relaxation takes the form of a monoexponential decay characterized by a single time constant “T₂” (Fig. 1):

$$S\left(t\right)=e^{-\frac{t}{T_2}}$$

In complex systems such as tissue, however, a spectrum of T₂ values may be present:

$$m\left(T_{2_i}\right);i=1,\dots ,p$$

The signal decay is then a sum of the individual T₂ components:

$$S\left(t\right)=\sum _{i=1}^{p}m\left(T_{2_i}\right)e^{-\frac{t}{T_{2_i}}}$$

Changes in the T₂ spectrum may reflect underlying structure and/or pathology within the tissue. For example, the T₂ spectrum of white matter contains a component at ~15 ms associated with myelin, and a second component ~90 ms associated with intra-/extra-cellular water [[1], [2], [3], [4], [5], [6], [7]] (Fig. 1). In diseases such as multiple sclerosis, myelin loss may be detected directly through a reduction in the short T₂ component [[8], [9], [10], [11], [12]]. Similar approaches haven been proposed for other anatomy [2,13,14].

In MRI, data is acquired in the time domain (Eq. (3)). To determine the T₂ spectrum, an inversion is required. Based on the form of Eq. (3), one plausible approach could be Inverse Laplace Transformation:

$$S\left(t\right)\equiv L^{-1}\left\{\mathrm{m}\left(T_2\right)\right\}$$

Unfortunately, MR signals of the form in Eq. (3) aren't appropriate for a strict use of the inverse Laplace Transform [15,16]. Rather, Eq. (3) can be considered as a Fredholm integral equation of the first kind [15]. To solve Eq. (3), most approaches employ some form of non-negative least squares (NNLS) fitting [17]. The most straightforward method is to directly apply the NNLS algorithm. Unfortunately, a difficulty arises when the number of free parameters significantly exceeds the number of data points – a situation typically encountered with in vivo T₂ decay experiments. Under these circumstances, the problem becomes ill-posed [16]. As a result, most approaches to inverting Eq. (3) employ some form of regularization [18,19]. This involves adding a smoothing term to the NNLS fitting equation [20]. The advantage of regularization is that it reduces the ill-possedness of the problem, and stabilizes the fit. However, the disadvantage is that the derived solution is not necessarily identical to the underlying true T₂ spectrum [21]. Techniques have been developed to optimize the smoothness-accuracy tradeoff [16,20]. Nonetheless, there is limited range of possible solutions within the constraints of a regularization term.

In this study, a new technique for T₂ spectroscopy is presented. It is based on a weighted linear combination of time domain data. Like regularization, this approach provides a spectrum that is smoothed relative to the true T₂ spectrum. However, the process of linear combination can be cast as a filter design problem. Therefore, unlike regularization, one has a large degree of control over the nature of the smoothing. Furthermore, we will demonstrate that the effects of smoothing may be at least partially removed through deconvolution.

In the sections below, the basic theory underlying this linear combination T₂ spectroscopy technique will be derived. Simulations will be presented to characterize its behaviour. Finally, the technique will be applied to experimental MRI phantom data to demonstrate feasibility.